- On 15 May 2001 at 10:48:39, Thierry Buclin (Thierry.Buclin.-at-.chuv.hospvd.ch) sent the message

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The following message was posted to: PharmPK

Dear PharmPK users,

In a clinical experiment about an oral drug, we have been able to

sample both the blood and a peripheral compartment, namely CSF. A

2-compartment model with 1st order absorption nicely fits the data.

The AUC in the peripheral compartment is expected to reflect the

brain drug exposure. However, I can't find a formula to express this

AUC in terms of model parameters (either micro or macroconstants).

Does anyone know the theoretical value of AUC in a peripheral

compartment ?

Thierry BUCLIN, MD

Division of Clinical Pharmacology

University Hospital CHUV

CH 1011 Lausanne - SWITZERLAND - On 15 May 2001 at 13:42:22, "He, Yan-Ling" (YHE.-a-.PARTNERS.ORG) sent the message

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The following message was posted to: PharmPK

Dear Dr. Buclin,

It sounds like a wonderful data set. However, I do not think that you can find

"a theoretical value of AUC" in a peripheral compartment. As you

mentioned, the

AUC is representative of brain drug exposure, but rather than finding "an

absolute value", the AUCs are very much useful when you compare the results

under various conditions, ex., at different doses etc. To

characterize the time

course in CSF compartment, I would suggest you to do the non-compartment

analysis. You may find the paper we published 10 years ago useful.

Good luck!

He et al.: Effect of Folinic Acid on Tissue Residence and Excretion of

Methotrexate in Rats. Drug. Metab. Dispos. 19: 729-734 (1991).

Yan-Ling He, Ph.D.

Department of Anesthesia and Critical Care

Massachusetts General Hospital

Harvard Medical School

55 Fruit St. Boston

MA 02114-2696

USA

E-mail: Yhe.-at-.partners.org - On 16 May 2001 at 10:47:04, David Bourne (david.-at-.boomer.org) sent the message

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[A few replies - db]

From: Nick Holford

Date: Wed, 16 May 2001 07:48:01 +1200

To: david.-a-.boomer.org

Subject: Re: PharmPK AUC in peripheral compartment

The following message was posted to: PharmPK

"Thierry Buclin (by way of David Bourne)" wrote:

> In a clinical experiment about an oral drug, we have been able to

> sample both the blood and a peripheral compartment, namely CSF. A

> 2-compartment model with 1st order absorption nicely fits the data.

> The AUC in the peripheral compartment is expected to reflect the

> brain drug exposure. However, I can't find a formula to express this

> AUC in terms of model parameters (either micro or macroconstants).

> Does anyone know the theoretical value of AUC in a peripheral

> compartment ?

The usual assumption when peripheral compartment concs are not

directly measured is that at steady state the conc in the peripheral

compartment (C2) is the same as the conc in the central compartment

(C1). Clearly over any period of time at steady state the concs in

both compartments are the same and so is the AUC in each compartment.

The integral ie. the AUC, from 0 to infinity is the same for any drug

entering the central compartment (single or multiple doses).

Generally:

C1*CLic = C2*CLic

or in microconstants

C1*K12*V1 = C2*K21*V2

At SS then C1=C2 by assumption so:

CLic=CLic

and

K12*V1 = K21*V2

The well known formula for AUC in the central compartment (AUCc) is:

AUCc=F*Dose/CL1

where CL1 is the exit clearance from the central compartment and

F*Dose is the integral of the amount of drug leaving the central

compartment.

By analogy, the AUC in the peripheral compartment is determined its

exit clearance and by the amount of drug that leaves it. CL2 is the

exit clearance from the peripheral compartment ie. V2*K21 or CLic.

For AUC 0-inf the amount leaving is the same as the amount that

enters which is determined by the amount in the central compartment

(F*Dose) and the fraction of this amount which exits to the

peripheral compartment (Fp). Fp is simply Clic/CL1. Therefore:

AUC2=F2*(F*Dose)/CL2

=K12*V1/CL1*(F*Dose)/(K21*V2)

=Clic/CL1*(F*Dose)/Clic

=F*Dose/CL1

=AUC1

i.e. AUC2=AUC1.

This result demonstrates why it is unnecessary to consider tissue

compartment concentrations if you believe that the drug effect you

are interested in is determined by the integral of concentration in

any tissue e.g. brain. Simply measuring central compartment concs is

sufficient to predict AUC in the central compartment which is the

same in any compartment.

