Multiscale Modeling of Follicular Ovulation as a Mass and Maturity Dynamical System

Abstract

International audienceDuring each ovarian cycle, numerous follicles are in competition for their survival, only a few of them ovulate and participate to the reproduction as fertilizable oocyte. Therefore, the development of ovarian follicles is crucial for the survival of mammals species and a better understanding of this selection process is required to improve the treatment of infertility or to improve the reproduction potential in domestic species. In this work, we introduce the system of nonlinear PDEs which modelize the growth and maturation of ovarian follicles. Then we study the dynamic of this system of nonlinear PDEs using analysis and dynamical systems tools. 1. Introduction. Reproduction ability and efficiency are major assets in the natural selection of species. It is studied to improve the reproduction potential of cattle or to treat the infertility in the human researches (see [28, 23, 22]). Even if the infertility seems to be due for 40 percent to the male and 40 percent to the female (10 percent both and 10 percent without any reason), treatments and studies of infertility are mostly focused on female infertility [22, 23, 36, 24, 14]. During each menstrual cycle, eggs are developed. They are the culmination of a growth and maturation process of ovarian follicles which are, in some sense, the basic unit of female reproductive biology. Ovarian follicle is composed of roughly spherical aggregations of cells (granulosa) found in the ovary and contain a single oocyte. In the ovarian cycle (growth, i.e., cell division and maturation process), numerous follicles are in competition for their survival (see [3, 4, 5, 6, 14, 13, 21, 24, 28, 22]). Only a few number of them ovulate and participate to the reproduction as fertilizable oocyte(s), the others degenerate (atresia) (see [5, 14, 12]). Therefore, the development of ovarian follicles is crucial for the survival of mammals species and this cell-scale competition process leads to a specie-scale competition process through the reproduction [14, 13]. Thus, a better understanding of this selection process is required to improve the treatment of infertility or to improve the reproduction potential in domestic species. The follicular development result from a complex process involving the Follicle Stimulating Hormone (FSH) which controls the proliferation rate, the apoptosis rate and the differentiation rate of follicles. The follicles secrete hormones such as estradiol exerting a negative feedback on FSH release. At the end of this process, the survival follicles are those who are the less sensitive to the fall of plasmatic level of FSH in other words the most mature follicles, the others undergo atresia. In this work, we are interested in two models of the ovarian follicles development. In the first one, H. M. Lacker [21] considers that follicles are defined by their maturity which follow a dynamical system of coupled Ordinary Differential Equations. The assumptions, done by the authors, are mostly imputable to mathematical considerations. In the second one ...

Dimensional Reduction of a Multiscale Model Based on Long Time Asymptotics

“…On the other hand, the same asymptotic behavior as in the smooth velocity case [26], tending towards a monokinetic distribution, is observed in numerical simulations [2,4]. Nevertheless, we will show that if we relax the hypotheses made in [26] on the velocity field, the reduced model can have only measure solutions in some cases, and is therefore of little use for practical and numerical purposes, while in other cases the monokinetic behavior [26] is preserved. The cell population model of interest [16,11] falls into the first case.…”

“…On the one hand, existence and uniqueness of the solution in our case is proven for bounded initial conditions, and hence as well the existence of zeroth and first order moments [30]. On the other hand, the same asymptotic behavior as in the smooth velocity case [26], tending towards a monokinetic distribution, is observed in numerical simulations [2,4]. Nevertheless, we will show that if we relax the hypotheses made in [26] on the velocity field, the reduced model can have only measure solutions in some cases, and is therefore of little use for practical and numerical purposes, while in other cases the monokinetic behavior [26] is preserved.…”

“…Alternative 2D set-up. In [26], a reduced model has been derived for a slightly different 2D model. The maturation velocity satisfies hypothesis (2.4), the cell cycle in the proliferation zone is no more subdivided into two phases G1 and SM and the model coefficients are assumed to be smooth, thanks to a regularization across the internal boundary between the cell cycle and differentiation domain, as shown in Figure 2.2.…”

“…The knowledge of a distribution function for the solution in asymptotically long time enables us to estimate higher order moments at intermediate times. For a specific smooth velocity field, this approximation yields a reduced model whose solution is defined for all times and can be used to design a measure solution starting from a measure initial condition [26].…”

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.…”