Section: Scientific Foundations

Parametrization of complex systems

The parametrization of complex systems in order to fit experimental results or to have a good qualitative behavior is a delicate issue since its require to simulate the complex systems for a large number of sets of parameters, which is very expensive.

In many medical contexts, the available data for one particular patient are rather poor (a few MRI for instance). However many patients are studied (20 to 100 or even more in frequent pathologies). Therefore it is difficult or even impossible to parametrize a model for a given patient (too many parameters with respect to the number of available clinical data). However, it is possible to infer the distribution of the parameters in the global population by using all the data of all the patients at the same time. This is the principle of populational parametrization: to look for the distribution of the parameters (Gaussian or log Gaussian) and not to try to study each patient individually.

Many algorithms have been developed for populational parametrization, in particular so called SAEM (Stochastic Approximation Expectation Maximization) algorithms, based on MCMC (Monte Carlo Markov Chain) algorithms. These algorithms are very expensive, and require hundreds of thousands of evaluations of the model. For ordinary differential equation based models, SAEM converge quickly (it takes ten to twenty minutes on a laptop for the Monolix implementation of SAEM. Monolix is developped by M. Lavielle at Inria).

However for PDE based models, the evaluation of one single model may be long (a few minutes, up to ten minutes), hence the evaluation of hundreds of thousands models is completely out of range. Moreover, SAEM can not be parallelized in an efficient way.

Numed has set a general strategy to allow populational approaches on complex systems or on PDE based models. It relies on a precomputation strategy, combined iteratively with SAEM algorithms.

With such a strategy, populational parametrization of a PDE like reaction diffusion equation (KPP) may be done on a few hours on a small cluster of cores (32 cores).