Section: New Results

Non conservative transport equations for cell population dynamics

We have considered a class of kinetic models for which a moment equation has a natural interpretation. We have shown that, depending on their velocity field, some models lead to moment equations that enable one to compute monokinetic solutions economically. We have detailed the example of a multiscale structured cell population model, consisting of a system of 2D transport equations. The reduced model, a system of 1D transport equations, is obtained from computing the moments of the 2D model with respect to one variable. The 1D solution is defined from the solution of the 2D model starting from an initial condition that is a Dirac mass in the direction removed by reduction. For arbitrary initial conditions, we have compared 1D and 2D model solutions in asymptotically large time. Finite volume numerical approximations of the 1D reduced model can be used to compute the moments of the 2D solution with proper accuracy, both in the conservative and non conservative framework. The numerical robustness is studied in the scalar case, and a full scale vector case is presented [29].

These results have been partly presented in a workshop on “Asymptotic behavior of systems of PDEs arising in physics and biology : theoretical and numerical points of view” ( ABPDE II), Lille, June 15-17, 2016.

We have designed and analyzed a simplified version of our multiscale model for the morphogenesis of ovarian follicles [6]. We have formulated both a stochastic model, in the framework of branching processes, and a deterministic one, in the framework of nonconservative transport equations. The simplifications result in linear models, in which the oocyte growth is uncoupled from the proliferation of the surrounding follicular cells. The cell population is distributed into concentric layers around the oocyte, and structured according to the cell age. Cells are subject to the process of cell division, which resets their age and allow them to possibly move to the adjacent outer layer. Since there is no symmetry in the cell displacements (the only allowed cell motion is centrifugal), we have faced the problem of the model irreducibility. To study the asymptotic behavior, we thus had to adapt the classical results based on entropy or the computation of stochastic moments. We have proved that there is, as expected, an exponential asymptotic growth led by a Malthus parameter, which can be computed analytically in the simplest (Markovian) case, or numerically. Interestingly, the value of this global parameter merges with one of the local Malthus-like parameters defined on the layer level. In both the deterministic and stochastic cases, we could derive accurate information on the time-varying mean cell number per layer and we also got additional information on the asymptotic age distribution.

This work has been undergone in the framework of the master thesis of Frédérique Robin (M2 Mathématiques du Vivant, Université Paris-Saclay), and pursued as a PhD subject. Preliminary results have been the matter of a presentation during the “Journées INRA-Inria” held in Mallemort (France) on October 6-7th: F. Clément, F. Robin, R Yvinec. Dynamiques de populations cellulaires structurées individus-centrées : Morphogenèse des follicules ovariens.