A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle

Abstract

In this survey, we review the current state of the art in mathematical modeling of the menstrual cycle. We compare and contrast different modeling choices and the benefits and limitations of various models. We investigate the sensitivity of these models to variations in parameter values, highlighting that these models display particular sensitivity to the growth rate of the reserve pool of follicular stimulating hormone. We also describe the incorporation of time delays in the model equations and discuss the ways in which these delays reflect the biological system and impact the dynamics. We explore the qualitative effects that introducing exogenous hormones into these models plays on inducing a contraceptive state. Through our comparative study of these models, we are able to highlight important areas of future work in the mathematical modeling of hormonal contraception and the menstrual cycle.

Lihong Zhao and Ruby Kim contributed equally to this work.

1 Introduction

Developing a mechanistic understanding of the menstrual cycle is important to human health and wellness. Roughly, half of people of reproductive age menstruate. About 5–8% of these individuals experience moderate to severe symptoms of premenstrual syndrome [1]. Many experience issues with their menstrual cycles that affect their health, wellness, and quality of life, including amenorrhea (absence of menstruation), dysmenorrhea (painful menstruation), menorrhagia (excessive menstruation), and diseases directly related to the reproductive system, such as uterine fibroids, endometriosis, and polycystic ovarian syndrome [2]. Some key features of the menstrual cycle are generally robust, with feedback interactions between hormones in the hypothalamus, pituitary gland, and ovaries driving mostly predictable rhythms. At the same time, the system can exhibit a great deal of variation both between different individuals and from cycle to cycle within a single individual.

Mathematical modeling of the menstrual cycle is a relatively recent area of focus, and the development and analysis of mathematical models of this system may still be considered in the early stages. The menstrual cycle is a potentially rich area of study from the perspective of mathematics and dynamical systems. The periodic nature of the hormone fluctuations makes it a natural candidate of study for those who are interested in oscillations and limit cycles. When analyzing a system that evidences naturally cycling behavior, Hopf bifurcations play an important role. A Hopf bifurcation is the process whereby periodic orbits (“self-oscillations”) emerge from a fixed point when a parameter crosses a critical value [3]. Identifying the types of bifurcations that are part of a mathematical system of equations yields insight into the structure and mechanisms of a system of equations. Determining how and when a Hopf bifurcation, in particular, may develop is therefore of particular interest in a mathematical model of the menstrual cycle. The existing mathematical work has already made strong contributions in this area. Several early mathematical models in the literature have helped quantify the complex interactions that drive the human menstrual cycle [4,5,6]. Selgrade and Schlosser [7, 8] worked on a series of models that drove more recent developments in the field, especially as more data became available [9,10,11,12].

The dynamics of this system are made both more interesting and more complicated by the introduction of hormonal contraceptives. An individual’s autonomy and ability to understand and control their own reproductive health is an important issue in healthcare and in modern society. Contraceptive use is linked to increases in economic empowerment, education, and labor force participation for women [13]. Hormonal contraceptives (including oral contraceptives) remain a leading form of birth control in the United States [14]. However, very few researchers have explicitly incorporated hormonal contraception into their mathematical models.

In this chapter, we provide a survey of the efforts in mechanistic modeling of the menstrual cycle, with a special eye toward the modeling of the effects of hormonal contraception. We begin in Sect. 2 with an overview of the biology of the menstrual cycle, including the different phases and key hormonal drivers of the system. We also describe how these hormones are impacted through hormonal contraception. In Sect. 3, we turn our attention toward existing mathematical models of the menstrual cycle, highlighting the goals and key results from these papers. We discuss strengths and limitations of each of these models. We provide a comparative analysis of these models in Sect. 4. First, we perform a sensitivity analysis that shows how variations in the parameter representing the growth rate of the reserve pool of follicular stimulating hormone have considerable impact on the cycle length across models. This analysis reveals interesting qualitative and quantitative differences on the impact of cycle length in each case. Second, we discuss the different ways in which time delays are incorporated into various differential equation models. We close the section by introducing a simple modification to the existing models to include hormonal contraception and discuss the qualitative responses of each model. To the best of our knowledge, this is the first such comparative analysis of mathematical models of the menstrual cycle. We conclude in Sect. 5 by highlighting existing challenges and important future directions.

2 Biological Background on the Menstrual Cycle

2.1 Stages of the Menstrual Cycle

Here we provide a brief introduction to the stages of the menstrual cycle. A more detailed background of the biology can be found in [15,16,17]. The expected length of the menstrual cycle is around 29 days [18], although the variability in cycle length is large, with normal cycle lengths ranging from around 15 to 50 days [16, 18]. Day 1 of the menstrual cycle is the first day of menstruation (the “period”), and a menstrual cycle ends at the start of the next period. Both the uterus and ovaries experience changes throughout the menstrual cycle.

The ovarian cycle consists of two primary phases: the follicular phase and the luteal phase. Based on the traditional assumption of a standard 28-day menstrual cycle, the first phase of the ovarian cycle, called the follicular phase, lasts approximately 14 days. The second phase, called the luteal phase, begins on day 15 and lasts until the end of the ovarian cycle. Studies show that follicular phase lengths vary more than luteal phase lengths [18]. At birth the fetal ovary contains a fixed amount of primordial follicles (small, fluid-filled sacs that contain ova or “eggs”). In each ovarian cycle, a small portion of primordial follicles are activated at the beginning of the follicular phase to begin maturation [19, 20]. During this phase, one of the follicles will develop more quickly (the primary or dominant follicle), and the others will die out. The primary follicle will continue to develop during the remainder of the follicular phase until it fully matures. The key process during this phase is meiosis, a special type of cell division that produces genetic variation and ensures that all the germ cells needed for sexual reproduction contain the correct number of chromosomes [21]. Two rounds of nuclear division (called meiotic division) are involved in meiosis; the primary follicle develops into a secondary follicle when the first division is completed, and it continues to develop until it fully matures. Around day 14, the fully matured follicle ruptures, and the ovum is released, i.e., ovulation occurs. After ovulation, the follicle that previously contained the egg transforms into the corpus luteum [22, 23]. This signals the beginning of the luteal phase. If the ovum is fertilized, the second meiotic division completes. If the ovum is not fertilized, the corpus luteum begins to degrade until the start of the next ovarian cycle.

