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Re: [NMusers] Simplified Bateman equation where Ka=Ke

From: Jakob Ribbing <>
Date: Mon, 15 May 2017 17:05:56 +0200

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:

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.




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 =
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.

Jakob Ribbing, Ph.D.

Senior Consultant, Pharmetheus AB

Cell/Mobile: +46 (0)70 514 33 77

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 individual or entity to which it is directed. It may contain =
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immediately. Please do not copy it or disclose its contents to any other =

On 15 May 2017, at 16:08, Rik Schoemaker <> =

> 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
> +31 20 441 6410

Received on Mon May 15 2017 - 11:05:56 EDT

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