Title Pharmacokinetics Author David Bourne, PhD Prologue About the Author Dr David Bourne, PhD is Professor Emeritus of Pharmaceutical Sciences, College of Pharmacy, University of Oklahoma Health Sciences Center. He has taught Pharmacokinetics and Biopharmaceutics for over 40 years at the University of Kentucky, the University of Queensland, the University of Oklahoma and the University of Colorado. He has authored or co-authored over 130 research and educational articles and a number of books and book chapters on pharmacokinetics and biopharmaceutics. He is author of the non-linear regression program Boomer. He has had an active presence on the WWW since 1994 with his complete course available for most of this time. The Book This book will cover the course material for a course in Basic Pharmacokinetics and a graduate course, Advanced Pharmacokinetics. The course material can be also found on the website referenced on the first page of each chapter. The book includes links to a number of interactive elements including video tutorials, simulations, calculators and tables. It is hoped that these elements will enhance your understanding of the material.Chapter 1: Introduction Definition: Pharmacokinetics is the study of drug and/or metabolite kinetics in the body. It deals with a mathematical description of the rates of drug movement into, within and exit from the body. It also includes the study of drug metabolism or biotransformation rates. The body is a very complex system and a drug undergoes many steps as it is being absorbed, distributed through the body, metabolized or excreted (ADME). See Figure 1.0.1. The drug may also interact with receptors and cause therapeutic and/or toxic responses. Although the details of drug kinetics can be quite complex it is fortunate that we can often approximate drug kinetic processes using “simple” mathematical models. Chapter 2: Background Mathematical Material Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- understand exponents and logarithms, algebraically and graphically use linear (Cartesian) and semi-log graph paper for the representation of data draw a 'best-fit' straight line through data on linear and semi-log graph paper understand and be able to use spreadsheets understand (a little) differential and integral calculus write differential equations given a compartmental modeling scheme as a diagram or description calculate the area under the plasma concentration versus time curve (AUC) using the linear trapezoidal rule 2.1 Exponents Definition: N = bx or N = b^x or N = b ** x where N is the number, b is the base, often 10 but also e ( = 2.7183... Napier's constant), and x is the exponent (or power term when an integer as in 102 is 10 to the power 2). Note use of ^ (common calculator or single line format) or ** (common computer language format) for general exponentiation. Exponents with base e may also be expressed as exp(x) [= ex] (a common computer function format). With the same base, exponents can be added or subtracted For example; ax x ay = a(x+y) to perform multiplication or ax / ay = a(x-y) to perform division. Some Example Calculations a) 100 = 102 = 10^2 = 10**2 b) 100 = e4.605 = e^4.605 = e**4.605 where e = 2.7183 !!! c) 10 x 100 = 101 x 102 = 101+2 = 103 = 1000 d) 10 x 100 = e2.303 x e4.605 = e6.908 = 1000 when the base is the same you can add exponents to multiply numbers Subtract exponents to divide e) 5.6/1.2 = e1.723 / e0.182 = e1.723 - 0.182 = e1.541 = 4.67 Use your calculator to check this answer In pharmacokinetics and the study of other rate processes we are interested in numbers expressed as a base 10 or e with a negative exponent. Exponential decay refers to the decrease in the value of the number as the negative exponent increases in magnitude. Exponential decay is illustrated in tabular form and graphical form in Table 2.1.1 and Figure 2.1.1 (also Interactive 2.1.1). 2.2 Logarithms If N = bx then x = logb N For example, 100 = 102 thus log (100) = 2 [base 10 assumed] or 100 is the antilog of 2. These are called common logs (log). Natural logs (ln) use the base e (=2.7183). Note the use of log and ln to denote common or natural logarithms, respectively. To convert from common log (base 10) to natural log (base e) use 2.303 x log10(N) = lne(N) that is, ln(10) x log(N) = ln(N) For example, ln (100) = 2.303 x log (100) = 2.303 x 2 = 4.606 Common logs are often used with equilibrium equations and buffer or pH calculations. Logarithms to base e are often used in pharmacokinetics and other kinetic processes. Before calculators, logarithms were used to multiply or divide numbers. The two numbers to be multiplied or divided would be converted to logarithms. For multiplication the logs are added and for division the logs are subtracted. Examples: 
a) 23.7 x 56.4 = x To find x take the natural log of both numbers, add and take the 'anti'-log (base e) ln(23.7) + ln(56.4) = ln(x) 3.1655 + 4.0325 = 7.1980 = ln(x) x = 1337 or 23.7 x 56.4 = e3.1655 x e4.0325 = e(3.1655 + 4.0325) = e7.1980 = 1337 b) 6.75 / 14.7 = y To find y take the common log of both numbers, subtract and take the 'anti'-log (base 10) log(6.75) - log(14.7) = log(y) 0.8293 - 1.1673 = -0.338 = log(y) y = 0.4592 If you can find a slide rule you will notice that the scale on the slide is proportional to the log of the number. By aligning numbers on the slide you can quickly add or subtract the length of each number and thus multiply or divide the numbers. When carefully preformed the result can be quite accurate. Later we will look at semi-log graph paper. The y-axis on this paper also has a scale proportional to the log of the number. Figure 2.2.1 illustrates the multiplication of 23.7 x 56.4 using a slide rule to give approximately 1300. The decimal point is determined by rough estimation, Figure 2.2.1. Example log x and ln x values are illustrated in Table 2.2.1 and Interactive 2.2.1. Logarithm at Wikipedia Search for Slide Rule via Google 2.3 Graphing Data on Linear and Semi-Log Graph Paper A very important part of pharmacokinetic analysis is the ability to graph data and interpret the resulting graphs. The graph can provide a useful picture of the data and provide an insight into the underlying mathematical model. Graphs should be drawn carefully. Straight lines should be straight so use a ruler!! Pharmacokinetic data involves models with rate processes. The most simple of these would be a single first order decline described with a single exponential term. We may use an equation such as Equation 2.3.1 Concentration (Cp) versus time to describe drug concentrations after an IV bolus dose. These data are drawn as three graphs. The first graph (Interactive 2.3.1) is a linear (or Cartesian) plot of the data versus time. Notice the smooth decline in concentration with time. The second graph (Interactive 2.3.2) is a plot of the natural log (ln) of the concentration values versus time. Now we get a straight line graph. You might try 'rearranging' the equation above to verify that a straight line is to be expected. The third graph (Interactive 2.3.3) is on different paper. This is semi-log graph paper. On this graph paper the scale on the y-axis is proportional to the log of the number not the number itself, like the scale on the slide rule in the previous section. Notice that the distance from 1 to 2 is the same as the distance from 2 to 4 or from 4 to 8. Again, we have a straight line but without the need to calculate the log of each number. For many calculations involving rate constants you will be determining the slope of these lines. The second and third graphs (Interactive 2.3.2 and Interactive 2.3.3) are a lot easier to use in terms of calculating slopes than Interactive 2.3.1, since they each represent a straight line. We will revisit the equation for this line later. Remember semi-log graph paper has a normal x- axis scaling but the y- axis scaling is proportional to the log of the number not the number itself. It saves you from taking the log of each number before you plot it. Note: Once you take the log of a number you loose the units. Example use of semi-log graph paper: Plot the data (Table 2.3.1), draw a line "through the data", and calculate the slope of the line. A best-fit line is drawn through the data points and extended to the extremes of the graph paper (for better accuracy, Figure 2.3.1). Values for Cp1 and Cp2 are read from the y-axis and t1 and t2 values are read from the x-axis (Table 2.3.2). These values can be used to estimate a value for the slope of the line and also the rate constant for the drug elimination. From the Y-axis the first point is 0, 22.8 and from the X-axis the second point is 12.8, 1 (in the format x-value, y-value). Always try to use the extremes of the line for the best accuracy. The slope can be calculated using Equation 2.3.2. This equation uses logarithms with base 10 for the calculation of slope. Equation 2.3.2 Equation for Slope Thus the slope of the line in Figure 2.3.1. is: [log(1.0) - log(22.8)]/(12.8 - 0) = (0.000 - 1.358)/12.8 = - 0.106 hr-1 In the study of pharmacokinetics first order rate processes and rate constants are common. When the rate constant, k, is calculated from the slope drawn on a semi-log plot, it is found to be equal to -slope • 2.303 [Note ln(10) = 2.303]. Thus, the rate constant is calculated as -slope x 2.303. The calculation of k is easier with calculators (etc.) if you use ln (logarithm with base 'e') in the calculation. At the same time you can change the sign by calculating the numerator as ln(Cp1) - ln(Cp2). Thus k can be calculated using Equation 2.3..3. Equation 2.3.3 Equation for k version 1 Thus, k = [ln(22.8) - ln(1.0)]/(12.8 - 0) = [3.127 - 0.000]/12.8 = 3.127/12.8 = 0.244 hr-1 We can perform this calculation using a different approach. Before hand-held calculators logarithms were used to perform multiplication and division. Here we can use that approach to re-arrange Equation 2.2. Therefore instead of subtracting the two logarithms we can divide one by the other and take the log of the quotient, Equation 2.3.4. Equation 2.3.4 Equation for k version 2 Thus, k = ln(22.8/1.0)/(12.8 - 0) = ln(22.8)/12.8 = 3.127/12.8 = 0.244 hr-1 The advantage of the approach in Equation 2.3.4 is that you don't have to deal with a double negative when Cp2 is less than one and you only need to take one ln value. For example, if Cp1 and Cp2 were 12.5 and 0.13 at 0 and 12 hours, respectively. Using Equation 2.3.3 gives k = [ln(12.5) - ln(0.13)]/12 = [2.526 - -2.040]/12 = [2.526 + 2.040]/12 = 4.566/12 = 0.381 hr-1. Using Equation 2.3.4 gives k = ln(12.5/0.13)/12 = ln(96.15)/12 = 4.566/12 = 0.381 hr-1. Note, each approach gives the same answer. Drawing a Line through the Data Drawing a line through the data doesn't mean through just two data points but through all the data points. Be especially careful about picking two adjacent data points. Sometimes the first and last point can work but the last point, the lowest concentration data point will probably be inaccurate. The best approach is to put the line through all the data. There should be points above and below the line. Maybe an equal number above and below. Drawing this 'best-fit' line is like averaging the data. That is why it is important to use points from the (extremes of the) line to calculate the slope. Any individual data point may have some assay or other error associated with it. If you look at adjacent pairs of data on a graph you will notice that a line drawn through these pairs of data can have quite different slopes. Drawing the 'best-fit' line averages all the data to give the best value for the slope and intercept. Don't ignore that line after you have drawn it, Movie 2.3.1. 2.4 Using Spreadsheets From Visicalc™, to Lotus 1-2-3™, to Excel™ to Numbers™, spreadsheets have provided powerful what-if capability to the personal computer user. Using a combination of numbers, labels, and formulas the user is able try out various equations and mathematical scenarios. The results of these calculations can be linked to graphical output in various forms of charts or graphs. A new spreadsheet document provides a worksheet of cells. The user can enter either numbers, labels, or functions into each cell. Cells are designated by letter and number. For example, Cell 'A1' in the top left corner contains the text 'Example Spreadsheet'. Cells A8:A20 contain the numbers 0 through 12. Cell B8 could be completed by entering the formula '=($B$3/$B$5)*EXP(-$B$4*A8)' which calculates Cp at time = 0. Cells with formulas can be copied (down for example) and thus values for Cp at other times can be quickly calculated. Notice the use of '$' to designate absolute cell references. When copied down these references stay the same and refer to the same cell, e.g. $B$3 = Dose. The reference A8 however is relative and changes for each cell to refer to the relative time in the 'A' column Once the formulas are entered changes to parameter values will be quickly reflected by new values in the appropriate cells. A simple spreadsheet illustrating these and other spreadsheet facilities is illustrated in Figure 2.4.1.2.5 Calculus Differential Calculus Pharmacokinetics is the study of the rate of drug absorption and disposition in the body. Thus differential calculus is an important stepping stone in the development of many of the equations used. These differential equations can be integrated using a variety of techniques including Laplace transforms. However, in this book we won't be describing integration. Differential calculus is involved with the study of rates of processes. The calculus part comes in when we look at these processes in detail, that is, during small time intervals. We may say that at time zero a patient has a concentration of 25 mg/L of a drug in plasma and at time 24 hours the concentration is 5 mg/L. That may be interesting in itself, but it doesn't give us any idea of the concentration between 0 and 24 hours, or after 24 hours. Using differential calculus we are able to develop equations to look at the process during the small time intervals that make up the total time interval of 0 to 24 hours. Then we can calculate concentrations at any time after the dose is given. In many cases the rate of elimination of a drug can be described as being dependent on or proportional to the amount of drug remaining to be eliminated. That is, the process obeys first order kinetics. This is illustrated with Figure 2.5.1. The lines on this plot can be represented mathematically by Equation 2.5.1. Equation 2.5.1 First Order Equation In this equation k is a proportionality constant which we call a rate constant and X is the amount remaining to be eliminated. Note the use of the symbol 'd' to represent a very small increment in X or t. Thus dX/dt represents the slope of the line (or rate of change) over a small region of the curved line in Figure 2.5.1. When data are collected at discrete times such as 4 and 6 hours the larger change in X and t can be represented by ΔX/Δt as shown in Figure 2.5.1. Note the slope changes as the value of X (y-axis) changes. Integration of Equation 2.5.1 (and other differential equations, Equation 2.5.2) provides the integrated equations such as Equation 2.5.3. Equation 2.5.2 Rate of Elimination Equation 2.5.3 Integrated Equation What we have done is convert the equation for rate of change of X versus time into an equation for X versus time. A differential equation is converted into an integrated equation. (Compare Equation 2.5.2 and Equation 2.5.3). We will work with both differential and integrated equations in this book. Integral Calculus Differentiation is the reverse of integration. With differentiation breaking a process down to look at the instantaneous process, integration sums up the information from small time intervals to give a total result over a larger time period. Another example of integration is the calculation of the area under the plasma concentration versus time curve. Later we shall learn that this summation or integration process can be used to evaluate dosage forms, that is it can be used as a measure of dosage form performance. In the section above we converted from the rate of change equation to an equation for X. We can also go further and calculate an area under the curve, which is a further integration step. Another example is the progression from distance, to speed (the rate of change of distance), to acceleration (the rate of change of speed), Figure 2.5.2. References Derivative at Wikipedia Integral at Wikipedia 2.6 Writing Differential Equations Rate processes in the field of pharmacokinetics are usually limited to first order, zero order and occasionally Michaelis-Menten kinetics. Linear pharmacokinetic systems consist of first order disposition processes and bolus doses, first order or zero order absorption rate processes. These rate processes can be described mathematically. First Order Equation Each first order rate process ("arrow") is described by a first order rate constant (k1) and the amount or concentration remaining to be transferred (X1), Equation 2.6.1. Equation 2.6.1 First Order Equation Zero Order Equation Zero order rate processes are described by the rate constant alone. Amount or concentration to the zero power is equal to 1, Equation 2.6.2. Equation 2.6.2 Zero Order Equation Michaelis-Menten Equation The Michaelis Menten process is somewhat more complicated with a maximum rate (velocity, Vm) and a Michaelis constant (Km) and the amount or concentration remaining, Equation 2.6.3. Equation 2.6.3 Michaelis-Menten Equation The full differential equation for any component of a pharmacokinetic model can be constructed by adding an equation segment for each arrow in the pharmacokinetic model. The rules for each segment: 1. Direction of the arrow If the arrow goes into the component the equation segment is positive If the arrow leaves the component the equation segment is negative. 2. Type of rate process If the rate process is first order multiply the (first order) rate constant by the amount or concentration of drug in the component at the tail of the arrow. If the process is zero order just enter the rate constant. For a Michaelis-Menten processes include the amount or concentration of drug in the component at the tail of the arrow in the Michaelis-Menten equation. An example In Figure 2.6.1 the rate process from one to two is zero order. The process from two to three is first order and the process from two to four follows Michaelis-Menten kinetics. We can now systematically write the differential equations for each component of the model. Component 1 There is one arrow leading out of component one so the rate process is negative. This process is zero order so we just write the rate constant. The equation for this component is shown in Equation 2.6.4. Equation 2.6.4 Component 1 Component 2 This component is more of a challenge. There are three arrows connected with this component. One arrow leads to the compartment so this equation segment is positive. The other two arrows lead away from the component and these equation segments are negative. Starting with the zero order process provides a positive k0 to the differential equation. The first order process is developed as the rate constant multiplied by the amount remaining in component 2. This is negative and is -k1 x X2. The final segment is the Michaelis Menten process from component 2 to component 4. This is also negative and is described as -Vm x X2/(Km + X2). The total differential equation for this component is shown in Equation 2.6.5. Equation 2.6.5 Component 2 Component 3 Here we have one arrow going to this component from component 2, thus the equation segment is positive. The rate process is first order so we multiply the rate constant by the amount remaining in the component at the tail of the arrow, component 2. The equation is shown in Equation 2.6.6. Equation 2.6.6 Component 3 Component 4 The fourth component is also described with one equation segment. The arrow leads to this component so the segment is positive. The rate process is a Michaelis Menten process. The equation for this component is shown in Equation 2.6.7. Equation 2.6.7 Component 4 2.7 Area under the Plasma Concentration versus Time Curve (AUC) The area under the plasma (serum, or blood) concentration versus time curve (AUC) has an number of important uses in toxicology, biopharmaceutics and pharmacokinetics. Toxicology AUC can be used as a measure of drug exposure. It is derived from drug concentration and time so it gives a measure how much and for how long a drug stays in a body. A long, low concentration exposure may be as important as shorter but higher concentration. Haber's law propose exposure k = C X t (Fritz Haber, 1868-1934) where k is essentially AUC. Some drugs are dosed using AUC to quantitate the maximum tolerated exposure (AUC Dosing). The efficacy of some antibiotics are related to AUC/MIC, thus maintaining a concentration above a minimum inhibitory concentration (MIC) is more important than peak concentrations. Biopharmaceutics The AUC measured after administration of a drug product is an important parameter in the comparison of drug products. Studies can be performed whereby different drug products may be given to a panel of subjects on separate occasions. These bioequivalency or bioavailability studies can be analyzed by comparing AUC values. Pharmacokinetics Drug AUC values can be used to determine other pharmacokinetic parameters, such as clearance or bioavailability, F. Similar techniques can be used to calculate area under the first moment curve (AUMC) and thus mean resident times (MRT). Calculation of AUC The area under the plasma concentration time curve (AUC) is very useful for calculating the relative efficiency of different drug products. It can used to calculate the total body clearance (CL) and the apparent volume of distribution. If we have a smooth line for concentration versus time or an equation for Cp versus time from a pharmacokinetic model we could slice the area into vertical segments. Each segment would be very thin, Δt and in extreme dt, in width (much smaller than the segment in Figure 2.7.1). The total AUC is calculated by adding these segments together. In calculus this would be the integral. Each very narrow segment has an area equal to concentration times dt. Thus the total area (AUC) is given by Equation 2.7.1. Equation 2.7.1 Total AUC as an Integral With a first order elimination process the integrated equation for plasma concentration as a function of time is given by Equation 2.7.2. Equation 2.7.2 Cp versus t This is essentially the same as the integrated equation (Equation 2.5.3) used for amount of drug remaining. The difference here is we use concentration not amount. We can substitute this equation for Cpt into Equation 2.7.1 to derive an analytical equation for AUC (Equation 2.7.3). Equation 2.7.3 AUC as an Integral. One Compartment Bolus In Math Tables we can find Equation 2.7.4. Equation 2.7.4 Integral from Math Tables To get the total AUC we can set a = k, t1 = 0, and t2 = ∞ With t = 0, e-k•t = 1 and at t = ∞, e-k•t = 0 which results in Equation 2.7.5 and 2.7.6. Equation 2.7.5 Total AUC as the Integral Equation 2.7.6 Total AUC as a Function of Cp0 and k This is analytical integration (exact solution, given exact values for Cp0 and k). This only applies for concentrations collected after an IV bolus and first order kinetics. Also, after most dosage regimens, once the concentrations follow a semi-log linear straight line we can use a similar equation to calculate the remaining AUC. The AUC of the last segment. With t = tlast to ∞, AUC is calculated as Cpt/k. We will use this result later in this section. This can be derived from Equation 2.7.4 resulting in Equation 2.7.7. Equation 2.7.7 AUC from Last Time to Infinity We could use Equation 2.7.6 to calculate the AUC value if we knew Cp0 and k but usually we don't do this. We calculate AUC directly from the Cp versus time data (Figure 2.7.2). Thus, we need to use a different approach. The Trapezoidal Rule The simplest, most common approach is a numerical approximation method called the trapezoidal rule. We can calculate the AUC of each segment if we consider the segments to be trapezoids. [Four sided figure with two parallel sides]. The area of each segment can be calculated by multiplying the average concentration by the segment width. For the segment from Cp2 to Cp3 we can use., for example, Equation 2.7.8. Equation 2.7.8 AUC for trapezoid 2 to 3 This segment is illustrated in Figure 2.7.3. The area from the first to last data point can then be calculated by adding the areas together as shown in Equation 2.7.9 Equation 2.7.9 AUC Cp1 to Cpn This is illustrated in Figure 2.7.4. To finish this calculation we have two more areas to consider. The first and the last segments. After a rapid IV bolus with a one compartment model, the first segment can be calculated after determining the zero time plasma concentration Cp0 by extrapolation. That is, by plotting the Cp versus time data on semi-log graph paper and extending the best-fit line back to the y-axis. The first segment can be calculated using Equation 2.7.10. Equation 2.7.10 AUC from Time Zero to First Data Point If we assume that the last few data points follow a single exponential decline (a straight line on semi-log graph paper) the area of the final segment can be calculated from tlast to infinity using Equation 2.7.11. Equation 2.7.11 AUC from Last Data Point to Infinity In this way the total AUC can be calculated using Equation 2.7.12. Equation 2.7.12 Total AUC from Time Zero to Infinity In the case of an IV administration, one compartment model there is only one slope on semi-log graph paper and thus only one rate constant, k. With other modes of administration and other pharmacokinetic models multiple rate constants may be represented in the concentration versus time curve. However, if the last few data points approximate a straight line on semi-log graph paper, representing the slowest rate constant this can be used to calculate the last AUC segment from the time of the last data point to infinity. In the case of oral administration there are two rate constants, ka and kel. Either rate constant may be the slower and therefore could used to calculate the last AUC segment. In Equation 2.7.13 k is the slower rate constant. Equation 2.7.13 AUC from Last Data Point In the Chapter on multi-compartment pharmacokinetics there are two rate constants after IV administration, α and β. By definition β is less than α so the last AUC segment would be calculated using this rate constant, Equation 2.7.14. Equation 2.7.14 AUC from Last Data Point using Terminal Rate Example Calculation This calculation can be explored further with the interactive table, Interactive 2.7.1. Another example of an AUC calculation presented graphically. Click on the green link above to try the calculator. Pressing Random Data to start. Plot the data on the semi-log plot. Select or deselect data to get a good value for the terminal slope, kel, using the graph and R2 value. Select all the data and press Calculate to determine the AUC. The last segment is calculated using the last data point and the value of kel. If the expected initial concentration is zero replace the value at the top of the table, as shown above, otherwise use the calculated extrapolated value determined when Plot is pressed. Dose/AUC may be clearance if after an IV bolus or CL/F if after an extravascular, e.g. oral, dose. Other methods have been described by Yeh and Kwan (Yeh, K.C. and Kwan, K.C. 1978 A comparison of numerical integrating algorithms by trapezoidal Lagrange and spline approximation, J. Pharmacokin. Biopharm., 6, 79-98) and Purves (Purves, R.D. 1992 Optimum numerical integration methods for estimation of area under the curve (AUC) and area under the moment curve (AUMC), J. Pharmacokin. Biopharm., 20, 211-226).Chapter 3: IV Bolus One Compartment Model Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- understand the separate assumptions associated with the one compartment model, rapid IV bolus dosing and linear elimination understand the properties of first order kinetics and linear models write the differential equations for a simple pharmacokinetic model define, use, and calculate the parameters: CL (total body clearance) V (apparent volume of distribution) kel (elimination rate constant) t1/2 (half-life) AUC (area under the concentration versus time curve) as they apply to a one compartment linear model use the integrated equations for a one compartment linear model to perform various dose and dosing regimen calculations Definition: Pharmacokinetics is the study of drug and/or metabolite kinetics in the body. It deals with a mathematical description of the rates of drug movement into, within and exit from the body. It also includes the study of drug metabolism or biotransformation rates. The body is a very complex system and a drug undergoes many steps as it is being absorbed, distributed through the body, metabolized or excreted (ADME). See Figure 3.0.1. The drug also interacts with receptors and causes therapeutic and/or toxic responses. Although the details of drug kinetics are complicated it is fortunate that we can often approximate drug kinetic processes using "simple" mathematical models.3.1 Assumptions We will start the course with a one compartment - linear model. Also, we will first consider drug kinetics after a rapid intravenous injection, an IV bolus injection. According to this model we will consider the body to behave as a single well-mixed container. To use this model mathematically we need to make a number of assumptions. Rapid Mixing We assume that the drug is mixed instantaneously in blood or plasma after an IV bolus dose. The actual time taken for mixing is usually very short, within a few of minutes, and in comparison with normal sampling times it is insignificant. We usually don't sample fast enough to see drug mixing in the blood. Linear Model We also assume that drug disposition and elimination follows first order kinetics. First order kinetics means that the rate of change of drug concentration by any process is directly proportional to the drug concentration remaining to undertake that process. Remember first order kinetics is an assumption of a linear model not a one compartment model. If we have a linear model if we double the dose, the concentration will double at each time point. One compartment The drug in the blood is in rapid equilibrium with drug in the extravascular tissues. The drug concentration may not be equal in each tissue or fluid however we will assume that they are proportional to the concentration of drug in the blood at all times. This is not an exact representation however it is useful for a number of drugs to a reasonable approximation.3.2 Linear Model - First Order Kinetics First-order kinetics To illustrate first order kinetics we might consider what would happen if we were to give a drug by IV bolus injection, collect blood samples at various times and measure the plasma concentrations of the drug. We might see a steady decrease in concentration as the drug is eliminated, as shown in Figure 3.2.1. Rate versus Cp If we measure the slope of this curve at a number of times we are actually measuring the rate of change of concentration at each time point, ΔCp/Δt, represented by the straight line tangents in Figure 3.2.2. Now if we plot this rate of change versus the plasma concentration, for each data point, we will get a straight line when first order kinetics are obeyed. This is shown in Figure 3.2.3. Note the y-axis is negative on this graph. This behavior can be expressed mathematically as shown in Equation 3.2.13.3 One Compartment Model Scheme or Diagram describing the Model The one compartment pharmacokinetic model can be represented schematically as shown in Figure 3.3.1. Developing the Differential Equation In the previous section, with Figure 3.2.2, we estimated the slope of the Cp versus time line at various times. Plotting ΔCp/Δ (the slope) versus Cp produced a straight line plot (Figure 3.2.3). Thus, the rate of change of Cp versus time is proportional to the concentration remaining to be eliminated, Cp. The slope of this line, the proportionality constant, can be defined as kel, the elimination rate constant. If we measure the slope over very small time intervals we are calculating the tangent to the line. We can now say that the rate of change of Cp versus time is the differential of the concentration with respect to time as Δt approaches 0; ΔCp/Δt approaches dCp/dt which gives Equation 3.3.1. Equation 3.3.1 Rate of Change of Cp versus Concentration Equation 3.3.1 is a differential equation for the one-compartment model after an IV bolus administration. By taking very small time steps we are going from the gross or large time interval term, ΔCp/Δt, to the continuously varying dCp/dt term. Note, the negative sign in front of the kel and CL/V term. This is because the slope or tangent is decreasing or negative for positive concentration values. Equation 3.3.1 relates the rate of change of concentration, Cp versus concentration. We can also relate the rate of elimination to the concentration remaining. Previously, we developed the required differential equations by looking at the arrows leaving or entering a component of the model. In Figure 3.3.1 there is one arrow leaving component one, X. Thus we can write the differential equation for the model shown here as Equation 3.3,1. 3.4 Integrated Equation The differential equation developed in the previous section provide concise descriptions of the rate of change of drug concentration (dCp/dt) [or the elimination rate (dX/dt)]. However, they can be difficult to use when trying to determine kel or CL. The tangent of the Cp versus time plot can not be determined accurately. Later in the Chapter dealing with the Analysis of Urine Data a method of measuring the rate of excretion directly from the data will be described. However, integrated forms of Equation 3.4.1 (Equation 3.4.2 are generally more useful. Equation 3.4.1 Rate of Change of Cp versus Cp Laplace transforms or other mathematical methods can be used to integrate these equations resulting in Equation 3.4.2. Equation 3.4.2 Concentration versus Time after an IV Bolus This equation describes the single exponential decline in drug concentration as a function of time. This fall in plasma concentration is called mono-exponential decay. If we know kel (or CL and V) and Cp0 we could calculate Cp at any time after a single IV bolus dose. However, it still isn't very convenient for estimating a value of kel from concentration versus time data. We could use a non linear regression program such as Boomer however for estimation purposes using graph paper we would prefer a straight line equation. A straight line equation can be achieved by taking the natural logarithm of both side of Equation 3.4.2 resulting in Equation 3.4.3. Equation 3.4.3 Ln(Cp) versus Time This integrated (logarithmic) form of the equation for Cp represents a straight line equation, that is an equation of the form: y = a - m • t with a = intercept = ln(Cp0) and m = slope = -kel or - CL/V. Plotting ln(Cp) versus t should give a straight line with a slope of - kel or -CL/V and an intercept of ln(Cp0) as shown in Figure 3.4.1. NOTICE, there are no UNITS for ln(Cp) in Figure 3.4.1. There are units of hour for time (X axis) so the slope has units of time-1 e.g. min-1, hr-1. Here we can measure kel by determining Cp versus time and plotting ln(Cp) versus time. Alternately we could use semi-log graph paper. As mentioned earlier the scale on the y-axis of semi-log graph paper is proportional to the log of the number, not the number itself. This plot (Figure 3.4.2) allows us to calculate the slope and kel given Cp versus time data. 3.5 Total Body Clearance, CL Total body clearance or clearance is an important pharmacokinetic parameter that describes how quickly a drug is eliminated from the body. Defining Equation and Units Clearance, CL, is often defined as the volume of blood or plasma completely cleared of the drug per time. However, it may be easier to view it as the proportionality constant relating the rate of elimination and drug concentration. The rate of elimination of a drug can be described by Equation 3.5.1. Equation 3.5.1 Rate of Elimination In this equation, the elimination rate, dX/dt, is related to the concentration of drug remaining. The proportionality constant for this relationship is Clearance. The symbol for clearance is CL (TBC, total body clearance) and the units are volume per time such as mL/min, L/hr. Equation Equation 3.5.1 can be rearranged to solve for clearance as shown in Equation 3.5.2. Equation 3.5.2 Clearance as Rate of Elimination divided by Cp Clearance can be calculated with this equation by measuring the amount of drug eliminated during some time interval and the drug concentration at the midpoint of this collection interval. The clearance of the endogenous material, creatinine, is determined by this method to provide a measure of renal function. The renal clearance of other compounds can be calculated in a similar fashion. Clearance can also be calculated using the integral of Equation 3.5.2. Integrating dX/dt and Cp with respect to time give Dose and AUC, respectively. The total amount that can be eliminated is the total amount administered, that is, the dose. Thus clearance can be calculated from the dose and the calculated area under the concentration versus time curve, AUC, Equation 3.5.3. Equation 3.5.3 Clearance Calculated as Dose divided by AUC In the case of the one compartment model and an IV bolus dose the rate of elimination can be expressed as a function of kel and amount, where amount is apparent volume times concentration, V x Cp, Equation 3.5.4. Equation 3.5.4 Rate of Elimination as a Function of Cp or X Comparison of Equation 3.5.1 and Equation 3.5.4 indicates that a value of CL can be estimated from kel and V from a one compartment model, Equation 3.5.5. Equation 3.5.5 Clearance - One Compartment Linear Model The clearance of a drug can be used to understand the processes involved in drug elimination, distribution and metabolism. Relating clearance to a patient’s renal or hepatic function can help in the determination of suitable drug dosage regimens. Although clearance may be calculated using Equation 3.5.5 a more fundamental calculation may be the determination of kel from CL and V, Equation 3.5.6. Equation 3.5.6 kel Calculated from CL and V The elimination half-life can also be calculated from CL and V using Equation 3.5.7. Equation 3.5.7 Half-life Calculated from CL and V The interplay between V, CL and kel (or t1/2) can be explored using the Interactive 3.5.1. Notice, as V increases the amount of drug per volume of plasma, i.e. the concentration, decreases. For a given value of clearance the rate of elimination and the elimination rate constant decreases (with a corresponding increase in half-life). Use the Change button in the lower left to select the CL and V or kel and V parameters. The interplay between V, CL and kel or t1/2 can be explored using Interactive 3.5.1 or Interactive 3.5.2. Notice, as V increases the amount of drug per volume of plasma, i.e. the concentration, decreases. Therefore for a given value of clearance the rate of elimination and the elimination rate constant decreases (with a corresponding increase in half-life). References Clearance has been discussed on the PharmPK listserv. 3.6 Apparent Volume of Distribution, V We can use Equation 3.6.1 to calculate the plasma concentration at any time when we know CL and Cp0. Equation 3.6.1 Concentration versus Time However, usually we don't know Cp0 ahead of time, but we do know the dose. A dose in mass units, often in mg. To calculate Cp0 we need to know the volume that the drug is distributed into. That is, the apparent volume of the mixing container, the body. This apparent volume of distribution, V, is not a physiological volume. It won't be lower than blood or plasma volume but for some drugs it can be much larger than body volume. It is a mathematical 'fudge' factor relating the amount of drug in the body and the concentration of drug in the measured compartment, usually plasma, serum or blood. Defining Equations and Units The apparent volume of distribution can be defined in terms of the amount of drug in the body and the concentration measured in plasma, serum or blood, Equation 3.6.2. Equation 3.6.2 Apparent Volume of Distribution We’ll use plasma for most of the rest of the book but most of the equations and other details could apply to other samples. The units for the apparent volume of distribution are volume units. Most commonly V is expressed in liters, L. On occasion the value for the apparent volume of distribution will be normalized for the weight of the subject and expressed as a percentage or more usually in liters/kilogram, L/Kg. Mathematically Equation 3.6.2 can be represented as Equation 3.6.3. Equation 3.6.3 Apparent Volume of Distribution Immediately after the intravenous dose is administered the amount of drug in the body is the IV dose so the amount is the dose and the concentration is the initial concentration, Cp0, Equation 3.6.4. Equation 3.6.4 Apparent Volume of Distribution from Dose and Cp0 Rearranging gives Equation 3.6.5 for Cp0. Equation 3.6.5 Cp0 from Dose and V Combining Equation 3.6.1 and Equation 3.6.5 provides an equation for concentration versus time given values for dose, kel or CL and V, Equation 3.6.6. Equation 3.6.6 Concentration versus Time The one compartment model assumption is that there is a rapid equilibration in drug concentrations throughout the body, however, this does not mean that the concentration is the same throughout the body. This is illustrated in Figure 3.6.1. In the first beaker the concentration throughout the beaker is the same and the apparent volume of distribution is the same as the size of the beaker. In the second beaker after a rapid equilibrium, distribution between the solution (representing plasma) and the charcoal (representing various tissues of the body) may be complete. However, drug concentrations within the beaker (representing the patient) are not uniform. Much of the drug is held with the charcoal leaving much smaller concentrations in the solution. After measuring the drug concentration in the solution the apparent volume of the patient is much larger, the apparent volume of distribution is much larger. Determining Values of V The usual method of calculating the apparent volume of distribution of the one compartment model is to extrapolate concentration versus time data back to the Y-axis origin. See Figure 3.3.2 for an example. This gives an estimate of Cp0. When the IV bolus dose is know the apparent volume of distribution can be calculated from Equation 3.6.4. The line in Figure 3.3.2 was calculated with Equation 3.6.6 using a dose of 450 mg, apparent volume of distribution of 15 L and clearance of 3.0 L/hr. Explore this calculation using Interactive 3.6.1. References Gibaldi, M. 1984 "Biopharmaceutics and Clinical Pharmacokinetics", 3rd ed., Lea & Febiger, Chapter 12, page 214 Activated carbon at Wikipedia Volume has been discussed on the PharmPK listserv 3.7 Elimination Rate Constant, kel The elimination rate constant (abbreviated as kel or k10) is the first order rate constant describing drug elimination from the body. This is an overall elimination rate constant describing removal of the drug by all elimination processes including excretion and metabolism. Metabolites are different chemical entities and have their own elimination rate constant. The elimination rate constant is the proportionality constant relating the rate of change of drug concentration and concentration OR the rate of elimination of the drug and the amount of drug remaining to be eliminated. Defining Equations and Units The parameter kel can be defined in terms of change in drug concentration (Equation 3.7.1) or change in drug amount (Equation 3.7.2). Equation 3.7.1 Rate of Change of Cp versus Cp Equation 3.7.2 Rate of Change of Amount, X, versus X By inspection of these equations it can be seen that the units for kel are time-1, for example hr-1, min-1, or even day-1. In both equations the rate expression is divided by Cp or X, respectively to provide units for kel. Units for kel are concentration per time divided by concentration if using the concentration equation or mass per time divided by mass if using the amount equation. In both cases the units are reciprocal time, Table 3.7.1. Determining Values of kel From the integrated equation presented in an earlier section, Equation 3.3.3, kel can be calculated from the negative slope of line drawn on a semi-log plot of Cp versus time. Thus with two value for Cp and time a value for kel can be determined directly from Equation 3.7.3. Equation 3.7.3 Calculation of kel from Two Data Points With more than two Cp versus time data points it is possible to plot the data on semi-log graph paper and draw a 'best-fit' line through the points. This plot was shown in the previous section. The best answer for kel can be calculated by taking points at either end of the 'best-fit' line. This approach has been covered in more detail in an earlier section, section 2.3. Equation 3.6.1 can be expanded to provide Equation 3.7.3 as means of calculating concentration at any time, t, after an IV dose with kel and V. Explore this calculation using Interactive 3.7.1. Use the change button to select kel and V versus CL and V. Equation 3.7.3 Concentration versus Time from Dose, kel and V Note: It is important to distinguish between the elimination rate and the elimination rate constant. The rate (tangent or slope, dCp/dt) changes as the concentration changes, however, for a first-order, linear process the rate constant (here kel) is constant, it does not change.3.8 Half-life of Elimination, t1/2 Another important property of first order kinetics is the half-life of elimination, t1/2. Defining Equation and Units The elimination half-life is the time taken for the plasma concentration to fall to half its original value. Units for this parameter are units of time such as hour, minute, or day. Thus if Cp is the concentration at time one, Cp/2 is the concentration after one half-life. An equation for half-life can be derived from Equation 3.8.1 for concentration versus time assuming linear or first-order elimination kinetics. Equation 3.8.1 Derivation of Equation for t1/2 Equation 3.8.2 kel from half-life Rearranging Equation 3.8.1 gives an equation for kel, Equation 3.8.2. Note: Half-life, t1/2, is independent of concentration. This a property of first order processes These equations can be used as an approximate method of calculating kel. If we look at a plot of Cp versus time on semi-log graph paper, Figure 3.8.1. The steps to take are: 1. Draw a line through the points (this tends to average the data, a best-fit line) 2. Pick any Cp and t1 on the line 3. Determine Cp/2 and t2 using the line 4. Calculate t1/2 as (t2 - t1) And finally kel = 0.693/t1/2 (Equation 3.8.2) You might also consider determining Cp/4 or Cp/8 after two half-lives or three half-lives, respectively. This should provide a more accurate answer as the differences in Cp and t will be larger. The line smooths out the bumps. There may be less accurate data points, so by drawing a line you average the data. Remember, the half-life is the same whether going from 40 to 20 or from 10 to 5 mg/L. This is a property of the first order process. Note: Going from: Cp - > Cp/2 in 1 half-life i.e. 50.0 % lost 50.0 %
Cp - > Cp/4 in 2 half-lives i.e. 25.0 % lost 75.0 %
Cp - > Cp/8 in 3 half-lives i.e. 12.5 % lost 87.5 %
Cp - > Cp/16 in 4 half-lives i.e. 6.25 % lost 93.75 %
Cp - > Cp/32 in 5 half-lives i.e. 3.125 % lost 96.875 %
Cp - > Cp/64 in 6 half-lives i.e. 1.563 % lost 98.438 %
Cp - > Cp/128 in 7 half-lives i.e. 0.781 % lost 99.219 % Thus over 95 % is lost or eliminated after 5 half-lives. Typically, with pharmacokinetic processes, this is considered completion of the process [Although in theory it takes an infinite time]. For other purposes you may wish to wait 7 half-lives where over 99% of the process is complete. The half-life describes the time it takes for a drug concentration (or other process) to fall to half the original value. For first order processes (as described and derived as above) this time is independent of concentration. When the kinetics are described by non-linear (not first order) kinetics, for example Michaelis-Menten kinetics, the half-life at one concentration may be quite different from the half-life at another concentration. In the pharmacokinetic area of study the half-life of a drug usually refers to the biological or terminal half-life. Generally both terms refer to the half-life measured for the terminal or slowest slope on the semi-log drug concentration versus time plot. At low concentration more processes tend to follow first order kinetics. However, at later times with lower concentrations assay sensitivity can be a serious problem. Also, if absorption is very slow the slowest slope may refer to the absorption process instead of drug disposition. References Ritschel, W.A. and Kearns, G.L. 2004 Handbook of Basic Pharmacokinetics ... including Clinical Applications, 6th ed., American Pharmaceutical Association, Washington, DC ISBN 1-58212-054-4 pp 369-401 Half-life has been discussed on the PharmPK listserv. 3.9 Some Example Calculations Example 1 Question: What is the concentration of drug 0, 2 and 4 hours after a dose of 500 mg. Known pharmacokinetic parameters are apparent volume of distribution, V is 30 liter and the elimination rate constant, kel, is 0.2 hr-1. Answer: Use Equation 3.9.1 to calculate the required concentrations shown in Equation 3.9.2. Equation 3.9.1 Concentration versus Time Equation 3.9.2 Calculated Concentrations at 0, 2 and 4 hours Example 2 We can also calculate the pharmacokinetic parameters kel and V if we know the dose given and the plasma concentrations at two (or more) times after an IV bolus administration. Question: If Cp2 hours is 4.5 mg/liter and Cp6 hours is 3.7 mg/liter after a 400 mg IV bolus dose what are the values of kel and V? Answer: This time we will use Equation 3.9.3. to calculate kel. This calculation is shown in Equation 3.9.4. Equation 3.9.3 Calculating kel with Two Data Points Equation 3.9.4 kel calculated from Two Data Points Now using either the 2 hour or the 6 hour data point calculate the concentration at time zero, see Equation 3.9.5. Equation 3.9.5 Calculating Cp0 extrapolating from Cp2 The last step is to calculate the apparent volume of distribution, V, as shown in Equation 3.9.6. Equation 3.9.6 V from Dose and Cp0 One additional step might be to calculate the clearance assuming a one compartment model with first-order elimination. This calculation is shown in Equation 3.9.7. Equation 3.9.7 Clearance from kel and V Example 3 Question: What IV bolus dose is required to achieve a plasma concentration of at least 2.4 µg/mL (= 2.4 mg/L) for as long as 6 hours after the dose is administered. The elimination rate constant, kel, is 0.17 hr- 1 and the apparent volume of distribution, V, is 25 L. Answer: Rearranging Equation 3.9.1 gives Equation 3.9.8 which can be used by entering the parameter values given in the problem and calculating the required dose, Equation 3.9.9. Equation 3.9.8 Dose for Cp at time t Equation 3.9.9 Dose Calculated for Specified Cp at Time 6 hr Example 4 Question: After an I.V. bolus dose of 500 mg the data in Table 3.9.1 were collected. Calculate kel and V. Plot the data on semi-log graph paper (Figure 3.9.1) and determine Cp1, Cp2, t1 and t2. The elimination rate constant, kel, apparent volume of distribution, V, and the clearance, CL, can be calculated as before for Example 2, Equation 3.9.10, Equation 3.9.11 and Equation 3.9.12. Equation 3.9.10 Elimination Rate Constant, kel Equation 3.9.11 Apparent Volume of Distribution, V Equation 3.9.12 Clearance, CL Explore this calculation further using Interactive 3.9.1. Chapter 4: Analysis of Urine Data Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- draw the scheme and write the differential equations for a one compartment pharmacokinetic model with elimination of drug and metabolite into urine (parallel pathways of elimination) use the appropriate integrated equations for this pharmacokinetic model to calculate amount of drug excreted into urine plot cumulative amount excreted versus time, A.R.E. versus time and rate of excretion versus time (midpoint) and use these graphs to calculate pharmacokinetic parameters define, use, and calculate the parameters: CLR and CLM (renal and non renal clearance) ke (excretion rate constant) km (metabolism rate constant) U∞ and M∞ fe and fm use fe, the fraction excreted, to calculate overall elimination rate constants in patients with impaired renal function So far we have looked at information we can get from plasma data following a rapid intravenous dose of a drug using a one compartment model. There is another part of the model which can be readily sampled. Sometimes it is not possible to collect blood or plasma samples but we may be able to measure the amount of drug excreted into urine. we may not want to take repeated blood samples from certain patient populations, for example very young, pediatric patients the apparent volume of distribution maybe so large that plasma concentrations are too small to measure accurately it may be important to determine the role of metabolism in the elimination of a drug. Analysis of urine data for unchanged drug and metabolite concentrations is essential to the quantitative study of drug metabolism If we collect data for amount of drug excreted into urine it may be possible to determine the elimination rate constant or half-life and other pharmacokinetic parameters. 4.1 Metabolism and Excretion - Parallel Pathways Although a few drugs are eliminated as unchanged drug into urine or alternately may be completely metabolized, most drugs are eliminated by excretion AND metabolism. There is often more than one excretion or metabolism pathway. Scheme or diagram Parallel elimination pathways are illustrated in Figure 4.1.1 where ke is the excretion rate constant and km is the metabolism rate constant. Here we have two pathways for elimination (with others as a shadow). We can write the differential equations for the four components shown in this diagram (X, U, M, Mu). There could be more pathways. It may be necessary to specify excretion by exhalation, in sweat, or as is commonly the case, more than one metabolic pathway. For now we will consider one excretion path (unchanged drug in urine) and metabolism, with the metabolite also excreted in urine. The Equations Drug in the Body, X For X = V • Cp, amount of drug in the body Equation 4.1.1 includes terms for excretion and metabolism. The number (and type) of these elimination processes can be changed to accommodate a variety of possible routes of excretion or metabolism. Some of these processes may not be first order, however many can be represented by first order parameters. Equation 4.1.1 Rate of Change of the Amount of Drug in the Body The elimination rate constant, kel, represents the sum of all the ('first-order') rate constants so we can substitute kel for (ke + km) in Equation 4.1.1 to give Equation 4.1.2. Equation 4.1.2 Rate of Change of the Amount of Drug in the Body Dividing by V gives Equation 4.1.3, which is the same as Equation 3.4.1 in Chapter 3. Equation 4.1.3 Rate of Change of Drug Concentration In Equation 4.1.4 the rate of elimination of drug from the central or plasma compartment is expressed in clearance terms, here, renal (CLR) and metabolic (CLM) clearance. The total body clearance (CL) is equal to the sum of the clearance terms in the model. In Equation 4.1.4 total body clearance, CL = CLR + CLM. Equation 4.1.4 Rate of Elimination using Clearance Parameters With more elimination pathways we sum all these process parameters to arrive at the elimination rate constant, kel, or total body clearance, CL. Thus the equation for rate of elimination, either with rate constant parameters or clearance parameters, is the same as before in Chapter 3. The integrated equation is given in Equation 4.1.5 and Equation 4.1.6. Equation 4.1.5 Cp versus Time with kel as the Elimination Parameter Equation 4.1.6 Cp versus Time with CL as the Elimination Parameter Drug Excreted into Urine, U The rate of excretion, dU/dt, can be derived from the model Figure 4.1.1, in terms of ke or CLR, Equation 4.1.7. Equation 4.1.7 Rate of Change of Cumulative Amount Excreted into Urine Substituting for Cp (= (Dose/V) • e-kel • t or = (Dose/V) • e-CL • t / V) we get Equation 4.1.8. Equation 4.1.8 Rate of Excretion of Unchanged Drug into Urine Integrating using Laplace transforms gives Equation 4.1.9, for the cumulative amounted excreted unchanged in urine, U, versus time. Equation 4.1.9 Cumulative Amount Excreted as Unchanged Drug versus Time Note: ke or CLR are in the numerator of Equation 4.1.9. As time approaches infinity the exponential term, e-k • t, approaches zero. Setting the exponential terms in Equation 4.1.9 to zero gives Equation 4.1.10 for the total amount of unchanged drug excreted in urine, U∞. Equation 4.1.10 Total Amount Excreted as Unchanged Drug into Urine Amount of Drug Metabolized in the body, M, and excreted into urine, Mu For M and Mu, the amount of drug which has been metabolized, the differential equations are Equation 4.1.11 and Equation 4.1.12. Equation 4.1.11 Rate of Change of Amount of Metabolite in the Central Compartment Equation 4.1.12 Rate of Excretion of Metabolite into Urine After integrating Equation 4.1.11 or Equation 4.1.12 using Laplace transforms we get Equation 4.1.13 for Mu. Equation 4.1.13 Cumulative Amount Excreted as Metabolite versus Time Setting each exponential term, e-k • t, in Equation 4.1.13 to zero gives Equation 4.1.14 for the total amount excreted into urine as the metabolite. With fm the fraction metabolized. Equation 4.1.14 Total Amount Excreted as Metabolite into Urine Adding Equation 4.1.10 and Equation 4.1.14 gives Equation 4.1.15. Equation 4.1.15 Mass Balance - Total Amount Eliminated equals Dose Notice the total amount of the drug excreted and metabolized adds up to the Dose. This is based on the assumption that we have information from all the pathways of elimination. In Figure 4.1.2 we have simulated amounts of drug in the body (X), unchanged drug in urine (U), metabolite in the body (M) and metabolite excreted in urine (Mu). The parameters used to create this figure were Dose = 500 mg, kel = 0.2 hr-1, fe = 0.25, fm = 0.75, and kmu = 0.5 hr-1. Note U∞ and Mu∞, 125 and 375 mg respectively, total 500 mg, the dose given. After all these equations have fun and explore the effect of changing the parameters, Dose, kel, fe and kmu with Interactive 4.1.1. 4.2 Plotting and Analyzing Urine Data Cumulative amount excreted versus time Urine data is usually collected as drug concentration in the urine sample and the volume of the sample at the end of the collection interval. Multiplying these numbers together gives the amount of drug excreted during the collection interval as ΔU. Accumulating these ΔU values gives the cumulative amount of drug excreted up to the specified time, U. We can plot U versus time as the cumulative amount excreted versus time plot. As we lose drug from the body it will appear in urine. Earlier we wrote the differential equation for U and presented the integrated equation for U. The linear plot of cumulative amount excreted into urine as unchanged drug versus time is shown below. Notice that the value of U∞ is NOT EQUAL to the dose, it is somewhat less than dose, unless all of the dose is excreted into urine as unchanged drug. The remaining portion of the dose should be found as metabolites and from other routes of excretion. The equation for the cumulative amount excreted versus time is shown below in Equation 4.2.1. Equation 4.2.1 Cumulative Amount Excreted into Urine versus Time This results in the plot shown in Figure 4.2.1. Notice that the total amount excreted as unchanged drug, U∞, is less than the dose administered in this plot. Cumulative amount excreted plots are basically descriptive in nature. At most, you can get a general sense of how much drug is excreted, an estimate of U∞ and an approximate estimate of t1/2. Rate of excretion (R/E) Equation 4.2.2 represents the rate of excretion of unchanged drug into urine as an exponential function. This equation was derived in the previous section. Equation 4.2.2 Rate of Excretion of Unchanged Drug into Urine Since urine data is collected over a relatively large intervals of time the data is better represented as ΔU rather than dU. Also, since ΔU is collected over a significant time interval the time point for this interval should be the midpoint of the interval, tmidpoint rather the time at the beginning or the end of the interval. Equation 4.2.2 can be replaced by Equation 4.2.3 to better represent the experimental data collected. It may look like a strange way of plotting the data, but actually it's quite convenient to use because urine data results are collected as an amount of drug excreted during a time interval. Equation 4.2.3 Rate of Excretion of Unchanged Drug versus Midpoint Time The amount excreted is the product of the volume of urine voided times the concentration of drug in the sample. This is already a measure of rate of excretion. Taking the ln of both sides gives Equation 4.2.4, an equation for a straight line. Plotting ln[ΔU/Δt] versus midpoint time provides kel from the slope and ke from the intercept divided by the Dose. Equation 4.2.4 Natural log of Rate of Excretion versus Midpoint Time Note: The intercept (on semi-log graph paper) is ke • DOSE, Figure 4.2.2. Also, NOTE: This plot is rate of excretion versus midpoint of the collection interval, time. Rate of excretion plots can be very useful in the determination of parameters such as kel, ke and fe. There can be a little more scatter than with the ARE plot, mentioned below, but the rate of excretion plot has significant advantages. Analysis of 'real' data may show considerable scatter in the rate of excretion plot. Thus, positioning the straight line on a semi-log plot may be difficult. In practical terms it is difficult to get a lot of early times points unless the subjects are catheterized and even then early times may be difficult to interpret. This means that this method can be difficult to use with drugs which have short half-lives. However, a significant advantage of the rate of excretion plot is that each data point is essentially independent, especially if the bladder is fully voided for each sample. A missed sample or data points is not critical to the analysis. Amount remaining to be excreted (ARE) The third plot is the amount remaining to be excreted, or ARE plot. The equation for this plot can be derived from Equation 4.2.1 rewritten in terms of U∞ (Equation 4.2.5) to give Equation 4.2.6 and in turn Equation 4.2.7. Rearranging Equation 4.2.7 gives Equation 4.2.8. Equation 4.2.5 Total Amount Excreted into Urine as Unchanged Drug, U∞ Equation 4.2.6 Cumulative Amount Excreted versus Time Equation 4.2.7 Cumulative Amount Excreted versus Time Equation 4.2.8 Amount Remaining to be Excreted (ARE) as Unchanged Drug versus Time Taking the log of both sides give the straight line equation. Plotting ln[U∞ - U] versus time provides kel from the slope and U∞ from the intercept, Equation 4.2.9, where fe is the fraction excreted as unchanged drug into urine. Equation 4.2.9 Ln(ARE) versus Time The intercept, U∞, can be used to determine fe and thus ke, Equation 4.2.9. Amount remaining to be excreted (ARE) plots (Figure 4.2.3) use the U∞ value to estimate each data point. Error in any data is accumulated into U∞ and thus each ARE value. This can lead to curved lines (instead of the expected straight lines) or an inability to use the method (with a missing sample). However, with good data these plots can be useful and may be somewhat smoother than rate of excretion versus time plots. These three plots, cumulative amount excreted versus end of interval time, rate of excretion versus midpoint time and amount remaining to be excreted can be further investigated using Interactive 4.2.1, Interactive 4.2.2 and Interactive 4.2.3. 4.3 Fraction Excreted or Metabolized, fe or fm The equations provided in the two previous sections provide two new parameters, the fraction excreted unchanged in urine, fe (Equation 4.3.1), and the fraction metabolized, fm (Equation 4.3.2). Equation 4.3.2 could be repeated for any number of metabolites. Note however: fe + fm1 + fm2 + ... = 1 Equation 4.3.1 Fraction Excreted into Urine as Unchanged Drug Equation 4.3.2 Fraction Excreted into Urine as Metabolite Renal function We can now use this information to start to understand dosage adjustments for patients with poor kidney function. These patients will have reduced ability to excrete some drugs. That is the ke value for a drug will be lower in these patients than in normal patients. Depending on the value of fe this may have a large effect on kel or it may be insignificant. Fortunately there are a number of clinical tests for renal function which can be used. One common one is creatinine clearance (CLCr). Creatinine is formed in the body and is excreted almost entirely by filtration in the kidney. The normal creatinine clearance value is similar to the glomerular filtration rate of 120 to 130 mL/min. We can measure the CLCr prior to drug treatment and adjust the dosage accordingly. For example. A drug with an fe = 0.95 (Dose 500 mg; V = 33L; Cp0 = 15 mg/L) The normal kel = 0.116 hr-1 (t1/2 = 6 hr) Equation 4.3.1, the equation for fe, leads to: ke = kelnormal x fe = 0.116 x 0.95 = 0.110 hr-1 km = kelnormal x fm = kelnormal x (1 - fe) = 0.116 x 0.05 = 0.006 hr-1 If we now consider a patient with a creatinine clearance of 12 to 13 mL/min. That is, about a tenth of the normal kidney function, ke should be about a tenth of normal in this patient. Therefore, kepatient = 0.011 hr-1 Now assuming km is unchanged kelpatient = kepatient + km = 0.011 + 0.006 hr-1 = 0.017 hr-1 (t1/2 = 41 hour) Thus the half-life changes from 6 hours to 41 hours in this patient with impaired renal function. It takes seven times longer for the body to eliminate half the dose. If repeated doses were given based on a normal half-life the levels in this patient would rapidly reach toxic concentrations. Compare Figure 4.3.1 with Figure 4.3.2. Another example For another drug with fe = 0.15 (Dose = 250 mg) the normal kel = 0.58 hr-1 (t1/2 = 1.2 hr) kenormal = kelnormal x fe = 0.58 x 0.15 = 0.087 hr-1 km = kelnormal x fm = kelnormal x (1 - fe) = 0.58 x 0.85 = 0.493 hr-1 If we now consider a patient with renal function and ke reduced by a tenth The kepatient = 0.009 hr-1 and again assuming that km is unchanged kelpatient = kepatient + km = 0.009 + 0.493 hr-1 = 0.502 hr-1 (t1/2 = 1.4 hour) Thus the half-life changes from 1.2 hour to 1.4 hours, a small change in the elimination rate despite the same large change in renal function. The fraction excreted, fe, provides a clue to whether or not dosing regimens need to be changes in patient with poor renal function. References Bennett, WM et al. 1977 Guidelines for Drug Therapy in Renal Failure, Annuals of Int. Medicine 86(6), 754-83 Bennett et al. 1974 A Guide to Drug Therapy in Renal Failure, J.A.M.A. 230(11), 1544-15534.4 Clearance Clearance can be defined as the volume of plasma which is completely cleared of drug per unit time but this isn't always a clear explanation of this important parameter. The symbol is CL and common units are ml/min, L/hr, i.e. volume per time. One way of looking at clearance is to consider the drug being eliminated from the body ONLY via the kidneys. [If we were to also assume that all of the drug that reaches the kidneys is removed from the plasma then we have a situation where the clearance of the drug is equal to the plasma flow rate to the kidneys. All of the plasma reaching the kidneys would be cleared of drug]. The amount cleared by the body per unit time is dX/dt (here equal to dU/dt), the rate of elimination (also the rate of excretion in this example). To calculate the volume which contains that amount of drug we can divide by Cp. That is the volume = amount/concentration. This is the same as Equation 3.5.2 in Chapter 3, Equation 4.4.1. Equation 4.4.1 Clearance as the Ratio of Rate of Excretion to Cp For this particular example where elimination = excretion and kel = ke we can derive another equation for clearance which may useful, Equation 4.4.3. Since Equation 4.4.2 Rate of Excretion Equation 4.4.3 Clearance Calculated from kel and V As we have defined the term here CL is the total body clearance. We have assumed that the drug is cleared totally by excretion in urine. Below we will see that the total body clearance can be divided into clearance due to renal excretion and that due to other processes such as metabolism. Clearance is a useful term when talking of drug elimination since it can be related to the efficiency of the organs of elimination and blood flow to the organ of elimination. It is useful in investigating mechanisms of elimination and renal or hepatic function in cases of reduced clearance of test substances. t The units of clearance, volume/time (e.g. ml/min) may be easier to visualize, compared with elimination rate constant (units 1/time, e.g. 1/hr) although half-life (in units of time) is probably even easier. Some people view clearance as a primary pharmacokinetic parameter along with the apparent volume of distribution although both these parameters can be related to more fundamental terms. As an example if the kidney removes all of the drug presented by blood flow then the renal clearance will be equal to renal blood flow, Qrenal (= Qkidney), Figure 4.4.1. In later chapters we will look at some of these more fundamental terms. When a drug is eliminated by more than one pathway total body clearance, CL, can be separated into various clearance terms describing these pathways. Thus, total body clearance might be split into clearance due to renal excretion, CLR, Equation 4.4.4 and clearance due to another pathway such as metabolism, CLM, Equation 4.4.5. Equation 4.4.4 Renal Clearance from ke and V Equation 4.4.5 Metabolic or Hepatic Clearance These are mathematical representation of CLR and CLM. Later we will see that renal clearance is functionally dependent on processes such as glomerular filtration rate, renal secretion and reabsorption. Also we will see that liver or hepatic clearance is functionally dependent on factors such as liver blood flow, fraction of unbound drug and intrinsic liver metabolism. To complete this sequence of equations total body clearance can be calculated from kel and V, Equation 4.4.6. Equation 4.4.6 Total Body Clearance Clearance Calculated from AUC Another more general method of calculating clearance can be derived from the basic definition, Equation 4.4.7. Equation 4.4.7 Total Body Clearance from Rate of Excretion and Cp For renal clearance we can write rate of excretion from renal clearance and plasma concentration, Equation 4.4.8. Equation 4.4.8 Rate of Excretion from Renal Clearance and Cp Integrating both sides gives Equation 4.4.9. Equation 4.4.9 Total Amount Excreted as Unchanged Drug Rearranging gives Equation 4.4.10. Equation 4.4.10 Renal Clearance from U∞ and AUC Hepatic or metabolic clearance can be derived in a similar fashion, Equation 4.4.11. Equation 4.4.11 Metabolic or Hepatic Clearance from Mu∞ and AUC Since the total amount eliminated is the IV dose (the total amount absorbed) total body clearance can also be calculated from the AUC, Equation 4.4.12. Equation 4.4.12 Total Body Clearance from Dose and AUC Unlike Equation 4.4.4, Equation 4.4.5 and Equation 4.4.6 which use model derived parameter values Equation 4.4.10, Equation 4.4.11 and Equation 4.4.12 use the model independent parameters AUC and Dose, U∞ or Mu∞ when the disposition (distribution and elimination) parameters are linear or first-order. Variable Renal Clearance Equation 4.4.10 can be useful in determining if renal clearance is consistent throughout or between dosing intervals. Plotting ΔU/Δt versus Cpmidpoint for the collection interval, Δt, should provide a straight line if renal clearance is constant. If clearance is NOT constant after the drug administration dividing ΔU/Δt by Cpmidpoint provides estimates of renal clearance for each of the collection periods (Equation 4.4.13). Equation 4.4.13 Renal Clearance from Rate of Excretion and CpMidpoint. 