There are three primary phases in the uterine cycle: the menstrual phase, the proliferative phase, and the secretory phase. The menstrual phase occurs in the first 5–7 days of the uterine cycle. The innermost lining of the uterine wall, called the endometrium, sheds through the cervix and vagina. This first phase is often commonly referred to as a “period.” The phase from the end of the period until ovulation is the proliferative phase. During this phase, the endometrium thickens, while the dominant follicle in the ovary develops. The phase after ovulation is called the secretory phase. If the egg released from the ovary is fertilized, the thickened endometrium is ready for the egg to implant and grow to support the pregnancy. If the egg is not fertilized, the thickened endometrium breaks down and menstruation occurs. At this point, the cycle begins again. The lower section of Fig. 1 provides a visualization of the timeline of these different phases over a standard 28-day menstrual cycle.

Fig. 1

A schematic of a normal 28-day menstrual cycle. Note that in the hormone/ovarian schematic, black dashed lines indicate transition between different stages in the ovarian cycle, black solid lines indicate hormone production, gray solid lines indicate hormone inhibition, and black dash-dotted lines indicate hormone stimulation. The following abbreviations are used in the schematic: GnRH, gonadotropin-releasing hormone; FSH, follicle stimulating hormone; LH, luteinizing hormone; InhA, inhibin A; InhB, inhibin B; E${ }_2$, estradiol; P${ }_4$, progesterone

2.2 Introduction to the Role of Hormones in the Menstrual Cycle

The menstrual cycle is regulated by several hormones. The key hormonal drivers of the menstrual cycle can be divided into two types: gonadotrophic hormones that are produced in the pituitary and hypothalamus and travel through the bloodstream and reproductive or ovarian hormones that are produced locally in the ovaries. We provide a brief overview of the dynamics and interactions of these hormones here; we refer the reader to [15, 17, 24] for more in-depth discussions of these hormones and their roles.

The primary gonadotrophic hormones regulating the menstrual cycle are luteinizing hormone (LH) and follicular stimulating hormone (FSH). There are three key reproductive hormones in the menstrual cycle system: estrogen (E${ }_2$), progesterone (P${ }_4$), and inhibin.

During the menstrual phase of the uterine cycle (roughly the first one-third of the follicular phase of the ovarian cycle), the levels of E${ }_2$ and P${ }_4$ are very low, which triggers the shedding of the endometrium. FSH enhances the development of the follicles, preparing an egg for ovulation during the follicular phase of the ovarian cycle. The primary follicle secretes E${ }_2$ and inhibin as it grows. Both of these hormones suppress further production of FSH via a negative feedback, i.e., increased E${ }_2$ and inhibin levels lead to inhibition of FSH [25]. After the period, E${ }_2$ increases over the proliferative phase and causes the endometrium to thicken. When E${ }_2$ levels are high enough, a signal is sent to the brain, which causes a rapid and significant rise in LH [22]. This surge in LH triggers ovulation [26] and, thus, is often used as a biomarker for ovulation. The corpus luteum produces E${ }_2$, P${ }_4$, and inhibin during the luteal phase. Right after ovulation, E${ }_2$ levels drop [26], and P${ }_4$ levels increase, which signals the endometrium to stop thickening and prepare for a fertilized egg. There is a negative feedback of P${ }_4$ on the further release of LH from the pituitary. If the ovum is fertilized, the individual becomes pregnant, beginning a new hormonal process. If the ovum is not fertilized, the corpus luteum begins to degrade, which results in a reduction of both E${ }_2$ and P${ }_4$. The corpus luteum degenerates into corpus albicans toward the end of the menstrual cycle, and the resulting sharp drop in E${ }_2$ and P${ }_4$ induces menstruation, where the endometrium breaks down. This decline in E${ }_2$ leads to an increase in FSH, which marks the beginning of the next menstrual cycle. A diagram showing the relationships of those key hormones and the key elements in the ovarian cycle is shown in Fig. 1. Note that both FSH and LH increase in the early stages of the menstrual cycle, with peaks occurring mid-cycle. The highest levels of E${ }_2$ are observed in the first half of the cycle (preceding the LH surge), with a lower secondary peak occurring in the second half. The highest levels of P${ }_4$ are observed in the second half of the cycle. Representative curves of hormone levels throughout the menstrual cycle are depicted in Fig. 2.

Fig. 2

Schematic time courses of hormone levels throughout the menstrual cycle based on data by Welt et al. [12]

The reproductive hormone inhibin is involved in negative feedback control of FSH [15]. Inhibin is divided into two types: inhibin A (InhA) and inhibin B (InhB). The contrasting profiles of InhA and InhB suggest that FSH and LH regulate these two inhibins differently [27]. InhA is secreted in the luteal phase by the corpus luteum and peaks in the second half of the menstrual cycle, while InhB is secreted in the follicular phase by developing follicles and peaks in the first half of the menstrual cycle [12]. According to [28], “the appearance of measurable inhibin A can be seen as a marker for a follicle having at least matured to a stage corresponding to the late follicular stage in adult women.” InhB has been shown to be useful in the evaluation of the ovarian reserve and the assessment of an individual’s reproductive capacity [29, 30].

2.3 Hormonal Contraception and Its Effect on the Menstrual Cycle

As a means to prevent pregnancy, hormonal contraception aims to inhibit ovulation or fertilization. The primary biomarkers for ovulation are large increases in P${ }_4$ and LH, so contraceptives that work to inhibit ovulation attempt to suppress levels of P${ }_4$ and LH [31, 32]. Keeping FSH and E${ }_2$ levels low is also important to attain a contraceptive state. If FSH level is low, follicular growth is limited, and follicles cannot mature. Low levels of E${ }_2$ mean that LH level cannot increase because of the positive feedback on LH.

Oral contraceptives are the most widely used variant of hormonal contraception in the United States [33]. There are two primary types of oral contraceptives: pills that are synthetic progesterone only (sometimes called “mini pills”) and pills that contain both synthetic estrogen and synthetic progesterone (“combination pills”) [34]. Synthetic progesterone directly reduces the synthesis of LH, although it also contributes to inducing a contraceptive state more indirectly through its effect on FSH (limiting follicle growth) and through thickening of the cervical mucus [11, 35]. Synthetic estrogen contributes to contraception via suppressing LH (thereby eliminating the LH surge) and by enhancing the effects of synthetic progesterone by increasing receptor effectiveness. Other hormonal contraceptive methods (e.g., vaginal rings, transdermal inserts, and intrauterine devices) also employ the mechanisms of reducing P${ }_4$ and LH [33].

