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The following message was posted to: PharmPK
Dear all,
As we all known, free concentration (Cu) is the key when regarding to membrane transport,
tissue distribution, target binding or pharmacological response or efficacy, etc. Ideally,
Cu is needed to be determined. But due to many reasons, total concentration (Ctot) is
still the most convenient for monitoring and widely used in PK analysis at current stage.
At the meantime, plasma protein binding is also routinely determined in vitro either by
dialysis or ultrafiltration. Then fu is obtained. Very commonly, by multiplying fu, we can
obtain Cu with the determined Ctot and AUCu with the AUCtot (AUC calculated based on
Ctot).
I like to regard this multiplying process with fu that determined in vitro to obtain Cu in
vivo as an in vitro to in vivo extrapolation. My question is: does this extrapolation from
in vitro determinant really reflect the in vivo plasma protein binding situation?
Why do I raise this question? What is the behind?
We know the fu is determined in an in-vitro closed system without elimination/clearance
pathway. We give a certain amount of drug into this closed system. Generally, the drug
amount we give is far less than the total protein amount. Given in a same system and the
volume is equal, we can talk about concentration. In this situation with total protein
concentration is "excessive" than the drug concentration, the fu is only determined by the
Ka (affinity constant, compound specific) and the total binding site (n\0x00[Ptot]), so fu can
be regarded as a constant. It can be expected the fu will no longer be a constant and
begins to increase if drug concentration continuously increases and fu finally approaches
to 1. Notice all the above description is based on in vitro closed system.
Let me switch to the in vivo steady state. We all know at this time Cu is only determined
by the dose input, time and the CLu (free drug clearance). For a given dose regime and
within a linear PK scenario which implicates the CLu is a constant, then Cu should be a
constant. I hope I am correct up to now.
For an in vivo steady state achieved by continuous IV infusion, it is easy to imagine that
in a C-T plot, Cu will be a horizontal line. And in practice and to my experience, the
Ctot which we really determined also keeps constant and is also a horizontal line, which
is higher than the Cu line, in a C-T plot. So, fu will be kept as a constant, which is
Cu/Ctot. If we assume the fu obtained in vitro is equivalent to the fu in vivo, then we
can calculate Cu if we only get Ctot in practice.
But for an in vivo steady state achieved by multiple IV bolus injections, the Ctot curve
during a dose interval in a C-T plot will not be a horizontal line but a downward
exponential curve. But Cu is still a constant line as described hereinbefore if I am
correct. Then, considering fu=Cu/Ctot, fu will not be a constant and will change every
time point as Ctot changes. Is this real?
So my question is here. Is fu in vivo really NOT a constant? If it is not, how can we use
a single fixed fu value obtained in vitro to calculate Cu in vivo?
More questions and thoughts are below.
What defines fu? What defines Cu? I find in the textbook "Pharmacokinetic &
Pharmacodynamic Data Analysis: Concepts and applications 4th edition" written by J.
Gabrielsson and D. Weiner, Page 147, they gave a equation (2:294), showing fu was
determined by the Ka and the total binding site (n\0x00[Ptot]), together with Cu. However, if
so, as Cu is a constant at steady state achieved by multiple IV bolus injections, Ka and
n\0x00[Ptot] will also not change, then fu is ought to be a constant. But it is exactly
contrary to my deduction above. But as we all admit at steady state Cu is only determined
by dose input and CLu and NOT determined by fu, so if Ctot keeps changing after multiple
IV bolus injections, then fu will also keep changing. I am very confused about this issue.
Furthermore, as many people simply multiply fu that determined in vitro to Ctot to obtain
Cu, they always think Cu is determined by fu. I think it is totally wrong. Firstly, I
question whether the fu determined in the in vitro closed system is really consistent or
comparable to the fu in vivo? Does anyone show any proof? Secondly, if fu in vivo is not a
constant, what is the rationale that most people use this multiplying method to estimate
Cu in vivo?