4.5 Example Calculations using Urine Analysis Plots After an IV dose of 300 mg, total urine samples were collected and assayed for drug concentration. Thus the data collected is the volume of urine collected and the drug concentration in urine during each interval. These are the data in columns 1, 2 and 3 of Interactive 4.5.1. Completing the Table Multiplying column 2 by column 3 gives column 4, the amount excreted during the interval, ΔU. Accumulating the ΔU amounts in column 4 gives the cumulative amount excreted, U, up to the end of the current time interval. The total amount excreted, U∞ is found at the bottom of column 5, that is 200.1 mg. The rate of excretion, ΔU/Δt, is calculated by dividing the ΔU amount by the length of the time interval, Δt. A.R.E. in the last column is calculated by subtracting the value for U in each row from U∞. The Data Plots Cumulative Amount Excreted into Urine Plot The plot in Figure 4.5.1 shows U rapidly increasing at first then leveling off to U∞ (= 200 mg). NOTE: U∞ ≠ DOSE for this set of data. Notice also that U∞/2 (100 mg) is excreted in about 3 hours which gives an estimate of the elimination half-life. Otherwise this plot is a qualitative representation of the data. Calculation Using Rate of Excretion Data The plots in Figure 4.5.2 and Figure 4.5.3 are more useful for calculating parameter values. A straight line can be drawn through the data on each semi-log plot. The elimination rate constant, kel, can be determined from the slope of this line and ke or fe determined from the intercept. Figure 4.5.2 provides a semi-log plot of ΔU/Δt versus tmidpoint. As you can see this gives a straight line plot. Estimating the intercept value to be 53 mg/hr and if the line crosses the axis at 23.6 hr where the rate of excretion is 0.1 mg/hr a value for kel can be estimated (Equation 4.5.1) and ke can be determined from the intercept (Equation 4.5.2). Equation 4.5.1 Calculation of kel from Rate of Excretion Plot Equation 4.5.2 Calculation of ke from the Intercept of the Rate of Excretion Plot Thus fe = ke/kel = 0.177/0.266 = 0.665. This plot can be used to estimate kel, ke and fe. A disadvantage of this type of plot is that the error present in "real" data can obscure the straight line and lead to results which lack precision. Also it can be difficult to collect frequent, accurately timed urine samples. This is especially true when the elimination half-life is small. Calculation Using ARE Data The ARE data are plotted on Figure 4.5.3. Estimating the intercept value to be 210 mg and if the line crosses the axis at 19 hr where the ARE is 1 mg a value for kel can be estimated from the slope (Equation 4.5.3) and fe can be determined from the intercept (Equation 4.5.4). Equation 4.5.3 Calculation of kel from the ARE Plot Equation 4.5.4 Calculation of fe from the Intercept of the ARE Plot Thus ke = fe x kel = 0.70 x 0.281 = 0.197 hr-1. One disadvantage of this approach is that the errors are cumulative, with collection interval, and the total error is incorporated into the U∞ values and therefore into each ARE value. Another problem is that total (all) urine collections are necessary. One missed sample means errors in all the results calculated. Try these calculations using Interactive 4.5.2 (Rate of Excretion) or Interactive 4.5.3 (ARE).Chapter 5: Intravenous Infusion One Compartment Linear Model Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- draw the schemes and write the differential equations for a one compartment pharmacokinetic model after IV infusion administration use the integrated equations for drug concentrations during and after an IV infusion administration to calculate parameter values and suitable dosing regimens including IV infusion alone, fast/slow IV infusion or infusion/bolus dosage regimens calculate kel, CL and V from data collected after a single IV infusion define, use, and calculate the parameters: k0 (infusion rate constant) D (infusion duration) Hospital patients will commonly receive drugs by intravenous infusion. The inconvenience of administering the drug over a long time is not a real problem with bedridden patients. Some may already be receiving intravenous fluids. If a drug is chemically stable and is compatible with the intravenous fluid it may be added to the fluid and thereby be given by slow or continuous infusion. Some drugs cannot be given by rapid intravenous injection. Therefore they may be given by slower IV infusion over 15 or 30 minutes.5.1 Continuous IV Infusion - Steady State The Model Giving the drug by infusion changes the drug concentration versus time curve. The equations used to describe the drug concentration are different. The model can be described schematically as Figure 5.1.1. In Figure 5.1.1 we have added a zero order infusion rate constant, k0, to the diagram presented earlier, (Figure 3.3.1). Since this is a zero order process the units for k0 are amount per time, for example 25 mg/min. The elimination process can be represented by kel or clearance. Differential and Integrated equation Equation 5.1.1 is the differential equation during the infusion period and it can be integrated to give Equation 5.1.2 using Laplace transforms. Dividing both sides by the apparent volume of distribution, V, gives Equation 5.1.3 for Cp after a continuous IV infusion. Equation 5.1.1 Differential Equation for Drug Amount during an IV infusion Equation 5.1.2 Integrated Equation for Drug Amount versus Time Equation 5.1.3 Integrated Equation for Concentration versus Time Equation 5.1.3 can be used to estimate the drug concentration at various times after an infusion is started OR to calculate the infusion rate needed to achieve a desired drug concentration. You may notice that Equation 5.1.3 for Cp is quite similar to Equation 4.1.9 that we used before for the cumulative amount of drug excreted into urine. As you might expect the plot of Cp would be similar in shape, Figure 5.1.2. If we continue the infusion indefinitely then we will approach a steady state plasma concentration when the rate of infusion will be equal to the rate of elimination. This is because the rate of infusion is constant whereas the rate of elimination will increase as the plasma concentration increases. At steady state the two rates become equal. We can determine the steady state concentration from the differential equation by setting the rate of change of Cp to zero, i.e. dCp/dt = 0. This leads to Equation 5.1.4 and Equation 5.1.5. This could also be calculated from the integrated equation by setting e- kel • t = 0 at t = ∞. Equation 5.1.4 At Steady State Concentration doesn’t Change Equation 5.1.5 Cpss Calculated from k0, kel and V or k0 and CL We can now calculate the infusion rate necessary to produce some desired steady state plasma level. An Example Calculation A desired steady state plasma concentration of a drug may be 15 mg/L. If the average half-life is about 4 hr and the apparent volume of distribution is about 25 liter. What infusion rate is necessary? First calculate kel from the t1/2, kel = 0.693/4 = 0.17 hr-1 then required k0 = kel • V • Cp = 0.17 x 25 x 15 = 63.8 mg/hr We could use an infusion rate of 60 mg/hr which would produce a slightly lower Cpss value. Cpss = k0/(kel • V) = 60/(0.17 x 25) = 14.1 mg/L5.2 Continuous IV Infusion - Time to Reach Steady State Another important factor is the time to reach the steady state concentration. The time to reach half the steady state concentration can be derived. Starting with Equation 5.2.1 and Equation 5.2.2 we can develop an equation relating kel and the time to half steady state, Equation 5.2.3. Equation 5.2.1 Cp at Steady State Equation 5.2.2 Cp at Half Steady State Equation 5.2.3 Equation for kel and t1/2 And thus Equation 5.2.4. Note the time to reach half of the steady concentration is the same as the elimination half-life. Equation 5.2.4 Time to Half Steady State is the Same as the Elimination Half-life Thus the approach to Cpss is exponential in nature and is controlled by the elimination process NOT the infusion process. NOTE however that the value of Cpss IS controlled by k0 and clearance (or k0, kel and V). For a drug with a t1/2 equal to 4 hours the time to reach 94% of steady state will be 16 hours, Table 5.2.1. We could calculate how long it might take to reach a therapeutic concentration. For this drug it might be 10 mg/L. Thus using Equation 5.2.5 and the values from before: Equation 5.2.5 Cp during an IV Infusion k0 = 60 mg/hr, kel = 0.17 hr- 1, V = 25 L and Cprequired = 10 mg/L or Equation 5.2.6 Calculating Time to 10 mg/L Taking the ln of both sides, Equation 5.2.6, gives -0.17 * t = -1.231 or t = 7.24 hr. This result is illustrated in Figure 5.2.1. Thus if we started an infusion to achieve a steady state plasma concentration of approximately 15 mg/L (actually 14.1 mg/L) it would take 7.25 hours to reach a therapeutic level of 10 mg/L. This is probably too long so another strategy might be explored. You can investigate these calculations using Interactive 5.2.1. Remember you can select CL/V or kel/V with the change button.5.3 Combined Infusion and Bolus Administration One reason we give a drug by IV is because we need a quick therapeutic response. One way to achieve a therapeutic concentration more quickly than a continuous IV infusion is to give a loading dose by rapid intravenous injection and then start the slower maintenance infusion. For Drugs which can be given as a bolus An example, our objective is to achieve Cpss = 14.1 mg/L. Previous calculations show that this concentration can be maintained with k0 = 60 mg/hr for a patient with the parameter values V = 25 L and kel = 0.17 hr- 1. A loading dose can be calculated by rearranging Equation 5.3.1. or Equation 5.3.1 Calculation of IV Bolus Dose DOSE = V • Cp0 = 25 x 14.1 = 353 mg The plasma concentration from the combined bolus and infusion regimen is shown as the black horizontal line in Figure 5.3.1 Explore the interplay of dose and parameter values using Interactive 5.3.1. An IV bolus and maintenance infusion is one way to achieve a steady state plasma concentration rapidly and maintain it. However, we may not be able to give a bolus dose intravenously so another approach may be necessary. 5.4 Combined Slow and Fast Infusion Another method to achieve a desired concentration quickly and maintain that concentration is to give a loading dose by rapid infusion and then give a slower maintenance infusion once the required concentration is achieved. For example, using the previous parameters; kel = 0.17 hr-1; V = 25 L and with a required Cp = 14.1 mg/L If give the loading IV infusion over 30 minutes we need to give the infusion at a rate which will produce Cp = 14.1 mg/L at the end of this time. Therefore:- Cp30 min = 14.1 mg/L Rearranging Equation 5.4.1 gives the required infusion rate of 735 mg/L. Therefore we need to give a dose of 367 mg over 30 minutes to achieve a plasma concentration of 14.1 mg/L at 30 minutes. or Equation 5.4.1 Calculation of the Fast Infusion Rate It is important to realize what the steady state plasma concentration would be if we didn't turn this fast infusion off. This could be quite toxic, Equation 5.4.2. Equation 5.4.2 Cpss if the Fast Infusion was not Stopped Consequently we would need to ensure that at 30 minutes the rapid infusion rate was slowed from 735 mg/hr to 60 mg/hr. One way to do this would be to only provide 367 mg (or 360 mg) in the infusion syringe at first. The dosing regimen (or controlled sequence of drug administration) to achieve the desired plasma concentration would be: a) a loading dose by IV infusion of 367 mg over 30 minutes followed by a maintenance IV infusion of 60 mg/hr. The expected result of this dosing regimen is shown in Figure 5.4.1 and can be explored by changing dosing regimen and parameter values in Interactive 5.4.1. 5.5 Post Infusion Before moving on we should look at the equation for plasma concentration after an infusion is stopped. Remember that the equation for plasma concentration versus time during an IV infusion is given by Equation 5.5.1. Equation 5.5.1 Cp versus Time During an IV Infusion If the infusion is continued indefinitely then the plasma concentration approaches a steady state plasma concentration, Equation 5.5.2. Equation 5.5.2 Steady State Concentration If however the infusion is stopped the plasma concentration is expected to fall, exponentially. Scheme or diagram Notice the scheme shown to represent 'after the infusion is stopped' is the same as that for after a bolus injection, Figure 5.5.1. More Equations The equation for drug concentration versus time during an IV infusion is shown in Equation 5.5.1. At the end of the infusion period when t = D the plasma concentration can be calculated using Equation 5.5.3. Equation 5.5.3 Concentration at the End of the Infusion Duration D Once the infusion is stopped all we have is first order elimination, a mono-exponential Equation 5.5.4. Equation 5.5.4 Concentration after the IV Fusion has Stopped (t > D) In Equation 5.5.4 t is time from the start of the infusion. Thus (t - D) is the time from the end of the infusion. Equation 5.5.5 can be used as shown when t is greater than D (that is for drug concentrations after the infusion has stopped). Also, if t is less than or equal to D can be set D = t before using the equation. In this way the term e-kel * (t-D) becomes equal to 1 and can be dropped from the equation and the equation reverts to Equation 5.5.1. Equation 5.5.5 Concentration During and After an IV Infusion. If t < D then D = t This result can be seen in Figure 5.5.2. If we use the previous example data, V = 25 L; kel = 0.17 hr-1; D = 0.5 hour; and k0 = 735 mg/hr, what would be the plasma concentration be at 4.5 hours (t = 4.5 hours). That is if we stop the loading infusion and don't start the maintenance infusion, Equation 5.5.6. Thus 4 hours after the infusion was stopped the drug concentration has fallen to half the value at the end of the infusion, Figure 5.5.2 and 5.5.3. Equation 5.5.6 Concentration Four Hours after the IV Infusion was Stopped Did you remember that the drug half-life was 4 hours? Example Calculation Following a two-hour infusion of 100 mg/hr plasma samples were collected and analyzed for drug concentration, Table 5.5.1. Calculate kel and V. The red line drawn through the data points and back to the Y-axis represents the best-fit line for these post-infusion data points, Figure 5.5.4. The elimination rate constant, kel, can be calculated with two concentration versus time points from this best-fit line (13, 2 and 1.9,24) using Equation 5.5.7. Equation 5.5.7 Calculation of kel Re-arranging Equation 5.5.3 allows the calculation of the apparent volume of distribution, V, Equation 5.5.8. Equation 5.5.8 Calculation of V Equation of 5.5.9 Calculation of CL This calculation can be explored using Interactive 5.5.1. Chapter 6: Routes of Drug Administration Student Objectives for this Chapter After completing the material in this chapter each student should:- be able to describe various routes of drug administration including the concentration versus time profile that might be expected from their administration be able to describe the biopharmaceutically relevant advantages and disadvantages of various routes of drug administration One method of classifying different routes of administration is as ENTERAL and PARENTERAL. Enteral means to do with the GI tract and includes oral, sublingual, buccal, and rectal. Parenteral means not through the alimentary canal and commonly refers to injections such as IV, IM, and SC; but could also include topical and inhalation. From a pharmacokinetic point of view if might be more important to distinguish IV administration from the other, extravascular routes of administration. After extravascular administration at least the drug must cross at least one membrane before reaching the central circulation. Except for local effects, an absorption process is involved in the administration and the pharmacokinetic model. The amount and rate of absorption can greatly influence the systemic concentration, exposure and effectiveness of drugs. Table 6.0.1. References Dosage Form definitions from the FDA Shargel, L. and Yu, A.B.C. 1999 Applied Biopharmaceutics and Pharmacokinetics, 4th ed., Appleton and Lange, Stamford, CT pp108-109, pp154-163. Gibaldi, M. 1984 Biopharmaceutics and Clinical Pharmacokinetics, 3rd ed., Lea and Febiger, Philadelphia, PA Chapters 3-7. Viswanathan, S. 2004 Advances in Drug Delivery, Pharmaceutical Formulation Quality, June/July, p20-28 6.1 Oral (PO) For many drugs absorption after oral administration can be quite variable in extent of absorption and rate of absorption. Dosage form design may also be used modify the rate and improve the extent of absorption. Advantages: The oral route is convenient with portable dosage forms, pain free and easy administration. It is relatively inexpensive with compact dosage forms, convenient multi-dose bottles and automated tablet or capsule machines that can produce tablets in large quantities. There a variety of dosage forms available including, fast release tablets, capsules, enteric coated tablets, layered tablets, slow release, suspensions, mixtures. Disadvantages: Sometimes the oral route inefficient, high dose or low solubility drugs may suffer poor availability with only part of the dose absorbed. Griseofulvin was reformulated about 1970 to include the drug as a micronized powder. The recommended dose at that time was decreased by a factor of two because of the improved bioavailability. Orally administered drugs may undergo first-pass metabolism since drugs absorbed orally are transported to the general circulation via the liver. Drugs that are extensively metabolized will be metabolized in the liver during absorption, Figure 6.1.1. For example the propranolol oral dose is somewhat higher than the IV and the same is true for morphine. Both these drugs and many others are extensively metabolized in the liver. This may result in some drug being ineffective when administered orally. Food and G-I motility can effect drug absorption. Often patient instructions include a direction to take with food or take on an empty stomach. Absorption can be slower or incomplete when tetracyclines and penicillins are given with food. However, for propranolol bioavailability is higher after food, and for griseofulvin absorption is higher after a fatty meal. There may be a unwanted local effect in the GI tract. Antibiotics may kill normal gut flora and allow overgrowth of fungal varieties. Thus, an anti-fungal agent may be included with an antibiotic. The oral route may be difficult with the unconscious patient. Generally a patient must be able to swallow solid dosage forms. Liquids or suspensions may be given by tube. 6.2 Buccal and Sublingual (SL) Some drugs are taken as smaller tablets which are held in the mouth or under the tongue. These are buccal or sublingual dosage forms. Buccal tablets are often harder tablets (4 hour disintegration time), designed to dissolve slowly. Nitroglycerin, as a softer sublingual tablet (2 min disintegration time), may be used for the rapid relief of angina. This route of administration is also used for some steroids such as testosterone and oxytocin. Chewing gum containing nicotine may be used as a cigarette smoking replacement. Advantages: When drugs are absorbed in the mouth they aren’t subject to first pass metabolism. Bioavailability can be much higher. There can be rapid absorption from the mouth because of the good blood supply and absorption is usually quite rapid, especially for drugs with good lipid solubility. Drugs may be more stable during absorption as the pH in the mouth is relatively neutral (compared with the stomach which is quite acidic). Disadvantages: Holding the dose in the mouth is inconvenient. If any part of the dose is swallowed that portion must be treated as an oral dose and could be subject to first pass metabolism. Usually more suitable for drugs with small doses. Bad drug taste may need to be masked. 6.3 Rectal (PR) Drugs given by the rectal route are most commonly given as a suppository or enema. Some drugs given by this route include aspirin, theophylline, chlorpromazine and some barbiturates. Advantages: Some (but not all) of the veins draining the rectum lead directly to the general circulation thus by-passing the liver. Therefore there may be a reduced first-pass effect. This route may be most useful for patients unable to take drugs orally or with younger children. Disadvantages: Drug absorption from a suppository is often incomplete and erratic. However for some drugs it can be quite useful. Absorption from solutions used as an enema may be more reliable. This route of administration may not be well accepted and there may be some discomfort. 6.4 Intravenous (IV) Drugs may be given into a peripheral vein over 1 to 2 minutes or longer by infusion. Rapid injections are used to treat acute conditions such as epileptic seizures, acute asthma, or cardiac arrhythmias. Advantages: A quick response is possible. Plasma concentration can be precisely controlled using IV infusion administration. The whole dose is delivered to the blood stream. That is the bioavailability is generally considered to be 100% after IV administration. Also, larger doses may be given by IV infusion over an extended time. Poorly soluble drugs may be given in a larger volume over an extended time period. Disadvantages: It may be difficult to find a suitable vein. There may be some tissue damage at the site of injection. Because of the rapid response, toxicity can be a problem with rapid drug administrations. For drugs where this is a potential problem the dose should be given as an infusion while monitoring for toxicity. Trained personnel are required to give intravenous injections. Intravenous dosage forms are relatively expensive. Sterility requirements, pyrogen testing and larger volume of solvent means greater cost for preparation, transport and storage. 6.5 Subcutaneous (SC) This involves administration of the drug by injection just under the skin. It is commonly used for insulin injection. Advantages: Can be self administered by the patient. Absorption can be fast from aqueous solution but slower with depot formulations. Absorption is usually complete and can be improved by massage or heat. A vasoconstrictor may be added to reduce the absorption of a local anesthetic agent used for dental applications, thereby prolonging its effect at the site of interest and reducing systemic effects. Disadvantages: This dosage form can be painful and finding suitable sites for repeat injection can be a problem. Irritant drugs can cause local tissue damage. A maximum of 2 mL per injection can limit its use to smaller doses. 6.6 Intramuscular (IM) Advantages: Larger volumes can be given by IM compared with SC. IM administration may be easier than IV injections. A depot or sustained release effect is possible with IM injections, for example procaine penicillin injections. Disadvantages: Trained personnel are required for injections. The site of injection will influence the absorption. Generally the deltoid muscle provides faster and more complete absorption. Absorption can be rapid from aqueous solution. However, absorption is sometimes erratic, especially for poorly soluble drugs such as diazepam or phenytoin. The solvent may be absorbed faster than the drug causing precipitation of the drug at the site of injection. Irritating drug may be painful. 6.7 Inhalation This route may be used for a local effect, for example as bronchodilators. Or it can be used for systemic effect such as general anesthesia. There will be rapid absorption with the drug absorbed directly into the central circulation, by-passing the liver. Absorption of gases is relatively efficient, however solids and liquids are excluded if larger than 20 micron and even then only 10 % of the dose may be absorbed. Solid doses may given as a powder with 50 % of the particles within the range of 2 to 6 micron. Larger than 20 micron and the particles impact in the mouth and throat. Smaller than 0.5 micron and they aren't retained. Some portion of the dose may be swallowed.6.8 Topical or Transdermal Many topical dosage forms are are used for a local response such as ear drops, eye drops or ointment, antiseptic creams and ointments, sunscreens, and callous removal products. A number of drugs may be administered topically for a systemic response such as nitroglycerin ointment. Generally absorption is quite slow. However, absorption through the skin especially via cuts and abrasions or from sites were the skin is quite thin can be quite marked. This can be a significant problem when handling toxic materials in the laboratory or pharmacy. This can also be a serious problem with garden chemicals. Gloves and other protective measures should be used. An occlusive dressing may be used to improve absorption. Transdermal patches can provide prolonged or controlled (e.g. using iontophoresis) drug delivery. There may be some skin irritation. Drug absorption will vary by site of administration, skin condition, age and gender. Systemic absorption (transdermal) is better with low dose, low molecular weight, lipid soluble drugs. 6.9 Other ROA’s There are many other routes of drug administration. These might include; nasal where systemic absorption has been demonstrated for propranolol and some low dose hormones; intra-arterial for cancer chemotherapy to maximize drug concentrations at the tumor site; and intrathecal directly into the cerebrospinal fluid. Others routes with limited systemic absorption but with local utility include ocular, aural, vaginal, urethral and intrasynovial Chapter 7: Pharmacokinetics of Oral Administration Student Objectives for this Chapter After completing the material in this chapter each student should:- be able to draw the scheme and write the differential equations for a one compartment pharmacokinetic model with first order absorption be able to use the integrated equations for this pharmacokinetic model to calculate parameter values and dosing regimens be able to define, use, and calculate the parameters: absorption rate constant, ka fraction absorbed, bioavailability, F time of peak concentration, tpeak maximum plasma concentration, Cpmax be able to describe the effect of changing ka and/or F values on plasma concentration versus time curves including with altered liver function on first-pass metabolism with improved drug absorption through reformulation with different dosage forms such solution, tablets and controlled release tablets So far we have considered the pharmacokinetics of intravenously administered drugs, either as a bolus or by infusion. If we know kel and V for a particular patient we can calculate appropriate doses or dosing rates (infusion rates) to produce the necessary therapeutic concentrations. In the previous Chapter we considered a number of routes of drug administration. Most of the routes of administration were extravascular; for example IM, SC, and most importantly oral. With these types of drug administration the drug isn't placed in the central compartment but must be absorbed through at least one membrane. This has a considerable effect on drug pharmacokinetics and may cause a reduction in the actual amount of drug which is absorbed. Most commonly the absorption process follows first order kinetics. Even though many oral dosage forms are solids, which must dissolve before being absorbed and absorption may occur at various parts of the GI tract, the overall absorption process can often be considered to be a single first order process. At least that's the assumption we will use for now. 7.1 Scheme or diagram A one compartment oral absorption pharmacokinetic model is shown in Figure 7.1.1. Here Xg is the amount of drug to be absorbed, Xp is the amount of drug in the body, and ka is the first order absorption rate constant. Differential Equations Drug Amount Remaining to be Absorbed, Xg The differential equation for Xg is shown in Equation 7.1.1 Equation 7.1.1 Rate of Change of Amount Remaining to be Absorbed from GI Tract This is similar to the equation for dCp/dt after an IV bolus administration. Using Laplace transforms it is possible to derive the integrated equation, Equation 7.1.2. Equation 7.1.2 Amount Remaining to be Absorbed versus Time Here F is the fraction of the dose which can be absorbed, that is, the bioavailability. We could therefore plot Xg (the amount remaining to be absorbed) versus time on semi-log graph paper and get a straight line with a slope representing ka, Figure 7.1.2. And as a linear plot as shown in Figure 7.1.3. Drug Amount in the Body, X The differential equation for X ( = V • Cp) the amount of drug in the body is shown in Equation 7.1.3. Equation 7.1.3 Rate of Change of Amount in the Body The first term, ka • Xg, represents absorption and the second term, kel • V • Cp or CL • Cp, represents elimination Even without integrating this equation we can get an idea of the plasma concentration time curve. Shortly after the dose is administered ka • Xg is much larger than kel • V • Cp (= CL • Cp) and the value of dCp/dt is positive, therefore the slope is positive and Cp will increase. With increasing time after the dose is administered, as Xg decreases, Cp is initially increasing, therefore there will be a time when ka • Xg will equal kel • V • Cp (= CL • Cp). At this time dCp/dt will be zero and there will be a peak in the plasma concentration. At even later times Xg will approach zero, and dCp/dt will become negative and Cp will decrease. It could be expected that the plasma concentration time curve will look like Figure 7.1.4. 7.2 Integrated equation We can also calculate the line (in Figure 7.1.4) using the integrated form of the equation which can be derived using Laplace transforms. If we use F • DOSE for Xg0 where F is the fraction of the dose absorbed, the integrated equation for Cp versus time is shown in Equation 7.2.1. Notice that the right hand side of this equation (Equation 7.2.1) is a constant multiplied by the difference of two exponential terms. A bi-exponential equation. We can plot Cp as a constant times the difference between two exponential curves (see Figure 2.1.1). We can plot each exponential separately, see Figure 7.2.1. Notice that the difference starts at zero, increases, and finally decreases again toward a value of zero. Plotting this difference multiplied by gives Cp versus time. We can calculate the plasma concentration at any time if we know the values of all the parameters of Equation 7.2.1 (see Figure 7.2.2). The parameters CL, kel and V are dependent on the drug and the patient. Different patients will have different values for CL, kel and V. This might depend on their age, weight, sex, genetic make-up (pharmacogenomics) and state of health. Different drugs can have quite different values of CL, kel and V, Figure 7.2.3. While dose is clearly a parameter associated with the dosage form, the parameters F and ka are partly determined by the drug and patient and also by the dosage form or route of administration. The rate, ka, and extent, F, of absorption can depend on the drug and patient with respect to transfer from the site of administration to the blood stream. The value of F may be reduced by poor solubility, drug instability, metabolism by intestinal flora, metabolism or reverse transport by various enzyme systems. The value of ka will be influenced by the drug dissolution rate and ability of the drug to move across any barriers between the site of administration and the blood stream. Both F and ka can also be influenced by the drug dosage form. Generally F is maximized but reduced values of ka may be desired to produce a sustained release effect. Time of Peak Concentration By setting the rate of change of Cp versus time, dCp/dt, to zero and after some rearranging an equation for the time of peak concentration can be derived, Equation 7.2.2. Equation 7.2.2 Time to Peak Concentration As an example we could calculate the peak plasma concentration given that F = 0.9, Dose = 600 mg, ka = 1.0 hr-1, kel = 0.15 hr-1, and V = 30 liter. Using Equation 7.2.3 we can calculate the time of peak concentration as 2.23 hr. Equation 7.2.3 Time to Peak Concentration Using Equation 7.2.1 we can calculate Cppeak or Cpmax for a single oral dose, Equation 7.2.4, as 12.9 mg/L. Equation 7.2.4 Time to Peak Concentration As another example we could consider what would happen with ka = 0.2 hr-1 instead of 1.0 hr-1. Using Equation 7.2.3 the time to peak concentration can be calculated, Equation 7.2.5, as 5.75 hr, and using Equation 7.2.1 again we can calculate Cppeak or Cpmax for a single oral dose with ka = 0.2 hr-1, Equation 7.2.6, to be 7.6 mg/L. Equation 7.2.5 Time to Peak Concentration Equation 7.2.6 Peak Concentration Note the peak drug concentration is lower and slower with the smaller ka value. Special Case - kel = ka From Equation 7.2.1 we can see that there is a problem if kel = ka. Dividing by ka - kel = 0 gives an answer that is undefined. A useful equation can be derived by starting with the differential equation, setting kel = ka = k, using Laplace transforms. Using the Convolution Machine we get Equation 7.2.7. Equation 7.2.7 Laplace of Amount in the Body/Plasma (Xp) where kel = ka = k Equation 7.2.7 was converted to an equation for amount versus time, Equation 7.2.8, using the Laplace table provided by Mayersohn and Gibaldi (1970). Equation 7.2.8 Amount of Drug in Plasma, Xp, versus Time Rearranging gives an equation for concentration versus time, Equation 7.2.9. Equation 7.2.9 was used when calculating concentration versus time for simulations when ka = kel. Equation 7.2.9 Concentration of Drug, Cp, versus Time where ka = kel The time of maximum concentration, tpeak, can be calculated with Equation 7.2.10, derived by setting dX/dt = 0 = k • Xg - k • Xp (Equation 7.1.3) and substituting in values for Xg (= F • Dose • e-k • t) and Xp (= F • Dose • k • t • e-k • t). Equation 7.2.10 Time of Maximum Concentration, peak, where ka = kel Reference Mayersohn, M. and Gibaldi, M. 1970 “Mathematical Methods in Pharmacokinetics. I. Use of the Laplace Transform in Solving Differential Rate Equations," Amer. J. Pharm. Ed., 34, p608-614 7.3 Bioavailability Parameters, ka and F Changing Absorption Rate Constant, ka Before going on to calculate the parameters ka, kel, and F from data provided we can look at the effect different values of ka and F have on the plasma concentration versus time curve. As ka changes from 3, 0.6 to 0.125 hr-1 the time of peak concentration changes to 1, 2.75 and 6.25 hour. Notice that with higher values of ka the peak plasma concentrations are higher and earlier, Figure 7.3.1. Changing Extent of Absorption or Bioavailability, F Changing F values is equivalent to changing the dose. Thus the higher the F value the higher the concentration values at each time point. Since the values of kel and ka are unchanged the time of peak plasma concentration is unchanged, Figure 7.3.2. Thus, tpeak = 1, 1, and 1 hour. The same tpeak in each case. Explore the effect of ka and F values on peak concentrations and time of peak concentrations using Interactive 7.3.1. Chapter 8: Calculation of Bioavailability Parameters Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- calculate ka using the method of Residuals including drawing the Cplate line estimating the residual values drawing the residual line (and possibly rescaling the time axis) the method of Wagner and Nelson and describe when each method may be more appropriate calculate F using plasma (AUC) or urine (U∞) data understand the difference between absolute and relative bioavailability and be able to convert between these values On many occasions you will be able to get the parameter values from tables and references. However, you should also know how to get these values from the data. The two parameters we will concentrate on in this Chapter are ka and F. These values can be used to compare dosage forms or brands. In the previous Chapter we saw the effect changing ka or F has on the plasma concentration time curve. In this chapter we will calculate ka and F from drug concentration versus time data. 8.1 Method of Residuals The equation for Cp versus time, Equation 8.1.1, can be re-written as shown in Equation 8.1.2 where A can be defined by Equation 8.1.3. Equation 8.1.1 Cp versus Time after Oral Administration Equation 8.1.2 Simplified Equation for Cp versus Time Equation 8.1.3 A as a Function of F, Dose, V, ka and kel If one of the rate constants (ka or kel, in Equation 8.1.2) is much larger than the other the later concentrations will fall on a straight line on a semi-log plot. This leads to the use of the method of residuals for the determination of the two rate constants, ka and kel. The method works best if the ratio between the two rate constants is greater than five. The faster exponential term will approach zero more quickly, and at later times can be ignored. A typical concentration versus time curve after oral administration is shown in Figure 8.1.1. Note the line is straight at later times on this semi-log plot. The equation for this straight line portion, Cplate, can be obtained from the equation for Cp by setting the faster term (which is usually e-ka • t) to zero. Cplate can be described by Equation 8.1.4. The plot of Cplate versus time is a straight line on semi-log graph paper, with a slope of -kel and intercept of A. This is shown on Figure 8.1.2 as the blue line. Equation 8.1.4 Cplate versus Time Now looking at the equation for Cp versus time again substituting for Cplate gives Equation 8.1.5.Rearranging Equation 8.1.5 gives Equation 8.1.6. Equation 8.1.5 Cp versus Time including Cplate Plotting the Residual) versus time on semi-log graph paper should give a straight line the same intercept as before, A, and slope which can provide a value for ka as shown in Figure 8.1.3. This is the method of residuals or "feathering". It can give quite accurate values for kel, ka, and V/F if: i) one rate constant is at least five times larger than the other and ii) both absorption and elimination are first order processes. An Example Calculation Using the Method of Residuals Example data collected after oral administration are shown in columns 1 and 2 of Table 8.1.1. These data are plotted in Figure 8.1.4. The plasma concentrations from column 2 are plotted as open red circles. The Cplate line drawn through the later concentration versus time data is shown as a red line back to the intercept, 5.5 mg/L. The Cplate values at early time points are shown in column 3. The residual values in column 4 were calculated as (Cplate - Cp), that is column 3 minus column 2. The residual values are plotted as filled blue circles. The best-fit line to the residual versus time is shown as a blue line in this figure. For these data kel is 0.2 hr-1 and ka is 2.05 hr-1 with an intercept of 5.5 mg/L. The method of residuals can be further explored using Interactive 8.1.1 Enter some data or click the ‘Random Data’ button. Plot the data with the first ‘Plot’ button. You can delete points that don’t belong on a straight line by unchecking the checkbox beside the data. Clicking the first ‘Calculate’ button draws a best-fit line through the selected data and provides parameter estimates. The Cplate and residual data are calculated after you press the ‘Calculate Residual’ button. Zero or data below zero are not included in the table. The residual data are plotted with the second ‘Plot’ button. The best-fit line through the residual data (and the intercept) are calculated with the second ‘Calculate’ button. Additional parameter values are shown in the table, calculated using semi-log linear regression. 8.2 Wagner-Nelson Method The value of the absorption rate constant, ka, can also be calculated using the Wagner-Nelson method. With this method you don’t need to assume a first order process for absorption so you can use the method to explore the absorption process. Advantages The absorption and elimination processes can be quite similar and accurate determinations of ka can still be made. The absorption process doesn't have to be first order. This method can be used to investigate the absorption process in detail. This method could be used to investigate data obtained after IM administration to find that two absorption steps maybe appropriate. Possibly a fast step from drug in solution and a slower step from drug precipitated at the injection site. The method can also provide very useful information about the absorption processes with different dosage forms. Disadvantages The major disadvantage of this method is that you need to know the elimination rate constant, from data collected following intravenous administration or know that the terminal slope after oral (or other extravascular) administration reflects first order elimination described by the elimination rate constant. The required calculations are more complex. Theory The working equations can be derived from the mass balance equation, Equation 8.2.1 or Equation 8.2.2. Equation 8.2.1 Mass Balance Equation Equation 8.2.2 Mass Balance Equation Differentiating each term, in Equation 8.2.2, with respect to time gives Equation 8.2.3. Equation 8.2.3 Differential Equation Replacing dX/dt and dE/dt with terms derived earlier gives Equation 8.2.4. Multiplying both sides by dt gives Equation 8.2.5. Equation 8.2.4 Rate of Change of Amount Absorbed Equation 8.2.5 Change in Amount Absorbed Integrating Equation 8.2.5 gives Equation 8.2.6. Equation 8.2.6 Amount Absorbed versus Time If we don’t know the apparent volume of distribution we can divide both sides by V and and solve for A/V, Equation 8.2.7. If we extend this equation to infinity the concentration becomes zero and the AUC is the value from zero to infinity, Equation 8.2.8. Equation 8.2.8 Maximum Amount Absorbed divided by Volume As a last step (Amax - A), the amount remaining to be absorbed, can be expressed as the amount remaining in the GI tract, Xg, Equation 8.2.9. We can use this equation to look at the absorption process. Equation 8.2.9 Amount Remaining to be Absorbed divided by Volume If the absorption step was a single first order process, the amount remaining to be absorbed divided by V can be described by Equation 8.2.10. OR Equation 8.2.10 Amount Remaining ro be Absorbed divided by V versus Time Thus a plot of ln (Amax - A) versus time will give a straight line for first order absorption and a value for ka can be determined from the slope. Note that linear or other types of plots may be used to explore other absorption behavior. For example a straight line on linear graph paper might suggest that absorption follows zero order kinetics, such as with an infusion step or a mimicking of zero order absorption with a topical patch or another slow release delivery device. The data (Amax-A)/V versus time, column 7, Table 8.2.1, can be plotted on semi-log and linear graph paper to explore the absorption process further. The value (Amax-A)/V is sometimes divided by Amax/V to give a fraction or percent remaining to be absorbed. In either case the slope or shape of the line can provide useful information about the absorption process. Plotting (Amax-A)/V versus time produces a straight line on semi-log graph paper, Figure 8.2.1, and a curved line on linear graph paper, Figure 8.2.2. This would support the determination that absorption can be described as a single first order process for these data. The first-order absorption rate constant, ka, can be calculated to be 0.306 hr-1 from the slope of the line on the semi-log graph paper using the data from zero to six hours. The later points fall significantly below the line. For practice you can try calculating the absorption rate constant, ka, with the Wagner-Nelson Method using the Interactive 8.2.1. Compare your answers with the computer! Enter some data or click the ‘Random Data’ button. Plot the data with the first ‘Plot’ button. You can delete points that don’t belong on a straight line by unchecking the checkbox beside the data. Clicking the first ‘Calculate’ button draws a best-fit line through the selected data providing an estimate of kel. A different value of kel can be calculated if available from IV data and entered here. The A/V and (Amax-A)/V data are calculated after you press the ‘Calculate A/V’ button. (Amax-A)/V data below zero or equal to zero are not included in the table. The (Amax-A)/V data are plotted with the ‘Plot’ button. The best-fit line through the residual data (and the intercept) are calculated with the second ‘Calculate’ button. The best-fit parameter value for ka is shown in the table. References Wagner, J and Nelson, E. 1964 Kinetic Analysis of Blood Levels and Urinary Excretion in the Absorptive Phase after Single Doses of Drug, J. Pharm. Sci., 53(11), 1392, J. Pharm. Sci., 53(11), 1392 8.3 Method of Inspection The method of Residuals and the Wagner-Nelson methods are useful technique for determining good estimates of ka. A computer program providing non-linear regression analysis may be able to provide even more accurate estimates of ka. Therefore, a quick, approximate method might be of interest. The method of inspection could be useful in this role. It is capable of providing a quick, approximate estimate of ka for checking the results obtained from other methods or as an initial estimate for more detailed analysis (Swintosky, J.V. et al., 1969). Requirements for the Method of Inspection We assume that ka is much larger than kel. That is, that ka is at least five time greater than kel. This is the same requirement as for the Method of Residuals when ka is greater than kel. Assume that absorption is complete (i.e. approximately 95 % complete) at the time of the peak concentration. This follows from the first assumption The Method The first step is to estimate the time of the peak drug concentration by inspection. If we assume that the time of peak concentration is approximately five time the absorption half-life, Equation 8.3.1 or Equation 8.3.2. Equation 8.3.1 Time of Peak Concentration or Equation 8.3.2 Drug Absorption Half-Life, t1/2 (absorption) From this value for t1/2 (absorption) we can estimate the absorption rate constant, Equation 8.3.3. Equation 8.3.3 Absorption Rate Constant An Example Considering the results illustrated in Figure 8.3.1 the time of peak can be estimated to be approximately 1.5 hours. With a tpeak of 1.5 hour the t1/2 (absorption) can be estimated as 1.5/5 = 0.3 hour. And ka can be estimated as ln(2)/0.3 = 0.693/0.3 = 2.3 hr-1. For comparison the ka value used to calculated these data was 2 hr-1. References Swintosky, J.V., Dittert, L.W. and Doluisio, J.T. 1969 Estimation of drug absorption rates from blood concentration profiles, Amer. J. Hospital Pharmacy, 26 (Sept), 519-522 8.4 Calculation of F So far we have looked at the equation for calculating Cp as a function of time, and methods of determining ka and kel. That is the method of residuals and the Wagner-Nelson method. Now to continue, we can look at methods of calculating F, the extent of absorption, i.e. the fraction of the dose which is absorbed. Returning to the equation for Cp as a function of time, Equation 8.4.1. Equation 8.4.1 Cp versus Time after Oral Administration We can calculate ka and ke given Cp versus time data. From the method of residuals the intercept, A, can be determined, Equation 8.4.2. Equation 8.4.2 Intercept Value A Since we know the dose and have calculated ka and kel, it is possible to calculate F/V (or V/F). However, with only data from a single oral administration that is all we can determine; we cannot separate V and F. Of course if we have a value for V from IV data, we could use this to determine F. Thus F must be determined by comparison with another dose administration. If the other dosage form is an intravenous dose then the F value is termed the absolute bioavailability. In the case where the reference dosage form is another oral or other non IV product , the value for F is termed the relative bioavailability. Using plasma data When a bioavailability study is conducted at least two dosage forms are administered to each subject. One dosage form is the product to be tested, while the other dosage form is a standard or reference dosage form. This may be an IV dose, oral solution or more commonly the original manufacturer's product. The doses are given with sufficient time between administrations for the drug to "washout" or be completely eliminated. We usually assume that each subject eliminates each dosage form at similar rates and use the estimate of the slowest rate to determine the wash-out period. During the derivation of the Wagner-Nelson equations we calculated Amax, the maximum amount absorbed, Equation 8.4.3 which can used to derive an equation for F, Equation 8.4.4. Equation 8.4.3 Total Amount Absorbed, Amax Now by giving two dosage forms A and B, and calculating AUC values for each we can calculate the relative bioavailability of dosage form A with respect to dosage form B, FA/FB, Equation 8.4.5. and since we get Equation 8.4.4 Bioavailability or Fraction Absorbed If we can assume that kelA = kelB and VA = VB then we can use Equation 8.4.6. The relative bioavailability, F, can be calculated. If dosage form B is an IV administration then FB = 1 and F = FA and where FA represents the absolute bioavailability. Equation 8.4.5 Bioavailability of Product A Relative to Product B Equation 8.4.6 Bioavailability of Product A Relative to Product B An Example using Drug in Plasma Data AUCA = 12.4 mg.hr/L [Dose = 250 mg] and AUCB = 14.1 mg.hr/L [Dose = 200 mg]. The relative bioavailability is 0.70. Using Unchanged Drug in Urine Data We can do the same type of calculation using urine data alone. Since fe can be defined in terms of U∞, F and Dose, Equation 8.4.7 we can rearrange this equation to solve for F, Equation 8.4.8. Equation 8.4.7 Fraction Excreted as Unchanged Drug, fe Equation 8.4.8 Bioavailability from Urine Data With data for two dosage forms we can we can calculate the relative bioavailability of product A versus product B, Equation 8.4.9. Equation 8.4.9 Bioavailability of Product A Relative to Product B If feA = feB Equation 8.4.9 simplifies to Equation 8.4.10. Equation 8.4.10 Bioavailability of Product A Relative to Product B An Example using Unchanged Drug in Urine Data 250 mg dose; U∞, A = 175 mg; U∞, B = 183 mg Using IV and Oral Plasma Data When both IV and oral data are available it is possible to calculate V from the IV data and V/F from the oral data (for example using the Method of Residuals). The value for F can be calculated from the ratio of V and V/F, Equation 8.4.11. Equation 8.4.11 Absolute Bioavailability from IV and Oral Data Chapter 9: Bioavailability Studies Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- define various terms relating to bioavailability studies describe some of the past problems with bioavailability describe a typical bioavailability study evaluate data derived from a bioavailability study Bioavailability studies are carried out to evaluate different dosage forms. These studies called bioavailability or drug product evaluation studies might compare: two (or more) dosage forms made by two (or more) different manufacturers, e.g. innovator and generic. These studies are called bioequivalence studies and they look at the similarity of F and ka values between the products studied. one type of dosage form with a 'standard' formulation, for example, tablet versus intravenous or capsule versus solution. These are bioavailability studies designed to determine values of ka and F for the product under study. Changes in ka may be intentional (for slow release dosage forms). The F value may need to be determined. Second brand or generic drug manufacturers are required to prove that their product is equivalent to previously marketed products which have demonstrated clinical efficacy. For most drugs, the second and subsequent manufacturer must show that their product is bioequivalent with the same ka and F, as the product(s) on the market. References Albert, K.S. 1980 Drug Absorption and Disposition: Statistical Considerations, Amer. Pharm. Assoc., Washington DC Dittert, L.W. et al. 1972 Guidelines for Biopharmaceutical Studies in Man, Amer. Pharm. Assoc., Washington DC U.S. Food and Drug Administration (FDA) FDA Enforcement Report Index FDA Recalls and Safety Alerts Electronic Orange Book: Approved Drug Products with Therapeutic Equivalence Evaluations Preface 9.1 Definitions [FDA, CFR] Definitions [Food and Drug Administration, Code of Federal Regulations] Bioavailability "This term means the rate and extent to which the active ingredient or active moiety is absorbed from a drug product and becomes available at the site of action. For drug products that are not intended to be absorbed into the bloodstream, bioavailability may be assessed by measurements intended to reflect the rate and extent to which the active ingredient or active moiety becomes available at the site of action." Pharmaceutical Equivalent "Drug products are considered pharmaceutical equivalents if they contain the same active ingredient(s), are of the same dosage form, route of administration and are identical in strength or concentration (e.g., chlordiazepoxide hydrochloride, 5 mg capsules). Pharmaceutically equivalent drug products are formulated to contain the same amount of active ingredient in the same dosage form and to meet the same or compendial or other applicable standards (i.e., strength, quality, purity, and identity), but they may differ in characteristics such as shape, scoring configuration, release mechanisms, packaging, excipients (including colors, flavors, preservatives), expiration time, and, within certain limits, labeling." Pharmaceutical equivalents are the same drug entity, the same type of dosage form, the same dose and meet the same compendial requirements. For example Aspirin Tablets, U.S.P. of a particular strength. Although the U.S.P. monograph includes dissolution and chemical assay requirements there are no bioavailability requirements (at least not in U.S.P. XX). Thus all Aspirin U.S.P. tablets of a particular dose would be pharmaceutical equivalents. Capsules of aspirin would not and neither would tablets of a different dose. Dosage forms containing different salt forms, esters or other chemical form are not pharmaceutical equivalents. Pharmaceutical Alternatives "Drug products are considered pharmaceutical alternatives if they contain the same therapeutic moiety, but are different salts, esters, or complexes of that moiety, or are different dosage forms or strengths (e.g., tetracycline hydrochloride, 250 mg capsules vs. tetracycline phosphate complex, 250 mg capsules; quinidine sulfate, 200 mg tablets vs. quinidine sulfate, 200 mg capsules). Data are generally not available for FDA to make the determination of tablet to capsule bioequivalence. Different dosage forms and strengths within a product line by a single manufacturer are thus pharmaceutical alternatives, as are extended-release products when compared with immediate- or standard-release formulations of the same active ingredient." Pharmaceutical alternatives are drug products that can provide the same therapeutic moiety. Different dosage forms, doses and even salts can be pharmaceutical alternatives. Therapeutic Equivalent "Drug products are considered to be therapeutic equivalents only if they are pharmaceutical equivalents and if they can be expected to have the same clinical effect and safety profile when administered to patients under the conditions specified in the labeling." Thus, pharmaceutical equivalents that have been shown to be bioequivalent (and the same by other determinations of clinical effect and safety profile) are therapeutic equivalents. Therapeutic equivalents would be expected to produce identical drug concentration time profiles and therapeutic response when administered under the same conditions. This is not the same as two pharmacologically similar (equivalent) compounds that may produce the same therapeutic response in some individuals (e.g., propoxyphene hydrochloride versus pentazocine hydrochloride for the treatment of pain). Bioequivalent Drug Products "This term describes pharmaceutical equivalent or alternative products that display comparable bioavailability when studied under similar experimental conditions. Section 505 (j)(7)(B) of the Act describes one set of conditions under which a test and reference listed drug shall be considered bioequivalent: the rate and extent of absorption of the test drug do not show a significant difference from the rate and extent of absorption of the reference drug when administered at the same molar dose of the therapeutic ingredient under similar experimental conditions in either a single dose or multiple doses; or the extent of absorption of the test drug does not show a significant difference from the extent of absorption of the reference drug when administered at the same molar dose of the therapeutic ingredient under similar experimental conditions in either a single dose or multiple doses and the difference from the reference drug in the rate of absorption of the drug is intentional, is reflected in its proposed labeling, is not essential to the attainment of effective body drug concentrations on chronic use, and is considered medically insignificant for the drug. Where these above methods are not applicable (e.g., for drug products that are not intended to be absorbed into the bloodstream), other in vivo or in vitro test methods to demonstrate bioequivalence may be appropriate." Bioequivalence Requirement [Code of Federal Regulations] means a requirement imposed by the Food and Drug Administration for the in vitro and/or in vivo testing of specified drug products which must be satisfied as a condition of marketing, Figure 9.1.1. Drug Names Brand Name [Shargel and Yu, 1985] is the trade name of the drug. Chemical Name [Shargel and Yu, 1985] is the name used by the organic chemist to indicate the chemical structure of the drug. The IUPAC Name. Drug Product [Federal Register 1977] means a finished dosage form, e.g., tablet, capsule, or solution, that contains the active drug ingredient, generally, but not necessarily, in association with inactive ingredients. Generic Name [Shargel and Yu, 1985] is the established, non proprietary or common name of the active drug in a drug product, Figure 9.1.2. References Shargel, L and Yu, A.B.C. 1985 Applied Biopharmaceutics and Pharmacokinetics, 2nd ed., Appleton-Century-Crofts, Norwalk, CT, p129-131 Billups, N.F. and Billups, S.M. 1986 American Drug Index, 30th ed., Lippincott, Philadelphia, PA FDA, CDER Guidances (Drugs) Search for bioavailability in the PharmPK listserv archive Name the Drug - a game9.2 Bioavailability Problems There are a number of examples of drugs products which have exhibited bioavailability problems in the past. These examples given here are all pre-1976 and as mentioned in the text were included in the earlier edition of the book with no further examples reported [Gibaldi, 1984]. This is an indication that more attention is now being given to formulation development during drug development. However, more recent example may be found by searching the FDA Enforcement Pages. Example Bioavailability Problems [via Gibaldi, 1984] Chlorpropamide. Three chlorpropamide formulations were tested and the peak plasma concentration after administration of one brand was less than half the peak concentration after the other two formulations, Figure 9.2.1. Digoxin. The text reports a number of bioavailability problems with digoxin. One example is particularly interesting. Doctors in Israel noticed 15 cases of digoxin toxicity between Oct-Dec 1975 with almost no reports for the same period the previous year. It was found that the local manufacturer had changed the formulation to improve dissolution without telling the physicians. Urinary data suggested a two-fold increase in availability of the new formulation. Phenytoin. Again there are a number of examples in the text. One report described an incidence of phenytoin intoxication in Australia in 1968 and 1969. Apparently the tablet diluent was changed from calcium sulfate to lactose. Later studies showed that the bioavailability was higher from the dosage form containing lactose. Other drugs with problems in the past include Acetazolamide, Aminosalicylate, Ampicillin, Aspirin, Ascorbic Acid, Chloramphenicol, Chlorothiazide, Diazepam, Furosemide, Iron, Levodopa + 10 [Gibaldi, 1984]. Bioavailability - Bioequivalence Studies Bioavailability studies are designed to determine either an absolute bioavailability, relative to an IV formulation, or relative bioavailability compared with an alternate reference dosage form with good absorption characteristics. They can be used to compare different routes of administration, for example oral versus IV or IP versus IM. Bioequivalence studies are designed to compare drug products. The objective is to determine if these products are bioequivalent. The dosage forms should be similar, especially the route of administration. For example, tablet versus tablet or maybe tablet versus capsule, given orally. These studies may be necessary before a generic product may be marketed. In general a relative bioavailability is determined which may be close to 100%. Reasons for Bioequivalence Requirements The FDA may decide to require bioavailability studies for a variety of reasons including: Results from clinical studies indicate that different drug products produce different therapeutics results. Results from bioavailability studies indicate that different products are not bioequivalent. The drug has a narrow therapeutic range. Low solubility and/or large dose. Absorption is considerably less than 100% The Biopharmaceutics Classification System (BCS) Biowaivers may be granted for drugs in Class 1 unless the drug has a narrow therapeutic range. For other drugs additional information including dissolution, perfusion and bioequivalence tests may be required. References Gibaldi, M. 1984. Biopharmaceutics and Clinical Pharmacokinetics, 3rd ed., Lea & Febiger, page 143-152. Monro, A. M. and Welling, P. G. (1974). The bioavailability in man of marketed brands of chlorpropamide. European Journal of Clinical Pharmacology, 7(1), 47–49. Ritschel, W.A. 1992 Handbook of Basic Pharmacokinetics, 4th ed, Drug Intelligence Publications, Hamilton, IL, p 539 U.S. Food and Drug Administration (FDA) FDA Enforcement Report Index FDA Recalls, Market Withdrawals and Safety Alerts 9.3 Bioavailability - Bioequivalence (BA/BE) study characteristics With recently introduced products properly conducted bioavailability studies will have been performed before the product is allowed to be marketed. However products which were approved some time ago may not have been tested as thoroughly. It is therefore helpful to be able to evaluate the testing which may have been undertaken. There are a number of situations where a pharmacist may be required to evaluate bioavailability study testing. When selecting drug products for a prescription, product performance should be a most important criteria. Review of the Electronic Orange Book provided by the FDA is an essential part of this process. Once it is established that two or more products are equivalent, then the choice of brand can be made on the basis of economic factors including cost and availability. The evaluation of a drug product bioavailability study involves the consideration of various factors. Drug The drug substance in each product must be the same. Bioavailability studies are conducted to compare two or more products containing the same chemical substance. We can't compare different chemical substances. The apparent volume of distribution and clearance or elimination rate constant can be quite different for different drug substances, thus no interpretation of the results is possible. The first rule of bioavailability testing is that you compare the drug products with the same drug in each dosage from. The only time that this rule may be relaxed is in the case of pro-drug administration. A pro-drug is a compound which will form the drug of interest in the body. In this case it may be appropriate to compare the delivery of a dosage form containing the drug with another dosage form containing a pro-drug. This testing is generally conducted to evaluate the usefulness of the pro-drug, rather than a strict comparison of the drug products. Once the usefulness of the pro-drug is demonstrated comparisons between dosage forms all containing the pro-drug should be undertaken to evaluate the drug product performance. In the case of bioequivalence studies the drug materials must be pharmaceutical equivalents or pharmaceutical alternatives. Drug product Usually the comparison is made between two (or more) similar products, containing exactly the same chemical substance. However, in bioavailability testing different dosage forms can be compared (when they contain the same drug). For example we could compare an IM dosage form with an IV dosage form, Figure 9.3.1. By calculating the AUC values we can determine the absolute bioavailability of the IM dosage form. In this case it appears to be close to 100%. The rate of absorption for the IM dose can be determined also, but of course no comparison is possible. Alternately we could compare brand A tablet with brand B tablet or capsule. By comparing the AUC and ka values we can make comparisons concerning both the extent and rate of absorption. In this case absorption of A appears to be faster than B but the extent of absorption doesn't appear to be all that different. These products would not be bioequivalent, Figure 9.3.2. Subjects A number of factors of concern include health, age, weight, enzyme status, number. Studies with humans must be carefully evaluated and approved by an Institutional Review Board (IRB). There must be an optimal risk/benefit ratio and given that in most bioavailability studies (with healthy volunteers) there is little direct benefit to the individual any risks should be minimal. All subjects must give informed consent that includes a requirement that they be provided with clear descriptions of their risks and benefit to participation. Health Usually a study is designed so that each subject takes each product in turn. Thus the effect of the individual subject can be eliminated or reduced. Such a study design is called a cross-over design, Table 9.3.1 or Table 9.3.2. Even though each subject will act as their own control it is usually best to have subjects of similar kinetic characteristic so that major variations are not introduced. Thus healthy volunteers are often preferred for drug product evaluation studies. Informed consent should be obtained from each volunteer and some biochemical and medical examinations will be used to confirm their medical state. For some drugs there may be special disease states which may cause the exclusion of some volunteers. For example, in one study we looked at propranolol products, and otherwise healthy volunteers with a history of asthma were excluded from the study. Age As you will see later, age can have a significant effect on drug pharmacokinetics. Elderly patients and young children can have quite different kinetics compared with young adults. In the interest of a better matched group, subjects between the ages of 18 to 35 years are preferred although a wider range may be appropriate. Pharmacokinetic changes usually aren't important until age greater than 60. Weight The apparent volume of distribution is usually proportional to weight in subjects of normal weight for height. However, in overweight (or underweight) subjects the apparent volume of distribution, in L/Kg, maybe somewhat different. Again to better match the subjects, normal weights are preferred. Life insurance weight tables may be useful. Enzyme status Smokers or subjects taking certain other drugs may have altered pharmacokinetics for the drug of interest. This can be caused by alteration of enzyme activity or by drug-drug interactions. These effects add complications to a study and an attempt is usually made to minimize these factors. Smokers may or may not be included but changes in such factors should be avoided. Number The number of subjects included in the study should be sufficient to see any real (maybe 20% variation) differences in bioavailability. Usually 10 to 20 subjects are used in these studies. In clinical studies where the end-point is a clinical response, much larger numbers are required because of the greater variability in clinical response. For drugs which are extensively metabolized there can be considerable between subject variability in AUC, tmax and Cppeak values and a larger number of subjects may be required to produce the required statistical power, the ability to distinguish between non-equivalent products. Assay The same assay method should be used for all phases of the study. It is not appropriate to use one assay for product A samples and another assay for product B samples. This wouldn't be done in a single study, however, if you were trying to compare the results from one study with those from another study, different assay methods may have been used. For example products A and B may have been compared with one assay method. In another study products B and C may have been compared with a second assay method. Comparison of products A and C may not be valid. One assay method may pick up an interference such as a metabolite which is not indicative of the drug concentration or the bioavailability. The assay method should be sensitive and specific (An, 2018). Design Usually a complete cross-over design is used, Table 9.3.1 or Table 9.3.2. With this design each subject receives all products with a wash-out period between each dose administration, Figure 9.3.3. Some example study designs When more than 3 or 4 products are involved it has been suggested that a different design is used whereby each subject will take maybe 3 or 4 products of a possible 8 to 12. This type of design, an incomplete block design, usually requires more subjects to get the same information, but it does mean that each subject is not required to take as many doses. It is harder to recruit subjects for longer studies that might be required for a complete cross-over design. Data analysis - Statistics The rate of absorption can be characterized by the absorption rate constant and also the time of peak concentration. The extent of drug absorption may be characterized by the F value, the peak concentration or the total AUC values. Any differences in the average values of these parameters can then be analyzed statistically to determine the significance of the differences. The 5 % confidence levels is usually used as the criteria of acceptance. The analysis of variance is a technique for separating the effect of product, subject, and sequence. The significance of each of these factors can be tested. In this example two effects are significant, Table 9.3.3. There appears to be a significant effect due to treatment and subject. This would indicate that the subjects are significantly different from each other and that the treatments are significantly different in terms of the parameter measured. It is quite common that Cpmax or AUC values are significantly different for different subjects, because of their different weights or size. The different treatments would appear to be NOT bioequivalent. By these studies the relative bioavailability of two or more products can be determined. Hopefully with proper testing we can ensure that drug products labeled to contain equivalent chemical amounts will be bioequivalent as well. References FDA Guidance Statistical Approaches to Establishing Bioequivalence An, G., Schmidt, R. L., Mock, D. M., Veng Pedersen, P., & Widness, J. A. (2018). Overlooked Issues on Pharmacokinetics Data Interpretation of Protein Drugs-a Case Example of Erythropoietin. The AAPS Journal, 21(1), 6. http://doi.org/10.1208/s12248-018-0269- Chapter 10: Physiological factors affecting oral absorption Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- describe membrane structure and how it effects drug transport describe the differences between passive, facilitated and active transport describe the effect of the parameters of Fick's first law on passive drug transport across membranes describe the relationship between GI physiology and drug absorption including changes in stomach emptying time and the presence of food Physiological, physical-chemical, and formulation factors which can influence the observed rate and extent of oral absorption will be discussed in the next few Chapters. First, we can look briefly at the overall picture of drug absorption, distribution, and elimination, Figure 10.0.1. The ultimate goal is to have the drug reach the site of action in a concentration which produces a pharmacological effect. No matter how the drug is given (other than IV) it must pass through a number of biological membranes before it reaches the site of action. We can start by looking at: Membrane Physiology Considering the structure of membranes Transport processes Gastrointestinal physiology Characteristics of gastrointestinal physiology Gastric motility and emptying Influence of food Other factors 10.1 Membrane physiology Membrane structure In 1900 Overton performed some simple but classic experiments related to cell membrane structure. By measuring the permeability of various types of compounds across the membranes of a frog muscle he found that lipid molecules could readily cross this membrane, larger lipid insoluble molecules couldn't and small polar compounds could slowly cross the membrane. He suggested that membranes were similar to lipids and that certain molecules (lipids) moved across membranes by dissolving in the membrane. These results suggest that the biologic membrane is mainly lipid in nature but contains small aqueous channels or pores, Figure 10.1.1. Other experiments involving surface tension measurements have suggested that there is also a layer of protein on the membrane. These results and others have been incorporated into a general model for the biological membrane. This is the Davson-Danielli model (1935), Figure 10.1.2. Later work (Danielli, 1975) suggested the presence of "active patches" and protein lining pores in the membrane, Figure 10.1.3. Work during the 1970s and 1980s suggested the model proposed by Singer and Nicolson 1972 called the fluid mosaic model. With this model the lipid bilayer is retained but the protein drifts between the lipid rather than forming another layer on either side of the lipid bilayer, Figure 10.1.4. The membrane then acts as a lipid barrier with protein formed pores. The protein within the membrane can act as transport enhancers in either direction depending on the protein. The barriers between various organs, tissues and fluids areas will consist of cells of different structure and membranes characteristics. In some cases the cells are loosely attached with extracellular fluid freely moving between the cells. Drugs and other compounds, lipid or not, may freely move across this barrier, Figure 10.1.5. In other cases there may be tight junctions between the cells which will prevent non lipid movement, Figure 10.1.6. These are general structures of the cellular layer. Layers in different parts of the body have somewhat different characteristics which influence drug action and distribution. In particular, membrane protein form and function, intracellular pore size and distribution is not uniform between different parts of the body. Examples of some barrier types Blood-brain barrier. The cellular barrier between the blood and brain have very tight junctions effectively eliminating transfer between the cells. Additionally there are specific transport mechanisms, such as P-glycoproteins which actively causes the removal of drugs and other compounds from the brain. This will prevent many polar (often toxic materials) materials from entering the brain. However, smaller lipid materials or lipid soluble materials, such as diethyl ether, halothane, can easily enter the brain across the cellular membrane. These compounds are used as general anesthetics. Renal tubules. In the kidney there are a number of regions important for drug elimination. In the tubules drugs may be reabsorbed. However, because the membranes are relatively non-porous, only lipid compounds or non-ionized species (dependent of pH and pKa) are reabsorbed. Hepatic blood vessels. The capillaries are lined with a basement membrane broken in part by sinusoids and fenestrations interspersed with cells held together with tight junctions. The result is a barrier that allows considerable transfer between the blood and hepatocytes. Blood capillaries and renal glomerular membranes. These membranes are quite porous allowing non-polar and polar molecules (up to a fairly large size, just below that of albumin, 69,000 Dalton) to pass through. This is especially useful in the kidney since it allows excretion of polar (drug and waste compounds) substances. Transport across the membranes Carrier mediated Active The body has a number of specialized mechanisms for transporting particular compounds; for example, glucose and amino acids. Sometimes drugs can participate in this process; e.g. 5-fluorouracil. Active transport requires a carrier molecule and a form of energy, Figure 10.1.7. the process can be saturated transport can proceed against a concentration gradient competitive inhibition is possible Facilitated A drug carrier is required but no energy is necessary. e.g. vitamin B12 transport. saturable if not enough carrier no transport against a concentration gradient only downhill but faster P-glycoprotein P-glycoprotein transporters (PGP, MDR-1) are present throughout the body including liver, brain, kidney and the intestinal tract epithelia. They appear to be an important component of drug absorption acting as reverse pumps generally inhibiting absorption. This is an active, ATP-dependent process which can have a significant effect on drug bioavailability. P-glycoprotein works against a range of drugs (250 - 1850 Dalton) such as cyclosporin A, digoxin, β-blockers, antibiotics and others. This process has been described as multi-drug resistance (MDR). Additionally P-glycoprotein has many substrates in common with cytochrome P450 3A4 (CYP 3A4) thus it appears that this system not only transports drug into the lumen but causes the metabolism of substantial amounts of the drug as well (e.g. cyclosporin). Clinically significant substrates of PGP include digoxin, cyclosporin, fexofenadine, paclitaxel, tracrolimus, nortriptyline and phenytoin (Humma 2003). A number of compounds can act as PGP inhibitors including atorvastatin (digoxin AUC increased), cyclosporine (increased paclitaxel absorption), grapefruit juice (increased paclitaxel absorption) and verapamil. Rifampin and St. John's wort have been reported to induce PGP expression (Ritschel and Kearns, 2004). The distribution of PGP polymorphism varies by race. The 'normal' 3435C allele is found in 61% African American and 26% in European American. The clinically important 3435T polymorph is found in 13% of African American and 62% of European American. The 3435T allele has been associated with reduced PGP expression (concentration) and consequently higher absorption. Digoxin levels were higher in healthy subjects with the 3435T allele compared with results in subjects with the 3435C allele (Humma, 2003). Passive Most (many) drugs cross biologic membranes by passive diffusion. Diffusion occurs when the drug concentration on one side of the membrane is higher than that on the other side. Drug diffuses across the membrane in an attempt to equalize the drug concentration on both sides of the membrane. If the drug partitions into the lipid membrane a concentration gradient can be established, Figure 10.1.8. The rate of transport of drug across the membrane can be described by Fick's first law of diffusion, Equation 10.1.1 Equation 10.1.1 Fick’s First Law of Diffusion The parameters of include: D: diffusion coefficient. This parameter is related to the size and lipid solubility of the drug and the viscosity of the diffusion medium, the membrane. As lipid solubility increases or molecular size decreases then D increases and thus rate of diffusion increases. A: surface area. As the surface area increases the rate of diffusion also increases. The surface of the intestinal lining (with villae and microvillae) is much