3 Mathematical Models of the Menstrual Cycle

There are a number of research groups that have developed mathematical models of the menstrual cycle over the years (cf. [5, 6, 36,37,38,39,40]). Of this collection of modeling efforts, the models first developed by Selgrade, Schlosser, and Harris Clark [7, 8, 41, 42] stand out because they have been used as the foundation for a significant number of subsequent models [43,44,45,46,47]. Another modeling approach was introduced more recently by Röblitz et al. [48] and built upon by George et al. [49]. The model structure created by Röblitz et al. diverges in several ways from the models in the Selgrade lineage. In the following subsections, we introduce the main features and contributions of each of these models. In Fig. 3, we provide a chart depicting the relationship among several of the Selgrade-based models, and we implement and test four of them in this chapter.

Fig. 3

Selgrade-based mathematical models of the human menstrual cycle. We implement and compare models (3), (5), (6), and (7)

3.1 Early Modeling Efforts

Very early efforts in mathematical modeling of the menstrual cycle date back to the 1970s. Shack et al. [4] used first-order differential equations to create a phenomenological computational model of the menstrual cycle. These authors were able to simulate periodic behavior that qualitatively captures some of the hormone fluctuations of the system. In two papers by Bogumil and colleagues [5, 6], the authors observed that short-duration, random events could significantly affect the qualitative behavior of the cycle and that rapid changes in hormone levels were likely responsible for regulating the cycle. In particular, computational simulations of this model were used to study how the introduction of estradiol (E${ }_2$) at various points in the cycle might contribute to phase shifts or “resetting” of the cycle, and it was observed in a contemporaneous study that this model did not produce results that agreed with physiologically observed behavior in this context [50]. Biological knowledge of the menstrual cycle has advanced a great deal since these early works; nevertheless, they provide an important foundation for more modern modeling efforts.

Motivated by the problem of exposure to environmental estrogen affecting reproduction, Selgrade and Schlosser developed two mechanistic models that tracked the follicular phase and the luteal phase of the ovarian cycle [7, 8]. The first model [7] used a system of nine linear ordinary differential equations (ODEs) to capture the stages of the cycle that drive the production of the gonadotropin hormones LH and FSH. The production of the ovarian hormones E${ }_2$, P${ }_4$, and inhibin was modeled as functions of LH and FSH. The ODE system presented in the second model was nonlinear [8] and introduced delays. This model focused on separating the processes of gonadotropin synthesis and gonadotropin release. In this second model, the functions for E${ }_2$, P${ }_4$, and inhibin were simplified to be functions of time only, with no direct dependence on LH or FSH [7]. While the first model was able to capture the dynamics of E${ }_2$, P${ }_4$, inhibin, FSH, and LH for normally cycling adult women with no external interference, the second model was extended to allow for the administration of exogenous ovarian hormones E${ }_2$ and P${ }_4$. The authors stated that their second model could be used as a testbed for exploring a variety of hormone-modulating scenarios. The authors suggested that future uses of this second model could include exploring the role of xenoestrogens in breast and ovarian cancer, better understanding anorexia’s role in the cessation of menstruation, and testing the effects of different methods of hormonal birth control [8]. Both the first and second models were fit to hormone-level data from McLachlan et al. [9]. Selgrade and Schlosser’s model systems [7, 8] together form the foundation for a number of subsequent modeling efforts [41,42,43, 51]. We explore several of these models in detail in the subsections below. A detailed comparison of models is provided in Table 1 located in supplementary information at the end of this chapter.

Table 1 Model comparison. Here we name the models in the format “First author + year of publication.” Diff.: differential; Aux.: auxiliary; Eq.: equation; Eqs.: equations; Exo. horm.: exogenous hormones. See text for definitions of symbols

3.2 Available Data Sets on Normal Menstrual Cycles

Two data sets are commonly used in models of the menstrual cycle: McLachlan et al. [9] and Welt et al. [12]. Each of the models we review in Sects. 3.3–3.7 was fit to one of these two data sets. Both studies followed approximately 40 women with a history of regular menstrual cycles (around 25–35 days). Daily blood samples were taken from participants during one complete menstrual cycle in order to measure bloodstream hormone levels (LH, FSH, E${ }_2$, P${ }_4$, and inhibin). The mean daily hormone levels were reported in each of these studies, with data centered around ovulation (the day of the LH surge). The Welt et al. data were normalized to a 28-day cycle using mean hormone levels in seven phases, including early, mid, and late follicular and luteal phases and the mid-cycle surge [12, 52]. An important distinction between the sets is that Welt et al. reported separate data for inhibin A and inhibin B, while McLachlan et al. reported a measurement of total inhibin since the separate assays for inhibin A and inhibin B were not yet available.

There are other more recent data sets on the normal menstrual cycle (published in or after the year 2000), but these published data sets do not provide the information on all the hormones that may be interesting from a mathematical modeling perspective. For example, Sehested et al. [53] collected daily blood samples from study participants and reported the mean daily levels of inhibin A, inhibin B, FSH, E${ }_2$, and LH, but they did not report levels of P${ }_4$. In another study, Stricker et al. [54] summarized the mean, median, 5th, and 95th percentiles of the daily levels of LH, FSH, E${ }_2$, and P${ }_4$ of study participants but did not report inhibin levels.

3.3 Harris Clark et al. Model

The Harris Clark et al. model was presented in both a manuscript and a Ph.D. dissertation [41, 42]. Harris Clark, Schlosser, and Selgrade [41] merged the two models of Schlosser and Selgrade [7, 8] to create a version of the model that included both a system of nonlinear delay differential equations to describe synthesis and release of gonadotropin hormones LH and FSH (as in [8]), as well as the more complicated functional forms for the ovarian hormones E${ }_2$, P${ }_4$, and inhibin that depend directly on the values of LH and FSH (as in [7]). This model was tuned to published data on normally cycling women (McLachlan et al. data [9]) to predict serum levels of these ovarian and pituitary hormones. The model also allowed for a stable abnormal cycle (in the sense of serum levels of ovarian and pituitary hormones), which can be used to fit for women with polycystic ovarian syndrome. Harris Clark et al. used this model to capture how exogenous administration of E${ }_2$ and P${ }_4$ impacts the menstrual cycle. They presented numerical experiments that show that hypothetical exogenous P${ }_4$ therapy can move a disrupted cycle into a normal cycle and that hypothetical exogenous E${ }_2$ can disrupt a normal cycle. We reproduced numerical solutions of this model, which are graphed in Fig. 4. The authors suggested that an exhaustive study of state space should be carried out to determine whether there are additional stable solutions. They pointed out that the following two changes would make the model more biologically realistic: future models should separately model inhibin A and inhibin B, instead of just generic inhibin, and should include a model component that explicitly captures the GnRH pulse frequency and amplitude, which can be affected by the ovarian hormones. The first of these two challenges was later taken up by Pasteur and Selgrade (described in Sect. 3.4).

