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

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1 Bulletin of Mathematical Biology (22), 1 17 oi:1.6/bulm Available online at on Multiple Stable Perioic Solutions in a Moel for Hormonal Control of the Menstrual Cycle LEONA HARRIS CLARK Department of Mathematics, North Carolina State University, Raleigh, NC , U.S.A. Harris.Leona@epamail.epa.gov PAUL M. SCHLOSSER CIIT Centers for Health Research, P.O. BOX 12137, Research Triangle Park, NC 2779, U.S.A. schlosser@ciit.org JAMES F. SELGRADE Department of Mathematics an Biomathematics Program, Box 825, North Carolina State University, Raleigh, NC , U.S.A. selgrae@math.ncsu.eu This stuy presents a nonlinear system of elay ifferential equations to moel the concentrations of five hormones important for regulation an maintenance of the menstrual cycle. Linear moel components for the ovaries an pituitary were previously analyze an reporte separately. Results for the integrate moel are now presente here. This moel preicts serum levels of ovarian an pituitary hormones which agree with ata in the literature for normally cycling women. In aition, the moel inicates the existence an stability of an abnormal cycle. Hence, the moel may be use to simulate the effects of external hormone therapies on abnormally cycling women as well as the effects of exogenous compouns on normally cycling women. Such simulations may be helpful in unerstaning the role of xenobiotics in fertility problems, in preicting successful hormone therapies, an for testing hormonal methos of birth control. c 22 Publishe by Elsevier Science Lt on behalf of Society for Mathematical Biology. Present Aress: U.S. Environmental Protection Agency, Research Triangle Park, NC 2779, U.S.A. Author to whom corresponence shoul be aresse /2/ $35./ c 22 Publishe by Elsevier Science Lt on behalf of Society for Mathematical Biology.

2 2 L. H. Clark et al. 1. INTRODUCTION Control an regulation of the menstrual cycle in ault women epens on the coorinate actions an reactions of ovarian an pituitary hormones. Follicle stimulating hormone (FSH) an luteinizing hormone (LH), which are prouce by the pituitary glan, initiate the evelopment of ovarian follicles an regulate the phases of the ovary an the prouction of ovarian hormones, [see Yen (198, 1999) Hotchkiss an Knobil (1994) an Zeleznik an Benyo (1994)]. Simultaneously, at least three ovarian hormones, estraiol (E 2 ), progesterone (P 4 ), an inhibin (I h), affect the synthesis an release of LH an FSH via the hypothalamus an the pituitary. We are eveloping a physiologically base moel which escribes the roles of these five hormones in this ual control system an which preicts serum concentrations of these hormones consistent with ata in the literature for normally cycling women [see Selgrae an Schlosser (1999), Schlosser an Selgrae (2), Harris (21) an Selgrae (21)]. McLachlan an Korach (1995) an Daston et al. (1997) have suggeste that the estrogenic activity of environmental substances may isrupt the sexual enocrine systems in both humans an animals. This activity may be contributing to the increase incience of breast cancer (Davis et al., 1993), to eclines in sperm counts (Sharpe an Skakkebaek, 1993), an to evelopmental abnormalities (McLachlan, 1985). In aition, many fertility problems in women are coincient with abnormal serum levels of the ovarian an pituitary hormones. For instance, polycystic ovarian synrome (PCOS) is usually accompanie by acyclic E 2 concentration an a higher than normal ratio of LH concentration to FSH concentration [see Yen (1999)]. Our mathematical moel can be use to investigate the existence an stability of abnormal cycles an to simulate the effects of external hormone therapies on abnormally cycling women as well as the effects of exogenous compouns on normally cycling women. Such simulations may be helpful in unerstaning the role of xenobiotics in fertility problems, in preicting successful hormone therapies, an in testing hormonal methos of birth control which function by suppressing the mi-cycle surge in LH. Here we present a moel consisting of 13 nonlinear, elay, ifferential equations. In Section 2, we briefly iscuss the moeling process an review two preliminary linear moels for the prouction of the ovarian hormones (Selgrae an Schlosser, 1999) an for the prouction of the pituitary hormones (Schlosser an Selgrae, 2). To valiate our new nonlinear moel, we show that it has an asymptotically stable perioic solution which closely approximates ata in McLachlan et al. (199) for 33 normally cycling women. This ata set contains aily averages of the five hormones for 31 consecutive ays an these averages were compute by centering ata from each iniviual woman about the ay of her LH surge. Using Hopf bifurcation theory an the software of Engelborghs et al. (2), DDE- BIFTOOL, we exhibit another stable perioic solution for the same parameter values for which the solution that approximates normal hormonal levels exists.