larger than the stomach. This is one reason absorption is generally faster from the intestine compared with absorption from the stomach. x: membrane thickness. The smaller the membrane thickness the quicker the diffusion process. As one example, the membrane in the lung is quite thin thus inhalation absorption can be quite rapid. (Ch -Cl): concentration difference. Since V, the apparent volume of distribution, is at least four liters and often much higher the drug concentration in blood or plasma will be quite low compared with the concentration in the GI tract. It is this concentration gradient which allows the rapid complete absorption of many drug substances. Normally Cl << Ch which leads to Equation 10.1.2. Thus the absorption of many drugs from the G-I tract can often appear to be first-order, Movie 10.1.1. Pinocytosis Larger particles are not able to move through membranes or interstitial spaces so other processes must be available. These processes involve the entrapment of larger particles by the cell membrane and incorporation into the cell, cytosis. A spontaneous incorporation of a small amount of extracellular fluid with solutes is called pinocytosis. Phagocytosis is a similar process involving the incorporation of larger particles. Examples include Vitamin A, D, E, and K, peptides in newborn. References Washington, N., Washington, C. and Wilson, C.G. 2001 Physiological Pharmaceutics, Barriers to Drug Absorption, 2nd ed., Taylor & Francis, London From the Lipid Bilayer to the Fluid Mosaic: A Brief History of Membrane Models Ritschel, W.A. and Kearns, G.L. 2004 Handbook of Basic Pharmacokinetics ... including Clinical Applications, 6th ed., American Pharmaceutical Association, Washington, DC ISBN 1-58212-054-4 Humma, L.M., Ellingrod, V.L. and Kolesar, J.M. 2003 Lexi-Comp's Pharmacogenomics Handbook, Lexi-Comp, Hudson, OH ISBN 1-59195-060-0 One hundred years of membrane permeability: does Overton still rule? Qais Al-Awqati 1999 Nature Cell Biology, 1(8) pp E201 - E202 10.2 Gastrointestinal (GI) Physiology A summary of gastrointestinal (GI) physiology and factors which may affect drug absorption are presented in Table 10.2.1. Significant absorption of low dose drugs can occur in the mouth. This can be important for drugs with high first pass metabolism and a need for rapid activity. Drugs are usually not absorbed from the esophagus. In the stomach the high pH can alter absorption rates and may also contribute to drug break down or instability. For some drugs, such as acetaminophen, the rate of stomach emptying correlates with the rate of absorption. The very large surface area of the duodenum and small intestine from the presence of villae and microvillae results in these regions being major sites of absorption. The large intestine may be a significant site of absorption from sustained or controlled release products. Absorption from the lower colon and rectum has some potential for avoiding first pass metabolism. Characteristics of GI physiology Gastric emptying and motility Generally drugs are better absorbed in the small intestine (because of the larger surface area) than in the stomach, therefore quicker stomach emptying can increase drug absorption, Table 10.2.2. For example, a good correlation has been found between stomach emptying time and peak plasma concentration for acetaminophen (paracetamol). The quicker the stomach emptying (shorter stomach emptying time) the higher the plasma concentration, Figure 10.2.1. Also slower stomach emptying can cause increased degradation of drugs in the stomach's lower pH; e.g. l-dopa. Effect of Food Food can effect the rate of gastric emptying. For example fatty food can slow gastric emptying and retard drug absorption. Generally the extent of absorption is not greatly reduced. Occasionally absorption may be improved. Griseofulvin absorption is improved by the presence of fatty food. Apparently the poorly soluble griseofulvin is dissolved in the fat and then more readily absorbed. Propranolol plasma concentrations are larger after food than in fasted subjects. This may be an interaction with components of the food, Figure 10.2.2. Other factors Intestinal Motility and Transit Time (Mayersohn, 1971) References Heading, R.C., Nimmo, J., Prescott, L.F. and Tothill, P. 1973. The dependence of paracetamol absorption on the rate of gastric emptying, Br. J. Pharmacol., 47, 415-421 Mayersohn, M. 1971. Physiological Factors Influencing Drug Absorption, Can. Pharm. J., 164-169 Melander, A., Danielson, K., Schersten, B. and Wahlin, E. 1977. Enhancement of the bioavailability of propranolol and metaprolol by food, Clin. Pharmacol. Ther., 22, 108-112 Petring, O.U. and Blake, D.W. Gastric Emptying in Adults: An Overview Related to Anaesthesia, Anaesth Intensive Care. 1993 Dec; 21(6):774-81 Washington, N., Washington, C. and Wilson, C.G. 2001 Physiological Pharmaceutics, Barriers to Drug Absorption, 2nd., Taylor & Francis Chapter 11: Physical-Chemical Factors Affecting Oral Absorption Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- describe the physical-chemical factors that affects oral absorption describe the pH-partition hypothesis as it applies to drug absorption and estimate values of Brodies D value describe Fick's first law as it applies to drug dissolution There are two major headings, which can be used to discuss this material as it affect oral drug absorption. These are: 1. the pH-partition theory and 2. the dissolution of drugs 11.1 pH - Partition Theory For a drug to cross a membrane barrier it must normally be soluble in the lipid material of the membrane to get into membrane and it has to be soluble in the aqueous phase as well to get out of the membrane on the other side. Many drugs have polar and non-polar characteristics or are weak acids or bases. For drugs which are weak acids or bases the pKa of the drug, the pH of the GI tract fluid and the pH of the blood stream will control the solubility of the drug and thereby the rate of absorption through the membranes lining the GI tract. Brodie et al. (Shore, et al. 1957) proposed the pH - partition theory to explain the influence of GI pH and drug pKa on the extent of drug transfer or drug absorption. Brodie reasoned that when a drug is ionized it will not be able to get through the lipid membrane, but only when it is non ionized and therefore has a higher lipid solubility. Brodie tested this theory by perfusing the stomach or intestine of rats, in situ, and injected the drug intravenously. He varied the concentration of drug in the GI tract until there was no net transfer of drug across the lining of the GI tract. He could then determined the ratio, D as Equation 11.1.1 or Equation 11.1.2 Equation 11.1.1 Brodie’s D Value Equation 11.1.2 Another Equation for Bodie’s D Value These values were determined experimentally, but we should be able to calculate a theoretical value if we assume that only non ionized drug crosses the membrane and that net transfer stops when [U]b = [U]g, Figure 11.1.1. Brodie found an excellent correlation between the calculated D value and the experimentally determined values. Review of Ionic Equilibrium The ratio [U]/[I] is a function of the pH of the solution and the pKa of the drug, as described by the Henderson–Hasselbalch equation Weak acids In Equation 11.1.3, HA is the weak acid and A- is the salt or conjugate base. The equilibrium constant for this process, the dissociation for a weak acid, is given by Equation 11.1.4 Equation 11.1.3 Equilibrium between Weak Acid and its Conjugate Base Equation 11.1.4 Dissociation Constant for a Weak Acid Taking the negative log of both sides of Equation 11.1.4 and rearranging gives Equation 11.1.5 and Equation 11.1.6 where pKa = -log Ka and pH = -log[H+]. Equation 11.1.5 -log Ka Henderson-Hasselbalch Equation for Weak Acids Weak Bases In Equation 11.1.7 B is the weak base and HB+ is the salt or conjugate acid. The equilibrium constant for the conjugate acid dissociation is given by Equation 11.1.8. Equation 11.1.7 Equilibrium between a Weak Base and its Conjugate Acid Equation 11.1.8 Dissociation Constant for the Conjugate Acid, HB+ Taking the negative log of both sides of Equation 11.1.8 and rearranging gives Equation 11.1.9 and Equation 11.1.10 where pKa = -log Ka and pH = -log[H+]. Equation 11.1.9 -log Ka Equation 11.1.10 Henderson-Hasselbalch Equation for a Weak Base This equilibrium, Equation 11.1.7, can also be described by the dissociation constant for the weak acid, Kb, Equation 11.1.11. Equation 11.1.11 Dissociation Constant for a Weak Base Taking the negative log of both sides of Equation 11.1.11 and rearranging gives Equation 11.1.12 and Equation 11.1.13. Equation 11.1.12 -log Kb Equation 11.1.13 Henderson-Hasselbalch Equation for a Weak Base Note that Ka x Kb = [H3O+] • [OH-] = Kw which is approximately 10-14, thus pKb - pOH = pH - pKa and thus Equation 11.1.13 can be rearranged to give Equation 11.1.10. D Values and Drug Absorption Even though the D values refer to an equilibrium state a large D value will mean that more drug will move from the GI tract to the blood side of the membrane. The larger the D value, the larger the effective concentration gradient, and thus the faster the expected transfer or absorption rate. Compare D for a weak acid (pKa = 5.4) from the stomach (pH 3.4) or intestine (pH 6.4) with blood pH = 7.4. Stomach The ratio of unionized to ionized drug can be calculated from the Henderson–Hasselbalch equation. For this compound the ratio in the stomach is 100, Figure 11.1.2 and Equation 11.1.14. Equation 11.1.14 Ratio of Unionized to Ionized - Weak Acid (pKa = 5.4) in the Stomach (pH = 3.4) Blood In the blood, pH equal 7.4, the ratio is 0.01, Equation 11.1.15. Equation 11.1.15 Ratio of Unionized to Ionized - Weak Acid (pKa = 5.4) in Blood (pH = 7.4) Therefore the calculated D value would be determined starting with Equation 11.1.16. Replacing the values for [I]blood and [I]stomach with the equivalent values in terms of [U] and setting [U]blood equal to [U]stomach gives a value of 100, Equation 11.1.17. Equation 11.1.16 Brodie Value for a Weak Acid in the Stomach Equation 11.1.17 Brodie D Value for a Weak Acid in the Stomach By comparison in the intestine where the pH is higher, say 6.4 the calculated D value is (100+1)/(10+1) = 9.2, Figure 11.1.3. From this example we could expect significant absorption of weak acids from the stomach compared with from the intestine. Remember however that the surface area of the intestine is much larger than the stomach. This approach can be used to compare a series of similar compounds with different pKa values. We have applied the pH - partition theory to drug absorption, later we will use this theory to describe drug re-absorption in the kidney. With this theory it should be possible to predict that by changing the pH of the G-I tract that we would change the fraction non ionized and therefore the rate of absorption. Thus kaobserved = ku x fu assuming that the ionized species is not absorbed, Figure 11.1.4. For some drugs it has been found that the intercept is not zero in the above plot, suggesting that the ionic form is also absorbed, for example, results for sulfaethidole, Figure 11.1.5. Maybe the ions are transported by a carrier which blocks the charge, a facilitated transport process. Interactives 11.1.1 and 11.1.2 can be used to explore the interplay between weak acid and bases pKa, pKb values, Ph and Brodie D value. References Acid Dissociation Constant at Wikipedia Shore, P.A., Brodie, B.B., and Hogben, C.A.M. 1957. The Gastric Secretion of Drugs: A pH Partition Hypothesis, J. Pcol., Exp., Therap., 119, 361-369 Hogben, C.A.M., Tocco, D.J., Brodie, B.B., and Schanker, L.S., 1959. On the Mechanism of Intestinal Absorption of Drugs, J. Pcol., Exp., Therap., 125, 275-282 Crouthamel, W.G., Tan, G.H., Dittert, L.W. and Dolusio, J.T. 1971 Drug absorption IV. Influence of pH on absorption kinetics of weakly acidic drugs, J. Pharm. Sci., 60, 1160-63 Martin, A. 1993 Physical Pharmacy, 4th ed., Lea & Febiger, Philadelphia 11.2 Drug Dissolution So far we have looked at the transfer of drugs in solution in the G-I tract, through a membrane, into solution in the blood. However, many drugs are given in solid dosage forms and therefore must dissolve before absorption can take place, Figure 11.2.1. If absorption is slow relative to dissolution then all we are concerned with is absorption. However, if dissolution is the slow, rate determining step (the step controlling the overall rate) then factors affecting dissolution will control the overall process. This is a more common problem with drugs which have a low solubility (below 1 g/100 ml) or which are given at a high dose, e.g. griseofulvin. There are a number of factors which affect drug dissolution. One model that is commonly used to describe drug dissolution describes this process as diffusion controlled movement through a stagnant layer surrounding each solid particle. A physical model is shown in Figure 11.2.2. First we need to consider that each particle of drug formulation is surrounded by a stagnant layer of solution. After an initial period we will have a steady state where drug is steadily dissolved at the solid-liquid interface and diffuses through the stagnant layer towards the bulk solution. If diffusion is the rate determining step we can use Fick's first law of diffusion to describe the overall process, Equation 11.2.1. Equation 11.2.1 Fick’s First Law of Diffusion applied to Dissolution through the Stagnant Layer If we measure drug concentration at various distances from the surface of the solid we would see that a concentration gradient is developed, Figure 11.2.3. Fick's first law By Fick's first law of diffusion, Equation 11.2.1 the rate of solution, dissolution, is controlled by the diffusion coefficient, D, surface area, A, the solubility, Cs, the bulk concentration, Cb and the thickness of the stagnant layer, h. If Cb is much smaller than Cs then we have so-called "Sink Conditions" and the equation reduces to Equation 11.2.2 Equation 11.2.2 Pick’s First Law - Sink Conditions Surface area, A The surface area per gram (or per dose) of a solid drug can be changed by altering the particle size, Figure 11.2.4. For example, a cube 3 cm on each side has a surface area of 54 cm2. If this cube is broken into cubes with sides of 1 cm, the total surface area is 162 cm2. Actually if we break up the particles by grinding we will have irregular shapes and even larger surface areas. Generally as surface area increases the dissolution rate will also increase. Improved bioavailability has been observed with griseofulvin, digoxin, etc. Methods of particle size reduction include mortar and pestle, mechanical grinders, fluid energy mills, solid dispersions in readily soluble materials (PEG's). Diffusion layer thickness, h This thickness is determined by the agitation in the bulk solution. In vivo we usually have very little control over this parameter. It is important though when we perform in vitro dissolution studies because we have to control the agitation rate so that we get similar results in vitro as we would in vivo. The apparent thickness of the stagnant layer can be reduced when the drug dissolves into a reactive medium. For example, with a weakly basic drug in an acidic medium, the drug will react (ionize) with the diffusing proton (H+) and this will result in an effective decrease in the thickness of the stagnant layer. The effective thickness is now h' not h. Also the bulk concentration of the drug is effectively zero, Figure 11.2.5. For this reason weak bases will dissolve more quickly in the stomach. Diffusion coefficient, D The value of D depends on the size of the molecule and the viscosity of the dissolution medium. Increasing the viscosity will decrease the diffusion coefficient and thus the dissolution rate. This could be used to produce a sustained release effect by including a larger proportion of something like sucrose or acacia in a tablet formulation. Drug solubility, Cs Solubility is another determinant of dissolution rate. As Cs increases so does the dissolution rate. We can look at ways of changing the solubility of a drug. Salt form If we look at the dissolution profile of various salts, Figure 11.2.6 we can see quite a variation in the rate of dissolution. The salts with the higher solubility will dissolve faster. This result might support the use of the benzathine (or procaine) forms for IM depot use. Salts of weak acids and weak bases generally have much higher aqueous solubility than the free acid or base, therefore if the drug can be given as a salt the solubility can be increased and we should have improved dissolution. One example is Penicillin V. This can lead to quite different Cp versus time results after oral administration, Figure 11.2.7. Note that the tpeak values are similar thus ka is probably the same. Cppeak should show a good correlation with solubility. Maybe this is an example of site limited absorption where only drug in solution by the time the drug gets to the 'window' is absorbed. Use the potassium salt for better absorption orally. The lower value for sodium penicillin G (sodium benzyl penicillin) may be due to its instability in the acid environment of the stomach as well as its lower solubility. Crystal form Some drugs exist in a number of crystal forms or polymorphs. These different forms may well have different solubility properties and thus different dissolution characteristics. Chloramphenicol palmitate is one example which exists in at least two polymorphs. The B form is apparently more bioavailable, Figure 11.2.8. The recommendation might be that manufacturers should use polymorph B for maximum solubility and absorption. However, a method of controlling and determining crystal form would be necessary in the quality control process. Shelf-life could be a problem as the more soluble (less stable) form may transform into the less soluble form. In time the suspension may be much less soluble with variable absorption. References Aguiar, A.J., Krc, J., Kinkel, A.W., and Samyn, J.C. 1967 Effect of Polymorphism on the Absorption of Chloramphenicol from Chloramphenicol Palmitate, J. Pharm. Sci., 56(7), 847-853 Bevill, R.F., Dittert, L.W. and Bourne, D.W.A. 1977 Disposition of Sulfonamides in Food-Producing Animals IV: Pharmacokinetics of Sulfamethazine in Cattle following Administration of an Intravenous Dose and Three Oral Dosage Forms, J. Pharm. Sci., 66, 619-23 Juncher, H. and Raaschou, F. 1957 The solubility of oral preparations of penicillin V, Antibiot. Med. Clin. Therap., 4, 497Chapter 12: Formulation Factors Affecting Oral Absorption Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- describe various dosage forms and the expected bioavailability and pharmacokinetic consequences of each dosage form describe formulation components which affect the oral absorption of drug products The role of the drug formulation in the delivery of drugs to the site of action should not be ignored. With any drug it is possible to alter its bioavailability considerably by formulation modification. With some drugs an even larger variation between a good formulation and a bad formulation has been observed. Since a drug must be in solution to be absorbed efficiently from the G-I tract, you may expect the bioavailability of a drug to decrease in the order solution > suspension > capsule > tablet > coated tablet. This order may not always be followed but it is a useful guide. One example is the results for pentobarbital. Here the order was found to be aqueous solution > aqueous suspension = capsule > tablet of free acid form. This chapter will briefly discuss each of these formulation types particularly in regard to the relative bioavailability. 12.1 Solution Dosage Forms Drugs are commonly given in solution in cough and cold remedies and in medication for the young and elderly. In most cases absorption from an oral solution is rapid and complete, compared with administration in any other oral dosage form. The rate limiting step is often the rate of gastric emptying. Since absorption will generally be more rapid in the intestine. When an acidic drug is given in the form of a salt, it may precipitate in the stomach. However, this precipitate is usually finely divided and is readily redissolved and thus causes no special absorption problems. There is the possibility with a poorly water soluble drug such as phenytoin that a well formulation suspension, of finely divided powder, may have a better bioavailability. Some drugs which are poorly soluble in water may be dissolved in mixed water/alcohol or glycerol solvents. This is particularly useful for compounds with tight crystal structure and higher melting points that are not ionic. The crystal structure is broken by solution in the mixed solvent. An oily emulsion or soft gelatin capsules have been used for some compounds with lower aqueous solubility to produce improved bioavailability. 12.2 Suspension Dosage Forms A well formulated suspension is second only to a solution in terms of superior bioavailability. Absorption may well be dissolution limited, however a suspension of a finely divided powder will maximize the potential for rapid dissolution. A good correlation can be seen for particle size and absorption rate. With very fine particle sizes the dispersibility of the powder becomes important. The addition of a surface active agent will improve dispersion of a suspension and may improve the absorption of very fine particle size suspensions otherwise caking may be a problem. As a suspension ages there is potential for increased particle size with an accompanying decrease in dissolution rate. Smaller particles have higher solubility and will tend to disappear with the drug coming out of solution as larger particles. 12.3 Capsule Dosage Forms In theory a capsule dosage form should be quite efficient. The hard gelatin shell should disrupt rapidly and allow the contents to be mixed with the G-I tract contents. The capsule contents should not be subjected to high compression forces which would tend to reduce the effective surface area, thus a capsule should perform better than a tablet. This is not always the case. If a drug is hydrophobic a dispersing agent should be added to the capsule formulation. These diluents will work to disperse the powder, minimize aggregation and maximize the surface area of the powder. Tightly packed capsules may have reduced dissolution and bioavailability. 12.4 Tablet Dosage Forms The tablet is the most commonly used oral dosage form. It is also quite complex in nature. The biggest problem is overcoming the reduction in effective surface area produced during the compression process. One may start with the drug in a very fine powder but then proceed to compress it into a single dosage unit. Ingredients Tablet ingredients include materials to break up the tablet formulation. Drug - may be poorly soluble, hydrophobic Lubricant - usually quite hydrophobic Granulating agent - tends to stick the ingredients together Filler - may interact with the drug, etc., should be water soluble Wetting agent - helps the penetration of water into the tablet Disintegration agent - helps to break the tablet apart Coated tablets are used to mask an unpleasant taste, to protect the tablet ingredients during storage, or to improve the tablets appearance. This coating can add another barrier between the solid drug and drug in solution. This barrier must break down quickly or it may hinder a drug's bioavailability. 12.5 Sustained Release Dosage Forms Another tablet formulation is the enteric coated tablet which is coated with a material which will dissolve in the intestine but remain intact in the stomach. Polymeric acid compounds have been used for this purpose with some success. This topic and the area of sustained release products will be discussed in more detail in other courses. Benefits for short half-life drugs, sustained release can mean less frequent dosing and thus better compliance. reduce variations in plasma or blood levels for more consistent result. Problems More complicated formulation, may produce more erratic results. A sustained release product may contain a larger dose, i.e. the dose for two or three (or more) 'normal' dosing intervals. A failure of the controlled release mechanism may result in release of a large potentially toxic dose. more expensive technology Types of products erosion tablets waxy matrix matrix erodes or drug leaches from matrix coated pellets different pellets (colors) have different release properties coated ion exchange osmotic pump insoluble semi permeable coat with small hole. Osmotic pressure pushes the drug out at a controlled rate. Results Reduced side effects 12.6 Quality Control Disintegration Disintegration time is the time required for the tablet to break down into particles which can pass through a sieve while agitated in a specified fluid. Indicates the time to break down into small particles. Not necessarily solution. In the process of tablet manufacturer the drug is often formulated into a granular state (that is small but not fine) particles. This is done as the granule often has better flow properties than the a fine powder and there is less de-mixing leading to better uniformity. The granules are then compressed to produce the tablet. The disintegration test may lead to an end point of tablet to granule only, although the granules may be larger than the sieve opening. Dissolution The time is takes for the drug to dissolve from the dosage form is a measure of drug dissolution. Numerous factors affect dissolution. Thus the dissolution medium, agitation and temperature are carefully controlled. The dissolution medium maybe water, simulated gastric juice, or 0.1M HCl. The temperature is usually 37°C. The apparatus and specifications may be found in the U.S.P. The U.S.P. methods are official however there is a wide variety of methods based on other apparatus. These are used because they may be faster, cheaper, easier, sensitive to a particular problem for a particular drug, or developed by a particular investigator. Dissolution tests are used as quality control to measure variability between batches which maybe reflected by in vivo performance. Thus the in vitro test may be a quick method of ensuring in vivo performance. Thus there has been considerable work aimed at defining the in vitro - in vivo correlation. Chapter 13: Multiple IV Bolus Dose Administration Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- understand and be able to describe drug accumulation after repeated dose administration use the integrated equations for drug concentration after multiple IV bolus doses calculate suitable dosing regimens including loading dose, maintenance dose, and dosing interval define, use, and calculate the parameters: dosing interval, τ fraction remaining at the end of the dosing interval, R (= e-kel* τ) accumulation factor, 1/(1-R) maximum plasma concentration, Cpmax minimum plasma concentration, Cpmin calculate suitable multiple dose regimen to achieve desired Cpmin and Cpmax values This Chapter will consider drug pharmacokinetics after multiple IV bolus dose administration. But, first a review of the equations for a single dose administration. 13.1 Single Dose Review IV Bolus The one compartment pharmacokinetic model after an IV bolus dose, Figure 13.1.1, can be defined using both differential and integrated equations, Equation 13.1.1, Equation 13.1.2 and Equation 13.1.3. Equation 13.1.1 Differential Equation for One Compartment IV Bolus Dose Model Equation 13.1.2 Concentration versus Time after IV Bolus using kel and V Equation 13.1.3 Concentration versus Time after IV Bolus using CL and V IV Infusion During an Infusion The one compartment pharmacokinetic model after an IV infusion, Figure 13.1.2, can be defined using both differential, Equation 13.1.4 and integrated equations, Equation 13.1.5. Equation 13.1.4 Differential Equation for One Compartment during an IV Infusion Equation 13.1.5 Concentration versus Time During an IV Infusion At Steady State, Equation 13.1.6 and can be use to calculate maintenance infusion rates or steady state concentrations. Equation 13.1.6 Differential and Integrated Equations at Steady State After an IV Infusion, has stopped the differential equation looks like that after an IV bolus dose, Equation 13.1.7. The integrated equation, Equation 13.1.8, includes the parameter, D,for infusion duration. This equation can be used for concentrations during an infusion if you set D = t. The second exponential term becomes one and disappears, the equation reverts to Equation 13.1.5. Equation 13.1.7 Differential Equation after an IV Infusion has stopped Equation 13.1.8 Concentration versus Time During and After an IV Infusion Oral (Extravascular) The one compartment model after oral administration, Figure 13.1.3, can be defined using both differential, Equation 13.1.9 and integrated equations, Equation 13.1.10. Equation 13.1.9 Differential Equation for One Compartment Model after Oral Administration Equation 13.1.10 Concentration versus Time after Oral Administration After a single IV bolus dose drug concentrations fall with an exponential decline on linear graph paper, Figure 13.1.4, and a straight line on semi-log graph paper, Figure 13.1.5. With an IV infusion of duration, D, there is a steady increase to CpD followed by an abrupt exponential decline in concentration. After a single oral administration the curve around the peak concentration is smoother. 13.2 Multiple IV Dose With this refresher, drug pharmacokinetics after multiple dose administration may be easier to understand. Aspirin given for a headache may be given as a single administration, whereas aspirin for arthritis will be given as a multiple dose. Antibiotics are usually given as a multiple dose regimen to produce and maintain effective plasma concentration. In fact, many drugs are given this way; e.g. anti-hypertensives, anti-epileptics etc. Multiple dose administration is a very common method of drug administration. Up to this point we can calculate the drug concentration in plasma at any time after a single dose. We will continue now by looking at the equations for multiple dose administration. After a single dose administration we assume that there is no drug in the body before the drug is given and that no more is going to be administered. However, in the case of multiple dose administration we are expected to give second and subsequent doses before the drug is completely eliminated. Thus ACCUMULATION of the drug should be considered. On repeated drug administration the plasma concentration will be repeated for each dose interval giving a PLATEAU or STEADY STATE with the plasma concentration fluctuating between a minimum and maximum value. We have already looked at the shape of the plasma concentration versus time curve following a single intravenous administration. If we assume instantaneous mixing we start off with an initial concentration, Cp0, calculated as Dose/V and then we have a fall in concentration with time controlled by the elimination rate constant or clearance. Independent Doses If the doses are given far enough apart then the concentration will have fallen to approximately zero before the next dose. There will then be no accumulation of drug in the body, Figure 13.2.1. 13.3 Drug Accumulation However if the second dose is given early enough so that not all of the first dose is eliminated then the drug will start to accumulate and we will get higher concentrations with the second and third dose. As an example we could consider a drug with a half-life of 6 hours. Giving a dose of 100 mg every six hours with an apparent volume of distribution of 25 liter, the Cp0 = 4 mg/liter (see Figure 13.3.1). After six hours the plasma concentration will fall to 2 mg/liter. If we give the same dose again the plasma concentration will increase by 4 mg/liter from 2 mg/liter to 6 mg/liter. Then after another half-life (6 hours) the plasma concentration will fall to 3 mg/liter. Again, another dose will increase the plasma concentration by 4 mg/liter to 7 mg/liter. After another half-life the plasma concentration will be 3.5 mg/liter. After repeated drug administration every six hours the plasma concentration will accumulate until it fluctuates between a maximum and minimum value of 8 mg/liter and 4 mg/liter. In this example the dose was given every drug elimination half-life of 6 hours. Note, the amount of drug eliminated or the drug concentration drop during one half-life increases as the concentration increases. Remember also that elimination is considered to be a first order process! With each dose, drug accumulated until the amount of drug eliminated during each dosing interval was equal to the amount of the dose. In the first interval plasma concentrations fall from 4 to 2 mg/L. Continuing for a number of doses gives the following table, Table 13.3.1. There is a limit to drug accumulation because as the plasma concentration increases the amount of drug eliminated during the dosing interval will also increase as the rate of elimination is equal to the amount of the drug in the body multiplied by the rate constant for a first order elimination. (Compare this with the case of a continuous infusion). So far we can see that if we give repeated doses before the body can eliminate the previous doses then we will get accumulation of the drug. We have also seen that when we have first order elimination this accumulation will not proceed indefinitely but will approach a steady state. 13.4 Development of General Equation We can now consider a general equation which could describe the plasma concentration at any time after multiple IV bolus drug administration. Concentration at the End of the First Dosing Interval If the concentration resulting from the first and subsequent doses is represented as Cp01 (= Dose/V) we can calculate the concentration at the end of the first interval by multiplying by the e-kel• τ, the term representing the fraction remaining at the end of the dosing interval, τ, Equation 13.4.1. Equation 13.4.1 Concentration at the End of the First Dosing Interval Concentration at the Start of the Second Interval If we add the concentration from the second dose to the concentration at the end of the first interval we get the concentration at the beginning of the second interval, Equation 13.4.2. Equation 13.4.2 Concentration at the Start of the Second Interval Concentration at the end of the second dose interval The next step is to multiply the concentration at the beginning of the second interval by the fraction remaining term, e-kel• τ, Equation 13.4.3. Equation 13.4.3 Concentration at the End of the Second Interval We can continue this process for the third, fourth and subsequent doses. It will help if we re-define the fraction remaining term replacing e-kel • τ with the term, R. This gives Equation 13.4.4 and Equation 13.4.5. Equation 13.4.4 Concentration at the End of the Second Interval Equation 13.4.5 Concentration at the Start of the Third Interval You might recognize this as the sum of a geometric series with each term R times the preceding term. We can generalize this to beginning and end of n dosing intervals, Equation 13.4.6 and Equation 13.4.7. Equation 13.4.6 Concentration at the Start of the nth Interval Equation 13.4.7 Concentration at the End of the nth Interval These equations represent the sums of a geometric series and they can be simplified to give Equation 13.4.8 and Equation 13.4.9. Equation 13.4.8 Concentration at the Start of the nth Interval Equation 13.4.9 Concentration at the End of the nth Interval Starting with Equation 13.4.8 we can calculate drug concentration in blood or plasma at any time, t, following uniform multiple IV bolus administration by multiplying by e-kel•t, Equation 13.4.10. Note carefully that in Equation 13.4.10 ‘t’ represents the time since the last (nth) dose and not the time since the beginning of the dosing regimen (the first dose). Expanding Equation 13.4.10 gives the general equation for drug concentration at any time, t, after any number, n, of uniform IV bolus doses given every t hours, Equation 13.4.11. Equation 13.4.10 Concentration at time, t, after the nth IV Bolus Dose Equation 13.4.11 Concentration at time, t, after nth IV Bolus Dose. The General Equation Interactive 13.4.1 can be used to explore concentration versus time plots (linear or semi-log) after uniform IV bolus dose and dosing interval.13.5 Cpmax and Cpmin More useful equations can be derived from this general equation. These are equations to calculate the maximum and minimum plasma concentration after many doses, at steady state. That is as n approaches ∞, Rn (= e-n • kel • τ) approaches 0. With t = 0 or t = τ we can calculate the maximum and minimum concentrations. These are the limits of the PLATEAU or STEADY STATE CONCENTRATIONS. The concentration at the beginning and end of a dosing interval at steady state, after ‘many’ IV bolus doses is given by Equation 13.5.1 and Equation 13.5.2. Equation 13.5.1 Cpmax, the Concentration at the Beginning of a Dosing Interval after Many Does, at Steady State Equation 13.5.2 Cpmin, the Concentration at the End of Dosing Interval after Many Doses, at Steady State An example may be helpful: t1/2 = 4 hr; IV dose 100 mg every 6 hours; V = 10 liter, therefore What are the Cpmax and Cpmin values when the plateau values are reached? Converting the given half-life into a value for the elimination rate constant. therefore Therefore at steady state the plasma concentration will fluctuate between 15.5 and 5.48 mg/liter during each dosing interval. Accumulation Factor In Equation 13.5.1 the ratio, Dose/V represents the initial concentration after the first dose, Cp0. Thus the ratio between the highest, initial concentration at steady state, Cpmax and the highest concentration after the first dose, Cp01 can be expressed as a ratio, Equation 13.5.3. Equation 13.5.3 Accumulation Factor The accumulation factor describes how much drug accumulates during a multiple dosing regimen and gives a direct measure of how much higher the concentrations are during a dosing interval at steady state compared with the concentrations during the first dosing interval. It is interesting to note that if the dose is given every drug elimination half-life the accumulation factor is two since R equal to e-kel • τ equals one half. If the dosing regimen used indicates that the dose is given every half-life Cpmax will be twice the Cp01 value and Cpmin will be equal to the Cp01 value. For the example above R is equal to 0.354 and the accumulation factor can be calculated. To complete the example above we can calculate the plasma concentration at any time following multiple IV bolus administration (using Equation 13.4.11 in the previous section) OR we can calculate the Cpmax and Cpmin values (using Equation 13.5.1 and Equation 13.5.2). Time to Steady State Just as in the case of a continuous infusion it takes some time to get to the plateau where the concentrations vary between Cpmax and Cpmin during each dosing interval, Figure 13.5.1. As before, it can be shown that the time to reach a certain fraction of the plateau concentration is dependent on the drug elimination half-life only, much the same as for the approach to steady state during an IV infusion. Thus we may again have a problem with an excessive time required to reach the plateau. And as before a loading dose can be used to achieve steady state concentrations rapidly. Loading Dose and Maintenance Dose The IV bolus loading dose to quickly achieve a required drug concentration, Cpmax, can be calculated as Cpmax • V. Note, this is the same calculation as the IV bolus dose calculated in Chapter 3, Equation 13.5.4. Equation 13.5.4 Loading Dose Rewriting Equation 13.5.1 with Dose expressed more explicitly as the Maintenance Dose gives Equation 13.5.5 for Cpmax. Equation 13.5.5 Cpmax at Steady State Rearranging gives Equation 13.5.6 for Maintenance Dose. Equation 13.5.6 Maintenance Dose In the previous example Cpmax was 15.5 mg/liter. A suitable loading dose can be calculated as Cpmax x V = 15.5 x 10 = 155 mg as a bolus which would give Cp = 15.5 mg/liter, Figure 13.5.1. This loading dose could be followed by 100 mg every 6 hours (as described above) to maintain the Cpmax and Cpmin values at 15.5 and 5.5 mg/liter, respectively, Equation 13.5.6. Using Cpmax and Cpmin to Calculate a Suitable Dosing Regimen We can try another example of calculating a suitable dosing regimen. Consider that we know V = 25 liter and kel = 0.15 hr-1 for a particular drug and for this drug we need to keep the plasma concentration between 35 mg/liter (maximum tolerated concentration, MTC) and 10 mg/liter (minimum effective concentration, MEC). What we need is the maintenance dose, the loading dose, and the dosing interval. Dividing Equation 13.5.1 by Equation 13.5.2 gives, Equation 13.5.7. Equation 13.5.7 Ratio of Cpmax to Cpmin gives a Value for R Using Equation 13.5.7 with this example gives Taking the ln of both sides gives Solving for τ gives A dosing interval of 8 hours would be more reasonable and keeps the concentration between the limits of MTC and MEC. Thus with τ' = 8 hr and kel = 0.15 hr-1 From Equation 13.5.6 we can derive Equation 13.5.8 for the required maintenance dose. Equation 13.5.8 Maintenance Dose For this example the maintenance dose can be calculated as 35 x 25 x (1 - 0.301) or 612 mg Again we can round the value to a more realistic value leading to a maintenance dose of 600 mg every 8 hours. This regimen should be quite suitable as the maximum and minimum values are still within the limits suggested. All that remains is to calculate a suitable loading dose. This loading dose could be round (down) to a more suitable 850 or 800 mg. Let's use 800 mg. The dosing regimen is then a loading dose of 800 mg followed by a maintenance dose of 600 mg every 8 hours. To check this regimen Concentrations from the loading dose NOTE the use of the loading dose in this equation and that the Cpmax is below the MTC (35 mg/L). Concentrations from the maintenance dose and Here we use the maintenance dose. Note, the Cpmax and Cpmin are below and above the MTC and MEC (10 mg/L), respectively. This answer can be expressed graphically, Figure 13.5.2.13.6 Multiple IV Infusion Dose A drug may be given as multiple intravenous (IV) infusion. For various reasons mentioned in Chapter 5 an infusion may be preferable to a bolus dose. Thus, multiple intravenous infusions may be an appropriate dosing regimen for longer term therapy. The equation for a single IV infusion was given previously in Chapter 5, Equation 5.5.5 and repeated here as Equation 13.6.1. Equation 13.6.1 Concentration versus Time During and After an IV Infusion By use of the superposition principle drug concentration over multiple IV infusions can be calculated. Time in each equation is offset by the previous dosing intervals. Explore this calculation using Interactive 13.6.1 Chapter 14: Multiple Oral Dose Administration Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- use the integrated equations for multiple oral dose administration to calculate plasma concentration or calculate appropriate multiple dose regimen define, use, and calculate the parameter: average plasma concentration, use the equation to calculate or adjust an appropriate dosing regimen use the superposition principle to calculate Cp after non uniform IV or oral dosing regimen 14.1 Multiple Oral Dose Administration So far we have looked at multiple IV bolus dose administration with uniform dose and dosing interval. In an analogous fashion, equations can be developed which enable you to calculate the plasma concentration achieved following multiple oral administration with uniform dose and dosing interval. We can start by looking at the plasma concentration achieved following a single oral dose versus time, Equation 14.1.1. Equation 14.1.1 Concentration after a Single Oral Dose General Equation This can be converted to an equation describing plasma concentration at any time following n equal doses with constant dosing interval, τ, using a "multiple dose function", Equation 14.1.2 to give Equation 14.1.3. Multiplying each exponential term by the multiple dose function provides the general equation for concentration versus time, Equation 14.1.3. Equation 14.1.2 ‘Multiple Dose’ Function Equation 14.1.3 Concentration versus Time, t, after Multiple Oral Doses - Uniform Dose and Interval, τ The plasma concentration versus time curve described by this equation is similar to the IV curve in that there is accumulation of the drug in the body to some plateau level and the plasma concentrations fluctuate between a minimum and a maximum value, Figure 14.1.1. The Cpmax value could be calculated at the time t = tpeak after many doses (as n approaches ∞) using Equation 14.1.3 but it is complicated by the need to determine the value for tpeak. However, using Equation 14.1.3 allows the calculation of Cp at any time after any number of doses. These calculations can be explored using Interactive 14.1.1. 14.2 Cpmin Equation As with the development of a multiple IV bolus dosage regimen it is important to start with a target. For multiple oral dosing the target concentration may be presented as a Cpmax and Cpmin value. Since calculation Cpmax after multiple oral administration is complicated by the need to determine tmax it is easier to determine Cpmin first. Starting with the general equation, Equation 14.1.3, we can derive an equation for Cpmin which can be more easily determined at t = 0 or t = τ. Thus at t = 0 and as n approaches infinity e-n•k•τ approaches 0. This leads to Equation 14.2.1, version1. Equation 14.2.1 Cpmin after Many Oral Doses - Version 1 This equation can be further simplified if we assume that the subsequent doses are given after the plasma concentration has peaked and e-ka • τ is close to zero. That is the next dose is given after the absorption phase is complete. Therefore if the next dose is given well after the time of peak concentration, Figure 14.2.1, the equation for Cpmin can be simplified, Equation 14.2.2. Equation 14.2.2 Cpmin after Many Doses - Version 2 The relationship between loading dose and maintenance dose and thus drug accumulation during multiple dose administration can be studied by looking at the ratio between the minimum concentration at steady state and the concentration at the end of the first dosing interval,τ, after the first dose. [Assuming e-ka • τ is close to zero], Equation 14.2.3. Equation 14.2.3 Ratio between Cp after First and Last Dose Equation 14.2.3 can be simplified to give Equation 14.2.4. Equation 14.2.4 Accumulation Factor as a Ratio of Cp after First and Last Dose This turns out to be the same equation as for the multiple IV bolus doses. Therefore we can estimate a loading dose just as we did for an IV multiple dose regimen. Equation 14.2.5 holds if each dose is given after the absorption phase of the previous dose is complete. Equation 14.2.5 Loading Dose We can further simplify Equation 14.2.2 when ka is high and ka >> kel then (ka - kel) is approximately equal to ka and ka/(ka - kel) is approximately equal to one. Equation 14.2.6 is an even more extreme simplification. However, it can be very useful if we don't know the ka value but we know that absorption is reasonably fast. Equation 14.2.6 will tend to give concentrations that are lower than those obtained with the full equation (Equation 14.2.1). Thus any estimated fluctuation between Cpmin and Cpmax will be overestimated using the simplified equation. Equation 14.2.6 Cpmin after Many Oral Doses - Version 314.3 Cpaverage Equation Another very useful concentration value for the calculation of oral dosing regimens is the average plasma concentration during a dosing interval at steady state, Cpaverage or . The average plasma concentration can be defined as the average concentration during a dosing interval and calculated as the area under the plasma concentration versus time curve during a dosing interval at steady state divided by the dosing interval, Equation 14.3.1, Figure 14.3.1. Equation 14.3.1 Average Cp for a Dosing Interval at Steady State It can be shown that the AUC from zero to infinity after the first dose is equal the AUC during one dosing interval at steady state, Equation 14.3.2. Equation 14.3.2 AUC after First Dose is Equal to AUC during One Interval at Steady State The AUC after the first dose was found during the Wagner-Nelson derivation and from the clearance (kel • V) equation as shown in Equation 14.3.3. The AUC from Equation 14.3.3 can be substituted in Equation 14.3.1 to give Equation 14.3.4. Equation 14.3.3 AUC Equation Equation 14.3.4 Average Concentration for a Dosing Interval at Steady State An interesting result of this equation, Equation 14.3.4, is that we get the same average plasma concentration whether the dose is given as a single dose every dosing interval, τ, or is subdivided into shorter dosing intervals. For example 300 mg every 12 hours will give the same average plasma concentration as 100 mg every 4 hours. However, the difference between the maximum and minimum plasma concentration will be larger with less frequent dosing. An Example - Part 1 With F = 1.0, V = 30 liter, t1/2 = 6 hours or kel = 0.693/6 = 0.116 hr-1, calculate the dose given every 12 hours that will achieve an average plasma concentration of 15 mg/L. Since We could now calculate the loading dose To get some idea of the fluctuations in plasma concentration we could calculate the Cpmin value. Assuming that ka >> kel and that e-ka • τ approaches 0 we can use Equation 14.2.6. Therefore the plasma concentration would probably fluctuate between 7 and 23 mg/L (very approximately) with an average concentration of about 15 mg/L. [23 = 15 + (15 - 7), i.e. high = average + (average - low), very approximate!]. An Example - Part 2 As an alternative we could give half the dose, 312 mg, every 6 hours to achieve: As shown in Figure 14.3.2, the is the same, that is 15 mg/L. Thus the plasma concentration would fluctuate between about 10.4 to nearly 20 with an average of 15 mg/L. References Hawkins Van Tyle, J. and Winter, M.E. 2004 Chapter 2 "Carbamazepine" in Basic Clinical Pharmacokinetics, 4th ed., Winter, M.E., Lippincott Williams & Wilkins, Baltimore, MD Aminimanizani, A. and Winter, M.E. 2004 Chapter 12 "Theophylline" in Basic Clinical Pharmacokinetics, 4th ed., Winter, M.E., Lippincott Williams & Wilkins, Baltimore, MD 14.4 Superposition Principle The superposition principle can be used when all the disposition processes are linear. The disposition processes are distribution, metabolism and excretion (DME). That is, the processes that occur after the drug is absorbed. Thus, the superposition principle can be used when the DME processes are linear or first-order. According to this approach concentrations after multiple doses can be calculated by adding together the concentrations from each dose. Also, doubling the dose will result in the concentrations at each time doubling. This is not true when disposition processes are non-linear. For example, calculate drug concentration at 24 hours after the first dose of 200 mg. The second dose of 300 mg was given at 6 hours and the third dose of 100 mg at 18 hours. The apparent volume of distribution is 15 L and the elimination rate constant is 0.15 hr-1. The concentration from the first dose at 24 hours after the administration of the first dose The concentration from the second dose at 24 hours after the administration of the first dose The concentration from the third dose at 24 hours after the administration of the first dose The total concentration from all three doses at 24 hours after the administration of the first dose. This method involved calculating the contribution from each dose at a time 24 hours after the first dose. The result of this calculation is shown graphically in Figure 14.4.1. Another approach is to work through the dosing regimen dose by dose. Total drug concentration just after the first dose Total drug concentration just before the second dose Total drug concentration just after the second dose Total drug concentration just before the third dose Total drug concentration just after the third dose Total drug concentration 6 hours after the third dose. This calculation is also illustrated in Image Gallery 14.4.1 using a spreadsheet. Non-uniform dosing intervals Prior to this Chapter the calculations we have looked at consider that the dosing intervals are quite uniform, however, commonly this ideal situation is not adhered to completely. Dosing three times a day may be interpreted as take with meals, the plasma concentration may then look like the plot in Figure 14.4.2. The ratio between Cpmax and Cpmin is seven fold (8.2/1.1 = 7.45) in this example. However this regimen may be acceptable if 1) the drug has a wide therapeutic index there is no therapeutic disadvantage to low overnight plasma concentrations, e.g., analgesic if patient stays asleep. There are interactive graphs for multiple IV bolus and oral administration after the first five doses using kel and V or CL and V and after enough doses to reach at steady state on the next few pages. Explore non-uniform, multiple dosing, after first dose or at steady state using Interactive 14.4.1 (IV Bolus) or Interactive 14.4.2 (Oral) Chapter 15: Excretion Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- describe the various routes by which a drug may be excreted from the body understand the parameter renal clearance and its relationship with the excretion processes in the kidney understand the influence of renal disease on drug elimination calculate suitable drug dosage regimen for patients with impaired renal function based on a or a Cpmin/Cpmax approach The major routes of drug excretion described in this Chapter are renal, biliary, pulmonary and salivary. 15.1 Renal Excretion The major organ for the excretion of drugs is the KIDNEY. The functional unit of the kidney is the nephron and components of the nephron include Bowman's capsule, Proximal Tubule, Loop of Henle, Distal Tubule and the Collecting Duct. Low molecular weight molecules are filtered in Bowman's capsule. Active secretion of weak electrolyte drugs (acids) and reabsorption of water occurs in the proximal tubules. Additional reabsorption of water occurs in the Loop of Henle. Passive reabsorption of water and lipid soluble drugs occur in the distal tubule, Figure 15.1.1. There are three major renal excretion processes to consider; 1) glomerular filtration; 2) tubular secretion; and 3) tubular re-absorption Glomerular Filtration In the glomerular all molecules of low molecular weight (< 60,000 Dalton) are filtered out of the blood. Most drugs are readily filtered from the blood unless they are tightly bound to large molecules such as plasma protein or have been incorporated into red blood cells. The glomerular filtration rate varies from individual to individual but in healthy individuals the normal range is 110 to 130 mL/min (≈ 180 L/day). About 10% of the blood which enters the glomerular is filtered. This filtration rate is often measured by determining the renal clearance of inulin. Inulin is readily filtered in the glomerular, and is not subject to tubular secretion or re-absorption. Thus inulin clearance is equal to the glomerular filtration rate. Again, most drugs are filtered from blood in the glomerular, the overall renal excretion however is controlled by what happens in the tubules. More than 90% of the filtrate is reabsorbed. 120 mL/min is 173 L/day. Normal urine output as you may realize is much less than this, about 1 to 2 liter per day. Tubular secretion In the proximal tubule there is re-absorption of water and active secretion of some weak electrolyte but especially weak acids. As this process is an active secretion it requires a carrier and a supply of energy. This may be a significant pathway for some compounds such as penicillins. Because tubular secretion is an active process there may be competitive inhibition of the secretion of one compound by another. A common example of this phenomena is the inhibition of penicillin excretion by competition with probenecid. When penicillin was first used it was expensive and in short supply, thus probenecid was used to reduce the excretion of the penicillin and thereby prolong penicillin plasma concentrations. Since then it has been shown that probenecid also alters the distribution of penicillins to various tissues causing more drug to distribute out of plasma, causing even less to be eliminated. This could also be used to reduce the excretion of cephalosporins. Drugs or compounds which are extensively secreted, such as p-aminohippuric acid (PAH), may have clearance values approaching the renal plasma flow rate of 425 to 650 mL/min, and are used clinically to measure this physiological parameter (Documenta Geigy, 1970). Tubular re-absorption In the distal tubule there is passive excretion and re-absorption of lipid soluble drugs. Drugs which are present in the glomerular filtrate can be reabsorbed in the tubules. The membrane is readily permeable to lipids so filtered lipid soluble substances are extensively reabsorbed. A reason for this is that much of the water, in the filtrate, is reabsorbed and therefore the concentration gradient is now in the direction of re-absorption. Thus if a drug is non-ionized or in the unionized form it maybe readily reabsorbed. Many drugs are either weak bases or acids and therefore the pH of the filtrate may influence the extent of tubular re-absorption for these drugs. When urine is acidic weak acid drugs tend to be reabsorbed. Alternatively when urine is more alkaline, weak bases are more extensively reabsorbed. Making the urine more acidic can cause less reabsorption of weak bases or enhanced excretion. These changes can be quite significant as urine pH can vary from 4.5 to 8.0 depending on the diet (e.g. meat can cause a more acidic urine) or drugs (which can increase or decrease urine pH). In the case of a drug overdose it is possible to increase the excretion of some drugs by suitable adjustment of urine pH. For example, in the case of pentobarbital (a weak acid) overdose it may be possible to increase drug excretion by making the urine more alkaline with sodium bicarbonate injection, Figure 15.1.2. This method is quite effective if the drug is extensively excreted as the unchanged drug (i.e. fe approaches 1). If the drug is extensively metabolized then alteration of kidney excretion will not alter the overall drug metabolism all that much. The effect of pH change on tubular re-absorption can be predicted by consideration of drug pKa according to the Henderson-Hesselbalch equation. Renal clearance One method of quantitatively describing the renal excretion of drugs is by means of the renal clearance value for the drug. Renal clearance relates the rate of excretion, ΔU/Δt, to drug concentration, Equation 15.1.1. Units are mL/min. Equation 15.1.1 Rate of Excretion Remember that renal clearance can be calculated as part of the total body clearance for a particular drug. Renal clearance can be used to investigate the mechanism of drug excretion. If the drug is filtered but not secreted or reabsorbed the renal clearance will be about 120 mL/min in normal subjects. If the renal clearance is less than 120 mL/min then we can assume that at least two processes are in operation, glomerular filtration and tubular re-absorption. If the renal clearance is greater than 120 ml/min then tubular secretion must be contributing to the elimination process. It is also possible that all three processes are occurring simultaneously. The drug renal clearance value can be compared with physiologically significant values, e.g. glomerular filtration rate (GFR) of approximately 120 mL/min or renal plasma flow of about 650 mL/min. Renal clearance is given by Equation 15.1.2. Equation 15.1.2 Renal Clearance Each of these rates can be explored further in terms of fraction unbound (fU), GFR, renal blood flow (QR), intrinsic secretion clearance (CLisec) and fraction reabsorbed (fR) (Bauer 2008), Equation 15.1.3. Equation 15.1.3 Renal Clearance ‘Expanded’ (Bauer, 2008) The influence of fU, GFR, (CLisec and fR can be explored using the interactive graph to calculate plasma concentrations after IV bolus or oral dosing. Renal clearance values can range from 0 mL/min, the normal value for glucose which is usually completely reabsorbed to a value equal to the renal plasma flow of about 650 mL/min for compounds like p-aminohippuric acid. We can calculate renal clearance using the pharmacokinetic parameters ke and V. Thus CLrenal = ke • V. Renal clearance can also be determined as U∞/AUC. We can calculate renal clearance by measuring the total amount of drug excreted over some time interval and dividing by the plasma concentration measured at the midpoint of the time interval, Equation 15.1.4 and Equation 15.1.5. Equation 15.1.4 Renal Clearance Equation 15.1.5 Renal Clearance To continue we can briefly look at some other routes of drug excretion. We will then return to the topic of renal excretion by considering drug dosage adjustments in patients with reduced renal function. References Bauer, L.A. 2008 Applied Pharmacokinetics, Second Edition, McGraw-Hill, New York, NY, p 14 The Kidney at Wikipedia Documenta Geigy, Scientific Tables. 7th ed., Geigy Pharmaceuticals, 197015.2 Hemodialysis Hemodialysis or 'artificial kidney' therapy is used in renal failure to remove toxic waste material normally removed by the kidneys from the patient's blood. In the procedure blood is diverted externally and allowed to flow across a semi-permeable membrane that is bathed with an aqueous isotonic solution. Small molecules including nitrogenous waste products and some drugs will diffuse from the blood, thus these compounds will be eliminated. Therefore in patients with kidney failure, hemodialysis may be an important route of drug elimination. This technique is particularly important with drugs which: are smaller (< 500) molecular weight; are not tightly bound to plasma protein; have a small apparent volume of distribution and have good water solubility; Conversely drugs which are tightly bound or extensively stored or distributed into tissues are only poorly removed by this process. A simulation of drug concentration with and without dialysis is shown in Interactive 15.2.1. Low flux hemodialysis will readily remove molecules smaller than 500 Dalton. As the molecular weight increases to approximately 1000 Dalton the amount removed steadily decreases to insignificant. Thus clearance by hemodialysis falls from a maximum of about 4 L/hr to near zero with higher molecular weight compounds. Protein binding, represented by the fraction unbound (fu) will cause a proportional decrease in the hemodialysis clearance. Hemodialysis clearance is converted to a rate constant by dividing by the apparent volume distribution. This rate constant is added to the patient's elimination rate constant to give the increased, apparent rate constant during hemodialysis. As can be seen in Interactive 15.2.1 this can cause an increased removal of a drug. This can be useful in cases where the concentration is too high but it can also complicate the maintenance of therapeutic concentrations. Hemodialysis can mean a significant increase in the amount of drug removed during the dialysis period. An additional dose maybe required. Later in this chapter we will adjust dosage regimens for reduced renal function according to or Cpmin/Cpmax requirements. References Shargel, L. and Yu, A.B.C. 1999 Applied Biopharmaceutics and Pharmacokinetics, 4th ed., Appleton-Century-Crofts, Norwalk, CT, page 556, table 18.6 15.3 Biliary Excretion The liver secretes 0.25 to 1 liter of bile each day. Some drugs and/or their metabolites are excreted by the liver into bile. Anions, cations, and non-ionized molecules containing both polar and lipophilic groups are excreted into the bile provided that the molecular weight is greater than about 300 Dalton. Molecular weights around 500 Dalton appears optimal for biliary excretion in humans. Lower molecular weight compounds are reabsorbed before being excreted from the bile duct. Conjugates, glucuronides (drug metabolites) are often of sufficient molecular weight for biliary excretion. This can lead to biliary recycling. Indomethacin is one compound which undergoes this form of recycling, Figure 15.3.1. Figure 15.3.2 illustrate plasma concentration versus time curve which may result from extensive enterohepatic recycling. Note, the presence of the second peak shortly after the dumping of bile into the small intestine in response to the presence of food. Other compounds extensively excreted in bile include cromoglycate (unchanged drug), morphine, and chloramphenicol (as glucuronide). At least part of the biliary secretion is active since bile/plasma concentrations maybe as high as 50/1. There can also be competition between compounds. The efficiency of this biliary excretion system can be assessed by use of a test substance, such as Bromsulphalein. 15.4 Pulmonary Excretion The lung is the major organ of excretion for gaseous and volatile substances. Most of the gaseous anesthetics are extensively eliminated in expired air. The breathalyzer test is based on a quantitative pulmonary excretion of ethanol. How a Breathalyzer works 15.5 Salivary Excretion Saliva volume is typically one to two liters per day with flow rates ranging from 0.5 mL/min to more than 10 times that in the presence or thought of food. Saliva pH commonly ranges between 7.4 and 6.2 although lower values are possible. Saliva also contains a number of enzymes including amylase, ptylin, lipase, and esterases. Salivary excretion is not really a method of drug excretion as the drug will usually be swallowed and reabsorbed, thus a form of 'salivary recycling'. Drug excretion into saliva appears to be dependent on pH partition and protein binding. In many cases salivary concentration represents the free drug concentration in plasma. This mechanism appears attractive in terms of drug monitoring, that is determining drug concentration to assist in drug dosage adjustment. For some drugs, the saliva/free plasma ratio is fairly constant. Therefore drug concentrations in saliva could be a good indication of drug concentration in plasma. For some drugs localized side effects maybe due to salivary excretion of the drug. Pharmacokinetic studies in special populations may be more feasible using saliva concentrations. References Washington, N., Washington C. and Wilson, C.G. 2001 Physiological Pharmaceutics, Barriers to Drug Absorption, Taylor and Francis, Inc., New York, NY ISBN 0-748-40562-3 Kambhampati, S.R.P., Vanapalli, S.R., Nimmagudda R., Berens K., Putcha, L., Cheung, J.V., and Bourne, D.W.A. 2000 A comparison of neural network and PK/PD prediction of core body temperature from saliva melatonin concentration, Intelligent Engineering Systems Through Artificial Neural Networks, 10, 795-800 15.6 Renal Disease Considerations Getting back to the renal excretion of drugs. If a drug is extensively excreted unchanged into urine (i.e. fe closer to 1), alteration of renal function will extensively alter the drug elimination rate. Fortunately creatinine or inulin clearance can be used as a measure of renal function. For most drugs which are excreted extensively as unchanged drug it has been found that there is a good correlation between creatinine or inulin clearance and drug clearance or observed renal clearance and elimination rate (since V is usually unchanged). Dose adjustment Creatinine clearance Creatinine is produced in the body by muscle metabolism from creatine phosphate. Creatinine production is dependent on the age, weight, and sex of the patient. Elimination of creatinine is mainly by glomerular filtration (> 90%) with a small percentage by active secretion. With the patient in stable condition the production is like a continuous infusion to steady state with the infusion rate controlled by muscle metabolism and the elimination controlled by renal function. Thus as renal function is reduced serum creatinine concentrations increases. Other compounds such as inulin are also used for GFR measurement. Although inulin GFR values are probably more accurate they involve administration of inulin and careful collection of urine for inulin determination. The major advantage of creatinine is that its formation is endogenous. Determination of creatinine clearance consists of collection of total urine and a plasma/serum determination at the mid-point time, Equation 15.6.1. Equation 15.6.1 Creatinine Clearance Serum creatinine is expressed as mg/100 mL and creatinine clearance as mL/min. Normal inulin clearance values are 124 mL/min for men and 109 mL/min for women (Documenta Geigy, 1970). Because of some small renal secretion of creatinine, normal values of creatinine clearance are slightly higher than GFR measured with inulin. Thus, normal creatinine clearance values are about 120 to 130 mL/min. Various investigators have developed equations which allow calculation of creatinine clearance using serum creatinine values. Thus a single serum level may used when renal condition is stable. One commonly used equation is that of Cockcroft and Gault. For males use Equation 15.6.2. For females use 85% of the value calculated for males. CsCr is the serum creatinine concentration in mg/dL. Equation 15.6.2 Creatinine Clearance (Cockcroft-Gault Equation) The original authors of this equation used actual body weight in Equation 15.6.2. More recently it has been recommended that ideal body (IBW) be used in this equation unless the actual body weight (ABW) is less (Murphy, 2001), Equation 15.6.3. This is consistent with earlier the recommendation to use lean body weight in Equation 15.6.2 as creatinine is formed in muscle (Shargel and Yu, 1985). Equation 15.6.3 Ideal Body Weight (Murphy, 2001) More recent equations for estimating creatinine clearance are those recommended by the National Kidney Disease Education Program (NKDEP) for adults (from the MDRD study) and children (Original Schwartz equation) calculation. Estimation of kel in a patient The relationship between creatinine clearance and overall drug elimination can be easily seen by looking at plots of kel observed versus creatinine clearance. These are often called Dettli plots. Figure 15.6.1 shows the situation with considerable excretion as unchanged drug. i.e. fe between 0.3 and 0.7 (here fe = 0.5). In Figure 15.6.2 drug is excreted entirely as unchanged drug. i.e. fe = 1  In Figure 15.6.3 drug is excretion only as metabolized drug. i.e. fe = 0. We can use this information to calculate initial dosage regimens for patients taking drugs with high (> 0.25) fe values. The first step is to estimate the creatinine clearance in the patient from their serum creatinine value. From a Dettli plot (kel versus CLcr) constructed from previous studies with this drug we can estimate the elimination rate constant in this patient. We can therefore calculate an optimum dose and dosing interval to achieve the desired average drug concentration or maximum or minimum drug concentrations. The Dettli plot, Figure 15.6.4, may be built into a computer program or nomogram. This is the plot shown before. In the references shown below there is information useful for calculating kel in patients with impaired renal function, Table 15.6.1. As an example these data could be used to calculate the kel for a patient with a CLcr of 10 mL/min compared a subject with normal renal function of 120 mL/min For kanamycin kelpatient = knr + b • CLcr = 0.01 + 0.0024 x 10 = 0.01 + 0.024 = 0.034 hr-1 Compare this with the value for the normal subject: kel = 0.01 + 0.0024 x 120 = 0.298 hr-1 For sulfadiazine kelpatient = 0.03 + 0.0005 x 10 = 0.03 + 0.005 = 0.035 hr-1 Compare this with the value for the normal subject: kel = 0.03 + 0.0005 x 120 = 0.09 hr-1 For tetracycline kelpatient = 0.008 + 0.00072 x 10 = 0.008 + 0.0072 = 0.0152 hr-1 Compare this with the value for the normal subject: kel = 0.008 + 0.00072 x 120 = 0.0944 hr-1 References Documenta Geigy, Scientific Tables, 7th ed., Geigy Pharmaceuticals, 1970, p531 Shargel, L. and Yu, A.B.C. 1985 Applied Biopharmaceutics and Pharmacokinetics, 2nd ed., Appleton-Century-Crofts, Norwalk, CT, page 312 Murphy, J.E. 2001 Clinical Pharmacokinetics, 2nd ed., ASHP, Bethesda, MD, p4-5 Wagner, J.G. 1975 Fundamentals of Clinical Pharmacokinetics, Drug Intelligence Publications, Inc., Hamilton, IL Bennett, W. M., Singer, I., Golper, T., Feig, P., and Coggins, C. J. (1977). Guidelines for drug therapy in renal failure. Annals of Internal Medicine, 86(6), 754–783. Chow, M. S., Ronfeld, R. A. 1975 Pharmacokinetic data and drug monitoring: I. Antibiotics and antiarrhythmics. Journal of Clinical Pharmacology, 15(5-6), 405–418. Dettli, L. C. 1974 Drug dosage in patients with renal disease. Clinical Pharmacology Therapeutics, 16(1), 274–280. Welling, P. G., Craig, W. A., and Kunin, C. M. 1975 Prediction of drug dosage in patients with renal failure using data derived from normal subjects. Clinical Pharmacology and Therapeutics, 18(1), 45–52. 15.7 Cpaverage Calculations For patients with poor renal function taking drugs with high fe values, dosage regimen adjustment is essential. Drugs that are eliminated via the kidneys will have reduced elimination in patients with impaired renal function. Steady state and average drug concentration will rise dangerously unless the dosage regimen is adjusted. One way to make this adjustment is to adjust the dose or dosing interval to maintain a required average drug concentration, Cpaverage or . For example consider the drug kanamycin. A 70 Kg patient with normal kidney function may receive 250 mg IM every six hours (about 3 half-lives; t1/2 = 2.3 hours). If F = 1.0 and V = 13.3 liter, kel = 0.693/2.3 = 0.30 hr-1 the average concentration can be calculated as 10.4 mg/L, Equation 15.7.1 Equation 15.7.1 Average Concentration If we assume that ka >> kel the Cpmin value can be calculated, Equation 15.7.2, where R = e-0.3 * 6 = 0.165 Thus, Cpmin = 3.7 mg/L Equation 15.7.2 Cpmin Calculation These are the results you should expect in a patient with a normal creatinine clearance value. However in a patient with a creatinine clearance of only 10 mL/min the elimination rate constant will be quite different and if the same dosage regimen were used quite different plasma concentrations would be achieved (see Figure 15.7.1). The elimination rate constant for this patient would be 0.034 hr-1 (t1/2 = 20 hr). The calculated average concentration using this same dosing regimen would be the toxic value of 92 mg/L, Equation 15.7.3. This average plasma concentration is well above the maximum recommended value of 35 mg/L. Clearly some dosage adjustment should be made to the dosage regimen. We could consider a) changing the dose b) changing the dosing interval or c) changing both the dose and the dosing interval. We can make these alterations easily using Equation 15.7.4. Equation 15.7.4 Average Concentration From this we can see that decreasing the dose or increasing the dosing interval will have the desired response. Altered dose Assuming that a Cp of 10.4 mg/L (the value obtained in the normal patient on a normal dosage regimen) is satisfactory we can calculate a dose to achieve this value using Equation 15.7.5. Assuming ka >> kel, R = 0.815 and Cpmin = 9.3 mg/L. Equation 15.7.5 Estimated Dose to Achieve the Desired Average Concentration Thus this new dosing regimen of 28 mg every 6 hours should keep the drug concentrations below toxic levels. Altered dose interval Instead of altering the dose we could calculate a new dosing interval, Equation 15.7.6. Equation 15.7.6 Estimated Dosing Interval to Achieve the Desired Average Concentration Therefore giving 250 mg every 53 hours could also achieve a safer plasma concentration profile. R = 0.165; Cpmin = 3.7 mg/L. We would expect greater fluctuations with this method and dosing every 53 hours is not all that convenient. Every 6 hours is not all that great either if a longer dosing interval would work. We might consider dosing every 24 hours. Altered dose and interval Using a dosing interval of 24 hours suggests a dose of 113 mg or 100 mg, Equation 15.7.7. R = 0.442; Cpmin = 6.7 mg/L. Equation 15.7.7 Estimated Dose Required every 24 hour The lines in Figure 15.7.2 were calculated to achieve an average concentration of 10.4 mg/L using these three dosing regimen, 28 mg q6h, 250 mg q54h or 113 mg q24h. Note the differences in peak and trough values with different dose and τ values. 