- On 10 May 2000 at 22:31:47, Chantal Le guellec (leguellec.aaa.med.univ-tours.fr) sent the message

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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 - On 11 May 2000 at 20:56:10, David_Bourne (david.at.boomer.org) sent the message

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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. - On 14 May 2000 at 22:29:16, exfamadu.-at-.savba.sk sent the message

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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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