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The following message was posted to: PharmPK
Hello!
I would like to draw a plot graph showing the blood concentration of a
drug after a short infusion. It's pharmacokinetic is based on a 3
compartment model. I have the following equation to calculate the
rate of infusion to maintain a certain level. k12, k21, k13, k31, k10
are the coefficients between compartments, Vc is the volume of
central comaprtment, C is the target concentration, R is the rate of
infusion, t is the time.
R= C*Vc(k10+[k12*e^(-k21*t)]+[k13*e^(-k31*t)]). If it is, C is be
equal R/Vc(xxxx), but this does not explain what happens after a
short infusion, because as soon as R goes zero, the C will be zero,
which is incorrect. Can anybody help me? Please do not suggest
softwares, because I know a few, but I would like to understand how
it mathematically works.
Thank you, Laszlo
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[Two replies - db]
From: "Rob Ariano"
Date: Mon, 15 Oct 2001 15:32:32 -0500
To: david.at.boomer.org
Subject: Re: PharmPK Equation for 3 compartment modelling
The following message was posted to: PharmPK
Where have you found this equation describing the serum-conc. time
profile for an infused drug in a 3-cmpt model? Notice for t=zero,
you actually achieve a value for C other than zero; which is
impossible in the absence of a load!
Regards,
Robert Ariano, Pharm.D.,BCPS
Clinical Pharmacist Critical Care
St.Boniface General Hospital; &
Associate Professor of Pharmacy,
& Medicine, University of Manitoba,
204-237-2050 Phone
204-237-2165 FAX
rariano.at.sbgh.mb.ca
www.sbgh.mb.ca
---
From: "Mike Makoid"
Date: Mon, 15 Oct 2001 16:08:37 -0500
To: david.-a-.boomer.org
Subject: RE: PharmPK Equation for 3 compartment modeling
The following message was posted to: PharmPK
>I would like to draw a plot graph showing the blood concentration of a
> drug after a short infusion. It's pharmacokinetic is based on a 3
> compartment model. I have the following equation to calculate the
> rate of infusion to maintain a certain level. k12, k21, k13, k31, k10
> are the coefficients between compartments, Vc is the volume of
> central comaprtment, C is the target concentration, R is the rate of
> infusion, t is the time.
>R= C*Vc(k10+[k12*e^(-k21*t)]+[k13*e^(-k31*t)]). If it is, C is be
> equal R/Vc(xxxx), but this does not explain what happens after a
> short infusion, because as soon as R goes zero, the C will be zero,
This is an incorrect assumption on your part. What happens after an IV
Bolus to the same model? If you can answer that question, you should
understand. Realize the body treats the drug the same way as an IV
bolus after the infusion is stopped. After the infusion is terminated,
the body doesn't care how the drug got, or how long it took. It just
wants to get rid of it. So at the end of the infusion, you have a new
question - the IV bolus question BUT with different initial conditions =
not all of the drug is in the central compartment at time right after
the infusion termination, some is in each of the compartments and
excreted. Treat it as Two questions, before and after termination. mm
Michael Makoid, Ph.D.
Professor and Chair
Department of Pharmacy Sciences
School of Pharmacy and Allied Health Professions
Creighton University
2500 California Plaza
Omaha, NE 68178
Voice 402 280 2952
Fax 402 280 1883
Cell 402 250 4618
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[Two replies - db]
From: "Edward Dennis Bashaw"
Date: Mon, 15 Oct 2001 21:57:52 -0400
To: david.aaa.boomer.org
Subject: Re: PharmPK Equation for 3 compartment modelling
The following message was posted to: PharmPK
Laszlo,
You are being too literal in your interpretation of time. In your equation
at time =0 C=0, this is the initial condition prior to the infusion. Once
the infusion begins the clock starts. Your error is that you are counting
DOWN time instead of it proceeding in a forward manner. At the end of a 5
min infusion the time is not "0" but "5". Also you are confusing rate of
infusion with dose. Rate is the measure of transfer of drug or amount into
the system. Rate in this equation is not an end unto itself as you have it
here, you need to re-consider the left side of the equation as your
simplification of "R" is the main problem here. Personally, I would
recommend you consult a copy of Shargel & Yu's (or other PK) textbook for a
more detailed explanation, using the two compartment model as a start and
then expanding it to fit your needs by adding in the appropriate exponents.
Dennis Bashaw, Pharm.D.
Team Leader, Pharmacokinetics
US Food and Drug Administration
---
From: "Hans Proost"
Date: Wed, 17 Oct 2001 10:09:43 MET
To: david.aaa.boomer.org
Subject: Re: PharmPK Equation for 3 compartment modelling
The following message was posted to: PharmPK
Dear Dr. Hollos,
As pointed out by others, a steady-state plasma concentration will
be obtained only if one starts with a bolus dose equal to Css.Vc
(i.e. a loading dose), followed by a continuously changing infusion
rate R according to your equation, which is known as BET (Bolus-
Elimination-Transfer) (Lauven PM, Der Anesthesist 1982;31:15-20,
in German):
R= Css*Vc(k10+[k12*e^(-k21*t)]+[k13*e^(-k31*t)])
During the infusion the plasma concentration will be exactly
constant (provided that your patient behaves according to this
model).
After stopping the infusion, the plasma concentration will decrease
in a similar way as after a single bolus, following a three-
exponential profile. The rate constants will be equal to that after
bolus injection, but the intercept will not! The intercepts are
dependent on the duration of the infusion (of course, asymptotically
approaching a constant value after long infusion).
These intercepts can be obtained from convolution of drug input I
(bolus dose and BET infusion) and the 'unit impulse response' UIR
(plasma concentration profile after bolus administration of a unit
dose):
C(t) = integral(0-t) [ I(tau) . UIR(t-tau) d tau ]
This equation for the plasma concentration profile C(t) results in
rather complicated derivations (taking into account the bolus dose
and the infusion from time zero to T_inf), which need quite a lot of
paper to write down. But it certainly works.
Best regards,
Hans Proost
Johannes H. Proost
Dept. of Pharmacokinetics and Drug Delivery
University Centre for Pharmacy
Antonius Deusinglaan 1
9713 AV Groningen, The Netherlands
tel. 31-50 363 3292
fax 31-50 363 3247
Email: j.h.proost.-at-.farm.rug.nl
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The following message was posted to: PharmPK
Dear All
I would like to say thank to all who responded to my original e-mail
regarding the mathematical modelling of three compartment
pharmacokinetics of iv administered drugs. The original equation
describing the serum level time profile was published in an
anesthesiology journal in 1998, which explained how the propofol
target-controlled infusion (Diprifusor) worked. Now I understand
from your letter, that it is not simple enough to be described by only
one simple equation.
Thanks again, Laszlo
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