15.8 Cpmax/Cpmin Calculations Another approach is to use the desired Cpmax and Cpmin to define the dosing regimen. The steps to be taken for this approach include: Define Cpmin and Cpmax. From information about the drug with reference to the patient's clinical requirements these concentration targets can be defined. For example the normal upper limit for an aminoglycoside peak concentrations might be 6 mg/L, however in case of life-threatening infection higher levels may be approached. Initial calculation might be based on a peak of 6 mg/L and a trough below 1 mg/L. (Use 1 mg/L as the trough and extend the interval when making the adjustment in τ). Determine CLCr. Probably from serum creatinine levels using the Cockcroft-Gault equation. Determine kel. Using the equation kel = km + b • CLCr with km and b values from the literature. Calculate Tau. Since and we know Cpmin, Cpmax and kel we can calculate the dosing interval, τ. Typically this will be some uneven time value. Round τ. A more usual dosing interval should now be chosen. For example a τ of 7.8 or 6.7 hour could be rounded to 8 hours, thus dosing three times a day. Recalculate R. A new value of τ results in a new value of R. Calculate a Maintenance Dose. The maintenance dose can be calculated from the minimum or the maximum plasma concentration. Maintenance dose = Cpmax • V • (1 - R) OR = Cpmin • V • (1 - R)/R h. Calculate a Loading DOSE. The loading dose can be calculated directly (for an IV bolus) by equating Cp0 and the Cpmax value. Loading dose = Cpmax • V Example A 75 kg, 65 year old male patient, serum creatinine concentration of 2.3 mg/100 ml, is to be given an aminoglycoside IV to achieve a peak plasma concentration of 6 mg/L and trough concentration below 1 mg/L. The apparent volume of distribution is reported to be 0.28 L/Kg and km and b values are 0.02 and 0.0028, respectively (Wagner, 1975). a. Cpmax = 6 mg/L and Cpmin = 1 mg/L b. c. kel = km + b • CLCr = 0.02 + 0.0028 x 34 = 0.115 hr-1 d. e. Since a longer dosing interval is needed to keep the trough level below 1 mg/L use a τ value of 18 hours. f. New R value. R = e-0.115 x 18 = 0.1262 g. Calculate maintenance dose using Cpmax = 6 mg/L as reference point. Maintenance dose = Cpmax • V • (1 - R) = 6 x 75 x 0.28 x (1 - 0.1262) = 110 mg Thus use 100 mg IV every 18 hours Cpmin = Cpmax • R = 5.45 x 0.1262 = 0.69 mg/L h. The loading dose can be calculated Loading dose = Cpmax • V = 6 x 75 x 0.28 = 126 mg. Using 125 mg would give a Cpmax = = 5.95 mg/L. Thus a loading dose of 125 mg followed by 100 mg every 18 hours should be satisfactory. References Wagner, J.G. 1975 Fundamentals of Clinical Pharmacokinetics, Drug Intelligence Publications, Inc., Hamilton, IL, p161, Table 3-9 Chapter 16: Metabolism Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- describe the various processes by which a drug may be metabolized including Phase 1 and 2 reactions describe the role of pharmacogenomics in drug metabolism and drug response understand the effect of induction of drug metabolism understand the role of inhibition of metabolism on drug interactions define the parameters: hepatic clearance hepatic or liver blood flow extraction ratio free intrinsic clearance understand the relationship between the parameters hepatic clearance, hepatic blood flow, fraction unbound, and free intrinsic clearance and be able to discuss the venous equilibration model discuss the differences between flow limited and capacity limit metabolism of drugs The body has another way of removing drugs, within the body. This method of elimination is metabolism or biotransformation. Metabolic processes, in general, have the overall effect of converting drug molecules into more polar compounds. Again, in general, the effect of this should be to decrease tubular re-absorption in the kidney and thus increase overall drug elimination. It can also means an immediate loss of pharmacological activity because transport into the site of action is hindered (less lipid soluble) or the molecule no longer fits into the receptor site. There are exceptions however, and a number of 'new' drugs have been discovered as active metabolites, 'pro-drugs'. Metabolism takes place by enzymatic catalysis (by reducing the activation energy of the reaction). Most metabolism occurs in the liver although other sites have been described, such as intestinal wall, kidney, skin and blood. Enzyme activity and concentration in an organ and tissue is can be quite variable, controlled by factors such as age, disease, sex, diet, co-administration of other drugs and genetics. Drugs that are extensively metabolized, where metabolism is a major route of elimination, usually have considerable between individual variability. With these drugs therapeutic drug monitoring and pharmacogenomics become important considerations. 16.1 Metabolic Processes Drugs may be metabolized by a wide variety of enzymes located throughout the body. There is a wide variety of reactions that can be called metabolism. These reactions may be grouped into Phase 1 and Phase 2 type reactions. Some have included Phase 0 and Phase 3 transport processes as part of the overall topic of metabolism. Commonly there are four types of reactions involved in drug metabolism. These are: 1. oxidation 2. reduction 3. hydrolysis 4. conjugation The first three are often lumped together as phase 1 reactions, while the fourth process, conjugation, is called phase 2 metabolism. A common scheme in the overall metabolism of drugs is that metabolites are metabolized. In particular a drug may be oxidized, reduced or hydrolyzed and then another group may be added in a conjugation step. A common cause of capacity limited metabolism is a limit in the amount of the conjugate added in the conjugation step. Phase 0 Phase 0 has been described as the transport of drug from the blood into the heptacytes in the liver, the basolateral (sinusoidal) uptake processes (see Chapter 11) (Ishikawa, 1992). Although not included in the Phase 0 designation, absorption of drugs from the intestinal lumen to the portal blood supply involves transport and metabolism enzyme processes. The reverse transport enzyme P-glycoprotein (PGP) is often accompanied by the metabolizing enzyme P450 3A (CYP3A). Both of these processes can significantly reduce drug bioavailability and provide a potential for drug interactions (Ritschel and Kearns, 2004). Phase 1 Phase 1 metabolic processes include oxidation, reduction and hydrolysis reactions which typically provide functional groups capable of undergoing Phase 2 reactions. The enzymes which catalyze Phase 1 reactions are found in a number of sub-cellular components including cytoplasm, mitochondria and endoplasmic reticulum. Although the liver is a major organ of metabolism, metabolic enzymes are found throughout the body. Oxidation (add O or remove H) Oxidation is the addition of oxygen and/or the removal of hydrogen. The cytochrome P450 enzymes are the most important of the oxidative enzymes. The cytochrome P450 or CYP family consists of a number of subfamilies such as CYP2C or CYP3A. The individual enzymes are numbered as CYP2C8 or CYP3A4. Hydroxylation is the introduction of an OH group by oxidation. The enzyme CYP3A4 is responsible for the oxidation of dapsone (N-hydroxylation), diazepam (3-hydroxylation), taxol (3'-hydroxylation), warfarin ((S)-4'-hydroxylation) and others. The enzyme CYP2D6 assists in the oxidation of alprenolol, amiodarone (aromatic hydroxylation), debrisoquine (4-hydroxylation), imipramine (2-hydroxylation), propranolol (4-hydroxylation), codeine (O-demethylation) and others. CYP2C9 is responsible for the oxidation of ibuprofen, phenytoin, tenoxicam, tolbutamide and warfarin (also CYP1A2). A few example reactions Hydroxylation, Figure 16.1.1 and Figure 16.1.2 Oxidation at S, Figure 16.1.3 and two step oxidation, Figure 16.1.4. Oxidation, Figure 16.1.5 and dehydrogenase, Figure 16.1.6 Reduction (add H or remove O), Figure 16.1.7 Hydrolysis Addition of water with further breakdown of the molecule. In blood plasma (esterases) and liver. Esters to alcohol and acid, Figure 16.1.8 Amides to amine and acid, Figure 16.1.9 Phase 2 Conjugation Conjugation reactions involve the addition of molecules naturally present in the body to the drug molecule. The drug may have undergone a phase 1 reaction. Glucuronidation This is the main conjugation reaction in the body. This occurs in the liver. Natural substrates are bilirubin and thyroxine. Aliphatic alcohols and phenols are commonly conjugated with glucuronide. Thus hydroxylated metabolites can also be conjugated. for example morphine. Acylation Acylation, especially acetylation with the acetyl group, e.g. sulfonamides Glycine Glycine addition (NH2CH2COOH) for example nicotinic acid Sulfate Sulfate (-SO4) for example morphine, paracetamol Phase 3 Elimination of the drug or metabolite into bile Excretion by ATP dependent transporter (e.g. MRP2) Metabolite is often more Polar In most cases the metabolite is formed by production of a more polar group, for example C-H -> C-OH, or addition of a polar group, for example acetyl (CH3COO-). Generally the resultant metabolite is more water soluble, and certainly less lipid soluble. Less drug is reabsorbed from the kidney. Occasionally the metabolite is less water soluble. A significant example is the acetyl metabolite of some of the sulfonamides. Some of the earlier sulfonamides are acetylated to relatively insoluble metabolites which precipitated in urine causing crystalluria. The earlier answer this was the triple sulfa combination but now the more commonly used sulfonamides have different elimination and solubility properties and exhibit less problems. Drug as a Pro-drug - Active Metabolite In most cases the metabolites are inactive, however, occasionally the metabolite is also active, even to the extent that the metabolite may be the preferred compound to be administered. The original drug may take on the role of a pro-drug. For example:- amitriptyline ---> nortriptyline codeine ---> morphine primidone ---> phenobarbital Drug metabolism can be quantitatively altered by drug interactions. This alteration can be an increase by induction of enzyme activity or a reduction by competitive inhibition. Pharmacogenomics - Pharmacogenetics Pharmacogenomics and the older term pharmacogenetics describe the interaction between drug pharmacokinetics or activity and genetic or genomic parameters. While pharmacogenetics deals with genetic difference between individuals, pharmacogenomics deals with the more specific interaction with genes and single nucleotide polymorphisms (SNPs). Genetic polymorphism will cause differences in enzymes, proteins, transporters and receptors. Responses to Pharmacogenomic Variation Alteration in enzyme activity may produce clinically significant differences in drug metabolism. Altered protein structure can cause altered drug protein binding Changes in drug transporters can alter drug absorption or distribution Drug receptor formation can be controlled genetically. Alterations in drug receptors may significantly change drug response. Some definitions (from Wikipedia or the references below) Chromosomes consists of a long strand of deoxyribonucleic acid (DNA). All non reproductive human (diploid) cells contain two pairs of 22 chromosomes plus two sex determining chromosomes for a total of 46 Each strand of DNA consists of a double chain of deoxyribose, pentose sugar, phosphate group and nitrogenous heterocyclic bases (adenine, cytosine, guanine or thymine). Specific sequences of base pairs in the DNA strands define the gene. a Gene is a specific section or location of the DNA strand of a chromosome that carries the coding information for some protein or structural RNA. It consists of at least forty base pairs. Genes take up only approximately 1% of a DNA strand of a chromosome. The cells transcribe the genetic information on the gene into RNA and the RNA is translated into a specific protein (enzymes, transporters or receptors). This process is called gene expression. Genetic or mutated variations in the base pair sequence in a gene are alleles also called polymorphism. Alleles are different forms of a gene. There may be numerous variations of alleles for any gene. The diploid cells contain two copies of each of the 22 non sex chromosomes. The 'normal' or more common allele is called the wild type. Modified or altered alleles may be called mutant. If the allele is copied one copy is inherited from the mother and from the father. If the alleles from each copy are the same the individual is homozygous for that genetic variation. Different alleles mean the individual is heterozygous. There may be many modifications or mutant alleles to an particular gene. Some alleles may be dominant where only one copy is needed for a specific expressions. In other case the allele may be recessive and both copies must be the same (homozygous) for this gene expression. In other cases the heterozygous situation may provide an intermediate expression (result). For example consider a gene that determines the activity of a particular enzyme. If the gene can exist as only two alleles (say A or B), with A being dominant, AA, AB, and BA will produce one (the more common) enzyme activity and only the BB case will provide the alternate enzyme activity. In other cases the two homozygous forms (AA and BB) will produce the extremes of enzyme activity and the heterozygous forms (AB and BA) will produce an intermediate activity. In other cases, there may be many more than two alleles so a wide range of enzyme (protein, transporter or receptor) may be expressed. Single nucleotide polymorphisms (SNPs) are alterations in a single base pair at a particular location on the DNA strand of a gene. SNPs may occur at any part of the DNA strand and commonly outside the ≈ 1% that include gene information. A few examples The muscle relaxant succinylcholine is usually rapidly deactivated by plasma butyrylcholinesterase within a few minutes. However, in some individuals genetic variation in the expression of this enzyme results in reduced enzyme activity, reduced metabolism and prolong drug activity. Drug activity may last up to an hour in these individuals (Kalow, 2004). During World War II it was observed that some African-American soldiers suffered hemolytic toxicities after usual doses of the anti-malarial primaquine. This was later identified as a higher frequency of genetically controlled lack of the enzyme glucose-6-phosphate dehydrogenase (G6PD) (Kalow, 2004). In a MASH episode (episode 210 - The Red/White Blues) Max Klinger (played by Jamie Farr), a regular character portraying a solder of eastern Mediterranean origin also exhibited symptoms of primaquine toxicity which was later attributed to a higher incidence of a genetic deficiency in this population as well. Fast and slow acetylators (N-acetyltransferase, NAT) of isoniazid have been identified in varying frequencies in different populations. Normal doses given to unidentified, slow acetylators results in toxicities such as numbness, pain and tingling (Kalow, 2004). drug transporters, MDR1 codeine metabolism to morphine - CYP 2D6 warfarin dosing References Kwon, Y. 2001 Handbook of Essential Pharmacokinetics, Pharmacodynamics and Drug Metabolism for Industrial Scientists, Chapter 8 Metabolism, Kluwer Academic, New York Ritschel, W.A. and Kearns, G.L. 2004 Handbook of Basic Pharmacokinetics ... including Clinical Applications, 6th ed., American Pharmaceutical Association, Washington, DC ISBN 1-58212-054-4 Humma, L.M., Ellingrod, V.L. and Kolesar, J.M. 2003 Lexi-Comp's Pharmacogenomics Handbook, Lexi-Comp, Hudson, OH ISBN 1-59195-060-0 Ishikawa T. 1992 The ATP-dependent glutathione S-conjugate export pump. Trends Biochem Sci., pp463-468 Licinio, J. and Wong, M.-L. 2002 Pharmacogenomics, Wiley-VCH, Weinheim, Germany ISBN 3-527-30380-4 Vavricka, S.R. et al. 2002 Interactions of rifamycin SV and rifampicin with organic anion uptake systems of human liver, Hepatology, Jul 36(1), pp164-72 Kalow, W. 2004 Pharmacogenetics: A Historical Perspective in Pharmacogenomics: Application to Patient Care, Amer. College Clin. Pharmacy, Kansas, MO ISBN 1-880401-80-0 P450 - Drug Table Cytochrome P450 (CYP) Allele Nomenclature Committee FDA Guidance for Industry Pharmacogenomic Data Submissions (Mar 2005) and Companion Guidance (Aug 2007) 16.2 Induction and Inhibition Metabolism based drug-drug and other interactions can have a significant influence on the use and safety of many drugs. Induction of drug metabolism can lead to an unexpected drop in drug concentration or the build-up of metabolites. The reverse can occur when there is inhibition of drug metabolism. Induction Enzyme induction is an increase in enzyme concentration caused by a drug or environmental compound. Induction may result from transcriptional activation (more common with CYP450 enzymes) or enzyme stabilization. A number of drugs can cause an increase in liver enzyme activity over time. This in turn can increase the metabolic rate of the same or other drugs. Phenobarbitone will induce the metabolism of itself, phenytoin or warfarin. Carbamazepine is another drug which can induce its own metabolism. Rifampin has been shown to cause up to a twenty times increase in midazolam metabolism. Cigarette smoking can cause increased elimination of theophylline (two fold increase) and other compounds. Dosing rates may need to be increased to maintain effective plasma concentrations. Inhibition Understanding drug inhibition potential is an important part of any new drug development. One preparation used in these studies are cDNA expressed CYP450 enzymes. Human liver microsomes may also be used for a broader enzyme exposure. Inhibition may be competitive (inhibitor binds to free enzyme), uncompetitive (inhibitor binds to enzyme-substrate complex), noncompetitive (inhibitor and substrate bind to different sites on the enzyme) or mixed. The parameters, Ki (inhibitory constant - the concentration of inhibitor that increases the 'apparent' Km twofold) and IC50 (inhibitor concentration causing a 50% inhibition) can be determined with these in vitro systems. A number of drugs can inhibit the metabolism of other drugs. An example is the 1998 withdrawal from the market of the drug mibefradil. Shortly after the 1997 FDA approval of this drug it was found to be a potent metabolic inhibitor of drugs such as simvastatin and other statins, cyclosporin and terfenadine resulting in serious toxicities. Warfarin inhibits tolbutamide elimination which can lead to the accumulation of drug and may require a downward adjustment of dose. Grapefruit juice has been shown to cause a two-fold increase in saquinavir AUC and reduced (inhibited) metabolism of other drugs such as midazolam and coumarin. References Krishna, R. 2004 Applications of Pharmacokinetic Principles in Drug Development, Kluwer Academic/Plenum Publishers, New York, NY ISBN 0-306-47766-1 Bonate, P.L. and Howard, D.R., ed. 2004 Pharmacokinetics in Drug Development: Clinical Study Design and Analysis, Volume 1, AAPS Press, Arlington, VA ISBN 0-9711767-4-4 Shargel, L. and Yu, A.B.C. 1999 Applied Biopharmaceutics and Pharmacokinetics, 4th ed., Appleton & Lange. Stamford, CT ISBN 0-8385-0278-4 Aminimanizani, A. and Winter, M.E. 2004 Chapter 12 "Theophylline" in Basic Clinical Pharmacokinetics, 4th ed., Winter, M.E., Lippincott Williams & Wilkins, Baltimore, MD 16.3 Hepatic Clearance The systemic or total body clearance clearance, CL, is a measure of the efficiency with which a drug is irreversibly removed from the body. One important component of this total body clearance is liver or hepatic clearance, CLH. There are a number of models used to describe hepatic clearance including the venous equilibration model. This model include a number of parameters which can be considered in the understanding of hepatic clearance and liver disease or altered physiological state. Venous equilibration model equation We can consider the organ clearance as it may be measured in an isolated organ system. Here we would have an isolated liver, perfused with blood containing the drug of interest. By measuring the drug concentration in the blood entering and leaving the organ at steady state, the organ clearance of the drug can be measured directly. In Figure 16.3.1, QH is the blood flow rate to the organ, CA is the concentration of drug in the blood entering the organ, and CV is the concentration of drug in the blood leaving the organ. The term E is the steady state extraction ratio. High E values mean high clearance by the liver and thus extensive metabolism. The sum of the individual organ clearance values are equal to the systemic clearance, CL. For a drug which is eliminated entirely via the liver, the hepatic clearance is equal to the systemic or total body clearance. From the equation, Figure 16.3.1, we can see that the organ clearance is a function of the liver blood flow and the extraction ratio of the drug. The liver blood flow is a physiological parameter which may be altered in disease states. The extraction ratio, we shall see shortly is a parameter dependent not only of the condition of the liver but also the nature of the drug. Both the hepatic clearance and the extraction ratio are empirical parameters which can be used as measures of the efficiency of the elimination process. They are dependent on three independent variables. total hepatic blood flow (QH), fraction unbound (fU) or the extent of drug binding to blood constituents. This may be saturable with high dose, polar compounds, and the free intrinsic clearance (CLint) or the rate-limiting step in drug uptake from blood, intracellular transport, metabolism, and where necessary biliary secretion. The free intrinsic clearance may be thought of as the clearance of drug from liver plasma water, devoid of the influence of blood flow or binding. Since a major part of this parameter is metabolism which is typically enzyme mediated this parameter may be saturated at higher doses, for some drugs The equation describing hepatic clearance in terms of these parameters using the venous equilibration model can be defined as (Wilkinson and Shand 1975), Equation 16.3.1. Equation 16.3.1 Hepatic Clearance With this equation it is possible to look at the influence of free intrinsic clearance, drug binding, and liver blood flow on the overall hepatic clearance of a drug using applets calculating plasma concentrations after iv bolus or oral dosing. Drugs can be classified into three types depending on the intrinsic clearance and binding. Flow limited, capacity limited, and others. Flow limited drugs, With high fU • CLint (= CLtotalint) value. [fU • CLint >> QH], Equation 16.3.2. Equation 16.3.2 Hepatic Clearance - Flow Limited For drugs with high total intrinsic clearance the extraction ratio, E, approaches 100%, the hepatic clearance approximates and is dependent of hepatic blood flow. Hepatic clearance is said to be FLOW LIMITED. Also, we can note that the hepatic clearance is not dependent on moderate changes in free intrinsic clearance or binding to blood constituents. Examples include; lidocaine, propranolol, morphine. Capacity limited drugs, Very low total intrinsic clearance. [fU • CLint << QH], Equation 16.3.3. Equation 16.3.3 Hepatic Clearance - Capacity Limited With drugs having very low intrinsic clearance, hepatic extraction is inefficient and hepatic clearance becomes independent of hepatic blood flow. Now changes in free intrinsic clearance and/or binding to blood constituents becomes very important in determination of the overall hepatic clearance. Hepatic clearance is said to be CAPACITY LIMITED as the intrinsic capacity of the liver controls the drug clearance. Examples include; phenytoin, warfarin, and quinidine. For such drugs it is possible that liver disease will cause a decrease in CLint but also an increase in fu. In this case the overall hepatic clearance doesn't reflect just the hepatic metabolic activity but also the drug binding. This is illustrated with tolbutamide. In patients with hepatitis there is an increase in fu but no change in CLint. As a result CL is increased and the elimination half-life decreases. The change in elimination half-life reflects changes in binding and not changes in drug metabolizing activity. Other drugs Between these two extremes. Capacity-limited but binding-insensitive drugs. The three parameters; QH, fU, and CLint are important determinants of drug elimination. Examples include; theophylline, antipyrine There are other models for liver metabolism besides the well-stirred (venous equilibration) model described above, such as the parallel-tube (sinusoidal perfusion) and the dispersion model. References Wilkinson, G.R., and Shand, D.G. 1975 A physiological approach to hepatic drug clearance, Clin. Pharmacol. Ther., 18, 377-90 16.4 Systemic Availability Even if we can assume that a drug is completely absorbed across the G-I tract, a proportion of the dose may be eliminated by the liver before reaching the systemic circulation because of the anatomical arrangement of the portal circulation. This pre-systemic or first-pass elimination can be determined from the extraction ratio, E, such that the fraction of the dose that is available to the central circulation is 1-E. This 1-E value becomes the maximum availability possible before allowing for reduced product performance. For drugs which are extensively metabolized, first pass metabolism can be quite important. It means that higher doses must be given orally compared with parenteral administration. For example morphine 30 mg PO compared with 5 mg IV and lidocaine not active orally. In liver disease there is potential for changing the systemic availability of high extraction drugs and thereby affecting steady state concentrations. If liver disease causes a modest reduction in the extraction ratio, from for example 0.95 to 0.9, the fraction of the orally administered drug reaching the systemic circulation (1-E) will be doubled. One of the consequences of the pathogenesis of chronic liver disease is the development of porta-systemic shunts that may carry drug absorbed from the G-I tract through the mesenteric veins directly into the systemic circulation. Thus in a disease where biochemical hepatic function is relatively well maintained (e.g., schistosomiasis), oral treatment with high clearance drugs such as morphine or propranolol can lead to high blood levels and an increase in adverse drug effects. For example, 30 mg morphine orally may act like 30 mg IV and lead to over dosage toxicity. Pharmacokinetics of Drugs in Patients with Liver Disease Liver disease can have a profound effect on the patient's physiology which in turn can influence drug pharmacokinetics. Using the venous equilibration model presented on the previous page we can expect changes in fu, QH and CLint to influence the overall pharmacokinetics of a drug. This can be due to changes in protein binding (Chapter 18), to reduced enzymatic activity of the liver cells or reduced ability of the drug to reach the enzymes present in liver cells. Decreased protein binding appears to be more common in chronic liver disease (such as cirrhosis) compared with more acute diseases (such as viral hepatitis). Some examples include morphine (15%), propranolol (38%), diazepam (70-200%), phenytoin (40%) and tolbutamide (30%) (Benet et al. 1984 t70). A number of high extraction ratio (flow limited) drugs exhibit increased oral bioavailability, decreased clearance or increased half-life. Some examples include chlormethiazole (F increased 1000% and decreased clearance), lidocaine (decreased clearance), meperidine (increased F and t1/2 with decreased clearance), propranolol (increased F and t1/2) and verapamil (increased F and t1/2 and decreased clearance) (Benet et al. 1984 t67). Capacity limited (poorly extracted) drugs also have altered pharmacokinetic parameters in patients with liver disease. Examples include ampicillin (increased t1/2), diazepam (increased t1/2 and decreased CL), theophylline (increased t1/2 and decreased CL) and tolbutamide (increased t1/2 and decreased CL) (Benet et al. 1984 t68-9). Pharmacogenomic Considerations Enzymes are produced according to the genetic make-up of the individual. This means that different individuals may produce more or less of a particular enzyme but it also means that different forms (allele variants) of the enzyme may be produced by different individuals. Enzymes control the metabolism of many drugs and different forms of these enzymes will cause differences in the pharmacokinetics of the drugs. Some enzymatic forms are more active, others less active. One example is the enzyme CYP2C9 and its influence on warfarin pharmacokinetics (that is, the five times more active S-form). There are three forms of CYP2C9; *1, *2 and *3. The CYP2C9*1 is the wild form and the most common. However, there are 10% and 8% of population with the *2 and *3 forms, respectively. Both these variants result in reduced enzymatic activity or reduced drug clearance. One study (Higashi et al 2002) indicated that the daily maintenance dose ranged from 5.6 mg (*1/*1) to 1.6 mg (*3/*3) with different individuals. These studies suggest the potential for more difficult stabilization of the required dose and higher incidence of bleeding. Genotyping can be performed by relatively sophisticated (expensive) techniques which should become more affordable and common in the future. Another approach is the use of a test compound to determine CYP2C9 (or other enzyme) activity in an individual. For CYP2C9, compounds such as diclofenac, phenytoin, tolbutamide and S-warfarin have been considered (Ritschel and Kearns, 2004 p359). A third approach is to recognize this potential for genetic (and other) sources of variability and carefully monitor the initial doses of the therapeutic compound, that is, apply therapeutic drug monitoring (TDM). References Humma, L.M., Ellingrod, V.L., and Kolesar, J.M. 2003 Lexi-Comp's Pharmacogenomics Handbook, Lexi-Comp, Hudson, OH ISBN 1-59195-060-0 Higashi, M.K., Vennestra, D.L., Kondo, L.M., Wittkowsky, A.K., Srinouanprachanh, S.L., Farin, F.M. and Rettie, A.E. 2002 Association between CYP2C9 genetic variants and anticoagulation-related outcomes during warfarin therapy, J. Amer. Med. Assoc., 287(13), 1690-98 Benet, L.Z., Massoud, N. and Gambertoglio, J.G. 1984 Pharmacokinetic basis for drug treatment, Raven Press, New York, NY ISBN 0-89004-874-6 Ritschel, W.A. and Kearns, G.L. 2004 Handbook of Basic Pharmacokinetics, 6th ed., APhA, Washington, DC ISBN 1-58212-054-4 Chapter 17: Distribution Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- understand and describe the processes by which drugs are distributed throughout the body understand the effect of protein binding on drug distribution and methods by which protein binding is measured 17.1 Drug Distribution Patterns Distribution can be thought of as following one of four types of patterns. 1) The drug may remain largely within the vascular system. Plasma substitutes such as dextran are an example of this type, but drugs which are strongly bound to plasma protein may also approach this pattern. 2) Some low molecular weight water soluble compounds such as ethanol and a few sulfonamides become uniformly distributed throughout the body water. 3) A few drugs are concentrated specifically in one or more tissues that may or may not be the site of action. Iodine is concentrated by the thyroid gland. The antimalarial drug chloroquine may be present in the liver at concentrations 1000 times those present in plasma. Tetracycline is almost irreversibly bound to bone and developing teeth. Consequently tetracyclines should only be given to young children or infants in extreme conditions as it can cause discoloration and mottling of the developing second set of teeth. Another type of specific concentration may occur with highly lipid soluble compounds which distribute into fat tissue. Another example is the distribution of a bone scan agent, 99mTc-MDP. A normal scan shows accumulation in bone, at the injection site and 'maybe' in organs of elimination (Saha, 1984). Polychlorinated biphenyls, PCB, are highly lipid soluble and extensively distributed into fat tissue, maybe, with very little leaving these tissues. DDT, dicophane, which is also very lipid soluble has very restricted use. Remember Silent Spring by Rachel Carson, 1962. 4) Most drugs exhibit a non-uniform distribution in the body with variations that are largely determined by the ability to pass through membranes and their lipid/water solubility. The highest concentrations are often present in the kidney, liver, and intestine usually reflecting the amount of drug being excreted. Pattern 4 is the most common being a combination of patterns 1, 2 and 3, Figure 17.1.1. A useful indicator of the type of pattern that characterizes a particular drug is the apparent volume of distribution. A value of V in the region of 3-5 liter (in an adult) would be compatible with pattern 1. This is approximately the volume of plasma. Pattern two would be expected to produce a V value of 30 to 50 liter, corresponding to total body water. Drugs or other compounds exhibiting pattern 3 would exhibit very large values of V if the drug concentration effect was acting on most of the dose. Chloroquine has a V value of approximately 17,000 liter. Drugs following pattern 4 may have a V value within a wide range of values, Table 17.1.1. These patterns of variation have been used to determine body fluid volumes, Table 17.1.2. References Ritschel, W.A. and Kearns, G.L. 2004 Handbook of Basic Pharmacokinetics ... including Clinical Applications, 6th ed., American Pharmaceutical Association, Washington, DC, p 369-401 and p 83-85 Saha, G.B. 1984 Fundamentals of Nuclear Pharmacy, 2nd ed. P239, Fig 12-26 17.2 Factors Affecting Drug Distribution There are a number of factors that affect drug distribution. These factors can be split into two categories. Those that affect the rate of distribution and those that affect the extent of distribution, Table 17.2.1. We have already covered some material about membrane permeability. The capillaries are typically lined with endothelium whose cells overlap, though to a lesser degree than epithelial cells. Also, the junctions between cells are discontinuous. Capillary walls are quite permeable. Lipid soluble drugs pass through very rapidly. Water soluble compounds penetrate more slowly at a rate more dependent on their size. Low molecular weight drugs pass through by simple diffusion. For compounds with molecular diameter above 100 Å transfer is slow. For drugs which can be ionized the drug's pKa and the pH of the blood will have a large effect on the transfer rate across the capillary membrane. There are two deviations to the typical capillary structure which result in variation from normal drug tissue permeability. i) Permeability is greatly increased in the renal capillaries by pores in the membrane of the endothelial cells, and in specialized hepatic capillaries, known as sinusoids which may lack a complete lining. This results in more extension distribution of many drugs out of the capillary bed. ii) On the other hand brain capillaries seem to have impermeable walls restricting the transfer of molecules from blood to brain tissue. Lipid soluble compounds can be readily transferred but the transfer of polar substances is severely restricted. This is the basis of the "blood-brain" barrier. Membrane permeability tends to restrict the transfer and distribution of drugs once they are delivered to the tissue. The other major factor which determines the rate of drug distribution is blood perfusion. Blood perfusion rate The rate at which blood perfuses to different organs varies widely, Table 17.2.2. Total blood flow is greatest to brain, kidneys, liver, and muscle with highest perfusion rates to brain, kidney, liver, and heart. It would be expected that total drug concentration would rise most rapidly in these organs. Certain smaller organs such as the adrenals (1.2 - 5.5 mL/min/mL or 0.2 - 1% CO) and thyroid (2.4 - 4 mL/min/mL or 1 - 2% CO) also have large perfusion rates. As an example; thiopental gets into the brain faster than muscle, whereas, penicillin gets into muscle more quickly than it gets into brain, Figure 17.2.1. i) Thiopental is only partly ionized and passes into the brain or muscle easily. Perfusion limits the transport. Since brain has a higher perfusion rate the thiopental can transfer in and out more quickly. ii) Penicillin is quite polar and is thus slowly permeable. Permeability limited transfer is faster in muscle as muscle capillaries are less restrictive. Thus transfer of penicillin is faster in muscle than brain. In brain, perfusion or membrane permeability limits drug transport or distribution. Thiopental diffuses readily, thus perfusion limits its distribution. Since perfusion is higher to the brain than to muscle, transport to the brain is faster. Penicillin less readily diffuses thus it is diffusion which limits penicillin distribution. Muscle diffusion is easier thus distribution into muscle is faster for penicillin than distribution into brain. References Rowland, M. and Tozer, T.N, 1995 Clinical Pharmacokinetics Concepts and Applications, 3rd ed., Williams & Williams. Media, PA Shargel, L., Wu-Pong, S. and Yu, A.B.C. 2005 Applied Biopharmaceutics and Pharmacokinetics, 5th ed., McGraw-Hill, New York, NY 17.3 Extent of Distribution We can now consider factors which alter the extent of drug distribution Plasma protein binding Extensive plasma protein binding will cause more drug to stay in the central blood compartment. Therefore drugs which bind strongly to plasma protein tend to have lower volumes of distribution. Proteins involved Although drugs are bound to many macromolecules, binding to plasma protein is the most common. Of these plasma proteins, albumin, which comprises 50 % of the total proteins binds the widest range of drugs. Acidic drugs commonly bind to albumin, while basic drugs often bind to α1-acid glycoproteins and lipoproteins. Many endogenous substances, steroids, vitamins, and metal ions are bound to globulins, Table 17.3.1. Forces involved Groups on the protein molecules that are responsible for electrostatic interactions with drugs include: the —NH3+ of lysine and N- terminal amino acids, the —NH2+— of histidine, the —S- of cysteine, and the —COO- of aspartic and glutamic acid residues. In order to achieve reasonably stable complexes, however, it is likely that in most cases the initial electrostatic attraction is reinforced at close range by van der Waal's forces (dipole-dipole; dipole-induced dipole; induced dipole-induced dipole) and hydrogen bonding. This is suggested by the frequently crucial role of protein configuration in the binding phenomenon. Agents which denature protein may cause the release of bound drug. Often there may be competition between drugs, in which agents that are bound very tightly, such as coumarin anticoagulants, are able to displace less tightly bound compounds from their binding sites. Slight changes in the binding of highly bound drugs can result in significant changes in clinical response or cause a toxic response. Since it is the free drug in plasma which equilibrates with the site of pharmacological or toxic response, a slight change in the extent of binding, such as 99 to 98 % bound, which can result in an almost 100 % change in free concentration, can cause very significant alteration in response. For a large number of drugs, including warfarin and phenytoin, drug response will be dependent on free drug concentration. Alteration of free concentration by drug interaction or disease state can alter the intensity of action of these drugs. Examples include phenylbutazone and salicylates displacing tolbutamide to give an increased effect, hypoglycemia. As you can see from Table 17.3.2, the extent of protein binding can vary considerably from one drug to another. Protein binding determination Spectral changes Most drugs have distinct UV spectra because of the conjugated chromophores in the molecule. When a drug interacts with a protein the UV or visible spectrum may be changed because of alterations in the electronic configuration. These alterations can be quantitated and used to determine the extent of binding. Changes in fluorescence spectra can be used in the same way for example with warfarin (Chignell 1973). Gel filtration This involves the use of porous gels that are molecular sieves. They separate components on the basis of size. Low molecular weight drugs are held on the gel whereas bound drug and protein are washed through. Equilibrium dialysis The protein solution (e.g. plasma) containing drug and a buffer solution are placed on opposite sides of a dialysis membrane, Figure 17.3.1. After a sufficient time (maybe 12-24 hours), free drug concentration will be the same on both sides of the membrane. Protein binding can be determined by measuring the concentration of drug on each side of the membrane. On left the concentration will involve free and bound drug, whereas on the right there is no binding and the concentration will equal to the free drug concentration. Ultrafiltration A quicker method of separating free and bound drug is the ultrafiltration method. Drug and protein solution are placed in a filter membrane device and liquid containing free drug is forced through the membrane by centrifugation, Figure 17.3.2. Protein binding equilibria With one type of binding site, protein binding can be described mathematically by Equation 17.3.1 Equation 17.3.1 Drug Protein Equilibrium, One Binding Site with [D] free drug concentration, [P] total protein concentration and 'n' binding sites per molecule. Thus [nP] is the total concentration of protein binding sites and [rP] = [DP] is the concentration of bound drug or bound protein sites with r drug molecules bound per protein molecule. Typically there may be 1 - 4 binding sites per protein molecule, Figure 17.3.3. The binding association equilibrium constant can be described by Equation 17.3.2 Equation 17.3.2 Protein Binding Association Constant where Plots Equation 17.3.2 can be rearranged to give Equation 17.3.3 Equation 17.3.3 Scatchard Plot Equation Thus plotting r/[D] versus r should give a straight line. This is called a Scatchard plot, Figure 17.3.4. An alternative rearrangement gives Equation 17.3.4. Thus a plot of 1/r versus 1/[D] should also give a straight line. This is the double reciprocal plot, Figure 17.3.5. Equation 17.3.4 Double ReciprocalPlot Equation With one type of binding site these plots produce straight lines which can be used to determine values of Ka and n. With more than one type of binding site, these plots are curved (Chignell, 1973). Tissue localization of drugs In addition to plasma protein binding, drugs may bind to intracellular molecules. Certain of these may be actual drug receptors, and the interaction that occurs may represent the molecular basis of the pharmacological action. The affinity of a tissue for a drug may be for any of several reasons, including binding to tissue proteins (such as albumin) or to nucleic acids, or in the case of adipose tissue, dissolution in the lipid material. The concentration of chloroquine in the liver is due to the binding of the drug to DNA. Barbiturates distribute extensively into adipose tissue, primarily because of their high lipid solubility. Tetracyclines bind to bone and should be avoided in young children or discoloration of permanent teeth may occur. Unlike plasma binding, tissue binding of a drug cannot be measured directly as handling of the tissue results in disruption of the binding. However, this doesn't mean that tissue binding and changes in tissue binding are not important. Although drug pKa and lipid solubility will influence the extent of drug distribution the influence of plasma and tissue binding can be summarized in Equation 17.3.5. In Equation 17.3.5 the apparent volume of distribution is determined from the differential binding in the central (plasma) and peripheral (tissue) compartments. A drug tightly bound to plasma protein (fraction unbound in plasma is very small) will have a small total volume, similar to blood or plasma volume, V1, whereas a drug with extensive tissue binding (fraction unbound in tissue is very small) will have a much larger apparent volume of distribution, Bauer, 2008. Equation 17.3.5 Apparent Volume of Distribution References Bauer, L.A. 2008 Applied Clinical Pharmacokinetics, 2nd ed., McGraw Hill, p 16 Chignell, C.F. 1973 Ann. New York Acad. Sci., 226, p49, 53 Niazi, S. 1979 Textbook of Biopharmaceutics and Clinical Pharmacokinetics, Appleton-Century-Crofts, New York, NY p101 Ritschel, W.A. and Kearns, G.L. 2004 Handbook of Basic Pharmacokinetics ... including Clinical Applications, 6th ed., American Pharmaceutical Association, Washington, DC, pp369-401 17.4 Other Distribution Considerations Weight considerations The apparent volume of distribution will often be proportional to the total body weight of a patient. In fact many V values found in the literature will be given as so many liter per kilogram total body weight. The assumption made is that the body composition is unchanged on a percentage basis, thus distribution will be identical no matter what the patient weighs. This works within some limits. For example body composition of the very young and the very old may be quite different from 'normal', that is the average subject in whom the parameter values may have been originally determined. The young and the old will be discussed in more detail later. Another group of patients in which body composition may be greatly altered from `normal' is the obese. These patients have a higher proportion of adipose tissue and lower percentage of water. Thus for drugs which are relatively polar, volume of distribution values may be somewhat lower than the total body weight may suggest. For example the apparent volume of distribution of antipyrine is 0.62 l/kg in normal weight subjects but 0.46 l/kg in obese patients (Abernethy et al., 1981). Other drugs such as digoxin and gentamicin are also quite polar and tend to distribute into water rather than adipose tissue. Protein binding interactions The role of protein binding in drug interactions can be quite involved (Rowland and Tozer, 1989). Although drugs may well displace each other from common binding sites, the clinical (and pharmacokinetic) importance of these interactions may require considerable investigation. For these effects to be important one drug must be extensively protein bound, while the displacer must have a high affinity for the same binding site. Therapeutically the major criteria is the free drug concentration. One result of a drug interaction is to tend to produce an increase in free drug concentration, however, that will cause an increase in elimination and thus an overall reduction in total drug concentration, potentially maintaining the free concentrations unchanged. References Abernethy, D.R., Greenblatt, D.J, Divoll, M. et al. 1981 Alterations in drug distribution and clearance due to obesity, J.P.E.T., 217, p681-85 Rowland, M and Tozer, T.N. 1989 Clinical Pharmacokinetics, Lea & Febiger, Philadelphia, pp 260-270 Chapter 18: Multi-Compartment Pharmacokinetic Models Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- draw the scheme and write the differential equations appropriate to a multi-compartment pharmacokinetic model recognize and use the integrated equations for these models to calculate parameter values and for dosage regimen calculations calculate the parameters of these models using the method of residuals So far we have talked about the pharmacokinetics of drugs in terms of a one compartment model. We have assumed that the drug, once administered is mixed instantaneously in the blood and that the drug distributes throughout the body rapidly reaching equilibrium throughout the tissue into which the drug enters. We have in essence considered that the body acts as a well mixed container. With first order drug elimination we found that the plasma concentration will fall mono-exponentially with time following IV bolus administration, Figure 18.0.1. And the log of the plasma concentration will fall as a straight line, Figure 18.0.2. Commonly we find with real data, especially if we have a number of early data points, that the log Cp versus time plot is not a straight line. We see an initial early deviation from the straight line, followed by a log-linear phase. The initial phase is a more rapid drop in plasma concentration before settling into the log-linear fall in plasma concentration, Figure 18.0.3. This suggests that the body is not behaving as a single well mixed compartment. There appears, mathematically, to be distribution between two (or more) compartments. That is we don't have instantaneous equilibrium between the drug in all the various tissues of the body. In the next approximation we can consider that the body is behaving as two distinct compartments. These compartments can be called the central compartment and the peripheral compartment. Exact anatomical assignment to these compartments is not always possible. However, generally the rapidly perfused tissues often belong in the central compartment.18.1 Intravenous Administration Compartmental Model In this book we will mostly use the parameters V1, k12, k21 and kel to represent the distribution processes of a two compartment model. As for the one compartment model, kel (or CL) can be split into various elimination pathways but for now we will stay with kel to describe all these processes (Ritschel and Kearns, 2004; Bauer, 2008; and Shargel, Wu-Pong and Yu, 2012), Figure 18.1.1. Alternately, these four parameters could be replaced with V1, V2, CL and CLD (Kong and Jusko, 1988). A third approach is to use the four parameters V1, V2, Kp and CL (Rowland and Tozer, 1995). Below, after the integration of Equation 18.1.1 to Equation 18.1.2 leads to the parameters A, B, α, and β to describe this model. Differential equation The differential equation for drug in the central compartment following intravenous bolus administration is Equation 18.1.1. Equation 18.1.1 Differential Equation for Amount of Drug in the Central Compartment The kel • X1 term describes elimination of the drug from the central compartment, while the k12 • X1 and k21 • X2 terms describe the distribution of drug between the central and peripheral compartments. Writing differential equations can be reviewed in Chapter 2. Integrated equation Integration of this equation (using Laplace transforms) leads to a bi-exponential equation for plasma concentration as a function of time, Equation 18.1.2 and Equation 18.1.3. Equation 18.1.2 Integrated Equation for Concentration versus Time Equation 18.1.3 Integrated Equation for Concentration versus Time including k21 and V1 The A, B, α, and β terms were derived from the micro-constants during the integration process. They are functions of the parameters k12, k21, kel and V1. The substitutions for the sum and product of α and β (developed during the integration process) are functions of kel, k12 and k21. α + β = kel + k12 + k21 α • β = kel • k21 Knowing the values of kel, k12 and k21 we can calculate α + β as well as α • β then using Equation 18.1.4 we can calculate α and β. Note, in this equation, α is calculated when '+' is used in the numerator and β is calculated when '-' is used in place of the '±'. Thus α is defined as greater than β. Equation 18.1.4 Converting from kel, k12 and k21 to α and β Once we have values for α and β we can calculate values for A and B using Equation 18.1.5. Equation 18.1.5 Calculate A and B An example calculation A drug follows first order (i.e. linear) two compartment pharmacokinetics. After looking in the literature we find a number of parameter values for this drug. These numbers represent the micro constants for this drug. In order that we can calculate the drug concentration after a single IV bolus dose these parameters need to be converted into values for the macro constants. The kel and V1 for this drug in this patient (90.5 Kg) are 0.192 hr-1 and 0.39 L/kg, respectively. The k12 and k21 values for this drug are 1.86 and 1.68 hr-1, respectively. What is the plasma concentration of this drug 1.5 hours after a 500 mg, IV bolus dose. In order to complete this calculation first calculate the appropriate A, B, α and β values. Since and Now: and The last step is Later in this chapter we will use equations for the reverse process of converting α, β, A and B into values for k12, k21, kel and V1. References Bauer, L.A. 2008 Applied Clinical Pharmacokinetics, 2nd ed., McGraw Hill, p 29 Kong, A.-N., and Jusko, W. J. 1988 Definitions and applications of mean transit and residence times in reference to the two-compartment mammillary plasma clearance model. Journal of Pharmaceutical Sciences, 77(2), 157–165. Ritschel, W.A. and Kearns, G.L. 2004 Handbook of Basic Pharmacokinetics ...Including Clinical Applications, 6th ed., APhA, p 118 Rowland, M and Tozer, T.N. 1995 Clinical Pharmacokinetics Concepts and Applications, 3rd Ed., Williams and Wilkins, p 144 Shargel, L., Wu-Pong, S. and Yu, A. 2012 Applied Biopharmaceutics & Pharmacokinetics, 6th ed., McGraw Hill, p 64 18.2 Parameter Determination Method of residuals Values for kel, k12, k12 and other parameters can be determined by first calculating A, B, α, and β. For this we can use the method of residuals (in a similar fashion to determining ka and kel for the one compartment model after oral administration). Starting with the equation for Cp, Equation 18.2.1. Equation 18.2.1 Concentration versus Time after an IV Bolius Dose, Two Compartment Model By definition α is greater than β then as t approaches ∞, e-α • t approaches 0 faster than e-β • t. Therefore if the ratio α/β is large enough (greater than 5) the terminal data points will fall on a straight line on semi-log graph paper. Equation 18.2.2 is similar to the equation for the late plasma concentration values after oral administration with a one compartment model. This will be a straight line if plotted on semi-log graph paper. Equation 18.2.2 Cplate versus Time Beta can be calculated from the slope of this line, Equation 18.2.3, Figure 18.2.1. Equation 18.2.3 Determining β from the Cplate Line The units for β and α (below) are reciprocal time, for example min-1 or hr-1. Biological half-life or Terminal half-life The t1/2 calculated as 0.693/β is often called the biological half-life or terminal half-life. It is the half-life describing the terminal elimination of the drug from plasma. [For the one compartment model the biological half-life is equal to 0.693/kel]. The difference between the Cplate values (red line, Figure 18.2.2) at early times and the actual data at early times is again termed the 'residual', Equation 18.2.4. Equation 18.2.5 Determining α from the Residual Line The slope of the residual line (green line, Figure 18.2.2) will provide the value of α and the value of A can be estimated as the intercept of the concentration axis (y-axis). A more accurate value for the α value can often be determined by expanding the scale on the time axis (Figure 18.2.3). However, don't forget to use the new time values when calculating α from the equation, Equation 18.2.5. Explore this calculation using Interactive 18.2.1. Converting macro constants to micro constants With A, B, α, and β determined using the method of residuals we can calculate the micro-constants from the equations, Equation 18.2.6. Equation 18.2.6 Converting from A, B, α and β to kel, k221 and k21 18.3 Effect of changing Distribution Parameters on Drug Concentration versus Time Changing the Ratio of k12 to k21 From the k12 and k21 values we can assess the extent of distribution of drug into the peripheral compartment. The higher the ratio k12/k21 the greater the distribution of drug into the peripheral compartment. The larger the individual values of k12 and k21 the faster is the transfer between the central and peripheral compartments and the more the body behaves as a single compartment. As the ratio increases the distribution phase is more pronounced. Conversely with the ratio 1/4 there is very little distribution phase. Also note that the β value or the slope of the terminal phase is changing even though the kel is fixed at 0.167 hr-1, Figure 18.3.1. Explore this relationship with Interactive 18.3.1. Changing the Magnitude of k12 and k21 (Same Ratio) With faster and faster distribution the initial drop in plasma concentration becomes quite rapid. If you were sampling every 30 minutes, the initial phase would be missed. The data would look just like a one compartment model. Redrawing the slow plot with k12/k21 (0.5/0.25) over 24 hours and gives a plot that is definitely still bi-exponential, Figure 18.3.2. Explore this relationship with Interactive 18.3.1. Changing the Magnitude of V2 With larger values of V2 there is more distribution into the peripheral compartment. This leads to a bigger drop in Cp at early time points and a slower beta elimination phase, Figure 18.3.3. Explore this relationship with Interactive 18.3.1. Changing the Magnitude of CL2 Larger values of CL2, the distribution clearance, produce a faster drop in the early Cp values, Figure 18.3.4. Explore this relationship with Interactive 18.3.1. 18.4 Apparent Volumes of Distribution The concentration of drug in the body is determined not only by the rate constant values but also by the apparent volume of distribution. In the case of the two compartment model a number of volume terms can been defined. V1 The apparent volume of the central compartment, V1 or Vc, can be calculated as shown in Equation 18.4.1. Equation 18.4.1 Apparent Volume of the Central Compartment This parameter is important because it allows the calculation of the highest plasma concentration or Cp0 after an IV bolus administration. This concentration may result in transient toxicity. V1 can also be used in dose calculations. Varea ( = Vß = Vz) Varea or Vß of Vz is defined as shown in Equation 18.4.2. Equation 18.4.2 Apparent Volume, Area Because of the relationship with clearance and β and with V1 and kel this parameter is quite useful in dosing calculations. This parameter can be readily calculated via AUC and β values from the 'raw' data and is therefore commonly quoted. Note, for a two compartment model, β and 𝜆z are the same parameter. In a more general, non compartmental model approach 𝜆z is slowest, or terminal rate constant. Vss Vss, V steady state, defined as shown in Equation 18.4.3. Equation 18.4.3 Apparent Volume, Steady State This term relates the total amount of drug in the body at 'steady state' with the concentration in plasma or blood. A pseudo-steady state can be seen in Figure 18.4.1 where the amounts or concentrations of drug in compartment 1 and 2 are falling in parallel. A true steady state would only be achieved after a continuous infusion of the drug. The relationship between these volume terms is that Varea > Vss > V1 Note: For a one compartment model the values for all these volume parameters would be equal. Example Calculation As an example we can look at the data in Table 18.4.1. The first two columns are the time and plasma concentration collected after IV bolus administration of 500 mg of drug. These data are plotted (blue squares) in Figure 18.4.2. At longer times, after 4 hours, out to 12 hours the data appears to follow a straight line on semi- log graph paper. Since α > β this terminal line is described by B • e-β • t. Following it back to t = 0 gives B = 10 mg/L. From the slope of the line β = 0.25 hr-1. Cplate values at early times are shown in column 3 and the residual in column 4, Table 18.4.1. The residual values are plotted (red circles) giving a value of A = 25 mg/L and α = 1.51 hr-1 Note that α/β = 6 so these values should be fairly accurate. B = 10 mg/L β = (ln 10 - ln 0.5)/12 = 2.996/12 = 0.25 hr-1 A = 25 mg/L α = (ln 25 - ln 0.27)/3 = 4.528/3 = 1.51 hr-1 Therefore Cp = 25 • e-1.51 • t + 10 • e-0.25 • t We can now calculate the micro-constants. The AUC by the trapezoidal rule + Cplast/β = 56.3 + 2.0 = 58.3 mg.hr.L-1, [Note the use of β] thus Notice that Varea > Vss > V1 [34.3 > 26.7 > 14.3] 18.5 Dosage Calculations Dosage calculations are complicated by the extra terms in the equations however some calculations are still reasonably straightforward. The dose required for a particular initial plasma concentration can be calculated if V1 is known, Equation 18.5.1. Equation 18.5.1 Loading Dose To achieve an initial Cp of 20 mg/L given V1 = 30 liter would require a DOSE = 20 x 30 = 600 mg. Alternately if a dose of 500 mg is given and the V1 value is 16 L, the expected Cp0 can be calculated. Cp0 = 500/16 = 31.3 mg/L If the A, B, α, and β values are known or calculated, then the plasma concentration at any time after a single IV dose can be calculated. The plasma concentration achieved after a continuous IV infusion is given by the same equation described for the one compartment model, Equation 18.5.2. Equation 18.5.2 Maintenance Infusion Rate If a plasma concentration of 30 mg/L is required and V1 = 15 L and kel is 0.2 hr-1 then the required infusion rate can be readily determined. k0 = 30 x 15 x 0.2 = 90 mg/hr This result can be seen as the green line in Figure 18.5.1. Since the time to reach the steady state concentration is controlled by the β value this could mean a slow approach to the desired value, thus an IV bolus loading dose may be useful. Unfortunately this calculation is not straight forward. With V1 = 15 L, kel = 0.2 hr-1, and required Cp = 30 mg/L Bolus DOSE = 15 x 30 = 450 mg and Infusion Rate = k0 = 30 x 15 x 0.2 = 90 mg/hr As you can see (Figure 18.5.1 and Interactive 18.5.1) this results in quite a dip in the Cp versus time curve. With IV bolus doses of either 600 or 300 mg (shown in Figure 18.5.2) the curves may or may not be better depending on the therapeutic range of the drug. Another approach is to give a fast infusion followed by a slower, maintenance infusion. Here 1200 mg was given over 4 hours (at 300 mg/hr) before switching to the slower 90 mg/hr maintenance rate, Figure 18.5.3. Calculations derived in part from Ritschel and Kearns, 2004. Explore various fast and slow IV infusions using Interactive 18.5.2. Other approaches include multiple IV bolus doses and more complex infusion regimens. References Ritschel, W.A. and Kearns, G.L. 2004 Handbook of Basic Pharmacokinetics ...including Clinical Applications, 6th ed., Amer Pharm Assoc., Washington, DC, p 205 18.6 Oral Administration The model for a two compartment model after oral administration is shown in Figure 18.6.1. The differential equation for the central compartment is Equation 18.6.1. The plasma concentration can be described by an equation with three exponential terms, Equation 18.6.2, where A + B + C = 0. Equation 18.6.1 Differential Equation for Drug Amount in the Body after Oral Administration Equation 18.6.2 Integrated Equation for Drug Concentration in Plasma after Oral Administration Depending on the values of α, ß and ka a distribution phase may, Figure 18.6.2, or may not, Figure 18.6.3, be seen. Bioavailability Bioavailability calculations are the same as for the one compartment model, i.e., by comparison of AUC or U∞. These apply for any linear system. Also if α, β, and ka are sufficiently separated the method of residuals could be applied (twice) to determine values for these three parameters. Average Plasma Concentration The average plasma concentration equation can also be used to calculate appropriate dosing regimens. For example if an average plasma concentration of 20 mg/L is required and V1 = 15 L, kel = 0.15 hr-1, F = 0.9 and a dosing interval of 12 hours is to be used then the required dose can be calculated from the equation for , Equation 18.6.3. Equation 18.6.3 Average Plasma Concentration The required dose can be calculated using Equation 18.6.4 and the parameter values provided above. Equation 18.6.4 Dose Required for an Average Concentration of 20 mg/L Chapter 19: Non Compartmental Analysis Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- understand and use the non compartmental approach to parameter estimation define, use, and calculate the parameters: AUMC (area under the first moment curve) MRT (mean residence time) MAT (mean absorption time) MDT (mean dissolution time) 19.1 Non Compartmental Analysis So far we have looked at one compartment pharmacokinetic models and two compartment models. With suitable data we may even consider three or four compartment models. This can lead to considerable modeling effort to determine the best model and the best values for the parameters of the best model. Another approach is to not be concerned about how many compartments are needed but to use a non compartmental approach. Non compartmental analysis can be used to determine certain pharmacokinetic parameters without having to decide on a particular compartmental model. This method requires that the disposition processes, distribution, metabolism and excretion (DME), are linear or first order. The method also tends to remove some of the detail that might be explored with a more detailed modeling approach. Unless a one compartment model would be appropriate it is not possible to reproduce an accurate concentration versus time profile using only non compartmental analysis parameters. However, the calculations can be faster and more straightforward. The parameters can be useful for understanding the drug pharmacokinetics and dose calculations. The method is based on the calculation of the area under the plasma concentration versus times curve (AUC, zero moment curve) and the area under the plasma concentration multiplied by time versus curve (AUMC, first moment curve). The AUC and AUMC can be calculated as before using the trapezoidal rule. Maybe best covered with an example. Consider a drug given both by IV and oral administration. Both the AUC and AUMC were calculated using the trapezoidal rule without making any assumption concerning the number of compartments. The final segment of the AUC curve is calculated as Cplast/k', where k' is the last exponential term (the slowest) calculated from the Cp versus time graph. The last segment of the AUMC curve is calculated using Equation 19.1.1 using the same k’ value. These calculations can be explored using example data after an IV bolus dose, Table 19.1.1. Equation 19.1.1 Last AUMC Segment The data from Table 19.1.1, Cp versus time, Figure 19.1.1, and Cp x time versus time, Figure 19.1.2, can be plotted on linear graph and various parameters can be calculated. From the AUC and AUMC values we can calculate the mean residence time, MRT. This is the average time that the drug stays in the body (or plasma as measured here). It can be related to the average elimination rate constant as 1/MRT. The values of AUC and AUMC from Table 19.1.1 give, Equation 19.1.2 Equation 19.1.2 Mean Residence Time, MRT MRT = 549.31/67.27 = 8.17 hr and Equation 19.1.3 for kel’. Equation 19.1.3 Apparent Elimination Rate Constant, kel’ kel' = 1/8.17 = 0.12 hr-1 Remember we can also calculate CL, Equation 19.1.4, Equation 19.1.4 Total Body Clearance CL = Dose/AUC = 100/67.27 = 1.49 L.hr-1 Finally a steady state volume can be calculated, Equation 19.1.5, as Vss = CL • MRT = 1.49 x 8.17 = 12.14 L Equation 19.1.5 Apparent Volume of Distribution, Steady State, Vss Oral data can be analyzed by these methods as well. The data from Table 19.1.2, Cp versus time, Figure 19.1.3, and Cp x time versus time, Figure 19.1.4, can be plotted on linear graph and additional parameters can be calculated (NOTE: We don't calculate Clearance or Vss using oral data). The data were calculated after a 250 mg oral dose of the same drug. From these data a value for MRT was calculated as MRT = AUMC/AUC = 1361/149.8 = 9.08 hr We can subtract MRTIV from this MRTPO to get information about the absorption process, the mean absorption time, MAT, Equation 19.1.6. Thus Equation 19.1.6 Mean Absorption Time MAT = MRTPO - MRTIV = 9.08 - 8.17 = 0.92 hr From this we can calculate an average ‘apparent’ absorption rate constant, Equation 19.1.7. Equation 19.1.7 Apparent Absorption Rate Constant, ka’ ka' = 1/MAT = 1/0.92 = 1.09 hr-1 We can also calculate the bioavailability of the oral dosage form using the dose adjusted AUC ratio, Equation 19.1.8. Thus Equation 19.1.8 Oral Bioavailability F = (149.70/67.27) x (100/250) = 0.89. Another Example using Data from Bevill, 1977 The pharmacokinetics of sulfamethazine in cattle following IV and three oral dosage forms. The dosage forms studied were: IV Bolus 107 mg/Kg Oral Solution 107 mg/Kg Oral Rapid Release Tablet 105 mg/Kg Oral Sustained Release Tablet 249 mg/Kg These data, Figure 19.1.5, were originally analyzed using non linear regression analysis with a compartmental model for the IV and oral dosage forms. Re-analysis of the data using non compartmental analysis, Table 19.1.3, provided interesting comparison results, Table 19.1.4. From the non compartmental analysis: kel' = 1/MRT and ka' = 1/MAT kd' = 1/MDT where MDT is the mean dissolution time for the fast and slow release tablets. References Bevill, R.F., Dittert, L.W. and Bourne, D.W.A. 1977 Pharmacokinetics of Sulfamethazine in Cattle following IV and Three Oral Dosage Forms, J. Pharm. Sci., 66, 619-23 Chapter 20: Non-Linear Pharmacokinetic Models Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- draw the scheme and write the differential equations for compartmental pharmacokinetic models with non-linear metabolism elimination understand the process of parallel pathways as it applies with one or more non-linear pathways define and use the parameters Vm and Km design and calculate appropriate dosage regimens when non-linear pharmacokinetics apply All of the rate processes discussed so far in this course, except for the infusion process, follow first order kinetics. In particular the elimination process has been assumed to follow first order kinetics. However, for some drugs it is observed that the elimination of the drug appears to be zero order at high concentrations and first order at low concentrations. That is 'concentration' or 'dose' dependent kinetics are observed. At higher doses, which produce higher plasma concentrations, zero order kinetics are observed, whereas at lower doses the kinetics are linear or first order. This is more commonly seen after overdoses have been taken but for a few drugs it is observed at concentrations considered therapeutic. This occurs with drugs which are extensively metabolized. A typical characteristic of enzymatic reactions and active transport is a limitation on the capacity of the process. There is only so much enzyme present in the liver, and therefore there is a maximum rate at which metabolism can occur. A further limitation in the rate of metabolism can be the limited availability of a co-substance or co-factor required in the enzymatic process. This might be a limit in the amount of available glucuronide or glycine, for example. Most of our knowledge of enzyme kinetics is derived from in vitro studies where substrate, enzyme, and co-factor concentrations are carefully controlled. Many factors are involved in vivo so that each cannot be easily isolated in detail. However, the basic principles of enzyme kinetics have application in pharmacokinetics. Dose dependent pharmacokinetics can often be described by Michaelis-Menten kinetics with the RATE of elimination approaching some maximum rate, Vm, Equation 20.0.1, with Km a Michaelis-Menten constant. Km can be considered as the concentration at which the rate of metabolism is half the maximum rate, Vm. Equation 20.0.1 Rate of Metabolism for a Saturable Process - Michaelis-Menten kinetic References Examples of Causes of Drug Showing Dose- and Time-dependent Kinetics, Table 14-1 Shargel, L. and Yu, A.B.C. 1985 Applied Biopharmaceutics and Pharmacokinetics, 2nd ed., Appleton-Century-Crofts, Norwalk, CT p254 20.1 Non-linear One Compartment Model Scheme or Diagram The one compartment pharmacokinetic model after an IV bolus with a single Michaelis-Menten, saturable elimination is shown in Figure 20.1.1. Differential Equation An equation for the rate of change of drug concentration with time can be derived using the techniques from Chapter 2, Writing Differential Equations. Keeping track of units becomes even more important with non linear kinetics. In Equation 20.1.1 the units for Vm are amount.volume-1.time-1 for example mg.L-1.day-1. Another approach is to derive the equation for the rate of change of drug amount with time. Equation 20.1.1 Rate of Change of Concentration with Time Equation 20.1.2 looks the same as Equation 20.1.1. The difference is the dX/dt on the left and the units for Vm on the right. The units for Vm are the same as dX/dt, i.e. amount.time-1 for example mg/day. Note the units for Vm are the same as the units for the differential term on the left hand side of Equation 20.1.1 and Equation 20.1.2. The Cp units cancel top and bottom. Dividing the rate of elimination (metabolism) by the drug concentration provides a value for the drug clearance. Equation 20.1.2 Rte of Change of Amount with Time Notice that Equation 20.1.3 includes a concentration term on the right hand side (in the denominator). Clearance is not constant but varies with concentration. As the concentration increases we would expect the clearance to decrease. Calculations based on an assumption of constant clearance, such as the calculation of AUC are no longer valid. A simple increasing of dose becomes an adventure. No longer can we increase the dose by some fraction, for example 25%, and expect the concentration to increase by the same fraction. The calculations are more complex and must be done carefully. The superposition principle can no longer be applied to concentration. Equation 20.1.3 Non Linear Equation for Clearance It is not possible to integrate Equation 20.1.1 explicitly but by looking at low and high concentrations we can get some idea of the plasma concentration versus time curve. Low Cp -> approximation to first order At low concentrations, where Km >> Cp, Km + Cp is approximately equal to Km where Vm/Km is a constant term and the whole equation now looks like that for first order elimination, with Vm/Km a pseudo first order rate constant for metabolism, km, Equation 20.1.4. Equation 20.1.4 Rate of Change of Concentration at Low Cp - Pseudo First Order Therefore at low plasma concentrations we would expect first order kinetics. Remember, this is the usual situation for most drugs. That is Km is usually larger than the therapeutic plasma concentrations. High Cp -> approximation to zero order For some drugs, higher concentrations are achieved, that is Cp >> Km, then Km + Cp is approximately equal to Cp and we now have zero order elimination of drug, that is, the rate of elimination is INDEPENDENT of drug concentration (remaining to be eliminated). At high plasma concentrations we have zero order or concentration independent kinetics, Equation 20.1.5. Equation 20.1.5 Rate of Change of Concentration at High Cp - Pseudo Zero Order In Figure 20.1.2 at high Cp, in the zero order part, the slope is fairly constant (straight line on linear graph paper) and steeper, that is, the rate of elimination is faster than at lower concentrations. [However, the apparent rate constant is lower. This is easier to see on the semi-log graph in Figure 20.1.3]. At higher concentrations the slope is equal to -Vm. At lower concentrations we see an exponential decline in plasma concentration such as we see with first order elimination. On semi-log graph paper we can see that in the zero order region the slope is more shallow, thus the apparent rate constant is lower. The straight line at lower concentrations is indicative of first order kinetics. This relationship can be explored using Interactive 20.1.1. Another way to use or look at Figure 20.1.3 is to consider the slope of the line as a measure of a pseudo or apparent first order rate constant, k'. If we start with Equation 20.1.2 since this includes Vm with the more usual units of amount/time (mg/day) we can derive equation Equation 20.1.6 for this pseudo first order rate constant. gives Equation 20.1.6 Pseudo First Order Rate Constant As for clearance described earlier (Equation 20.1.3) this pseudo rate of elimination changes with concentration. As the concentration increases the elimination process slows down. We can take this one step further by looking at a 'half-life' for the elimination. Earlier when we talked about linear kinetics we talked about the time it takes to get to steady state