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I am in search of equations alongwith its derivation for Value of Tmax (also Cmax if
possible) for Two Compartment Open Pharmacokinetic Model for Oral Administration, which
is not available in majority of standard text books of Biopharm & Pharmacokinetics
One equation i obtain from internet is Tmax = (Ln (a/b))/(a-b) , which may be true but i
am not sure.
where, a and b (also described as Alpha and Beta as well as Lamda1 and Lamda2 in other
literatures) are Distribution Rate Constant & Elimination Rate Constant respectively
b is also obtained from Elimination (terminal) half life
Kindly inform me that whether the above equation for Tmax is true or not.
if not then what is right equation
and if True then please guide me that from where i will find the derivation of entire
equation.
Anticipating your kind co-operation and prompt response.
Thanks & Regard
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The following message was posted to: PharmPK
Hi,
Tmax is defined as the time to reach Cmax, the maximal PLASMA concentration. For
one-compartment first-order absorption model, Tmax can be written as ln(ka/ke)/(ka-ke)
where ka is the first-order absorption rate constant and ke is the first-order
elimination rate constant. The equation for plasma concentration at time t is described
below and Tmax was derived by finding t when dCp/dt =0.
Cp(t)=F*(Dose/Vd)*(ka/(ka-ke))*{exp(-ke*t)-exp(-ka*t)}
dCp/dt=F*(Dose/Vd)*(ka/(ka-ke))*{-ke*exp(-ke*t)+ka*exp(-ka*t)}
(Try to see if you get ln(ka/ke)/(ka-ke), then you will understand easily for the Tmax of
two-compartment first-order absorption model)
If Tmax is written as ln(alpha/beta)/(alpha-beta), then it implies the model of interest
is two-compartment intravenous bolus model. Alpha and beta are hybrid terms for those
three micro constants of the two-compartment model (k10, k12, k21) and the hybrid terms
were created for the purpose of solving Laplace transformation. I do not see its
practical value in this case because the Tmax has been derived for the time to reach
maximal PERIPHERAL concentration following an intravenous bolus dose. Would you need it?
What you need is the time to reach the maximal PLASMA concentration for two-compartment
first-order absorption model, then the Tmax will be time t when dCp/dt=0 from the
equation below.
dCp/dt=F*(Dose/Vc)*ka{-ka*((k21-ka)/(alpha-ka)*(beta-ka))*exp(-ka*t)-alp ha*((k21-alpha)/(ka-alpha)*(beta-alpha))*exp(-alpha*t)-beta*((k21-beta)/(k a-beta)*(alpha-beta))*exp(-beta*t)}
where ka is the first-order absorption rate constant, k21 is the first-order transfer
rate constant from peripheral compartment to central compartment, Vc is central volume of
distribution, alpha and beta are hybrid terms for k12 (the first-order transfer rate
constant from central compartment to peripheral compartment), k21, and k10 (the
first-order elimination rate constant from central compartment).
It seems difficult to simplify this equation like the case of one-compartment model, thus
I do not see its practical utility unless you want to compare some extreme cases.
Cmax can be obtained by simple plug-in of the Tmax obtained above into the equation below
for the plasma concentration at time t.
Cp(t)= F*(Dose/Vc)*ka{((k21-ka)/(alpha-ka)*(beta-ka))*exp(-ka*t)+((k21-alpha)/(ka -alpha)*(beta-alpha))*exp(-alpha*t)+((k21-beta)/(ka-beta)*(alpha-beta))*ex p(-beta*t)}
I hope this helps even though it does not give you a simple solution.
I attempted to derive these equations for the first time, so any comments for my possible
mistakes will be greatly appreciated.
Thanks,
Jee Eun
Disclaimer: This is my personal opinion and does not represent my position at FDA.
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The following message was posted to: PharmPK
The equation you list is for a one compartment model where a and b are the absorption and
elimination rate constants.
Tmax can not be solved by an equation following absorption in a two compartment model.
It can only be determined by simulation with explicit PK parameters.
Leslie Z. Benet, Ph.D.
