Differential Equation Models for the Hormonal Regulation of the Menstrual Cycle

Abstract

There are growing concerns about the effects of environmental substances on the sexual endocrine system. It is believed that estrogenic substances may disrupt the sexual endocrine system by initiating or promoting such adverse effects as cancer, developmental disorders, and the reduction of fertility [17,40]. While these effects appear to be more imminent during high levels of exposure to estrogenic substances, concerns are increasing because low levels of exposure to estrogenic substances occur more frequently for longer periods of time; in our diets (phytoestrogens), in the environment (pesticides), and in contraception (spermicides and birth control pills) and hormonal therapies [17,40]. These effects might have a profound effect on the menstrual cycle. Therefore, mathematical models that accurately predict the serum levels of hormones that control the menstrual cycle would be useful tools in evaluating the effects of environmental substances.

The human menstrual cycle is controlled by the pituitary hormones, luteinizing hormone (LH) and follicle-stimulating hormone (FSH), and the ovarian hormones, estradiol, progesterone, and inhibin. The pituitary hormones stimulate the growth of ovarian follicles that secrete hormones and work to produce a fertilized ovum. The mathematical models to be presented in this work predict the blood levels of these five hormones as they interact to regulate and maintain the menstrual cycle.

The unmerged model has a pituitary component and an ovarian component consisting of linear systems of ordinary differential equations with time dependent coefficients. The pituitary systems describe the synthesis, release, and clearance of LH and FSH during the menstrual cycle, based on their response to estradiol, progesterone, and inhibin. Functions representing the ovarian hormones are used as inputs into these systems. The ovarian system describes the roles of FSH and LH in the development of ovarian follicles and the production of estradiol, progesterone, and inhibin during the menstrual cycle. Functions representing the pituitary hormones are used as inputs into this system. The merged model is formed by merging the pituitary and ovarian systems together. The merged system is a highly nonlinear system of delay differential equations that describes the interactions between the five hormones throughout the menstrual cycle. This model predicts reasonably accurate blood levels of these hormones observed in normally cycling women as reported in the literature.

The merged system is shown to have two stable periodic solutions for the same parameter set, a large amplitude solution that fits data found in the literature for normally cycling women and a small amplitude solution arising from Hopf bifurcation in the system parameters. The small amplitude cycle possesses many similarities to the menstrual cycle disorder referred to as polycystic ovarian syndrome (PCOS). Hormonal treatments for this abnormality are simulated and the large amplitude cycle fitting the data for normally cycling women is successfully recovered. In addition, simulations of exogenous estrogen exposure show that the large amplitude cycle can be perturbed into the small amplitude cycle. Therefore, in this modeling environment, an exogenous estrogen input disrupts the normal menstrual cycle.