- On 15 Oct 2001 at 11:01:00, "Laszlo Hollos" (laci.-a-.hollos.net) sent the message

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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 - On 15 Oct 2001 at 23:39:01, David_Bourne (david.at.boomer.org) sent the message

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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 - On 17 Oct 2001 at 10:54:38, David Bourne (david.-a-.boomer.org) sent the message

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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 - On 28 Oct 2001 at 14:56:24, "Laszlo Hollos" (laci.-at-.hollos.net) sent the message

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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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