Fig. 4

Numerical solutions to the Harris Clark et al. [41] model plotted against data from McLachlan et al. [9]. The root mean square error (RMSE) for each variable is shown, and the period is computed to be approximately 30.1 days. Note that the units for the hormone levels mirror the units reported in the McLachlan et al. data set

3.4 Pasteur and Selgrade Model

Prior to the mid-1990s, separate bioassays for inhibin A and inhibin B were not available [55], so hormone models typically incorporated their effects in a single term. In 2011, Pasteur and Selgrade published the first hormone model to incorporate separate inhibin terms [43]. This model first appeared as part of Pasteur’s dissertation [44]. Pasteur and Selgrade estimated parameters for the pituitary and ovarian systems individually and then merged the models. In their multi-inhibin model, one of the key changes was to the functional form of FSH synthesis function. They asserted that because the circulating level of each inhibin has a negative effect on FSH synthesis, they replaced what they called an “unrealistic” quadratic inhibition term (E${ }_2^2$) that inhibits FSH with a linear term (E${ }_2$). They also modified the governing equation for the reserve pool of FSH. Incorporating multiple inhibins required two separate delay terms: one for inhibin A and one for inhibin B. The authors expanded the ovarian model by adding two new stages at the beginning of the follicular phase and an additional one around the time of ovulation to capture both an early and a mid-cycle peak by inhibin B. They considered the effects of exogenous hormones, noting that increasingly large amounts of exogenous E${ }_2$ suppressed the LH surge to an increasingly large degree. After treatment, the hormone concentrations returned to a stable normal cycle after a few months.

Aside from the multi-inhibin model, Pasteur’s dissertation [44] also included a five-hormone model that is almost identical to the Harris Clark et al. [41] model, but it was fit to the Welt et al. data instead of the McLachlan et al. data. We do not include Pasteur’s five-hormone model in this review. Instead, we point the readers to [56], where Selgrade et al. thoroughly compared these two models and discussed their sensitivity to the data used for parameter fitting.

3.5 Margolskee and Selgrade Model

Margolskee and Selgrade [45] carried out a bifurcation analysis of the Harris Clark et al. [41] model and focused, in particular, on the size of the time delay parameter. In [41], three separate time delay values were used—one for each of the three corresponding ovarian hormones. Margolskee and Selgrade concluded that it is sufficient to include only a delay of $\tau = 1.5$ days for the effect of the inhibin on the pituitary’s secretion of FSH. In the Harris Clark et al. [41] model, the delay for the inhibin term was $d_{Ih} = 2.0$. Margolskee and Selgrade found that the delays required for E${ }_2$ and P${ }_4$ were less than 1 day and could be set to zero. The authors said that this change to the delay values improves the fit to the data for normally cycling women from Welt et al. [12], and they provided extensive bifurcation analysis exploring the effects of changing the length of the inhibin delay. They discussed why the shorter inhibin delay is consistent with biological evidence and how it permits increased ovarian development during the follicular phase of the cycle. We reproduced numerical solutions of this model, which are graphed in Fig. 5.

Fig. 5

Numerical solutions to the Margolskee and Selgrade [45] model plotted against data from Welt et al. [12]. The root mean square error (RMSE) for each variable is shown, and the period is computed to be approximately 28 days. The units for the hormone levels in these plots reflect the units reported in the Welt et al. data set

We note that when the authors fit the model to this new data set [12], a mismatch in units occurred. Unfortunately, all descendants of this model seem to inherit this unit mismatch.

3.6 Wright et al. Model

The focus of the Wright et al. [46] model was to explore the effects of oral contraceptive drugs in hormonal control of the menstrual cycle of adult women. This model also included a system of nonlinear delay differential equations with four auxiliary equations representing the ovarian hormones E${ }_2$, P${ }_4$, and inhibin A. Hormonal contraceptive treatments via oral administration of ethinyl estradiol and progestin were modeled by modifying state variables for blood concentrations of E${ }_2$ and P${ }_4.$ The authors assumed that a contraceptive state is attained if model simulations show a reduction in the LH surge to non-ovulatory levels or a reduction in P${ }_4$ levels throughout the cycle. The model extended the Margolskee and Selgrade [45] and Harris Clark et al. [41] models by adding autocrine mechanisms to describe how exogenous estrogen and progesterone could push a normal cycle into a contraceptive state. Model parameters were generally kept at the values used in [45], except for the changes needed for the new model components.

While this model did include more realistic mechanisms to account for exogenous hormones, there were still several simplifying assumptions. The authors assumed that the effect of estrogen on progesterone can be combined into one term P${ }_{app}$ that does not differentiate between the neuroendocrine and the ovarian systems. They also assumed that P${ }_{app}$ cannot be larger than P${ }_4$. As with the previous works of Harris Clark et al. [41] and Margolskee and Selgrade [45], this model only tracked inhibin A and did not incorporate inhibin B.

Wright et al. showed that the administration of synthetic progesterone, synthetic estrogen, or a combination of these can have a contraceptive effect by preventing ovulation. They concluded that a low dose of both treatments given together is most effective at achieving contraception. They provided numerical experiments that illustrate how the combined contraceptive treatment pushes the system into a non-ovulatory menstrual cycle fairly quickly and that the cycle also returns to normal in short order after the treatment ends. Future work mentioned coupling the model with a model for absorption and metabolism of oral contraceptive drugs. Wright et al. suggested that the addition of absorption and metabolism could help discover minimal effective doses of contraceptive drugs and may lead to patient-specific dosing strategies through pharmacokinetic/pharmacodynamic (PK/PD) modeling based on individual hormone data. We reproduced numerical solutions of this model, which are graphed in Fig. 6.

