Multiscale Modeling of Follicular Ovulation as a Reachability Problem

Abstract

During each ovarian cycle, only a definite number of follicles ovulate, while the others undergo a degeneration process called atresia. We have designed a multi-scale mathematical model where ovulation and atresia result from a hormonal controlled selection process. A 2D-conservation law describes the age and maturity structuration of the follicular cell population. In this paper, we focus on the operating mode of the control, through the study of the characteristics associated with the conservation law. We describe in particular the set of microscopic initial conditions leading to the macroscopic phenomenon of either ovulation or atresia, in the framework of backwards reachable sets theory.

Adaptive mesh refinement strategy for a non conservative transport problem

“…Model and numerical method. In this section we apply the strategy presented above to the numerical simulation of a biological model dedicated to the cell dynamics in ovarian follicular development introduced in Echenim et al [8] and references therein. The model is multiscale to account for the selection process of ovulatory follicles which involves the cellular, follicular and ovarian levels.…”

“…has already been computed during the loop on the finer level • if the right hand side neighbor of mesh D 1,2k+1 is the coarser mesh D 0,k+1 , no reconstruction is required to compute F n (1,2k+1)+1/2 for the simple first order scheme (8). In the case of a higher order scheme,φ n 1,2k+2 would have to be reconstructed • if the right hand side neighbor of mesh D 0,k is the finer mesh D 1,2k+2 , then…”

“…For each level, a first loop computes the numerical fluxes between meshes active at the current level using (8), possibly requiring local reconstruction if neighboring meshes entering the flux estimation are coarser than the current level. Then a second loop updates the solution on the meshes of the current level using (7).…”

Coupled Somatic Cell Kinetics and Germ Cell Growth: Multiscale Model-Based Insight on Ovarian Follicular Development

“…cumulus) remain in close interactions with it up to the ovulation event [16,17,41]. Coupling the stochastic IBM model presented in this paper with the already existing models of terminal follicular development [10,11,36] would be a natural way of embedding the oocyte dynamics in those models. In the law describing oocyte growth, the oocyte radius depends on both the number of its surrounding granulosa cells (and thus indirectly, on their cell cycle duration λ) and their average intensity of secretion κ.…”

Multiscale modelling of endocrine systems: new insight on the gonadotrope axis

“…. The well-posedness of the whole model is discussed in [9], in the case where the control values are in steadystate (which just amounts to define them as function of time on successive small time intervals) and regularised expressions for the velocities. The reasoning is based on considering a sequence of subproblems defined on each Ω i,k{k=0,...,N−1} , and tracking the traces of the solutions on the outward boundaries of the Ω i,k domains.…”