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Emergent Oscillations in Mathematical Model of the Human Menstrual Cycle Natalie L. Rasgon, MD, PhD, Lara Pumphrey, BA, Paolo Prolo, MD, Shana Elman, MA, Andre B. Negrao, MD, Julio Licinio, MD, and Alan Garfinkel, PhD Findings: Two distinct temporal patterns of oscillatory behavior have been demonstrated for both pituitary and gonadal steroids in the early follicular phase: first, rapid oscillations in gonadotropin releasing hormone, follicle stimulating hormone, and luteinizing hormone (Q~1 hour) that were an immediate consequence of the programmed equations. Second, there were slower, undulating, emergent rhythms in luteinizing hormone and follicle stimulating hormone, and also in estrogen, having oscillatory periods of 2–12 hours. There was also a longer-period (Q2–3 days) emergent rhythm in progesterone. In the mid-luteal phase, estrogen and progesterone rhythms were correlated, and all hormones showed an ~6-hour periodicity. Conclusion: To our knowledge, the oscillatory behavior of peripheral sex steroids in the follicular phase has not been previously noted. CNS Spectr 2003;8(11):805-814
FOCUS POINTS • The aperiodicity of the hypothalamo-pituitarygonadal (HPG) axis may be evidence of “deterministic chaos” and not randomness. • Nonlinear modeling may be used to explain the chaotic organization of the HPG axis, which may be expanded by introducing additional physiologic variables into the system, such as neuromediators or other hormones. • Reproductive hormones modulate hormonal, neurotransmitter, and biological clock mechanisms, all of which have been implicated in the pathophysiology of mood disorders. • Nonlinear modeling of biological and psychological variables contributing to the pathogenesis of complex behavior aberrations may contribute to create an integral model of brain function, predict course of the illness, and eventually, offer novel strategies for therapeutic intervention. • Further studies are required to evaluate the extent to which alterations in the ultradian pattern of hormonal release in HPG axis actually contribute to various clinical disorders of menstrual cyclicity, fertility, or sexual function.
INTRODUCTION The relationship between affective illnesses and endocrine systems has generated considerable interest in recent years.1,2 Reproductive hormones modulate hormonal, neurotransmitter, and biological clock mechanisms, all of which have been implicated in the pathophysiology of mood disorders. 3 Further development of models of the endocrine system may lead to better understanding of the relationship between mental illness and physiological cycles, such as the female reproductive cycle. The phenomenology of reproductively influenced mood disorders, such as postpartum depression and rapid-cycling bipolar disorder, permits the hypothesis that during specific
ABSTRACT Background:The aim of this study was to develop a mathematical model of the hypothalamo-pituitary-gonadal axis that would reflect available data in humans. Methods: A model of hormonal relationships at the early follicular and midluteal phases of the human menstrual cycle is proposed.
Dr. Rasgon is associate professor of psychiatry in the Department of Psychiatry & Biobehavioral Sciences, and associate director of the Women’s Wellness Program at the Stanford School of Medicine in Palo Alto, California. Ms. Pumphrey is a PhD candidtate in the Department of Obstetrics and Gynecology at the David Geffen School of Medicine at the University of California, Los Angeles (UCLA). Dr. Prolo is assistant researcher at the UCLA School of Dentistry Laboratory of Human Oral and Molecular Immunology in the Section of Diagnostic Sciences.Ms. Elman is research assistant in the Department of Psychiatry and Biobehavioral Sciences at the David Geffen School of Medicine at UCLA. Dr. Negrao is research associate at the National Institutes of Health in Bethesda, Maryland. Dr. Licinio is professor of psychiatry at the Neuropsychiatric Institute in the Department of Psychiatry and Behavioral Sciences at the David Geffen School of Medicine at UCLA. Dr. Garfinkel is professor of medicine in the Departments of Medicine and Physiological Sciences at UCLA. Disclosures: Dr. Rasgon has received financial support from Abbott, Forest, GlaxoSmithKline, Eli Lilly, the National Institute of Mental Health, Novartis, Pfizer, and Stanley Foundation. Dr. Prolo has received funding from Fondazione Cassa di Risparmio di Cuneo, Italy. Dr. Licinio has received grant support for this work from the NIMH Intramural Research Program (CNE, NIMH, NIH) and National Institute of Health grants RR017611, RR016996, HL004526, DK058851, DK063240, HG002500, GM061394, and RR000865. Paper submitted on February 12, 2002, and accepted on June 2, 2003. Please direct all correspondence to: Natalie L. Rasgon, MD, PhD Stanford University, Department of Psychiatry and Biobehavioral Sciences, 401 Quarry Road, Room 2360, Palo Alto, CA 94305-5723; Tel: 650-724-6689; E-mail:
