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Dear All,
I just have a simple question. Do all drugs follow exponential decay in the
elimination phase? Can you have a fairly flat curve for some time in this
phase? Are there any exceptions?
Thanks for your help.
Chandrani Gunaratna, Ph.D.
Senior Research Chemist
Bioanalytical Systems
2701 Kent Avenue
West Lafayette, IN 47906
Phone: (765)463-4527
E-Mail: prema.aaa.bioanalytical.com
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[Two replies - db]
Date: Wed, 15 Sep 1999 02:06:21 -0700 (MST)
X-Sender: ml11439.at.pop.goodnet.com
To: PharmPK.at.boomer.org
From: ml11439.aaa.goodnet.com (Michael J. Leibold)
Subject: Re: PharmPK Elimination phase kinetics
Hello Dr.Gunaratna,
Most drugs obey "linear pharmacokinetics" which manifests as
exponential decay in the plasma concentration versus time curve.
However, some drugs obey "nonlinear pharmacokinetics" where the
plasma concentration versus time curve is not "log linear" or
exponential in nature. Nonlinear pharmacokinetics is also called
Michaelis-Menten pharmacokinetics and is described by the equation:
dc/dt= -C(Vmax)/(Km + C) (equation 1) (nonlinear differential
equation)
Vmax= maximal metabolic rate
Km= Michaelis-Menten constant
At high concentrations (C>>Km), the plasma concentrations
decline at constant rate equal to Vmax (the maximal rate of metabolism).
dc/dt= -Vmax (equation 2)
At very low concentrations(C<in a log linear, exponential fashion described by the first order
differential equation:
dc/dt= -C(Vmax)/(Km) (equation 3) (linear differential
equation)
dc/dt= -CK where C= Coe-kt
At intermediate concentrations, the plasma concentrations
decline at a variable rate as function of the varying plasma
concentrations themselves (equation 1).
It has been suggested that all drugs which are hepatically
metabolized are subject to the same Michaelis-Menten enzyme
pharmacokinetics. That is, at some high concentration all drugs
will exhibit enzyme saturable, nonlinear, Michaelis-Menten
pharmacokinetics. Theophyline has been found to exhibit nonlinear
pharmacokinetics at higher concentrations, but linear pharmacokinetics
at lower concentrations. However, most drugs in the therapeutic range
are the bottom of the Michaelis-Menten curve and obey linear
pharmacokinetics governed by equation 3, and this is why their
plasma concentration versus time curves are exponential.
The plasma concentrations of phenytoin (a drug which obeys
Michaelis-Menten pharmacokinetics in the therapeutic range) appears
"bowed" on a log plasma concentration versus time curve relative to
the straight line appearance of a drug with linear pharmacokinetics.
Phenytoin is the classic Michaelis-Menten drug and various dosage
schemes have been devised based on this concept.
I hope that this was helpful!
Mike Leibold, PharmD, RPh
ML11439.aaa.goodnet.com
---
From: "Barnes, Edward"
To: "'PharmPK.at.boomer.org'"
Subject: RE: PharmPK Elimination phase kinetics
Date: Wed, 15 Sep 1999 09:25:19 -0400
Not all drugs follow an exponential decay (1st order).
In a saturated system you can have a linear decay (0 order). This is also
referred to as Michaelis-Menten kinetics. Alcohol (C2H5OH) is an example of
a drug that exhibits 0 order elimination, especially when consumed in large
quantities.
Ed Barnes
TRI
3202 Tower Oaks Blvd.
Rockville, MD 20852
ebarnes.-a-.tech-res.com
301-230-4793
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The concept of zero-order elimination is OK as an mathematical entity
(See Mike Leibold's equation 2) but it is not a good way to characterize
real concentration time profiles. Ethanol kinetics have been extensively
studied and a variety of models tested (Holford NHG. The clinical
pharmacokinetics of ethanol. Clinical Pharmacokinetics. 1987;
13:273-292). The zero-order fiction might be OK for those who like to
make money crystal ball gazing for drunk driving defendants but there is
very strong evidence that ethanol kinetics are not properly described by
a zero-order model. A combination of a mixed order and first order
elimination pathway is required to describe ethanol PK over a wide range
of concentrations and this combined model should probably always be
considered for any drug which appears to have capacity limited kinetics.
