Impact of Pharmacokinetic–Pharmacodynamic Model Linearization on the Accuracy of Population Information Matrix and Optimal Design

Influence of experimental design on hyperparameter estimates precision when performing a population pharmacokinetic–pharmacodynamic (PK–PD) analysis has been shown by several studies and various approaches have been proposed for optimizing or evaluating such designs. Some of these methods rely on the optimization of a suitable scalar function of the population information matrix. Unfortunately for the nonlinear models encountered in pharmacokinetics or pharmacodynamics the latter is particularly difficult to evaluate. Under some assumptions and after a linearization of the PK–PD model a closed form of this matrix can be obtained which considerably simplifies its calculation but leads to an approximation. The aim of this paper is to evaluate the quality of the latter and its potential impact, when comparing or optimizing population designs and to relate it to Bates and Watts curvature measures. Two models commonly used in PK–PD were considered and nominal hyperparameter values where chosen for each one. Several population designs were studied and the associated population information matrix was computed for each using the approximate procedure and also using a reference method. Design optimizations were calculated under constraints for each model from the reference and approximate population information matrix. Nonlinearity curvatures were also computed for every model and design. The impact of model linearization when calculating the population information matrix was then examined in terms of lower bound accuracies on the hyperparameter estimates, design criterion variation, as well as D-optimal population designs, these results being related to nonlinearity curvature measures. Our results emphasize the influence of the parameter effects curvature when deriving the lower bounds of the hyperparameter estimates precision for a given design from the approximate population information matrix especially for hyperparameters quantifying the PK–PD interindividual variability. No discrepancies were detected between the population D-optimal designs obtained from the approximate and reference matrix despite some minor differences in criterion variation with respect to the design. More pronounced differences were, however, observed when comparing the amplitudes of criterion variation which can lead to errors when calculating design efficiencies. From a practical point of view, a strategy easily applicable by the pharmacokineticist for avoiding such problems in the context of population design optimization or comparison is then proposed.