Mixture models and model mixtures with MONOLIX
Marc Lavielle (1), Hector Mesa (1), Kaelig Chatel (1), An Vermeulen (2)
(1) INRIA Saclay, (2) J & J Pharmaceutical R & D
Objectives: A patient population is usually heterogeneous with respect to response to drug therapy. In any clinical efficacy trial, patients who respond, those who partially respond and those who do not respond present very different profiles. Then, diversity of the observed kinetics cannot be explained adequately only by the inter-patient variability of some parameters and mixtures are a relevant alternative in such situations:
- Mixture models are useful to characterize underlying population distributions that are not adequately explained by the observed covariates. Some non observed "latent" categorical covariates assign the individual patients to the components of the mixture.
- Between-subject model mixtures (BSMM) also assume that there exist subpopulations of patients. Here, different structural models describe the response of each subpopulation and each patient belongs to one subpopulation.
- Within-subject model mixtures (WSMM) assume that there exist subpopulations (of cells, of virus,...) within the patient. Different structural models describe the response of each subpopulation and proportions of each subpopulation depend on the patient.
Our objective is to develop a methodology for analyzing these different models, to implement it in MONOLIX and to apply it to some simulated and real viral kinetic data.
Method: We have extended the SAEM algorithm for mixture models and model mixtures. The algorithms were first evaluated using simulated PK data.
We then applied the proposed methodology for analyzing viral load data arising from 578 HIV infected patients. The randomized, controlled, partially blinded POWER studies were conducted by TIBOTEC and comprised 3 studies of up to 144 weeks, performed in highly treatment experienced patients, using darunavir/ritonavir (DRV/RTV) or an investigator-selected control PI, combined with an optimised background regimen (OBR), consisting of nucleoside reverse transcriptase inhibitors with or without the fusion inhibitor enfuvirtude.
We propose to describe these viral load data with a mixture of three models. Indeed, the data seem to exhibit three different typical profiles: responders, non-responders and rebounders.
Results: The between-subject model mixture (BSMM) is able to properly assign each patient to one of the three subpopulations. The conditional probabilities to belong to each group are computed for each patient. Nevertheless, the boundary between these different subpopulations is not obvious and several profiles seem to be "somewhere in-between". The within subject model mixture (WSMM) decomposes each profile into a linear combination of the three typical profiles. The proportions of the mixture are computed for each patient. This can well describe the profile of each individual. Furthermore, the BIC criteria clearly selects the WSMM model: BIC(WSMM)=14 668, whereas BIC(BSMM)=15029.
Conclusion: Between-subject and within-subject mixtures are relevant alternatives to mixture models for describing different profiles in a whole population. The SAEM algorithm is shown to be efficient for estimating mixture models and model mixtures in a general framework. These algorithms are now implemented in MONOLIX.
References:
[1] Wang X., Schumitzky A. and D'Argenio A. "Nonlinear random effects mixture models : Maximum likelihood estimation via the EM algorithm", Comp. Stat. & Data Anal., vol 51, 6614-6623, 2007.
[2] Lemenuel-Diot A., Laveille C., Frey N., Jochemsen R., Mallet A. "Mixture Modelling for the Detection of Subpopulations in a PK/PD analysis", PAGE 2004.
[3] Kuhn E and Lavielle M. "Maximum likelihood estimation in nonlinear mixed effects model", Comp. Stat. & Data Anal., vol 49, 1020-1038, 2005.