Fig. 6

Numerical solutions to the Wright et al. [46] model plotted against data from Welt et al. [12]. The root mean square error (RMSE) for each variable is shown, and the period is computed to be approximately 28.7 days. The units for the hormone levels in these plots reflect the units reported in the Welt et al. data set

3.7 Gavina et al. Model

The aim of the Gavina et al. [47] model was to employ optimal control techniques to minimize the total exogenous estrogen or progesterone doses required to enter a contraceptive state. This work was novel in that it considered nonconstant dosage as well as constant dosage of exogenous hormones. This model was also built on the normal menstrual cycle models of Harris Clark et al. [41] and Margolskee and Selgrade [45]. As with those models and the model of Wright et al. [46], this work incorporated only inhibin A (denoted as Inh in their model). The authors justified this assumption by stating that inhibin B is more important when studying reproductive aging. In addition to the system of nonlinear delay differential equations, these authors used auxiliary equations to track E${ }_2$, P${ }_4$, and Inh. In order to make the optimal control analysis more tractable, the authors made some simplifications from previous models. For example, instead of using the nonlinear term for inhibition and the additional equations employed in Wright et al. [46], these authors opted to use a linear inhibitory term and fewer additions to the Margolskee and Selgrade model; they found that these simplifications reduced computation time when running the optimization code.

The authors showed that the baseline simplified model without the administration of exogenous hormone reasonably matches the 28-day data from Welt et al. [12] and then proceeded to explore optimal dosing schedules. They observed a reduction in dosage of about 92% in estrogen monotherapy and 43% in progesterone monotherapy. Their simulations showed that delivering the estrogen contraceptive in the mid-follicular phase is the most effective. In addition, they showed that combination therapy significantly lowers doses even further, which is in line with the findings of [46]. While this work made promising steps by studying nonconstant dosing of exogenous hormones, it is important to note that the authors did not discuss whether the optimal dosing schedules are biologically feasible or would be possible to incorporate from a pharmacological perspective. For example, they presented an optimal dosing strategy for delivering synthetic estrogen that requires a large mid-follicular spike in this treatment. While this regimen minimizes the total amount of estrogen delivered over the entire cycle, it is unclear whether the magnitude of estrogen applied at this mid-follicular stage would be plausible to administer or if it would create adverse effects for the user. We reproduced numerical solutions of this model, which are graphed in Fig. 7.

Fig. 7

Numerical solutions to the Gavina et al. [47] model plotted against data from Welt et al. [12]. The root mean square error (RMSE) for each variable is shown, and the period is computed to be approximately 28 days. The units for the hormone levels in these plots reflect the units reported in the Welt et al. data set

3.8 Models Incorporating GnRH

The models discussed in the previous subsections focus on modeling the hormones that are produced in the hypothalamus and pituitary (FSH and LH) and the ovaries (P${ }_4$, E${ }_2$, InhA, and InhB). They are potentially well-suited to the incorporation of exogenous hormones. However, these models do not include the gonadotropin-releasing hormone GnRH, which controls the synthesis of FSH and LH. We are aware of two recent models that do incorporate GnRH, which we describe briefly here: Röblitz et al. [48] and George et al. [49].

In 2013, Röblitz et al. [48] published a hormone model with the aim of explicitly exploring GnRH-receptor binding. The focus of this work was to develop a detailed model of GnRH to explore GnRH-receptor binding in response to the introduction of GnRH analogs. Motivated by model equations from Harris Clark [42], Pasteur [44], and Reinecke [51], this model included several novel features. Unlike other models discussed here, this work developed explicit terms for the GnRH system. Moreover, the model eliminated time delays by introducing “effect” compartments, so that the effects of a given hormone could be delayed without incorporating time delays into the differential equations. Finally, the model included separate terms to account for the effects of InhA and InhB. Importantly, their model correctly predicted cycle variations due to doses of a GnRH agonist (which causes the ovaries to stop making estrogen and progesterone) and a GnRH antagonist (which impedes ovulation in in vitro fertilization treatment). The authors also communicated an explicit goal to model individual patients, instead of an idealized patient from aggregate data.

In a more recent model by George et al. [49], the authors proposed a simplified model of six nonlinear differential equations to explore the effects of GnRH on the dynamics of the key hormonal drivers in the menstrual cycle. While the model was dramatically simplified from that of Röblitz, the authors were still able to capture some of the qualitative effects of GnRH on the gonadotrophic hormones, including that increasing the width of a GnRH pulse can affect the timing of the release of LH, thereby affecting the timing of when the ovarian hormones are produced.

4 Comparisons Between Existing Menstrual Cycle Models

In this section, we provide a comparative analysis of four menstrual cycle models discussed in Sect. 3: Harris Clark et al. [41], Margolskee and Selgrade [45], Wright et al. [46], and Gavina et al. [47]. For ease of comparison, we selected these four models from the Selgrade lineage, Fig. 3. Our analysis focuses on three areas of interest: a sensitivity analysis of the shared parameter $V_{FSH}$, a comparison of the ways modelers represent inhibitory delays and the disparate effects of inhibin in the system, and the models’ responses to exogenous estrogen and progesterone. The MATLAB codes for the four models we implemented here are available on GitHub: https://github.com/rubyshkim/menstrual-cycle_models.

4.1 Sensitivity to $V_{FSH}$

By investigating the sensitivity of the models by Harris Clark et al. [41], Margolskee and Selgrade [45], Wright et al. [46], and Gavina et al. [47] to variations in parameter values, we discover that the maximum growth rate of the FSH reserve pool ($V_{FSH}$) significantly influences the length of the menstrual cycle.

