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

Mathematical Biosciences

Volumn 234, Issue 2, 2011, Pages 95-107

  Margolskee, Alison ^a   Selgrade, James F ^a  

^a NORTH CAROLINA STATE UNIVERSITY   (United States)

Author keywords

Degenerate Hopf bifurcation; Estradiol; Follicle; Hill function

Indexed keywords

BIFURCATION DIAGRAM; CLINICAL DATA; ESTRADIOL; FOLLICLE; FOLLICLE STIMULATING HORMONE; HILL FUNCTION; HORMONAL CONTROL; HORMONAL REGULATION; MENSTRUAL CYCLE; OVARIAN DEVELOPMENT; PARAMETER VALUES; PERIODIC SOLUTION; TRANSCRITICAL; TWO STAGE;

FUNCTIONS; ORDINARY DIFFERENTIAL EQUATIONS; TIME DELAY;

HOPF BIFURCATION;

ESTRADIOL; FOLLITROPIN; INHIBIN; LUTEINIZING HORMONE;

BIFURCATION; BIOLOGICAL ANALYSIS; HORMONE; MASS TRANSFER; MATHEMATICAL ANALYSIS; NUMERICAL MODEL;

ARTICLE; CONTROLLED STUDY; FOLLITROPIN RELEASE; HORMONAL REGULATION; HORMONE SYNTHESIS; MATHEMATICAL ANALYSIS; MATHEMATICAL COMPUTING; MATHEMATICAL MODEL; MENSTRUAL CYCLE; OVARY CYCLE; OVARY DEVELOPMENT; PROTEIN FUNCTION;

COMPUTER SIMULATION; ESTRADIOL; FEMALE; FOLLICLE STIMULATING HORMONE; HUMANS; INHIBINS; MENSTRUAL CYCLE; MODELS, BIOLOGICAL; OVARIAN FOLLICLE; OVULATION;

EID: 82255192557     PISSN: 00255564     EISSN: 18793134     Source Type: Journal    
DOI: 10.1016/j.mbs.2011.09.001     Document Type: Article

References (42)