3 Perioic Solutions for Menstrual Cycle 3 This new solution may escribe some biologically feasible abnormal conition in women an has some similarities to PCOS. Exogenous progesterone treatments are presente which perturb the system from the abnormal cycle to the normal cycle. In aition, we emonstrate exogenous estrogen inputs which cause isruption to the normal cycle an which ultimately result in abnormal cycling. Hence, this moel illustrates the possibility of environmental enocrine isruption. 2. MODEL DEVELOPMENT A normal menstrual cycle for an ault woman ranges anywhere from 25 to 35 ays in uration (Ojea, 1992) an consists of two phases, the follicular phase an the luteal phase, separate by ovulation. The pituitary, responing to signals from the hypothalamus, synthesizes an releases the gonaotropin hormones, LH an FSH. Although these hormones have a pulsatile secretion pattern, we assume that the ovary respons to average serum levels of LH an FSH (Oell, 1979), so our moel tracks aily average hormone concentrations. Concurrently, the ovary prouces E 2, P 4, an I h, which control the pituitary s synthesis an release of the gonaotropin hormones uring the various stages of the cycle. Chávez-Ross (1999) reviewe much of the literature on mathematical moels of the menstrual cycle an the estrus cycle of roents, incluing moels of follicle growth an selection as well as cycle regulation. Previous moels of cycle regulation [e.g., Schwartz (197), Bogumil et al. (1972a,b), McIntosh an McIntosh (198), Plouffe an Luxenberg (1992)] have useful components but also contain elements which are not base on biological mechanisms. For instance, they may contain a switch to turn on the LH surge or convolution integrals which weight the effects of E 2 concentrations over time. We try to link the terms in our ifferential equations moel to physiological mechanisms. Our moeling approach is ivie into three steps. The first two steps evelope linear, time-epenent systems for the prouction of the pituitary hormones [see Schlosser an Selgrae (2)] an for the prouction of the ovarian hormones [see Selgrae an Schlosser (1999)]. The thir step carrie out in this work creates a 13-imensional, highly nonlinear, autonomous system by merging these two simpler components. Using the linear biiagonal structure of the pituitary an ovarian systems, Selgrae an Schlosser (1999) showe that if each linear system has perioic hormone inputs of the same perio then the system has a unique, globally asymptotically stable perioic solution of that perio. However, this strong result for both linear systems oes not imply any stability properties for the nonlinear system forme by merging these linear component systems. In fact, here we exhibit two locally, asymptotically stable perioic solutions for the merge system Pituitary moel. Firstly, Schlosser an Selgrae (2) erive two systems of linear orinary ifferential equations which escribe the pituitary s syn-

4 4 L. H. Clark et al. synthesis release clearance Hypothalamus/Pituitary Serum P 4 Ih E 2 Figure 1. Control of the pituitary s synthesis an release of LH an FSH. Compartments represent the brain an the bloo. The plus or minus arrows inicate stimulatory or inhibitory effects of ovarian hormones on synthesis an release. thesis an release of LH an FSH as controlle by the ovarian hormones. Data from McLachlan et al. (199) were use to obtain input functions for serum levels of E 2, P 4, an I h. The state variables are pituitary an serum levels of LH an FSH an the ifferential equations are linear in these variables but, since the inputs are functions of time, the systems are nonautonomous. To obtain the LH an the FSH systems we assume that gonaotopin synthesis occurs in the pituitary, an the hormones are hel in a reserve pool for release into the bloo stream [see Wang et al. (1976)]. The stimulatory an inhibitory effects of the ovarian hormones on this process are inicate in Fig. 1. Let RP LH enote the state variable which represents the amount of LH in the reserve pool an let LH enote the serum concentration of LH. The system of ifferential equations governing the synthesis (s LH ), release (r LH ) an clearance (c LH ) of LH has the form where t RP LH = s LH (E 2, P 4 ) r LH (E 2, P 4, RP LH ) t LH = 1 v r LH(E 2, P 4, RP LH ) c LH (LH) (1) V 1,LH[E 2(t)]8 [K m LH ] 8 + [E 2 (t)] 8 s LH (E 2, P 4 ) = V,LH +, (1a) 1 + P 4 (t P )/K i LH,P r LH (E 2, P 4, RP LH ) = k LH[1 + c LH,P P 4 (t)]rp LH, (1b) 1 + c LH,E E 2 (t) an c LH (LH) = a LH LH. (1c) In (1), E 2 (t) an P 4 (t) are the inputs which are explicit functions of time approximate from the ata for serum concentrations of E 2 an P 4. The term s LH (E 2, P 4 )

