# [NMusers] Simplified Bateman equation where Ka=Ke

From: Rik Schoemaker <rik.schoemaker_at_occams.com>
Date: Mon, 15 May 2017 14:08:42 +0000

Dear fellow NMusers,

My previous submission to the forum had Word equations, and I think the ema=
il 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 sys=
tem breakdown. The consequence however is a very useful equation governed o=
nly by Cmax and Tmax. The Bateman function describes the biexponential equa=
tion associated with a kinetic system with first order absorption and linea=
r 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 bie=
xponential equation breaks down to a single exponential equation (see Eqn 2=
5 in Garret  or Eqn 2 in Bialer )

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  or Eqn 3 and 4 in Bialer ). Substituting K by 1/Tmax an=
d 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 publ=
ished it somewhere: can anyone help me out?

Cheers and thanks,

Rik

 Garrett ER. The Bateman function revisited: a critical reevaluation of =
the quantitative expressions to characterize concentrations in the one comp=
artment body model as a function of time with first-order invasion and firs=
t-order elimination. J Pharmacokinet Biopharm (1994) 22(2):103-128.
 Bialer M. A simple method for determining whether absorption and elimin=
ation 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:rik.schoemaker_at_occams.com

Received on Mon May 15 2017 - 10:08:42 EDT

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