Besides, all the in vivo scenarios I mention are at steady state when Cu is a constant.
How about the fu in vivo after a single dosing which does not guarantee steady state?
These questions have bothered me for quite a long time. I really tried to seek answers,
but failed. So I appreciate any comment or suggestion from you. I am not a native English
speaker and it seems I am still in the confused state about this issue, so if there is
anything I did not expressed clearly, please let me know.
Thank you in advance.
Yi Gu, PhD
Manager, Drug Metabolism & Pharmacokinetics
Hutchison MediPharma Limited
Building 4, 720 Cai Lun Road, Zhangjiang Hi-Tech Park
Shanghai, China
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Yi Gu,
I agree with almost all of your description and theory except for the idea that Cu somehow
remains constant during steady state with intermittent dosing.
"But for an in vivo steady state achieved by multiple IV bolus injections, the Ctot curve
during a dose interval in a C-T plot will not be a horizontal line but a downward
exponential curve. But Cu is still a constant line as described hereinbefore if I am
correct. "
I suggest you think of the system as being driven by unbound concentrations. These will
rise and fall with intermittent dosing. Unbound drug then binds to plasma proteins and
creates bound drug. Because of this the total concentration then goes up and down with
unbound drug. Because binding to plasma proteins is usually rapid the total concentration
can be simply predicted from unbound drug concentration i.e. Ctot=Cu + Cu*Bmax/(Cu+Kd).
[Bmax is the maximum binding capacity of plasma protein and Kd is the equilibrium
dissociation constant]
Note the causal property implied by this relationship. Ctot is determined by Cu - not the
other way around. I approach all PK problems where the data allows Cu and Ctot to be
distinguished by considering the PK only in terms of unbound drug. Then I use a binding
model with Cu to predict Ctot.
If you think about the system this way then I think you will no longer have to deal with
the paradox that you describe.
Best wishes,
Nick
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The following message was posted to: PharmPK
Hi Nick,
I was unclear about one of your comments (given below). Can you please
elaborate further?
"Note the causal property implied by this relationship. Ctot is determined
by Cu - not the other way around."
If I am administering, lets say, an IV injection, knowing the dose amount and
the volume of the central comportment, I know the C (in this case also
Ctot). Then part of the drug binds to plasma proteins (can be given by the
Langmuir type relationship as described in your reply) and the rest remains
free, the sum of which is the original Ctot. Thus knowing Ctot, Bmax and Kd
and given the equation you have described, one has to calculate Cu from it
(the inverse mapping of the Langmuir type function leads to a quadratic
equation).
Thanks in advance,
Ray
Siladitya Ray Chaudhuri
Senior Scientist II
Simulation Technologies (GastroPlus)
Simulations Plus, Inc.
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Hi,
What you describe is just algebra -- it is not the physicochemical basis for drug binding.
If there is no Cu then there will be no Cbound and therefore no Ctot. You need to consider
the causal mechanism of binding. Drug molecules enter and leave the system in the unbound
state. Binding to plasma proteins only happens because of unbound concs. It is largely
just a nuisance phenomenon that causes many misunderstandings about PK. The world would be
much simpler if only unbound concentrations were measured :-)
The causal relationship is
Ctot = Cu + Bmax*Cu/(Kd + Cu)
Of course, if you measure Ctot and Cbound then you can estimate Cu by subtraction.
However, this naive approach necessarily increases the error in Cu (subtracting two
quantities with error will increase the error of the result).
Binding parameters are better estimated using the causal relationship. Some simple algebra
allows you to solve for Cbound in terms of Ctot and the binding parameters. See
http://holford.fmhs.auckland.ac.nz/docs/ligand-binding.pdf
slide 15. All serious attempts
to describe ligand binding predict Cbound using this method ever since the pioneering work
of Munson & Rodbard and their LIGAND program published in 1980.
