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Gauss Hermite Quadrature in Population Parameters Estimation. Application to the Detection of Subpopulations

A. Diot(1,2), C. Laveille(2), N. Frey(2), R. Jochemsen(2) & A. Mallet(1)

(1) INSERM U436, Dept Biomathematics, CHU Pitié Salpetrière, 91 bd de l’Hôpital, 75013 Paris, France; (2) Institut de Recherches Internationales Servier, 6 place des Pléiades, 92415 Courbevoie Cedex, France

Several methods have been proposed for estimating the different parameters entering the non-linear mixed effects models in population pharmacokinetics/pharmacodynamics. This work presents a methodological approach which allows to obtain a better approximation of the likelihood function. Its principal aim is to compute this likelihood up to a satisfactory and adjustable degree of approximation. Our hypothesis is that the accuracy of this approximation is likely to point out subtle features such as multimodality. Indeed, one feature of the method is the detection of heterogeneities or subpopulations with the introduction of mixture in the distribution of the random parameters.

In order to obtain good estimations and detection of heterogeneities, we propose an approach using the Gauss Hermite numerical quadratures. Thanks to this quadrature, we can express the integral-based individual likelihood according to an easy way to compute expression: weighted values of the individual likelihood function calculated for tabulated nodes (Stroud and Secrest,1966). Starting from the basic quadrature formula, we show that several adaptations are necessary to render this approach efficient in population pharmacokinetic/pharmacodynamic settings : it is necessary to shift and scale the space of the random parameters to obtain accurate estimates of the likelihood. Proposed methods are based on arguments given by Liu and Pierce (1994) and Pinheiro and Bates (1995).

In this work, we point out how this method is able to detect subpopulations in clinical studies. It relies upon the example of a pharmacokinetic model of an anti-diabetic drug for which we study the problem of different classes of clearance. To understand this classification we try to explain it using the covariates of the patients. A second example presents a pharmacodynamic model of this drug, the salient question being the detection of responder and non responder patients.