Mechanistic model of hormonal contraception.

Contraceptive drugs intended for family planning are used by the majority of married or in-union women in almost all regions of the world. The two most prevalent types of hormones associated with contraception are synthetic estrogens and progestins. Hormonal based contraceptives contain a dose of a synthetic progesterone (progestin) or a combination of a progestin and a synthetic estrogen. In this study we use mathematical modeling to understand better how these contraceptive paradigms prevent ovulation, special focus is on understanding how changes in dose impact hormonal cycling. To explain this phenomenon, we added two autocrine mechanisms essential to achieve contraception within our previous menstrual cycle models. This new model predicts mean daily blood concentrations of key hormones during a contraceptive state achieved by administering progestins, synthetic estrogens, or a combined treatment. Model outputs are compared with data from two clinical trials: one for a progestin only treatment and one for a combined hormonal treatment. Results show that contraception can be achieved with synthetic estrogen, with progestin, and by combining the two hormones. An advantage of the combined treatment is that a contraceptive state can be obtained at a lower dose of each hormone. The model studied here is qualitative in nature, but can be coupled with a pharmacokinetic/pharamacodynamic (PKPD) model providing the ability to fit exogenous inputs to specific bioavailability and affinity. A model of this type may allow insight into a specific drug's effects, which has potential to be useful in the pre-clinical trial stage identifying the lowest dose required to achieve contraception.

A model of ovulatory regulation examining the effects of insulin-mediated testosterone production on ovulatory function.

Polycystic ovary syndrome (PCOS), a common cause of infertility in women, is often accompanied by abnormal reproductive and metabolic hormone levels. Specifically, androgens such as testosterone are elevated in many PCOS women, and the syndrome itself is frequently associated with insulin resistance, which leads to hyperinsulinemia, i.e., elevated insulin. Although the precise role of insulin in ovulatory function is unclear, its role in ovulatory dysfunction is often linked to the effects of increased ovarian androgen production. We present a mathematical model of the menstrual cycle that incorporates regulation by the pituitary-ovarian axis and mechanisms of ovarian testosterone production. We determine a physiological role for testosterone in the normal ovulatory cycle and study the role of hyperinsulinemia in pathological regulation of the cycle. Model results indicate increased ovulatory disruption with elevated insulin-mediated testosterone production and suggest that variations in the response of ovarian follicles to essential signals can alter the degree to which hyperinsulinemia disrupts the ovulatory cycle. The model also provides insight into the various PCOS phenotypes and the severity of ovulatory dysfunction.

A delay differential equation model of follicle waves in women.

This article presents a mathematical model for hormonal regulation of the menstrual cycle which predicts the occurrence of follicle waves in normally cycling women. Several follicles of ovulatory size that develop sequentially during one menstrual cycle are referred to as follicle waves. The model consists of 13 nonlinear, delay differential equations with 51 parameters. Model simulations exhibit a unique stable periodic cycle and this menstrual cycle accurately approximates blood levels of ovarian and pituitary hormones found in the biological literature. Numerical experiments illustrate that the number of follicle waves corresponds to the number of rises in pituitary follicle stimulating hormone. Modifications of the model equations result in simulations which predict the possibility of two ovulations at different times during the same menstrual cycle and, hence, the occurrence of dizygotic twins via a phenomenon referred to as superfecundation. Sensitive parameters are identified and bifurcations in model behaviour with respect to parameter changes are discussed. Studying follicle waves may be helpful for improving female fertility and for understanding some aspects of female reproductive ageing.

Modeling endocrine regulation of the menstrual cycle using delay differential equations.

This article reviews an effective mathematical procedure for modeling hormonal regulation of the menstrual cycle of adult women. The procedure captures the effects of hormones secreted by several glands over multiple time scales. The specific model described here consists of 13 nonlinear, delay, differential equations with 44 parameters and correctly predicts blood levels of ovarian and pituitary hormones found in the biological literature for normally cycling women. In addition to this normal cycle, the model exhibits another stable cycle which may describe a biologically feasible "abnormal" condition such as polycystic ovarian syndrome. Model simulations illustrate how one cycle can be perturbed to the other cycle. Perturbations due to the exogenous administration of each ovarian hormone are examined. This model may be used to test the effects of hormone therapies on abnormally cycling women as well as the effects of exogenous compounds on normally cycling women. Sensitive parameters are identified and bifurcations in model behavior with respect to parameter changes are discussed. Modeling various aspects of menstrual cycle regulation should be helpful in predicting successful hormone therapies, in studying the phenomenon of cycle synchronization and in understanding many factors affecting the aging of the female reproductive endocrine system.

