- On 20 Jan 2006 at 17:32:26, cedric.vinson.-a-.fr.netgrs.com sent the message

Back to the Top

The following message was posted to: PharmPK

Dear all,

I have 2 questions concerning PBPK organ models

* on the well-sirred model :

CLh= Qh*fub*CLint/ (Qh+fub*CLint)

- question 1 : is fub the fraction unbound in blood or in plasma?

I was convinced it was fu in blood (=fu plasma/Rb), but the book of

M.Rowland and T.Tozer states p.166 that is fu in plasma. I'm a bit

confused

now... Could someone clarify this point?

- question 2 : if the fub is fu in blood, it is possible in theory to

have a

fub superior to 1 (with for exemple with fup=1 and a Rb=0.7)

I don't know if this case can happen in reality or not, but from a

mathematical point of view, it's possible In the well stirred equation,

should I use in this case the value (fup/Rb)>1 or should I limit the

fub to

1?

I ask that because for products in early development, you can have a

high fu

(predicted or measured) without any idea of the blood/plasma ratio

(Rb). And

i've read somewhere that Rb is more often close to (1-haematocrit, so

let's

say 0.6) than to 1, so 0.6-0.7 is my default value when i don't know the

Rb...

* on the parallel tubes model :

Could someone know some references that contains this model expressed in

ordinary differential equations (to use in a numerical simulation

software)

? In all papers I found, the model is expressed in the integrated/

analytical

form (I'm not sure of the correct english word for that), but i didn't

managed to find it in ODE form...

Best regards

C.Vinson

[The text "Applied Pharmacokinetics and Pharmacodynamics", 4th ed.,

page 133 has equations for CL(H) for both models. They look like

integrated (analytical) equations but they are just 'defining' a

value for the parameter which can be used in the differential

equation. i.e. CL = ...

If I have some parts of the recent clearance discussion correct you

can write the differential equation as:

dX/dt = CL(H) x Cx (where Cx is concentration in blood or plasma

depending on the answers to question 1 and 2.)

Since Cx = X/Vx

dX/dt = CL(H) x X/Vx (where Vx is ...). Be careful of your units and

get a clear idea about the answers to questions 1 and 2 - db] - On 26 Jan 2006 at 16:31:01, cedric.vinson.aaa.fr.netgrs.com sent the message

Back to the Top

The following message was posted to: PharmPK

Thanks for your answer David

I think i have the answer for question 1 : it could be some sort of

typo in

the book of Rowland, because in the same book (chapter "Definition of

symbols"), the fub is fup/Rb. So i'm quite confident again that I

should use

the fu blood in the well-stirred model. But the question 2 is still

unresolved...

Concerning your feeling on the differential form of PT model, i'm not

sure

you're right, because in the differential form of the well-stirred

model,

dX/dt is not equal to CL(H) x Cx = (Qh*fub*CLint/ (Qh+fub*CLint))*Cx

what i have understood is that the formula Qh*fub*CLint/(Qh

+fub*CLint) is an

integrated formula describing the *blood* clearance due to hepatic

elimination (cf. chapter 9 of Gibaldi and perrier, S Blood

clearance : in

this chapter, the authors states that the formula is an integrated

form of a

PBPK model...)

But after the long debate on clearance and elimination rate, i'm not

sure

that someone want to discuss that sort of question anymore now... ^_^'

[Interesting follow-up. If you are looking at the whole body you can

use CL(H) - model of your choice equation in a differential or

integrated equation.

For example C = (Dose/V) x exp(-CL(H) x t/V) (did I get that right? i

would use the k version)

or dC/dt = - CL(H) x C/V

However if you are interested in the fine detail of how the CL(H)

equation is derived you should look in the primary literature. For

example the book by Kwon (http://www.boomer.org/pkin/book.html)

p90-93 discusses three different models. I'm sure there are plenty of

differential equations included in those primary references ;-)

As an aside I have started to develop an applet to display Cp versus

time using either the well stirred or parallel tube model - the

current version seems to have a bug ;-( but should be fixed tonight db] - On 30 Jan 2006 at 17:29:16, "Hans Proost" (j.h.proost.aaa.rug.nl) sent the message

Back to the Top

The following message was posted to: PharmPK

Dear Cedric,

You wrote:

> * on the well-sirred model :

> CLh= Qh*fub*CLint/ (Qh+fub*CLint)

> - question 1 : is fub the fraction unbound in blood or in plasma?