concentrations. With linear kinetics this time was independent of concentration and could be calculated as 3, 4 or 5 half-lives. With non-linear kinetics, this time will increase with concentration just as this pseudo half-life increases with concentration, Equation 20.1.7. This is very important later when we use steady state concentrations to make parameter estimates. If we don't wait long enough our determination of steady state concentration will be in error and so will the parameter estimates. This time to steady state might change from a few days to weeks as the dose is increased. Equation 20.1.7 Pseudo Half-Life for Elimination Averaging Equation 20.1.2 over a dosing interval at steady state where the dose is equal to the drug metabolized during the interval leads to Equation 20.1.8. Equation 20.1.8 Dose Required to Achieve an Average Concentration Rearrangement of Equation 20.1.8 to solve for Cpaverage can be used to illustrate the problem of arbitrarily increasing the dose for a drug that exhibits non linear, Michaelis-Menten (MM), kinetics, Equation 20.1.9. Equation 20.1.9 Average Concentration at Steady State. Note Dose < Vm The presence of saturation kinetics can be quite important when high doses of certain drugs are given, or in the case of over-dose. In the case of high dose administration the effective elimination rate constant is reduced and the drug will accumulate excessively if saturation kinetics are not understood. Phenytoin is an example of a drug which commonly has a Km value within or below the therapeutic range. The average Km value is about 4 mg/L. The normally effective plasma concentrations for phenytoin are between 10 and 20 mg/L. Therefore it is quite possible for patients to be overdosed due to drug accumulation. At low concentration the apparent half-life is about 12 hours, whereas at higher concentration it may well be much greater than 24 hours. Dosing every 12 hours, the normal half-life, could rapidly lead to dangerous accumulation. At concentrations above 20 mg/L elimination maybe very slow in some patients. Dropping for example from 25 to 23 mg/L in 24 hours, whereas normally you would expect it to drop from 25 -> 12.5 -> 6 mg/L in 24 hours. Typical Vm values are 300 to 700 mg/day. These are the maximum amounts of drug which can be eliminated by these patients per day. Giving doses approaching these values or higher would cause very dangerous accumulation of drug. Figure 20.1.4 is a plot of Cpaverage versus dose calculated using Equation 20.1.9. This relationship can be explored using Interactive 20.1.2. Oral Administration An equation for the rate of change of drug amount in plasma with time can be derived as before. Keeping track of units becomes even more important with non linear kinetics. In Equation 20.1.10 the units for Vm are amount.time-1 for example mg.day-1. Equation 20.1.10 Rate of Change of Amount with Time after Oral Administration At higher concentrations the slope is equal to -Vm. At lower concentrations we see an exponential decline in plasma concentration such as we see with first order elimination. On semi-log graph paper we can see that in the zero order region the slope is more shallow, thus the apparent rate constant is lower. The straight line at lower concentrations is indicative of first order kinetics. This relationship can be explored using Interactive 20.1.3. The lines in Interactive 20.1.1 and Interactive 20.1.3 are calculated using a 4th order Runge-Kutta method with a small but fixed step-size. This may not work well in some circumstances.20.2 Parallel Pathways Another drug with saturable elimination kinetics is aspirin or maybe more correctly salicylate. In the case of aspirin or salicylate poisoning the elimination may be much slower than expected, Figure 20.2.1, because of Michaelis-Menten kinetics as described in Figure 20.2.2. This should be considered in any aspirin or salicylate poisoning case. With aspirin we have parallel first order and Michaelis-Menten elimination processes. Therefore as dose and consequently the concentration increases proportionally more drug would be removed by the first order processes rather than the saturable one. This is illustrated in Figure 20.2.3. At low doses the t1/2 could be calculated from km and Vm/Km. t1/2 = 0.693/(0.15 + 7/10) = 0.82 hr At high dose the contribution from the saturable metabolism is less significant and the t1/2 can be estimated from km alone. t1/2 = 0.693/0.15 = 4.6 hr The phenomena of non-linear pharmacokinetics is of great importance in multiple dose therapy in which more significant changes in the plateau levels are produced by the accumulation of drug in the body than can be expected in single dose studies. This accumulation will result in toxic responses especially when the therapeutic index of the drug is low. This relationship can be explored using Interactive 20.2.1. References Levy, G. 1965. Pharmacokinetics of Salicylate Elimination in Man, J. Pharm. Sci., 54, 959-967 Niazi, S. 1979 Textbook of Biopharmaceutics and Clinical Pharmacokinetics, Appleton-Century-Crofts, New York, NY, Fig 7.14 p 181 20.3 Dosing Approaches First dose One approach is to use the population values for Vm and Km. For phenytoin we could use values of Vm = 7 mg/kg/day and Km = 5 mg/L as population estimates. Consider consulting the literature for more appropriate, specific population values. Aiming at 15 mg/L for Cpaverage with a patient weight of 80 kg, Equation 20.3.1 can be used to estimate the 'first' dose. Equation 20.3.1 Dose versus Cpaverage Probably better to start out low since toxicity is more probable above 20 mg/L. Second dosing regimen After giving a continuous dose regimen to steady state, measure the plasma concentration and adjust the dose. For example if after 420 mg/day, Cpaverage is 20 mg/L then a downward adjustment would be necessary. If we assume that the Km is close to the average value of 5 mg/L we can estimate Vm from Equation 20.3.2 or Equation 20.3.3. Equation 20.3.2 Estimating Vm Equation 20.3.3 Calculating Vm with another Version of the Equation With a new value for Vm, closer hopefully to the value in this patient, a new dose regimen can be calculated, Equation 20.3.4. This new dose is approximately 400 mg/day. Note: A reduction in dose of 20 mg/day (5 %) is calculated to give a 5 mg/L change (25 %) in Cpaverage. Equation 20.3.4 Second Daily Dose Another approach at this point could be the use of the "Orbit Graph" method described by Tozer and Winter, 1992, from Vozeh, S. et al., 1981. This method uses data previously derived from a patient population to construct shapes, orbits, representing 50, 75, 90, 95 and 97.5% of the population values of Vm and Km. Plotting this information with one data point allows the estimation of a second dosing regimen. Third dosing regimen In another case, if we have two steady state plasma concentrations after two different dose rates, collected shortly before the next dose, we can solve Equation 20.3.1 for the two parameters, Vm and Km, by solving simultaneous equations. AND This assumes that the patient is fully compliant and steady state has been reached during each dosing regimen. For non steady state situations a Bayesian or other non linear regression analysis method may be useful, see Chapter 21. With Cpaverage, 1 = 8.0 mg/L and Cpaverage, 2 = 27.0 mg/L for R1 = 225 mg/day and R2 = 300 mg/day AND Multiplying Equation (1) x 300 300 • 225 • Km + 300 • 225 • 8 = 300 • 8 • Vm (3) and multiplying Equation (2) x 225 300 • 225 • Km + 300 • 225 • 27 = 225 • 27 • Vm (4) calculating Equation (4) minus Equation (3) 300 • 225 • (27 - 8) = (225 • 27 - 300 • 8) • Vm and With these Vm and Km values we can now calculate a new, better dosing regimen using Equation 20.3.1. Graphical methods There are a number of graphical methods which have been described for use when you have data from two or more dosing intervals. Basically these rely on converting the equations mentioned above into a straight line form which can be plotted to give the Vm and Km as a function of the intercept and/or slope. Again, steady state, average concentrations are required with the patient fully compliant. Some of these methods have been reviewed by Mullen and Foster, 1979. They found that iterative computer analysis (non linear regression analysis) was the best method, followed by a plot of Cpaverage versus Cpaverage/Dose (rearranging Equation 20.3.1 gives Equation 20.3.5), and this was followed by a direct linear plot method. Equation 20.3.5 Cpaverage versus Cpaverage/Dose Direct Linear Plot Method Since the direct linear plot method was the least complicated and required no calculations these authors thought it should be quite useful. This method, described by Mullen, 1978, involves plotting Dose and Cpaverage data points on linear graph paper. The y-axis represents Dose and Vm while the x-axis represents Km in the positive direction and Cpaverage in the negative direction. This is shown in Figure 20.3.1. For the example in Figure 20.3.1, after a dose of 256 mg/day and 333 mg/day the steady state Cpaverage values were measured to be 7 and 20 mg/L, respectively. These data are represented as the red and blue lines, respectively. These lines intercept at 4, 400 which provide a value of Km and Vm of 4mg/L and 400 mg/day, respectively. If in Figure 20.3.1 we were aiming for a Cpaverage value of 15 mg/L this point on the x-axis (represented as -15 on the x-axis) could be connected with the intersection of the red and blue lines by drawing the green line. The intersection of the green line with the y-axis provides the required dose of 316 mg/day. Various doses and expected Cpaverage values could be estimated from the intersection of the red and blue lines. References Tozer, T.N. and Winter, M.E. 1992 Chapter 25, "Phenytoin" in Applied Pharmacokinetics, 3rd. ed., Evans, W.E., Schentag, J.J., and Jusko, W.J. ed., Applied Therapeutics, San Francisco, CA, figure 25-11, p25-31 Vozeh, S., Muir, K.T., Sheiner, L.B. and Follath, F. 1981 Predicting individual phenytoin dosage, J. Pharmacokin. Biopharm., 9, 131-146 Mullen, P. W. 1978 Optimal phenytoin therapy: a new technique for individualizing dosage. Clin. Pharmacol. Ther., 23(2), 228–232 Mullen, P.W. and Foster, R.W. 1979 Comparative evaluation of six techniques for determining the Michaelis-Menten parameters relating phenytoin dose and steady-state serum concentrations, J. Pharm. Pharmacol., 31, 100-104 Chapter 21: Pharmacodynamic and PBPK Models Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- understand different types of concentration - effect relationships understand the mathematical relationships involved with direct reversible pharmacological effect kinetics understand the development and use of physiologically based pharmacokinetic (PBPK) models The method of data analysis or modeling should match the data available and potential applications of the data and models, Figure 21.0.1. Compartmental models allow summarization of data and understanding of drug processes and the calculation of dose regimens and doses to effective drug concentrations. Model independent analysis allows summarization of data but reduced understanding of the time or dose dependent processes. Calculation of dosage regimens to an effective drug concentration is possible. Physiologically based and pharmacodynamic models require more data but allow increased understanding of drug distribution and response. These models allow dose adjustment to target tissue concentration or to a required drug response. With more data, more samples or sampling sites, the models can be expanded or extended, Figure 21.0.2. Drug concentrations in various tissues allows the understanding and quantization of drug distribution in more detail. Drug response, either therapeutic or toxic, can be incorporated into useful models. Patient covariate information provides information for customizing dosing regimens to a particular patient. 21.1 Pharmacodynamic Models Concentration Effect Relationships The basic premise for the clinical utility of pharmacokinetics is that there is a clearly defined relationship between drug concentration in readily available samples and drug response. Application of pharmacokinetic methods allows us to account for the variability in an individual's ability to absorb, distribute, metabolize and excrete drugs. The objective of these methods is to control the drug concentration in blood or plasma and potentially other fluids and tissues. For this approach to be effective there needs to be a relationship between these concentrations and the response to the drug. In some cases the response is direct and reversible and the Hill equation or some variation of it maybe be applied. In other cases the reversible response may be indirect, often with a sequence of events required. It is also possible that a drug will produce an irreversible response. Direct Reversible Effects Examples of direct reversible effects might include blood pressure control or muscle relaxation. For many drugs the response is directly related to the drug concentration at the receptor site which in turn is related to the concentration in plasma. As the receptor concentration increases so does the response and with lower concentration the response drops, Equation 21.1.1. Equation 21.1.1 Drug Receptor Interaction In general the receptor might be 'mathematically' within the central compartment, a peripheral compartment or a separate effect compartment. In each case the relationship between concentration at the receptor and the response might be described with a Sigmoid Emax model (Hill equation), Equation 21.1.2 and Equation 21.1.3. Equation 21.1.2 Sigmoid Max Response versus Concentration, Hill Equation Equation 21.1.3 Sigmoid Max Response versus Concentration, Hill Equation, with Baseline Response In Equation 21.1.2 and Equation 21.1.3, Emax is the maximum response, CγR, 50% is the concentration which produces a 50% maximum response (often called EC50%) and γ is a slope factor. For responses which increase from zero, Equation 21.1.2 is more appropriate. In other cases where the response is an increase or decrease from a baseline value and Equation 21.1.3 may be a better choice. Another version of Equation 21.1.2 is the Emax model but this is the same except that γ is set to 1. Equation 21.1.2 can be illustrated in Figure 21.1.1 as a sigmoid curve. Here Emax was 80, CR, 50% was 1 and γ was 0.5, 1 or 2. With data such as response versus concentration we could readily fit the data with a Hill equation model using Boomer or some other non-linear regression program. We could also add appropriate weight to the data and include a second response to a fitted model. Another approach is to rearrange the Hill equation to produce a Logarithmic model, Equation 21.1.4. Equation 21.1.4 Linear version of the Hill Equation Plotting log(Response/(Emax - Response)) versus log(CR) should produce a straight line graph with a slope of γ and intercept of -log CγR, 50%. Another more commonly used Logarithmic model can be derived empirically from the Hill equation if we look at the response versus log concentration plot between 20 to 80% maximum response. Notice the straight line portion, Figure 21.1.2. This region of the full response concentration curve can be described by Equation 21.1.5. Equation 21.1.5 Logarithmic Model for the Middle Part of the Curve This might be useful if the concentration isn't high enough to estimate Emax. It is also interesting to consider concentrations after an IV bolus, one compartment model in this log form, Equation 21.1.6, equating plasma concentration and receptor concentration assuming the receptor is close to the plasma or central compartment. Note that with this scenario the response decreases linearly with time, Equation 21.1.7. One example of this is the results after the last dose of a one week regimen of labetalol (Derendorf and Hochhaus, 1995). Equation 21.1.6 Log(Cp) versus Time Equation 21.1.7 Response versus Time With each of these variations we see a direct, immediate relationship between response and concentration. The higher the concentration the higher the response. Note, the receptor may be within the central compartment or a peripheral compartment. It might be in a region where there is little drug amount and thus doesn't show up as a distinct 'pharmacokinetic' compartment. For this situation we might need to include a 'small' effect compartment. We can explore this by looking at the relationship between response and concentration in various compartments, Figure 21.1.3. One way to explore this aspect of the model, that is, where is the receptor, is to plot response versus concentration. In Figure 21.1.4 we plot response versus the concentration in the central compartment. Since there is a significant difference in response between the increasing concentration and decreasing concentration (a counterclockwise hysteresis curve) we might assume that the receptor is not in the central compartment. We could construct the same plot for the concentration in the tissue or peripheral concentration and explore the possibility that the receptor might be linked to this compartment. Wagner et al. (1968) found the response to LSD could be correlated with peripheral drug concentration. A response to digoxin was also correlated with peripheral drug concentrations (Reuning, 1973). In other case there may not be a correlation between central or peripheral concentrations. Here we might include a hypothetical effect compartment as shown in Figure 21.1.5 (Sheiner et al., 1979). Indirect Reversible Response The bodies response to anticoagulants or anti-diabetic medication might result from a cascade of processes and thus be considered indirect but reversible. One example is the pharmacodynamics of warfarin described by Nagashima et al. (1969). Although peak concentrations of warfarin occur after a few hours the maximum response to this drug can take a few days. The model used to describe this included degradation and synthesis of prothrombin complex activity. Irreversible Response The response from antibiotics and anti-cancer drugs may be considered irreversible responses but the details are often more complex (Jusko 1971). An example is the irreversible effect of aspirin on platelet aggregation which last the life of the affected platelets (which have a 7-10 day life span, Lalonde, 2006). References Lalonde, R.L. "Pharmacodynamics" Chapter 5 in Applied Pharmacokinetics and Pharmacodynamics: Principle of Therapeutic Drug Monitoring, 4th ed., ed. Burton, M.E., Shaw, L.M., Schentag, J.J. and Evan,s W.E., LWW, Baltimore, MD, 2006 Hill, A. V. 1910 The possible effects of the aggregation of the molecules of hemoglobin on its dissociation curves. J. Physiol. (Lond.), 40, iv-vii. Derendorf, H. and Hochhaus, G. Handbook of Pharmacokinetic/Pharmacodynamic Correlation, CRC Press, 1995 page 207 Sheiner, L.B., et al. 1979 Simultaneous modeling of pharmacokinetics and Pharmacodynamics: application to d-tubocurarine, Clin. Pharmacol. Therap., 25, 358-371 Reuning, R.H. et al. 1973 Role of pharmacokinetics in drug dosage adjustment I Pharmacologic effect kinetics and apparent volume of distribution of digixon, J. Clin. Pharmacol., 13, 127-41 Wagner, J.G. et al. 1968 Correlation of performance test scores with "tissue concentrations" of lysergic acid diethylamide in human subjects, Clin Pharmacol. Therap., 9, 635-8 Nagashima, R., O'Reilly, R.A., Levy, G. 1969 Kinetics of pharmacologic response in man: the anti-coagulant action of warfarin, Clin. Pharmacol. Therap., 10, 22-35 Jusko, W.J. 1971 Pharmacodynamics of chemotherapeutic effect: dose-time-response relationships for phase-specific agents, J. Pharm. Sci., 60, 892-5 21.2 Physiologically Based Pharmacokinetic Models Another approach to modeling drug and metabolite concentrations is physiologically based pharmacokinetic modeling (PBPK). This is in away the opposite of the one compartment model. Each organ or tissue may be included as separate parts of the model. Some of these may be lumped together (hybrid models) but often each organ or tissue is treated separately. Some components 0f the models are physiological values. For example, tissue or organ volumes or mass and blood flow values to and from these organs and tissues. In many cases the drug distribution within the organ or tissue is considered rapid, Figure 21.2.1, thus blood flow may be considered the limiting determinant for drug distribution. Transfer and distribution within the organ or tissue may be result in equilibrium during the blood transfer through the organ or tissue. Thus, mathematically the model will consider the drug concentration leaving the organ or tissue is in equilibrium with the concentration in the organ or tissue described by a partition coefficient, R, for the drug between blood and tissue. The parameters of these models include organ or tissue volumes, V, blood flows, Q, and drug partition coefficients. Two of the equations describing these models can be derived using a mass balance approach, Equation 21.2.1 and 21.2.2. Additional parameters include renal excretion and metabolic clearance. Each tissue and organ in the models is represented by an equations for transfer into and out of the model component, Figure 21.2.2. Equation 21.2.1 Rate of Change of Amount in Plasma Equation 21.2.2 Rate of Change of Amount in Muscle PBPK models have been used for a number of drugs including thiopental, Figure 21.2.3 (blood), Figure 21.2.4 (lean tissue), Figure 21.2.5 (liver tissue) and Figure 21.2.6 (adipose tissue). There are significant advantages to understanding more fully administration, distribution, excretion and metabolism of drugs throughout the body and not just readily sampled fluids. Careful use of PBPK models allow the more complete study of drug ADME in animals species. Volume and blood flow parameters are well understood in many laboratory animals and humans. Drug partition coefficients can also be determined in laboratory animals. Metabolism and excretion data can be determined in humans using blood and urine samples. Thus models can be developed in animals and humans that can provide important information about tissue and organ distribution in humans. References Bischoff, K. B., and Dedrick, R. L. 1968 Thiopental pharmacokinetics. J. Pharm. Sci., 57(8), 1346–1351. Chapter 22: Pharmaceutical Analysis Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- describe the need and techniques for separation of a drug from the sample matrix describe methods used to quantitate drug concentrations understand the Advantages and Disadvantages of some of these methods Topics to be covered in this chapter Sample Preparation Spectroscopic Analysis Chromatographic Separation High Performance (Pressure) Liquid Chromatography (HPLC) Gas Liquid Chromatography (GLC) Radioimmunoassay (RIA) Enzyme Multiplied Immunoassay (EMIT) Fluorescence Polarization Immunoassay (FPIA) Comparison of Clinical Analytical Assay Methods Pharmaceutical Analysis generally involves two steps; a) separation of the compound of interest and b) quantitation of the compound. The better the separation the easier the quantitation. 22.1 Sample Preparation Separation (from Dosage Form or Biological Sample) High concentrations of a drug in a sample means milligrams of drug in a liter or kilogram of sample matrix. The needle (1 mg) in a haystack (1 kg) means looking for and accurately measuring 1 part in 1,000,000. Low drug concentrations make this ratio 10 to 1000 times larger. Separating the drug from most of the background material can be very useful in the determination of drug concentrations. The presence of similar compounds such as precursors, degradation products or metabolites makes the separation even more of a challenge. Some useful techniques include: Centrifugation - removing excipients and/or macromolecules A significant separation can be achieved at times by simple centrifugation. Centrifugation of whole blood after clot formation produces serum with no red blood cells and fewer plasma proteins. Centrifugation of whole blood pretreated with anticoagulants such as heparin, EDTA or citrate produces plasma. Protein can be removed from plasma by centrifugation after the addition of trichloroacetic acid or acetonitrile. Extraction - using pH/pKa partitioning or lipid/aqueous solubility differences Extraction can be used to remove unwanted interfering compounds or to concentrate the compound(s) of interest. The use of various organic solvents of differing polarity and/or aqueous buffers of various strength and pH values can provide excellent resolution based on the solubility of the free compounds of interest and/or their salt forms. Extraction may be used to remove lipophilic interfering compounds or to move the desirable compounds into a cleaner environment. Ultrafiltration One method of separating free drug from plasma samples is ultrafiltration. The sample can be forced (by centrifugation or pressure) through a membrane filter. The filtrate is a protein free solution of the compound. Chromatography Adsorption, partitioning, size, or charge Chromatography can be used in a clean-up mode or in a sensitive analytical mode. In the clean-up mode the compound of interest may be adsorbed tightly onto a column while the interfering compounds are washed through it and subsequently the drug of interest is flushed from the column with a stronger eluting solvent. In the analytical mode the challenge is to choose the right column (stationary phase) and mobile phase so that the compound is eluted within a reasonable time and well resolved from other components in the sample. Immunoassay The very tight, strong binding between a compound and an antibody can be very useful as a method drug analysis. Using controlled, limited amounts of antibody and labelled drug it is possible to separate the drug before subsequent quantification. Other separation methods used in drug analysis might include dialysis and electrophoresis. 22.2 Drug Quantitation The other part of the drug analysis process is recognizing (the right needle) and counting or measuring the number of drug molecules. Each drug has physical and chemical properties that can be exploited. In some cases a radio-labeled drug may be used. Some useful quantitation techniques: Absorbance Most compounds include chromophores within their molecular structure. Thus these compounds absorb electromagnetic energy in the visible (350 - 700 nm) and/or ultraviolet range (200 - 350 nm). Within specific concentration ranges the amount of energy absorbed is proportional to the concentration of the compound in the sample. Fluorescence A number of compounds which absorb light energy are also able to re-emit some of that energy as light at a higher wavelength (lower energy). The emitted energy can be measured and correlated with the concentration of the compound. Mass Spectrometry After careful separation from the mobile phase of HPLC or GLC the drug to be measured can be introduced into a mass spectrometer. This method can be very specific with high resolution based on molecular weight. It is also quite sensitive although more complex. Radioactivity Radioactive atoms can be chemically incorporated into a compound of interest and subsequently used to quantitate the compound. The quality of the method depends on how well the radio-label remains with the compound of interest. Electrochemical Various compounds will undergo oxidation or reduction under the influence of an electrical potential. These electrochemical reactions result in an electrical charge which can be detected and used as a measure of the drug concentration. Other techniques include optical activity, conductivity and refractive index Absorption Spectroscopy Theory Beer-Lambert's Law Instrumentation Single Beam Double Beam Application Theory Absorption of light energy by drug molecules in solution produces changes in electronic transitions as well as vibrational and rotational changes. For example the bonds of the carbonyl group contain sigma and pi electrons. These electrons may transition from bonding to anti-bonding levels, Figure 22.2.1. Each of these transitions would result in a single peak in the absorbance / wavelength spectrum except for the broadening effect of the rotational and vibrational transitions, Figure 22.2.2. As light passes through a compound in solution the intensity is reduced, Figure 22.2.3. The longer the path-length the more light is absorbed. Also, the higher the concentration of the compound in solution the more light is absorbed. Absorbance is proportional to path-length and the concentration (Beer-Lambert's law), Equation 22.2.1, where a = absorptivity (ε, epsilon - molar absorptivity includes path length and wavelength). Equation 22.2.1 Beer-Lambert Law for Light Absorption b = path length (commonly 1 cm) and c = concentration (molar if molar absorptivity). If b is 1 cm and c is in g/100ml the absorptivity is given as A1%1 cm at wavelength (lambda). Absorptivity may also be called the extinction coefficient or absorption coefficient Instrumentation Single Beam Spectrophotometer The Turner model 330 single beam spectrophotometer has a cell holder for the sample and dials for zero adjustment, 100% transmission and wavelength. Absorbance is read from the upper scale on the meter, Figure 22.2.4. Double Beam Spectrophotometer, Figure 22.2.5 The double beam spectrophotometer uses a beam splitter to produce two beams of light. One goes through the sample and the other goes through the reference or blank. The light from each beam is compared and used to quantitate the amount of light absorbed and thus the concentration in the sample. Application Analysis of Drugs by Visible Spectroscopy Standard solutions of the drug to be measured are prepared and their absorbance measured. Plotting the absorbance versus concentration should give a straight line through the origin, Figure 22.2.6, according to Beer-Lambert's law. The absorbance of the unknown sample can be used to determine the concentration in the sample. Fluorescence Spectroscopy Molecule absorbs energy and immediately (10-6 to 10-8 sec) emits energy at a higher wavelength (lower energy), Figure 22.2.7. Phosphorescence is similar but involves a slower emission step (> 10-4 sec). Excitation and emission wavelengths specific to the compound Emission measured at 90° to the excitation light path, Figure 22.2.8. The detector sensitivity can be increased as the emitted fluorescent energy is measured against a black, dark background. Emission proportional to drug concentration References Bauer, H.H., Christian, G.D., and O'Reilly, J.E. 1978 Instrumental Analysis 22.3 Chromatographic Separation Chromatography is based on the separation of substances of interest by their differing affinity between a mobile phase and stationary phase. The mobile phase is usually a liquid or a gas while the stationary phase is usually a solid but may be an immobilized liquid. Relative affinity may be based on relative solubility, adsorption, size or charge. Differences in solubility are expressed by partitioning between the mobile and stationary phases. Adsorption differences cause the separation of molecules in a non aqueous environment. Permeation (gel permeation) chromatography is based on smaller molecules being retained by inclusion within smaller pores of the gel. Separation by ion-exchange chromatography is based on the exchange of ions in the mobile phase with ions on the stationary phase. As such it is better suited to purification than separation between similar materials. 22.4 High Performance (Pressure) Liquid Chromatography (HPLC) The General Approach Collect blood sample Separate plasma (partial clean-up) Extraction into organic solvent (polarity/pH) Injection onto column Separation on column Quantitation with detector Record and analyze results HPLC Instrumentation Figure 22.4.1 and Figure 22.4.2 Solvent Reservoir Pump High pressure - 1000 to 5000 psi Injector Low pressure - stop flow High pressure value Column Normal Phase - organic (water-free) mobile phase Silica gel - non-aqueous Adsorption Reverse phase (C8, C18) - aqueous mobile phase Partitioning Ion-exchange - aqueous mobile phase Molecular sieve - aqueous mobile phase Size Quantification - Detector Specific Absorbance Fluorescence Mass Spectrometry Electrochemical Radioactivity Non-specific Refractive index Conductivity Recorder or Integrator or Computer System The output from the HPLC instrument is a chromatogram which present the intensity of the measurement versus time, Figure 22.4.3. The peak height or area can be measured and plotted versus concentration, Figure 22.4.4. Typically a linear plot is the result over a useful range of concentration. Reference Pieper and Rutledge, Laboratory Techniques for Pharmacists, Upjohn 1989 22.5 Gas Liquid Chromatography (GLC) GLC Instrumentation Figure 22.5.1 Gas cylinder and regulator Nitrogen Helium Argon Hydrogen Injection port Column Gas-solid - adsorption Gas-liquid - High boiling point stationary phase Glass, steel, capillary glass Column oven, Figure 22.5.2 Detector Flame ionization (FID) general purpose - modest sensitivity Nitrogen phosphorus FID specific for N and P Electron capture (EC) Mass spectrometry (MS) Recorder or Integrator or Computer System, Figure 22.5.3 References Pieper and Rutledge, Laboratory Techniques for Pharmacists, Upjohn 1989 22.6 Radioimmunoassay (RIA) RIA involves the separation of the drug using the specificity of antibody - antigen binding and quantitation using radioactivity. Components of RIA Assay Kit Drug Antibody Labelled Drug General Procedure for Performing a RIA Analysis Mix sample containing drug with fixed quantity of labelled drug and antibody Allow to equilibrate - incubate Separate drug bound to antibody from unbound drug Charcoal adsorption of antibody (and bound drug) Antibody - antibody binding precipitates bound drug Antibody bonded to container Measure radioactivity associated with bound labelled drug low drug concentration means more bound radioactivity and higher measurement high drug concentration means less bound radioactivity and lower measurement Determine standard curve Non-linear plot of radioactivity versus concentration Logit-log concentration plot is linear An example can be described with blank and three standard samples. Figure 22.6.1 presents the samples before and after incubation to achieve equilibrium. Notice the different amounts of labeled drug bound to the antibody in each sample. After separation of the antibody and bound drug the amount of radioactivity can be measured, Table 22.6.1. The blank has the highest activity since there is no unlabeled drug in this sample. The activity in each sample can be determined and plotted versus known concentration, Figure 22.6.2. Using the logit of these results provides a straight line standard curve, Figure 22.6.3. 22.7 Enzyme Multiplied Immunoassay (EMIT) The EMIT method relies on separation using the specificity of the antibody - antigen binding and quantitation using an enzyme reaction Components of the EMIT Assay Method Figure 22.7.1 Drug Antibody Substrate Enzyme bound to drug General Procedure for EMIT Assay Mix sample containing drug with fixed quantity of enzyme bound drug, and antibody Add substrate Measure absorbance at 15 and 45 seconds after substrate addition Quantitate by measuring enzyme-substrate reaction (by UV - visible spectroscopy) Δ Absorbance from Reaction rate from Drug concentration Non linear relationship between Δ Absorbance and Concentration Determine standard curve References Pieper and Rutledge, Laboratory Techniques for Pharmacists, Upjohn 1989 
22.8 Fluorescence Polarization Immunoassay (FPIA) FPIA Procedure Figure 22.8.1 Fluorescein-labelled drug competes with unlabeled drug for antibody Sample excited with plane polarized light (490 nm) Fluorescein emits plane polarized light (520 nm) Small, free drug-fluorescein, rotates faster leading to less emission Larger, antibody-drug-fluorescein, rotates slower and produces more emission Drug in sample competes for antibody with fluorescein bound drug More drug in the sample; less fluorescein labelled drug bound to antibody; lower emission of plane polarized light Higher drug concentration results in lower light emission values Advantages Available for a variety of drugs Rapid turnaround times, sensitivity, ease of operation Disadvantages Background interference in serum sample (requires blank measurement) References Pieper and Rutledge, Laboratory Techniques for Pharmacists, Upjohn 1989 22.9 Comparison of Clinical Assay Methods Basis for Comparison of Methods Table 22.9.1 Sample (Type and size) Analysis Time Sensitivity, Specificity Accuracy, Precision Ease of Use, Versatility Cost of Equipment Cost of Supplies Chapter 23: Clinical Applications of Pharmacokinetics Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- understand the basics of a Therapeutic Drug Monitoring service describe and understand how changes in physiology effect the pharmacokinetics of drugs in the very young and the elderly 23.1 Therapeutic Drug Monitoring We can start this topic by talking about the Clinical Pharmacy service, Therapeutic Drug Monitoring. This involves the measurement and interpretation of plasma/serum/blood concentrations in patients. Why Therapeutic drug monitoring (TDM) becomes important when: a) the drug has a narrow therapeutic-toxic range, b) there is a large variability in pharmacokinetic parameter values between patients, c) the therapeutic effect is not readily assessed (e.g. antibiotics) or clinical symptoms are to be avoided (e.g. seizure). Not as useful for blood pressure lowering (can measure B.P. directly) or anticoagulants (can measure clotting time directly), d) there is a direct relationship between Cp or concentration in other biological sample (e.g. saliva) and pharmacological effect, e) an appropriate (accurate, short turn around, inexpensive) analytical method is available for the drug, f) the expected or desired therapeutic effect is not observed (may be absorption or compliance problem), g) a drug with high first pass effect is involved, or h) a patients has altered and/or variable renal state and the drug is eliminated mostly as unchanged drug in urine (fe less than 1) Typical drugs Table 23.1.1 Procedure Pharmacist and physician develop initial dosing recommendations Information required Patient - Age, weight, sex, height, smoker Clinical - Drug requirements, clinical status (renal - serum creatinine; cardiac - cardiac output, liver, etc.) Calculate initial loading dose or maintenance regimen and make recommendations Organize sample collection and analysis Accurate Timing in necessary, Figure 23.1.1. Evaluate the analytical result using pharmacokinetic principles and recalculate dosing regimen recommendations Organize further samples if necessary, repeat as necessary Calculations Computer or calculator programs can be used to help the bedside development of dosing regimen. Other more sophisticated programs are available to calculate values for the drug pharmacokinetics and make further recommendations. 23.2 Pediatric Considerations The physiologic processes that determine drug disposition undergo radical changes during biological maturation (Morseli 1976, Rane and Wilson 1976, and Shirkey 1973). Thus, the processes of drug absorption, distribution, metabolism, and excretion are modified throughout infancy and childhood. These changes mean that a) drug disposition both changes during maturation and differs from adult norms, and b) a large inter patient variability in drug disposition is observed for many drugs in this patient population. Further complications are that little data is available concerning the disposition of drugs in infants before the general release of new drugs. Also studies in infants and young children are difficult to perform because of the limited amount of sample which can be collected. The most dramatic changes occur in the first year. For children older than one year, dose adjustments can often be made on a weight or surface area basis. Pharmacokinetic changes Absorption Infants after the newborn period have a relative achlorhydria; with gastric acid secretions increasing to reach adult levels at age 3. The bioavailability of acid labile penicillins is increased in newborns. Delayed gastric emptying and irregular intestinal peristalsis leads to slower absorption of some drugs in infants and young children. Distribution Total body water as a fraction of body weight decreases throughout the first year of life (see Table 23.2.1). Extravascular fluid is proportionately higher at an earlier age as well. In general, distribution volumes expressed as volume per body weight tend to be larger in neonates than in adults and decrease towards adult values during later childhood. This has been observed for ampicillin, ticarcillin, and amikacin. Binding to plasma proteins appears to be less in newborn infants compared with older children and adults. This appears to be true for both acidic and basic drugs. The presence of competing substances, such as bilirubin in premature infants, complicates the picture. Metabolism The various pathways of drug metabolism mature at different rates, and therefore the ability of the newborn to metabolize drugs differs both quantitatively and qualitatively from that of older patients. No general rules can be developed and a few examples can illustrate the variety of effects observed. Caffeine is very slowly metabolized in newborns. During the first month almost no metabolism occurs, with half-lives of about 4 days resulting from renal elimination, normally a minor pathway. Between 3 and 7 months, caffeine is metabolized similarly to adults and the half-lives change to adult values during this period. For the similar compound theophylline the half-life was 13 to 29 hours for 8 low birth weight infants (Hale 2004). Glucuronidation is quite inefficient at birth, thus with chloramphenicol which is normally glucuronidated in adults and has no major alternate metabolic pathway, the overall elimination is much slower in newborns compared with adults. Sulfate conjugation is normally well developed at birth thus acetaminophen elimination in newborn, predominantly sulfation, is not greatly different from that of adult elimination. For drugs which undergo M-M or saturable metabolism the effect of age is interesting. For phenytoin, Km is not changed with age, but the maximum metabolism rate, Vm is progressively lower with younger patients. Excretion Glomerular filtration and renal tubule function in premature infants and newborns is somewhat immature. GFR, normalized for body surface area, increases gradually reaching adult values at about 6 months, Table 23.2.2. Renal tubular capacity, measured by renal clearance of p -aminohippurate, achieve adult values 1 to 2 months later. Therefore drugs which depend primarily on the renal route of elimination, such as gentamicin, ampicillin, and furosemide, have prolonged elimination times in neonates and young infants. Examples Some examples of changed pharmacokinetic parameters in younger patients in Table 23.2.3.Dosing recommendations For some drugs detailed pharmacokinetics development of dosage regimens for pediatric patients is not practical. Either there isn't enough good data to make an accurate estimate of dose regimen or there is too much intra-patient variability in parameter values. Alternately, dosage regimens can be determined from various reference texts such the Pediatric Dosage Book (Shirkey 1973). For very young infants and neonates the primary literature should be consulted. For older children minimal adjustments can be based on weight, surface body area, or age. Dosing based on weight, e.g. per mg doses or Clark's rule, Equation 23.2.1 Equation 23.2.1 Clarks’ Rule Dosing based on surface area, e.g. per m2 Dosing based on age, e.g. Young's rule for children older than 2 years, Equation 23.2.2 Equation 23.2. Yong’s Rule References Morseli, P.L. 1976 Clinical pharmacokinetics in neonates, Clin. Pharmacokin., 1, 81-98 Rane, A., and Wilson, J.T. 1976 Clinical pharmacokinetics in infants and children. Clin. Pharmacokin., 1, 2-24 Shirkey, H.C. 1973 Pediatric Dosage Handbook, Amer. Pharm. Assoc., Washington, DC Hilligoss, D.M. 1980 Chapter 5 "Neonatal Pharmacokinetics" in Applied Pharmacokinetics, Evans, W.E., Schentag, J.J., and Jusko, W.J. ed., Applied Therapeutics, San Francisco, CA Table 1, p 88 Miles, M.V. 1983 "Pediatric Pharmacokinetics" in Applied Clinical Pharmacokinetics, ed Mungall, D.R., Raven Press, New York, NY, pp 367-388 Taketomo, C.K., Hodding, J.H. and Kraus, D.M. 2004 Lexi-Comp's Pediatric Dosage Handbook, 11th ed., Lexi-Comp., Hudson, OH Hale, T.W. 2004 Medications and Mother's Milk, 11th ed., Pharmasoft Publishing, Amarillo, TX, p112-3 23.3 Geriatric Considerations Geriatric patients constitute incur a disproportionate percentage of the total costs of drugs. In future decades even higher proportions of the total drug cost is expected to be needed for the elderly patient. Although the elderly are a major group of drug users, most drug studies are performed on patients or volunteers aged 55 years or less. There is a significant increase in drug toxicities as one ages from 20 to 79 years. Part of this increase may well be due to multiple medication and drug interactions. Additionally a significant part of this increase in drug toxicities could be due to incomplete understanding of changes in the ADME processes of drug disposition with aging. We will consider some of the physiological changes which occur in aging and then look at the ADME processes in turn (Garnett and Barr 1984, Triggs and Nation 1975, Vestal 1978, Sjoqvist and Alvan 1983, and Ho and Triggs 1984). Physiologic changes with age There are a number of physiological changes which may effect drug pharmacokinetics in the elderly. These factors are listed on Table 23.3.1 (Massoud, 1984a). Pharmacokinetic changes Absorption As noted in the table there are a number of physiologic changes which potentially alter drug absorption. For example, GI motility and pH changes. There has been little evidence, however, to suggest that this is of major consequence. Reduced absorption in the elderly has been observed for some compounds which are actively absorbed (e.g. galactose, calcium, thiamine, and iron). The absorption of most drugs by passive processes is generally not affected. Only tmax was reduced for tolbutamide in the elderly versus young. For other drugs studied, including l-DOPA, metoprolol, propranolol, cimetidine, and digoxin no changes were observed which could be ascribed to absorption alone. Higher Cpmax values were observed in a number of cases, however, most of these changes could be explained by changes in distribution or clearance. Distribution Changes in body composition may occur as the patient ages. Body fat may increase from significantly and lean body weight may decrease in proportion to total body weight. This should give lower values of the apparent volume of distribution for drugs which stay in the central compartment, while lipid soluble drugs would have somewhat larger apparent volume of distribution values. The apparent volume of distribution for diazepam and chlordiazepoxide in the elderly is larger, whereas, the volumes for lorazepam and oxazepam were relatively unchanged. The lipid solubility of the first two drugs is much higher than for the second pair. Although total plasma protein concentrations remain relatively constant, albumin concentrations are lower in the older patient. The fraction of unbound phenytoin increases 25 to 40% in the older patient, however, as earlier described this would also lead to increased clearance. In the case of diazepam it has been found that the percentage unbound could be correlated with age for females but not males. Drug interactions based on protein binding, and other factors, can be more pronounced in the elderly because they tend to be taking more drugs. Cardiac output in the elderly is reduced, thus distribution to the kidneys and liver are expected to be reduced. For high extraction drugs this could alter the overall elimination of the drug. Metabolism The liver is the major organ involved in metabolism and as shown in the table, liver blood flow and liver mass tend to decrease with age. Protein binding also is reduced, especially to albumin, as mentioned in the previous section. The third determinant of drug metabolism, intrinsic clearance is quite variable and dependent on the metabolic pathway. Acetylation appears to be unchanged with age, isoniazid clearance is not altered. It appears that for some drugs which undergo Phase I metabolism (oxidations, reductions) metabolism reduces with increasing age. Examples are lidocaine, phenytoin, propranolol and theophylline. For other drugs which undergo Phase II metabolism (conjugations) the metabolism does not appear to change greatly with age. Some example drugs are isoniazid (acetylation) and temazepam (glucuronidation). Elimination With increasing age the glomerular filtration process is reduced by a reduction in kidney size, reduction in the number of nephrons, reduction in the number of functioning glomeruli, and a decrease in renal blood flow. Serum creatinine is also decreased with age because of the reduced muscle mass. Drug which are renally excreted and for which dosage adjustments should be made in elderly patients include; the aminoglycosides, digoxin, lithium, methotrexate, quinidine, and the tetracyclines (except doxycycline). Examples of the type of changes in pharmacokinetic values with age are shown in Table 23.3.2. Dosing recommendations The effects of age on drug disposition depend on the particular compound in question and the characteristics of the population being studied. When evaluating geriatric studies it is important to distinguish between long-term-care patients who are not considered to be healthy and the active who are older (age > 65 years) and living in the community. Some of the changes ascribed to the elderly may be due to immobility of the patient or an underlying disease or diseases. Also because of changing body composition between males and females it is often important to distinguish between these groups. In terms of dosage regimen adjustment, for some drugs, such as the aminoglycosides dose adjustment is progressive with a steady change in creatinine clearance with age reflecting the similar change in the clearance of the drugs under question. References Semla, T.P., Beizer, J.L. and Higbee, M.D. 2003 Lexi-Comp's Geriatric Dosage Handbook, 9th ed., Lexi-Comp, Huson, OH ISBN 1-59195-067-8 Garnett, W.R. and Barr, W.H. 1984 Geriatric Pharmacokinetics, The Upjohn Company, Kalamazoo, MI (Publication #8800-86) Triggs, E.J. and Nation, R.L. 1975 Pharmacokinetics in the Aged: A Review. J. Pharmacokin. Biopharm., 3, 387-418 Vestal, R.E. 1978 Drug Use in the Elderly: A Review of Problems and Special Considerations. Drugs, 16, 358-382 Sjoqvist, F. and Alvan, G. 1983 Aging and Drug Disposition - Metabolism. J. Chron. Disease, 36, 31-37 Ho, P.C. and Triggs, E.J. 1984 Drug Therapy in the Elderly. Austral. NZ J. Med., 14, 179-190 Massoud, N. 1984a Chapter 15 "Pharmacokinetic Considerations in Geriatric Patients" in Pharmacokinetic Basis for Drug Treatment ed Benet, L.Z., Massoud, N., and Gambertoglio, J.G., Raven Press, New York, NY, Table 2, page 286. Massoud, N. 1984b Chapter 15 "Pharmacokinetic Considerations in Geriatric Patients" in Pharmacokinetic Basis for Drug Treatment ed Benet, L.Z., Massoud, N., and Gambertoglio, J.G., Raven Press, New York, NY, Table 3, page 297 Chapter 24: Non-Linear Regression Analysis of Pharmacokinetic Data Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- understand the reasons why models are developed and used understand how models can summarize or 'compress' data understand how models can be used to study pharmacokinetic mechanisms understand how models can be used to predict concentrations or dosage regimens understand the use of formulas as 'Mathematical models' understand the criteria of least squares understand how parameter adjustment changes the fit to data understand the use of computer programs such as Boomer for non-linear regression analysis of pharmacokinetic data consider Bayesian analysis of clinical data understand the use of computer programs such as PhoenixTM WinNonlin for non-linear regression analysis of population pharmacokinetic data With no patient data but patient information, data analysis is not possible, Figure 24.0.1. Dosing calculations can be based on nomograms, package inserts or other published information. If a few data points are available from a single patient and population parameter values are available it may be possible to perform a Bayesian analysis. This analysis method can combine patient data with population information. If a few data points are available for each subject and data are available for a large number of subjects a population analysis may be possible. This can also provide population parameter values with measures of the uncertainty in these values. When more data in each subject is available a 'traditional' non-linear regression analysis using graphical methods or a computer program is possible. When more data from multiple subjects is available a population analysis is again very useful. Alternately, in this last case, a two step approach of analyzing each subject's data separately using non-linear regression analysis and combining these results may be applied. A third approach might be to combine the data from multiple subjects in a naive pool analysis. 24.1 Modeling of Pharmacokinetic Data Why Model Data Summarize Data Pharmacokinetic models are very useful for summarizing data. A suitable model with good parameter value estimates and estimates of their uncertainty can be helpful. Thus, a model with population mean and standard deviation data could summarize pages and pages of data from many subjects or patients. During the development of new drugs and new dosage forms numerous pharmacokinetic studies in animals (pre-clinical) and humans (clinical) are performed. These and other studies will produce large amounts of data. Even a simple six subject study will provide considerable data. A full page of data, Table 24.1.1 and numerous plots of the data. Figure 24.1.1 illustrates one of these plots of the data from just one subject. Also portrayed in Figure 24.1.1 is a simple one compartment model with two parameters, V and kel or CL. We can start the data analysis with a semi-log plot of the data in Table 24.1.1, shown in Figure 24.1.2. Values for V and kel can be determined from the intercept and slope of the best-fit line. If we model all the data in Table 24.1.1 we can summarize these data with the model and averaged parameter values, Equation 24.1.1. Equation 24.1.1 Simple Model with Parameter Values Thus the data from six subjects can be summarized with an equation (model) and parameter values for the model. Explore Mechanisms Developing models is an important step in understanding how drugs are absorbed, distributed, metabolized and excreted. After developing good models it is possible to explore correlations between pharmacokinetic parameter values and clinical parameters such as measures of renal, hepatic, cardiac or other patient characteristics. Developing and testing pharmacokinetic (and other models) is an important basis of scientific enquiry. When we quantitate observations and model data we can better understand what is happening in the system under study. Correlation can be explored between the parameter values and other observations that may be collected. Table 24.1.2 provides pharmacokinetic parameters from a number of subject as well as some of the data that might be collected from a patient's hospital chart. With these data we could explore some of these correlations. For example plotting the kel measured in these patient versus the clinical parameter creatinine clearance may result in a plot such as Figure 24.1.3. Figure 24.1.3 suggests that there is a significant correlation between elimination of this drug and renal function as expressed by the creatinine clearance. A large fraction of the drug dose must be excreted into urine. If renal function is poor, elimination would be impaired and the drug dosage regimen should be adjusted appropriately. We could also explore the relationship between apparent volume of distribution and creatinine clearance. In Figure 24.1.4 we see that there is little correlation between the apparent volume of distribution and creatinine clearance. In another study we might look at the effect of drug dose and pharmacokinetic parameters. Some data are shown in Table 24.1.3. Plotting these data on semi-log graph paper provides three lines with different slope and shape, Figure 24.1.5. It would appear that these data represent nonlinear or saturable pharmacokinetics (which were discussed in more detail in Chapter 20). A model which could explain these data are shown in Figure 24.1.6 along with a plot of AUC versus dose. This is another representation of these and more data collected after additional dose values which illustrates the nonlinear model. These and other mechanisms can be explored by modeling pharmacokinetic data. Make Predictions Once a satisfactory pharmacokinetic model and parameter values have been determined we can make predictions such as anticipated drug concentrations after a particular drug dosage regimen. Alternately we could calculate suitable dosage regimens to produce and maintain optimal drug concentrations as shown in earlier Chapters. For example we can determine the dose required to achieve a certain drug concentration six hours after an IV Bolus dose, Equation 24.1.2. Equation 24.1.2 Dose Required to Achieve a Required Concentration at 6 Hours Using this information we can calculate (or predict) the drug concentrations at various time up to and including the six hours requested, Figure 24.1.7. We can also predict drug concentrations after a specific multiple dose regimen, Equation 24.1.3. Equation 24.1.3 Concentration after Multiple Uniform IV Bolus Doses With this information we can predict drug plasma concentrations after multiple 100 mg IV bolus doses every 12 hours, Figure 24.1.8. With more extensive models even more involved predictions or calculations can be performed. General Approach Modeling involves a number of steps, Figure 24.1.9. Ideally the pharmacokinetic modeler is involved in the design of the experiment. The study is designed and data are collected. The modeler will then develop suitable models consistent with the data, route of administration and dosage regimen. The data will be modeled using appropriate computer programs and the results evaluated. Problems at this point might lead to more modeling or even more studies and data collection. Finally we might use the model to make predictions such as useful dosage regimen design. Mathematical models as equations What is a mathematical model, a pharmacokinetic model? It can be useful to describe a model as a diagram with components to represent drug amount and arrows to represent rate processes. However, every pharmacokinetic model needs to be represented as a formula or an equation. Understanding these equations and the parameters in these equations is an important part of understanding pharmacokinetics. As mentioned earlier, pharmacokinetic models are described as equations or formulas. In general, there is a dependent variable (y variable) expressed as a function of independent variable(s) (x variable) with various constants and/or parameters, Equation 24.1.4. Constants and parameters may be interchangeable or considered very similar. From a modeling point of view parameters have values that are determined by the computer program. Constants are terms that are held fixed during the modeling process. Mathematical models take many forms. The simplest form is the equation for a straight line. Foe example: In Figure 24.1.10 peak height ratio is the dependent variable and concentration is the independent variable. Slope and intercept are parameters. This is an equation that is very useful for standard curves used in drug analysis. A pharmacokinetic model is the next example. This is a very simple example which is a not a straight line unless it is transformed. In this exponential equation there are two parameters, kel and V, with dose as the constant, Figure 24.1.11. A third example is a pharmacological equation relating drug effect to drug concentration using a form of the Hill equation. The parameters in this model are EMax, EC50% and γ, Figure 24.1.12. We'll see more of this type of model in the next chapter Exploring Potential Models The first step towards successfully modeling pharmacokinetic data is to consider the route of administration and the data available. Data collected after an intravenous (IV) bolus may be the easiest to analyze as there are no absorptions steps to consider. The bolus dose should be well defined and it should not be necessary to estimates its value. An IV infusion adds an administration step, a rate of infusion and possibly a duration if the infusion is stopped. Again, both of these parameters should be well know. A dose given by mouth or as a intramuscular (IM) injection require the addition of an administration step. This might be a simple one compartment process or it could be more complex. Solubility, stability and site of absorption can add to the complexity of the absorption step and the overall model. Samples other than blood or plasma may be available. Unchanged drug and metabolite in urine samples can add another dimension to the model selection process. The absorption, distribution, metabolism and excretion (ADME) processes may not be all first order. Data collected after different doses can be useful, as seen in the figures earlier, Figure 24.1.5 and Figure 24.1.6. Data collected after multiple dosing also adds to the model detail. At this point we could envisage potential models with rapid distribution and a single compartment representing the body. We might now move to the consideration of data plotted on linear and semi-log graphs. These graphs should confirm our thoughts regarding the administration and elimination, metabolism and excretion, of the drug and metabolite. A distribution phase may suggest a multi compartment pharmacokinetic model. Even after extravascular administration such as oral dosing a distribution phase may be evident in the semi-log plot. Compare the plots in Figure 24.1.13, Figure 24.1.14, Figure 24.1.15 and Figure 24.1.16. The early data in the second semi-log plot indicate a deviation from the terminal straight line at early time points, leading one to consider a two or three compartment distribution as part of the model. Initial Estimates Fitting any model to pharmacokinetic data with any computer software is more efficient and more likely to succeed if you can provide good initial estimates. A number of techniques can be useful. For a simple one compartment model after an IV bolus the equation for concentration versus time can be expressed in logarithmic form as a straight line as illustrated in Figure 24.1.16. The slope and intercept can provide estimates of V and kel, see IV Bolus - Example 4. Estimating the area under the concentration versus time curve (AUC) can provide an estimate of clearance. Initial estimates for the parameters of a two compartment model can be determined by the method of residual (aka: curve striping or feathering the curve). In a similar fashion the absorption and elimination rates constant for oral administration, one compartment model can be estimated using the method of residuals. Another approach that can be quite useful is to perform a non compartmental analysis (NCA) of the data and derived estimates in the process. Criteria of least squares We need to decide on a criteria for a best fit when analyzing data and finding the best parameters values. If we put a line through data drawn on a piece of graph paper we can put the line where we think it looks best. However, if we want the computer program to find the best parameter values we need to have a well defined criteria. A commonly used criteria is the least squares criteria. Least squares criteria refers to the formula used as a measure of how well the computer generated line fits the data. Thus it is a measure of the total of the differences between the observed data and the calculated data point, Figure 24.1.17. Most commonly with pharmacokinetic modeling these differences are measured in the vertical direction. That is, in the y axis values. Usually time is the x or independent variable and it should be possible to measure time accurately. The y axis or dependent variable, usually concentration, often involves an assay method which means there may be error (or variation) in each result, Figure 24.1.17. Again, usually the residual or error is assumed to be in the vertical direction although there are programs available that are capable of looking at oblique error in both the x and y direction. For the rest of our modeling discussion we will assume that the error is in the y axis variable only. Looking at an individual data point and the calculated value with the same x value the residual can be expressed as a simple subtraction, Equation 24.1.5. Equation 24.1.5 Residual in the Y Direction The problem is that over all the data points there might be high positive and high negative residuals that could cancel out. An absolute difference would solve this problem but squaring the residual is better statistically and achieves the same result, Equation 24.1.6. Equation 24.1.6 Residual in the Y Direction Squared This gives us an equation of the residual for one data points. To complete the calculation we need to include the residuals for all the data points. This is called the sum of the squared residuals (SS), Equation 24.1.7. Equation 24.1.7 Sum of the Squared Residual Finally we need to take the error in each data point as a separate value. That is the error may be different for each measured, observed data point. We can compensate for this by applying a weight to each residual thus the usual criteria for a best fit is a minimum sum of the weighted, squared residuals (WSS), Equation 24.1.8. Equation 24.1.8 Weighted Sum of the Squared Residuals The job of the computer program is to produce a minimum value for WSS. This is also called an objective function. The fit with the minimum value of the WSS or objective function represents the best fit according to the least squares criteria. Inspection of Equation 24.1.8 leads to the conclusion that this can be achieved by changing the calculated values (Ycalculated,i) by changing the parameter values. Other approaches such extended least squares, iterative reweighted least squares, Bayesian analysis and population analysis methods use modifications of this objective function. Another minimization approach uses the log-likelihood, -2LL. Changing parameters to fit to the data Once we have a suitable criteria we can have the computer program change the parameters value to achieve a best fit to the data. The computer program will systematically alter the each parameter until the least square criteria value is minimized. These systematic steps are called optimization algorithms. These algorithms are described later. The data analysis computer program must change the parameter values to achieve a minimum value for the weighted sum of the squared residuals (WSS). This can be illustrated by changing the slope and intercept of the equation for a straight line. The calculated WSS changes with each change in the parameter values, Figure 24.1.18. Another more involved example is the calculation of the best fit to data collected after oral administration. Two of the parameters involved in this model are ka and kel. Adjusting the values of these parameters provide different values for the WSS, Figure 24.1.19. 24.2 Non-Linear Regression Analysis of Individual Subject Data Analysis using Non-linear Regression Programs In this section I will briefly describe how to set up two non-linear regression programs, Boomer (https://www.boomer.org/boomer/) and Phoenix WinNonlin (https://www.certara.com/software/pkpd-modeling-and-simulation/phoenix-winnonlin/), which can be used in the analysis of pharmacokinetic data. Other useful pharmacokinetic programs are listed on the Software page on the PharmPK website. Although graphical methods, such as we have used throughout this book, can be quite useful there are some significant limitations. A major limiting feature is that it must be possible to represent the data with a straight line. It may be possible to transform the data, for example by taking the log of the y value, but a straight line is still necessary. A problem with transforming the data is that the variance or error in each data point can be distorted inappropriately. It is fortunate that taking the log of the y-value (usually concentration) and performing a linear regression of ln(y-value) results in a reasonable representation of the variance. This is not generally the case as for example with ARE plots for urine data analysis. Using non-linear regression analysis it is possible to assign a specific variance (or weight) to each data point that is appropriate for the measurement. Thus data that are very accurately known can be analysed with data that are less accurately known by assigning a higher weight to the better data. Generally a weight equal to the reciprocal of the variance is applied to the data, Equation 24.2.1. Equation 24.2.1 Ideally the Weight for Each Data Point will be Equal or Proportional to the Variance of Data Point Another very useful feature of non-linear regression analysis is that we can fit more than one line simultaneously. For example we may give a drug to a subject and collect blood and urine samples at various times. The blood can be assayed for drug plasma concentration and the urine data can be assayed for excreted drug amounts, Figure 24.2.1. These data, with appropriate weights, can be analyzed together to give a comprehensive fit to the data. With non-linear regression analysis the shape of data curve doesn't matter. It can be a curve or straight line. There can be multiple lines (as above). The non-linear regression program simply adjusts the parameters of the model until the calculated line(s) best represents the data. Thus data collected during an IV infusion as well after the infusion has been stopped can be analyzed readily using non-linear regression analysis, Figure 24.2.2. Using graphical methods only the data after the infusion has finished could be analyzed. Program Set-Up - General approach Model Visualization The first step in the running a nonlinear regression problem is to visualize the model. The two factors which need to be considered are the route of drug administration and the data that has been collected. Designing the experiment and planning the samples involves identifiability and optimal sampling; here we are concerned with modeling data already collected. From the route of administration the modeler can include details in the model that describe the administration. This might be single or multiple dose regimens. Drug may be given by IV bolus injection, IV infusion, or an extravascular route such as oral, topical or inhalation. The extravascular routes may include multiple steps such dissolution and absorption in the case of oral administration or diffusion from a patch and through the skin in the case of a topical dose. The structure of this part of the model may be derived from an understanding of the dosage form and prior knowledge however certain parameter values may need to be determined by nonlinear regression. Dissolution rate constants, multiple absorption rate constants, lag times and extent of absorption may be among the parameters to be estimated. Consideration of the data plotted on linear and semi-log graph paper will provide more information about a suitable model. Distribution may be described with more than one compartment. More data types, such drug excreted in urine or metabolite concentrations, provide information useful in extending the model to better describe drug metabolism or excretion. With some nonlinear programs it is possible to draw a sketch of a proposed models or select a pre-drawn model from a library (WinNonlin). Users of other programs such as Boomer may wish to draw this visualization by hand. Having a clear idea of what the model 'looks' like can be very useful in correctly defining the model with any nonlinear regression program. Derive the Mathematical Model A number of nonlinear regression program relieve the modeler from the chore of deriving the equations required to describe the pharmacokinetic model. These programs include Boomer and WinNONLIN, although further model definition is possible if the equations are known. ADAPT II users and users of WinNONLIN with models that go beyond the built-in library will need to use suitable equations to define the model. Differential equations can be derived directly from the model diagram. Integrated equation for most pharmacokinetic models can be derived using Laplace transforms. This method is discussed elsewhere in an introduction and more information. Defining the Model within the Nonlinear Regression Program Each program is somewhat different in the way one defines the model but there are three different approaches. Details of how models are define within any given program should be available within the program manual. Selection from a library Some programs have a collection of predefined models within a library. The user simply selects the appropriate model from this library. WinNonlin is an example of this approach. It is probably the easiest for the user IF their model is in the library. JGuiB provides a GUI which allows the user to select a model for use with Boomer. Selection of Model Parameters Programs like Boomer provide the user with a collection of parameters, rate constants, volumes, components (compartments) and many more to provide a toolbox from which the user can build their model. This provides the user with considerable flexibility without the need to derive the equations required for the model. If the parameter toolbox contains all the required parts this can be very convenient. Describe the Model using Computer Code For maximum flexibility the modeler can describe their model using a computer language. WinNonlin provides this flexibility with the Phoenix Modeling Language (PML). Although this approach is flexible it does also require more coding experience from the modeler. Program Set-up - An Example using Boomer Analysis Type METHOD OF ANALYSIS 0) Normal fitting 1) Bayesian 2) Simulation only 3) Iterative Reweighted Least Squares 4) Simulation with random error 5) Grid Search Types of Analysis Non-linear regression analysis of a good number of data from one subject can be analyzed as a 'Normal Fitting'. Model Specification With Boomer the model is defined by selecting appropriate parameters from the types available. MODEL Definition and Parameter Entry * Allowed Parameter Types * -5) read model -3) display choices -2) display parameters 0) Time interrupt 1) Dose/initial amount 2) First order rate 3) Zero order 4-5) Vm and Km of Michaelis-Menten 6) Added constant 7) Kappa-Reciprocal volume 8-10) C = a * EXP(-b * (X-c)) 11-13) Emax (Hill) Eq with Ec(50%) & S term 14) Second order rate 15-17) Physiological Model Parameters (Q, V, and R) 18) Apparent volume of distribution 19) Dummy parameter for double dependence 20-22) C = a * SIN(2 * pi * (X - c)/b) Special Functions for First-order Rate Constants 23-24) k = a * X + b 25-27) k = a * EXP(-b * (X - c)) 28-30) k = a * SIN(2 * pi * (X - c)/b) 31,32-33) dAt/dt = - k * V * Cf (Saturable Protein Binding) 34-36) k * (1 - Imax * C/(IC(50%) + C)) Inhibition 0 or 1st order 37-39) k * (1 + Smax * C/(SC(50%) + C)) Stimulation 0 or 1st order 40) Uniform [-1 to 1] and 41) Normal [-3 to 3] Probability 42) Switch parameter 43) Clone component 44-47) Four parameter logistic model 48-51) Four parameter Weibull model Parameter types available with Boomer (Mar 2022) For example the one compartment model with a 2 hour infusion, Figure 24.2.3, can be used to analyze the data in Figure 24.2.2, This model is described in the Boomer output in tabular format. Notice the use of parameters type 2 and 18 for kel and V, respectively. The infusion rate, a type 3 parameter, is turned off (stopped) at the value of the Duration (2 hr), a type 0 parameter. Model and Parameter Definition # Name Value Type From To Dep Start Stop 1) kel = 0.9139E-01 2 1 0 0 0 0 2) V = 13.47 18 1 1 0 0 0 3) Duration = 2.000 0 0 0 0 0 0 4) k0 = 100.0 3 0 1 0 0 1 The Model Included in the Boomer Output Data and Weight Specification The x and y values (Time and Concentration) for each data set (line) are entered from the keyboard or from a data file already stored on the computer hard-drive. A weight can then be assigned by equation or independently entered by the user. More detail regarding these weight equations can be found in the Boomer manual and online. Weighting function entry for [Drug] 0) Equal weights 1) Weight by 1/Cp(i) 2) Weight by 1/Cp(i)^2 3) Weight by 1/a*Cp(i)^b 4) Weight by 1/(a + b*Cp(i)^c) 5) Weight by 1/((a+b*Cp(i)^c)*d^(tn-ti)) Choice of Weights provided in Boomer Program Output After successful completion of the Boomer run a detailed output file will provide information about the fit to the data using the model and selected weighting scheme. This output file will provide tabular, statistical and graphical output which can aid the analyst in the interpretation of the results. Tabular and Statistical Section 1 ** FINAL OUTPUT FROM Boomer (v3.1.5) ** 25 July 2005 --- 10:52:41 am Title: Fit to data before and after termination of an IV infusion Input: From Fig2202.BAT Output: To Fig2202.OUT Data for [Drug] came from keyboard (or ?.BAT) Fitting algorithm: DAMPING-GAUSS/SIMPLEX Weighting for [Drug] by 1/Cp(Obs )^2 Numerical integration method: 2) Fehlberg RKF45 with 1 de(s) With relative error 0.1000E-03 With absolute error 0.1000E-03 DT = 0.1000E-02 PC = 0.1000E-04 Loops = 1 Damping = 1 DT = 0.1000E-02 PC = 0.1000E-04 Loops = 1 Damping = 1 Section 2 ** FINAL PARAMETER VALUES *** # Name Value S.D. C.V. % Lower <-Limit-> Upper 1) kel 0.91394E-01 0.431E-02 4.7 0.0 1.0 2) V 13.473 0.618 4.6 1.0 0.10E+03 Final WSS = 0.886379E-01 R^2 = 0.9671 Corr. Coeff = 0.9834 AIC = -17.8088 Log likelihood = 8.02 Schwartz Criteria = -17.4143 R and R^2 - jp1 0.9856 0.9714 R and R^2 - jp2 0.9856 0.9714 Tabular and Statistical Output Provided by Boomer Section 1: Preliminary Output describing the input/output details, the fitting (optimization) algorithm, integration method and weighting scheme. Section 2: Best-fit parameter values with statistical information are provided. The parameter CV values, the WSS, AIC and other values provide information about the model and how well the data have been fit to the model.Section 3 Model and Parameter Definition # Name Value Type From To Dep Start Stop 1) kel = 0.9139E-01 2 1 0 0 0 0 2) V = 13.47 18 1 1 0 0 0 3) Duration = 2.000 0 0 0 0 0 0 4) k0 = 100.0 Section 4 Data for [Drug] :- DATA # Time Observed Calculated (Weight) Weighted residual 1 0.000 0.00000 0.00000 0.00000 0.00000 2 0.5000 3.80000 3.62770 0.263158 0.453408E-01 3 1.000 7.40000 7.09336 0.135135 0.414372E-01 4 1.500 10.8000 10.4042 0.925926E-01 0.366464E-01 5 2.000 0.00000 13.5672 0.00000 0.00000 6 3.000 12.0000 12.3822 0.833333E-01 -0.318495E-01 7 5.000 9.00000 10.3137 0.111111 -0.145965 8 9.000 8.00000 7.15558 0.125000 0.105553 9 12.00 5.00000 5.43962 0.200000 -0.879240E-01 10 18.00 3.90000 3.14352 0.256410 0.193969 11 24.00 1.70000 1.81662 0.588235 -0.686003E-01 WSS for data set 1 = 0.8864E-01 R^2 = 0.9671 Corr. Coeff. = 0.9834 R and R^2 - jp1 0.9856 0.9714 R and R^2 - jp2 0.9856 0.9714 Maximum value for [Drug] is 12.000 at 3.000 Tabular and Statistical Output Provided by Boomer (continued) Section 3: The model definition section provides the opportunity to confirm that the model has been described correctly. Section 4: The data are provided next as observed x and y values, calculated y values and weight and residual information. The observed data in this table should be checked against the correct values. Systematic differences between observed and calculated values may be detected in this section if the data analysis is incorrect. Section 5 Calculation of AUC and AUMC based on trapezoidal rule AUC and AUMC for [Drug] using Observed data Time Concentration AUC AUMC 0.00000 ( 0.00000 ) 0.500000 3.80000 0.950000 0.475000 1.00000 7.40000 3.75000 2.80000 1.50000 10.8000 8.30000 8.70000 2.00000 0.00000 11.0000 12.7500 3.00000 12.0000 17.0000 30.7500 5.00000 9.00000 38.0000 111.750 9.00000 8.00000 72.0000 345.750 12.0000 5.00000 91.5000 543.750 18.0000 3.90000 118.200 934.350 24.0000 1.70000 135.000 1267.35 153.601 1917.29 Secondary Parameters MRT = 12.482 Half-life values for each first order rate constant Parameter 1 has a half-life of kel is 7.58 Dose/AUC (= Clearance/F) Parameter 4 gives k0/AUC (CL/F) of -0.651 Tabular and Statistical Output Provided by Boomer (continued) Section 5: The program can calculate the AUC and a number of 'secondary' parameters including MRT, half-life, and Dose/AUC. A zero time point must be entered for accurate estimates of AUC. Graphical Output Section 1 Plots of observed (*) and calculated values (+) versus time for [Drug] . Superimposed points (X) 13.57 Linear 13.57 Semi-log | | + | + | | | X | + | | * | X + | | | | * | X | | + | * | | * | | + + | * | | | | * | | * | + | + + | * | | | | | + | | * |* * | |+ | | |X * | + | | | + | | | | + | | * | | | | | |X * | X |_____________________________________ |X__*_________________________________ 0.000 1.700 0 <--> 24. 