Dynamics and bifurcation of a model for hormonal control of the menstrual cycle with inhibin delay
Open accessAbstract
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.
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As with the previous works of Harris Clark et al. and Margolskee and Selgrade, the Wright et al. model only tracked inhibin A and did not incorporate inhibin B.
“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.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
The Auxiliary Equations (A), equation A3, model a single inhibin concentration, Inh, and equation S3 uses that concentration to inhibit FSH synthesis; there is no separate inhibin B variable. However, the paper calls this variable “Inh,” not “inhibin A,” so our copy does not establish that the model tracked inhibin A specifically. (LLM, conf=0.83)
Experimental — accuracy not yet measuredMargolskee and Selgrade [45] 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.
“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.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
Section 3 says the delay model better approximates Welt et al.’s 28-day data, and the Figure 2 caption shows a single modeled inhibin measure, “Inh,” alongside Welt data points. The Introduction describes the data as coming from normally cycling women, but our copy does not identify it as the younger group or explain how inhibin A and B measurements were handled. (LLM, conf=0.82)
Experimental — accuracy not yet measuredMargolskee and Selgrade found that the delays required for E2 and P4 were less than 1 day and could be set to zero.
“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.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
The Wright et al. model extended the Margolskee and Selgrade and Harris Clark et al. models by adding autocrine mechanisms to describe how exogenous estrogen and progesterone could push a normal cycle into a contraceptive state.
“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.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
The models first developed by Selgrade, Schlosser, and Harris Clark stand out because they have been used as the foundation for a significant number of subsequent models.
“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 [43-47].”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
Margolskee and Selgrade carried out a bifurcation analysis of the Harris Clark et al. model and focused on the size of the time delay parameter.
“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.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
Margolskee and Selgrade provided extensive bifurcation analysis exploring the effects of changing the length of the inhibin delay.
“and they provided extensive bifurcation analysis exploring the effects of changing the length of the inhibin delay.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
Margolskee and Selgrade 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.
“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.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
Margolskee and Selgrade concluded that it is sufficient to include only a delay of τ = 1.5 days for the effect of the inhibin on the pituitary's secretion of FSH.
“Margolskee and Selgrade concluded that it is sufficient to include only a delay of . τ = 1 . 5 days for the effect of the inhibin on the pituitary's secretion of FSH.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
The change to the delay values improves the fit to the data for normally cycling women from Welt et al.
“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]”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
A 25% reduction in VFSH 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.
“A 25% reduction in . VFSH 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.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
Many other authors [43, 45-47] parameterized their models using the Welt et al. data set [12], but the ways they handled the inhibin data are slightly different.
“Many other authors [43, 45-47] parameterized their models using the Welt et al. data set [12], but the ways they handled the inhibin data are slightly different.”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
The models of [8, 41, 45-47] incorporate explicit time delays in strategic parts of the models.
“The models of [8, 41, 45-47] incorporate explicit time delays in strategic parts of the models”
From A Survey of Mathematical Modeling of Hormonal Contraception and the Menstrual Cycle (2024) · cites this paper as [45]
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- Authors
- Margolskee, A., & Selgrade, J. F.
- Affiliations
- North Carolina State University (US)
- Corresponding author
- Selgrade, James F.
- Venue
- Mathematical Biosciences
- Publisher
- Elsevier BV
- Published
- 2011-12
- Volume
- 234
- Issue
- 2
- Pages
- 95-107
- ISSN
- 0025-5564 (ISSN-L)
- OpenAlex
- W2127415102
- Cited by
- 18
- References
- 43
- Citations by year
- 2026: 2 · 2025: 1 · 2024: 1 · 2023: 2 · 2022: 2 · 2020: 1 · 2018: 1 · 2017: 1 · 2016: 1 · 2015: 1 · 2014: 2 · 2013: 2 · 2012: 1
- FWCI
- 1.77
- Type
- journal article
- Language
- en
- Open access
- Licence
- Locations
- Mathematical Biosciences (publishedVersion)
- PubMed (publishedVersion)
- Research Explorer (The University of Manchester) (submittedVersion, other-oa)
- Funders
- National Science Foundation (DMS-0920927)
- Topics
- Health Sciences › Medicine › Reproductive Medicine › Hypothalamic control of reproductive hormones
- Health Sciences › Medicine › Public Health, Environmental and Occupational Health › Mathematical and Theoretical Epidemiology and Ecology Models
- Physical Sciences › Mathematics › Geometry and Topology › Advanced Differential Equations and Dynamical Systems
- Keywords
- menstrual cycle, hormonal regulation, inhibin, follicle-stimulating hormone, luteinizing hormone, estradiol, time delay, bifurcation analysis, Hopf bifurcation, transcritical bifurcation, saddle-node bifurcation, degenerate Hopf bifurcation, periodic solutions, ovulation, folliculogenesis, ovarian development
- MeSH terms
- Computer Simulation, Estradiol, Female, Follicle Stimulating Hormone, Ovarian Follicle, Humans, Inhibins, Menstrual Cycle, Models, Biological, Ovulation
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