Best wishes,
Nick
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The following message was posted to: PharmPK
Hi Nick,
I am not sure I have get your idea correctly. Do you mean Cu does not remain constant
during steady state with intermittent dosing? I am afraid it may be a contradiction to the
PK principle that Cu=Dose/(CLu*tau) considering CLu is a constant at steady state. Could
you please show me some literatures to support your idea?
You showed a equation: Ctot=Cu + Cu*Bmax/(Cu+Kd). If Bmax, Kd and Cu are all constants,
then Ctot will also be a constant. Obviously, it is not the case in the steady state with
intermittent dosing. So my question still remains. Could anyone demonstrate that Cu at
steady state achieved by intermittent dosing will not a constant? If yes (Cu is not a
constant), how is it related to the equation Cu=Dose/(CLu*tau), and then what defines Cu?
If not (Cu is a constant), then is fu really keep changing? In this scenario, I guess the
equation part Cu*Bmax/(Cu+Kd), which is derived from in vitro closed system, is perhaps
not applicable to the in vivo situation. Am I right? Then, my question is still here, how
is the in vivo and in vitro correlation/exploration of the plasma protein binding?
Anyway, thank you Nick.
Yi
Yi Gu, PhD
Manager, Drug Metabolism & Pharmacokinetics
Hutchison MediPharma Limited
Building 4, 720 Cai Lun Road, Zhangjiang Hi-Tech Park
Shanghai, China
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The following message was posted to: PharmPK
Hi Nick,
I understand that at any given time, the equilibrium between bound and
unbound drug is given by the Langmuir relationship (in fact we use this to
represent saturable binding within our ocular drug delivery module). I also
understand that it is the unbound drug that participates in processes like
transit, metabolism etc. However, if you think of an unbound drug leaving
compartment A and going over to compartment B (which lets say has no drug to
begin with), then as soon as it enters compartment B, it becomes the total
drug, part of which then gets bound and the rest is unbound. So, the unbound
drug is the difference between the total and bound drug at any given time.
Thus, I see Cu as the derived quantity mathematically, even though
mechanistically it may be the most important. I don't think this depends on
how we measure things or what should ideally be measured.
Regards,
Ray
Siladitya Ray Chaudhuri
Senior Scientist II
Simulation Technologies (GastroPlus)
Simulations Plus, Inc.
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Yi Gu,
You need to understand the difference between the average steady state concentration which
is predicted by Dose/(CLu*tau) and the time course of drug concentration at steady state
which will go up and down after each dose.
Nick
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The following message was posted to: PharmPK
Hello,
Still discovering the PK field and its concepts, please correct me if
I understood something wrong, but I do not see whye you assume Cu is a
constant in case of " steady state " achieved by multiple bolus
injections...
Assuming that binding to plasma proteins is much quicker than drug
elimination and other PK process, that is the equilibrium between
bound and unbound drug is (almost) instantaneous, you have by
definition, with a very simplified model
[P-D]
Kb = -------
[P] [D]
where [P] is protein concentration in plasma, [D] = Cu, [P-D] is drug
bound to protein concentration and Kb the (apparent) binding constant;
apparent, because the formula uses concentrations and not activities;
all of this implicity at time t.
Hence, Ctot = [D] + [P-D] = [D](1 + Kb [P]) = [D] fu
where fu only depends on Kb and plasma protein concentrations, hence
is constant as far as the plasma composition does not change.
If I well understood, the equation Ctot = Cu + Bmax Cu /( Kd + Cu ) is
derived from the previous one, in the aim to remove the nuisance
parameter [P] by introducing the notion of maximal occupancy of these
proteines by the drug (leading to Bmax), am I right?
Hence, since with multiple bolus injections, Ctot is not constant, Cu
is not constant --- speaking of " steady state " in this case seems to
be abusive, since none concentrations is constant, despite very usual,
hence I do not see why the real steady-state formula when continuous
perfusion is performed would apply " as is ", as for instance the Cu dose/(Clu * tau).