Bifurcation analysis of a menstrual cycle model reveals multiple mechanisms linking testosterone and classical PCOS.

A system of 16 differential equations is described which models hormonal regulation of the menstrual cycle focusing on the effects of the androgen testosterone (T) on follicular development and on the synthesis of luteinizing hormone (LH) in the pituitary. Model simulations indicate two stable menstrual cycles - one cycle fitting data in the literature for normal women and the other cycle being anovulatory because of no LH surge. Bifurcations with respect to sensitive model parameters illustrate various characteristics of polycystic ovarian syndrome (PCOS), a leading cause of female infertility. For example, varying one parameter retards the growth of preantral follicles and produces a "stockpiling" of these small follicles as observed in the literature for some PCOS women. Modifying another parameter increases the stimulatory effect of T on LH synthesis resulting in reduced follicular development and anovulation. In addition, the model illustrates how anovulatory and hyperandrogenic cycles which are characteristic of PCOS can be reproduced by perturbing both pituitary sensitivity to T and the follicular production of T. Thus, this model suggests that for some women androgenic activity at the levels of both the pituitary and the ovaries may contribute to the etiology of PCOS.

Modeling endocrine control of the pituitary-ovarian axis: androgenic influence and chaotic dynamics.

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.

A lifelong model for the female reproductive cycle with an antimüllerian hormone treatment to delay menopause.

A system of 16 non-linear, delay differential equations with 66 parameters is developed to model hormonal regulation of the menstrual cycle of a woman from age 20 to 51. This mechanistic model predicts changes in follicle numbers and reproductive hormones that naturally occur over that time span. In particular, the model illustrates the decline in the pool of primordial follicles from age 20 to menopause as reported in the biological literature. Also, model simulations exhibit a decrease in antimüllerian hormone (AMH) and inhibin B and an increase in FSH with age corresponding to the experimental data. Model simulations using the administration of exogenous AMH show that the transfer of non-growing primordial follicles to the active state can be slowed enough to provide more follicles for development later in life and to cause a delay in the onset of menopause as measured by the number of primordial follicles remaining in the ovaries. Other effects of AMH agonists and antagonists are investigated in the setting of this model.

Dynamics and bifurcation of a model for hormonal control of the menstrual cycle with inhibin delay.

A system of 13 ordinary differential equations with 42 parameters is presented to model hormonal regulation of the menstrual cycle. For an excellent fit to clinical data, the model requires a 36 h time delay for the effect of inhibin on the synthesis of follicle stimulating hormone. Biological and mathematical reasons for this delay are discussed. Bifurcations with respect to changes in three important parameters are examined. One parameter represents the level of estradiol adequate for significant synthesis of luteinizing hormone. Bifurcation diagrams with respect to this parameter reveal an interval of parameter values for which a unique stable periodic solution exists and this solution represents a menstrual cycle during which ovulation occurs. The second parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. The third parameter is the time delay. Changes in the second parameter and the time delay affect the size of the uniqueness interval defined with respect to the first parameter. Saddle-node, transcritical and degenerate Hopf bifurcations are studied.

Bifurcation analysis of a model for hormonal regulation of the menstrual cycle.

A model for hormonal control of the menstrual cycle with 13 ordinary differential equations and 41 parameters is presented. Important changes in model behavior result from variations in two of the most sensitive parameters. One parameter represents the level of estradiol sufficient for significant synthesis of luteinizing hormone, which causes ovulation. By studying bifurcation diagrams in this parameter, an interval of parameter values is observed for which a unique stable periodic solution exists and it represent an ovulatory cycle. The other parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. Changes in this parameter affect the uniqueness interval defined with respect to the first parameter. Hopf, saddle-node and transcritical bifurcations are examined. To attain a normal ovulatory menstrual cycle in this model, a balance must be maintained between healthy development of the follicles and flexibility in estradiol levels needed to produce the surge in luteinizing hormone.

Multiple stable periodic solutions in a model for hormonal control of the menstrual cycle.

This study presents a nonlinear system of delay differential equations to model the concentrations of five hormones important for regulation and maintenance of the menstrual cycle. Linear model components for the ovaries and pituitary were previously analyzed and reported separately. Results for the integrated model are now presented here. This model predicts serum levels of ovarian and pituitary hormones which agree with data in the literature for normally cycling women. In addition, the model indicates the existence and stability of an abnormal cycle. Hence, the model may be used to simulate the effects of external hormone therapies on abnormally cycling women as well as the effects of exogenous compounds on normally cycling women. Such simulations may be helpful in understanding the role of xenobiotics in fertility problems, in predicting successful hormone therapies, and for testing hormonal methods of birth control.