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phases of the reproductive cycle ovarian steroids mediate particular vulnerabilities among women for affective changes.4 Research of affective disorders has established multiple lines of evidence supporting the dysregulation of the hypothalamus, pituitary, and various end organ systems as the neuroendocrinological basis for bipolar disorder and major depression. A causal and reciprocal relationship between certain endocrine and mood disorders is supported by an increased occurrence of mood disorders in Cushing’s syndrome,5 well-documented frequency of hypercortisolemia in patients with major depression,6-8 and an association between depression and elevated androgen levels in women.2,9,10 A high prevalence of menstrual abnormalities has been reported among women with bipolar disorder, in many cases preceding the diagnosis of bipolar disorder.11-13 Endocrine systems are characterized by a variety of oscillatory behaviors, at time scales ranging from minutes to months. The sources or causes of these rhythmic secretions are often unknown. It is tempting to think that for any given rhythm, there must be “pacemaker” cells responsible for it. Sturis and colleagues14 challenged this fundamental assumption, demonstrating that an observed 2-hour rhythm in insulin-glucose dynamics did not require a postulated intrapancreatic pacemaker, but rather emerged from the coupling together of physiologic components, none of which are oscillatory. This represents a significant alternative paradigm for explaining rhythmic behaviors. Here we propose a model of the normal human menstrual cycle based upon published data and our own observations of endocrine reproductive system dynamics. We model hormonal relationships at two specific phases of the human menstrual cycle: the early follicular phase (EFP) and midluteal phases (MLP). We found that several long-period rhythms emerged from the dynamics of the models. Since these longer-period rhythms strongly affect hormone levels, they must be taken into account in designing rational clinical interventions. These hormone levels are known to modulate mechanisms that have been implicated in the biology of affective disorders, such as seasonal affective disorder, rapid-cycling bipolar disorder, and postpartum depression.15 Consequently, an understanding of the endocrine rhythms underlying these disorders may aid clinical practice. Mathematical models of endocrine systems have appeared in the literature since the 1960s. These models reflected the limited physiological data available at that time. 16,17 For example, episodic secretion of luteinizing hormone (LH) was not described until the Volume 8 – Number 11
early 1970s.18 The modern view of the dynamics of the reproductive endocrine system was not fully described until the works of Knobil19 and Pohl and Knobil.20 Currently, the view of the dynamics of the hypothalamo-pituitary-gonadal (HPG) axis is that there is periodic secretion of various hormones, exerting positive (ie, amplifying) and negative (ie, inhibiting) feedback on each other. Numerous studies have found that LH is released by the anterior pituitary in hourly pulses, called circhoral oscillations. This release is in response to pulsatile secretion of gonadotropin-releasing hormone (GnRH) from the hypothalamus.21-25 GnRH pulses, in turn, are under complex control of peripheral steroids and central neuromediators (ie, corticotropin releasing hormone, endogenous opioids, etc). 21-25 GnRH pulse frequencies of one per hour induce release of both LH and follicle stimulating hormone (FSH) in monkeys, sheep, and humans.22 FSH stimulates the secretion of estrogen from the ovarian follicles. Estrogen, and another ovarian hormone, inhibin, exert negative feedback on FSH release from the pituitary. Estrogen together with progesterone exerts negative feedback on the hypothalamus, influencing the frequency and amplitude of GnRH. Early Follicular Phase