The assumption that C1=C2 at steady state can be relaxed so that

P*C1=C2 where P might be a partition coefficient and/or reflect

elimination of drug from the peripheral compartment that does not

return to the central compartment. In that case, AUC2=P*AUC1, which

doesn't change anything except for a scale factor.

--

Nick Holford, Divn Pharmacology & Clinical Pharmacology

University of Auckland, 85 Park Rd, 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

---

From: Michael Leibold

Date: Wed, 16 May 2001 04:28:06 -0700

To: david.aaa.boomer.org

Subject: Re: PharmPK AUC in peripheral compartment

The following message was posted to: PharmPK

Dr. Buclin,

In the basic linear compartmental pharmacokinetic sense, the AUC of

the concentration curve in the peripheral compartment could be obtained

by integrating the equation describing concentrations in the peripheral

compartment.

The Laplace transformed expressions for amounts in the central and

peripheral compartment following first order absorption into the central

compartment are:

X1s= [KaFXo(s+k21)]/[(s+Ka)(s+ a)(s+ b)]

X2s= [K12KaFXo[/[(s+ Ka)(s+ a)(s+ b)]

The inverse Laplace transform the amount in the peripheral compartment

yields:

X2= K12KaFXo[ Ae-Kat + Be-at + Ge-bt ]

A= 1/[(a - Ka)(b - Ka)]

B= 1/[(Ka- a)(b - a)]

G= 1/[(Ka- b)(a - b)]

The equation for the concentration in the peripheral compartment could

be obtained by dividing the above equation by Vp. The integral of this equation

would represent the AUCp.

AUCp(o->inf)= [K12KaFXo/Vp][ A/Ka + B/a + G/b ]

The above equations can be easily modified to the multiple dose case if

necessary.

Mike Leibold, PharmD, RPh

ML11439.-at-.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

---

From: "Paul S. Collier"

Date: Wed, 16 May 2001 14:11:41 +0100

To: david.-a-.boomer.org

Subject: Re: PharmPK AUC in peripheral compartment

The following message was posted to: PharmPK

Thierry,

For a two compartment model with elimination from the central

compartment:

V2 = V1*k12 / k21

A = K12*D / (V2*(beta-alpha))

B = K12*D / (V2*(alpha-beta))

C(2) = A*exp(-alpha*t) + B*exp(-beta*t)

AUC =( A / alpha) + (B / beta)

The problem however is that this estimate of C(2), i.e. the

concentration in the peripheral compartment, is based on V2 which is an

apparent volume and not the real volume. The predicted concentrations

are therefore unlikely to reflect the observed concentrations of drug

in CSF.

Paul

Dr P.S. Collier

School of Pharmacy

The Queen's University of Belfast

97 Lisburn Road

Belfast BT9 7BL

N. Ireland, U.K

Tel: +44 (0)28 90 272009

FAX: +44 (0)28 90 247794

Email: p.collier.aaa.qub.ac.uk

---

From: Bert.L.Lum.-a-.Stanford.EDU, "Pharm.D."

Date: 16 May 2001 07:26:24 -0800 (PST)

To: david.aaa.boomer.org

Subject: Re: PharmPK AUC in peripheral compartment

The following message was posted to: PharmPK

Thierry Buclinsaid:

For a compartmental model with CSF parameters, you might look at this paper

which did an IV drug with a CNS compartment.

Cancer Chemother Pharmacol 1996;37(3):195-202

Cerebrospinal fluid pharmacokinetics and penetration of continuous

infusion topotecan in children with central nervous system tumors.

Baker SD, Heideman RL, Crom WR, Kuttesch JF, Gajjar A, Stewart CF.

http://www.qub.ac.uk/pha/index.html

--- - On 16 May 2001 at 16:46:32, Nick Holford (n.holford.aaa.auckland.ac.nz) sent the message

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The following message was posted to: PharmPK

> From: Nick Holford

> The assumption that C1=C2 at steady state can be relaxed so that

> P*C1=C2 where P might be a partition coefficient and/or reflect

> elimination of drug from the peripheral compartment that does not

> return to the central compartment. In that case, AUC2=P*AUC1, which

> doesn't change anything except for a scale factor.

I should also have noted that the factor P can reflect the situation

when intercompartmental clearance is not the same in both directions.

