Optimal design for population PK/PD models (Q2913241)

A model \(y_{ij}=\eta \left (x_{ij},\gamma _i\right)+v_{ij},\,\,\,i=1,\dots ,N,\,\,\,j=1,\dots ,k_i\), motivated by pharmacokinetics measurements is considered. \(N\) is the number of patients, the \(i\)th one observed at times \(x_{ij}\), \(\,j=1,\dots ,k_i\), \(\gamma _i\) is the vector of his individual parameters, which should be estimated. The random error \(v_{ij}\) has two independent components, an additive and a component proportional to the response \(\eta \left (x_{ij},\gamma _i\right)\). Due to this, the unknown parameters \(\gamma _i\) enter also in the error variances. The vectors \(\gamma _i\), \(i=1,\dots ,N\), are supposed to be sampled from a normal or a lognormal random vector \(\gamma \) characterizing the population of patients. An example of a compartmental modeling is illustrating this set-up. A design point \(x\) is here defined as a sequence of times \(\left \{x_{ij}\right \}_{i,j}\) and a design \(\xi \) is given in a standard way by the relative frequencies of replications \(\xi \left (x\text{}\right)\) of each design point \(x\). The optimality of a design is considered according to design criteria applied on the Fisher information matrix, which is computed at the point \(\gamma ^0\), the mean of the distribution of \(\gamma \) or of exp\(\left \{\gamma \right \}\). The proposed optimization algorithm goes back to a known procedure by \textit{C. L. Atwood} [Ann. Stat. 1, 342--352 (1973; Zbl 0263.62047)]. The paper presents essentially a survey of different approximations of the information matrix, including some new improvements under various assumptions, and also a detailed survey, with comments, on many softwares and implementations of this design problem. It is oriented mainly towards practitioners.