0 <--> 24. Graphical Output Provided by Boomer Section 1: A linear and semi-log printer-type plot of the observed and calculated y-values versus the x-values. Systematic deviations, indicating a poor weighting scheme or model selection, may be apparent from these graphs. Section 2 Plot of Std Wtd Residuals (X) Plot of Std Wtd Residuals (X) versus time for [Drug] versus log(calc Cp(i)) for [Drug] 1.955 1.955 | X | X | | | | | | | | | X | X | | |X | X | XX | X X | | 0X==X================================= 0====================================X | X | X | | | X |X | X | X | | | X | X | | -1.471 -1.471 0.0 <--> 24. 1.8 <--> 14. Graphical Output Provided by Boomer Section 2: Standardized weighted residuals versus x-value or log(calculated y value) are very useful tools for detecting poor model or weighting scheme selection. Any obvious pattern in these plots should be explored as potential evidence of a poor fit to the data. Program Set-up - An Example using PhoenixTM WinNonlin Analysis Type The previous Boomer example was an analysis of data collected after an IV infusion of 200 mg over 2 hours. In WinNonlin this could be analyzed as a non Population PK analysis using parameters specified as micro constant, that is kel and V. The first step is to create a new project, Figure 24.2.4. Next a new worksheet for the data, Figure 24.2.5. Set up two columns for Time and Concentration with units and enter the data, Figure 24.2.6 and Figure 24.2.7. We can have a look at the data first as a linear and semi-log plot to see what model might be appropriate, Figure 24.2.8. Map Time to X and Concentration to Y and execute the plot, Figure 24.2.9. Starting with a linear plot, Figure 24.2.10, double-click on the y-axis, set it to logarithmic, Figure 24.2.11, and we have a semi-log plot Figure 24.2.12. After looking at the semi-log plot we might plan on starting with a one compartment model. We can send the Data Worksheet to a Maximum Likelihood Model, Figure 24.2.13. Uncheck Population as this is single subject data. Use Micro Parameterization although we could choose Clearance. Dosing by Intravenous and One compartment model. Check Infusion and Duration. For most data a Multiplicative Error model works well. This is similar to weighting by the reciprocal of the concentration squared as was done with the Boomer example. Here extended least squares is used and an initial value of 0.1 (or 10%) is entered for the fractional standard deviation, Figure 24.2.14. We can now enter the dosing information. Two hundred milligram given over two hours starting at time zero, Figure 24.2.15. Initial parameter values can be entered and confirmed and/or adjusted, Figure 24.2.16 and Figure 24.2.17. Here we selected the model from a library within WinNonlin but we could build the model with a graphical or Phoenix Modeling Language (PML) code. The same model can be represented by each of these methods, Figure 24.2.18 and Figure 24.2.19. The WinNonlin project file for this example can be downloaded here. The columns have been mapped as before and it is time for the program to fit the data with the model by adjusting the parameter values. Below is another example of a graphical and PML code description for a two compartment model with two covariates, one continuous (Weight) and the other categorical (Sex), Figure 24.2.20. In Figure 24.2.21 lines 2-4 define the differential equations for the central compartment (A1), output (A0) and the peripheral compartment (A2). Amounts are converted to the observed quantity, concentration, on line 5. Line 6 describes the dosing into the central compartment. The initial error value, Eps, and the error model, here multiplicative are defined on lines 7 and 8. The adjustable parameters in the model are described on lines 9-12. Note the inclusion of Weight (on V) and Sex (on Ke) as two covariates on lines 9 and 10. On line 9 Weight has been normalized by the mean of the study observed weights. mean(Weight) could be replaced by 70 (as a fixed estimate for a population) when the study weights might not be typical. The covariates are listed in lines 13-14. Initial estimates for the adjustable parameters are included on lines 15-18. Lower and upper limits are typically not required. Lines 19-20 provide initial estimates for covariates. The last line, 21, describes initial estimates for the diagonals for the adjustable parameters. Program Output After successful completion of the WinNonlin run a number of items will provide information about the fit to the data using the model and selected weighting scheme. This output file will provide tabular, statistical and graphical output which can aid the analyst in the interpretation of the results. Tabular and Statistical Overall fit parameters including RetCode, -2LL and AIC are provided in the Overall Table, Figure 24.2.22. The best fit values are provided along with Stder and CV%. Note that the CV% values are quite low. These are similar values to those provided by Boomer, Figure 24.2.23. Graphical We can also look at linear and semi-log plots of observed and calculated concentrations versus time. These plot are also indicative of a good fit to the data, Figure 24.2.24 and Figure 24.2.25. Another diagnostic plot that is important to consider is the weighted residual plot. This plot can help confirm not only the selected model but also the weighting scheme, Figure 24.2.26. 24.3 Bayesian Analysis of Clinical Data Monitoring drug disposition in patients can be challenging. Drug regimens may not be constant and only a small number of drug concentrations may be available. Fortunately, the drug registration process requires extensive pharmacokinetic studies. Combining prior population pharmacokinetic data with a few data points from the patient allows for a better understanding of the drug disposition in this patient. Bayesian pharmacokinetic analysis allows the integration of population information with patient data. The parameters of the pharmacokinetic model are adjusted to best-fit both the population and patient data. A different objective function is required to include both of these observations, Equation 24.3.1. Equation 24.3.1 Objective Function for Bayesian Optimization Note the use of variance instead of weight in this objective function, Equation 24.3.1. Realistic values for the data variance (and thus weight) must be balanced with variance of each population parameter. The inclusion of patient data and population data provides the ability to estimate parameters in the patient for improved drug regimen recommendations. In this section we will briefly describe how to set up one non-linear regression program, Boomer. Program Set-up As an example we can consider a patient given a drug by IV infusion over 30 minutes. Population values for CL and V may be found in the text or other reference. Values of these parameters might vary in patients with various disease states but for a normal patient (70 Kg) the values in Table 24.3.1 may be used as an example. After a dose of 1000 mg/hr for 30min (500 mg IV) samples were collected at 1 hr and 9 hr after the start of the infusion. These samples were assayed and found to contain 15.6 and 5.8 mg/L, respectively. Assay standard deviations were estimated to be 5% of the value measured. This patient’s data and the population parameters from Table 24.3.1 were analyzed with Boomer using the Bayesian method, Figure 24.3.1. Figure 24.3.2 illustrates the resulting 'best-fit' to these two data points. Analysis Type The Bayesian method is specified early in the Boomer input steps. METHOD OF ANALYSIS 0) Normal fitting 1) Bayesian 2) Simulation only 3) Iterative Reweighted Least Squares 4) Simulation with random error 5) Grid Search -5) To perform Monte Carlo run (Only once at the start of BAT file) -4) To perform multi-run (End of BAT file only) -3) To run random number test subroutine -2) To close (or open) .BAT file -1) To finish Enter choice (-3 to 5) 1 Specifying the Bayesian Analysis Method Model Specification Boomer doesn't include clearance as a parameter type so it must be entered as a type 19 (dummy) parameter. The elimination rate constant kel is specified as CL/V. Since volume (type 18) parameters cannot be specified before rate constants a dummy parameter Vd is specified first. The model parameter V is simply set equal to Vd. Model and Parameter Definition # Name Value Type From To Dep Start Stop 1) CL = 3.613 19 0 0 0 0 0 2) Vd = 29.33 19 0 0 0 0 0 3) kel = 0.1232 2 1 0 7001002 0 0 4) V = 29.33 18 1 1 1002000 0 0 5) Duration = 0.5000 0 0 0 0 0 0 6) k0 = 1000. 3 0 1 0 0 1 Model Specification from the .OUT File Data and Weight Specification The two data points were entered from the keyboard with the weight specified as a 5% standard deviation. A 5% standard deviation translates into a variance of 0.0025 x Observed Value2. Thus the 'a' and 'b' values entered are 0.0025 and 2, respectively. Weighting function entry for [Theophylline] 0) Equal weights 1) Weight by 1/Cp(i) 2) Weight by 1/Cp(i)^2 3) Weight by 1/a*Cp(i)^b 4) Weight by 1/(a + b*Cp(i)^c) 5) Weight by 1/((a+b*Cp(i)^c)*d^(tn-ti)) Data weight as a function of Cp(Obs) Enter choice (0-5) 3 Enter a value 0.0025 Enter b value 2 Specifying the Weight for the Data Points by Equation Program Output Tabular and Statistical Output Section 1 ** FINAL OUTPUT FROM Boomer (v3.1.5) ** 26 July 2005 --- 2:21:06 pm Title: Bayesian Fit to Data Input: From Keyboard Output: To Fig2206.OUT Data for [Drug] came from keyboard (or ?.BAT) Fitting algorithm: DAMPING-GAUSS/SIMPLEX Weighting for [Theophylline] by 1/a*Cp(Obs )^b With a = 0.2500E-02 and b = 2.000 Bayesian Fitting to data: Numerical integration method: 2) Fehlberg RKF45 with 1 de(s) With relative error 0.1000E-03 With absolute error 0.1000E-03 DT = 0.1000E-02 PC = 0.1000E-04 Loops = 1 Damping = 1 Section 1: Preliminary Output describing the input/output details, the fitting (optimization) algorithm, integration method and weighting scheme. Section 2 ** FINAL PARAMETER VALUES *** # Name Value S.D. C.V. % Lower <-Limit-> Upper Population mean S.D. (Weight) Weighted residual 1) CL 3.6134 0.904E-02 0.25 1.0 10. 3.610 1.470 0.6803 0.2309E-02 2) Vd 29.330 0.822E-01 0.28 1.0 0.10E+03 30.10 4.200 0.2381 -0.1833 Final WSS = 0.385777E-01 R^2 = 1.000 Corr. Coeff = 1.000 AIC = -2.51016 Log likelihood = 1.11 Schwartz Criteria = -5.12387 R and R^2 - jp1 1.0000 1.0000 R and R^2 - jp2 1.0000 1.0000 Section 2: Best-fit parameter values with statistical information are provided. The parameter CV values, the WSS, AIC and other values provide information about the model and how well the data have been fit to the model. Section 3 Model and Parameter Definition # Name Value Type From To Dep Start Stop 1) CL = 3.613 19 0 0 0 0 0 2) Vd = 29.33 19 0 0 0 0 0 3) kel = 0.1232 2 1 0 7001002 0 0 4) V = 29.33 18 1 1 1002000 0 0 5) Duration = 0.5000 0 0 0 0 0 0 6) k0 = 1000. 3 0 1 0 0 1 Section 3: The model definition section provides the opportunity to confirm that the model has been described correctly. Section 4 Data for [Drug] :- DATA # Time Observed Calculated (Weight) Weighted residual 1 0.000 0.00000 0.00000 0.00000 0.00000 2 0.1250 0.00000 4.22917 0.00000 0.00000 3 0.2500 0.00000 8.39372 0.00000 0.00000 4 0.3750 0.00000 12.4946 0.00000 0.00000 5 0.5000 0.00000 16.5329 0.00000 0.00000 6 0.7500 0.00000 16.0314 0.00000 0.00000 7 1.000 15.6000 15.5452 1.28205 0.702675E-01 8 1.500 0.00000 14.6165 0.00000 0.00000 9 2.000 0.00000 13.7433 0.00000 0.00000 10 4.000 0.00000 10.7420 0.00000 0.00000 11 6.000 0.00000 8.39608 0.00000 0.00000 12 9.000 5.80000 5.80182 3.44828 -0.625972E-02 13 12.00 0.00000 4.00914 0.00000 0.00000 WSS for data set 1 = 0.4977E-02 R^2 = 1.000 Corr. Coeff. = 1.000 R and R^2 - jp1 1.0000 1.0000 R and R^2 - jp2 1.0000 1.0000 Section 4: The data are provided next as observed x and y values, calculated y values and weight and residual information. The observed data in this table should be checked against the original data to verify that they have been entered correctly. Systematic differences between observed and calculated values may be detected in this section if the data analysis is incorrect. Graphical Output Section 1 Plots of observed (*) and calculated values (+) versus time . Superimposed points (X) 16.53 Linear 16.53 Semi-log | + | + | + | + | X | X | | + | + | + | | | + | + | | | + | | | | | + | + | | | | | | | |+ + |+ + | | | | | | | | | X | | | | | |+ + | X | | | | | | | | | | | |+ |X** * * * * * | + |_____________________________________ |X**_*_*_____*_____*_________________* 0.000 4.009 0 <--> 12. 0 <--> 12. Section 1: A linear and semi-log printer-type plot of the observed and calculated y-values versus the x-values. Section 2 Plot of Std Wtd Residuals (X) Plot of Std Wtd Residuals (X) versus time versus log(calc Cp(i)) 1.290 1.290 | X | X | | | | | | | | | | | | | | | | | | | | | | | | | | | | 0XXX=X=X=====X=====X=================X 0XX================X======X==X==XX==XX | X | X | | -0.1149 -0.1149 0.0 <--> 12. 4.0 <--> 17. Section 2: Standardized weighted residuals versus x-value or log(calculated y value) are very useful tools for detecting poor model or weighting scheme selection. Any obvious pattern in these plots should be explored as potential evidence of a poor fit to the data. Patterns are not as obvious with a typical Bayesian analysis since there are fewer observed data points. 24.4 Non-Linear Regression Analysis of Population Data Analysis using of Population Data using PhoenixTM NLME During the development of new drug entities (NDE) the manufacturer will study the disposition of the drug in numerous volunteeers healthy and otherwise. The early, Phase I studies are typically conducted in healthy volunteers with the objective of determining the disposition characteristics of the NDE. These studies usually involve the collection of numerous blood and other samples from each subject. The data analysis techniques described earlier in this course may be quite useful for these studies. Later studies during Phase III involve the administration of the NDE to numerous patients who may be expected to have some therapeutic benefit from the NDE. These subjects may provide a wide range of covariates such as age, weight, sex, genetic characteristics, co-administration of other drugs and various clinical or pathological conditions. In these studies the protocol may provide for the collection of only one or two blood samples during various dosage regimens. It is difficult to analyze data separately to determine good estimate of the pharmacokinetic parameters. Fortunately these studies provide data from a large number of subjects and there are a number of computer programs which are capable of analyzing these data simultaneously. One of these population pharmacokinetic (PopPK) computer programs is PhoenixTM NLME. Other PopPK may be found here. NLME will allow the analyst to include data from many subjects in one analysis and provides estimates of the best-fit pharmacokinetic parameter values and estimates of their variability (standard deviation or variance) between the subject as well as relationships with covariate values. The output from these analyses may be useful in the Bayesian estimate of clinical pharmacokinetic data as described earlier. The PopPK approach can also be used in cases of data rich sources, such as bioavailability studies, or sparse data information that might be available post-marketing during therapeutic drug monitoring. A Typical (Initial) Workflow After starting the program a new Project can be selected from the File Menu, Figure 24.4.1. Use File > Import to select a suitable data file, maybe a .csv file. This one includes a Dose column, Figure 24.4.2. Once the data file has been imported we can create a XY plot. Control-click on the Data file and Send to > Plotting > XY Plot. Map the X and Y to Time and CObs and Group by ID, Figure 24.4.3. Executing this object produces the first graphical look at the data. On this semi-log plot the data after an IV bolus appear to follow a straight line for each subject, Figure 24.4.4. Let try a simple one compartment model using kel and V as parameters. Control-click on the Data file and Send To... > Modeling > Maximum Likelihood Models. Map the Time_1 and Cobs values to Time and CObs. ID to ID. Dose to A1, Figure 24.4.5. Down below check Population? Under the Structure Tab select Type: PK, Parameterization: Micro, Num Compartments: 1 and Residual Error: Multiplicative with the default Stdev: 0.1, Figure 24.4.6. Under the Parameters: Fixed Effects Tab enter 100 for tvV and 0.1 for tvKe, Figure 24.4.7. Executing this object provides tabular and graphical output, Figure 24.4.8, 24.4.9 and Figure 24.4.10. And a plot of conditional weighted residuals (CWRES) versus Time (TAD, Figure 24.4.11) This is just the start. Maybe try a two compartment model. Consider different weighting schemes although multiplicative error is a good place to start. Covariates, weight and sex, could be considered for inclusion in the model. Chapter 25: Integrating Differential Equations using Laplace Transforms Student Objectives for this Chapter After completing the material in this chapter each student should be able to:- Write differential equations for components of a pharmacokinetic model To use the steps involved in integrating differential equations using Laplace Transforms 25.1 Writing Differential Equations Rate processes in the field of pharmacokinetics are usually limited to first order, zero order and occasionally Michaelis-Menten kinetics. Linear pharmacokinetic systems consist of first order disposition processes and bolus doses, first order or zero order absorption rate processes. These rate processes can be described mathematically. First Order Equation Each first order rate process ("arrow") is described by a first order rate constant (k1) and the amount or concentration remaining to be transferred (X1), Equation 25.1.1. Equation 25.1.1 First Order Rate Equation Zero Order Equation Zero order rate processes are described by the rate constant alone. Amount or concentration to the zero power is 1, Equation 25.1.2. Equation 25.1.2 Zero Order Rate Equation Michaelis Menten Equation The Michaelis Menten process is somewhat more complicated with a maximum rate (velocity, Vm) and a Michaelis constant (Km) and the amount or concentration remaining, 25.1.3. Equation 25.1.3 Michaelis-Menten Equation The full differential equation for any component of a pharmacokinetic model can be constructed by adding an equation segment for each arrow in the pharmacokinetic model. The rules for each segment: Direction of the arrow If the arrow goes into the component the equation segment is positive If the arrow leaves the component the equation segment is negative. Type of rate process If the rate process is first order multiply the (first order) rate constant by the amount or concentration of drug in the component at the tail of the arrow. If the process is zero order just enter the rate constant. For a Michaelis Menten processes include the amount or concentration of drug in the component at the tail of the arrow in the Michaelis-Menten equation. An example In Figure 25.1.1 the rate process from one to two is zero order. The process from two to three is first order and the process from two to four follows Michaelis-Menten kinetics. We can now systematically write the differential equations for each component of the model. Component 1, X1 There is one arrow leading out of component one so the rate process if negative. This process is zero order so we just write the rate constant. The equation for this component is Equation 25.1.4. Equation 25.1.4 Differential Equation for Component One, X1 Component 2, X2 This component is more of a challenge. There are three arrows connected with this component. One arrow leads to the compartment so this equation segment is positive. The other two arrows lead away from the component and these equation segments are negative. Starting with the zero order process provides is a positive k0 to the differential equation. The first order process is developed as the rate constant multiplied by the amount remaining in component 2. This is negative and is -k1 x X2. The final segment is the Michaelis Menten process from component 2 to component 4. This is also negative and is described as -Vm x X2/(Km + X2). The total differential equation for this component is Equation 25.1.5. Equation 25.1.5 Differential Equation for Component Two, X2 Component 3, X3 OK; downhill from here. Now we have one arrow going to this component from component 2, thus the equation segment is positive. The rate process is first order so we multiply the rate constant by the amount remaining in the component at the tail of the arrow, component 2. The equation is Equation 25.1.6. Equation 25.1.6 Differential Equation for Component Three, X3 Component 4, X4 The fourth component is also described with one equation segment. The arrow leads to this component so the segment is positive. The rate process is a Michaelis Menten process. The equation for this component is Equation 25.1.7. Equation 25.1.7 Differential Equation for Component Four, X4 More examples of pharmacokinetic models and writing and integrating differential equations using Laplace Transform are described in the next section. 25.2 Integrating Differential Equations using Laplace Transforms Laplace transforms are a convenient method of converting differential equations into integrated equations, that is, integrating the differential equation. It is similar to the use of logarithms to multiple or divide numbers. To multiple two numbers we convert each number into their respective logarithm and add. The sum is converted into the 'anti'-logarithm and the product of the original numbers is the result. For division, the logarithm of the two numbers are subtracted before obtaining the answer by taking the anti-logarithm. Differential equations can be converted into the integrated form using Laplace transforms by following a similar number of straight forward steps. Write the differential equation. Using the approach presented in the previous section you need to write the differential equation for the system of interest. Take the Laplace transform of each differential equation using a few transforms. We transform the equation from the t, time, domain into the s domain. For most pharmacokinetic problems we only need the Laplace transform for a constant, a variable and a differential. Use some algebra to solve for the Laplace of the system component of interest. Often the Laplace of a component 'up-stream' will need to be solved first and substituted into the equation of interest. Finally the 'anti'-Laplace for the component is determined from tables referenced below or by using the 'finger-print' method described in a later section. In step two above, only three Laplace transforms are necessary for most of the linear of pharmacokinetic systems that you might encounter. These are: The Laplace of a constant, Equation 25.2.1: Equation 25.2.1 Laplace of a Constant The Laplace of a variable (i.e. amount in a component of the model; e.g. X1, Equation 25.2.2: Equation 25.2.2 Laplace of a Variable The Laplace of the differential of a variable (e.g. dX1/dt), Equation 25.2.3: Equation 25.2.3 Laplace of the Differential of a Variable Time for an example IV Bolus - Linear one compartment model The IV bolus dose is placed in component one at zero time. Elimination is by a single first order process described by the rate constant kel (elimination rate constant). The scheme can be describe by Figure 25.2.1. Write the differential equation for X1. There is only one component. Equation 25.2.4 The Differential Equation for X1 Transform the equation into the Laplace form Equation 25.2.5 Transformed into the s domain Rearranging and solving for L(X1). Substitute Dose for X1. Equation 25.2.6 Solve for L(X1) The final result can be determined from the Laplace Transform table (below - line 3 with A == Dose: a == kel). Equation 25.2.7 Integrated Equation for X1 Additional problems and answers are available References Mayersohn, M. and Gibaldi, M. 1970 Mathematical Methods in Pharmacokinetics. I. Use of the Laplace Transform in Solving Differential Rate Equations, Amer. J. Pharm. Ed., 34:608-614 Table of Laplace Transforms in PDF format An addition to the Table of Laplace Transforms in PDF format Laplace transforms at Wikipedia Search for Laplace transforms at DuckDuckGo 25.3 More about the Laplace Transform Method It is time to explore some more techniques related to the Laplace transform method of integrating differential equations. In the previous page we saw how we could integrate differential equations using Laplace transforms. At that point we wrote the differential equation, took the Laplace transform of each equation, solved for the model components of interest and used Tables of Laplace transforms to take the back-transform. In this Chapter we will use the 'finger print' method to take the back transform [1]. We will also extend the method Laplace transforms to integrate differential equations derived from multi-compartment pharmacokinetic models, in the first instance a two compartment model [2]. Finally, we will look at a convolution method to develop the Laplace transform of more complex models [3] Finger-Print Method for Inverse Laplace Transform [1, 3] This method has been described by Benet and Turi [1] and Benet [3]. If the requirements for this method are meet the inverse Laplace transform can be written almost by inspection. The second paper by Benet [3] provides a more complex method where the requirements of no repeating factors in not met. Extension to Multi-compartment Pharmacokinetic Models [2] Using the previous methods and by making judicious substitutions it is possible develop the Laplace equations for models representing two and more compartment models. Rate constants such as k12 and k21 (micro constants) are substituted with macro constants such as α and β. This has been well described by Mayersohn and Gibaldi [2]. Convolution Method of Deriving Laplace Transforms [3] Benet [3] has presented a method of developing the Laplace transform for a pharmacokinetic model as the product of the input function and the disposition function. The input function is derived from the route of administration, while the disposition function depends on the complexity of the distribution and elimination processes. References Benet, L.Z. and Turi, J.S. 1971. Use of the General Partial Fraction Theorems for Obtaining Inverse Laplace Transforms in Pharmacokinetic Analysis, J. Pharm. Sci., 60: 1593-1594 Mayersohn, M. and Gibaldi, M 1971 Mathematical Methods in Pharmacokinetics. II. "Solution of the Two Compartment Open Model, Amer. J. Pharm. Ed., 35:19-28 Benet, L.Z. 1972 General Treatment of Linear Mammillary Models with Elimination from any Compartment as Used in Pharmacokinetics, J. Pharm. Sci., 61:536-541 25.4 Finger Print Method Inverse Laplace Transform The so-called finger print method provides a quick and easy method for the back transformation of many of the Laplace equations found in pharmacokinetics. With a few limitations the method can be commonly applied. This method is derived from the explanation of the general partial fraction method presented by Benet and Turi (1971). General Partial Fraction Method The general partial fraction method is summarized with the equation, Equation 25.4.1. Equation 25.4.1 Equation Demonstrating the General Partial Fraction Theorem The function in 's' on the left is transformed into the function on the right in terms of 't' (time). The 𝜆 terms are the roots of the polynomial term in the denominator on the left. In pharmacokinetic equations these are usually zero or negative. Limitations or Requirements There are two requirements for this method to be applicable. The degree, in s, of the polynomial in the denominator must be higher than the polynomial in the numerator. There must be no repeating terms in the denominator Examples Equation 25.4.2 Fractions which Don’t Comply with the Requirements Equation 25.4.3 Fractions which Do Comply with the Requirements General Procedure The general procedure for this method is to: Check the Limitations of Requirements Determine the Roots in the Denominator. After solving for the Laplace of the amount or concentration of interest the denominator should be of the form: Equation 25.4.4 The General Form of the Denominator In finding the root(s) of the polynomial in the denominator each of the factors can be set to zero and used to find a value (or root) for s. Thus, from Equation 25.4.4 the equations: s = 0 s + a = 0 s + b = 0 ... can be derived and the roots for the denominator are thus s = 0, s = -a, s = -b, etc. Write the back transform using the finger print method There may be one or more roots to the denominator. The next step is to cover the part corresponding to each root in turn and replace all instances of 's' in the remaining equation with the current root. This term is then multiplied by eroot*t. The final result is the sum of all the terms from each root. Note: The roots for pharmacokinetic problems are usually negative (or zero) and thus the exponential term usually ends up being negative. Simplify the result as necessary Reference Benet, L.Z. and Turi, J.S. 1971. "Use of the General Partial Fraction Theorems for Obtaining Inverse Laplace Transforms in Pharmacokinetic Analysis", J. Pharm. Sci., 60: 1593-159425.5 Finger Print Method - Example 1 The differential equation for amount of drug in body according to the one compartment model after a single oral dose (F * Dose) can be written as: Equation 25.5.1 Laplace Transform of Amount in the Body after Oral Administration Note that the denominator has a power of 2 in s and no repeating terms. The numerator has a power of 0 in s. Therefore the fingerprint method can be applied. Setting the denominator to zero: (s + ka) * (s + kel) = 0 gives the two roots -ka and -kel. The finger print method can then be applied as illustrated in the video below, Video 25.5.1. The solution, Equation 25.5.2. Equation 25.5.2 Integrated Equation for Amount in the Body after Oral Administration Dividing both sides by the apparent volume of distribution yields an Equation for concentration versus time, Equation 25.5.3. Equation 25.5.3 Concentration versus Time after Oral Administration 25.6 Finger Print Method - Example 2 The differential equation for amount of drug in body according to the one compartment model during an IV infusion (k0) can be written as, Equation 25.6.1. Equation 25.6.1 Laplace Transform of Amount in the Body during an IV Infusion Note that the denominator has a power of 2 in s and no repeating terms. The numerator has a power of 0 in s. Therefore the fingerprint method can be applied. Setting the denominator to zero: s * (s + kel) = 0 gives the two roots 0 and -kel. The finger print method can then be applied as illustrated in the video below, Video 25.6.1. The solution, Equation 25.6.2. Equation 25.6.2 Integrated Equation for Amount in the Body As before dividing both sides by the apparent volume of distribution give the integrated equation for the concentration versus time, Equation 25.6.2 Equation 25.6.2 Integrated Equation for Concentration versus Time during an IV Infusion 25.7 Multi Compartment Pharmacokinetic Models The Laplace transform method can be extended to multi compartment pharmacokinetic models. The algebra gets more involved or maybe you can use the convolution technique. Using the algebraic approach we can develop the differential equations and thus the integrated equation for the two compartment model (Mayersohn, M. and Gibaldi, M 1971), Figure 25.7.1. Following an IV bolus dose the drug is placed in component 1. Drug concentration in this component achieves equilibrium quickly. More slowly the drug either distributes into and out of various tissues and body fluids designated component 2 or is eliminated from the body as described by the first order rate constant, kel. The distribution processes are usually described using first order rate constants. Alternately, the elimination and distribution could be describe by clearance terms. For the purposes of this derivation we will confine the mathematical model to use rate constants. The differential equations for both components are shown below, Equation 25.7.1. Equation 25.7.1 Differential Equations for a Two Compartment Model Using the rules on a previous section it is possible to write the Laplace transform of each of the equations, Equation 25.7.2. Equation 25.7.2 Laplace Transforms from Equation 25.7.1 Since the amount of drug in component 2 is zero at time 0 the term X02 can be set to 0 giving Equation 25.7.3. Equation 25.7.3 Laplace of Component 2 With X01 the Dose we can substitute the value in Equation 25.7.3 for the Laplace of the amount in component 2 into Equation 25.7.2 to continue the derivation of the Laplace of component 1, Equation 25.7.4. Equation 25.7.4 Deriving the Laplace of the Amount in Comment 1 We can make a substitution to simplify Equation 25.7.4. Looking at Equation 25.7.5 and the denominator in Equation 25.7.4 we can make the substitutions, Equation 25.7.6, to derive the final equation for the Laplace of the amount in component 1, Equation 25.7.7. Equation 25.7.5 Introducing the terms α and β Equation 25.7.6 Substitutions Equation 25.7.7 Laplace of the Amount in Component 1 The next sections describes the back transform process using the finger print method. Reference Mayersohn, M. and Gibaldi, M 1971 Mathematical Methods in Pharmacokinetics. II. Solution of the Two Compartment Open Model, Amer. J. Pharm. Ed., 35:19-28 25.8 Finger Print Method - Example 3 The differential equation for amount of drug in body according to the two compartment model after an IV bolus (Dose) can be written as Figure 25.8.1. Equation 25.8.1 Laplace of Amount in the Body after an IV Bolus - Two Compartment Model Note that the denominator has a power of 2 in s and no repeating terms. The numerator has a power of 1 in s. Therefore the fingerprint method can be applied. Setting the denominator to zero: (s + 𝛼) * (s + β) = 0 gives the two roots -𝛼 and -β. The finger print method can then be applied as illustrated in the video below, Video 25.8.1. The solution, Equation 25.8.2. Equation 25.8.2 Integrated Equation for Amount in the Body after an IV Bolus - Two Compartment Model Dividing both sides by V1 provides the more familiar values A and B in the equation for Cp versus time, Figure 25.8.3. A and B were defined earlier in Equation 18.2.1. Equation 25.8.3 Integrated Equation for the Concentration versus Time after an IV Bolus - Two Compartment Model 25.9 Convolution The convolution method described by Benet (Benet, 1972) allows the development of the Laplace equation for the amount of drug in the central compartment by a simple multiplication of the input function and the disposition function. A ke/s function can be included to derive the amount of drug in urine. The input function describes the route of administration. The disposition function describes the first order distribution and elimination processes, and on this page, with elimination from the central compartment. Basically the equation is Equation 25.9.1 Equation 25.9.1 Basic Convolution Equation A general linear pharmacokinetic model with elimination via excretion into urine (ke), metabolism (km) or other processes (kother) is shown below, Figure 25.9.1. In Figure 25.9.1 the plasma or central compartment is red, the tissue compartments are brown, and the urine component of the model is represented by an orange border. Drug is represented by the filled green circles and metabolite by the blue circle. The overall elimination rate constant, kel, is the sum of all the renal excretion, metabolism and other elimination processes, thus kel = ke + km + kother. In general the choices we have for the input or route of administration function are shown in Table 25.9.1 Functions for more complex absorption processes could be developed. Disposition functions are included in Table 25.9.2. An additional Sample Site multiplier can be applied for other sample sites, beyond drug in the central compartment. Sample Site functions are included in Table 25.9.3. Try it out using Interactive 25.9.1. Reference Benet, L.Z. 1972 General Treatment of Linear Mammillary Models with Elimination from any Compartment as Used in Pharmacokinetics, J. Pharm. Sci., 61:536-541 Chapter 26: Numerical Integration Student Objectives for this Chapter Understand the Process of Numerical Integration using Euler's method Understand some of the other Numerical Integration Methods (algorithms) Consider the Reasons for Choosing One Method over Another Pharmacokinetics involves rate processes. Rate process are readily described with differential equations. With a diagram for a pharmacokinetic model with its circles and arrows you should now be able to write the corresponding differential equations. Thse differential eqautions can be integrated analytical by various methods. We have used the Laplace transform method as it can be applied to many of the equations found in pharmacokinetics. There are a few types of differential equations which can't be integrated and often it is just easier to let the computer do the integration for you. The computer takes the slope or rate of change or differential equation for a component and a starting point to get to a new value. Euler's method is a simple point and slope method that is relatively easy to understand. We will start by exploring this method in some detail before moving on to some other, more efficient methods. These methods include Point-slope Methods Euler's Method Runge-Kutta Methods RKF45 Method Multi-steps Methods Adams-Bashford Method Predictor-Corrector Methods Adams-Moulton Method Gear's Method 26.1 Euler's Method Euler's method is a point-slope numerical integration method. Using a single point (initial condition, X1(0) or Dose for example) and the slope (the differential equation) it is possible to estimate a new point value. This new value and slope can be used to calculate the bext value. Seems simple enough and really is quite simple. The method can be presented graphically for a one compartment model, Figure 26.1.1. We known from an earlier Chapter that the 'real' curve is not a straight line. Thus, Euler's method provide only an approximation to the real answer. However, if you use small enough steps the approximation can be useful. Euler's method is a point slope method The point (starting point) is the initial value for the first calculation The slope (differential equation is the result of calculating (evaluating) the differential equation value at the starting point. Thus, for this example of a one compartment model after an IV Bolus dose. Point = X1(0) and Slope = - k1 x X1(0) Thus, with X1(0) = 100 mg, k1 = 0.25 hr-1 and step size = 0.1 the new value is 97.5 mg. For comparison, the concentration using the exact equation of Dose x e-k1 x 0.1 = 100 x e-0.25 x 0.1 = 97.53. Not too bad, about 0.03% error, Figure 26.1.2. We can extend the calculation out for as long as we like. Table 26.1.1 provides the calculation out for 0.5 hours. Note the exact answer is Dose x e-k1 x 0.5 = 100 x e-0.25 x 0.5 = 88.25. An error of 0.17%. Euler's method calculations are based on the equation for the differential equation, the slope. Equation 26.1.1 can be repeated out to what ever time is required. Remember this is a straight line approximation to a line that is probably curved. The method is only accurate if the step size is not too big. We can see this with another example. Using parameters Cp0 = 100 and kel = 0.3 hr-1 we can explore the effect of step size. In the table below either 1, 2, 4, or 10 steps are taken to get from time 0 to time 1. Equation 26.1.1 Calculation using Euler’s Method Note the error drops from over 5 % to 0.05 % by decreasing the step-size by 10. After 10 steps the error with the smaller step-size would be approximately 0.5 %. A 10 fold improvement for a 10 fold increase in computational cost, Table 26.1.2. Note, the effect of the larger step-size, in this case 1, on the calculation, Figure 26.1.3. Notice the divergence between the slope at each time point and the exact curve, as time progresses. Euler's method is quite straight forward and easy to program but it requires small sizes for an accurate calculations. The smaller the step-size the large the number of steps and time and computational cost required to achieve a satisfactory answer. Other point-slope methods, such as the Runge-Kutta methods, tend to be more efficient, giving more accurate answers with relatively fewer steps. Euler's method is not very efficient but it is easy to understand as a simple example of a point-slope method. For this method to be useful small steps sizes are required and an adaptive method to adjust the step size. For example, comparing results from two half-step sizes with a full step. Comparing the result after two 0.05 steps with the result after one 0.1 step, halving the steps if the error or difference is too large. Reference Wikipedia entry for Euler's method. Numbers Example to Explore Step-size Adjustment using Euler's Method. 26.2 Runge-Kutta Method Runge-Kutta methods are also point slope methods but were designed to provide improved accuracy with larger stepsizes and without the need to higher differentials (beyond the first derivative) of the function of interest. There are a few more calculations. There is a whole family of Runge-Kutta methods. A commonly used method (in pharmacokinetic programs) is the fourth order. There are four steps in the calculation, that is, for each ‘step’, Figure 26.2.1. Notice the time at which the first derivatives (f terms, i.e. differential equations) are determined. Calculations are made at the initial time, two at half the step-size beyond the initial time and at the final time. These four calculations allow the use of larger overall step-sizes with good accuracy. The method can be represented graphically Figure 26.2.2. Compare the accuracy using the fourth order Runge-Kutta with the accuracy achieved with Euler's method. As with the previous Euler's method example the initial value is 100 and the rate constant is 0.3 hr-1, Table 26.2.1. This seems to be an efficient method of numerically integrating differential equations. However, what step-size should we use? We could repeat the calculation with a smaller step-size, say half the previous value. If the two estimates were within limits we could confirm that they were both satisfactory. The problem is that this would require 8 more calculations. That is the original four calculation plus the additional 8 for the re-calculation with the step-size half the previous value. If the accuracy is not sufficient even more calculations would be needed. One solution to this inefficiency is the Runge-Kutta-Fehlberg method. Runge-Kutta-Fehlberg Method (RKF45) The Runge-Kutta-Fehlberg method (Fehlberg, 1969) takes one additional calculation per step and uses it to determine the appropriate step-size. This makes the method very efficient for ordinary problems of numerical integration. This is the default method I would recommend for use with Boomer. Reference Fehlberg, E. 1969 Low-order Classical Runge-Kutta Formulas with Stepsize Control and Their Application to Some Heat Transfer Problems, NASA Technical Report, NASA TR R-315. Runge-Kutta methods at Wikipedia Runge-Kutta-Fehlberg methods at Wikipedia 26.3 Multi-Step Methods Unlike the point-slope methods, the multi-step methods require more than one previous point to start but they allow even larger step sizes. Although the calculations are more involved they can be quite efficient for certain calculations. The Adams-Bashford method uses the 0, -1, -2 and -3 data point to calculate the +1 value, Figure 26.3.1. Thus, quite large step-sizes can be used accurately. An additional advantage is that only one additional calculation is needed for each new step. The earlier points must be calculated with another method, such as the Runge-Kutta method. A refinement of this method is the Adams-Moulton method which includes a back calculation step to confirm the accuracy of the first step. These methods are called predictor-correcter methods (forward-backward calculations). These method work well with ordinary differential equations. They are very useful for most of problems encountered when integrating differential equations derived from compartmental pharmacokinetic models. However, there are occasions in pharmacokinetics where these previously described methods become very inefficient. This situation is stiff systems. Stiff systems are models where the ratio between the fastest and slowest rate constants is greater than 500 (stiffness ratio > 500). That is kfast/kslow > 500. This is common with physiologically based pharmacokinetic models as described in Chapter 21. Gear's method is an answer to this problem (1,2). Gear's method, a predictor-correcter method, is very efficient for stiff systems. It is quite capable of efficiently working with systems with a stiffness ratio greater than 106. References Gear, C.W. 1971 Algorithm 407 DIFSUB for Solution of Ordinary Differential Equations, Comm. ACM., 14, pp 185-190 Gear, C.W. 1971 The Automatic Integration of Ordinary Differential Equations, Comm. ACM., 14, pp 176-179 26.4 Comparison Between Integration Methods The results in Table 26.4.1 were determined using Boomer some time ago when computers were much slower. All the methods except Gear's method slowed considerably or fail (*) when the stiffness ratio is above 100. Example Boomer .BAT files (and the resulting .OUT files) demonstrating a simulation of a multi-compartment version of the model shown in Table 26.4.1 can be downloaded (Macintosh and Windows). The Runge-Kutta version (rk1.BAT) fails with too many steps when ka > 50. The Runge-Kutta-Fehlberg version (rkf100.BAT) starts to show problems with error messages when ka > 500. Gear's method (gear100.BAT) works with ka = 1.0 x 1012. Notes on the Use of Different Integration Methods with Boomer When differential equations are used in Boomer, by specifying rate constants as a parameter, the user is presented with the selection of an integration method. Method of Numerical Integration 0) Classical 4th order Runge-Kutta 1) Runge-Kutta-Gill 2) Fehlberg RKF45 3) Adams Predictor-Corrector with DIFSUB 4) Gears method for stiff equations with PEDERV 5) Gears method without PEDERV Enter choice (0-5) For ordinary compartmental problems choice 2 (Fehlberg RKF45) should be chosen as the most efficient. With stiff systems choice 5 (Gears method without PEDERV) is best. Each of these numerical integration methods requires that the user specify an upper limit of the acceptable error, either as a relative or as an absolute error. In the case of the Fehlberg RKF45 method both types of error are requested. Enter Relative error term for Numerical integration (0.0001) Enter Absolute error term for Numerical integration (0.0001) For each of the other methods only one error term is required (relative for Classical 4th order Runge-Kutta and the Runge-Kutta-Gill method and absolute for Adams Predictor-Corrector or Gears method). This can be important when editing a Boomer .BAT file. An additional line may need to be added if switching to the Fehlberg RKF45 method or a line may need to removed if switching from the Fehlberg RKF45 method. A final point may need to be made regarding the specification of the absolute error term. At times the default value of 0.001 may be too small. Note that even though the concentration calculated may be quite small, for example, 1 - 10 mg/L the number calculated, internally, may be somewhat higher. For example, with a dose expressed as 1000 mg an absolute error of 0.0001 would mean a change in the 7th significant figure. This may stress the method and thus a larger absolute value may be more appropriate and necessary. The important thing is to consider how realistic the error value might be for the problem being studied. Numerical Integration Methods in Phoenix™ NLME After unchecking the Closed form? Option in the Structure tab, Figure 26.4.1, Phoenix™ NLME provides a number methods for solving differential equations specified using Max ODE in the Run Options tab, Figure 26.4.2. Matrix Exponent Stiff Option using LSODE Auto-detect using LSODA Non-stiff Options using DVERK or DOPRI5 The Advanced Run Options pop-up menu provides the ability to adjust the values of relative tolerance (RTOL), the absolute tolerance (ATOL) and the Maximum number of steps (MAXSTEP). The Matrix Exponent method is the default, when applicable, and is a very efficient method for ordinary pharmacokinetically relevant systems of differential equations. The LSODE method is similar to Gear's method and is most appropriate with stiff systems. The LSODA method switches between the non-stiff Adam's method and a stiff system method similar to Gear's method. Two non-stiff methods are available. The DVERK method is a Runge-Kutta method based on Verner's 5-6 method and the DPRRI5 method is another Runge-Kutta method of order 4-5. References Gear, C.W. 1971 Algorithm 407 DIFSUB for Solution of Ordinary Differential Equations, Comm. ACM., 14, pp 185-190 Gear, C.W. 1971 The Automatic Integration of Ordinary Differential Equations, Comm. ACM., 14, pp 176-179 Hindmarsh, A C, and Petzold, L R. LSODA, Ordinary Differential Equation Solver for Stiff or Non-Stiff System. NEA: N. p., 2005. Web. LSODA, the R Package https://rdrr.io/cran/deSolve/man/lsoda.html based on Fortran code by Linda R. Petzold and Alan C. Hindmarsh. LSODE, the R package https://rdrr.io/cran/deSolve/man/lsoda.html based on Fortran code by Alan C. Hindmarsh and Andrew H. Sherman. Radhakrishnan, K. and Hindmarsh, A. C. 1993 Description and use of LSODE, the Livermore solver for ordinary differential equations. doi:10.2172/15013302. Chapter 27 Optimization Student Objectives for this Chapter Understand the process of optimization Understand how some of the optimization methods (algorithms) work Understand the Advantages and Disadvantages of some these methods How does the computer draw the best-fit line through the data. The objective of nonlinear regression analysis is to reduce the objective function (WSS) by making adjustment in the values of the Parameters of the Model. The advantages and disadvantages of any optimization routine or algorithm can be defined in terms of robustness and speed. A method is robust if it avoid getting lost. Some methods are slower than others. Fast and robust would be best.27.1 Optimization Examples The process of optimization can be illustrated with these spreadsheet examples, Figure 27.1.1, Figure 27.1.2 and Figure 27.1.3. You can download each of these worksheets (actually all in one file) and try your own version of optimization. Change the parameter values and watch the value for the objective function, WSS, change. With a little 'fiddling' you should be able to get close to a best-fit. Try the straight line example first. Download the Numbers - Spreadsheet or Excel - Spreadsheet Note, in contrast to your adjusting the parameter values and seeing the effect the digital computer does this best-fit optimization blind-folded. Before we move onto how the digital computer does this magic lets look briefly at another type of computer. My first computer assisted pharmacokinetic modeling at the University of Kentucky, as a post-doc mentored by LW Dittert, was performed with an analog computer, similar to these Electronic Associates Inc. (EAI TR-20 and TR-10) analog computers, Figure 27.1.4. The pharmacokinetic model was constructed by connecting components with patch cords, not shown. Adjustment of the potentiometers (top on each image) was equivalent to adjusting the rate constants of the model. Output was to an oscilloscope. Optimization was performed by 'twiddling' with the potentiometers and watching the oscilloscope for a match with the data. Great fun! Numerical integration at the speed of light…electrons. 27.2 Optimization Algorithms The digital computer program must move across the Weighted Sum of Squares (WSS) surface to reach the Global Minimum using a predefined method or algorithm without knowing ahead of time the shape of this WSS surface. Note that the WSS is a function of the observed data, x value (time), data weight, constants from the model, and adjustable parameters. For a model with two parameters such as a one compartment model after an IV bolus dose the WSS surface can be represented by a three dimensional surface, Figure 27.2.1. The objective of nonlinear regression analysis is to move from an initial estimate of the parameters to a global minimum value. When can see the WSS surface it is easy to move to the minimum by inspection, however, this ignores the calculations needed to construct the image of the WSS surface. The nonlinear regression program will look for this minimum without constructing the whole surface. The program will use the initial value and information around this value to move to a point closer to the minimum, lower WSS value., Figure 27.2.2. The computer can use a number of algorithms or methods to reach the finish point. Steepest Descent Methods Gauss-Newton Methods Damping Gauss Newton Marquardt Method Nelder Mead (Simplex) Method Grid Search Method 27.3 Steepest Descent Method The steepest descent method use the slope at the initial point and moves down hill according to Equation 27.3.1. Equation 27.3.1 New parameter value Calculated from WSS Surface The computer program can estimate the rate of change of WSS with respect to each parameters (δWSS/δP) by making a small change in each parameter and determining the new WSS. This gives a direction. A linear search in this direction provides the value of h for the lowest WSS. This point becomes the old value and a new search is undertaken. With each iteration the program moves closer to the minimum value of the WSS, Figure 27.3.1.. Advantages of this method Always 'downhill' Avoids 'saddle points' Efficient further from the minimum Disadvantages of this method Slower close to minimum Linear search may cause problems Might 'zigzag' down valleys The steepest descent method is not commonly used on its own to perform a nonlinear least squares best-fit but it does form the basis of another more useful method, Marquardt's method. The linear search, zigzagging down valleys and being so slow near the minimum mean that other methods are more efficient. 27.4 Gauss-Newton Methods The Gauss-Newton method uses a first and second derivative of the change in WSS with parameter value to estimate the direction and distance the program should to go to reach a better point on the WSS surface, Equation 27.4.1 and Equation 27.4.2. Equation 27.4.1 Equation for the next Parameter Value according to the Gauss-Newton Method - One Parameter Equation 27.4.2 Equation for the next Parameter Value according to the Gauss-Newton Method - Multiple Parameters The nonlinear regression program usually performs the single and double differentiations numerically. (For special problems it would be useful to analytically determine equations for these differentiations but this is not generally performed). Numerical differentiation requires the calculation of the WSS with two values of each parameter. The relative step-size between these values is represented by dT in Boomer. The default value is usually satisfactory. Each pass through the equations above provides a single iteration, hopefully getting closer to the global minimum with each iteration. The program needs a criteria to decide when it has converged on the solution and it is time to stop. This convergence criteria, dC, can be expressed as either the smallest fractional change in WSS or in the change in the parameter values. Advantages Relatively efficient (direction and step size determined) Works especially well near the minimum Disadvantage May become lost with poor initial estimates Damping Gauss-Newton Method A major disadvantage of the Gauss-Newton method is that it might get lost. Further away from the minimum, the shape of the WSS surface may be irregular and the differentiation equation may point in the wrong direction or distance. A local shallow slope might suggest the next best point is further away than would be best. A local minimum might even have a slope in the wrong direction leading the method even further away from the best answer. One way to compensate for this possible error is to damp the step-size, reducing the distance by half for each damp. The damping may still not solve all the problems but it can be useful, Figure 27.4.1. Marquardt Method The Marquardt method is another variation of the Gauss-Newton method, Equation 27.4.3. Equation 27.4.3 Equation for Next Parameter value using Marquardt's Method The Marquardt equation is similar to the Gauss-Newton equation above with the addition of the µI term. This term has the effect of allowing the method to act like the steepest descent method further from the global minimum where it is good and like the Gauss-Newton method closer to the answer where this method is better, Figure 27.4.2. All of these Gauss-Newton methods can be quite efficient but the calculations are involved and they can get lost or cause numerical instability. The Damping-Gauss-Newton and the Marquardt methods can be useful and are included in a number of nonlinear regression methods. 27.5 Nelder-Mead (Simplex) Method A totally different method that is quite commonly used in nonlinear regression programs is the Nelder-Mead or Simplex method. It is computationally quite simple, other than the calculation of the WSS. The method works with a number of rules. The starting point is used to construct a simplex, a shape with m+1 points, where m is the number of parameters. Thus for a two parameter problem there are three points, a triangle. The program calculates the WSS at each point of the simplex on the WSS surface. The Rules Reflect the point with the highest WSS through centroid (center) of the simplex If this produces the lowest WSS (best point) expand the simplex and reflect further If this is just a good point start at the top and reflect again If this the highest WSS (worst point) compress the simplex and reflect closer These rules are repeated until the convergence criteria are meet. The simplex moves over WSS surface and should contracts around the vminimum, Figure 27.5.1. The simplex method is relatively robust and numerically less complicated but it can be inefficient (slow) for simple problems. This method is recommended as a starting point with Boomer. Actually the method I use most of the time involves starting with the simplex method and (automatically) continuing with the Damping-Gauss-Newton method. It seems work well much of the time. 27.6 Grid Search Method The grid search method simply determine the WSS at each point on the grid of parameter values. It may offer some protection against local minima but it is not very efficient. As more parameters are included in the model the number of determinations can be excessive. With three parameters and 10 points per grid, 10 x 10 x 10 determinations are required. Example Grid Search Result Using Boomer and gnuplot (installed via https://brew.sh) the following grid search animated gif could be created, Video 27.6.1. Using Boomer (https://www.boomer.org/boomer/) select Grid Search as the optimization method and set upper and lower limits on each parameters to optimize and the number of grids. Boomer will output a .WSS file containing the WSS values at each grid point and a .DAT file containing the same information in a format for gnuplot. The .dem file can be used with gunplot to create an animated gif for each pair of parameters using the command gnuplot abc0x_0y.dem The .dem plot can be edited to adjust options available to experienced gnuplot users. 27.7 Optimization Methods Notes on Optimization with Boomer When setting up an optimization with Boomer upper and lower limits are required for each adjustable, fitted parameter. With good data and model you might start with lower limits half or a tenth of the initial estimate and upper limits twice or tens time the initial estimate. For apparent volumes of distribution a value of zero during optimization will produce a divide by zero error so a non-zero lower limit should be entered. For rate constants a lower limit of zero might be quite useful. You should start with relatively narrow limits and expand them only when appropriate and a warning of 'close to limits' is displayed in the output. With more difficult problems it might be useful to constrain the more well estimated parameters with very tight limits, recognizing that you will hit limits on this run but may get useful initial values for less well defined parameters to use on the next run. The upper and lower limits are hard limits, rather like the walls of a box. They are not constraints until a limit is hit. You should also consider using different initial estimates to avoid acceptance of convergence to a local minimum. The simplex or simplex-DGN method described below can be useful in helping to find a global minimum. You should also carefully review the output plots for evidence of local, non-global minima. The damping Gauss-Newton method provides progress output. Loop = 1 Damp = 1 P ( 1) = .5808 P ( 2) = .1106 P ( 3) = 2.101 P ( 4) = 39.64 WSS = 55.5363 Loop = 2 Damp = 1 P ( 1) = .4375 P ( 2) = .1319 P ( 3) = 2.199 P ( 4) = 42.54 WSS = 10.6959 Each iteration is represented as a loop number. Damp numbers higher than 2 or 3 indicate that the surface may be irregular. The Nelder-Mead (Simplex) method also provides progress output. Loop 1 - 1> 717.5 2> 643.2 3> 386.5 4> 973.2 5> 797.9 Loop 2 - 1> 717.5 2> 643.2 3> 386.5 4> 127.4 5> 797.9 Loop 3 - 1> 717.5 2> 643.2 3> 386.5 4> 127.4 5> 120.6 Loop 4 - 1> 58.02 2> 643.2 3> 386.5 4> 127.4 5> 120.6 Loop 5 - 1> 58.02 2> 38.82 3> 386.5 4> 127.4 5> 120.6 From line to line there should be one value, the highest, WSS reduced from the previous line. If selecting the Simplex or Simplex->Damping GN method a convergence criteria (PC) must be entered. If the Gauss-Newton, Damping Gauss-Newton, or Marquardt method is selected a step-size (DT value) must entered as well. When editing a .BAT file there is one more line with these methods. The DT value specifies the step-size used to create the partial differentials used in the optimization process (Jacobian). In both cases (PC and DT) the default values usually work well. FITTING METHODS 0) Gauss-Newton 1) Damping Gauss-Newton 2) Marquardt 3) Simplex 4) Simplex->Damping GN Enter Choice (0-4) 4 Enter PC for convergence (0.00001) .000 or FITTING METHODS 0) Gauss-Newton 1) Damping Gauss-Newton 2) Marquardt 3) Simplex 4) Simplex->Damping GN Enter Choice (0-4) 1 Enter DT for Jacobian (0.001) .000 Enter PC for convergence (0.00001) .000 The first three methods are numerically intensive, relative to the simplex method, thus there may be occasions when poorly defined calculations may produce 'fatal' errors, for example divide by zero or other math error. Better initial estimates or use of the less numerically intensive simplex method may be useful. Repeated Runs using the Simplex Method Another advantage of the simplex method as implemented in Boomer is that once the initial estimates (first point on the simplex) are specified the other points on the simplex are randomly generated. This means that each time a problem is run a different initial simplex is used and effectively different initial estimates are used with each run. This can be set-up automatically by adding a line after the first and entering the number of runs to be attempted. Consider, Boomer Batch File 5 wls,bayes,sim,irwls,sim+error,grid 1 Screen, diskfile outfname 1 Parameter type Dose 100.0 Parameter value If the -2 on the last line is replaced with a -4 and followed, on a new line, with the name of the xxxx.BAT file without the .BAT, Boomer will rerun the .BAT file repeatedly using a different initial simplex each time until the user stops the program with a control-C (or similar). Boomer Batch File 20 5 wls,bayes,sim,irwls,sim+error,grid 1 Screen, diskfile outfname 1 Parameter type Dose 100.0 Parameter value The .OUT file will now contain the results of multiple runs and the WSS (and other results) from each run can be examined. Typically, hopefully, there will be a majority of runs that converge to the same 'global' minimum with only a few that might stops with higher WSS values. Notes on Optimization with Phoenix™ The optimization methods available in Phoenix™ include: FOCE Lindstrom-Bates (FOCE L-B) FOCE ELS FO engine Naive pooled engine. For population data this method treats all the data as from a single subject. Laplacian QRPEM Adaptive Gaussian quadrature NonParametric engine The optimization method is specified on the Run Options tab, Figure 27.7.1. More details can be found in the Maximum Likelihood Models User's Guide. With Phoenix™, the parameters are transformed so upper and lower limits are normally not required. In some limited cases such as negative "best-fit" values for clearance or rate constant a lower limit of zero may be needed. When using the QRPEM or Naive pooled methods a lower limit of zero is set by the program. Chapter 28: Weighting Data Student Objectives for this Chapter To understand why all data are not equal To understand measures of data accuracy and how they relate to weighting To understand how to use different weighting schemes To understand why different weighting schemes might be used Not all data are the same. Not even data values within an experiment or set are the same. I don't mean the same value that is obvious but the uncertainty or error in the data. Each data point has a specified measured value but for that value to have any meaning we must also know the uncertainty or error in the data value. An estimate of the standard deviation or variance. In this way we have an estimate of the precision of the data value. When these data are modeled the measure of uncertainty should be included in the analysis. This uncertainty is expressed by means of a weight (or emphasis) given to each data point.28.1 Weighting Methods We often don't have a good estimate of the variance (or standard deviation) for each data point. However, we may have a general idea of way the variance changes with the magnitude of the data. This provides a way to estimate the variance for each data point before or during non linear regression analysis. Weighting schemes Equal Weight Reciprocal Variance Weighting Schemes Iteratively Reweighted Least Squares (IRWLS) Extended Least Squares (ELS) With some data sets the variance is very similar and equal weights might be applied to all the data points. In most cases however, each data point estimate will have an estimate of its variance. This can be simplified by selection of a weighting scheme which may apply to the data set under consideration. The weighting scheme can be used to adjust the weight for each data point during the analysis using the iteratively reweighted least squares. The actual parameters of the weighting scheme are evaluated along with the model parameters with extended least squares. 28.2 Equal Weight Although the variance of each data point should be considered there are situations when an equal weight should or could be applied to each data point. When the data values are similar it can be expected that the variance is similar. Thus, if the spread of data values is small the variance may be similar and an equal weight scheme may be useful, Figure 28.2.1. Alternately, if the data can be determined very precisely the choice of weighting scheme may not be as important and an equal weight scheme may be useful, Figure 28.2.2. The range in concentration values in Figure 29.2.1 is quite modest. These may be analyzed successfully using equal weight for each data point. The deficiencies associated with the equal weight scheme are more pronounced when trying to fit more than one data set simultaneously. In these cases, careful consideration of weighting between and within the data sets is important. 