> From: ml11439.-at-.goodnet.com (Michael J. Leibold)
> Subject: Re: PharmPK Elimination phase kinetics
> At high concentrations (C>>Km), the plasma concentrations
> decline at constant rate equal to Vmax (the maximal rate of metabolism).
> dc/dt= -Vmax (equation 2)
> From: "Barnes, Edward"
> Subject: RE: PharmPK Elimination phase kinetics
> In a saturated system you can have a linear decay (0 order). This is also
> referred to as Michaelis-Menten kinetics. Alcohol (C2H5OH) is an example of
> a drug that exhibits 0 order elimination, especially when consumed in large
> quantities.
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford.at.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.htm
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Another way of viewing the 1.order/0.order concepts is simply to view them
as special cases of saturable kinetics. In its simples case saturable
kinetics can be described by the familiar squared hyperbola (as in classical
Michaelis-Menten kinetics. Now if the speed of the process in question (like
elimination) is decribed by:
dX/dt = (Vmax/(Km + [X])) * [X] and Clearance =
Vmax/Km (a constant)
Then in the case that [X]<
dX/dt ~ (Vmax/Km) * [X]
and thus aproaches 1.order kinetics. So any facilitated process will at low
saturation (of course) be satisfactorily described as a 1.order process.
The other special case of saturation kinetics when [X]>>Km when (Km+[X])}~
[X]
of course gives dX/dt ~ Vmax thus 0.0rder kinetics. Clearance approaches
zero.
The point is that those two cases can best be viewed as special cases of
saturation: When the two extreme situations are not present then we have to
describe clearance as a variable dependent on ([X] = Vmax/Km + [X]).
At least in teaching PK to undergraduates I have found this approach an
effective way of structuring those concepts.
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The mathematical demonstration that the mixed order model
approximates to either a first-order process or a zero-order process
is not in doubt. However, in real life I would argue that one cannot
satisfactorily describe full concentration profiles by using the zero
order eliminaton model because the approximation will necessarily
fail when concentrations become low (in the region of the Km). The
first order approximation on the other hand can work very well, and
has obviously done so for countless drugs, because it is not uncommon
for the highest concentrations to be well under the Km.
A similar situation happens with the limiting case of the organ
clearance model (discussed recently on this list) which predicts
organ clearance is only determined by blood flow when CLint>>Q. This
approximation is very hard to satisfy in real life because it
requires extremely high enzyme activity (quantitated as CLint) in
relation to blood flow. In practice, the organ clearance of high
extraction ratio drugs depends on both blood flow and intrinsic
clearance so that inhibitors (or even inducers) of CLint can be
expected to modify organ clearance. On the other hand, the assumption
that CLint<liver have many different enzymes served by the same blood flow and
it not hard for enzyme activity to be low in relation to Q. The
finding of organ clearance being apparently independent of blood flow
is therefore not unusual.
Fortunately a double dose of foolish approximation (zero-order
elimination and organ clearance only dependent on blood flow) won't
happen because, as Nils points out, when C is much greater than Km,
CLint approaches zero and so the organ clearance cannot be dependent
on blood flow as CLint has to become less than Q.
--
Nick Holford, Dept Pharmacology & Clinical Pharmacology
University of Auckland, Private Bag 92019, Auckland, New Zealand
email:n.holford.aaa.auckland.ac.nz tel:+64(9)373-7599x6730 fax:373-7556
http://www.phm.auckland.ac.nz/Staff/NHolford/nholford.html
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I have to take issue with the implication that zero-order elimination is
not seen clinically. I have taken care of patients who have overdosed
with phenytoin who have concentrations in the 50 - 60 mcg/ml range. I
can attest that zero order elimination does occur clinically.
Ron Kavanagh, BS Pharm, PharmD, PhD
Office of Clinical Pharmacology and Biopharmaceutics
Food and Drug Administration
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Copyright 1995-2010 David W. A. Bourne (david@boomer.org)