CSF has a bulk flow return to the systemic circulation (lymphatic

fluid). CSF secretions do not necessarily have the same conc as in

plasma water (pw) because of active secretion processess. In this

case CLic pw->csf can be different from CLic csf-> plasma and the

steady state ratio of CSF/pw is not necessarily 1 (sometimes observed

to be quite a lot less than 1).

By definition at steady state:

Cplasma x CLic pw->csf = Ccsf x CLic csf->pw

therefore if the steady state ratio of Ccsf/Cpw is P then

P= Ccsf/Cpw->CLic pw->csf/CLic csf->pw

If the component of CLic pw -> csf that is via CSF secretion is

negligible then CLic pw->csf is approximately first order and

AUCcsf=P*AUCpw.

If CSF secretion of drug is not negligible in comparison to simple

first-order input by diffusion then the kinetics are no longer

linear. The only way I know to predict AUCcsf is to integrate Ccsf

numerically e.g.

dCcsf/dt = (Cpw*(CLdiffusion + Vmax/(Km+Cpw)) - Ccsf*(CLdiffusion +

CLbulkflow)/Vcsf

dAUCcsf/dt = Ccsf

--

Nick Holford, Divn Pharmacology & Clinical Pharmacology

University of Auckland, 85 Park Rd, 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 - On 18 May 2001 at 10:56:07, Michael Leibold (ML11439.at.goodnet.com) sent the message

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The following message was posted to: PharmPK

Dr. Buclin,

As per my previous communication, the inverse Laplace transform of the

amount in the peripheral compartment of a two compartment model yields:

X2= K12KaFXo[ Ae-Kat + Be-at + Ge-bt ]

A= 1/[(a - Ka)(b - Ka)]

B= 1/[(Ka- a)(b - a)]

G= 1/[(Ka- b)(a - b)]

The equation for the concentration in the peripheral compartment could

be obtained by dividing the above equation by Vp. The integral of this equation

would represent the AUCp.

AUCp(o->inf)= [K12KaFXo/Vp][ A/Ka + B/a + G/b ]

I thought this might be interesting if taken a bit further. The

multiple dose

form of the above equation is:

AUCp(o->inf)= [K12KaFXo/Vp][A(1 - e-nKaT)/Ka(1 - e-KaT) +

B(1 - e-naT)/a(1 - e-aT)

G(1 - e-nbT)/b(1 - e-bT) ]

A three compartment version would be:

KaFXoe-Kat

\

k12 k13

Cpt2<------->Cpt1<------->Cpt3

k21 \ k31

\->k10

This is a linear mammillary model with elimination from a central

compartment which 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 matrix 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:

[(s+k13+k10+k12) -k21 -k31][X1s] [(KaFXo/(s+ka))]

[ -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. Solving for the third compartment

results in:

X3s= [KaFXoK13(s+k21)]/[(s+ka)(s+a)(s+b)(s+g)]

Taking the inverse Laplace transform and dividing by Vp results

in an equation describing the concentration in the third compartment

as a function of time:

C3= [KaFXoK13/Vp]*[(k21-Ka)e-Kat/(a-Ka)(b-Ka)(g-Ka) +

(k21-a)e-at/(Ka-a)(b-a)(g-a) +

(k21-b)e-bt/(Ka-b)(a-b)(g-b) +

(k21-g)e-gt/(Ka-g)(a-g)(b-g)

The AUCp can then be derived from taking the integral of above:

AUCp(o->inf)= [KaFXoK13/Vp]*

[(k21-ka)/ka(a-ka)(b-ka)(g-ka) +

(k21-a)/a(ka-a)(b-a)(g-a)

(k21-b)/b(ka-b)(a-b)(g-b)

(k21-g)/g(ka-g)(a-g)(b-g) ]

This can be converted to the multiple dose case by inserting the

multiple dose multipliers as done previously for the two compartment

model.