For each model, we vary all parameters one at a time by 25% to determine which parameter values create the most sensitive behaviors when perturbed. Since periodic rhythms are an essential part of the menstrual cycle, we use period length (numerically approximated) as our metric. Of the parameters generating the most sensitivity, we chose to focus on a shared parameter $V_{FSH}$, which represents the maximum rate of growth of the reserve pool of FSH. Note that although FSH is a measurable hormone, $V_{FSH}$ represents a maximum growth rate. $V_{FSH},$ therefore, is a parameter that is useful in a modeling context, but experimental data on the physiological ranges of $V_{FSH}$ are sparse. Since $V_{FSH}$ is not typically measured in an individual, we focus on the qualitative (rather than the quantitative) model response to changing the values of the rate $V_{FSH}$. In the models of Selgrade, Schlosser, and Harris Clark [7, 41], the authors cite Odell [58] and state that the nominal values of the follicle growth rates during the follicular phase are chosen by assuming they are proportional to the FSH serum levels. FSH is regulated through negative feedback, and its baseline level can change depending on the availability of regulatory hormones. FSH stimulates the growth of follicles, which modulate inhibins, which in turn then inhibit FSH secretion. Female reproductive aging is accompanied by an increase in circulating FSH due to the loss of follicles, which lowers inhibin levels [12]. In the mathematical models, increasing $V_{FSH}$ from its nominal values shortens the period of the menstrual cycle; see Fig. 8 where the nominal value of $V_{FSH}$ for each model is identified by an orange point. This relationship is consistent with findings that suggest that lower circulating FSH levels lengthen the menstrual cycle [57]. Lowering $V_{FSH}$ from the nominal values generally increases the period in all four models.

Fig. 8

Sensitivity of the model simulations to $V_{FSH}$. We approximate the period of numerical solutions over 100 cycles for the models by Harris Clark et al. [41], Margolskee and Selgrade [45], Wright et al. [46], and Gavina et al. [47] for values of the parameter $V_{FSH}$ between 75% and 125% of its nominal value indicated by the orange point in each model. $V_{FSH}$ is the maximum growth rate of reserve pool FSH and generally has an inverse relationship with the period in our simulations. This result is consistent with the findings in [57]. There is notable sensitivity to variations in $V_{FSH}$ in all four models. The axes for each model are chosen to center the nominal value for $V_{FSH}$

However, there is some notable sensitivity in the model behaviors. The model by Harris Clark et al. [41] undergoes a non-smooth change in the approximated period. When $V_{FSH}$ is below just 0.966 of its nominal value of 5700 $\mu$g/day, E${ }_2$ loses its secondary peak during the luteal phase; example time courses with $V_{FSH} = 5415$ $\mu$g/day are provided in Fig. 9. Our simulations of the Wright et al. [46] and Margolskee et al. [45] models also show a sharp transition in approximated period at particular values of $V_{FSH}$ associated with a loss of the E${ }_2$ secondary peak. In addition, the length of a regular menstrual cycle is considered to range from 26 to 35 days [16]. A 25% reduction in $V_{FSH}$ results in a loss of periodicity in our simulations of the Gavina et al. [47], Margolskee and Selgrade [45], and Harris Clark et al. [41] models.

Fig. 9

Example time series from Harris Clark et al. [41] model simulations with $V_{FSH}=5415$ $\mu$g/day or 95% of its nominal value. Compared with Fig. 4, concentrations of all four hormones are reduced, and FSH and E${ }_2$ no longer have a secondary peak during the luteal phase

4.2 Variation in Incorporating Delays in the Models

In many of the models, discrete time delays are key components that represent the delays between changes in blood hormone levels and their effects on synthesis rate. Among all models we reviewed in this manuscript, only the models of Röblitz et al. [48] and George et al. [49] do not involve any delay differential equations. Röblitz et al. argued that the delayed inhibitory effect of inhibin B on FSH synthesis in their model was incorporated by the mechanisms in which low GnRH frequencies would stimulate FSH synthesis with adjusted rate constants. To avoid the use of delay differential equations, they introduced a new compartment for effective inhibin A, IhA${ }_e$, to account for the delayed inhibitory effect of inhibin A on FSH synthesis.

In the original Selgrade and Schlosser [8] model of the pituitary component of the menstrual cycle that five of the models we reviewed were based on, they included three discrete time delays—one corresponding to each ovarian hormone. First, $d_E=0.42$ day is the delay in the input function $E_2(t)$, which appears in the LH synthesis term in order to simulate both the rapid rise and fall of LH during the surge, as well as the time difference between the peaks of E${ }_2$ and LH. There is a $d_P=2.9$ day delay in the input function $P_4(t)$, which appears in the LH synthesis term to correctly simulate the timing of the surge in LH synthesis following the changes in serum levels of P${ }_4$. Finally, there is a $d_{Ih}=2$ day delay in the input function $Ih(t)$, which appears in the FSH synthesis term to capture the period of time between the changes in the inhibin blood levels and the FSH synthesis rates.

The Harris Clark et al. [41] model removed the delay in the input function E${ }_2(t)$ in the LH synthesis term, as their parameter identification indicated that it was insignificant; they kept the delay $d_{Ih}=2$ day but modified the other delay $d_P$ to be 1 day. The Margolskee and Selgrade [45] model reduced the number of delays to one, incorporating only a $\tau = 1.5$ day delay in the effect of the peptide inhibin on FSH synthesis. This led to an improvement in the fit to the Welt et al. data. They set the other two delays to 0 as the values were less than a day and did not contribute significant additional improvement on data fitting. Similarly, both the Wright et al. [46] and Gavina et al. [47] models kept the discrete time delay, $\tau =1.5$ day, in the input function $InhA(t)$, which appears in the FSH synthesis term. The Pasteur and Selgrade [43] multi-inhibin model includes four time delays: a $d_E=0.5086$ day delay in the input function $E_2(t)$ in the LH synthesis term, a $d_P=0.9156$ day delay in the input function $E_2(t)$ in the LH synthesis term, a $d_{IhA}=2.5$ day delay in the input function $IhA(t)$ in the FSH synthesis term, and a $d_{IhB}=1$ day delay in the input function $IhB(t)$ in the FSH synthesis term.

In this collection of models, we see that there are two main approaches taken to capturing the timing of the cascade of events from the onset of changes in blood hormones to the subsequent effect on synthesis: using delays or incorporating more model detail. The models of [8, 41, 45,46,47] incorporate explicit time delays in strategic parts of the models, and variations in the way the delays are included lie in the fine-tuning of the lengths of the delays. The models of [48, 49] have taken the approach of including more detailed steps in the hormone-synthesis cascade that in turn implicitly produce the necessary time delays. Although software for the numerical solution of delay differential equations is readily available, the delay-free approach may be preferable when implementing numerical solutions because traditional numerical ODE solution methods tend to be stable and easy to implement.