1. Alvarez-Blasco F., Botella-Carretero J.I., San Millan J.L., Escobar-Morreale H.F. Prevalence and characteristics of the polycystic ovary syndrome in overweight and obese women. Arch. Intern. Med. 2006, 166:2081.
2. Azziz R., Woods K.S., Reyna R., Key T.J., Knochenhauser E.S., Yildiz B.O. The prevalence and features of the polycystic ovary syndrome in an unselected population. J. Clin. Endocrinol. Metab. 2004, 89:2745.
3. Boer H.M.T., Stötzel C., Röblitz S., Deuflhard P., Veerkamp R.F., Woelders H. A simple mathematical model of the bovine estrous cycle: follicle development and endocrine interactions. J. Theor. Biol. 2011, 278:20.
4. Bogumil R.J., Ferin M., Rootenberg J., Speroff L., Vande Wiele R.L. Mathematical studies of the human menstrual cycle. I: Formulation of a mathematical model. J. Clin. Endocrinol. Metab. 1972, 35:126.
5. Bogumil R.J., Ferin M., Vande Wiele R.L. Mathematical studies of the human menstrual cycle. II: Simulation performance of a model of the human menstrual cycle. J. Clin. Endocrinol. Metab. 1972, 35:144.
6. Derry G.N., Derry P.S. Characterization of chaotic dynamics in the human menstrual cycle. Nonlinear Biomed. Phys. 2010, 4:5.
7. Doedel E.J. AUTO: a program for the automatic bifurcation analysis of autonomous systems. Congr. Numer. 1981, 30:265.
8. Engelborghs K., Luzyanina T., Roose D. Numerical bifurcation analysis of delay differential equations. J. Comput. Appl. Math. 2000, 125:265.
9. Ermentrout B. Simulating, Analyzing, and Animating Dynamical Systems 2002, SIAM, Philadelphia, PA.
10. Franchimont P., Henderson K., Verhoeven G., Hazee-Hagelstein M.T., Charlet-Renard C., Demoulin A., Bourguignon J.P., Lecomte-Yerna M.J. Inhibin: mechanisms of action and secretion. Intragonadal Regulation of Reproduction 1981, 167. Academic Press, London. P. Franchimont, C.P. Channing (Eds.).
11. Golubitsky M., Schaeffer D.G. Singularities and Groups in Bifurcation Theory 1985, vol. 1. Springer-Verlag, New York.
12. Hale G.E., Zhao X., Hughes C.L., Burger H.G., Robertson D.M., Fraser I.S. Endocrine features of menstrual cycles in middle and late reproductive age and the menopausal transition classified according to the staging of reproductive aging workshop (STRAW) staging system. J. Clin. Endocrinol. Metab. 2007, 92:3060.
13. L.A. Harris, Differential Equation Models for The Hormonal Regulation of The Menstrual Cycle, Ph.D. Thesis, North Carolina State University, Raleigh, North Carolina, 2001. http://www.lib.ncsu.edu/theses/available/etd-04222002-153727/unrestricted/etd.pdf.
14. Harris-Clark L., Schlosser P.M., Selgrade J.F. Multiple stable periodic solutions in a model for hormonal control of the menstrual cycle. Bull. Math. Biol. 2003, 65:157.
15. Hotchkiss J., Knobil E. The menstrual cycle and its neuroendocrine control. The Physiology of Reproduction 1994, 711. Raven Press, Ltd., New York. second ed. E. Knobil, J.D. Neill (Eds.).
16. Karsch F.J., Dierschke D.J., Weick R.F., Yamaji T., Hotchkiss J., Knobil E. Positive and negative feedback control by estrogen of luteinizing hormone secretion in the rhesus monkey. Endocrinology 1973, 92:799.
17. Keener J., Sneyd J. Mathematical Physiology I: Cellular Physiology 2009, Springer-Verlag, New York. second ed.
18. Klein N.A., Harper A.J., Houmard B.S., Sluss P.M., Soules M.R. Is the short follicular phase in older women secondary to advanced or accelerated dominant follicle development?. J. Clin. Endocrinol. Metab. 2002, 87:5746.
19. Liu J.H., Yen S.S.C. Induction of midcycle gonadotropin surge by ovarian steroids in women: a critical evaluation. J. Clin. Endocrinol. Metab. 1983, 57:797.
20. McIntosh J.E.A., McIntosh R.P. Mathematical Modeling and Computers in Endocrinology 1980, Springer-Verlag, Berlin.
21. McLachlan R.I., Cohen N.L., Dahl K.D., Bremner W.J., Soules M.R. Serum inhibin levels during the periovulatory interval in normal women: relationships with sex steroid and gonadotrophin levels. Clin. Endocrinol. 1990, 32:39.
22. Odell W.D. The reproductive system in women. Endocrinology 1979, 1383. Grune & Stratton, New York. L.J. DeGroot (Ed.).
23. Ojeda S.R. Female reproductive function. Textbook of Endocrine Physiology 1992, 134. Oxford University, Oxford. second ed. J.E. Griffin, S.R. Ojeda (Eds.).
24. R.D. Pasteur, A Multiple-inhibin Model for the Human Menstrual Cycle, Ph.D. Thesis, North Carolina State University, Raleigh, North Carolina, 2008. http://www.lib.ncsu.edu/theses/available/etd-06102008-194807/.
25. Plouffe L., Luxenberg S.N. Biological modeling on a microcomputer using standard spreadsheet and equation solver programs: the hypothalamic-pituitary-ovarian axis as an example. Comput. Biomed. Res. 1992, 25:117.
26. Ramaswamy S., Pohl C.R., McNeilly A.S., Winters S.J., Plant T.M. The time course of follicle-stimulating hormone suppression by recombinant human inhibin A in the adult rhesus monkey. Endocrinology 1998, 139:3409.
27. Reinecke I., Deuflhard P. A complex mathematical model of the human menstrual cycle. J. Theor. Biol. 2007, 247:303.
28. Robertson D.M., Prisk M., McMaster J.W., Irby D.C., Findlay J.K., de Kretser D.M. Serum FSH-suppressing activity of human recombinant inhibin A in male and female rats. J. Reprod. Fertil. 1991, 91:321.
29. Robertson D.M., Hertan R., Farnworth P.G. Is the action of inhibin mediated via a unique receptor?. Rev. Reprod. 2000, 5:131.
30. Robertson D.M., Hale G.E., Jolley D., Fraser I.S., Hughes C.L., Burger H.G. Interrelationships between ovarian and pituitary hormones in ovulatory menstrual cycles across reproductive age. J. Clin. Endocrinol. Metab. 2009, 94:138.
31. Robinson R.C. An Introduction to Dynamical Systems: Continuous and Discrete 2004, Pearson Prentice Hall, New Jersey.
32. Schlosser P.M., Selgrade J.F. A model of gonadotropin regulation during the menstrual cycle in women: qualitative features. Environ. Health Perspect. 2000, 108(Suppl. 5):873.
33. Scott R.S., Burger H.G. Mechanism of action of inhibin. Biol. Reprod. 1981, 24:541.
34. Selgrade J.F. Bifurcation analysis of a model for hormonal regulation of the menstrual cycle. Math. Biosci. 2010, 225:108.
35. Selgrade J.F., Harris L.A., Pasteur R.D. A model for hormonal control of the menstrual cycle: structural consistency but sensitivity with regard to data. J. Theor. Biol. 2009, 260:572.
36. Selgrade J.F., Schlosser P.M. A model for the production of ovarian hormones during the menstrual cycle. Fields Inst. Commun. 1999, 21:429.
37. Treloar A.E., Boynton R.E., Behn B.G., Brown B.W. Variation of the human menstrual cycle through reproductive life. Int. J. Fertil. 1967, 12:77.
38. Wang C.F., Lasley B.L., Lein A., Yen S.S.C. The functional changes of the pituitary gonadotrophs during the menstrual cycle. J. Clin. Endocrinol. Metab. 1976, 42:718.
39. Welt C.K., McNicholl D.J., Taylor A.E., Hall J.E. Female reproductive aging is marked by decreased secretion of dimeric inhibin. J. Clin. Endocrinol. Metab. 1999, 84:105.
40. Yen S.S.C. Polycystic ovarian syndrome (hyperandrogenic chronic anovulation). Reproductive Endocrinology. Physiology, Pathophysiology and Clinical Management 1999, 436. W.B. Saunders Co., Philadelphia. fourth ed. S.S.C. Yen, R.B. Jaffe, R.L. Barbieri (Eds.).
41. Yen S.S.C. The human menstrual cycle: neuroendocrine regulation. Reproductive Endocrinology. Physiology, Pathophysiology and Clinical Management 1999, 191. W.B. Saunders Co., Philadelphia. fourth ed. S.S.C. Yen, R.B. Jaffe, R.L. Barbieri (Eds.).
42. Zeleznik A.J., Benyo D.F. Control of follicular development, corpus luteum function, and the recognition of pregnancy in higher primates. The Physiology of Reproduction 1994, 751. Raven Press, Ltd., New York. second ed. E. Knobil, J.D. Neill (Eds.).