In the early follicular phase of the menstrual cycle, levels of both gonadal steroids are low and their influence on the central target tissues is less pronounced than in the luteal phase. In this phase, progesterone is particularly low, and its influence on the hypothalamus and pituitary are negligible. FSH levels are high in the early to mid-follicular phase and decline up until ovulation. In contrast, LH levels are low in the early follicular phase, and begin to increase by midfollicular phase and are markedly increased in response to GnRH by late follicular phase.26 Luteal Phase
During the luteal phase, large amounts of progesterone and, to a lesser extent, estrogen, are secreted by the ovaries. The secretion of progesterone during the luteal phase is episodic and the pulses correlate with LH pulses.27 The amplitude of GnRH pulses increases and their frequency greatly decreases under the influence of endogenous opioids. In the MLP of the menstrual cycle, rhythms of both pituitary and ovarian hormones correlate positively.24 Slower GnRH frequencies, one pulse every 3–4 hours, produce an increase in serum FSH concentrations, while LH declines.22 The fall in LH may be explained by the short half-life of LH, but the rise of FSH suggests that a slow frequency of GnRH stimulation may favor FSH 806
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release over LH release by the pituitary.28 These physiologic relationships formed the basis of our mathematical model. Figure 1 describes the structural relations among the hormones in the EFP and MLP.
back of estrogen. The concentrations for the variables in the models were IU/L for LH and FSH, pmol/L for estrogen, and nmol/L for progesterone. For each variable, metabolic clearance rates have been included in the differential equations as first-order, or level-dependent losses. Where possible, these rates were taken from the literature, but in some cases (eg, slope of sigmoidal feedback regulations), had to be estimated from observed dynamics (Figure 1A). The dynamics of FSH secretion were modeled as a function of both estrogen and GnRH levels. GnRH exerted a positive effect on FSH, represented by an upward sigmoidal function, while estrogen and another ovarian hormone, inhibin, (not accounted for in this model) had a negative feedback (downward sigmoid) on FSH release from the pituitary. There is also a negative term in the equation, modeling FSH metabolic clearance. Estrogen production was accounted for by the positive influence of FSH (increasing sigmoid function) and its own metabolic clearance, while estrogen exerted a negative feedback on GnRH and FSH. Progesterone levels were modeled as linearly depending on LH and its own metabolic clearance.
METHODS Differential Equations From the structural diagram and the observations above, a set of differential equations were derived (Appendix A). These equations were integrated numerically by a fourth-order Runge-Kutta method, with a time step of 0.25 minutes. The models were each run for a simulated 5.5 days (8,000 minutes). Early Follicular Phase
In this phase, the period of the GnRH pulsatile release was modeled as an increasing function of estrogen; thus, higher levels of estrogen inhibited GnRH secretion. The amplitude of the GnRH pulses was also controlled by estrogen levels: as estrogen amplitude increases, both the amplitude of the GnRH pulses decreases. LH pulsatility was modeled as following GnRH secretion linearly. The differential equations for LH represent the positive effect of GnRH pulses and the negative effect of metabolic clearance, in turn modulated by the negative feedA EARLY FOLLICULAR PHASE
B MID-LUTEAL PHASE
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LH
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FIGURE 1. The Structural Relations Among the Hormones in the Early Follicular Phase and the Mid-luteal Phase E2=estrogen; GnRH=gonadotropin releasing hormone; LH=luteinizing hormone; FSH=follicle stimulating hormone; P4=progesterone. Rasgon NL, Pumphrey L, Prolo P, Elman S, Negrao AB, Licinio J, Garfinkel A. CNS Spectr. Vol 8, No 11. 2003.
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interactions between the variables reflected the main physiologic components of the menstrual cyclicity described above.
