Context: We have proposed Splus and R functions, PFIM and PFIMOPT, for respectively designs evaluation and optimization in nonlinear mixed effects models (NMEM) [1]. These functions rely on an approximation of the Fisher information matrix using a first order linearization of the model [2]. Optimisation is based on the D-optimality criterion and uses a simplex algorithm. More recently, we have extended the expression of the Fisher matrix for models including the influence of covariates [3] and have implemented the Fedorov-Wynn algorithm for optimisation.

Objective: Our objectives were to apply and to illustrate this method to the example of a biexponential model of HIV viral load decrease under antiretroviral treatment [4]. This model involves four fixed effects, four additive random effects and an additive homoscedastic error. An additional fixed effect of the antiretroviral treatment on the first rate-constant is also considered.

Methods: We evaluate with PFIM a design of two groups of 100 patients with the same 6 sampling times per group and compare the empirical standard errors (SE) found with simulations with the SE predicted either with the nlme function of Splus or with MONOLIX, the new SAEM algorithm for NMEM estimation without any linearization [5-6]. We also use MONOLIX with one simulation of 5 000 patients to estimate the variance matrix and thus the expected Fisher information matrix under asymptotic convergence assumptions; we then derive the expected SE for smaller data sets. We apply the Fedorov-Wynn algorithm to optimise a design for a model without treatment effect and for a model where the treatment effect is estimated. We compare the optimised designs to those found with the Simplex algorithm. Last, from the predicted SE we compute and compare the power of a Wald test for the treatment effect under an alternative hypothesis for this parameter; this is perform for several empirical and optimised designs with either different total numbers of patients or different numbers of observations per patient.

Results: Regardless of the method, the SE were all very close which illustrates the usefulness of PFIM. For instance, for a treatment effect of 30%, the SE predicted for this parameter is 0.079 with PFIM and 0.078 with MONOLIX. The power computed from the SE given by PFIM is 92% for 100 patients per group and is reduced to 57% for 40 patients per group as in [4]. Optimisation with the Fedorov Wynn algorithm was faster and more robust than with the Simplex algorithm and led to a similar group structure and efficiency. Designs with fewer samples per patient but still reasonable power were optimised. For example, we showed that for a total number of patients of 100 per group, the power of an optimised design with 3 samples per patient divided into 4 sub-groups can be nearly as good as that of a design of 6 identical samples per patient chosen empirically: 87% versus 92%. This illustrates the consequence of the choice of the design on the number of samples and patients needed for a given power.

Conclusion: We illustrated the usefulness of PFIM and PFIMOPT on this new example and we showed that the Fedorov-Wynn algorithm is a good algorithm for design optimisation.

References:
[1] Retout, S. and F. Mentré, Optimisation of individual and population designs using Splus. Journal of Pharmacokinetics and Pharmacodynamics, 2003. 30(6): p. 417-443. http://www.bichat.inserm.fr/equipes/Emi0357/download.html
[2] Mentré, F., A. Mallet, and D. Baccar, Optimal design in random-effects regression models. Biometrika, 1997. 84(2): p. 429-442.
[3] Retout, S. and F. Mentré, Further developments of the Fisher information matrix in nonlinear mixed effects models with evaluation in population pharmacokinetics. Journal of Biopharmaceutical Statistics, 2003. 13(2): p. 209-27.
[4] Wu, H. and A.A. Ding, Design of viral dynamics studies for efficiently assessing potency of anti-hiv therapies in AIDS clinical trials. Biometrical Journal, 2002. 44(2): p. 175-196.
[5] Kuhn and Lavielle. Maximum likelihood estimation in nonlinear mixed effects model, Computational  Statistics and Data Analysis, to appear.
[6] http://mahery.math.u-psud.fr/~lavielle/monolix/