5 Perioic Solutions for Menstrual Cycle 5 captures the effects of E 2 an P 4 on LH synthesis. Clinical experiments [e.g., Tsai an Yen (1971), Swerloff et al. (1972), Karsch et al. (1973a,b), Liu an Yen (1983), Clarke an Cummins (1984)] have inicate that LH exhibits a biphasic response to E 2 concentrations of various strengths an urations. To moel this response, Schlosser an Selgrae (2) assume that the effect of E 2 on LH synthesis is ifferent than the effect on LH release, i.e., E 2 inhibits release [see the enominator in (1b)] but at high levels E 2 significantly promotes synthesis [see the Hill function in the numerator of (1a)]. The release term r LH (E 2, P 4, RP LH ) is the prouct of RP LH an a function which inclues the inhibitory effect of E 2 on the release of LH into the bloo. This term ivie by bloo volume v appears in the ifferential equation for the state variable LH, which also contains a linear clearance term. The parameters in (1a) (1c) are name accoring to the traitional scheme for chemical reactions, e.g., V represents the velocity of the reaction [see Keener an Sney (1998)]. The time-elay parameter P, which is assume only for the synthesis term, escribes the perio between the time when changes in serum levels of P 4 occur an the time when subsequent changes in LH synthesis rates occur. A similar elay for E 2 was initially inclue (Schlosser an Selgrae, 2) but recent parameter ientification (Harris, 21) inicate that it was insignificant. The pair of ifferential equations for synthesis an release of FSH (see M3 an M4) have a form similar to (1) an are iscusse in etail in Schlosser an Selgrae (2) where s FSH (I h) = V FSH 1 + I h(t I h )/K i FSH,I h, (2a) r FSH (E 2, P 4, RP FSH ) = k FSH[1 + c FSH,P P 4 (t)]rp FSH 1 + c FSH,E [E 2 (t)] 2, (2b) an c FSH (FSH) = a FSH FSH. (2c) Both the LH an FSH systems are linear an time-epenent ifferential equations with 17 parameters in total. The clearance rates an the volume of istribution v were foun in the literature but the other parameters were cruely estimate in Schlosser an Selgrae (2) using the ata from McLachlan et al. (199). Harris (21) applie a Neler-Meae minimizer in a least squares routine to reparameterize the LH an FSH systems an to obtaine the improve values liste in Table Ovarian moel. To escribe the prouction of E 2, P 4, an I h in the ovary, Selgrae an Schlosser (1999) erive a linear, time-epenent system of nine orinary ifferential equations [see (M5) through (M13) in the merge system (M1) (M13) below] to represent nine istinct stages of the ovary base on the

