Mathematical Studies of the Human Menstrual Cycle. I. Formulation of a Mathematical Model¹

Abstract

Thirty-four equations are presented which describe in a qualitative and quantitative fashion the hormonal and morphological components of the human menstrual cycle. (/ Clin Endocrinol Metab 35: 126, 1972) T HE cyclic morphological changes necessary to achieve fertility in women are controlled by the endocrine glands of the human female reproductive system. Lamport, in 1940 (1), described the interaction between steroids and gonadotropins with a linear second order differential equation. His model was unable to account for the cyclicity of the hormones, and he suggested that some other component must be involved. The critical oversight was the assumption of instantaneous equilibrium between plasma gonadotropin level and steroid production rate. Omission of the intervening morphological process eliminated the source of the long time constants which characterize this system.Endocrine system models have appeared in the literature in subsequent years, but until the recent work of Thompson (2) and Schwartz (3), no intensive efforts were made to describe the particular characteristics of the menstrual cycle.Thompson (2) developed a model which included an expression of ovarian morphological development and was able to demonstrate cyclic behavior.

Age Dependence of the Menstrual Cycle Correlation Dimension

“…We are presently engaged in modeling the system to obtain further insight into these questions. Previous mathematical models of the menstrual cycle offer little insight, because they have assumed either periodic solutions or stochastic processes [18][19][20], which are inconsistent with empirical data and with the results presented here.…”

Multiple Stable Periodic Solutions in a Model for Hormonal Control of the Menstrual Cycle

“…2. Since clearance from the blood of the ovarian hormones is on a fast timescale, we assume that serum levels of E 2 , P 4 , and I h are at quasi-steady state [see Keener and Sneyd (1998) as did Bogumil et al (1972a)]. Hence, we take these concentrations to be proportional to the tissue masses during the appropriate stages of the cycle giving the following three auxiliary equations for serum levels of E 2 , P 4 and I h as functions of time:…”

A delay differential equation model of follicle waves in women

“…Parameters for the three-wave ovarian component model Equations (9)- (20) fitting the McLachlan data [23]. 0.01 (U/L)/(μ g) h 3 0.95 (U/L)/(μ g) h 4 0.18 (U/L)/(μ g) Note: All of the following optimized parameters were truncated to two significant decimal digits. Table A4.…”

“…Applying the quasi-steady state assumption as did Bogumil et al [4,5], we write ovarian hormone concentrations as linear combinations of the stages that secrete these hormones:…”

“…Deterministic, mathematical models consisting of systems of differential equations have been used to describe hormonal regulation of the human menstrual cycle, for example, [4,5,8,15,29,30,32,33]. This physiological system may be considered dual control because hormones secreted by the hypothalamus and the pituitary glands affect the ovaries and the ovarian hormones affect the brain, see Figure 1.…”