Mid-Luteal Phase
In this phase, GnRH pulses were modeled as a linear function of estrogen (as in EFP), together with a ‘self-priming’ effect, had an increased pituitary responsiveness to subsequent stimulation by GnRH (ie, an increased secretion of LH in response to the second stimulus). 29 The self-priming effect was enhanced in the presence of estrogen and was more evident in the MLP. We modeled this by postulating that each pulse of GnRH initiates two pulses of LH (Figure 1B). FSH pulses followed the GnRH stimuli in a linear fashion, and were inhibited by estrogen (a negative sigmoidal relationship). LH followed the GnRH pulses as well, but received negative feedback from estrogen and progesterone.29 Estrogen was modeled as positively affected, again in a sigmoidal relationship, by FSH. progesterone followed LH pulses, as in EFP; however, levels were higher than in EFP, as was the strength of its negative feedback on the hypothalamus and pituitary. As in EFP, all variables had their respective metabolic clearance rates. The modeled
Ovariectomized
As an additional model, both gonadal steroids were removed from the structure of the model, to simulate the ovariectomized condition (Figures 2A and 2B). Frequency Analysis
Time series output from the model was frequencyanalyzed by Fast Fourier transform. Statistical significance of peaks was determined by resampling techniques: each time series was resampled to its autocorrelation length (to eliminate spurious autocorrelation), and a coarse-grained Fourier spectrum was computed. The resampled time series was randomly reshuffled and another spectrum was calculated. Spectral peaks that were higher than 95 out of 100 of the spectra of the reshuffled time series were deemed to be statistically significant at the 0.05 level. FINDINGS
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Although GnRH is programmed to exhibit a pulse every 60 minutes in the absence of feedback in the EFP, the presence of feedback from estrogen makes the rhythm significantly irregular and slows its mean period slightly, to ~1.1 hours (Figure 3A). Both LH and FSH follow GnRH pulsatility and have significant peaks at the same periods as GnRH release (~1.1 hours). These are high-frequency oscillations in Figures 3B and 3C. Several rhythmic behaviors emerge, in the sense that they are not passive responses to other, preprogrammed, rhythms. Both LH and FSH exhibit low-frequency modulations (Figures 3B and 3C). Nothing in the mathematics of either of the two hormones separately would suggest the existence of this rhythm. In this phase, FSH shows high amplitude peaks every 2–3 hours (Figure 3B). This has been suggested to be a result of a longer half-life of FSH and its lesser sensitivity to GnRH feedback.30 LH has a large amplitude modulation at a longer period, ~12 hours (Figure 3C). Neither estrogen nor progesterone show hourly oscillations. There is a significant difference between the two ovarian steroids: estrogen has an emergent 4–5-hour rhythm (Figure 3D), whereas progesterone has a very slow oscillation, with a period of 2–3 days (Figure 3E). In the EFP of the menstrual cycle, both the pituitary and the ovarian hormones exhibit slow, somewhat irregular oscillations. In the pituitary, these are
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FIGURE 2. Secretions of (A) GnRH patterns upon removal of estrogen and progesterone from the model; and (B) LH and FSH patterns upon removal of estrogen and progesterone from the model.
GnRH=gonadotropin releasing hormone; FSH=follicle stimulating hormone; LH=luteinizing hormone.
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superimposed on the higher, (q 1 hour), frequency oscillations due to GnRH.
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In the MLP, the period of the GnRH pulses decreases to ~120 minutes (Figure 4A). FSH and LH show the high-frequency effects of GnRH, as well as an emergent slower rhythm, at about a 6-hour period (Figure 4B and 4C). FSH pulses go in and out of synchrony with LH pulses. Both estrogen and progesterone exhibit rhythms at ~6 hour periods (Figure 4D and 4E). These rhythms are much less “spiky” than the secretions of LH and FSH. They are slower and contain much more area under the curve, that is, amount of hormone released. estrogen and progesterone rhythms correlate with each other in MLP, unlike in the EFP.
Time (hours)
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Upon removal of ovarian steroids from the model structure, patterns and frequencies of GnRH and both gonadotropins returned to periodic behavior (Figures 2A and 2B).