6 6 L. H. Clark et al. Table 1. Parameter values for the LH an FSH equations. LH equation (1) k LH 2.49 ay 1 a LH 14. ay 1 V,LH µg ay 1 V 1,LH 91 µg ay 1 K m LH 36 ng L 1 K i LH,P nmol L 1 c LH,E.49 L ng 1 c LH,P.7 L nmol 1 P 1. ay FSH equation V FSH 57 µg ay 1 a FSH 8.21 ay 1 k FSH 7.29 ay 1 I h 2. ays c FSH,E.16 (L/ng) 2 K i FSH,I h 641 U L 1 c FSH,P 644 L nmol 1 v 2.5 L capacity of each stage to prouce hormones. The epenence on time appears in the gonaotropin input functions, LH(t) an FSH(t), which were obtaine from the ata in McLachlan et al. (199). The capacity to prouce hormones is assume proportional to the mass of each stage, so the state variables represent the masses of the active follicular or luteal tissues uring the corresponing stages of the cycle (see Fig. 2). The follicular phase of the cycle consists of the follicle recruitment stage, RcF, the seconary follicular stage, SeF, an the preovulatory follicular stage, PrF. The transitional perio between the follicular phase an the luteal phase is ivie into two stages referre to as ovulatory scars, Sc 1 an Sc 2. The luteal phase consists of four stages, Lut i for i = 1,..., 4. The gonaotropins promote tissue growth within a stage an the transformation of tissue from one stage to the next as inicate in Fig. 2. Since clearance from the bloo of the ovarian hormones is on a fast timescale, we assume that serum levels of E 2, P 4, an I h are at quasi-steay state [see Keener an Sney (1998) as i Bogumil et al. (1972a)]. Hence, we take these concentrations to be proportional to the tissue masses uring the appropriate stages of the cycle giving the following three auxiliary equations for serum levels of E 2, P 4 an I h as functions of time: E 2 (t) = e + e 1 SeF(t) + e 2 PrF(t) + e 3 Lut 4 (t), (3a) P 4 (t) = p 1 Lut 3 (t) + p 2 Lut 4 (t) an (3b) I h(t) = h + h 1 PrF(t) + h 2 Lut 3 (t) + h 3 Lut 4 (t). (3c) The first term on the right in (M5), bfsh, represents the pituitary s stimulation of premature follicles an initiates the cyclic changes within the ovary. Follicle growth rates uring the follicular phase are assume proportional to the FSH an LH serum levels, see Oell (1979). The transition from the seconary follicular stage to the preovulatory follicular stage correspons to the selection of the ominant follicle an epens on LH. The ominant follicle secretes large amounts

7 Perioic Solutions for Menstrual Cycle 7 FSH Inactive Follicles Recruite Follicles RcF LH Corpus Luteum Lut1,Lut2, Lut3, Lut4 Seconary Follicles SeF LH Ovulatory Scars Sc1, Sc2 Preovulatory Follicle PrF LH Figure 2. Diagram of the stages of the ovary. Compartments represent follicular or luteal tissue in each stage. Soli arrows inicate transformation of tissue from one stage to another or growth within a stage when pointing back to the same compartment. Open arrows inicate inuction by gonaotropins. Dotte arrows inicate synthesis of ovarian hormones. of E 2 as reflecte in (3a), see Bair (1976). Ovulation an luteinization are not instantaneous events (Oell, 1979) an are represente by two stages referre to as ovulatory scars. Little hormone synthesis is assume uring this perio. The transition is promote by LH as reflecte by the first term in the equation for Sc 1. Then, in the moel the capacity to prouce hormones cascaes through four luteal stages. The primary source of P 4 an I h is the corpus luteum as inicate in (3b) an (3c). To improve on the parameter estimation for the ovarian system an the auxiliary equations in Selgrae an Schlosser (1999), Harris (21) use Neler Meae with least squares to obtain Table Merge moel. The final step of this moeling process is to merge the pituitary an ovarian systems into a single 13-imensional system of nonlinear, elay ifferential equations (M1) (M13) with the three auxiliary equations (3). While the two moel components were escribe previously, results for this merge system are presente for the first time here. In the merge system the functions LH an FSH are state variables an the functions E 2, P 4, an I h are linear combinations of the ovarian state variables via the auxiliary equations. Hence, system (M1) (M13) is autonomous since there are no time-epenent inputs as there were in the pituitary an ovarian systems. In aition, (M1-M13) is highly nonlinear, e.g., the first term in (M1) contains a rational function of egree 8 in the state variables. Discrete elays are present in the FSH an LH synthesis terms. The inhibitory effect of P 4 on LH synthesis present in (M1) results in elay