4.3 Subtle Differences in Modeling Inhibins

As mentioned in Sect. 3.2, data from McLachlan et al. [9] and Welt et al. [12] are commonly used in models of the menstrual cycle. It is important to note that these two data sets are different with respect to inhibin, and this impacts modeling choices. Separate bioassays for inhibin A and inhibin B were not available until the mid-1990s. As such, many of the models discussed here represented the total effects of inhibin using one compartment, despite the fact that they peak at different times in the cycle. In this section, we discuss the variations in choices of modeling inhibin and how it connects to these data sets.

Harris Clark et al. [41] fit their model to the McLachlan et al. data set [9], which only reported total inhibin values. Many other authors [43, 45,46,47] parameterized their models using the Welt et al. data set [12], but the ways they handled the inhibin data are slightly different. Pasteur and Selgrade [43] used the data from the younger age group with 23 women aged 20–34 for parameterization to model both inhibin A and inhibin B. Note that Pasteur et al. parameterized their five-hormone model using both the McLachlan et al. and Welt et al. data in [44, Chapter 3], but they used the inhibin A data from the Welt et al. data only, as they concluded that its profile matched the total inhibin profile from the McLachlan et al. data. Both Gavina et al. [47] and Wright et al. [46] included only inhibin A in their model and used the data from the younger age group (note that in [47] the state variable Inh denotes inhibin A). Margolskee and Selgrade [45] also fit their model to the data from the younger group in the Welt et al. data, but it is unclear how they handled the inhibin data as they did not distinguish inhibin A and inhibin B explicitly in their model.

4.4 Incorporation of Hormonal Contraception in Models

Perhaps the simplest way to incorporate hormonal contraception into a menstrual cycle model is with a constant dosing of exogenous estrogen (e.g., estradiol) and/or progesterone (e.g., progestin). In this section, we compare the qualitative behaviors of each of the models under this simple strategy for modeling hormonal contraception. As discussed in Sect. 2.3, a contraceptive state is marked by the absence of an LH surge. However, contraception may refer to total contraception, where LH levels are low and roughly constant, or biological contraception, where LH levels increase and decrease in an oscillatory manner but remain low. We capture the dynamics of response to hormonal contraception in each of these models by measuring (a) the amplitude of the LH peak (a biomarker for whether or not ovulation will occur) and (b) the period of P${ }_4$ oscillations (a proxy for whether total contraception is achieved).

As expected, with no exogenous progesterone and no exogenous estrogen (i.e., no hormonal contraceptives), all of the models we discuss in this manuscript have a relatively high LH peak and a mean cycle length between 28 and 30 days. This is consistent with the biology of the menstrual cycle in a non-contraceptive state. To visualize the response of each model to the addition of hormonal contraception, we create heatmaps in Fig. 10 for the effect of exogenous estrogen (horizontal axis) and exogenous progesterone (vertical axis) on the maximum value of the LH peak (colors, top panels) and the period of the P${ }_4$ oscillations (colors, middle panels) computed using findpeaks in MATLAB. It is advantageous to use the time course of P${ }_4$ to estimate the period because it only has one peak (or local maximum) per cycle, while the other variables have multiple peaks. Note that these models have some differences in units and baseline concentrations of variables. For each model, we administer exogenous estrogen up to the mean endogenous estradiol concentration in that model and exogenous progesterone up to 0.3 of the mean endogenous progesterone concentration in that model. These choices are based on the exogenous hormone concentrations used in the models which consider exogenous hormones [41, 46, 47].

Fig. 10

LH peak concentration and menstrual cycle period for the various models [41, 45,46,47] with daily administration of exogenous hormones, i.e., hormonal contraceptives. We vary the amount of exogenous estrogen $e_{\text{dose}}$ up to its mean concentration (horizontal axis) and exogenous progesterone $p_{\text{dose}}$ up to 30% of its mean concentration (vertical axis) in each model. In panels A–D, we visualize the effects of $e_{\text{dose}}$ and $p_{\text{dose}}$ on LH peak concentrations, with yellow representing high peak concentrations of LH and blue representing low peak concentrations of LH. These blue values represent the parameter ranges where the model is in a contraceptive state. In panels E–H, we visualize the effects of $e_{\text{dose}}$ and $p_{\text{dose}}$ on the period of P${ }_4$ oscillations, with yellow representing a longer cycle length and red representing a shorter cycle length. The white regions in these plots correspond to the absence of oscillations, which indicates that the model is in a total contraceptive state. Schematic plots of example 28-day hormone trajectories corresponding to contraceptive states for the Wright et al. [46] model are provided in panels i ($p_{\text{dose}}=0.6$) and ii ($p_{\text{dose}}=1.3$), which correspond to the dots indicated in panel C. Simulations were run for 100 cycles before each computation to help eliminate the effects of transient initialization behavior

First, we explore the behavior of the Harris Clark et al. [41] model to the introduction of exogenous hormones (Fig. 10A and E). We note that this model was not designed primarily with questions of hormonal contraception in mind. In Fig. 10A, we notice that, in the case without any exogenous estrogen, the LH peak increases as the level of exogenous progesterone increases. From a biological perspective, we know that progesterone-only hormonal birth control methods do induce contraceptive states [35], and this is not captured by this model. However, we do see that increasing exogenous estrogen induces a contraceptive state. Biological contraception is achieved for values of $e_{\text{dose}}$ larger than 20, while total contraception is achieved only for very large doses of exogenous estrogen. This model has a smaller variance in cycle length (as measured by the period of oscillations in LH) in this parameter space as compared to other models we tested, and we observe a non-monotonic response in cycle length to exogenous estrogen.

As noted in Sect. 3.5, Margolskee and Selgrade [45] modified the model of Harris Clark et al. [41] to perform bifurcation analyses to explore the effects of delays and to fit the data of normally cycling women more closely than previous efforts. The level of the LH peak in this updated model does depend on both exogenous estrogen and exogenous progesterone, but the effect of exogenous estrogen is stronger (Fig. 10B and F). We further observe that contraception is not achieved with progesterone-only treatment, and this model did not achieve total contraception for any values of parameters we tested. As with the model of Harris Clark et al. [41], the authors did not design this model to explore the effects of hormonal contraception, so it is perhaps not surprising to see that these effects are not strongly captured.