By the way, this last in case of continuous
perfusion would also give Ctot = dose/(Cltot * tau), also constant,
and also not working here.
However, I assume they work with the _average_ concentrations during
the multiple bolus injections, with average as usual defined by= 1/tau * integral from t to t + tau C(t) dt
and the average concentration is really a constant when " equilibrium
" is achieved, since the functions C(t) becomes periodic of period
tau. And these averages should also satisfy that Ctot/Cu = fu
Hope this helps,
Best regards,
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Hello,
I am not sure that my understanding is correct,
In the following, I thought
[P-D]
Kb = --------
[P] [D]
where [P] should be free protein concentration but not total protein concentration in
plasma.
So, Ctot = [D] + [P-D] = [D](1 + Kb [P]) = [D]/fu
then fu = 1/(1 + Kb [P]), since [P] is variable depending the [D], fu varies with the
changes of concentrations of drugs.
Using Ctot = Cu + Bmax*Cu/(Kd + Cu)
fu = Cu / Ctot = (Kd + Cu)/ (Kd + Cu + Bmax), when Cu << Kd, fu is closer to a constant.
when Cu approach Kd or even higher, drugs start to saturate plasma proteins, fu will be
eventually 1.
Please correct me.
Thanks,
Tricia
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The following message was posted to: PharmPK
Hello all,
Thanks for the discussion.
Dear Emmanuel Curis,
You said "[P] is protein concentration in plasma". I think here you already made a
assumption that the free drug concentration [D] is far less than the total protein
concentration. So fu is a constant in this situation. But as Tricia pointed out, in the
equation
[P-D]
Kb = --------
[P] [D]
[P] should be the protein concentration that unbound. As [D] (Cu) increases or even being
comparable to [P]. then fu will change as Cu changes. So at this point I agree with Tricia
who has already mentioned this.
But I still question the eqaution above. In my oppinion, this equation is established in
the in vitro closed system and it is applicable only for the in vivo "initial phase" when
the drug is just introduced into the plasma, and as Emmanuel Curis mentioned the binding
to plasma proteins is much quicker than drug elimination and other PK process. But at the
ideal steady state, Cu in plasma also get equilibrium with the Cu in tissues and in
clearance organs. So I think the above equation is limited as it does not take into
consideration the equilibrium with the Cu in tissues and in clearance organs. I can not
give out a equation I like as I still am not confident. But I welcome and appreciate
anyone could either point out my mistake or help me with some reference support. And even
at non-steady state, for example single dosing, the Cu in plasma will also try to achieve
towards the equilibrium with the Cu in tissues and in clearance organs (maybe the
equilibrium can not be achieved indeed). So I am very questionable about the traditional
fu deduction, like in above equation, which is only from the in vitro closed system.
If this kind of fu deduction is not applicalbe for in vivo, how could we expect Ctot and
Cu change simultaneously while keeping fu a constant?
Dear Emmanuel Curis, you also mentioned "the _average_ concentrations during the multiple
bolus injections". I guess it is the Css,av. At steady state with intermittent dosing,
Css,av=AUCss/tau. In my humble opinion, in this situation, Css,av is only a mathematic
expression and does not have physiological meaning. Actually, the body deals with Ctot and
Cu at very time point. That's why I did not talk about this parameter before. You also
mentioned "the average concentration is really a constant when 'equilibrium' is achieved,
since the functions C(t) becomes periodic of period tau. And these averages should also
satisfy that Ctot/Cu = fu". I agree that Css,av is a constant, but do not agree it
satisfies that Ctot/Cu = fu. If Css,av is a constant and fu is also a constant as you
pointed out, then Cu will also be a constant. There is also a contradiction remained.
Many thanks to you all for the discussion.
Yi
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The following message was posted to: PharmPK
Hello,
I agree I was a little too quick for the [P] interpretation, which can
be seen constant when Ctot changes only if the total protein
concentration is much higher than the total drug concentration. I
assume it depends on the exact kind of protein involved in reality...