8 8 L. H. Clark et al. Table 2. Parameter values for ovarian an auxiliary equations. Ovarian equations (M5) (M13) b.4 L ay 1 c 1.58 L µg 1 ay 1 c 2.48 ay 1 c 3.4 ay 1 c 4.61 ay 1 c ay ay ay 1 k ay 1 k 2.69 ay 1 k ay 1 k ay 1 α.7736 β.1566 γ.22 Auxiliary equation (3) e 48. ng L 1 e kl 1 e kl 1 e kl 1 p 1.5 nmol L 1 µg 1 p 2.5 nmol L 1 µg 1 h U L 1 h U L 1 µg 1 h U L 1 µg 1 h U L 1 µg 1 affecting two state variables via (3b) an a similar effect of I h on FSH in (M3) affects three state variables via (3c). To stuy the ynamical behavior of (M1) (M13), we use the elay ifferential equation solver DDE23 of Shampine an Thompson (21). V,LH + V 1,LH E2 8 t RP K m LH = 8 LH +E8 2 k LH[1 + c LH,P P 4 ]RP LH (M1) 1 + P 4 (t P )/K i LH,P 1 + c LH,E E 2 t LH = 1 k LH [1 + c LH,P P 4 ]RP LH a LH LH (M2) v 1 + c LH,E E 2 t RP V FSH FSH = k FSH[1 + c FSH,P P 4 ]RP FSH 1 + I h(t I h )/K i FSH,I h 1 + c FSH,E E2 2 t FSH = 1 k FSH [1 + c FSH,P P 4 ]RP FSH v 1 + c FSH,E E2 2 a FSH FSH t RcF = bfsh + [c 1FSH c 2 LH α ]RcF t SeF = c 2LH α RcF + [c 3 LH β c 4 LH]SeF t PrF = c 4LH SeF c 5 LH γ PrF t Sc 1 = c 5 LH γ PrF 1 Sc 1 (M3) (M4) (M5) (M6) (M7) (M8)

9 Perioic Solutions for Menstrual Cycle 9 4 LH, µg/l t ays E 2, ng/l t ays Figure 3. The soli curves (normal cycle) represent serum concentrations of LH an E 2 for normally cycling women as preicte by the merge system (M1) (M13) an s represent the aily mean serum levels of LH an E 2 for 33 women in McLachlan et al. (199). t Sc 2 = 1 Sc 1 2 Sc 2 t Lut 1 = 2 Sc 2 k 1 Lut 1 t Lut 2 = k 1 Lut 1 k 2 Lut 2 t Lut 3 = k 2 Lut 2 k 3 Lut 3 t Lut 4 = k 3 Lut 3 k 4 Lut 4. (M9) (M1) (M11) (M12) (M13) 3. RESULTS Simulations of system (M1) (M13) were run using the parameter values in Tables 1 an 2 without further ajustment to the parameters. The initial conitions for FSH an LH were chosen to correspon to the initial ata values in McLachlan et al. (199) an the initial conitions for the other state variables were chosen as a result of numerical experiments with the pituitary an ovarian systems. For these initial conitions, we observe a locally asymptotically stable perioic solution (Fig. 3) which approximates the ata of McLachlan et al. (199) an we refer to this

10 1 L. H. Clark et al. 2 FSH, µg/l t ays 4 LH, µg/l t ays Figure 4. Pituitary hormones, FSH an LH, for the simulate normal cycle (soli curve) an the simulate abnormal cycle (ashe curve). solution as the normal cycle. The perio of this normal cycle is roughly 29.5 ays as compare to the perio of the ata which we assume is 31 ays. For this same set of parameter values, there exists another locally asymptotically stable perioic solution of perio 24 ays, which we call the abnormal cycle. We iscovere this solution using the theory of Hopf bifurcations an the software of Engelborghs et al. (2), DDE-BIFTOOL, to track perioic solutions as parameters change from the values where Hopf bifurcation occurs. Numerical simulations inicate that the omain of attraction of this solution is smaller than that of the normal cycle [see Harris (21)]. A etaile stuy of the omains of attraction will be the topic of future work. The abnormal cycle may escribe some abnormal conition in women an, in fact, has some similarities to PCOS. Figures 4 an 5 compare the hormone profiles of the normal an abnormal cycles for 15 ays. For the abnormal cycle, the E 2 levels vary only slightly throughout a perio, FSH an P 4 concentrations are lower than those of the normal cycle, an the average LH concentration is slightly higher except at the time of the surge. Hence, the ratio of LH to FSH is elevate above that for the normal cycle. These characteristics are present in many PCOS iniviuals, see Yen (1999) an Marshall et al. (21). P P 4 treatment. Since PCOS is the leaing cause of female infertility in the Unite States (Nestler et al., 1998), clinical an experimental treatments have been implemente to correct hormonal imbalances in PCOS women, e.g., Petsos et al. (1986), Anttila et al. (1992), Buckler et al. (1992), Fia et al. (1996) an Nestler et al. (1998). One cause of these imbalances may be that