In contrast to previous models, Wright et al. [46] developed their model with the intention of including hormonal contraception. In Fig. 10C and G, we see that as both exogenous estrogen ($e_{\text{dose}}$) and exogenous progesterone ($p_{\text{dose}}$) increase, the LH peak decreases, indicating that the system is in a contraceptive state. This occurs for relatively low values for both parameters. We notice that the cycle length, as measured by the period of the oscillations in LH, is relatively stable over a large range of parameter values. Total contraception is achieved for large enough values of $e_{\text{dose}}$ and $p_{\text{dose}}$, as indicated by the white region in Fig. 10G. Mathematically, a transition to this parameter regime corresponds to a Hopf bifurcation.

The model of Gavina et al. [47] was also created to explore the effects of hormonal contraception, although from a different perspective than [46]. We notice that the modifications made in this model introduce different qualitative dynamics with constant exogenous hormonal contraception (Fig. 10D and H) than for the Wright et al. [46] model. While we do see the expected behavior of a reduction of LH peak when either exogenous hormone is increased initially, we notice that the LH peak again begins to increase for large values of $e_{\text{dose}}$. We see that LH undergoes a Hopf bifurcation in $e_{\text{dose}}$, with periodic behavior only occurring for small parameter values. Combining these observations, we can conclude that for large values of $e_{\text{dose}}$, LH is being held constant at a high level, which is not conducive to the expected contraceptive state.

5 Outlook and Future Directions

In this survey, we describe the biological mechanisms behind the menstrual cycle and hormonal contraception and review the existing mathematical models of this system. In a comparative analysis of several of these models, we highlight the ways different models are sensitive to parameters, focusing in particular on the maximum growth rate of the reserve pool of follicular stimulating hormone. We discuss the variation in incorporating delays and inhibins into these models. Finally, we explore the qualitative behavior of these models under the incorporation of exogenous hormones to model hormonal contraception. We conclude this chapter by discussing some of the existing challenges in this area and potentially promising future directions.

The complexity of the biological system may seem like a daunting challenge to mathematical modelers. Indeed, even models that try to simplify while including crucial mechanisms can lead to relatively large systems of differential equations with many parameters. The models by Harris Clark et al. [41], Pasteur et al. [43], Margolskee and Selgrade [45], Wright et al. [46], and Gavina et al. [47] all contain on the order of 13 ODEs, 4 auxiliary equations, and 50 parameters. The model of Röblitz et al. [48] considers the system in even more detail, including gonadotrophin-releasing hormone, which facilitates the synthesis and secretion of the gonadotrophic hormones FSH and LH. This model contains 33 ODEs and 114 parameters. These detailed models strive to accurately capture the range of temporal and spatial scales represented in this system. Furthermore, detailed models may provide more flexibility for use in preliminary drug development or testing interventions beyond what the model was originally built for, which is not usually possible in overly simplified phenomenological models. More detailed models do have some limitations, however. Parameter identification and overfitting can be concerns in large systems of differential equations; thus care must be taken when applying or extending these models. Large models are often less amenable to mathematical analysis, which may limit the possible insight into qualitative behaviors or general properties of the models. These different styles of models each have roles to play in gaining a deeper understanding of the menstrual cycle and hormonal contraception. There is still ample opportunity for developing simpler, more analytically tractable models of the menstrual cycle in the style of George et al. [49] that are nevertheless grounded in biological mechanisms.

The parameters in the existing models can present a challenge from the perspective of data fitting and model analysis. While some parameters in the model may be inferred from data, many cannot. When comparing or working with the models of the menstrual cycle discussed in this chapter, it is important to note that even parameters that share the same name or represent the same quantity across different models may not be the same order of magnitude or even the same units. There can also be challenges with a lack of consistency of units and dimensions, even within a single model. For readers who are interested in generalizing, extending, or applying the models we discuss in this chapter, we provide a detailed dimensional analysis on our public GitHub repository (https://github.com/rubyshkim/menstrual-cycle_models/blob/main/SupplementalFiles/SupplementalMaterial.pdf).

One of the interesting features of the menstrual cycle is that it has a high level of variability. Several of the models discussed in this chapter use data from Welt et al. [12], and the models are fit to the mean hormone levels reported across 23 subjects. In the experiments, there is a high level of variability in hormone levels both within and between individuals. For example, in [12, Fig. 5], the authors show hormone levels from two subjects, aged 36 and 47, of two menstrual cycles approximately 10 years apart. These data demonstrate that hormone levels can vary greatly between different subjects: In particular, inhibin A/B and P${ }_4$ levels are quite different in individuals of different ages. Even the same subject demonstrates variability of their hormone levels in two cycles. Unfortunately, this variability is not currently incorporated with any of the existing modeling approaches we have reviewed here. While variability can be challenging from the perspective of trying to fit models to data, we believe it also presents an excellent opportunity for further modeling. For example, it may be interesting to study the addition of intrinsic and/or extrinsic sources of noise into these models.

Incorporation of hormonal contraception into mathematical models of the menstrual cycle is still in the early scientific stages. We discuss very few models of the menstrual cycle that incorporate the impact of hormonal contraceptives on the menstrual cycle in Sect. 4.4; of these, only [47] explores time-varying exogenous hormones. This means that at the time of writing, there are no existing mathematical models of the effects of hormonal contraception on the menstrual cycle that incorporate the dynamics of the on/off dosing regimens or the metabolism of the exogenous hormones, even though methods from differential equations and dynamical systems are well-positioned to investigate these questions. One potential area of inquiry would be to develop a model of contraceptive dosing regimens on the menstrual cycle, either by creating a new model or by generalizing an existing model discussed here. This would allow for the exploration of the stability of the contraceptive state achieved by oral hormonal contraceptives using a mechanistic mathematical model of the menstrual cycle. Such a model could provide insight into when a contraceptive state is lost due to inconsistency or changes in hormonal birth control use, which may further inform the advisement of care providers and the choices of birth control users. Another interesting direction of study would be to explore the impact of hormonal stimulation used in cases of infertility. While infertility treatments have increased in their use and success over the last few decades, there are still substantial risks, including high-risk multiple pregnancies and ovarian hyperstimulation [59]. A mathematical model that can successfully incorporate the impact of hormonal contraceptives on the menstrual cycle could be leveraged to explore open questions in this budding field.

Supplementary Information

Table 1 provides a summary of six of the models we reviewed in this chapter. It contains a brief overview of the characteristics of each model (such as the number of differential equations/auxiliary equations/parameters, whether constant time delays are implemented, and how the exogenous hormones are modeled), study focus, some of the contributions, and limitations of each model.