Cauchy problem for multiscale conservation laws: Application to structured cell populations

TL;DR

This paper develops a mathematical model using a coupled hyperbolic system to describe the growth and selection of ovarian follicles, proving existence and uniqueness of solutions for the associated multiscale conservation laws.

Contribution

It introduces a novel vector conservation law framework for ovarian follicle dynamics, incorporating local and nonlocal maturity velocities, with rigorous proof of well-posedness.

Findings

Existence and uniqueness of weak solutions established.

Model captures the biological process of follicle selection and atresia.

Mathematical framework applicable to multiscale biological systems.

Abstract

In this paper, we study a vector conservation law that models the growth and selection of ovarian follicles. During each ovarian cycle, only a definite number of follicles ovulate, while the others undergo a degeneration process called atresia. This work is motivated by a multiscale mathematical model starting on the cellular scale, where ovulation or atresia result from a hormonally controlled selection process. A two-dimensional conservation law describes the age and maturity structuration of the follicular cell populations. The densities intersect through a coupled hyperbolic system between different follicles and cell phases, which results in a vector conservation law and coupling boundary conditions. The maturity velocity functions possess both a local and nonlocal character. We prove the existence and uniqueness of the weak solution to the Cauchy problem with bounded initial and boundary data.