Exploring inductive linearization for pharmacokinetic–pharmacodynamic systems of nonlinear ordinary differential equations

Abstract

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. The primary aim of this study was to explore the performance of an inductive approximation which iteratively converts nonlinear ODEs to linear time-varying systems which can then be solved algebraically or numerically. The inductive approximation is applied to three examples, a simple nonlinear pharmacokinetic model with Michaelis-Menten elimination (E1), an integrated glucose-insulin model and an HIV viral load model with recursive feedback systems (E2 and E3, respectively). The secondary aim of this study was to explore the potential advantages of analytically solving linearized ODEs with two examples, again E3 with stiff differential equations and a turnover model of luteinizing hormone with a surge function (E4). 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. In circumstances when either a linear ODE is particularly desirable 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.

Automated Scale Reduction of Nonlinear QSP Models With an Illustrative Application to a Bone Biology System

“…Standard linearization methods exist for such systems (e.g., a Taylor expansion), but their application is known to typically incur a relatively high degree of error . As an alternative, an inductive linearization has been shown to be applicable to nonlinear systems, and worked with arbitrary accuracy also for the relatively big system in this study ( Figure ).…”

“…The dimensions of y [0] depend on the number of ODEs in the nonlinear system (see ref. for further details). The linearization is performed for the solution y [n] ( t ) inductively by:…”

“…We note that the inductive approximation was applied in order to enable the direct use of proper lumping for nonlinear QSP models. Other possible benefits of using the inductive approximation can be found elsewhere …”

“…An inductive approximation was used to obtain the linearized system from a nonlinear model. 15 The method consists of two steps: (i) linearization to create a linear version of the nonlinear ODE; and (ii) then solving the linear ODE. Here, we briefly cover the main concepts of the linearization.…”

“…Other possible benefits of using the inductive approximation can be found elsewhere. 15 Because the PK of denosumab and the QSP bone biology model were expressed with nonlinear ODEs, they were linearized sequentially. A two-compartment PK model with a quasi-steady-state approximation of the target-mediated drug disposition model was used to describe denosumab PK.…”