Extending Monolix to use models with Partial Differential Equations
Résumé
This is a joint work with Emmanuel Grenier and Violaine Louvet. Objectives: To develop a methodology able to include models made of Partial Differential Equations (PDE), instead of Ordinary Differential Equations (ODE), in SAEM algorithms like the one used by the Monolix Software[1]. Methods: We design a general methodology to make parameters estimation within the Monolix software, for models involving PDE (see [2]). Indeed, Monolix is known to be a very efficient tool but, up to now, it is only able to work with models made of ODE (see e.g. [3]). In particular, this means that spatial effects can not be included in the associated dynamical models (only the time is included). The idea is (i) to translate to PDE model in terms of times series, by spatial integration and (ii) to build a computational time decreasing algorithm. This can be applied generically to a broad range of PDEs. As a specific illustration, we here present the application of our method to the Kolmogorov-Petrovsky-Piskounov (KPP) PDE which is the canonical model for reaction-diffusion phenomena[4]. We build a noisy population of individuals in silico. Monolix was used to estimate the population and individual parameters. Namely, these parameters were : the reaction an d diffu sion coefficients and the location of the initial condition of the model. Results: The method correctly predicted the individual parameters, including the initial condition locations, a parameter typically associated with the spatial effects of the underlying model (KPP). Furthermore the total computational cost of the Monolix run to make the parameters estimation is of the same order as for an ODE model, namely in less than 30 minutes. This is a significant achievement since without our time deacreasing algorithm this total computational time is around 23 days. Conclusions: To our knowledge, this is the first time that parameters estimation with a population approach, a Stochastic Approximation Expectation Maximization algorithm as used in Monolix, is performed with a PDE model. The method is precise and computationally efficient: it can open the path to the use of numerous new types of models and applications in population approaches. References: [1] The Monolix software. Analysis of mixed effects models. LIXOFT and INRIA, http://www.lixoft.com/. [2] Paul Vigneaux, Violaine Louvet, Emmanuel Grenier. Parameter estimation in non-linear mixed effects models with SAEM algorithm: extension from ODE to PDE. Submitted, 2012. Available at : http://hal.inria.fr/hal-00789135 [3] Ribba et al. A tumor growth inhibition model for low-grade glioma treated with chemotherapy or radiotherapy. Clin Cancer Res. (2012) Sep 15; 18(18):5071-80. [4] A. Kolmogoroff, I. Petrovsky, N. Piscounoff. Etude de l'equation de la diffusion avec croissance de la quantite de matiere et son application a un probleme biologique. Bulletin de l'universite d'Etat a Moscou. Section A, I(6):1-26, 1937. Available at : http://www.page-meeting.org/default.asp?abstract=2666