Section: Research Program

Non conservative transport equations for cell population dynamics

Models for physiologically-structured populations can be considered to derive from the so-called McKendrick-Von Foerster equation or renewal equation that has been applied and generalized in different applications of population dynamics, including ecology, epidemiology and cell biology. Renewal equations are PDE transport equations that are written so as to combine conservation laws (e.g. on the total number of individuals) with additional terms related to death or maturation, that blur the underlying overall balance law [33]. Renewal equations can be deployed only in contexts where a deterministic and continuous formalization is suitable to describe the populations. In the case of low or very low number of individuals, the probabilistic nature of the events driving the population dynamics has to be accounted for. In that context, the formalism initially developed in the framework of ecological modeling (see e.g. [30]) is particularly interesting since it can bridge the gap between branching processes and renewal equations for structured populations.

The development of ovarian follicles is a tightly-controlled physiological and morphogenetic process, that can be investigated from a middle-out approach starting at the cell level.

To describe the terminal stages of follicular development on a cell kinetics basis and account for the selection process operated amongst follicles, we have developed a multiscale model describing the cell density in each follicle, that can be roughly considered as a system of weakly-coupled, non conservative transport equations with controlled velocities and source term. Even if, in some sense, this model belongs to the class of renewal equations for structured populations, it owns a number of specificities that render its theoretical and numerical analysis particularly challenging: 2 structuring variables (per follicle, leading as a whole to 2$n$D system), control terms operating on the velocities and source term, and formulated from moments of the unknowns, discontinuities both in the velocities and density on internal boundaries of the domain representing the passage from one cell phase to another. On the theoretical ground, the well-posedness (existence and uniqueness of weak solutions with bounded initial data) has been established in [12], while associated control problems have been studied in the framework of hybrid optimal control [6]. On the numerical ground, the formalism dedicated to the simulation of these hyperbolic-like PDEs is that of finite volume method. Part of the numerical strategy consists in combining in the most efficient way low resolution numerical schemes (such as the first-order Godunov scheme), that tend to be diffusive, with high resolution schemes (such as the Lax Wendroff second-order scheme), that may engender oscillations in the vicinity of discontinuities [2], with a critical choice of the limiter functions. The 2D finite volume schemes are combined with adaptive mesh refinement through a multi-resolution method [4] and implemented in a problem-specific way on parallel architecture [1].

To describe the first stages of follicular development, which only involve a few cells, we call to a stochastic and discrete formalism. We have designed an individual-based, stochastic model [7] embedding specific laws of morphodynamics, which leads to a multiscale model where individual cells are endowed with a non-zero size and occupy space partitions of predefined sizes, which can bear a limit rate of overcrowding.