Exploring Inductive Linearization for simulation and estimation with an application to the Michaelis-Menten model

Abstract

Nonlinear ordinary differential equations (ODEs) are common in pharmacokinetic-pharmacodynamic systems. Although their exact solutions cannot generally be determined via algebraic methods, their rapid and accurate solutions are desirable. Thus, numerical methods have a critical role. Inductive Linearization was proposed as a method to solve systems of nonlinear ODEs. It is an iterative approach that converts a nonlinear ODE into a linear time-varying (LTV) ODE, for which a range of standard integration techniques can then be used to solve (e.g., eigenvalue decomposition [EVD]). This study explores the properties of Inductive Linearization when coupled with EVD for integration of the LTV ODE and illustrates how the efficiency of the method can be improved. Improvements were based on three approaches, (1) incorporation of a convergence criterion for the iterative linearization process (for simulation and estimation), (2) creating more efficient step sizes for EVD (for simulation and estimation), and (3) updating the initial conditions of the Inductive Linearization (for estimation). The performance of these improvements were evaluated using single subject stochastic simulation-estimation with an application to a simple pharmacokinetic model with Michaelis-Menten elimination. The reference comparison was a standard non-stiff Runge-Kutta method with variable step size (ode45, MATLAB). Each of the approaches improved the speed of the Inductive Linearization technique without diminishing accuracy which, in this simple case, was faster than ode45 with comparable accuracy in the parameter estimates. The methods described here can easily be implemented in standard software programme such as R or MATLAB. Further work is needed to explore this technique for estimation in a population approach setting.

Keywords: (PKPD) model development; Adaptive step size algorithm; Inductive Linearization; Nonlinear ordinary differential equations; Numerical methods; Optimization.

Fig. 1

Concentration–time profile for simulated data by typical profile from MATLAB ode45 and $\sigma =0.1$

Fig. 2

Comparison of relative difference of estimated parameters from their nominal simulation values (%) for a ode45, b IndLin (ss = 0.1), c IndLin (ss = 0.01) and d IndLin (adaptive step size, α = 0.01)

Fig. 3

Comparison of speed (seconds) among ode45, IndLin ($ss=0.1$), IndLin ($ss=0.01$) and IndLin (adaptive step size, $\alpha =0.01$) (a in linear scale and b in log scale)

Fig. 4

Comparison of the objective function value (OFV) among ode45, IndLin (ss = 0.1), IndLin (ss = 0.01) and IndLin (adaptive step size, α = 0.01) (a in linear scale and b in log scale)