11 Perioic Solutions for Menstrual Cycle 11 3 E 2, ng/l t ays 6 P 4, nmol/l t ays 15 Ih, U/L t ays Figure 5. Ovarian hormones, E 2, P 4 an I h, for the simulate normal cycle (soli curve) an the simulate abnormal cycle (ashe curve). the persistent rapi pulses of gonaotropin releasing hormone (GnRH) uring the luteal phase of the cycle favor LH synthesis instea of FSH synthesis (Marshall et al., 21). Sustaine levels of P 4 uring the luteal phase of the normal cycle inhibit the amplitue an frequency of GnRH secretion. Hence, the aministration of exogenous progesterone has been use to elevate serum P 4 to normal luteal levels an, subsequently, to reuce the LH/FSH ratio [see Petsos et al. (1986), Anttila et al. (1992), Buckler et al. (1992) an Fia et al. (1996)]. In our moel, the aministration of exogenous progesterone may be implemente easily by aing a term to the progesterone auxiliary equation (3b). Our P 4 therapy (see the mile graph of Fig. 6) as 8 nmol L 1 to (3b) for 5 ays at the beginning of the luteal phase of the abnormal cycle. Because of a slight rise in LH aroun ay 8 (Fig. 4) an ecreasing E 2 at that time (Fig. 5), we assume that the luteal phase of the abnormal cycle begins on ay 8 an we aminister P 4 from ay 8 to ay 13. This treatment increases serum P 4 by 8 nmol L 1 for those 5 ays an results in normal P 4 concentrations in the next cycle (see the bottom graph of Fig. 6), as well as normal levels of the other hormones. Figure 7 shows that, uring the treatment perio, LH initially spikes because P 4 promotes LH release but then ecreases to normal luteal phase levels for the uration of the treatment because P 4 inhibits LH synthesis. This behavior is consistent with what has been observe in clinical experiments, see Buckler et al. (1992). Mathematically, a perturbation to the abnormal cycle cause by the aition of 8 nmol L 1 of P 4 results in the solution leaving the omain of attraction

12 12 L. H. Clark et al. P 4 (w/o therapy) t ays therapy 5 P 4 (w/ therapy) 5 15 t ays t ays Figure 6. Progesterone treatment restores the normal circulation of hormones. The top graph epicts P 4 serum levels for the abnormal cycle. Aministering progesterone for the first 5 ays of the luteal phase (mile graph) of the abnormal cycle so that P 4 levels are elevate by 8 nmol L 1 rapily recovers the normal cycle (bottom graph). of the abnormal cycle an rapily approaching the normal cycle. If a treatment of 5 nmol L 1 for 5 ays is aministere then it takes approximately 12 ays (four normal cycle lengths) for the normal cycle to be attaine. With a treatment of only 3 nmol L 1 for 5 ays, the normal cycle will not be recovere. A complete ose response analysis for P 4 aministration will be carrie out in the future. E E 2 isruption. There is increasing concern that environmental substances with estrogenic activity may isrupt the sexual enocrine system in humans an animals, [e.g., see McLachlan an Korach (1995) or Daston et al. (1997)]. The effects of exogenous estrogen on the ynamical behavior of our moel may be teste by aing a term to the estrogen auxiliary equation (3a). The mile graph in Fig. 8 epicts an exogenous E 2 exposure of 5 ng L 1 for one complete cycle ( 3 ays), followe by a cycle with no exposure an then followe by another 3 ay perio of exposure. This perioic isruption is applie at the beginning of the normal cycle. After the first 3 ay exposure, E 2 levels return to near normal ranges for a complete month (see the thir graph in Fig. 8). However, the secon E 2 exposure results in an E 2 profile which never reaches a level sufficient to elicit an LH surge. Hence, after about 9 ays from the beginning of exposure, the isrupte solution lies in the omain of attraction of the abnormal cycle but it takes another 18 ays (6 normal cycles) for this isrupte solution to become quite close

