Dear fellow NMusers,
My previous submission to the forum had Word equations, and I think the email
server choked on that so I'm submitting a new text-only version :-)
I've been going insane trying to search for a reference to which I assumed was
a very common equation. It is the simplification of a Bateman function where
absorption cannot be distinguished from elimination, resulting in system
breakdown. The consequence however is a very useful equation governed only by
Cmax and Tmax. The Bateman function describes the biexponential equation
associated with a kinetic system with first order absorption and linear
elimination [1,2]:
C(Time)=(F*Dose*Ka/(V*(Ka-Ke)))*(exp(-Ke*Time)- exp(-Ka*Time))
In the case where Ka and Ke cannot be distinguished (Ka=Ke=K), this
biexponential equation breaks down to a single exponential equation (see Eqn 25
in Garret [1] or Eqn 2 in Bialer [2])
C(Time)=(F*Dose*K*Time/V)*(exp(-K*Time))
For this equation, Tmax can be derived to be given by 1/K and Cmax is given by
F*Dose/(V*e) where e is the base of natural logarithms (see Eqn 26 and 27 in
Garret [1] or Eqn 3 and 4 in Bialer [2]). Substituting K by 1/Tmax and V by
F*Dose/(Cmax*e) gives:
C(Time)=(Cmax*e*Time/Tmax)*(exp(-Time/Tmax))
Extremely useful for describing disease progression profiles, and I assumed it
to be widely know. Perhaps it still is, but then someone must have published it
somewhere: can anyone help me out?
Cheers and thanks,
Rik
[1] Garrett ER. The Bateman function revisited: a critical reevaluation of the
quantitative expressions to characterize concentrations in the one compartment
body model as a function of time with first-order invasion and first-order
elimination. J Pharmacokinet Biopharm (1994) 22(2):103-128.
[2] Bialer M. A simple method for determining whether absorption and
elimination rate constants are equal in the one-compartment open model with
first-order processes. J Pharmacokinet Biopharm (1980) 8(1):111-113
Rik Schoemaker, PhD
Occams Coöperatie U.A.
Malandolaan 10
1187 HE Amstelveen
The Netherlands
http://www.occams.com
+31 20 441 6410
mailto:[email protected]
Simplified Bateman equation where Ka=Ke
Dear Rik,
Thanks, theses were indeed convenient equations!
For multiple dosing Matt Hutmacher derived the explicit equation for the ka=ke
case of the Bateman function, and this effort is available as supplemental
material here:
https://static-content.springer.com/esm/art%3A10.1007%2Fs10928-012-9274-0/MediaObjects/10928_2012_9274_MOESM1_ESM.docx
I am sure this was not easy to derive, but it is still just a tiny example
among the vast number of contributions that Matt patiently made to the
community.
I will always miss him for that, and for being a friendly face and a good chat.
However, it is good to see that his spirit lives on in so many others.
Sincerely
Jakob
PS.
The above link is one of the online supplements to this publication, on a KPD
model that used the ka=ke assumption for (single and) multiple dosing:
J Pharmacokinet Pharmacodyn. 2012 Dec;39(6):619-34. doi:
10.1007/s10928-012-9274-0. Epub 2012 Sep 23.
Longitudinal FEV1 dose-response model for inhaled PF-00610355 and salmeterol in
patients with chronic obstructive pulmonary disease.
Nielsen JC, Hutmacher MM, Cleton A, Martin SW, Ribbing J.
DS.
Jakob Ribbing, Ph.D.
Senior Consultant, Pharmetheus AB
Cell/Mobile: +46 (0)70 514 33 77
[email protected]
www.pharmetheus.com
Phone, Office: +46 (0)18 513 328
Uppsala Science Park, Dag Hammarskjölds väg 52B
SE-752 37 Uppsala, Sweden
This communication is confidential and is only intended for the use of the
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is privileged and exempt from disclosure under applicable law. If you are not
the intended recipient please notify us immediately. Please do not copy it or
disclose its contents to any other person.
Quoted reply history
On 15 May 2017, at 16:08, Rik Schoemaker <[email protected]> wrote:
> Dear fellow NMusers,
>
> My previous submission to the forum had Word equations, and I think the email
> server choked on that so I'm submitting a new text-only version :-)
>
>
> I've been going insane trying to search for a reference to which I assumed
> was a very common equation. It is the simplification of a Bateman function
> where absorption cannot be distinguished from elimination, resulting in
> system breakdown. The consequence however is a very useful equation governed
> only by Cmax and Tmax. The Bateman function describes the biexponential
> equation associated with a kinetic system with first order absorption and
> linear elimination [1,2]:
>
> C(Time)=(F*Dose*Ka/(V*(Ka-Ke)))*(exp(-Ke*Time)- exp(-Ka*Time))
>
> In the case where Ka and Ke cannot be distinguished (Ka=Ke=K), this
> biexponential equation breaks down to a single exponential equation (see Eqn
> 25 in Garret [1] or Eqn 2 in Bialer [2])
>
> C(Time)=(F*Dose*K*Time/V)*(exp(-K*Time))
>
> For this equation, Tmax can be derived to be given by 1/K and Cmax is given
> by F*Dose/(V*e) where e is the base of natural logarithms (see Eqn 26 and 27
> in Garret [1] or Eqn 3 and 4 in Bialer [2]). Substituting K by 1/Tmax and V
> by F*Dose/(Cmax*e) gives:
>
> C(Time)=(Cmax*e*Time/Tmax)*(exp(-Time/Tmax))
>
> Extremely useful for describing disease progression profiles, and I assumed
> it to be widely know. Perhaps it still is, but then someone must have
> published it somewhere: can anyone help me out?
>
> Cheers and thanks,
>
> Rik
>
> [1] Garrett ER. The Bateman function revisited: a critical reevaluation of
> the quantitative expressions to characterize concentrations in the one
> compartment body model as a function of time with first-order invasion and
> first-order elimination. J Pharmacokinet Biopharm (1994) 22(2):103-128.
> [2] Bialer M. A simple method for determining whether absorption and
> elimination rate constants are equal in the one-compartment open model with
> first-order processes. J Pharmacokinet Biopharm (1980) 8(1):111-113
>
>
>
>
> Rik Schoemaker, PhD
> Occams Coöperatie U.A.
> Malandolaan 10
> 1187 HE Amstelveen
> The Netherlands
> http://www.occams.com
> +31 20 441 6410
> mailto:[email protected]
>
>
>
>