Mike Leibold, PharmD, RPh

ML11439.-at-.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) Wagner, J., Fundamentals of Clinical Pharmacokinetics, Hamilton, Drug

Intelligence Publications 1975 - On 20 May 2001 at 22:01:26, Michael Leibold (ML11439.-a-.goodnet.com) sent the message

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The following message was posted to: PharmPK

Dr.Buclin,

In writing the equation for AUCp(o->inf) for the third compartment,

I inadvertently omitted some of the + signs:

"The AUCp can then be derived from taking the integral of above":

AUCp(o->inf)= [KaFXoK13/Vp]*

[(k21-ka)/ka(a-ka)(b-ka)(g-ka) +

(k21-a)/a(ka-a)(b-a)(g-a) +

(k21-b)/b(ka-b)(a-b)(g-b) +

(k21-g)/g(ka-g)(a-g)(b-g) ]

Mike Leibold, PharmD, RPh

ML11439.at.goodnet.com - On 21 May 2001 at 12:25:34, Walt Woltosz (walt.-at-.simulations-plus.com) sent the message

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This elegant approach makes the asssumption that Ka is a constant. We

know that sometimes this can be assumed, but sometimes it cannot.

When it can, this approach should work well.

But . . .

The industry likes to treat Ka as a constant. Ka is most certainly

not a constant - ever - for any drug - even one that undergoes only

simple passive diffusion. We often use average Ka and get away with

it, but when we do so, it should be with our eyes open and an

awareness that we are making a simplifying assumption that will not

always hold.

Ka varies with location within the gut, and with time at any

location. At any given time, for the simplest case of passive

diffusion (the majority of drugs), the instantaneous rate of drug

transfer across the apical membrane into the enterocytes is a

function of the concentration gradient across the membrane - which

changes with time and location. It can go in either direction.

In addition, the rate of drug transfer across the basolateral

membrane is a function of the concentration gradient between the

enterocyte and the blood, and will not be the same as that across the

apical membrane. It can also go in either direction. In fact,

depending on the pharmacokinetic characteristics of the drug

(especially terminal half life and plasma protein binding), you can

actually have drug moving from drug to enterocytes to lumen - a net

secretion - even from an IV dose. We usually fit IV PK parameters

ignoring this, because we usually can - but for some drugs, it is an

oversimplification. We assume absorption affects pharmacokinetics,

but not vice versa. But pharmacokinetics can actually affect

absorption.

When you add carrier-mediated influx and/or efflux transport,

pH-dependent permeability and solubility (and dissolution), and you

have a very complex situation. We believe that only through a

sophisticated absorption/pharmacokinetics simulation that accounts

for all of these interacting effects can one properly assess the

behavior of drugs that experience them.

Of course, if you're lucky, and your drug is one of the easy ones,

then you can use the simplified analysis - but do you really know if

your drug is not affected by any of these factors?

Walt Woltosz

Chairman & CEO

Simulations Plus, Inc. (SIMU)

1220 W. Avenue J

Lancaster, CA 93534-2902

U.S.A.

http://www.simulations-plus.com

Phone: (661) 723-7723

FAX: (661) 723-5524

E-mail: walt.aaa.simulations-plus.com - On 21 May 2001 at 14:15:43, Thierry Buclin (Thierry.Buclin.-a-.chuv.hospvd.ch) sent the message

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The following message was posted to: PharmPK

Hi Nick

Many thanks for your explanations, which appeared crystal-clear, as

many of your interventions on PharmPK.

I was slowly moving towards the conclusion that AUC2 = AUC1 if

concentrations in compartment 1 were only available, which is the

usual case.

>> The assumption that C1=C2 at steady state can be relaxed so that

>> P*C1=C2 where P might be a partition coefficient and/or reflect

>> elimination of drug from the peripheral compartment that does not

>> return to the central compartment. In that case, AUC2=P*AUC1, which

>> doesn't change anything except for a scale factor.

As concentrations measured in compartment 2 are available, I feel

your suggestion of introducing a partition term P appropriate, as

there may be a substantial difference between the peripheral

concentrations extrapolated from plasma (ie Z2/V2, where Z2 = amount

in cpt 2) and the concentrations really measured in CSF; moreover,

this difference has many chances to correspond to a proportionality

factor. Now a point of detail : would it be equivalent to introduce

in the model a V2 term, instead of simply stating that V2=V1*K12/K21

and using the term P ?

Best regards

Thierry Buclin

Thierry BUCLIN, MD

Lecturer, consulting physician and clinical researcher

Division of Clinical Pharmacology

University Hospital CHUV - Beaumont 633

CH 1011 Lausanne - SWITZERLAND

Tel: +41 21 314 42 61 - Fax: +41 21 314 42 66 - On 21 May 2001 at 21:52:31, Nick Holford (n.holford.-at-.auckland.ac.nz) sent the message

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The following message was posted to: PharmPK

Thierry,

"Thierry Buclin (by way of David Bourne)" wrote:

>

> I was slowly moving towards the conclusion that AUC2 = AUC1 if

> concentrations in compartment 1 were only available, which is the

> usual case.