13 Perioic Solutions for Menstrual Cycle 13 LH mg/l t ays Figure 7. Effect of P 4 treatment on the LH of the abnormal cycle. LH spikes between ays 8 an 9 an then ecreases to normal luteal phase levels for the uration of the treatment because P 4 inhibits LH synthesis. to the abnormal cycle (ay 27 in Fig. 8). Surprisingly, a continuous E 2 exposure of 5 ng L 1 for 6 ays (two cycles) will not isrupt the normal cycle. However, continuous exposures at higher levels will cause cycle isruption. A strength an uration stuy for this phenomenon will be the subject of future work. 4. SUMMARY AND CONCLUSIONS Here we have presente a moel for hormonal control of the menstrual cycle which involves five hormones essential to this process. The system of 13 elay ifferential equations (M1) (M13) has 42 parameters (Tables 1 an 2) which were ientifie using ata from the literature. For these parameters, the moel exhibits at least two locally asymptotically stable perioic solutions. One solution approximates the ata an we suggest that the other solution represents some abnormal conition in women, possibly PCOS. An exhaustive stuy of state space nees to be one to etermine if there are other stable solutions. Also, the omains of attraction of stable solutions nee to be mappe. The abnormal cycle may be obtaine from a supercritical Hopf bifurcatioin by varying a single parameter. Hence, if only one parameter in Tables 1 an 2 is change, system (M1) (M13) has a stable equilibrium. We inten to investigate the physiological significance of this equilibrium solution. In clinical experiments, progesterone treatment has been use to normalize the gonaotropin levels of PCOS patients. We illustrate a P 4 therapy which perturbs the abnormal cycle to the normal cycle within 1 month. Also, we present an exogenous estrogen exposure which isturbs the normal cycle enough that the system ultimately oscillates abnormally. Other P 4 treatments an E 2 isruptions are possible an will be the subjects of future stuies.

14 14 L. H. Clark et al. E 2 (w/o isruption) t ays isruption t ays E 2 (w/ isruption) t ays Figure 8. Exogenous estrogen exposure can lea to abnormalities in the normal menstrual cycle. The top graph epicts E 2 serum levels for the normal cycle. Exposing the normal cycle to 5 ng L 1 of exogenous estrogen for a full cycle, followe by a full cycle with no exposure an then another full cycle of exposure (mile graph), isrupts the normal circulation of hormones (bottom graph). Finally, this moel may certainly be improve by incluing greater biological realism. For instance, the ata of McLachlan et al. (199) were collecte before the assay istinguishing inhibin A an inhibin B was available. With ata for inhibin A an B now in the literature (Groome et al., 1996) both shoul be inclue in the moel. The inhibin profile in the present moel is similar to inhibin A while inhibin B is prominent uring the follicular phase of the cycle. The present moel lumps together the pituitary an hypothalamus an oes not escribe the role of GnRH. However, we have mentione that the ovarian hormones may affect gonaotropin secretion by moulating the GnRH pulse frequency an amplitue. Hence, the present moel may be improve by eveloping an integrating a specific moel for GnRH. ACKNOWLEDGEMENTS The authors thank Dr Claue L Hughes for helpful conversations an for first suggesting that the abnormal cycle may represent PCOS. The authors woul also like to thank two referees whose suggestions have improve the clarity of the paper. The results on the merge moel presente here are a portion of the octoral thesis (Harris, 21) written by L Harris Clark who was supporte by a fellowship from the National Physical Science Consortium.

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