Solutions for nonlinear ordinary differential equations in pharmacokinetic-pharmacodynamic systems


Pharmacokinetic-Pharmacodynamic systems are often expressed with nonlinear ordinary differential equations (ODEs). While there are numerous methods to solve such ODEs these methods generally rely on time-stepping solutions (e.g. Runge–Kutta) which need to be matched to the characteristics of the problem at hand. 


To explore

l  the performance of an inductive approximation which iteratively converts nonlinear ODEs to linear time-varying systems which can then be solved algebraically or numerically (primary)

l  the potential advantages of analytically solving linearized ODEs (secondary)



The inductive approximation was applied to three examples, a simple nonlinear pharmacokinetic (PK) model with Michaelis-Menten elimination (E1), an integrated glucose-insulin (IGI) model and an HIV viral load model with recursive feedback systems (E2 and E3, respectively) [1,2]. In order to address our second aim, we also used two examples, E3 and a turnover model of luteinizing hormone (LH) with a surge function (E4) [3].



The inductive linearization coupled with a matrix exponential solution provided accurate predictions for all examples with comparable solution time to the matched time-stepping solutions for nonlinear ODEs. The time-stepping solutions however did not perform well for E4, particularly when the surge was approximated by a square wave.



Out of the analytical methods tested, a matrix exponential (ME) solution provided the best results from the view point of accuracy and runtimes for all examples. The ME solution also provides the benefit of being directly applicable to future model manipulation, for example the scale reduction (“lumping”) of complicated PKPD systems. In circumstances when either a linear ODE is particularly desirable (e.g. lumping) or the uncertainty in matching the integrator to the ODE system is of potential risk, then the inductive approximation method coupled with an analytical integration method would be an appropriate alternative.



1.        Jauslin PM, Frey N, Karlsson MO (2011) J Clin Pharmacol 51(2):153-164

2.        Lavielle M, Samson A, Karina Fermin A, Mentré F (2011) Biometrics 67(1):250-259

3.        Nagaraja NV, Pechstein B, Erb K, Klipping C, Hermann R, Locher M, Derendorf H (2003) J Clin Pharmacol 43(3):243-251