Professor
Department of Bioengineering & Therapeutic Sciences Schools of Pharmacy & Medicine
University of California San Francisco
533 Parnassus Avenue, Room U-68
San Francisco, CA 94143-0912
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The following message was posted to: PharmPK
Dear All:
About this model with Ka, V, and Ke, it is interesting to see that
the tmax also depends on the amount or concentration (however you wish to
write it) of drug present at the time the dose is given. For example, if you
wish to compute a dosage regimen to hit a desired peak during each of
several dosing intervals, when the initial condition is zero, as with the
first dose, the Tmax is latest, and the trough after that first loading dose
is the highest. Because of this, the second dose is always the smallest, and
the peak is the earliest. After that, the doses get a little bigger and
things begin to settle down and stabilize. So the initial condition at the
time of the dose must also be taken into account, I forget just how.
Very best regards,
Roger Jelliffe
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The following message was posted to: PharmPK
Dear Dhaivat!
> I am in search of equations alongwith its derivation for Value of Tmax
> (also Cmax if possible) for Two Compartment Open Pharmacokinetic Model
> for Oral Administration, which is not available in majority of standard
> text books of Biopharm& Pharmacokinetics
>
Right. Not missing in the majority of textbooks, but in all of them (in
other words, stop searching).
> One equation i obtain from internet is Tmax = (Ln (a/b))/(a-b) , which
> may be true but i am not sure.
>
Correct only for a one-compartment model (a = absorption rate constant,
b = elimination rate constant).
> if not then what is right equation
> and if True then please guide me that from where i will find the
> derivation of entire equation.
>
You are already at the right site. Derivation of the one-compartment here:
http://www.boomer.org/c/p4/c08/c0802.html and followings
Now for the bad news. Cmax/Tmax is the point of the curve where the
first derivate (i.e. the slope) is zero. In order to calculate Tmax, you
have to get the first derivate of C(t) and find the root of this
function. Before Tmax dC/dt >0, at Tmax dC/dt = 0, and after Tmax dC/dt
<0. For more than two exponential terms (i.e. > one-compartments) the
derivative of C(t) cannot be solved in closed form. That's why I wrote
above "stop searching". You can only estimate Cmax/Tmax by an iterative
(numeric) algorithm.
Helmut
-
Ing. Helmut Schuetz
BEBAC - Consultancy Services for
Bioequivalence and Bioavailability Studies
Neubaugasse 36/11
1070 Vienna, Austria
e-mail helmut.schuetz.-a-.bebac.at
web http://bebac.at
contact http://bebac.at/Contact.htm
forum http://forum.bebac.at
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The following message was posted to: PharmPK
Dear Dhaivat and All
The analytical derivation for tmax comes from equating the first derivative of the C(t)
expression to zero and solving for time, as done to estimate the extrema (minima or
maxima) of any function. The distinction depends on the sign of the second derivative,
but since in PK often C(t) profiles are biphasic, thus the generalization about tmax and
Cmax. However, not all expressions, i.e. compartmental models, have an analytical
solution. This was extensively discussed by JG Wagner when he coined the term 'vanishing
exponentials' (JPB (1976) 4:5,395; I recommend). Basically, for a single dose, the time
to reach the highest concentration depends on the overall rate limiting step, i.e. for a
two compartment disposition, the relative magnitude of ka (assuming first order
absorption), alpha and beta. They are often not all identifiable with real data, and thus
tmax does not have a single universal expression, unlike with the single exponential
disposition case. However, even in that case, tmax will only be larger than zero if ka>ke
which is an assumption, not a fact (careful with flip-flop, controlled release, etc.).
For multiple doses, depending on the accumulation factors (AF) for both exponentials, in
this case, tmax will be different for each dose, and at steady-state instead of
ln(ka/ke)/(ka-ke) it becomes ln(ka.AFabs/ke.AFel)/(ka-ke) which is always a smaller
number. The expression ln(alpha/beta)/(alpha-beta) does correspond to a tmax but for C2,
the estimated concentration in the peripheral compartment, again assuming first order
kinetics.
Hope this helps,
Cheers
--
Luis Pereira, PhD
Anesthesia Department
Children's Hospital Boston
Harvard Medical School
[Wagner. Linear pharmacokinetic models and vanishing exponential terms: implications in
pharmacokinetics. J Pharmacokinet Biopharm (1976) vol. 4 (5) pp. 395-425 - db]
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