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Dear all,
Could someone indicate me equations describing plasma concentration of a
drug following 3-exponential pharmacokinetics and administered as
IV-infusion, expressed as a function of intercompartmental constants and
volumes of each compartment.We were requested by anesthesiologists of
our institution who wish to implement such a model for
computer-controlled infusion of anesthetic drugs.
Many thanks for your help.
Chantal Le Guellec
Laboratoire de pharmacologie et toxicologie
CHU de Tours, France
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[Two replies - db]
Date: Thu, 11 May 2000 00:48:47 -0700 (MST)
X-Sender: ml11439.aaa.pop.goodnet.com
To: PharmPK.-at-.boomer.org
From: ml11439.-at-.goodnet.com (Michael J. Leibold)
Subject: Re: PharmPK 3-compartmental model
Chantal,
This is actually an answer from a previous discussion, but
also applies to your question:
Triexponential refers to the presence of three slopes in the
log plasma concentration versus time curve.
Cp = Ae-at + Be-bt + Ge-gt
In linear compartmental pharmacokinetics this is interpreted as a
three-compartment model:
Xo
\
k12 k13
Cpt2<------->Cpt1<------->Cpt3
k21 \ k31
\->k10
This is a linear mammillary model with elimination from a central
compartment wich usually represents the plasma compartment. The system
is described by linear first order differential equations describing
the change in compartmental amounts of drug with time:
dX1/dt= -(k13+k10+k12)X1 + k21X2 +k31X3
dX2/dt= k12X1 - k21X2
dX3/dt= k13X1 -k31X3
This system of differential equations can be solved by Laplace
transforms and matrix algebra. The matix representation of the Laplace
transformed system of differential equations is:
[SI-A][Xs]= [Us]
Where the Laplace transform of the system of differential equations
above is equal to the Laplace transformed vector of dose input, for
example: (Ko/s)(1-e-Ts).
[(s+k13+k10+k12) -k21 -k31][X1s] [(Ko/s)(1-e-Ts)]
[ -k12 (s+k21) 0 ][X2s] = [ 0 ]
[ -k13 0 (s+k31)][X3s] [ 0 ]
This can be solved by matrix algebra to yield Laplace transformed
quantities of each compartment, most important of which is the central
compartment quantity (X1s):
X1s= [(Ko/s)(1-e-Ts)(s+k21)(s+k31)]/[(s+a)(s+b)(s+g)]
The inverse Laplace transform of the above results in an equation
describing the amount in the central compartment as function of time,
and dividing this by the Vc (volume of the central compartment) results
in the equation for the plasma concentration over time:
Cp= Ko(k21-a)(k31-a)(1-e-aT)(e-at')/[a(b-a)(g-a)Vc] +
Ko(k21-b)(k31-b)(1-e-bT)(e-bt')/[b(a-b)(g-b)Vc] +
Ko(k21-g)(k31-g)(1-e-gT)(e-gt')/[g(a-g)(b-g)Vc]
The above equation describes the triexponential plasma concentration
curve of a drug being administered by a an intermittent infusion, where
T= infusion time and t'= time after infusion.
The multiple dose form of the above equation is:
Cp= Ko(k21-a)(k31-a)(1-e-aT)(1-e-naTau)(e-at')/[a(b-a)(g-a)Vc(1-e-aTau)] +
Ko(k21-b)(k31-b)(1-e-bT)(1-e-nbTau)(e-bt')/[b(a-b)(g-b)Vc(1-e-bTau)] +
Ko(k21-g)(k31-g)(1-e-gT)(1-e-ngTau)(e-gt')/[g(a-g)(b-g)Vc(1-e-gTau)]
Examples of drugs which exhibit triexpoential plasma concentration
curves are vancomycin and aminoglycosides. However, this is difficult to
detect as the curves really appear two compartment. Vancomycin has an
initial rapid distribution phase (T1/2~=7min) which can be detected with
numerous plasma samples and careful analysis. Aminglycosides have a long
washout phase due to renal tissue binding and release which can be detected
with plasma samples taken long after the drug has been administered to
detect a slow decline (T1/2~=100-200 hours) in plasma concentrations as drug
is released from renal tissue.
Mike Leibold, PharmD, RPh
ML11439.-a-.goodnet.com
References
1) Gibaldi, M., Perrier, D., Pharmacokinetics, New York, Marcel Dekker
1975
2) Gibaldi, M., Perrier, D., Pharmacokinetics 2nd ed, New York, Marcel Dekker
1982
3) Schumacher, G.E., Therapeutic Drug Monitoring, Norwalk, Appleton&
Lange 1995
4) Evans, W.E., Schentag, J.J., Jusko, W.J., Applied Pharmacokinetics 3rd ed,
Vancouver, Applied Therapeutics 1992
5) Wagner, J., Fundamentals of Clinical Pharmacokinetics, Hamilton, Drug
Intelligence Publications 1975
---
Reply-To: "David Foster"
From: "David Foster"
To:
Subject: Re: PharmPK 3-compartmental model
Date: Thu, 11 May 2000 07:54:18 -0700
X-Priority: 3
Are you interested in fitting a sum of three exponential to the data and
using this to estimate plasma concentration, or fitting a three compartment
model to the data? If you are using sums of exponentials and fitting this
to concentration data, the equation is:
c(t) = A0 - A1*exp(-a1*t) - A2*exp(-a2*t) - A3*exp(-a3*t)
where A0 - A1 - A2 - A3 = 0 (assuming there is no drug in the system at time
zero).
There are occasions, even when using a constant infusion directly into
plasma, where
this constraint may have to be relaxed. In addition, there are occasions
when a slight
delay is required. IN this case, t in the equation for c(t) is replaced by
c - tlag where tlag is the time of the delay.
If you are using a three compartment model, you just need to structure your
model according to how you want the compartments hooked together, then
simulate the experiment on the model. Software such as SAAM II can do
either of these quite easily.
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Dear Dr. Le Guellec
I recommend to use a model describing the behavior of the anesthetic
drug during and after the end of a short- or long-time IV-infusion (1),
instead of a model describing only a post infusion part of the
concentration-time profile of the drug. In general, such a model can be of
any order (lower or higher than a 3rd-order model). This model allows to
utilize the information about the behavior of the drug contained in the
measurements over the whole time interval from the beginning of the
infusion to the end of the sampling and that even in the earliest phase
representing intravascular mixing. Moreover this model and a computer-
controlled pump allow to adjust an adequate flow of the drug into the
patient body.
1. Durisova M., et al., Meth. Find. Exp. Clin. Pharmacol., 1998, 20, 217.
With best regards,
Maria Durisova
Dipl. Engineer Maria Durisova D.Sc.
Senior Research Worker
Scientific Secretary
Institute of Experimental Pharmacology
Slovak Academy of Sciences
SK-842 16 Bratislava
Slovak Republic
http://nic.savba.sk/sav/inst/exfa/advanced.htm
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Copyright 1995-2010 David W. A. Bourne (david@boomer.org)