Speaker
Description
Follicular phase in the ovarian development involves two key mechanisms: follicle maturation (via cellular growth and mass increase) and competition among follicles. Capturing both remains challenging. Compartmental ODE models (e.g., \cite{Hendrix}) describe maturation through discrete stages and reproduce macroscopic dynamics, but neglect follicular competition and cellular maturation. Conversely, models like \cite{Fischer} include competition but lack maturity structure. Physiologically structured population models (PSPMs) (e.g., \cite{Monniaux}), formulated as transport PDEs, account for both maturity growth and follicular competition; numerical simulations exist (e.g., \cite{Aymard}). Yet, they remain computationally intensive and less directly connected to established ODE frameworks.
We introduce a class of transport equations bridging these approaches via perturbative moment closures. These PDEs allow the application of standard numerical procedures for solving them. We present numerical simulations showing how the resulting PDEs reproduce cellular development and follicular competition consistent with the macroscopic ODE models. The structured PDEs preserve the key nonlinear mechanisms governing recruitment, selection, and atresia, providing a computationally efficient framework for multiscale modelling of ovarian follicle development for different physiological and pathological conditions.
Bibliography
@Article{Hendrix,
author={Hendrix, Angelean O.
and Hughes, Claude L.
and Selgrade, James F.},
title={},
journal={Bull Math Biol },
year={2014},
volume={76},
pages={136-156},
abstract={Mathematical models of the hypothalamus-pituitary-ovarian axis in women were first developed by Schlosser and Selgrade in 1999, with subsequent models of Harris-Clark et al. (Bull. Math. Biol. 65(1):157--173, 2003) and Pasteur and Selgrade (Understanding the dynamics of biological systems: lessons learned from integrative systems biology, Springer, London, pp. 38--58, 2011). These models produce periodic in-silico representation of luteinizing hormone (LH), follicle stimulating hormone (FSH), estradiol (E2), progesterone (P4), inhibin A (InhA), and inhibin B (InhB). Polycystic ovarian syndrome (PCOS), a leading cause of cycle irregularities, is seen as primarily a hyper-androgenic disorder. Therefore, including androgens into the model is necessary to produce simulations relevant to women with PCOS. Because testosterone (T) is the dominant female androgen, we focus our efforts on modeling pituitary feedback and inter-ovarian follicular growth properties as functions of circulating total T levels. Optimized parameters simultaneously simulate LH, FSH, E2, P4, InhA, and InhB levels of Welt et al. (J. Clin. Endocrinol. Metab. 84(1):105--111, 1999) and total T levels of Sinha-Hikim et al. (J. Clin. Endocrinol. Metab. 83(4):1312--1318, 1998). The resulting model is a system of 16 ordinary differential equations, with at least one stable periodic solution. Maciel et al. (J. Clin. Endocrinol. Metab. 89(11):5321--5327, 2004) hypothesized that retarded early follicle growth resulting in ``stockpiling'' of preantral follicles contributes to PCOS etiology. We present our investigations of this hypothesis and show that varying a follicular growth parameter produces preantral stockpiling and a period-doubling cascade resulting in apparent chaotic menstrual cycle behavior. The new model may allow investigators to study possible interventions returning acyclic patients to regular cycles and guide developments of individualized treatments for PCOS patients.},
issn={1522-9602},
doi={10.1007/s11538-013-9913-7},
url={https://doi.org/10.1007/s11538-013-9913-7}
}
@ARTICLE{Monniaux,
title = " ",
author = "Monniaux, Danielle and Michel, Philippe and Postel, Marie and
Cl{\'e}ment, Fr{\'e}d{\'e}rique",
abstract = "In this review, we present multi-scale mathematical models of
ovarian follicular development that are based on the embedding of
physiological mechanisms into the cell scale. During basal
follicular development, follicular growth operates through an
increase in the oocyte size concomitant with the proliferation of
its surrounding granulosa cells. We have developed a
spatio-temporal model of follicular morphogenesis explaining how
the interactions between the oocyte and granulosa cells need to
be properly balanced to shape the follicle. During terminal
follicular development, the ovulatory follicle is selected
amongst a cohort of simultaneously growing follicles. To address
this process of follicle selection, we have developed a model
giving a continuous and deterministic description of follicle
development, adapted to high numbers of cells and based on the
dynamical and hormonally regulated repartition of granulosa cells
into different cell states, namely proliferation, differentiation
and apoptosis. This model takes into account the hormonal
feedback loop involving the growing ovarian follicles and the
pituitary gland, and enables the exploration of mechanisms
regulating the number of ovulations at each ovarian cycle. Both
models are useful for addressing ovarian physio-pathological
situations. Moreover, they can be proposed as generic modelling
environments to study various developmental processes and cell
interaction mechanisms.",
journal = "Biol Cell",
volume = 108,
number = 6,
pages = "149--160",
month = feb,
year = 2016,
address = "England",
keywords = "Cell cycle; Follicle; Germ cell; Mathematical model; Ovary",
language = "en"
}
@article{Fischer,
title = { },
journal = {J Theor Biol},
volume = {547},
pages = {111150},
year = {2022},
issn = {0022-5193},
doi = {https://doi.org/10.1016/j.jtbi.2022.111150},
url = {https://www.sciencedirect.com/science/article/pii/S0022519322001485},
author = {Sophie Fischer-Holzhausen and Susanna Röblitz},
keywords = {HPG axis, Reproductive hormones, Mathematical modelling, Stochastic dynamical system},
abstract = {We present a modelling and simulation framework for the dynamics of ovarian follicles and key hormones along the hypothalamic-pituitary–gonadal axis throughout consecutive human menstrual cycles. All simulation results (hormone concentrations and ovarian follicle sizes) are in biological units and can easily be compared to clinical data. The model takes into account variability in follicles’ response to stimulating hormones, which introduces variability between cycles. The growth of ovarian follicles in waves is an emergent property in our model simulations and further supports the hypothesis that follicular waves are also present in humans. We use Approximate Bayesian Computation and cluster analysis to construct a population of virtual subjects and to study parameter distributions and sensitivities. The model can be used to compare and optimize treatment protocols for ovarian hyperstimulation, thus potentially forming the integral part of a clinical decision support system in reproductive endocrinology.}
}
@article{Aymard,
author = {Aymard, B. and Cl\'{e}ment, F. and Monniaux, D. and Postel, M.},
title = {Cell-Kinetics Based Calibration of a Multiscale Model of Structured Cell Populations in Ovarian Follicles},
journal = {SIAM J Appl Math},
volume = {76},
number = {4},
pages = {1471-1491},
year = {2016},
doi = {10.1137/15M1030327},
URL = { https://doi.org/10.1137/15M1030327},
eprint = { https://doi.org/10.1137/15M1030327},
abstract = { In this paper, we present a strategy for tuning the parameters of a multiscale model of structured cell populations in which physiological mechanisms are embedded into the cell scale. This strategy allows one to cope with the technical difficulties raised by such models that arise from their anchorage in cell biology concepts: localized mitosis, progression within and out of the cell cycle driven by time- and possibly unknown-dependent and nonsmooth velocity coefficients. We compute different mesoscopic and macroscopic quantities from the microscopic unknowns (cell densities) and relate them to experimental cell kinetic indexes. We study the expression of reaching times corresponding to characteristic cellular transitions in a particle-like reduction of the original model. We make use of this framework to obtain an appropriate initial guess for the parameters and then perform a sequence of optimization steps subject to quantitative specifications. We finally illustrate realistic simulations of the cell populations in cohorts of interacting ovarian follicles. }
}