> I was convinced it was fu in blood (=fu plasma/Rb), but the book of

> M.Rowland and T.Tozer states p.166 that is fu in plasma. I'm a bit

> confused now... Could someone clarify this point?

and in a next message:

> I think i have the answer for question 1 : it could be some sort of

> typo in

> the book of Rowland, because in the same book (chapter "Definition of

> symbols"), the fub is fup/Rb. So i'm quite confident again that I

> should use the fu blood in the well-stirred model.

In my opinion the equation in the book of Rowland and Tozer (Clinical

Pharmacokinetics, in my view the best book in the field) is correct. The

well-stirred model describes the situation with respect to blood, and

all

volume terms refer to blood: CL_b,H and Q_H. Please note that

according to

the definition on page xiii of the book of Rowland and Tozer, fub is

defined

as the ratio of the unbound concentration in plasma and the total drug

concentration in blood, thus fub = Cu / Cb. This may look strange,

but makes

perfectly sense in equations 13 and 14 on page 166. This can be shown

after

multiplying numerator and denominator of the right side of equation

14 by

Cb, i.e. the concentration in the blood leaving the liver, yielding

E_H = Cu * CL_int / ( Cb * Q_H + Cu * CL_int)

CL_int is the intrinsic clearance that relates rate of metabolism to

unbound

concentration at the enzyme site; in the well-stirred model this unbound

concentration is assumed to be the same as the unbound concentration

leaving

the liver.

So, the product 'Cu * CL_int' is the rate (amount / time) of

elimination in

the liver. The product 'Cb * Q_H' is the rate (amount / time) of drug

passing the liver unaltered. The extraction ratio E_H is the ratio of

the

rate of elimination in the liver divided by the rate of drug entering

the

liver, which equals the sum of the rate of elimination and the rate

of drug

leaving the liver, as shown in figure 11-1 on page 159.

In conclusion, Rowland and Tozer are fully correct.

> - question 2 : if the fub is fu in blood, it is possible in theory to

> have a

> fub superior to 1 (with for exemple with fup=1 and a Rb=0.7)

> I don't know if this case can happen in reality or not, but from a

> mathematical point of view, it's possible In the well stirred

equation,

> should I use in this case the value (fup/Rb)>1 or should I limit the

> fub to 1?

As stated above, fub = Cu / Cb, and the value of fub can exceed 1,

since Cb

can be smaller than Cu, in case of no or low plasma protein binding

(so Cu

close to C), and no or low distribution into red blood cells (so, Cb

lower

than C).

> Could someone know some references that contains this model

expressed in

> ordinary differential equations (to use in a numerical simulation

> software)

and in a next message, answering to David:

> dX/dt is not equal to CL(H) x Cx = (Qh*fub*CLint/ (Qh+fub*CLint))*Cx

Why is this equation not correct? Since CL_b,H is defined as a blood

clearance, this equation is correct only if Cx refers to the blood

concentration. And since the extraction refers to the amoun entering the

liver, Cx refers to the concentration of drug in the blood entering the

liver. If you include this in your equation, I would say that you

have the

correct formula.

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.aaa.rug.nl - On 1 Feb 2006 at 16:44:35, cedric.vinson.at.fr.netgrs.com sent the message

Back to the Top

The following message was posted to: PharmPK

Dear Hans,

Thanks for your long and precise answer.

First I would like to say that I never intended to mean that there

was an

error in the model described in the Rowland and Tozer's book (i fully

agree

that it is one of the best book in the field). It's just that the

statement

p.166, second line after equation 13, "and fub is the fraction

unbound in

plasma" was just confusing for me. But one of my colleagues (thanks

Pascal!)

pointed me out after reading my message that there was another

definition of

fub page xiii (fub = Cu / Cb which fully agree with mine since Cu/Cb=fu

plasma /Rb).

Concerning the question on differential equations, i will try to

implement

your suggestion in my simulation software.

But i'm a bit worried to not be able to conciliate the equations of

the well

stirred model i use on a daily basis and yours.

