Calculating D-optimal designs for compartmental models with a Michaelis–Menten elimination rate

Abstract

The optimal design of experiments for the parameters of the differential equations arising in many kinetic and pharmacodynamic models requires the numerical maximization of a function of the numerical solutions of sets of differential equations, the solutions depending on the experimental design.This problem in the design of experiments is solved as a dynamic optimization problem, using a simultaneous approach. The time horizon of the experiments is discretized in finite elements and orthogonal collocation on finite elements is used to parameterize the solution. The solution in each finite element is described by cubic Lagrange polynomials and the control action is represented by piecewise constant polynomials.We find D-optimum designs for two- and three-compartment models with Michaelis–Menten elimination rate kinetics. We consider three different design problems including both dynamic and static experiments. Single- and multi-experiment designs are addressed. Importantly, our algorithm allows the calculation of designs with constraints not only on the design region but also on the concentrations of the components in the compartments and on the rates of change of the design variables and of the resulting concentrations.