Seems reasonable it would take time to reach a final conclusion. AUC1

is only equal to AUC2 at steady state and that takes at least 4

half-lives :-)

> As concentrations measured in compartment 2 are available, I feel

> your suggestion of introducing a partition term P appropriate, as

> there may be a substantial difference between the peripheral

> concentrations extrapolated from plasma (ie Z2/V2, where Z2 = amount

> in cpt 2) and the concentrations really measured in CSF; moreover,

> this difference has many chances to correspond to a proportionality

> factor.

Its not clear to me what you mean here. You should not try to fit the

plasma data to a 2 cpt model and then hope that V2 or CLic for the

plasma conc model will correspond to the kinetics of drug in CSF. You

should treat CSF like an effect compartment e.g. as a DE model:

Assuming the plasma conc data supports a 2 cpt model:

dC1/dt= (Ratein + CLIc*C2 - (CL + CLic)*C1)/V1

dC2/dt= CLic*(C1-C2)/V2

Then fit the CSF concs with this:

dCcsf/dt=CLcsf*(P*C1-Ccsf)/Vcsf

CLcsf is the intercompartmental clearance between plasma and CSF and

Vcsf is the apparent volume of CSF.

You can either assume an anatomically reasonable value for Vcsf or

preferably just estimate the half-life of equilibration of plasma and

CSF (Teq=ln(2)*VCsf/CLcsf):

dCcsf/dt=ln(2)/Teq*(P*C1-Ccsf)

> Now a point of detail : would it be equivalent to introduce

> in the model a V2 term, instead of simply stating that V2=V1*K12/K21

> and using the term P ?

You can parameterize as you wish but I strongly recommend using

clearances and volumes rather than micro-rate constants.

See: Anderson BJ, Holford NHG, Woolard G. Paracetamol plasma and

cerebrospinal fluid pharmacokinetics in children. Br J Clin

Pharmacol. 1998; 46: 237-243

for a specific example.

--

Nick Holford, Divn Pharmacology & Clinical Pharmacology

University of Auckland, 85 Park Rd, 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 - On 22 May 2001 at 11:08:01, Michael Leibold (ML11439.-a-.goodnet.com) sent the message

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The following message was posted to: PharmPK

Dr.Buclin,

Reducing the AUCp equation in the textbook fashion results in the following

equation:

AUCp(0->inf)= k12*FXo/[Vp*k21*k10]

Comparing this to the reduced form of the AUC equation for the central

compartment, indicates that the AUCp is related to AUC by a constant.

AUC(o->inf)= F*Xo/Vc*k10

AUCp/AUC= [k12*Vc]/[Vp*k21]

From this I can see that you could calculate AUC from plasma concentrations

and then use the last equation to determine AUCp. However, your question was

originally concerned with an equation for AUCp involving two compartment

parameters. Alternatively, you could fit the plasma concentrations to

the two or

three compartment model and use these parameters to calculate the AUC in

the peripheral compartment, and calculate the AUCp this way.

Mike

Leibold, PharmD, RPh

ML11439.-at-.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) Wagner, J., Fundamentals of Clinical Pharmacokinetics, Hamilton, Drug

Intelligence Publications 1975

4) Godfrey, K. Compartmental Models and Their Application, New York, Academic

Press 1983 - On 24 May 2001 at 16:25:15, Michael Leibold (ML11439.-a-.goodnet.com) sent the message

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The following message was posted to: PharmPK

Dr. Buclin,

I see how the AUCp/AUC equation simplifies at steady state.

At steady state the equation relating Vss to Vc is:

Vss= [(k12 + k21)/k21]* Vc

Vss= Vc + Vp

As a result, at steady state:

Vp= [(k12 +k21)/k21]*Vc - Vc

Vp= (k12/k21)*Vc

This also indicates that the AUCp/AUC ratio is unity at steady state:

AUCp/AUC= [k12*Vc]/[Vp*k21]= [k12*Vc]/[(k12/k21)*Vc*k21]= Vc/Vc= unity

Mike Leibold, PharmD, RPh

ML11439.at.goodnet.com

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