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]
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]
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
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
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
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
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
PharmPK Discussion List Archive Index page
Copyright 1995-2010 David W. A. Bourne (david@boomer.org)