1 Introduction
Nonlinear mixed effects models (NLMEM) are widely used to analyze various biological processes described by longitudinal data. Since the primary models developed by Sheiner et al. [1] in pharmacokinetic (PK) and pharmacodynamic (PD), NLMEM are become widely used for modelling of biological processes. NLMEM, also called the population approach, allow estimation of the mean value of the parameters in the studied population and their interindividual variability, or population characteristics. NLMEM are also now commonly used for the joint modelling of several biological responses such as the PK of parent drugs and of their active metabolite. NLMEM allow a sparse sampling design with few data points per individual in a large set of individuals. This can be particularly useful in studies in specific populations such as children or patients with serious diseases, where classical studies with a large number of samples are often limited for ethical or physiological reasons.
Estimation of the parameters in NLMEM is commonly performed by maximum likelihood. However, due to the nonlinearity of the regression function, an analytical expression of the log-likelihood in nonlinear mixed effects models cannot be provided. To solve this issue several methods for estimating the parameters have been proposed, based on an approximation of the log-likelihood such as the First Order method (FO) or the First Order Conditional Estimate (FOCE) method proposed by Linsdstrom and Bayes [2]. Both methods use a linearization of the structural model either around the expectation of the random effects parameter (FO) or around individual estimates of the random effects (FOCE). These methods have been implemented in the NONMEM software [3, 4] but also in the nlme function of Splus and R software [5]. Compared to FO, the FOCE method provides less biased estimates and, in the context of joint modelling of multiple responses, is more appropriate with fewer problems of convergence or of inter-individual variance estimation [6, 7]. Alternative methods have also been proposed to maximize the likelihood using a stochastic approximation of the integrals, such as the Gaussian quadrature [8] or the Adaptative Gaussian quadrature methods implemented in the NLMIXED procedure of SAS. Recently, the Stochastic Approximation Expectation–Maximization algorithm (SAEM) has been developed and implemented in the MONOLIX software [9, 10]. It uses a stochastic approximation version of the standard expectation–maximization (EM) algorithm [11, 12]. The convergence and the consistence of the estimates have been proved by the authors. In this algorithm, the EM algorithm is used for finding maximum likelihood estimates of parameters
inserm-00371363, version 1 - 27 Mar 2009