Multi-scale modeling of follicular ovulation as a reachability problem

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 of 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.