I tried to manipulate on paper your equation (EQ. 1) to find the

equation i

use (EQ.2), but without any success :

dX / dt = (Qh*fub*CL_int/ (Qh + fub*CL_int)) * Cart EQ.1

dX/dt = Qh*Cart - Qh*Ctb - CL_int*fub*Ctb EQ.2

with

Cart = concentration in blood entering the liver

Ctb = concentration in blood leaving the liver (= total

concentration in liver / Kp)

fub = fu plasma / Rb

CL_int and Qh = as defined before

Qh*Cart = rate of input in the liver

Qh*Ctb = rate of output from the liver

CL_int*fub*Ctb = rate of elimination

I thought the (Qh*fub*CLint/ (Qh + fub*CLint)) equation was an

integrated

form of EQ.2, used to describe the systemic clearance caused by the

liver

metabolism... but after the explanations given by you and David, i'm now

wondering if i'm completely wrong. I'm also wondering was clear on

the fact

that my questions concern full PBPK models. The equation 2 is used to

describe the variations of the compound's amounts/concentrations in the

liver compartment only, not in the blood compartment.

Best regards, and thanks again for answering by basic questions

Cedric - On 1 Feb 2006 at 13:20:52, Jorge Duconge (jduconge.-at-.yahoo.com.mx) sent the message

Back to the Top

Dear Cedric,

Concerning your query on differential equation for hepatic clearance

(well-stirred model), please try this way:

Starting from your equation2

dX/dt = Qh*Cart - Qh*Ctb - CL_int*fub*Ctb [EQ.2]

assuming steady-state conditions:

dX/dt =0; so the right-side of your equation will become:

Qh*Cart - Qh*Ctb - CL_int*fub*Ctb = 0

and after a straightforward manipulation,

Cart = Ctb*(CL_int*fub +Qh)/Qh

thereafter and using the well-known expression CLh = Qh*Eh

where, Eh means hepatic extraction ratio = (Cart - Ctb)/Cart

substituting and canceling out,

Eh=CL_int*fub/(CL_int*fub +Qh),

Finally, you will get equation1 component

Clh=(Qh*fub*CL_int/ (Qh + fub*CL_int))

I hope this help you,

Regards,

Jorge Duconge - On 3 Feb 2006 at 09:34:35, cedric.vinson.at.fr.netgrs.com sent the message

Back to the Top

The following message was posted to: PharmPK

Dear Jorge,

Thanks for your clear demonstration. So the equation 1 is a

particular case

(steady-state) of equation 2, isn't it?

My main error was to think EQ.2 was an integrated form of EQ.1, but

it is in

fact equivalent to EQ.1 in the particular case of steady-state (since

you

fix dX/dt=0).

To get back to my original question concerning the parallel tube

model : i'd

like to express it the same way than EQ.1. , i.e. dX/dt = Rate_in -

Rate_out

- Rate_elimination

--> a form that does not imply steady state (am I wrong on this point?)

I will try to express it this way starting from David and Hans's

equation

(dX/dt=CLh*Cart, with CLh=Qh*[1-exp-(fub*CLint/Qh)]) and doing your

reasoning in an inverse way. I will get back on the forum if my limited

skills in mathematics prevent me to do this ^_^'

Thanks for your help

Best regards

Cedric - On 3 Feb 2006 at 14:27:03, "Hans Proost" (j.h.proost.-at-.rug.nl) sent the message

Back to the Top

The following message was posted to: PharmPK

Dear Cedric,

Your equations

> dX/dt = (Qh*fub*CL_int/ (Qh + fub*CL_int)) * Cart EQ.1

> dX/dt = Qh*Cart - Qh*Ctb - CL_int*fub*Ctb EQ.2

are indeed different.

Eq. 1 describes the loss of drug from the arterial blood flow through

the

liver. The only variable at the right side is Cart, as the driving

force. It

does not describe how the concentration in the blood changes, and so Ctb

remains to be calculated from:

Eh = fub*CL_int/ (Qh + fub*CL_int)) = (Cart - Ctb) / Cart

Please note that this approach does not take into account the volume

or Kp

of the liver.

Eq. 2 is a mass balance equation. If the volume of the liver is not

taken

into account, dX/dt = 0 at any time. As pointed out by Jorge Duconge,

this

equation can be rearranged to the equation for Eh and CLb,h.

> The equation 2 is used to

> describe the variations of the compound's amounts/concentrations

in the

> liver compartment only, not in the blood compartment.

If you want to take the volume and Kp of the liver into account to

make a

full PBPK model, you need additional equations to describe the change of

Cart to Ctb, in order to solve Eq. 2, and the condition dX/dt = 0

cannot be

assumed.

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-.rug.nl

Want to post a follow-up message on this topic? If this link does not work with your browser send a follow-up message to PharmPK@boomer.org with "Questions on PBPK organ models" as the subject

PharmPK Discussion List Archive Index page

Copyright 1995-2010 David W. A. Bourne (david@boomer.org)