28.3 Reciprocal Variance With most data sets, specification of appropriate weight for each data point or weight scheme for a data set are recommended. The best weight is the reciprocal of the variance of the data point. This value may be available for each point or may be available as an a equation for a data set. Large range of data values, variable error in the data or relatively large error in the data are all good reasons for considering applying a weight to each data point. It is essential to apply an appropriate weight to data points when different data sets are being fit simultaneously. This is especially true when the data sets have different values. There is a considerable range of concentration values in Figure 29.4.1. Weighting these data points properly during non linear regression analysis would be an important part of the process. When data values in different data set (fit simultaneously) are quite different in magnitude appropriate weighting is essential. In Figure 29.4.2 the magnitude of the urine data is much higher than the plasma data. If all the data were weighted equally the urine data would be fit well at the detriment of the fit to the plasma data. we could reverse this result by re-scaling the urine data to units of gram instead of mg. The now, smaller magnitude of the urine data would cause these data to be undervalued during any non linear regression analysis. The answer is to appropriately weight each data set, each data point. If we know the variance of each data point we can simply calculate the weight for each data point, Equation 28.3.1. However, this is not always possible but we might know the way in which the variance varies with the value of the data. That is, the relationship or equation relating the value of the data point to the variance and this the weight. These relationships or weighting schemes may take a number of forms. Equation 28.3.1 Ideally the Weight for each data point will be equal or proportional to the Variance of the Data Point 28.4 Weighting Schemes Much of the material on this page is Boomer specific but most of the weighting schemes are available with other non linear regression programs. Equal Weight In its simplest form equal weighting mean that each data point is assigned a weight of 1. Weight Proportional to the Inverse of the Value In this case the variance in the data point is proportional to the value of the data point. This approach or method is quite useful when the data has been determined by measuring radioactivity counts, Equation 28.4.1. Equation 28.4.1 Variance Proportional to the Observed Data Weight Proportional to the Inverse of the Square of the Value More commonly data are measured by methods, such as HPLC, that provide standard deviation values that are proportional to the value. That is, the coefficient of variation (CV) of these data are similar. This can be a reasonable approach as long as the limits of assay sensitivity are not approached too closely, Equation 28.4.2. Equation 28.4.2 Variance Proportional to the Square of the Observed Data The effect of different weighting schemes on the result of non linear regression are illustrated in Figure 28.4.1. None of the fitted lines in Figure 28.4.1 are particularly good, maybe another model is required, but the effect of weighting scheme can be readily determined. The equal weight scheme fits the high data points well. In contrast the weight scheme proportional to the value squared fit the low data points. A better model with a suitable weighting scheme should provide a much better fit. Variance Equal to a x Observed Valueb A more general approach to defining the weighting scheme is to use the scheme shown as Equation 28.4.3. This approach was described by Wagner (1975). Equation 28.4.3 Variance as a Function of Observed Data Values of the parameter a and b may be determined from the data or knowledge of the assay method. A plot of data variance versus observed value on log-log graph paper should reveal a straight line. The slope and intercept of this line can be used to determine suitable values for a and b. Alternately, if this information is not available, that is the variance of each data point, it might be possible estimate a value. (Of course if this information was available either as a variance value or standard deviation an appropriate weight could be calculated directly). One approach to estimating a suitable weight might be to fit the data with an arbitrary polynomial or other suitable empirical function. The objective would be to put a smooth line through the data without regard to the 'theoretical' model. Deviations from this arbitrary line might be used to estimate the standard deviation and thus the variance of the data with respect to the data value. This approach can be explored with the spreadsheet above in Figure 28.4.2. Download as a NumberTM spreadsheet. Variance Equal to cb + a x Observed Valueb Although the weighting scheme where variance is a x Observed Datab can be very useful in describing the constant CV part of assay standard curve with smaller data values the CV can rise dramatically. The inclusion of an additional term representing the assay sensitivity (c) can compensate for this increase in CV Equation 28.4.4. Equation 28.4.4 Variance as a Function of Observed Value with an Assay Sensitivity Term This weighting scheme provides information about the error in the data at both low and high concentrations. Further extension of this type of scheme could include other polynomial terms and further refinement. Variance Estimate taking the 'Age' of the Data into Account When data are collected over a period of time, such clinical samples collected over a few days or weeks it may be useful to discount older samples. This can be achieved by including sample time in the variance equation or weighting scheme, Equation 28.4.5. Equation 28.4.5 Variance as a Function of Observed Data with Sample Time Information As the difference between a sample time and the last sample time, tlast, becomes larger the weight for that data point becomes smaller. Therefore older data points, collected when patient status may be different, are given less emphasis in the current analysis. A value 1.002 for c (with t in hours) can serve as a starting point and discounts samples older than 24 hours by 5%. The above equations are just a collection of possible forms for the variance and thus weight equation that might be useful. Various software packages will have similar and other options for weighting the data. Interactive 28.4.1 illustrates some of these options. References Wagner, J.G. 1975 Fundamentals of Clinical Pharmacokinetics, Drug Intelligence Publications, Hamilton, IL, page 289 28.5 Iteratively Reweighted Least Squares (IRWLS) On the previous page all of the weighting schemes involved variance as a function of the value of the observed data point, except for the equal weight scheme. This is computationally efficient since each weight only needs to be calculated once at the beginning of the optimization step. It also ties the weight to the variance of the observed data points. However, when a weighting scheme is applied to a series of data points, data points with very low values may be given more emphasis than is appropriate. An alternative weighting has been used to try to overcome this disadvantage. This method is the iteratively reweighted least squares method. Effectively, this is identical to the methods on the previous pages except that observed data is replaced with calculated data. Thus, the weight is recalculated during each phase of the optimization process. Thus very low observed data points would not have the emphasis on the overall analysis. However, it is possible that the optimization may drive calculated values low, giving these points more emphasis and potentially distorting the final analysis. Weighting Schemes Based on IRWLS Equation 28.5.1 Variance Proportional to Calculated Data Equation 28.5.2 Variance Proportional to the Square of Calculated Data Equation 28.5.3 Variance as a Function of Calculated Value Equation 28.5.4 Variance as a Function of Calculated Value with an Assay Sensitivity Term Equation 28.5.5 Variance as a Function of Calculated Value with Sample Time Information The above equations are just a collection of possible forms for the variance and thus weight equation that might be useful. Various software packages will have similar and other options for weighting the data. References Peck, C.C., Sheiner, L.B. and Nichols, A.I. 1984 The Problem of Choosing Weights in Nonlinear Regression Analysis of Pharmacokinetic Data, Drug Metabolism Reviews, 15, pp 133-148 28.6 Extended Least Squares (ELS) Another approach to choosing the most appropriate weighting scheme is to fit parameters of the variance model during the optimization process. The form of the variance versus data equation needs to be determined. Once the form and parameters involved are determined the optimization process can determine the variance and model parameters given good initial estimates and enough good data. There are some differences. Since there are more parameters more data are necessary. Different algorithms are necessary for the optimization. A different objective function, the function minimized during the optimization, is needed and different optimization algorithms are necessary, Equation 28.6.1. The programs Phoenix™ and ADAPT include these algorithms. Equation 28.6.1 Objective Function for Extended Least Squares Optimization Note the inclusion of the lnV term in this Equation. This will prevent the optimization algorithm from driving the value of V high and thus the weight low to minimize the objective function (WSS) without regard for the model parameter values. vp is variance parameter value. The equation for V, the variance of each data point, may take a number of forms. Some of these are shown in Equation 28.6.2. Equation 28.6.2 Example Variance Equations References Peck, C.C., Sheiner, L.B. and Nichols, A.I. 1984 The Problem of Choosing Weights in Nonlinear Regression Analysis of Pharmacokinetic Data, Drug Metabolism Reviews, 15, pp 133-148 Peck, C.C., Beal, S.L., Sheiner, L.B. and Nichols, A.I. 1984 Extended Least Squares Nonlinear Regression: A Possible Solution to the 'Choice of Weights' Problem in Analysis of Individual Pharmacokinetic Data, J. Pharmcokin. Biopharm., 12, pp 545-558 28.7 Combining Individual and Population Data Once a drug is on the market and even before release pharmacokinetic models and parameter values should be well developed. Linear model or nonlinear. Numerous covariates would have been studied. This information can be very useful for developing suitable dosing regimens. In some cases customized for individual patients. One example is the development of dosing regimens for warfarin on warfarindosing.org [Apr23]. Other examples are included in resources such as FDA: Find Information about a Drug [Apr23] and NIH-NIAID: Search for Package Inserts [Apr23]. A drug nomogram may be used to determine an initial dosing regimen. The calculation of this initial dosing regimen may be based on the patient weight or height. Other covariates such as renal function, cardiac function, liver status or smoking status might be considered. Mullen (1978) proposed a direct linear plot method (Figure 20.3.1) using data collected after two phenytoin steady state dosing regimens. Another approach is the use of therapeutic drug monitoring (TDM). Briefly, this involves collecting one or more blood samples from the patient after their initial dose or dosing regimen. These samples are quickly assayed and analyzed for pharmacokinetic parameters. These parameter values may be supported by Bayesian estimation which includes population parameter values and their uncertainty (standard deviation or variance) as well as the individual subject data in the analysis. Bayesian Analysis Bayesian estimation of pharmacokinetic parameters values is similar to ordinary nonlinear regression with a few differences. Prior information of the pharmacokinetic model Populations values and their uncertainty and Some data from the individual patient of interest is necessary A modified objective function is used for this analysis containing components related to the data and the population values, Equation 28.7.1. Equation 28.7.1 Objective Function - Bayesian Estimation where CalcD and ObsD are the calculated and observed concentration values. CalcP and PopP are the calculated and observed population parameter values. DVariance and PVariance are the variance in the concentration values and the population parameter values. Pharmacokinetic programs such as Boomer optimizes the objective function by calculating concentration values by adjusting the pharmacokinetic parameter values, CalcP. Note, the calculated concentration values are a function of the parameter values and time. The optimization procedure minimizes both the concentration residual between observed and calculated concentration and parameter value residual between the estimated parameter and the population parameter values. References Mullen, P. W. 1978 Optimal phenytoin therapy: a new technique for individualizing dosage. Clin. Pharmacol. Ther., 23(2), 228–232 Chapter 29: Selection of the "best" model Student Objectives for this Chapter Evaluate various graphical plots produced by non-linear regression program Evaluate various output tables including parameter values and variability produced by non-linear regression program Calculate and evaluate statistical parameters such as AIC and F-test Evaluate various criteria and determine the best most appropriate model Modeling of pharmacokinetic data does not mean a single run of a non-linear regression model and automatic acceptance of the output. The output must be evaluated carefully. There may be errors or there may be systematic deviations from a 'best'-fit. The wrong weighting scheme may have been used. [See Weighting Scheme for details about weighting schemes.] The chosen model may not be the best for the data set(s) under investigation. Certain systematic deviations in the graphical output from non-linear regression programs may support or refute a particular model. The reported parameter variability can provide useful information. There are a number of statistical 'tests', such as AIC and F-test, which can be very informative. 29.1 The Best Model The best model may be developed from theoretical consideration of the system under study. In other case the model may be empirical, that is based on the data provided. In other case there may be a mixture such as a general theoretical description with the detail defined by the available data. Many pharmacokinetic models may be considered to be mixtures. 'Classical' compartmental models, such as the one compartment model, have some theoretical basis in terms of drug absorption, distribution, metabolism and excretion but the detail will depend on the data collected. Early data collected after an IV bolus may support a multi-compartment pharmacokinetic model (see Chapter 19 for more information). Without the early data or if the data had more error then a simpler model may be all that can be supported. Graphical, parameter variability and statistical criteria can be used to decide on the 'best' model. Occam's razor, the principle of parsimony or the KISS principle suggest that the smaller model is better. The model with the least adjustable parameters. This is the basis of the statistical tests, AIC and F-test. Does the more complicated model provide statistically significant improvements in the fit? Can the parameters be determined accurately, consistently? With insufficient data (samples/sites) or more complex models it may not be possible to identify, that is determine all the adjustable parameters in the model. Identifiability is described in more detail in the next chapter. The best model also depend on how it is to be used. A simple model may be satisfactory. A more complex model may be required so the details can be explored. Is the model too big? Too small? 29.2 Evaluation of Graphical Output Subjective evaluation of graphical output such as the calculated and observed data versus time plot and the weighted residual plots can be very useful in the determination of the best model. Calculated and Observed Data versus Time Plots Look for systematic deviations. Could the deviations be explained with an addition to the model? Is a distribution phase obvious. Are there straight lines portion on semi-log plots, linear plots? Combining data from different sample sites, such as plasma and urine data, in one analysis will increase the model size and complexity, Figure 29.2.1. Note the systemic, significant deviations between the line labeled 'Calculated (1c)'. This line represents the 'best' fit to a one compartment model. The two compartment (2c) model represents a much better fit to the observed data. Weighted Residual Plots Another plot that can be very useful in determining the 'best' model is the weighted residual plot. Figure 29.2.2. A pattern in the weighted residual plot may indicate that the model used is too small. In the case of Figure 29.2.2 there is a definite 'U' or 'V' shape suggesting that a bigger model (with two more parameters) should be investigated. A diagonal line would suggest a model with one additional adjustable parameter could be useful. 29.3 Parameter Variability Another criteria that can help in deciding if you have the right model is the reported variability in the parameter values. Most non linear regression model will provide some indication of parameter variability. This might include standard deviation, confidence interval or coefficient of variation. These numbers may be reported as zero, some reasonable number or a very large number or even NaN. Anything other than a reasonable number may mean a serious problem with the model. A value of zero is not good. A low number is OK but 0 usually means some computational problem, including wrong model, weighting scheme, insufficient data. If the coefficient of variation (CV) is below 10% for all your parameters then your model is probably doing quite well. Occasionally one or two or the parameters may have higher CV values. This indicates that the program is not able to provide more precise estimates for these parameters. Maybe these parameters are not necessary. As the CV increases above 15%, above 25%, greater than 100% you are provided with the evidence that the model may be too large. You should consider looking at a smaller model, removing or replacing parameters with larger CV values. Some non linear regression programs provide a table of correlation coefficients between the adjustable parameters of the model. A high correlation (> 0.90 or > 0.95) between two parameters suggests that one parameter may replace both or that an otherwise smaller model maybe worth exploring. 29.4 Statistical Criteria A number of statistical criteria can be used to determine the 'best' model. These may appear to be superior to graphical and parameter variability criteria because they are less subjective, however all these criteria should be used together in making your decision about the best model. Akaike's Information Criterion, AIC An AIC value can be calculated from the final fit data and used as a measure of which model is best. In Equation 30.4.1, N is the number of data points, excluding any data with zero weight (that might be used as place markers at time zero or when a dose is given). M is the number of adjustable parameters and 2LL is twice the loglikelihood. Equation 29.4.1 Akaike's Information criterion, AIC When comparing two or more models the lowest number is 'best'. That is, the LOWEST number on the number line from plus to minus infinity. For example, 1.0 is better than 10.0 0.1 is better than 1.0 -1.0 is better than 0.1 -10.0 is better than -1.0 Notice the inclusion of WSS or weighted sum of squared residual in Equation 29.4.1. It is VERY important that when a comparison is made that the weighting scheme used in the analysis of each model is the same. If you change the weighting scheme the weights and thus the WSS value is changed too. Use the same weighting scheme! This does not mean you can't try other weighting scheme but you must be careful with the comparison. Consider the matrix, Table 29.4.1. Comparison of weighting schemes can be made down the table on either side. Determination of the best model must be made across the table, NOT down. As an example the analysis of a data set containing 12, Table 29.4.2 data points using three different models with the same weighting scheme produced the results. In this example the lowest AIC is -22.8 for the two compartment model. F-Test A second statistical criterion. I idea is to test whether the increase in the number of parameters has produced a significant improvement in the fit. Again, the same weighting scheme must be used in this comparison. The F value is calculated and compared with tabled values usually at the 5% significance level. Equation 29.4.2 F Value In Equation 29.4.2 df is the degrees of freedom calculated as N - M (see above). The indexes j and k refer to the two models being compared. The model with the higher number of parameters is indexed as k. Where n = dfj - dfk and m = dfk The idea is to compare the calculated value with the tabled value. A calculated value higher than the table value supports the larger model as being significantly better. In Table 29.4.3 the two compartment model is better than the one compartment model since 101 is larger than 4.46. In comparison, the three compartment offers no significant improvement over the two compartment model since the calculated value, 0, is less than the tabled value, 5.14. Bayesian Information Criteria (BIC) or Schwarz's Criteria (SC) Another criteria that has been used for the comparison between pharmacokinetic models is the BIC or SC. This criteria can be estimated as a function of N, M and WSS, Equation 29.4.3. Equation 29.4.3 Bayesian Information Criteria, BIC Selecting the best model takes care and experience. There are a variety of criteria that must be considered. Not just the numerical criteria but also the graphical output and the biological appropriateness of the model and parameters should be considered. References Akaike, H. 1974 A new look at the Statistical Model Identification, IEEE Trans. Automat. Control, 19, 716-723 Bonate, Peter H. 2006 Pharmacokinetic-Pharmacodynamic Modeling and Simulation, Springer, NY, NY. Especially Chapter 1 The Art of Modeling. Chapter 30: Identifiability Student Objectives for this Chapter To understand the problem of model identifiability To recognize some common examples and types of identifiability problems To use some of the techniques that could be used to recognize identifiability problems In the earlier Chapter we considered the selection of the best model. We were able to compare model using the results from analysis using non-linear regression programs. If the model is too big, too many parameters, then we expect increased uncertainty in the parameter values determined by the program. It is possible to build model that are big enough that some parameters values can not be determined at all. These parameters can be called non identifiable. Parameters that can be determined, even if with some considerable variability, are essentially identifiable. This topic will be discussed by considering definitions, some examples, and numerical and analytical techniques that can be used to explore the identifiability of model parameters. 30.1 Identifiability - Definitions The identifiability problem or question is: Can all the parameters in the model be estimated accurately with the data provided? Parameters can be described as identifiable, non-identifiable or non-observable. Identifiable parameters are those which effect the value of the data and can be estimated with some degree of certainty. Non-identifiable parameters are those which effect the value of the data but which cannot be estimated accurately Non-observable parameters are those which don't have an effect on the data. Parameters can be defined using the model shown in Figure 30.1.1. Note that only one sample site is used to provide data. Drug concentration in plasma (or blood). The apparent volume of distribution, V, is identifiable. We can determine a good value of V from the Y-intercept of the data plotted on semi-log graph paper. The parameter, kel (not shown but equal to ke plus km) is identifiable from the slope of line on the semi-log plot. Note that changing either V or kel will change the value of the data collected, Cp. Although kel can be readily determined, neither ke nor km can be determined separately. Thus these parameters can be termed non-identifiability. They are observable because if we change the value of either of these parameters the data, Cp, will be altered. There are two other parameters in this model. Both kmu and Vm are non-identifiable and they have not effect on the data. They are both non-observable. Identifiability can also be define in terms of global or local identifiability. A parameter is globally identifiable or non-identifiable if this is independent of dose or scale. Local identifiability refers to identifiability dependent on dose or scale. For example, if a parameter is identifiable at one dose level but is non-identifiable at another dose level then this would be a local identifiability.30.2 Identifiability - Examples Identifiability problems arise from a number of sources: The selected model has too many parameters Data is selected from limited sample sites The dose or concentration levels are less than ideal Samples times are not appropriate Problems 1 and 2 are related. As the model becomes more complex samples from additional sites may be necessary to maintain the identifiability of the model parameters. Problems 3 and 4 can also be prevented with more careful development of the experimental design. A major advantage of considering identifiability before the experiments are carried out. Thus, the experimental design can be confirmed or improved before conducting expensive experiments. After the fact, consideration of identifiability allows the analyst to reject complex models for data provided. It can let you know why 'the wheels are spinning' and you aren't getting anywhere with a modeling exercise. Too Many Parameters Consider the model in Figure 30.2.1. In Figure 30.2.1 only the drug concentrations in the central (plasma or blood) compartment is collected, represented by '*'. An equation which describes this model the bi-exponential equation, Equation 30.2.1 Equation 30.2.1 Concentration after Oral Administration With good data it is possible to estimate ka and kel from the terminal slope and the residual line as described in Chapter 8. The intercept from both these lines provides A, Equation 30.2.2. Equation 30.2.2 Intercept from the Method of Residuals From Equation 30.2.2 we can see that all the components of the right hand side are known (Dose, kel and ka) except V and F. Thus, the ratio V/F or F/V can be determined or identified but not either parameter separately. If we are to fit drug concentration in plasma or blood after oral administration data with this model we could identify ka, kel, and V/F but not V or F. Additional data is needed. Maybe drug concentration data collected after IV administration. Sample Site Selection Consider the model in Figure 30.2.2. Note the samples collected are indicated by the '*'. For example, if only blood was collected but analyzed for drug and metabolite there would be data for these two components of the model. This will limited the number of parameters which can be identified. This can be illustrated by looking at drug concentrations or amounts calculated using this model. Note the data shown in Table 30.2.1 and Table 30.2.2. In both tables the data calculated for Cp and Cm are identical even though quite different values of ke were used. The values of km and Vm were different to compensate but we can see that different values of ke could be determined from these data points. The only difference is in the U values. This indicates that a number of parameters including ke, km and Vm may not be identifiable when drug and metabolite concentration in blood or plasma are determined. However, collection of drug amounts in urine has the potential of making these parameters identifiable. This can be further illustrated by plotting these data, Figure 30.2.3 and Figure 30.2.4. Note that in Figure 30.2.3 there is only one line for Cp and Cm, respectively. In Figure 30.2.4 there are two lines for U for the two values of ke. If only Cp and Cm are collected ke is non-identifiable and non-observable. Dose Level Selection Data collected after different dose levels can provide different information about pharmacokinetic parameters. This can lead to local non-identifiability. An example of this behavior can be seen with data collected (or simulated) with drugs which exhibit non-linear pharmacokinetics. In the simplest case the rate of elimination can be expressed using Michaelis-Menten parameters, Vmax and Km, Equation 30.2.3. Equation 30.2.3 Rate of Elimination by Michaelis-Menten Kinetics In Figure 30.2.5 three lines are shown after three different doses. This is a semi-log plot. Note the lowest line appears to be a straight line looking like linear kinetics. It would be expected that only a kel value and not Vmax and Km values could be identified from this line of data points. The data collected at higher doses may be more useful in the determination of the Michaelis-Menten parameters. data from all three doses modeled simultaneously would be even better. Sample Time Selection Samples must be calculated at the best times to gain the best information about pharmacokinetic parameter values. The chapter on Optimal Sampling (Chapter 32) provides more information on determining the best time to sample to obtain good estimates of parameter values. For example if early data points are not available it can be difficult to determine good values of faster rate constants. The equation for drug concentration after oral administration contains to rate constants, kel and ka, Equation 30.2.1. Equation 30.2.1 Concentration after Oral Administration If ka is faster than kel then missing early time points can mean that the parameter ka is not locally identifiable. For example, if samples are only collected from four hours on as shown in Figure 30.2.6 ka will be poorly estimated. ka/kel Flip-Flop with Oral Dosing - Determining V and F While analyzing concentration versus time data after oral administration we often assume that the absorption rate constant, ka, is larger than the elimination rate constant, kel. When using the method of residuals to estimate ka and kel we expect that ka is at least five times larger than kel. This is not always the case, especially with slow release dosage forms or drugs with rapid elimination. Given only oral dose concentration versus time data it is not possible to determine which is faster ka or kel. This is another example of global non-identifiable parameters. Notice in Tables 30.2.3 and 30.2.4 the calculated concentrations are the same at each time point despite the differences in ka and kel values and the value for V. It should also be noted that with only oral data as presented in these two tables only a value for V/F is identifiable. Neither V or F can be determined. These identifiability problems can be easily solved with prior knowledge of the kel and F values from analysis of intravenous data or simultaneously analyzing iv and oral data at the same time. 30.3 Identifiability - Numerical Approaches There are a number of approaches to detecting or confirming identifiability problems with a model or sample selection. These methods can be grouped as numerical and analytical.. Empirical Method The idea behind this method is that you should be able to fit simulated data and determine the original parameter values well if the parameters are identifiable. Repeated fitting of these data should result in the same parameter values if the parameters are identifiable. The method includes: simulation of data for the sample sites and dose levels expected include small errors include more sample times within the expected sampling scheme refit the model with the simulated data fit the model multiple times use different initial estimates review the output for consistent results Example 1 - Drug and Metabolite Model Consider the model shown in Figure 30.3.1. Data were simulated using Boomer for drug and metabolite concentrations are shown in Table 30.3.1. The data were then fitted to the model in Figure 31.4.1 using Boomer. Table 30.3.2 shows just one result from fitting the data in Table 30.3.2. Note that the CV% values look good for most of the parameters. This generally indicates a good fit and well defined parameters. However, the value of zero for V2 (= Vm) is not a good sign. The real problem becomes apparent when the model is fit multiple times with the same data and different starting points. Now we see a variety of values for ke, km and Vm. Curiously if the best-fit ke and km values are plotted it can be seen that they fall on a line indicating a common value of kel (= ke + km), Figure 30.3.2. The blue squares represent the starting points used for each fit and the red circle represents the final, best-fit, values. Note that all the red circles fall on a straight line representing the value of kel (= ke + km). This indicates the neither ke nor km are identifiable but does suggest that kel is identifiable. Example 2 - Michaelis-Menten Model We can repeat the exercise above, with the Michaelis-Menten model described in the previous section, one dose at time. The results from high (500 mg) and low dose (5 mg) data are shown below in Figure 30.3.3 and Figure 30.3.4. Notice that the CV% for the low dose are quite large (720%) indicating that Vm (=Vmax) and Km are not identifiable with these low dose data. Note also that the parameters have hit reasonable upper limits. Increasing the limits resulted in the values increasing to the new limits. The value for the parameter CV% using the high dose data are much better. The final parameter values for all three estimated parameters are close to the starting values. This indicates that if data as good as the simulated data, at the higher dose, were available then these parameters would be identifiable. Example 3 - Oral Administration Model The final example demonstrates the effect of missing early data points after oral administration, shown in Figure 30.2.6. Notice the high value for CV% for the ka parameter value compared with the CV% for kel and V, in Figure 30.3.5. Clearly ka is not well defined. 30.4 Identifiability - Analytical Approaches As with the numerical methods there are a number of analytical approach to investigating identifiability problems. There is a Laplace transform method and a Taylor series approach. Although the Taylor series method is more general in application it is more cumbersome and won't be described and further at this time. Laplace Transform Method The Laplace transform method is less complex to apply but it doesn't work with time-varying or non-linear systems. Fortunately, most pharmacokinetics systems contain parameters that do not vary with time and linear models are more common. Using a model previously explored, the excretion and metabolism model, Figure 30.4.1, we can work through the Laplace transform method. First let us consider what we can determine (identify) if we measure just the drug concentration in plasma or blood. Starting with the differential equation for this model component and taking the Laplace provides Equation 30.4,1. Equation 30.4.1 Laplace of Drug Concentration in Plasma or Blood Note that this equation includes what could be called an intensity term (Dose/V1) and one 's' term (s + ke + km). The intensity term can be evaluated from an intercept and the 's' term from a slope. Since we know 'Dose' the parameter V1 can be identified from the intercept. The slope provides information about ke + km (= kel) but not ke or km on their own. Thus, if we measure C1 two parameters are identifiable, V1 and kel. The parameters ke, km, kmu and V3 are either non-identifiable or non-observable. Next consider if we collect another sample, metabolite concentration in blood or plasma (C3), what additional parameters can be identified from the Laplace of this sample, Equation 30.4.2? Equation 30.4.2 Laplace of Metabolite Concentration in Plasma or Blood The intensity term is (Dose • km)/V3 but because we don't know km or V3 separately we can't identify either parameter. There are two 's' terms, (ke + km) and kmu. We still can't separate ke or km, however kmu can now be identified, from the slope of C3 versus time. If we collect one more sample, metabolite amount in urine (X4), more information may be available from Equation 30.4.3. Equation 30.4.3 Laplace of Metabolite Amount in Urine The intensity factor (km • kmu • Dose) provides information about km since kmu and Dose are known. There are three 's' terms, s, (s + ke + km) and (s+ kmu). Since we now know km we can identify ke from the second 's' term. Finally we can go back to the metabolite in blood or plasma data and use the intensity factor to estimate V3 since we now know km. With good drug in plasma and metabolite in plasma and urine data we should be able to estimate all the parameters of the model shown in Figure 30.4.1. A similar approach could be followed with other models to determine the samples that must be collected to identify or estimate the model parameters. References Godfrey, K.R. and Fitch, W.R. 1984 The deterministic identifiability of nonlinear pharmacokinetic models, J. Pharmacokin. Biopharm., 12, 177-191 Wang, Y.M.C. and Reuning, R.H. 1992 An experimental design strategy for quantitating complex pharmacokinetic models - enterohepatic circulation with time varying gallbladder emptying as an example, Pharm. Res., 9, 169-177 Chapter 31: Optimal Sampling Student Objectives for this Chapter To understand the idea of optimal sampling To understand the graphical and analytical methods of determining optimal sampling times Once we have developed a pharmacokinetic model for a drug and obtained an understanding of the expected parameter values we can start to apply this information to patient care. This information could be applied directly for dosage regimen. This is how various dosage regimen nomograms or other recommendations are developed. For some drugs more control is required and a therapeutic drug monitoring (TDM) approach may be appropriate. During the development of a TDM procedure selecting the best times to sample drug concentration is critical. This is where optimal sampling can play an important role. Why Optimal Sampling Limited samples, minimize sample numbers Sampling requiring needle stick, heel stick is painful Indwelling cannula may be used but as easy to set-up Each sample involves some blood loss Sample Collection and Analysis Sample organization and time required Assay costs Despite having limit data we want the Best Estimate of parameter values Optimal Sampling Optimal sampling at its most basic provides one best time for each parameter. For example, assuming a one compartment intravenous model when would be the best sample time for determining a value for the elimination rate constant (kel). If the expected value was 0.2 hr-1 the best time to sample can be calculated as 5 hours. According to this approach if you can repeat the sample a second (or third) sample at the same time would be suggested. While repeat samples aren't a bad idea it might not be the best approach if the value for kel is different from the expected value. A range of sample times might be better. For example if we expect the value of kel to be in the range of 0.1 to 0.3 hr-1 we should pick sample times in the range of 3.3 to 10 hours. How did we get these sample time estimates. There are a number of approaches. Graphical and analytical. This topic will be discussed by considering these techniques that can be used to explore the optimal sampling time for best model parameter estimation. 31.1 Graphical Approach One Compartment - IV Bolus Let's start with the one compartment pharmacokinetic model after an intravenous (IV) bolus. The equation concentration versus time is given by Equation 31.1.1. Equation 31.1.1 Concentration versus time after an IV Bolus - One Compartment Model The question is: What are the best sampling times to estimate kel and V. Using the graphical approach the steps are: Simulate data using the known model and the expected parameter values Adjust one parameter (at a time) by a small amount (± 1, 5, 10%) in both directions Plot ΔCp/ΔParameter Value versus Time Determine the time of maximum change Starting with the concentration versus time curve, Figure 31.1.1. Next vary the value of kel, higher and lower, while holding V constant and repeat the calculation as seen in Figure 31.1.2. Notice the values of Cp are close to equal early in the plot and at the end, near 24 hours. The next step is to calculate the difference between the upper and lower curves, dividing the differences by the difference in kel. Note this difference is greater towards the middle, maybe left of middle in Figure 31.1.2. The following plot represents ΔCp/Δkel versus time. The time of maximum ratio is indicated at time five hours. Here kel was 0.2 hr-1. Now we can do the same thing for V, the apparent volume of distribution. Plotting concentration versus time with values of V above and below the expected value of 15 L gives the concentrations in Figure 31.1.4. Notice the biggest difference between concentration values is near zero time. Figure 31.1.5 confirms this observation. This indicates that the best time to sample for V is close to time zero. That is, as early as possible in the sampling schedule. This approach can be used for any parameter of a pharmacokinetic model with know or expected values for the parameters. One Compartment - Oral Another example might be to determine the best times to sample for three parameters on the one compartment model after oral administration, i.e. kel, ka and V. The equation concentration versus time is given by Equation 31.1.2. Equation 31.1.2 Concentration versus Time - One Compartment Oral For this model the question is: What are the best sampling times to estimate kel, ka and V. Using the method described above, the steps are: Simulate data using the known model and the expected parameter values Adjust one parameter (at a time) by a small amount (± 1, 5, 10%) in both directions Plot ΔCp/ΔParameter Value versus Time Determine the time of maximum change What is the best Sample Time for kel? Starting with the concentration versus time curve, Figure 31.1.6. Now vary the value of kel higher and lower while holding ka and V constant and repeat the calculation as seen in Figure 31.1.7. The biggest difference between concentration values is after the peak, around six hours. Figure 31.1.8. This indicates that the best time to sample for kel is approximately 5.8 hr. Now we can do the same thing for ka, the first order absorption rate constant. Plotting concentration versus time with values of ka above and below the expected value of 1.5 hr-1 gives the concentrations in Figure 31.1.9. Notice the biggest difference between concentration values is near the peak concentration. Figure 31.1.10 confirms this observation. This indicates that the best time to sample for ka is early, approximately 0.6 hour. Finally we can do the same thing for V. Plotting concentration versus time with values of V above and below the expected value of 15 L gives the concentrations in Figure 31.1.11. Notice the biggest difference between concentration values is near the peak concentration. Figure 31.1.12 confirms this observation. This indicates that the best time to sample for V is near the peak concentration, approximately 1.6 hour. 31.2 Analytical Approach The analytical method is somewhat more calculus intensive and therefore maybe some what limited. However, the simple example of a one compartment model after an IV bolus is somewhat straightforward. The steps are: Differentiate Cp versus each Parameter, Pi, to determine dCp/dPi Differentiate dCp/dPi versus time to determine d2Cp/dPi.dt Set d2Cp/dPi.dt to zero to find the time for the maximum value of dCp/dPi Solve for time Starting with the equation for concentration versus time curve, Equation 31.2.1. Equation 31.2.1 Concentration versus Time - One Compartment IV Bolus Differentiating Cp versus kel gives Equation 31.2.2 Equation 31.2.2 dCp/dkel versus time Next, differentiating dCp/dt versus time gives Equation 31.2.3 Equation 31.2.3 d2Cp/dkel.dt versus time Rearranging Equation 31.2.3 with kel equal to 0.2 hr-1 gives a value of 5 hour as the best sample time. Equation 31.2.4 Best sample time when kel is 0.2 hr-1 31.3 Numerical Approach The optimal sample times can also be calculated numerically using a number of pharmacokinetic programs. Boomer has this facility as does ADAPT with the SAMPLE module. One Compartment - IV Bolus Administration: Boomer An example .BAT file is shown here as Figure 31.3.1. Boomer Batch File 4 wls,bayes,sim,irwls,sim+error,grid Note 1 1 Screen, diskfile f2121kelV 1 Parameter type Dose 100.0 Parameter value 0 Fixed,adjust,depend1,depend2 1 To 0 F-dependence ? 0 happy or not 2 Parameter type kel 0.2000 Parameter value 1 Fixed,adjust,depend1,depend2 Note 2 0.19990 0.20001 2 0.01 0 To 1 From 0 happy or not 18 Parameter type V 15.00 Parameter value 1 Fixed,adjust,depend1,depend2 Note 2 14.999 15.001 2 0.01 1 To Cp 1 From 0 happy or not -1 Parameter type 2 Integration method 0.000 Relative error 0.000 Absolute error Simulation kel varying 1 Data from disk or keyboard 0.000 X value 0.000 Y value 1.000 X value 0.000 Y value 2.000 X value 0.000 Y value 3.000 X value 0.000 Y value 4.000 X value 0.000 Y value 5.000 X value 0.000 Y value 6.000 X value 0.000 Y value 7.000 X value 0.000 Y value 8.000 X value 0.000 Y value 9.000 X value 0.000 Y value 10.00 X value 0.000 Y value -1.000 X value 0 Accept, correct, delete, insert, of 0 Continue or save data 0 Error type for line 0 Weight type Note 3 0 AUC line number 10 Continue,save,plot,supplemental,sen Note 4 -1 wls,bayes,sim,irwls,sim+error,grid Figure 31.3.1 Input .BAT file for optimal sampling calculation Note 1 Select 'Simulation with random error or sensitivity analysis' 
Note 2 Set parameters to be adjustable. Enter 'known' parameter values and set close upper and lower limits 
Note 3 Select equal weight. Other weighting is possible but interpretation is curious 
Note 4 Select option 10 for graphs and sensitivity The output of this run is shown in Figure 31.3.2. ** FINAL OUTPUT FROM Boomer (v3.4.7) ** 16 February 2021 --- 11:36:35 am Title: Simulation kel varying Input: From f2121kelv.BAT Output: To f2121kelV.OUT Data for Cp came from keyboard (or ?.BAT) Fitting algorithm: Simulation with Error Note 1 Weighting for Cp by Equal weight Numerical integration method: 2) Fehlberg RKF45 with 1 de(s) With relative error 0.1000E-03 With absolute error 0.1000E-03 This simulation took 0.3800 seconds # Name Value Low Original High 1) kel 0.20001 0.200 0.20 0.20 Note 2 2) V 15.000 15.0 15. 15. Model and Parameter Definition # Name Value Type From To Dep Start Stop 1) Dose = 100.0 1 0 1 0 0 0 2) kel = 0.2000 2 1 0 0 0 0 3) V = 15.00 18 1 1 0 0 0 Data for Cp :- DATA # Time Observed Calculated (Weight) Weighted residual 1 0.000 0.00000 6.66644 1.00000 0.00000 2 1.000 0.00000 5.45798 1.00000 0.00000 3 2.000 0.00000 4.46859 1.00000 0.00000 4 3.000 0.00000 3.65854 1.00000 0.00000 5 4.000 0.00000 2.99534 1.00000 0.00000 6 5.000 0.00000 2.45236 1.00000 0.00000 7 6.000 0.00000 2.00781 1.00000 0.00000 8 7.000 0.00000 1.64384 1.00000 0.00000 9 8.000 0.00000 1.34586 1.00000 0.00000 10 9.000 0.00000 1.10189 1.00000 0.00000 11 10.00 0.00000 0.902141 1.00000 0.00000 Plots of observed (*) and calculated values (+) versus time for Cp. Superimposed points (X) 6.666 Linear 6.666 Semi-log |+ |+ | | | | | | | | + | | | + | | | + | | | | | + | + | | | | | | + | + | | | | | + | + | | | | + | + | | | + | | | + | + | | + | | | + | + + | | | | | + | | |* * * * * * * * * * * | + |_____________________________________ |*__*___*__*___*___*__*___*__*___*___* 0.000 0.9021 0 <--> 10. 0 <--> 10. Sensitivity, Optimal Sampling Analysis Section For kel Max for line (A) Cp is 12.26 at 5.000 Note 3 Max 12.26 | AAAAAAAAAAAAAAAAAAA | AAAA AAAAA | AAA AAA | AA AAA | AA AAAA | AA AAAA | AA AAAA | A AA | AA AAA | A AAA | A AA | A | A | A | A | A | A | A | A | | A | A | | A | A | | A | A | | A | | A | | A | A | | A | | |A------------------------------------------------------------------------------ Min 0.000 0 10.00 For V Max for line (A) Cp is 0.4444 at 0.000 Note 3 Max 0.4444 |AA | A | A | A | A | AA | A | AA | A | A | AA | A | AA | A | AA | AA | AA | AA | AAA | AA | AAA | AA | AAA | AAA | AAA | AA | AAA | AAAA | AAAA | AAAA | AAAAA | AAAA | AAAAAAA | AAAAAA | | | | | |------------------------------------------------------------------------------- Min 0.000 0 10.00 Figure 31.3.2 Output .OUT file for optimal sampling calculation run Note 1 Equal weight selected Note 2 Parameter values constrained at specified values Note 3 Best time for kel is 5 hr and 0 hr for V. The resolution of this calculation is 1/100 the maximum time, here 0.1 hr. This approach and result are essentially the same as for the graphical approach shown earlier. One Compartment - Oral Administration: Boomer An example .BAT file is shown here as Figure 31.3.3. Boomer Batch File 4 wls,bayes,sim,irwls,sim+error,grid Note 1 1 Screen, diskfile f2121kelVka 1 Parameter type Dose 100.0 Parameter value 0 Fixed,adjust,depend1,depend2 1 To 0 F-dependence ? 0 happy or not 2 Parameter type ka 1.5000 Parameter value 1 Fixed,adjust,depend1,depend2 Note 2 1.49999 1.50001 2 0.01 2 To 1 From 0 happy or not 2 Parameter type kel 0.2000 Parameter value 1 Fixed,adjust,depend1,depend2 Note 2 0.19999 0.20001 2 0.01 0 To 2 From 0 happy or not 18 Parameter type V 15.00 Parameter value 1 Fixed,adjust,depend1,depend2 Note 2 14.999 15.001 2 0.01 1 To Cp 2 From 0 happy or not -1 Parameter type 2 Integration method 0.000 Relative error 0.000 Absolute error Simulation kel varying 1 Data from disk or keyboard 0.000 X value 0.000 Y value 1.000 X value 0.000 Y value 2.000 X value 0.000 Y value 3.000 X value 0.000 Y value 4.000 X value 0.000 Y value 5.000 X value 0.000 Y value 6.000 X value 0.000 Y value 7.000 X value 0.000 Y value 8.000 X value 0.000 Y value 9.000 X value 0.000 Y value 10.00 X value 0.000 Y value -1.000 X value 0 Accept, correct, delete, insert, of 0 Continue or save data 0 Error type for line 0 Weight type Note 3 0 AUC line number 10 Continue,save,plot,supplemental,sen Note 4 -1 wls,bayes,sim,irwls,sim+error,grid Figure 31.3.3 Input .BAT file for optimal sampling calculation Note 1 Select 'Simulation with random error or sensitivity analysis' 
Note 2 Set parameters to be adjustable. Enter 'known' parameter values and set close upper and lower limits 
Note 3 Select equal weight. Other weighting is possible but interpretation is curious 
Note 4 Select option 10 for graphs and sensitivity The output of this run is shown in Figure 31.3.4. ** FINAL OUTPUT FROM Boomer (v3.4.7) ** 16 February 2021 --- 12:06:23 pm Title: Simulation kel varying Input: From f2121kelvka.BAT Output: To f2121kelVka.OUT Data for Cp came from keyboard (or ?.BAT) Fitting algorithm: Simulation with Error Weighting for Cp by Equal weight Note 1 Numerical integration method: 2) Fehlberg RKF45 with 2 de(s) With relative error 0.1000E-03 With absolute error 0.1000E-03 This simulation took 0.1000E-01 seconds # Name Value Low Original High 1) ka 1.5000 1.50 1.5 1.5 Note 2 2) kel 0.20000 0.200 0.20 0.20 3) V 15.000 15.0 15. 15. Model and Parameter Definition # Name Value Type From To Dep Start Stop 1) Dose = 100.0 1 0 1 0 0 0 2) ka = 1.500 2 1 2 0 0 0 3) kel = 0.2000 2 2 0 0 0 0 4) V = 15.00 18 2 1 0 0 0 Data for Cp :- DATA # Time Observed Calculated (Weight) Weighted residual 1 0.000 0.00000 0.00000 0.00000 0.00000 2 1.000 0.00000 4.58151 1.00000 0.00000 3 2.000 0.00000 4.77330 1.00000 0.00000 4 3.000 0.00000 4.13613 1.00000 0.00000 5 4.000 0.00000 3.43726 1.00000 0.00000 6 5.000 0.00000 2.82554 1.00000 0.00000 7 6.000 0.00000 2.31588 1.00000 0.00000 8 7.000 0.00000 1.89665 1.00000 0.00000 9 8.000 0.00000 1.55296 1.00000 0.00000 10 9.000 0.00000 1.27148 1.00000 0.00000 11 10.00 0.00000 1.04101 1.00000 0.00000 Plots of observed (*) and calculated values (+) versus time for Cp. Superimposed points (X) 4.773 Linear 4.773 Semi-log | + | + | | + | + | | | + | | | + | | | | | + | | | + | | | | | + | | | + | | | | | + | + | | | | | | + | + | | | + | | | | + | + | + | | | | | | | + | | | | |X * * * * * * * * * * | + |_____________________________________ |X__*___*__*___*___*__*___*__*___*___* 0.000 1.041 0 <--> 10. 0 <--> 10. Sensitivity, Optimal Sampling Analysis Section For ka Max for line (A) Cp is -1.498 at 0.6000 Note 3 Max 0.2804 | AAAAAAAAAAAAAAAAAAAA | AA AAAAAAAAAA | AA AAAAAAAAAAA | A AAAAAAAAAAAA | A | AA |A------------------A----------------------------------------------------------- | | A | A | A | | A | A | | A | | A | A | A | | A | | | A | | A | A | A | | | A | | A A | | A | A | A | A A | A Min -1.498 0 10.00 For kel Max for line (A) Cp is 12.13 at 5.800 Note 3 Max 12.13 | AAAAAAAAAAAAAAAAA | AAA AAAAA | AAA AAAAA | AA AAAA | AA AAAA | AA AAA | A AAA | AA AA | A A | A | AA | A | A | A | A | A | A | A | A | A | | A | A | A | A | | A | A | A | | A | A | A | A | A | | AA | A | AA |A------------------------------------------------------------------------------ Min 0.000 0 10.00 For V Max for line (A) Cp is 0.3259 at 1.600 Note 3 Max 0.3259 | AAAAAAA | A A | A AA | A AA | AA | A AA | A | A AA | AA | A AA | A | A | A AA | AA | AA | AA | A AAA | AA | AAA | A | A AAA | AA | AA | AAA | AA | A AAA | AAA | AAA | AAAA | AAAA | AAAA | A | | | | | | | |A------------------------------------------------------------------------------ Min 0.000 0 10.00 Figure 21.3.4 Output .OUT file for optimal sampling calculation run 
Note 1 Equal weight selected 
Note 2 Parameter values constrained at specified values 
Note 3 Best time for ka is 0.6 hr, for kel is 5.8 hr and 1.6 hr for V. The resolution of this calculation is 1/100 the maximum time, here 0.1 hr. This approach and result are essentially the same as for the graphical approach shown earlier.Epilogue Copyright 2014-23 David Bourne This book includes a number of links to webpages on the Internet, the WWW. Note, some of these links may change from time to time. The University of Colorado is a Certara Center of Excellence. The Center of Excellence program supports leading institutions with Certara’s state-of -the-art, model-informed drug development software. V2.0.1 David Bourne, Ph.D. Boomer Manual Table of Contents Boomer Manual Non-linear Regression Program for the Analysis of Pharmacokinetic and Pharmacodynamic Data Copyright © 1986-2022 David W.A. Bourne References Bourne, D.W.A. 1986 MULTI-FORTE, a microcomputer program for modeling and simulation of pharmacokinetic data. Computer Methods and Programs in Biomedicine, 23, 277-281 Bourne, D.W.A. 1989 BOOMER, a simulation and modeling program for pharmacokinetic and pharmacodynamic data analysis. Computer Methods and Programs in Biomedicine, 29, 191-195Background MULTI-FORTE started as a direct FORTRAN 77 translation of the MULTI program written in BASIC by Yamaoka et.al (1981). MULTI was written for a microcomputer in BASIC and allowed for the nonlinear least squares analysis of integrated equations. Four fitting algorithms were included in this original program. Later versions of MULTI incorporated Bayesian fitting and numerical integration of differential equations as separate programs (Yamaoka, 1983, 1985). MULTI-FORTE includes normal fitting, Bayesian estimation, or simulation only with integrated or differential equation models in the one program. Also, the selection of weighting schemes and methods for numerical integration have been expanded. With MULTI-FORTE the pharmacokinetic model is written in FORTRAN 77, compiled and linked with the rest of the program. BOOMER was developed with the same fitting and numerical integration routines found in MULTI-FORTE. However, with BOOMER the model is described using model parameters (based on the approach used by SAAM, Berman and Weiss, 1967, Berman et al., 1962) and thus the user doesn’t need to have a FORTRAN compiler. References Berman, M. and Shahn, E. and Weiss, M.F. 1962 The Routine Fitting of Kinetic Data to Models: A Mathematical Formalism for Digital Computers, Biophysical J., Berman, M. and Weiss, M.F. 1967 SAAM Manual., U. S. Public Health Service Publication No. 1703. U. S. Government Printing Office, Washington, D. C. Yamaoka, K., Tanigawara, Y., Nakagawa, T., and Uno, T. 1981 A Pharmacokinetic Analysis Program (MULTI) for Microcomputer. J. Pharmacobio-Dyn., 4, 879-885 Yamaoka, K. and Nakagawa, T. 1983 A Nonlinear Least Squares Program Based on Differential Equations, MULTI(RUNGE), for microcomputers. J. Pharmacobio-Dyn., 6, 595-606 Yamaoka, K., Nakagawa, T., Tanaka, H., Yasuhara, M., Okumura, K. and Hori, R. 1985 A nonlinear multiple regression program, MULTI2(BAYES), based on Bayesian algorithm for microcomputers. J. Pharmacobio-Dyn., 8, 246-256 Fitting Algorithms Gauss-Newton This is a method based on expanding the model equation(s) of interest in a Taylor series (Hartley, 1961). Damping Gauss-Newton The correction term applied to the 'old' parameter values is reduced by a factor of two if the 'new' parameter values result in a worse value for WSS. This is allowed up to 25 times during each iteration, although rounding error may become significant above 15 reductions. Modified Marquardt The Marquardt method is intermediate between the Taylor series (Gauss-Newton) method and the method of steepest descent (gradient method). In the method of steepest descent the new parameter values are calculated in the direction of the negative slope of the WSS with respect to each of the parameters. Marquardt has suggested that the direction of change in parameters calculated by these two methods is not always in the same direction. Marquardt's method attempts to find a best compromise (Marquardt, 1963). The method used in MULTI-FORTE and Boomer is that further modified by Flecher (Flecher, 1971). Simplex The simplex method of Nelder and Mead is the fourth option. This method involves the progression of a moving simplex across the WSS surface. Typically the worst point of the simplex is projected across the center (centroid) of the simplex to a better value. There are a series of simple rules to move the simplex to the WSS minimum. It is a relatively robust (but slower) method (Nelder and Mead, 1965). The user provides the initial parameter estimates while the other points on the simplex are generated randomly. Simplex -> Damping Gauss-Newton The Simplex method is quite robust and can be used with less accurate initial parameter estimates. However, it is not possible to estimate parameter value variances by this method. Thus it may be useful to automatically proceed from the final Simplex values to the input stage of another method. Presently that other method is the Damping Gauss-Newton. This option is especially attractive when used with the multi-run approach described later. Each run starts with a different simplex with only the initial parameter estimate point fixed. The others are randomly generated. Thus, each run is started from a different place on the WSS surface. This increases the possibility of converging on the global minimum. Grid Search By specifying upper and lower parameter limits as well as the number of grids, it is possible to calculate the weighted sum of squares at each grid intersection. The results of this analysis can be output to disk for incorporation into a 3-D graphing program. One Parameter Optimization When only one adjustable parameter is specified another fitting method is automatically selected. This method is essentially a damped Newton-Raphson method. References Flecher, R. 1971 A Modified Marquardt Subroutine for Non-Linear Least Squares. Tech. Report AERE R6799, U.K. Atomic Energy Authority Research Establishment, Harwell, UK Hartley, H.O. 1961 The modified Gauss-Newton method for the fitting of non-linear regression functions by least squares, Technometrics, 3(2), 269-280 Marquardt, D.W. 1963 An Algorithm for Least-Squares Estimation of NonLinear Parameters. J.Soc. Indust. Appl. Math., 11(2), 431-441 Numerical Integration Algorithms Classical Fourth Order Runge-Kutta This method is an implementation of the fourth order Runge-Kutta method. A relative error term is used to adjust the step-size. Error control is achieved by simply halving the step-size until the change in calculated values is smaller than the error requested (Kutta, 1901). Runge-Kutta-Gill This is the numerical integration method used in MULTI(RUNGE) by Yamaoka et.al. (1983) It has also been modified to give automatic step-size control. Again, error control is achieved by halving the step-size until the desired accuracy is achieved. In MULTI(RUNGE) the step-size was adjusted manually (Gill, 1951). Fehlberg RKF45 This is a further modification of the Runge-Kutta fourth order method which uses a fifth order calculation to determine the appropriate step-size given values for the required relative and absolute error. This is the subroutine written by Watts and Shampine and published as Subroutine RKF45 in the book by Forsythe and Moler (Forsythe and Moler 1967; Fehlberg, 1969; Watt and Shampine, 1977 and Shampine and Watt, 1977). This method is very efficient and appears to be the method of choice for non-stiff systems of differential equations. Adams Predictor-Corrector This is one of the options of the subroutine written by Gear. This method is also for non-stiff systems of differential equations. Step-size and order are automatically controlled to achieve a user defined absolute error (Gear 1969; Gear 1971a and Gear 1971b). Gear with PEDERV subroutine This is the most efficient option for use with stiff equations. The subroutines DECOMP and SOLVE were modified from those presented in the text by Forsythe and Moler (Forsythe and Moler, 1967). Gear without PEDERV subroutine The partial derivatives are calculated by numerical differencing thus a user-supplied PEDERV subroutine is not required. References Fehlberg, E. 1969 Low-order classical Runge-Kutta formulas with stepsize control and their application to some heat transfer problems. NASA Technical Report, NASA TR R-315 Forsythe, G.E. and Moler, C. 1967 Computer Solutions of Algebraic Systems. Prentice-Hall, Englewood Cliffs, NJ 68-69 Gear, C.W. 1969 DIFSUB for solution of ordinary differential equations. Algorithm 407. Collected Algorithms from CACM Gear, C.W. 1971a Numerical Initial Value Problems in Ordinary Differential Equations. Prentice-Hall, Englewood Cliffs, NJ 158-166 Gear, C.W. 1971b The automatic integration of ordinary differential equations. Comm. A.C.M., 14(3), 176-179 Gill, S. 1951 A process for the step-by-step integration of differential equations in an automatic digital computing machine. Proc. Cambridge Philos. Soc., 47, 96-108 Kutta, W. 1901 Beitrag zur naherungsweisen Integration totaler Differentialgleichunge. Zeit. Math. Physik, 46, 435-453 Shampine, L.F. and Watts, H.A. 1977 The art of writing a Runge-Kutta code, Part 1. in Mathematical Software III, ed. Rice, J.R. Academic Press, New York, NY Watt, H.A. and Shampine, L.F. 1977 Subroutine RKF45 in Computer Methods for Mathematical Computations, Forsythe, G.E., Malcolm, M.A. and Moler, C.B. Prentice-Hall, Englewood, NJ 133-147Getting Started with Boomer Installing Boomer Download Boomer from the Boomer website. Macintosh OS X Version Double-click on the installer package. Boomer, cDAT, eBest, eOUT will be installed in /usr/local/bin without your $PATH. This manual as a pdf file will be installed in /Applications/boomer. Boomer should now run from any directory with the Terminal program. The latest, v3.6, only works well with macOS 12.3. You might try installing gfortran from fxcoudert’s Github repository. Windows Version Unzip the downloaded packet and put the files in a folder in a convenient place, maybe the Desktop. It is easier if you keep the data files within this same directory. Starting Boomer Boomer is run in the Macintosh OS X environment using the utility application Terminal (Found in /Applications/Utilities). Start Terminal and change directory to where you want the data, control and output files (using the ‘cd’ command, a UNIX command). The Mac offers the ability to drag and drop a folder onto the terminal window (after ‘cd ‘) to change the directory. With Boomer installed in your PATH (default behavior ) you can start Boomer by typing ‘boomer’. The command line instruction can take up to three arguments. Thus you might type boomer filename quiet mrun where filename is the BAT filename, quiet (or 3, q or Q) indicates quiet mode, and mrun is the maximum number of runs in multi run mode (maximum value 100). Running Boomer in the Windows environment is a little different. It is very useful to move the boomer.exe file into a folder where you plan to store your data, control and output files. Boomer can be started by double-clicking on the boomer.exe icon. A command line window will open. Don’t double-click on a .BAT file as this would run as a DOS/Windows system command. After Boomer starts you should see the title, copyright information, and references to the original MULTI. The method of data entry is then chosen from the menu option. DATA ENTRY 0) From KEYBOARD 1) From .BAT file -1) From .BAT file (with restart) 2) From KEYBOARD creating .BAT file 3) From .BAT file (quiet mode) -3) From .BAT file (quiet mode-with restart) 4) to enter data only 5) to calculate AUC from a .DAT file -9) to quit -8) Registration Information Enter choice (0-5, -1, -3, -8 or -9) Data can be entered from the keyboard or from a batch file (verbose (1) or quiet (3) modes). Alternately a batch file can be prepared while entering the data from the keyboard. Data Entry 0) From keyboard All subsequent input will be entered from the keyboard. 1) From .BAT file Subsequent input will be read from the batch file (named xxx.BAT). Only the xxx part of the name should be entered as the program will add the .BAT extension. As with all filename entries the extension is not to be entered. Boomer is looking for -> xxxx.BAT <- Do you have the right file name? PAUSE Filename incorrect - try again? statement executed To resume execution, type go. Other input will terminate the job. If a batch file (named xxx.BAT) has been prepared the data can be read from this file instead of from the keyboard. Only the xxx part of the name should be entered as the program will add the .BAT extension. -1) From .BAT file with Restart Option If a numerical integration method (especially RKF or Gear) is used with fitting problems it is possible that parameter values may be used which cause 'unstable' integration with the program stopping. With Restart ON (choice -1 or -3) it is possible for the program to restart automatically using parameter values which are stored to disk during the iteration process (in a file called Scratch.PAR). This option is especially useful with unattended operation using large systems of differential equations although less important with faster, modern computers. The current best parameter values are stored periodically in the file Scratch.PAR. 2) From keyboard creating .BAT file Again a .BAT file name is requested. Data will be entered from the keyboard and used both in the analysis and to create a .BAT batch file. Be careful not to use the name of a .BAT file previously created and needed later. The program will check to confirm that you want to over-write files having the same name. 3) From .BAT file (quiet mode) Subsequent data will be read from the batch file (named xxx.BAT). In this mode subsequent data entry is not displayed on the screen. This can result in faster operation. -3) From .BAT file with Restart Option (quiet mode) 4) to enter data only This choice allows the user to enter data into a xxx.DAT file for subsequent analysis. 5) to calculate AUC from a .DAT file This choice allows the user to calculate AUC from data in a xxx.DAT file. Method of Analysis Boomer will ask for your method of analysis. METHOD OF ANALYSIS 0) Normal fitting 1) Bayesian 2) Simulation only 3) Iterative Reweighted Least Squares 4) Simulation with random error or sensitivity analysis 5) Grid Search -5) To perform Monte Carlo run (Start of BAT file) -4) To perform multi-run (End of BAT file) -3) To run random number test subroutine -2) To close (or open) .BAT file -1) To finish Enter choice (-3 to 5) 0) Normal This option allows a normal weighted non-linear regression analysis of the data set. 1) Bayesian A Bayesian analysis is also possible. The observed data may be weighted. During data entry population mean and standard deviations are required for each adjustable parameter. 2) Simulation only This option is useful for models consisting of large systems of differential equations. That is, systems taking a long time to solve or for which fitting is not required. This option could be useful in the analysis of systems containing fast and slow rate constants (so-called 'stiff equations'). These are commonly observed with physiological models. 3) Iteratively Reweighted Least Squares (IRWLS) With normal fitting, option 0, the weight applied to each data point doesn’t change during the analysis. However with IRWLS the weighting functions use the calculated 'y' value at each stage of the fitting rather than the observed 'y' values. For example using a 1/value weighting function with this option results in a weight calculated as 1/ycalc rather than 1/yobs. This can reduce some of the strange fitting that results when very small 'y' observed values are included in a data set. These values may be given an emphasis much larger that they really deserve (Peck, 1984a and Peck, 1984b). This could also be modeled with an appropriate weighting scheme which reduces the influence of small numbers. 4) Simulation with random error or sensitivity analysis This option allows the simulation of data sets with added error. The error can be calculated as proportional or independent of the y-value and can be calculated as uniformly distributed or normally distributed. Log normal or exponential error can also be added to data. This same uniform, normal, log normal and exponential error can be randomly applied to any of the parameter values. See section Q, below, for more details. A repeat (Monte Carlo) option will allow a large number of simulations to be run, creating different data sets with each run. In the sensitivity (or optimal sampling) mode the program generates graphs of y-value sensitivity to each of the adjustable parameters. The point of maximum sensitivity (within 100 step increments) is also reported. The program calculates the sensitivity value using the equation below for each parameter. The time when this value is maximum could be used as an optimal sampling time. 5) Grid search method This option allows a systematic search of a specified parameter space. The space is defined by specifying upper and lower limits and the number of steps in the grid. Output can be in the normal form or as disk files of parameter values and weighted sum of squares values for input into a graphing program for 3-D surface plotting. -5) Monte Carlo run This is similar to the multi run option below with one significant difference. If data files are input or output they will be created with unique (incremental) names. These files can be useful in the generation of many multiple data sets that might be useful as a clinical trial simulation. A Monte Carlo run .BAT file can also be created from a 'ordinary' .BAT file by adding the number of runs (>10) as the second line in the .BAT file. Monte Carlo run Option It is possible to run repeated analyzes of 10 to 99999 data sets (memory and time permitting). This could be be useful in a Monte Carlo type analysis to simulate data sets for other purposes such as clinical trial simulations. An additional output data file (extension .NDT) is generated with the same name as the 'normal' output (.OUT) file but in a format suitable for use with NONMEM, pMetrics and Phoenix NLME. These runs can be set-up during the creation of the .BAT file as above or by creating a single run batch files as normal with -2 at the end. Open the .BAT file in a text editor and insert above the first data line the number of repeats required (must be greater than 10). Boomer Batch File 4 wls,bayes,sim,irwls,sim+error,grid Top of the .BAT file before editing and Boomer Batch File 25 4 wls,bayes,sim,irwls,sim+error,grid after editing to simulate 25 data sets. Save the file and select batch file run. When simulating data sets use the simulate with error option to create different data sets on each run. When saving the calculated data set (the first time through when creating the batch file) use any reasonable, short file name (e.g. test: A maximum of 4 characters are used with the DOS/Windows version). In repeat mode the program will create file names with a numbered extension. For example test0001.DAT, test0002.DAT, test0003.DAT ... in sequence. When reading the data sets in using the repeat option the same naming convention is used. That is specify 'test' and the program will expect to read files test0001.DAT, test0002.DAT, test0003.DAT etc. [There are four columns in the number field to accommodate file names from test0001.DAT to test9999.DAT]. The auxiliary program cDAT can read these multiple DAT files and combine them into a tab delimited summary file suitable for importing into a spreadsheet program. The cDAT program is installed by the Mac OS X boomer installation package in /usr/bin and provided in the DOS/Windows zip folder. The program can be invoked by typing cDAT and then providing the DAT file name and the number of files. It can also be invoked as a one line command. cDAT test nnn where 'test' is the first part of the DAT file name (can be entered as test0001 or the complete name test0001.DAT) and 'nnn' is the number of DAT files. The auxiliary program eOUT can extract specified lines from .OUT files (especially useful for Monte Carlo or Multi runs), optionally converting spaces to tabs. The output file is suitable for importing into a spreadsheet program. The eOUT program is installed by the Mac OS X boomer installation package in /usr/bin and provided in the DOS/Windows zip folder. The program can be invoked by typing eOUT and then providing the OUT file name and other parameters. It can also be invoked as a one line command. eOUT test.OUT key nskip nline tflag where test.out is the .OUT file name (test should also work), key is the string to search (enclose with '' if it contains spaces), nskip is the number of lines to skip after key is found, nline is the number of lines to export and tflag is 0 or 1 (if equal to 1 converts run of spaces to one tab). It is useful to examine the .OUT file before running eOUT to determine the best values of the parameters, especially key, nskip and nline. This could be useful in extracting the final fit parameter values from a multi-run .OUT file. The auxiliary program eBEST can extract the best run from a multi-run .OUT file. The eBEST program is installed by the Mac OS X boomer installation package in /usr/bin and provided in the DOS/Windows zip folder. The program can be invoked by typing eBEST and then providing the OUT file name. It can also be invoked as a one line command. eBEST test.OUT where test.out is the .OUT file name. The program then asks Enter percent from Best WWW Enter a percent (x) and the program displays the number of runs with a WSS within x% of the best. The best run is placed in a file BEST.OUT which is opened by BBEdit (Macintosh if installed) or NotePad (Windows) for review and saving. eBest also provides the file eBestP.OUT containing the WSS and parameter values for each run. -4) Multi run This option allows either repeated runs of the BAT file or chaining to another BAT file. It will not be selected initially but only at the end of the first run. Repeatedly running a BAT file is useful when fitting data using the Simplex method as each run starts with a randomly generated simplex. A multi run .BAT file can be created from a 'ordinary' .BAT file by changing the -1 or -2 on the last line to -4 and entering the .BAT file name on the following line. This could be used to chain a number of control files for multiple subjects. -3) Random Number Test The Macintosh toolbox routine, random, is used to generate random numbers for the simplex method (to create the starting simplex), and the simulation with error values. This option (-3) gives the user the opportunity of testing these random numbers. The numbers are output to disk files. Random produces uniformly random numbers (URN) in the range -32,768 to 32,767 which are scaled to real numbers between the value of 0.0 and 1.0. Normally distributed random numbers (NDRN) with mean of zero and standard deviation of one are calculated using the equation (Abramowitz and Stegun, 1964) -2) To close (or open) batch files This option allows the user to break a batch file run down into smaller parts. Each batch file could then be combined in the editor. Alternately the first data set could be analyzed using the make batch file mode and closed with a -2 without exiting the program. This can be useful with the DOS/Windows version as it will prevent the command line window to close preventing further review of the output. The editor could then be used to edit this newly created batch file to analyze the remaining data sets. A -2 is put on the end of any batch file created. Batch files can be chained by manually editing the .BAT file with the editor. Replace the -1 or -2 at the end of the batch file with a -4. On the next line put the name of the batch file you want to run next. Enter the name without the .BAT extension. Make sure that there is a carriage return after the .BAT file name. All output will go to the .OUT file specified in the first .BAT files. -1) To finish the program All normally finishing runs will come back to this choice. If no further analyses are required a -1 is entered. With a batch file run this -1 is put on the last line of the batch file in the expected sequence. If editing a batch file remember to ensure that a carriage return must be present in the batch file after the -1 for it to be correctly interpreted. Actually all batch, data, or model files must end with a carriage return to be interpreted correctly. Output options Where do you want the output? 