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 jakob.ribb...@pharmetheus.com 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 individual or entity to which it is directed. It may contain information that 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. On 15 May 2017, at 16:08, Rik Schoemaker <rik.schoema...@occams.com> 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:rik.schoema...@occams.com > > > >