[P-D]
However, the equation Kd =3D ------- comes from the thermodynamic
definition
[P] [D]
of an equilibrium, hence in my opinion is valid whatever the
conditions are, in vitro or in vivo, but of course the exact value of
Kd can change, at least because it is an apparent constant when
expressed with concentrations and depends on ionic strength,
temperature...
The fact that the drug may bind to several different proteins or sites
may change the exact formula or add terms, but does not change the
idea I think. For instance, with two different proteins, we would have
Ctot =3D Cu + Kd(1)[P1]Cu + Kd(2)[P2]Cu
The equilibrium with tissues, and protein binding in tissues, also add
similar equations, one by tissue, and as pointed out in other mails
what really controls the system is the plasma Cu and tissular Cu. But
finally, that can always be wrotten as Cu / Ctot =3D fu (I took the
inverse in my previous mail by mistake, sorry) --- in fact, I guess
this last equation is the definition of fu and other equations are
models to explain why there is this fu; as always, models make
simplifications.
All of this assumes that binding equilibria are much faster than other
processes involved.
As for the average concentration, I agree it is a mathematical
convenience and will not discuss its physiological interpretation. I'm
not yet completely clear with traditionnal notations in PK, but
looking at the formula, yes, I thought about Css,av for Ctot,av and
its equivalent for Cu,av.
Assuming fu is constant, if at any time t we have Cu(t) / Ctot(t) fu, =
then we also have for the averages Cu,av(t) / Ctot,av(t) =3D fu, so
I do not see the contradiction.
Obviously, if fu changes with time, the situation is not so clear, and
the value of fu obtained by the ratio of Cu,av(t) / Ctot,av(t) may be
different from the average value of Cu(t) / Ctot(t) =3D fu(t).
Best regards,=
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The following message was posted to: PharmPK
Hello,
I agree I was a little too quick for the [P] interpretation, which can
be seen constant when Ctot changes only if the total protein
concentration is much higher than the total drug concentration. I
assume it depends on the exact kind of protein involved in reality...
[P-D]
However, the equation Kd =3D ------- comes from the thermodynamic
definition
[P] [D]
of an equilibrium, hence in my opinion is valid whatever the
conditions are, in vitro or in vivo, but of course the exact value of
Kd can change, at least because it is an apparent constant when
expressed with concentrations and depends on ionic strength,
temperature...
The fact that the drug may bind to several different proteins or sites
may change the exact formula or add terms, but does not change the
idea I think. For instance, with two different proteins, we would have
Ctot =3D Cu + Kd(1)[P1]Cu + Kd(2)[P2]Cu
The equilibrium with tissues, and protein binding in tissues, also add
similar equations, one by tissue, and as pointed out in other mails
what really controls the system is the plasma Cu and tissular Cu. But
finally, that can always be wrotten as Cu / Ctot =3D fu (I took the
inverse in my previous mail by mistake, sorry) --- in fact, I guess
this last equation is the definition of fu and other equations are
models to explain why there is this fu; as always, models make
simplifications.
All of this assumes that binding equilibria are much faster than other
processes involved.
As for the average concentration, I agree it is a mathematical
convenience and will not discuss its physiological interpretation. I'm
not yet completely clear with traditionnal notations in PK, but
looking at the formula, yes, I thought about Css,av for Ctot,av and
its equivalent for Cu,av.
Assuming fu is constant, if at any time t we have Cu(t) / Ctot(t) fu, =
then we also have for the averages Cu,av(t) / Ctot,av(t) =3D fu, so
I do not see the contradiction.
Obviously, if fu changes with time, the situation is not so clear, and
the value of fu obtained by the ratio of Cu,av(t) / Ctot,av(t) may be
different from the average value of Cu(t) / Ctot(t) =3D fu(t).
Best regards,=
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