0) Terminal screen 1) Disk file Enter choice (0-1) 0) Terminal screen All output is directed to the computer terminal screen. This is useful for a quick check of the input data or for rapid analysis which doesn't need to be kept. 1) Disk file A disk file name is requested after selecting this option. The final results and requested plots can be directed to an .OUT file output file for later printing or enhancement. The output is saved as a text type file which could be read into a word processor for editing or other visual enhancement. The disk receiving a large output file will need to have enough space. Model definition - Boomer One limitation of the original implementation of MultiForte was that the user had to have and know how to use the Fortran compiler in order to study different pharmacokinetic models. Boomer allows the user to describe a model in terms of time interrupts (e.g. dose times or lag times), dose, first order rate constants, zero order rate constants, constants, Michaelis-Menten kinetics elimination rate systems (Portmann, 1970; Michaelis and Menten, 1913), multipliers (i.e. inverse apparent volumes of distribution), sums of exponentials, Hill equation components (Rubinow, 1975; Hill, 1910), second order rate constants, or simple physiological models (Duddy et al. 1984). This building of a model is similar to the approach taken with the mainframe simulation and modeling program, SAAM (Berman and Weiss, 1967). MODEL Definition and Parameter Entry * Allowed Parameter Types * -5) read model -3) display choices -2) display parameters 0) Time interrupt 1) Dose/initial amount 2) First order rate 3) Zero order 4-5) Vm and Km of Michaelis-Menten 6) Added constant 7) Kappa-Reciprocal volume 8-10) C = a * EXP(-b * (X-c)) 11-13) Emax (Hill) Eq with Ec(50%) & S term 14) Second order rate 15-17) Physiological Model Parameters (Q, V, and R) 18) Apparent volume of distribution 19) Dummy parameter for double dependence 20-22) C = a * SIN(2 * pi * (X - c)/b) Special Functions for First-order Rate Constants 23-24) k = a * X + b 25-27) k = a * EXP(-b * (X - c)) 28-30) k = a * SIN(2 * pi * (X - c)/b) 31,32-33) dAt/dt = - k * V * Cf (Saturable Protein Binding) 34-36) k * (1 - Imax * C/(IC(50%) + C)) Inhibition 0 or 1st order 37-39) k * (1 + Smax * C/(SC(50%) + C)) Stimulation 0 or 1st order 40) Uniform [-1 to 1] and 41) Normal [-3 to 3] Probability 42) Switch parameter 43) Clone component 44-47) Four parameter logistic model 48-51) Four parameter Weibull model Enter type# for parameter 1 (-5 to 51) With this approach the model is constructed step-by-step during the data entry process. Once a useful model is chosen it may be more convenient to use the batch mode, by first preparing a batch file, editing, and running the program using the edited batch file (see Data entry section 1 and 2). The types of parameters available are shown above. -5) Read model from .mdl file The model, described with adjustable or fixed constants, can be defined using a .mdl file. If a file has been previously defined it could be read at this point of the data entry. -4) Write model from .mdl file (available after the model is defined) The model, described with adjustable or fixed constants, can be defined using a .mdl file. Once the model has been defined a .mdl file can be written for later use. 0) Time interrupt These are times during a simulation at which you may wish to make a change. For example, give a second or subsequent dose, start or stop an infusion, or incorporate a lag time into the calculation. The information requested for this parameter is simply a name and a value. It is necessary to specify any interrupt time before it can be used (e.g. to turn off an infusion). Also, when you request a list of time interrupt values they will be numbered excluding other parameter types. It is easier to specify all the parameters that don't change first then the time interrupt values, and finally the doses or other parameters that depend on a time. As with any of the parameter types these time interrupt values may be fixed, adjustable, or dependent. 1) Dose/initial amount This would typically be the dose or it could be an initial concentration. Given as a dose it would generally be fixed, however, as an initial concentration it may be more appropriate to 'best-fit' this value. Name, value, and component to receive the dose should be specified. 2) First order rate constant By specifying various rate constants it is possible to build a system of differential equations for simulation of the model required. The model, shown above, would involve three first order rate constants, k10, k12, and k21. Each of these first order rate constants could be specified by name (k10), value, component to receive flux (0) and component to lose flux (1). In this model drug would flow from the central compartment (Component 1) to the outside world (the 'sink' designated as Component 0). The remaining rate constants can be specified in a similar manner. The data entry for k12 is shown below. Connection between the components of the model (specifically component 1) and the data to be fitted or simulated is made with the volume term. In this case the plasma data is equal to the amount in component 1 divided by the volume (of distribution). Enter -3 to see choices, -1 or -4 (save model) to exit this section Enter type# for parameter 2 (-5 to 43) 2 Enter parameter name k12 Enter k12 value 0.13 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 1 Enter lower limit 0 Enter upper limit 1.0 Enter component to receive flux 2 Enter component to lose flux 1 Input summary for k12 (type 2) Initial value 0.1300 float between 0.000 and 1.000 Transfer from 1 to 2 Enter 0 if happy with input, 1 if not, 2 to start over 3) Zero order rate constant Zero order rate constants would normally be used to describe an infusion dose from one time interrupt value to another. A start time of 0 specifies that the infusion starts at the beginning of the simulation. A stop time of 0 represents an infusion which continues until the end of the simulation. As with the first order rate constant, components to receive and lose flux are specified. Typically, the infusion comes from a component which does not need to be modeled, thus a value of 0 (the outside world) can be used for the "from" component. 4-5) Michaelis-Menten Kinetic Parameters, Vm and Km Rate processes can be specified as Michaelis-Menten processes, for example the rate of elimination may be as shown in the equation below. where Vm is the maximum rate and Km is the Michaelis-Menten constant. X(1) is the amount (or concentration) in the from component. The Vm and Km are entered as two separate parameters, however, the to/from information remains the same for both. Be careful of the units. If doses (and thus the values in the components of the model) are given as mg or g, then, Vm units should be mg/hr or g/hr (mass/time). Dose/initial concentrations given as mg/L will mean Vm units of mg/L/hr (mass/volume/time). The second half of the Michaelis-Menten entry process is the Km value. Only a value need be entered, unless the parameter is adjustable or dependent. Again, be careful of units. With a dose (in g or mg) entered the units of Km should also be g or mg. However if a Cp(zero) is used with mass/volume units, Km units should be the same. 6) Constant This is the first of the summing or intensity terms. It is necessary to distinguish between the components (and their numbers) used to describe the system of differential equations and the observed data sets (and their numbers). For example in the pharmacokinetic model above there are two components not including the 'sink'. As shown, there is one data set, Plasma. We could also specify another, second data set called Tissue, and a third called Rate of Change. When a constant is specified only the name, value and data set line is required. We might use this to offset a fixed (or adjustable) background level? Or it could be used as a constant term in a Hill equation sequence or a linear fit sequence. 7) Kappa (Reciprocal volume of distribution) Typically, if doses are entered in mg or g, then the kappa term becomes equivalent to the reciprocal of the apparent volume of distribution (see parameter type 18). That is: OR where X(1) is the amount in component 1 at the time specified. The units of kappa are reciprocal volume. If the dose/initial concentration is entered as an adjustable parameter the kappa term should probably be 1 and fixed. We could also use this type of parameter with a constant term to fit a straight line. For example X(observed) is specified by entering 0 for the component to be added. Specifying a negative value for the 'from' component allows the simulation of rates of change. For example, urine data may be collected (and analysed) as rate of excretion. In this case Kappa would probably be fixed with a value of 1 but with the component from specified as -x, where x is the component representing the cumulative amount in urine. 8-10) Sum of Exponential Terms Rather than specify a model in terms of components and rate constants (differential equations) it may be convenient (and computationally faster) to use a sequence of exponential terms. The pre-exponential term (a) is the parameter before the exponent term. A two exponential equation could be: The pre-exponential terms are 'A' and 'B'. Again, the Xobserved is specified by entering 0 for the from component. The lambda or exponent terms (b) in the above equation are α and β. A lag time can be specified for each exponential term. Usually these lag times would be the same. Second and subsequent lag times could be made dependent on the first if adjustable. 11-13) Emax (Hill Equation) Parameters Drug effect versus concentration data can modeled using this sequence of parameters. The concentration data could be entered as Xobserved values, or they could be calculated from a system of differential equations. The full equation (Hill equation) is where X(i) is the concentration in component 'i'. Specify component i = 0 if concentrations are entered as X-values (shown as time). Ec(50%) is the concentration which produces an effect of 50% of maximum. The third parameter, γ, is the power or slope term. You may want to try giving γ a value of 1, fixed, before doing a full fit with all three parameters adjustable. When plotting the Hill equation as Effect versus log(Concentration) it has been noticed that for the 20 - 80 % of maximum effect region a straight line may describe the Effect versus log(Concentration) well. Thus the simpler equation: may be useful. Since version 2.7.9 Boomer will allow this equation by modifying the way parameter type 11 works. The third parameter, slope parameter, is used as the switch. If slope is greater than 0 the Hill equation is used in the calculation. If slope is -1 (remember to not let this parameter be adjustable) then a is entered instead of Emax, and b is entered instead of EC(50%). For both versions of parameter type 11 a baseline 'effect' may be worth considering. 14) Second order rate constant Second order reaction processes can be described using this parameter type. Input includes one or two components for reactants or products (to or from). 15-17) Physiological Flow Parameters Physiologically based models can be defined by specifying blood flow rates between organs or vein/artery blood pools. A flow balance is calculated and displayed if any blood flow parameters are specified. Generally, blood flows in and out of an organ will need to be specified. Volume and extraction ratio data which are subsequently entered describe characteristics of the source of flow. Drug movement from this source is calculated using the differential equation segment: where X(i) is the amount of drug in the source (component i), X(i)/V(i) is the concentration of drug in the source tissue or blood pool, and X(i)/(V(i)*R(i)) is the drug concentration in the blood leaving the source tissue. In the case of vein or artery blood pool the value of R(i) should be 1. In addition to a flow term (15 above) a volume term is specified for the source organ or blood pool. Thus for blood flow from kidney to vein, the volume term would be for the kidney tissue. This volume term is used in the differential equation above and also to convert the amount values calculated internally to calculated concentration values. If you don't need to calculate values for a particular tissue you can enter 0 for the data set number. The extraction ratio, R, for the source tissue is the final term specified. For vein or artery blood pools the value of R should be 1. 18) Apparent volume of distribution Typically, if doses are entered in mg or g, the volume term can be used to relate the amount term, dose, with the concentration term measured. That is: where X(1) is the amount in component 1 at the time specified. If the dose/initial concentration is entered as an adjustable parameter the volume term should probably be 1 and fixed. 19) Dummy parameter for double dependence This parameter is not used in any calculations of y-value except for the calculation of double dependent parameters. One example might be the estimation of the bioavailability factor F. The initial amount administered may be expressed as F * DOSE where F is unknown and is to be estimate while DOSE is fixed and known. Thus F and DOSE can both be entered as dummy parameters, one adjustable and the other fixed. The term F * DOSE (a DOSE or Initial amount, type 1 parameter) would be specified as double dependent with double dependence type 5 (multiply, see below). Enter -3 to see choices, -1 or -4 (save model) to exit this section Enter type# for parameter (-5 to 51) 1 Enter parameter name F*Dose Enter F*Dose value 200 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 3 Two parameter dependence 1) + Par 1 + Par 2 2) + Par 1 - Par 2 3) - Par 1 + Par 2 4) - Par 1 - Par 2 5) + Par 1 x Par 2 6) - Par 1 x Par 2 7) + Par 1 / Par 2 8) - Par 1 / Par 2 9) + Par 1 ^ Par 2 0) for list of parameters Enter choice (0-8) 20-22) SIN Function These parameters (type 20-22) can be used to incorporate diurnal variation in a line function. Parameters include Amplitude, Period and Offset. The result of these three parameters is the addition of the value: Amplitude x SIN (2 x π x (X - Offset)/Period) to the selected data set (line). This would result in a value varying between +/- Amplitude. Normally a constant type 6 parameter (possibly adjustable, etc) would be also be used for the same data line so that the value would vary between Constant +/- Amplitude. With both parameters defined the equation for line 1 becomes: Enter type# for parameter (-5 to 51) 20 Enter parameter name Value Enter Amplitude Value value 20 Enter data set (line) number 1 Enter line description Pressure Enter component number (0 for obs x) 0 --- Enter Period Value value 24.0 --- Enter Offset Value value 2.0 23-24) First Order Rate Constant as a Linear Function On occasion it may be useful to add a rate constant to a model which varies with time (or the value in a component of the model) in a linear fashion. Thus a first order rate constant can be defined as: a * X + b. The `a' parameter is a slope term and `b' is an intercept. The X term can be the observed x, usually time or the value (or reciprocal - entered as a negative number) in any component. Thus zero or second order rate constant can be entered with this parameter type. For example, a zero order rate constant of 5.0 could be entered as a(SL) = 5.0 with drug from component 1 to 0, b = 0.0 and the F-dependence equal -1, the from component. The differential equation for this process is then: k is now a zero order rate constant. Enter type# for parameter (-5 to 51) 23 Enter parameter name k Enter a(SL) k value 5.0 Enter component to receive flux 0 Enter component to lose flux 1 -- Enter b k value 0.0 Enter component for F-dependence ( 1 to - 1 or 0 for no dependence) -1 25-27) First Order Rate Constant as an Exponential Function Under some circumstances the value for a rate constant may change progressively with time. For example an elimination half-life may change from 8 hours (kel = 0.0866 hr-1) to 6 hours (kel = 0.1155 hr-1) over a period of 60 hours (with a half-life of 12 hours; k = 0.05776 hr-1) because of enzyme induction. This could be simulated with two rate constants. The first would be a type 1 parameter with a value of 0.1155 hr-1. The second parameter would be type 25 with a = -0.0289 (=-( 0.1155 - 0.0866)), b = 0.05776, and c = 0). Thus the rate constant would change from 0.0866 to 0.1155 hr-1 over a period of 60 hours (5 half-lives). The differential equation would become: or F-dependence is also possible with this type of parameter. Specifying a positive (or negative) component for F-dependence multiplies (or divides) the above term by the amount in the specified component. Enter type# for parameter (-5 to 51) 25 Enter parameter name k Enter a(EXP) k value -0.0289 Enter component to receive flux 0 Enter component to lose flux 1 --- Enter b(EXP) k value 0.05776 Enter component for F-dependence ( 1 to - 1 or 0 for no de --- Enter c(EXP) k value 0 28-30) First Order Rate Constant as a SIN Function On occasion it may be necessary to include diurnal variation in the rate constants of a model. Again by using two rate constants, one type 1 and the other type 28, it is possible to simulate or analyze this phenomenon. With just the type 28 parameter the differential equation would be: F-dependence is also possible with this type of parameter. Specifying a positive (or negative) component for F-dependence multiplies (or divides) the above term by the amount in the specified component. Enter type# for parameter (-5 to 51) 28 Enter parameter name k Enter Amplitude k value 0.2 Enter component to receive flux 0 Enter component to lose flux 1 --- Enter Period k value 24.0 Enter component for F-dependence ( 1 to - 1 or 0 for no dependence) 0 --- Enter Offset k value 0 31-33) Saturable Protein Binding Elimination Many drugs are bound to plasma protein. With large dose drugs with small apparent volumes of distribution this protein binding may be saturated. Further, for drugs which also have low extraction ratios, that is, clearance is not flow limited, elimination of total drug is directly related to free drug concentrations (Bourne et al., 1981; Bevill et al., 1982). Because of the complexity of this model its specification is relatively rigid. For example, considering IV bolus administration with saturable protein binding controlled elimination a dose, amount, would be added to component 1. The first-order rate constant is entered with units of reciprocal time. The association constant, Ka, has units of reciprocal concentration. The total protein, Pt, is entered as a concentration. The amount of drug is converted into total concentration with a single apparent volume of distribution (type 18) linked to the protein binding parameter, type 31, through the `from' specification for both parameters. If required a second data set line equal to the free drug concentration can be specified after entering the Ka value. By specifying parameter type 32, Ka and Pt parameters can entered and thus Cfree versus Ctotal data sets can be analyzed. This can be useful for the analysis of in vitro data sets. Enter type# for parameter (-5 to 51) 31 Enter parameter name PB Enter k(PB) PB value 0.2 Enter component to receive flux 0 Enter component to lose flux 1 --- Enter Ka PB value 100 Enter data set (line) number --- Enter Pt PB value 0.5 34-36) First or zero order rate constants subject to inhibition This parameter can replace or add to a first order or zero order rate constant with inhibition determined by the amount in a component of the model. For example, a dummy, fixed value or a dynamic value controlled by separate rate processes. In the model below the inhibitor, I, in component 2 controls the rate constant k’ for drug, X, leaving component 1. The differential equation for component 1 is: This can be specified as Enter type# for parameter 6 (-5 to 51) 34 Enter parameter name k' Enter k' value 0.2000 Is k' zero(0) or first(1) order? 1 Enter inhibiting component # 2 Input summary for k' (type 34) Fixed value is 0.2000 Parameter is first order with link from component 2 Enter 0 if happy with input, 1 if not, 2 to start over 0 Enter IC(50%) k' value 5.000 Enter component to receive flux 0 Enter component to lose flux 1 Input summary for IC(50%) k' (type 35) Fixed value is 5.000 Transfer from 1 to 0 Enter 0 if happy with input, 1 if not, 2 to start over 0 Enter Imax k' 0<>1 value 0.5000 Input summary for Imax k' 0<>1 (type 36) Fixed value is 0.5000 37-39) First or zero order rate constants subject to stimulation This parameter can replace or add to a first order or zero order rate constant with inhibition determined by the amount in a component of the model. For example, a dummy, fixed value or a dynamic value controlled by separate rate processes. In the model below the inhibitor, I, in component 2 controls the rate constant k’ for drug, X, leaving component 1. The differential equation for component 1 is: This can be specified as Enter type# for parameter 6 (-5 to 51) 37 Enter parameter name k' Enter k' value 0.2000 Is k-S zero(0) or first(1) order? 1 Enter link component # 2 Input summary for k' (type 37) Fixed value is 0.2000 Parameter is first order with link from component 2 Enter 0 if happy with input, 1 if not, 2 to start over 0 Enter SC(50%) k' value 5.000 Enter component to receive flux 0 Enter component to lose flux 1 Input summary for SC(50%) k' (type 38) Fixed value is 5.000 Transfer from 1 to 0 Enter 0 if happy with input, 1 if not, 2 to start over 0 Enter Smax k' value 0.5000 Input summary for Smax k' (type 39) Fixed value is 0.5000 40) Uniform [-1 to 1] Probability This dummy parameter can be used for single dependence and provides the value 0 or 1 for a dependent parameter. If the parameter value (in the range -1 to 1) is greater than a random number generated uniformly distributed between -1 and 1 it provides the value 1, otherwise it provides a value of 0. One use for this parameter might be the simulation of compliance. For example, if 20% of the doses are not taken you might use a parameter value of 0.6 (= 2 * (1 - 0.2) - 1 = 2 * (1-x) -1) where x is the 'fractional' probability for 0 or non-compliance. With the parameter space from -1 to 1, i.e. width 2.0, 20% would be 0.4, thus set the parameter to 0.6. The type 40 parameter, Uniform, is set to 0.6, fixed. The second type 19, dummy parameter is entered as single dependent on Uniform, here the 4th parameter. Enter type# for parameter 4 (-5 to 51) 40 Enter parameter name Uniform Enter Uniform value 0.6000 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 0 Input summary for Uniform (type 40) Fixed value is 0.6000 Enter 0 if happy with input, 1 if not, 2 to start over 0 Enter -3 to see choices, -1 or -4 (save model) to exit this section Enter type# for parameter 5 (-5 to 51) 19 Enter parameter name Dummy Enter Dummy value 0.000 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 2 Enter parameter for dependence (+, 0 or -) 4 Input summary for Dummy (type 19) Value 0.000 dependent on parameter 4 Multiple runs, using the MonteCarlo option, will produce on average 20% of the runs with Dummy equal 1.0 41) Normal [-3 to 3] Probability This dummy parameter can be used for single dependence and provides the value 0 or 1 for a dependent parameter. If the parameter value (in the range -3 to 3) is greater than a random number, normally distributed with mean 0 and standard deviation of 1, it provides the value 1, otherwise it provides a value of 0. One use for this parameter might be the simulation of compliance. For example, if 84% of the doses are taken you might use a parameter value of 1 (1 standard deviation above the mean = 50 + 68/2 = 84%). The type 41 parameter, Normal, is set to 1.0, fixed. The second type 19, dummy parameter is entered as single dependent on Normal, here the 4th parameter. Enter type# for parameter 4 (-5 to 51) 41 Enter parameter name Normal Enter Normal value 1.000 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 0 Input summary for Normal (type 41) Fixed value is 1.000 Enter 0 if happy with input, 1 if not, 2 to start over 0 Enter -3 to see choices, -1 or -4 (save model) to exit this section Enter type# for parameter 5 (-5 to 51) 19 Enter parameter name Dummy Enter Dummy value 0.000 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 2 Enter parameter for dependence (+, 0 or -) 4 Input summary for Dummy (type 19) Value 0.000 dependent on parameter 4 Multiple runs, using the MonteCarlo option, will produce on average 84% of the runs with Dummy equal 0.0 and 16% equal 1.0. 42) Switch parameter Similar to parameter type 40 or 41 except change depends on the value of two specified control parameters. This parameter, Switch, is set equal to a threshold value. The dependent type 19 parameter, Dummy, is set 0 or 1 depending if the type 19, trigger, parameter value is smaller or greater than the Switch value. In the example below the trigger value is randomly determined with uniform error. Enter type# for parameter 4 (-5 to 51) 19 Enter parameter name Trigger Enter Trigger value 10.00 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 1 Enter lower limit 0.000 Enter upper limit 20.00 Simulation with random ERROR 0) no error for parameter ( 1), Trigger 1) error = uniform error 2) error = normal error 3) error = uniform error x true value 4) error = normal error x true value 5) error = log normal error about true value 6) error = EXP(normal error) Enter choice (0-6) 1 Enter absolute error factor (Std dev) 5.000 . Using 7.405 Input summary for Trigger (type 19) Initial value 7.405 float between 0.000 and 20.00 Enter 0 if happy with input, 1 if not, 2 to start over 0 Enter -3 to see choices, -1 or -4 (save model) to exit this section Enter type# for parameter 5 (-5 to 51) 42 Enter parameter name Switch Enter Switch value 10.00 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 0 Enter parameter # (1 to 5) for Switch 4 Input summary for Switch (type 42) Fixed value is 10.00 Enter 0 if happy with input, 1 if not, 2 to start over 0 Enter -3 to see choices, -1 or -4 (save model) to exit this section Enter type# for parameter 6 (-5 to 51) 19 Enter parameter name Dummy Enter Dummy value 0.000 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 2 Enter parameter for dependence (+, 0 or -) 5 Input summary for Dummy (type 19) Value 0.000 dependent on parameter 5 43) Clone component Clone a component using this parameter. The new component will have the same numerical value as the 'donor' component but will not influence the donor system. 44-47) Four parameter logistic model This selection allows the addition of the four parameter logistic model as describe by Kraupp et al., 1986. Normally this would be the only parameter selection for the analysis of radio-immunoassay assay data. The equation is shown below. The parameter A represents Bo, B is a shape factor, C is the midpoint of the curve (ED50 and D represents non specific binding (Nsb). 48-51) Weibull function A four parameter Weibull function similar to Equation 13 in Costa and Sousa Lobo, 2001 can be fit using these parameters. When amount released is expressed as a percentage Ymax will be fixed equal to 100. When Tlag is fixed equal to zero the two remaining parameters are alpha and beta. Alpha represents a measure of the rate of dissolution while beta is considered a shape factor. Specifying Parameter Values and Limits 0) Fixed Parameters If the method selected is a simulation, then all the parameters will be fixed and you won't be asked to specify fixed, adjustable, dependent, or double dependent. Alternatively, if a normal or bayesian fitting is to be performed some parameters can be specified as fixed. Often dose and time interrupt values will be fixed. You may wish to hold some rate constants fixed for one run and then allow them to be adjustable in a subsequent run. Enter 0 (or press return) to specify fixed parameters. 1) Adjustable parameters At least two adjustable parameters must be specified for a normal fitting problem. If 1 is entered to specify adjustable, a lower and upper limit are requested. The fitting process will then keep the adjustable parameter values between these limits. To reduce the influence of parameter limits you could specify a very wide range between the upper and lower limit. 2) Dependent parameters Occasionally a parameter (value) may be used more than once in a complete model definition. If these values are adjustable it is appropriate to let the first value be adjustable but set subsequent parameters equal to the first parameter (or the negative of the first parameter). This entry is shown in the figure below. Entering 0 will produce a listing of the parameters already specified. Enter type# for parameter (-5 to 51) 7 Enter parameter name 1/V Enter 1/V value 1 0) fixed, 1) adjustable, 2) single dependence or 3) double dependence 2 Enter parameter for dependence (+, 0 or -)0 # Name Value 1) Dose 100.0 2) kel 0.2500 Enter parameter for dependence (+, 0 or -)1 Enter data set (line) number 1 Enter line description Plasma Enter component number (0 for obs x) 1 3) Double Dependent Parameters Parameters can be dependent on two other parameters. This is usually accomplished using at least one dummy parameter, see above for an example. After entering all the information required for each parameter the program will ask if you wish to make any changes. Entering 0 will bring up the list of parameter types on the screen for the next entry. Numerical Integration Method If the number of differential equations in the model is greater than zero, then the next question is choice of numerical integration method. Method of Numerical Integration 0) Classical 4th order Runge-Kutta 1) Runge-Kutta-Gill 2) Fehlberg RKF45 3) Adams Predictor-Corrector with DIFSUB 4) Gears method for stiff equations with PEDERV 5) Gears method without PEDERV Enter choice (0-5) 0) Classical 4th order Runge-Kutta This is a relatively slow method but reasonably robust. The relative error is controlled by decreasing the step-size and reevaluation. 1) Runge-Kutta-Gill This is another fourth order method. Again it is slow, and the relative error is controlled by adjustment of step-size. This is the method used in MULTI(RUNGE) but with automatic control of step-size added in Multi-Forte and included in Boomer.2) Fehlberg RKF45 This method involves a fourth order evaluation with a fifth order used for efficient selection of optimal step-size. It is quite efficient and requires relative and absolute error values. 2) Adam Predictor-Corrector This method is incorporated into the DIFSUB subroutine and can be used for non-stiff systems of differential equations. The overhead involved in this subroutine makes it less efficient for small systems of differential equations. Possibly it would be better with bigger systems. An absolute error term is requested for error control. 3) Gear with S/R Pederv The Gear method is a very good method for solving systems of stiff equations. Again an absolute error value is required. Also with this option a subroutine PART must be described by the user for use by the subroutine PEDERV. The partial derivative of each equation with respect to each dependent variable is entered in subroutine PART. Thus if there are n differential equations and therefore n dependent variables, there will be n*n terms defined in PART. Commonly many of these terms will be zero, but they must all be defined. 4) Gear without Pederv This is also a good method for solving stiff equations. The subroutine PART is not required for this method as the partial derivatives are determined by numerical differencing. An absolute error term is also used in the step-size control process. This method may be slower than method 4 (with Pederv) but the PART subroutine is not required. Boomer automatically includes the PART subroutine so either method can be used. Numerical Integration - Error Terms Relative and/or absolute error values are requested if numerical integration is required. Since all calculations are performed in single precision specifying a relative error of less than 0.00001 may cause problems. For the same sort of reason very small values for the absolute error are not recommended and may in fact cause a failure of the program. The default value for each parameter is 0.0001 which can be entered by pressing return. Larger values for the error term should also be avoided especially if the program is used to fit to a data set. With a tight convergence criteria (low PC value, see below) and a larger error term the iteration process may oscillate erratically. This can be readily induced using a non-stiff system of differential equations and Gears method of numerical integration for a fitting problem. A large absolute error term and low PC value also helps. Enter Relative error term for Numerical integration (0.0001) Enter Absolute error term for Numerical integration (0.0001) Fitting Algorithm FITTING METHODS 0) Gauss-Newton 1) Damping Gauss-Newton 2) Marquardt 3) Simplex 4) Simplex->Damping GN Unless the simulation method of analysis is chosen, the next request is for the specification of the fitting algorithm. These algorithms are as presented in the original MULTI programs, simply translated to FORTRAN. 0) Gauss-Newton This is generally the fastest method, but it may diverge giving much slower times, or get lost giving incorrect answers. With good initial estimates for the parameters this can be a very useful method. 1) Damping Gauss-Newton This is also a quick method with the tendency to diverge more or less under control. It may still get lost if given poor initial estimates. Rerunning the same analysis with different initial estimates can give more confidence in a consistently achieved minimum sum of squares. This method would generally be the method of choice. 2) Marquardt In this method the Gauss-Newton approach has been altered to give better performance for oddly shaped sums of squares surfaces. For most problems it is slower than options 0 or 1. 3) Simplex This method can be quite slow, although not too bad. It can be especially useful in a case with poorer initial estimates. It is supposed to be more robust. Better data and/or better initial estimates will also help. This method will occasionally fail to produce a best fit. As the initial simplex is produced randomly rerunning the analysis with the same initial estimates will produce a different analysis, with possibly the same end-point. 4) Simplex -> Damping G-N With this analysis the simplex method is used at first because of its robustness. The final results from the simplex method are then used with the damping Gauss-Newton method for further refinement of the analysis. One parameter method When only one adjustable parameter is specified another fitting method is automatically selected. This method is essentially a damped Newton-Raphson method. Fitting Criteria DT value If a fitting method (i.e. not a simulation) other than the simplex method is chosen the program will request a value for DT. Entering 0.0 (or pressing return without an entry) will cause the default value of 0.001 to be used. The DT value is used in the calculation of the partial derivatives of the WSS with respect to the parameter values. This value is used as a fractional change in the parameter value during the calculation of the partial. (If the parameter value is zero, the DT value is used as an absolute change in value). Too large a number may lead to a distorted search for the minimum WSS, too small a value may over emphasize rounding errors. PC value If a fitting method is chosen (i.e. not a simulation) a value for PC will be requested. For the Gauss-Newton/Marquardt methods a default value of 0.00001 is suggested, whereas for the simplex method the default value is 0.0001. The PC value for the Gauss-Newton methods controls the convergence exit during the iteration process. If the relative change in the WSS is smaller than PC the iteration process is terminated normally. For the simplex method the convergence criteria is based on the relative change in the parameter values compared with the value of PC. Data Set Title This is used to identify the data set. Up to 60 characters can be entered. The data set title is printed with the final output. Individual Data Value Entry Enter data from 0) Disk file 2) ...including weights 1) Keyboard 3) ...including weights 0) Disk File If the data has been previously entered and stored in a data file it may be read by the program. This request is followed by a request for the .DAT filename. The .DAT part of the name should not be entered. Data files consist of two column data with a tab character between the `x' and the `y' value (and ‘weight’ if appropriate). A carriage return separates each data pair. This format is consistent with spreadsheet and graphing program formats. The current versions of Boomer will automatically read data in the older Boomer and MultiForte formats. 1) Keyboard With this option the individual time/concentration data are to be entered from the keyboard. Entering -1 for time will end this data entry. 2-3) … including Weights These are the same except the user enters estimates of the weight to be applied to each data point. This should usually be the reciprocal of the variance of the data point. After data entry from the keyboard or after reading the data from a file, the data set may be viewed or edited. Editing is quite complete with options to change, delete, insert an individual data point or add an offset to the x-values of the data line. After data entry or editing the data set can be saved as a .DAT file by specifying the disk file name. Again reusing a file name will result in the loss of the older data. Weighting Scheme If no weight is entered during the data entry process the program will ask for the user to select from various formulas. The relative error in each data point may or may not be known. The user may have some information about the relative error in each measurement. This may be independent of the value or proportional to some function of the value. The fitting process can be made aware of this information by use of a weighting scheme. Boomer allows a number of different weighting schemes to be used, including specifying the weight for each data point using option 2 or 3 in the previous section. Weighting function entry for Plasma 0) Equal weights 1) Weight by 1/Cp(i) 2) Weight by 1/Cp(i)^2 3) Weight by 1/a*Cp(i)^b 4) Weight by 1/(a + b*Cp(i)^c) 5) Weight by 1/((a+b*Cp(i)^c)*d^(tn-ti)) 0) Equal weight An unweighted least squares fit can be specified by giving each data point an equal weight. 1) Weight by 1/Cp(Obs) or 1/Cp(Calc) Each data point can be weighted by the reciprocal of the observed or calculated data value. This may be a reasonable alternative to the unweighted least squares. 2) Weight by 1/Cp(Obs)2 or 1/Cp(Calc)2 The square of the observed or calculated value is used in the weighting calculation. 3) Weight by 1/(a*Cp(Obs)b) or 1/(a*Cp(Calc)b) This option allows non-integer powers of the observed or calculated concentration to be used in the weighting process. The values of a and b could be determined from the data by plotting log(average concentration) versus log(variance) from multiple subject data sets as proposed by Wagner, 1975. 4) Weight by 1/(a + b*Cp(Obs)c) or 1/(a + b*Cp(Calc)c) This is an even more flexible version of the above scheme. 5) Weight by 1/((a + b*Cp(Obs)c)*d(tlast - ti)) or 1/((a + b*Cp(Calc)c)*d(tlast - ti)) This allows incorporation of estimated assay sensitivity (a may equal sensitivity2) and coefficient of variation (b may equal CV2) as well as reducing emphasis on older samples (d, typically 1.01 to 1.05) as may be appropriate in a clinical setting. This type of weighting scheme has been incorporated into some clinically useful Bayesian estimation programs (Peck et al., 1984). For simulations a weight of 1 is arbitrarily assigned to each data point. Observations with a value of zero are given a weight of zero and thus ignored during any fitting process. Selection of a 'best' weighting scheme is aided by consideration of the weighted residual plots which can be output by Boomer. Entry of Initial Parameter Estimates 1) Parameter values With all analyses (except simulations with no parameters) initial values for each parameter are required. Generally the better the estimate the better the analysis. 2) Parameter limits For all analyses (except for simulations) parameter limits, upper and lower, are requested. These limits can be wide or very close, it's up to the user. For those preferring no limits, very wide limits will suffice. These values are not used in anyway to change the parameter domain, but simply act as absolute limits for the parameter values. More like a box than a curved dish. When limits are specified some care may be necessary. For example, a lower limit for a volume of distribution value entered as zero may cause problems if the limit is approached during the early part of an iterative run. A value for volume of zero will give infinite calculated plasma concentrations and can cause an early exit to the fitting. A lower limit of zero for rate constants is usually better tolerated. 3) Population values If the Bayesian estimation procedure is chosen then the user must supply estimates of the population mean parameter value. Also required is an estimate of the uncertainty in the population value. This is entered as the population standard deviation. If a larger uncertainty is to be used a larger value for the population standard deviation would be entered. Simulation with Error Input Simulation with random ERROR 0) no error for line ( 1) 1) error = uniform error 2) error = normal error 3) error = uniform error x true value 4) error = normal error x true value 5) error = log normal error about true value 6) error = EXP(normal error) Enter choice (0-6) 6 Enter error intensity factor If a simulation is chosen, parameter limits are not required. When simulation with error is specified information about the type and magnitude of the error is requested. Uniform, normally distributed, log normal and exponentially distributed error can be specified. This error may be simply added to the calculated value (choices 1 and 2) or multiplied by and added to the calculated value (choices 3 and 4). The intensity factor refers to the magnitude of the error. With choices 1 and 2 this would be the maximum error or the standard deviation of the error added. For choices 3, 4, and 6 this is a fractional values. For example, a value of 0.1 with choice 3 would mean that the added error would have a standard deviation of 10% about the calculated value. 0) no error for line 1) Valn = Valo + Uni x Int 2) Valn = Valo + Nor x Int 3) Valn = Valo x (1.0 + Uni x Int) 4) Valn = Valo x (1.0 + Nor x Int) 5) Valn = 10log(Valo) + Nor x Int 6) Valn = Valo x eNor x Int where Valn = new value Valo = original value Int = intensity factor (standard deviation) Uni = uniformly distributed random number between -1 and + 1 Nor = normally distributed random number with mean zero and standard deviation equal to the intensity value Log = log normally distributed random number Exp = exponentially distributed random number AUC, AUMC, and MRT The program will automatically calculate the area under the curve (AUC), the area under the first moment curve (AUMC), and the mean residence time (MRT) of any data set. For accurate numbers a data value at time zero (dummy or otherwise) is needed. The program uses the trapezoidal rule to calculate the areas. The program uses the last two calculated data points to calculate a terminal slope for extrapolation to zero. Calculated zero time values are used if an observed zero time value is zero. If no zero time value is given a warning is issued. Zero time values with zero value are given zero weight in any fitting analysis. Enter 0 to exit this section. Additional Output 1) Saving Data, Graphs, Extra Files and Sensitivity Output Additional Output Enter -> 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Save Data x x x x x x x x Graphs x x x x x x x x Extra files x x x x x x x x Sensitivity x x x x x x x x Save calculated data files, Produce linear, semi-log and weighted residual plots, save extra data files or perform sensitivity or optimal sampling analysis Enter 0 - Continue without additional output This leads to the next analysis without any plotting. 2) Save data on disk for later graphing Enter 1, 3, 5, 7, 9, 11, 13, or 15 Select FORMAT for x-value 0) Don't save data 1) G14.1 2) G14.2 3) G14.3 4) G14.4 5) G14.5 6) G14.6 7) F14.0 8) F14.1 9) F14.2 10) F14.3 11) F14.4 12) F14.5 13) F14.6 This option allows writing the calculated values and/or the standardized residuals to disk files (.DAT files) for later analysis. Suitable file name(s) are requested as before. These data files are output as x-value, tab, y-value, return. This format should be compatible with a variety of Macintosh programs. Many programs will read the .DAT files directly. The data format can be specified in the dialog box. Choices include a `G' general format with 1 through 6 digits or a `F' floating point format with 0 through 6 decimal places. [Overflow with the `F' format will produce 14 *'s]. The file name (without extension) should also be provided. 3) Produce a printer plot now Enter 2, 3, 6, 7, 10, 11, 14, or 15 Printer-type plots (not high resolution) can be prepared for quick evaluation of the fit to the data. Four types of plots are produced. Linear and semi-log observed versus time, standardized residual versus time and standardized residual versus log(observed y value). These plots can be useful in the evaluation of the model or the selected weighting scheme (Draper, 1966). When there is more than one line of data the program also prints a combined plot for standardized residuals versus time or observed y value. 4) Example Plot Output 5) Example Standardized Weighted Residual Plots 6) Extra output files Enter 4, 5, 6, 7, 12, 13, 14, or 15 Choosing this option produces extra output files. ***.htm, ***.csv and ***.mdl files. The htm file is a html formatted file for web display or input as external data by OpenOffice. The csv file can be read by a spreadsheet program such as Numbers, Excel or OpenOffice. The mdl file contains details of the model with parameter values. The mdl file may be used for input or edited. 7) Sensitivity or Optimal Sampling Output Enter 8, 9, 10, 11, 12, 13, 14, or 15 With this option the program break the x scale (time) into 100 segments and calculates the sensitivity value according to the formula. These values are graphed versus the x value and the maximum value is output. The weight scheme for these data were: Weighting for Cp by 1/a*Cp(Obs)^b With a = 0.2500E-02 and b = 2.000 Weighting for Heart_Rate by 1/a*Cp(Obs)^b With a = 4.000 and b = 0.000 8) Three Dimensional WSS Output Table If the grid search method is chosen the program may output the WSS at each point on the specified grid with the parameter values to two disk files per pair of parameters. One disk file is in SYLK format and the other file is in tab-return format for entry into spreadsheet programs. With this output WSS surface plots could be generated. 9) Batch Files Creating a batch file on the run could be useful if you have a large number of data sets requiring similar analyses. Alternately, a single data set may be analyzed by different weighting schemes. You could enter the first set of information from the keyboard, creating a batch file as you go. The batch file could then be edited and the section containing data repeatedly duplicated using the copy and paste commands. An entry of -1 is used to terminate the data entry. This will need to be changed to 0 at the end of each set except the last. (This final -1 must be followed by a carriage return or it will not be properly recognized.) When editing the batch file, be sure to change data file names and initial estimates for parameter values. Do not change the integer or real number description starting in column 40. References Abramowitz, M and Stegun, I.A. 1964. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Applied Mathematics Series 55, National Bureau of Standards, 953 Bevill, R.F., Koritz, G.D., Rudawsky, G., Dittert, L.W., Huang, C.H., Hayashi, M., and Bourne, D.W.A. 1982 Disposition of Sulfadimethoxine in Swine: Inclusion of Protein Binding Factors in a Pharmacokinetic Model, J. Pharmacokin. Biopharm., 10, 539-550 Bourne, D.W.A., Bialer, M., Dittert, L.W., Hayashi, M., Rudawsky, G., Koritz, G.D., and Bevill, R.F. 1981 Disposition of Sulfadimethoxine in Cattle: Inclusion of Protein Binding Factors in a Pharmacokinetic Model, J. Pharm.Sci., 70, 1068-1072 Costa, P., and Sousa Lobo, J.M. 2001 Modeling and comparison of dissolution profiles. Eur. J. Pharm. Sci. 13 (2) 123-33 Draper, N.R. and Smith, H. 1966 Applied Regression Analysis. Wiley, New York, NY pp 86-100 Duddy, J., Hayden, T.L., Bourne, D.W.A., Fiske, W.D., Benedek, I.H., Stanley, D., Gonzalez, A., and Heierman, W. 1984 Physiological Model for Distribution of Sulfathiazole in Swine, J. Pharm. Sci., 73, 1525-1528 Hill, A.V. 1910 The possible effects of the aggregation of the molecules of haemoglobin on its dissociation curves. J. Physiol., 40 IV-vii Kraupp, M., Marz, R., Legenstein, E., Knerer and Szekeres, T. 1986 Evaluation of Radioimmunoassay Calculations by Four Parameter Logistic and Spline functions, J. Clin. Chem. Clin. biochem., 24, 1023-1028 Michaelis, L. and Menten, M.L. 1913 Die Kinetik der Invertinwirkung, Biochem. Z., 49, 333 Peck, C.C., Beal, S.L., Sheiner, L.B., and Nichols, A.I. 1984 Extended Least Squares Nonlinear Regression: A possible solution to the "Choice of Weights" problem in analysis of individual pharmacokinetic data. J. Pharmacokin. Biopharm., 12, 545-558 Peck, C.C., Sheiner, L.B., and Nichols, A.I. 1984 The problem of choosing weights in nonlinear regression analysis of pharmacokinetic data. Drug Metab. Rev., 15, 133-148 Portmann, G.A. 1970 "Pharmacokinetics," Chapter 1 in Current Concepts in the Pharmaceutical Sciences - Biopharmaceutics, Swarbrick, J. ed., Lea & Febiger, Philadelphia, PA p 37 Rubinow, S.I. 1975 Introduction to Mathematical Biology. Wiley, New York, NY, p 65 Wagner, J.G. 1975 Fundamentals of Clinical Pharmacokinetics. Drug Intelligence Publications, Hamilton, IL p 289 Output Information Boomer can output the final results to the terminal screen or a disk file (.OUT file). Intermediate results may also be output to the terminal screen. [Be careful, command-C or command-. will cause an exit from the program]. The output is headed by the version number and the system date and time. The title, fitting algorithm, weighting information, DT (if appropriate), PC, number of iteration loops, and damping (if appropriate) are printed next. For simulations, the elapsed time is displayed for comparison purposes. The next section displays the final parameter values. The number, name, final fitted value, standard deviation (SD), percent coefficient of variation (CV), inputted lower limit and upper limit are displayed for each parameter. A warning is given if any of the parameter values are within 5% of the upper or lower limit. A coefficient of variation of greater than 50% can indicate that the model selected has too many parameters, or that the data are insufficient or too 'noisy'. The SD or CV information is more an indication of how good the fit is or how noisy the data appears and does not give any information about the population statistics of the parameters. If a Bayesian analysis was selected the population mean and standard deviation inputted, the square root of the weight for the parameter (1/standard deviation), and the weighted residual are printed. The weighted residual is calculated as (parameter value-population mean value) times the square root of the weight. Thus the weighted residual squared is the contribution of this parameter to the overall WSS. The model fit can be evaluated in part by looking at the next output. The sum of the weighted squared residuals (WSS), the unweighted coefficient of determination (R2), and the unweighted correlation coefficient (R) between the observed and calculated y-values. The mean error (ME), mean absolute error (MAE) and root mean squared error (RMSE) are useful parameters for determining how useful the model might be at predicting future results. Akaike's information criterion (AIC), Schwarz criteria (SC), and log likehood (LL) are useful criteria for determining which model is best (Note always use the same weighting scheme when making this determination. AIC = N • ln (WSS) + 2 • M SC = N • ln (WSS) + M • ln(N) LL = -N/2 • [ln (2 • π) + ln (WSS/N) + 1] (from http://www.xycoon.com/) where N = number of data points (weighted); WSS = the weighted sum of squares; and M = the number of adjustable parameters. The lowest value of AIC, SC, the smallest value of WSS, and the value of R2 closest to 1.000 indicate a best fit (Akaike, 1973; Yamaoka et al., 1978). The AIC value can be quite useful in evaluating the number of terms in a given model type (e.g. two or three exponentials, 4 or 6 parameters). If the same weighting scheme is used the model with the lower AIC value is generally the more significant. This result can also be determined by calculation of the F value obtained by comparing the two models (Mandel, 1964; Boxenbaum et al., 1974). where WSSj is the sum of the weighted squared residuals obtained with the jth set of parameters. For example, j may refer to the one compartment model and k to the two compartment fit. In Boomer, the Model and Parameter Definition section is next. In this section all the constants and parameters are displayed with name, value, type, from, to, dep, start, and stop information in a compact tabular form. Careful analysis of this section should help to ensure that the model has been entered correctly. This is followed by information about the fitted data. Data for each line is presented as data number, time (x or independent value), calculated and observed concentration (y or dependent value), (weight), and weighted residual. The weight value is actually the square root of the weight used for each data point during the final calculation. The weighted residual is calculated as (observed value - calculated value) times the square root of the weight. Squaring and summing these values will give the WSS for this data set. At the end of each data set or line the calculated WSS, weighted R2, and unweighted R value are printed. This allows the analyst to determine which line is contributing most to the overall WSS in multi-line data sets. In the case of Bayesian analysis it allows the analyst to see the contribution of the concentration data and contrast this with the contribution of the fitted parameter values. Data can be optionally output in printer plots. These plots include linear and semi-log plots of observed and calculated y values versus x values, standardized weighted residuals versus x, and standardized weighted residuals versus log(calculated y values). The standardized weighted residuals are calculated as (observed y value - calculated y value) times the square root of the weight divided by the standard error. The standard error is calculated as the square root of WSS divided by the number of data points minus the number of parameters (or divided by just the number of point with a Bayesian analysis). The first two plots give the analyst a general 'feel' for the fit. Outliers may become obvious, systematic deviation between the observed and calculated data points may be obvious. The two residual plot can be very useful in the evaluation of the weighting scheme selected or the model selected. Non-random residuals may well mean a poor weighting scheme or a model with insufficient parameters (Draper and Smith, 1966). Again, all of the above output can be directed to a disk file for inclusion, as a text file, into a report etc. Calculated values versus time or the standardized residuals versus time data can be saved as disk files for later enhancement with a plotting program. The data is saved as a text file with x and y values separated by a tab and each data pair separated with a carriage return. Reference Akaike, H. 1973 A new look at the statistical model identification, IEEE Trans. Automat. Control., 19, 716-723 Boxenbaum, H.G., Riegelman, S., and Elashoff, R.M. 1974 Statistical Estimation in Pharmacokinetics. J. Pharmacokin. Biopharm., 2, 123-148 Draper, N.R. and Smith, H. 1966 Applied Regression Analysis. Wiley, New York, NY pp 86-100 Mandel, J. 1964 The Statistical Analysis of Experimental Data, Interscience, New York, p 164 Yamaoka, K., Nakagawa, T., and Uno, T. 1978 Application of Akaike's information criterion (AIC) in the evaluation of linear pharmacokinetic equations. J. Pharmacokin. Biopharm., 6, 165-175 Boomer Examples 1) Straight Line 2) Sum of exponentials 3) Two compartment pharmacokinetic model, with infusion dose The above examples indicate the entries required to define the models specified. If the run is for a fit to data, additional information as described in the 'Specifying Parameter Values and Limits' would be necessary. 4) Normal Fitting - Two Compartment IV Bolus Model Example data collected after a 250 mg IV bolus administration. The data (‘X’) plotted on semi-log graph. Semi-log linear regression of the last five data points gives; B = 5.8 mg/L and beta = 0.089 hr-1. The residual line results in; A = 9.3 mg/L and alpha = 2.3 hr-1. Manipulation of these results gives the values; k21 = 0.938 hr-1, k10 = 0.218 hr-1, k12 = 1.233 hr-1, and V1 = 16.6L. The next step is to define the model on paper. Number each component (circles) of the model in sequence. The central compartment is represented by component 1. The tissue compartment is component 2. There is one data set or data line. This is Data Set Number 1. The model is further defined by three first order rate constants (arrows) and one apparent volume of distribution (triangle). The apparent volume of distribution relates the amount of drug in the central compartment to the plasma concentrations in the data set. Including the dose there are five parameters needed to fully define the model. These are summarized in the table. Start Boomer and select Keyboard entry. Select normal fitting (0), screen output (0) and proceed to parameter definition and entry. Enter each of the parameters from the table. Enter -1 for parameter type to complete this section. Choose Runge-Kutta-Fehlberg (2) as the numerical integration technique and press return twice to accept the default values for relative and absolute error. Choose Damping Gauss-Newton (1) as the fitting algorithm and again press return twice to accept the default values of DT and PC. Enter a title for this run `Two compartment Fit to Data' and select keyboard data entry. Enter the x-value and y-value data. Enter -1 for x-value to finish the data entry. Press return (for 0), unless you need to correct some of the data points. Press return if you don't want to save the data. If you want to save the data enter a file name and choose the required format for the x and y-values. The available formats are G14.1 to G14.6 and F14.0 to F14.6. All of these formats produce data within 14 columns. The G format changes from fixed format to scientific format as necessary and is better if a wider range of values are expected. The number after the decimal point indicates the number of digits saved. The F format is fixed format with the number after the decimal point indicating the number of digits after the decimal point in the number. Examples include G14.2 12. 1.2 .12E+10 .12E-04 G14.4 1234. 12.34 .1234E+10 .1234E-05 G14.6 123456. 12.3456 .123456E+10 .123456-06 F14.0 12. 1. 0. F14.2 12.34 1.234 .12 F14.4 12.3456 1.2345 .1234 F14.6 12.345678 1.234567 .123456 ******.****** The last entry indicates any number greater than 9999999.999999 in F14.6 format. A variety of data weighting schemes is possible. Enter 0 to select equal weights. The program will now undertake the data analysis and finally produce the output information. This includes the model specification, final parameter values and calculated data. Choose 0 to skip the AUC calculation and enter 2 to produce a linear and semi-log plot of the observed and calculated data, and weighted residual plots. This completes the analysis. An alternative approach to the analysis of these data is to fit the data with a sum of exponential equation such as: This is carried out in the same way as the first example except for the model definition and the lack of numerical integration routine (and error terms) specification. 5) Bayesian Analysis - IV Infusion with kel function of Creatinine Clearance This example set involves the analysis of plasma concentrations measured after an IV infusion of 200 mg/hr for 30 minutes to a patient with a creatinine clearance of 30 ml/min. The measured concentrations were: Plasma concentration 1 hour after the start of the infusion = 4.0 mg/L Plasma concentration 12 hours after the start of the infusion = 1.7 mg/L From previous experiments with this drug in similar patients it had been found that: where a = 0.0028 +/- 0.00028 and b = 0.02 +/- 0.002 and V(1) = 21 +/- 2.1 L We can set up the model first on paper. Component 1 represents the one compartment model with a first order elimination rate constant k10. Linking the amount of drug in component 1 with the concentration in data set 1 is the apparent volume of distribution, V1. By putting the value of creatinine clearance in component 2 without any rate constants (it will stay fixed) we can now calculate k10 as a function of creatinine clearance (using parameter type 23). Start Boomer and select Keyboard entry. Select Bayesian fitting (1), screen output (0) and proceed to parameter definition and entry. Enter each of the parameters from the table below. After entering all the parameters enter -1. Choose integration method 2 and choose the default for the relative and absolute error terms by pressing return. Use the Marquardt fitting algorithm and the default DT and PC values. After entering the title for this analysis choose data from keyboard (1). Enter time and Cp values of 1 hr, 4 mg/L and 12 hr, 1.7 mg/L, respectively. Accept the data `as is' unless you need to make a correction. Continue without saving the data. With a Bayesian analysis a little more attention needs to be taken regarding the weighting of the data. When the parameters were entered a standard deviation value for each parameter is entered and then used as a weighting factor during the fitting process. We must also use a `suitable' weighting scheme for the plasma data. One scheme might be to assume that the standard deviation for each data point is 10% of the data value. Thus Standard Deviation = 0.1 x Value or Variance = 0.01 x Value2 And if the weight is to be 1/Variance a weighting scheme type 3 may be most useful where Weight = 1/(a x Value2) with a = 0.01 and b = 2. 6) Working with Batch (.BAT) files - Simulation In the two previous tutorials all the program entries were made from the keyboard. There are two other possibilities when working with Boomer. A batch file can be created while you are entering the instructions from the keyboard or a batch file can be used as the source of all the instructions. Once a batch file is created it is possible to edit the batch file and perform a sequence of similar runs. For example, fitting data from several subjects to the same model can be accomplished by creating a batch file during the analysis of the first subject. This batch file can be duplicated for as many subjects as available and edited appropriately. Another example might be to perform a sequence of simulations with different values for selected parameters of the model. Thus you could simulate drug effect after an IV bolus administration using a two-compartment model. By editing the resultant batch file the effect of varying some of the parameters can be determined. For example the effect of ke0 on the Effect versus time curve can be explored. Start Boomer and select Keyboard --> .BAT entry. Select Simulation (2), file output (1) and enter Tutorial_3 as the output file name. Proceed to parameter definition and entry. Enter each of the parameters from the table below. After entering all the parameters enter -1. Choose numerical integration method 2 and select the default values for relative and absolute errors. After entering a suitable title enter 1 for data from keyboard and enter time values of 0, 0.5, 1, 2, 3, 4, 6, 9, and 12 hours for each data set. Enter zero for the y-values. Enter 2 to plot the simulated data for concentration in the central compartment and the effect. The batch file Tutorial_3.BAT includes all the entries needed for this simulation. After the first simulation, enter -2 to quit the run without exiting the program. Using an editor it is possible to alter the value of ke0, save the new batch file and then choose batch file run to the new simulation. 7) Simultaneous Fit to IV and Oral Data - IRWLS Boomer will allow the simulation and analysis of up to 20 data set lines. Boomer also allows setting a parameter equal to another parameter or pair of parameters. Thus fitting IV and oral data simultaneously is not difficult. Again, the first step is to develop the model on paper. For example a model with separate IV and oral administration is shown in the figure below. There will be two types of parameter dependencies used to define this model. First, we can set the kel_po and V_po equal to kel_iv and V_iv, respectively. Thus the values of kel_iv and V_iv will be adjustable and the oral parameter values will follow along during the analysis. The second type of dependence is a double dependence. The oral dose and F value can be set-up as `dummy' parameter and a type 1 (dose/initial condition) can be set equal to F x Dose_po. This way the dose_po can be fixed at the dose given and the bioavailability, F, can be adjustable. For this tutorial we will use the iteratively reweighted least squares method. This is very similar to the normal fitting method except that the data weight is recalculated during the fitting process as the calculated concentration changes. Start Boomer and select Keyboard entry. Select IRWLS (3), file output (1) and enter Tutorial_4 as the output file name. Proceed to parameter definition and entry. Enter each of the parameters from the table above. After entering all the parameters enter -1. Choose numerical integration method 2 and select the default values for relative and absolute errors. Choose the simplex fitting algorithm (3) and select the default value for PC. After entering a suitable title enter 0 for data from disk file. Enter disk filenames for the IV data and the oral data, respectively (or select these files from the `Open' dialog). Choose weight type 2, (1/val2), for both data set lines. Once the fitting is complete choose 1 then 2 to calculate the AUC, AUMC, and MRT for the IV and oral data sets. 8) Simulation with Error and Monte-Carlo Simulations On occasion it may be useful to simulate models and be able to add error to the data. On other occasions it may be useful to simulate data where the parameters have different, random values. This is possible with the Simulation with Error option. Also, by editing a previously created batch file it is possible to perform many repeated simulations or analyses. In this tutorial you will simulate a two-compartment model after IV bolus administration and add a random error to the three rate constants. Once the initial batch file is created it will be edited to allow repeated simulation and production of a series of data files. Start Boomer and select Keyboard entry creating a .BAT file (2). Name this .BAT file, Tutorial_5. Select Simulation with random error (3), screen output (0). Proceed to parameter definition and entry. Enter each of the parameters from the table below. After entering all the parameters enter -1. Choose numerical integration method 2 and select the default values for relative and absolute errors. After entering a suitable title for the run enter the times 0, 0.5, 1, 2, 3, 4, 6, 9, and 12 with zero for the y-value from the keyboard. Accept the data (0) and don't save the data at this point. Enter error type 0 so no error is added to the data calculated. After the simulation enter 0 for the AUC line and 1 to save the calculated data. Enter Tutorial_5 as the data file name (use tut5 with DOS/Windows version) and -2 to finish the simulation run. This gives us an initial simulation run with one data set. The next step is to edit the file, Tutorial_5.BAT by adding a new second line with ` 10'. The first four lines are now: Boomer Batch File 10 4 wls,bayes,sim,irwls,sim+error,grid 0 Screen, diskfile, printer This will cause Boomer to run the simulation 10 times with different values for k10, k12, and k21, each time producing a new Tutorial_5 x.DAT file where the x is the sequence number. 9) Generating Phoenix Data Sets with One Categorical Covariate One use of the simulation with error method is the generation of multiple data sets that could be used for population pharmacokinetic (PopPK) projects. Another use might be clinical trial simulations. In this example we will generate 25 data set after a single IV bolus with a one compartment model described using dose, volume of distribution (V) and elimination rate constant (kel). The pharmacokinetic model is a one compartment linear model. The best way to start this process is to have a clear idea of the model and the type of parameters required. The pharmacokinetic model is shown above. Dose is a type 1, kel a type 2 and V is volume. Concentration, Cp, is to be calculated at specified times. Here the elimination rate constant is different for male versus female. In this example kelmael < female so these is a delta kel for female simulated subjects. A type 40 parameter, Uniform, and a type 19 parameter, Sex, are used to randomly switch between male and female values. The parameters V, Male (kelmale) and dFemale are adjustable parameters for the simulation with error run. The value of Sex is randomly 0 or 1 depending on whether the random number is above or below the value of Uniform, here 0, thus 50% male and female. Thus SexdF is zero or equal to dFemale. Finally the value of kel is calculated as Male plus SexdF. The 25 on the second line requests 25 randomly generated results. The second line is added to a single run batch file. It may be necessary to determine best values using a single run batch file. Once a single run works edit the batch file and add the number of ‘subjects’ you want to simulate. The batch file. Boomer Batch File 25 4 wls,bayes,sim,irwls,sim+error,grid 1 Screen, diskfile CovSex 1 Parameter type Dose 100.0 Parameter value 0 Fixed,adjust,depend1,depend2 1 To 0 F-dependence ? 0 happy or not 18 Parameter type V 10.00 Parameter value 1 Fixed,adjust,depend1,depend2 7.000 Lower limit 15.00 Upper limit 2 Parameter error type 1.000 Parameter error intensity 1 To CObs 1 From 0 happy or not 40 Parameter type Uniform 0.000 Parameter value 0 Fixed,adjust,depend1,depend2 0 happy or not 19 Parameter type Sex 0.000 Parameter value 2 Fixed,adjust,depend1,depend2 3 Dependence-para 0 happy or not 19 Parameter type Male 0.1000 Parameter value 1 Fixed,adjust,depend1,depend2 0.8000E-01 Lower limit 0.1200 Upper limit 2 Parameter error type 0.1000E-01 Parameter error intensity 0 happy or not 19 Parameter type dFemale 0.5000E-01 Parameter value 1 Fixed,adjust,depend1,depend2 0.3000E-01 Lower limit 0.7000E-01 Upper limit 2 Parameter error type 0.2000E-02 Parameter error intensity 0 happy or not 19 Parameter type SexdF 0.000 Parameter value 3 Fixed,adjust,depend1,depend2 5 Double dependence type 4 Dependence-para1 6 Dependence-para2 0 happy or not 2 Parameter type kel 0.1000 Parameter value 3 Fixed,adjust,depend1,depend2 1 Double dependence type 5 Dependence-para1 7 Dependence-para2 0 To 1 From 0 happy or not -1 Parameter type 2 Integration method 0.000 Relative error 0.000 Absolute error Categorical Covariate 1 Data from disk or keyboard 0.000 X value 0.000 Y value 2.000 X value 0.000 Y value … 12.00 X value 0.000 Y value -1.000 X value 0 Accept, correct, delete, insert, of 0 Continue or save data 2 Parameter error type 0.1000 Parameter error intensity 0 Weight type 0 AUC line number 2 Continue,save,plot,supplemental,sen -1 wls,bayes,sim,irwls,sim+error,grid One of the output files is suitable for input into the NLME Phoenix population pharmacokinetic software. Part of the output file, PhCovSex.csv The columns Male, dFemale and SexdF can be suppressed by leaving the ‘name’ blank. 10) Generating Phoenix Data Sets with Two Covariates (Categorical and Continuous) For this example we have the same pharmacokinetic model but using clearance, CL, instead of kel. This requires the internal calculation of kel as CL/V. With this example we have CL dependent on the categorical parameter Sex and V dependent on the continuous variable Weight. The batch file Boomer Batch File 25 4 wls,bayes,sim,irwls,sim+error,grid 1 Screen, diskfile CovSW 1 Parameter type Dose 100.0 Parameter value 0 Fixed,adjust,depend1,depend2 1 To 0 F-dependence ? 0 happy or not 19 Parameter type Weight 100.0 Parameter value 1 Fixed,adjust,depend1,depend2 75.00 Lower limit 125.0 Upper limit 2 Parameter error type 25.000 Parameter error intensity 0 happy or not 19 Parameter type Male 10.00 Parameter value 1 Fixed,adjust,depend1,depend2 5.000 Lower limit 20.00 Upper limit 2 Parameter error type 0.2500 Parameter error intensity 0 happy or not 19 Parameter type dFemaleCL 2.000 Parameter value 1 Fixed,adjust,depend1,depend2 1.000 Lower limit 4.000 Upper limit 5.000 Lower limit 20.00 Upper limit 2 Parameter error type 0.2500 Parameter error intensity 0 happy or not 40 Parameter type Uniform 0.000 Parameter value 0 Fixed,adjust,depend1,depend2 0 happy or not 19 Parameter type Sex 0.000 Parameter value 2 Fixed,adjust,depend1,depend2 5 Dependence-para 0 happy or not 19 Parameter type SexdFemale 0.000 Parameter value 3 Fixed,adjust,depend1,depend2 5 Double dependence type 4 Dependence-para1 6 Dependence-para2 0 happy or not 19 Parameter type CL 50.00 Parameter value 3 Fixed,adjust,depend1,depend2 1 Double dependence type 3 Dependence-para1 7 Dependence-para2 0 happy or not 19 Parameter type V/Wt 1.000 Parameter value 1 Fixed,adjust,depend1,depend2 0.500 Lower limit 2.000 Upper limit 2 Parameter error type 0.2500 Parameter error intensity 0 happy or not 18 Parameter type V 500.0 Parameter value 3 Fixed,adjust,depend1,depend2 5 Double dependence type 9 Dependence-para1 2 Dependence-para2 1 To CObs 1 From 0 happy or not 2 Parameter type kel 0.1000 Parameter value 3 Fixed,adjust,depend1,depend2 7 Double dependence type 8 Dependence-para1 10 Dependence-para2 0 To 1 From 0 happy or not -1 Parameter type 2 Integration method 0.000 Relative error 0.000 Absolute error Covariate with Wt and Sex 1 Data from disk or keyboard 0.000 X value 0.000 Y value 2.000 X value 0.000 Y value 4.000 X value 0.000 Y value 6.000 X value 0.000 Y value 8.000 X value 0.000 Y value 10.00 X value 0.000 Y value 12.00 X value 0.000 Y value -1.000 X value 0 Accept, correct, delete, insert, of 0 Continue or save data 2 Error type for line 0.0080 Error factor 0 Weight type 0 AUC line number 2 Continue,save,plot,supplemental,sen -1 wls,bayes,sim,irwls,sim+error,grid This took a little trial and error, thus working with a single run batch at first is recommended. Adjust the parameters in single runs and check the output is what you expect. The number of simulations can be easily increased or decreased by changing the value on the second line. An example NLME data file. The batch file can be edited by removing the names of the columns (type 19) that aren’t required. The NLME data file will no longer include these columns. 11) Grid Search Method The final tutorial will use the grid search method of fitting a set of data. This method breaks the range from lower to upper limit into a number of steps. With more than one parameter a number of grids are produced and the program will methodically calculate the WSS at each point on each grid. This can be quite slow unless a coarse grid is used. Again using the IV bolus two compartment model you can explore the grid search method by allowing the parameters k12 and k21 to be `grid searched'. Start Boomer and select Keyboard entry. Select Grid Search Method (5) and screen output (0). Proceed to parameter definition and entry. Enter each of the parameters from the table below. After entering all the parameters enter -1. Choose numerical integration method 2 and select the default values for relative and absolute errors. After entering a suitable title enter the data from the keyboard and -1 at the end of data entry. Accept the data and continue without saving. Enter weight type 1 then 1 to save a WSS file to disk. Next enter 20 as the maximum WSS for the grid. Enter a filename for the WSS file, however, only the first three letters are significant. After the run is complete the file Tut01_02.WSS contains the WSS data in a comma delimited format. Tut01_02.SFF, a SYLK format file, is also produced. The batch file, Tutorial_6.BAT, includes the steps required to perform this analysis. More Tutorials - Online Acknowledgments Dr Yamaoka et.al. for writing the original MULTI programs and showing that 'real' pharmacokinetic analysis can be performed on microcomputers. 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