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We compared the simulation results with previously published data from human subjects.31 LH and estradiol secretion was measured in six heathy women with an average age of 25.5±1.6 years who were studied in the follicular phase of the menstrual cycle. Blood was collected every 7 minutes for 1,442 minutes, for a total of 207 samples per subject, starting at 8:00 AM on one day and ending at 8:00 AM the next day. Subjects were acclimated to the study environment for 48 hours before the procedure. All subjects were non-smokers, non-obese, and free of any psychiatric or medical illness. They had no history of substance abuse. For the 30 days before the study, as well as during the study, no prescription or over the counter medications, hormones, or dietary supplements were permitted. During the blood collection, subjects were allowed only minimal physical activity. LH was measured by radioimmunoassay and assay sensitivity was 0.1 units/L; intra-assay coefficient of variation was 2.6%; and interassay coeffecient of variation was 5.4%. Estradiol was measured by radioimmunoassay and assay sensitivity was 8 pg/mL; intra-assay CV was 7.0%; and inter-assay CV was 8.1%. The simulated data compared remarkably well with the actual data (Figures 5A and 5B). In the simulation, estradiol showed small peaks at a 1-hour frequency and larger peaks less frequently. This pattern was also seen in the human subjects. Note also the
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FIGURE 3. Mathematical model for the EFP displaying hormonal feedback. (A) GnRH in EFP. (B) FSH feedback in EFP. (C) LH feedback in EFP. (D) Estradiol feedback in EFP. (E) Progesterone feedback in EFP. GnRH=gonadotropin releasing hormone; FSH=follicle stimulating hormone; LH=luteinizing hormone; E2=estradiol; P4=progesterone; EFP=early follicle phase. Rasgon NL, Pumphrey L, Prolo P, Elman S, Negrao AB, Licinio J, Garfinkel A. CNS Spectr. Vol 8, No 11. 2003.
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rounded, symmetric waveform of estradiol pulses, which is seen in both simulation and subject data. LH shows a waveform that was asymmetric with a “sawtooth” appearance (Figure 5B). This was seen in simulation and actual data. Similar to estrogen, LH showed smaller peaks with a 1-hour frequency and infrequent larger peaks in simulation and subject data.
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DISCUSSION The aim of this study was to develop a mathematical model of the HPG axis that would closely reflect available data in humans. We chose to use only human data, because there are significant differences in ovarian cyclicity among mammalian species. Computer simulation of these hormonal relationships produced several novel emergent phenomena. Two distinct temporal patterns of oscillatory behavior have been demonstrated for both pituitary and gonadal steroids in EFP: first, rapid oscillations in GnRH, FSH, and LH (Q~1 hour) that were an immediate consequence of the programmed equations. Second, there were slower, undulating, emergent rhythms in both LH and FSH, and also in estrogen, having oscillatory periods of 2–12 hours. There was also a longer-period (Q2–3 days) emergent rhythm in progesterone. In the MLP, estrogen and progesterone rhythms are correlated, and all hormones show an ~6-hour periodicity. To our knowledge, the oscillatory behavior of peripheral steroids in the follicular phase was not previously noted. We also show that even though the hormonal relationships were programmed at two discrete time-points of the menstrual cycle, thus “portrayed as a snapshot,” rather than as evolving functions, the behavior of the variables in the model in both snapshots was not static. When peripheral steroids were removed from the model, the patterns of GnRH and gonadotropin secretion became periodic. When they were put back in the model, the patterns became irregular. Because these are completely deterministic differential equations, it follows that any aperiodicity is evidence of “deterministic chaos” and not randomness.33 This is, therefore, an alternative to postulating a randomly activated endocrine pulse generator, as has been proposed for LH, FSH, and other pituitary hormones.33-35
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FIGURE 4. Mathematical model for the MLP; (A) GnRH in MLP; (B) LH feedback in MLP; (C) FSH feedback in MLP; (D) Estrogen feedback in MLP; (E) Progesterone feedback in MLP.
Limitations
The most important limitation of our model is that it is not fully quantitative. The types of datasets needed to fully validate all aspects of this model are not yet available. Only qualitative dynamics can be modeled, and it is not possible to attach meaningful numerical values to the hormone levels in the model. A number
GnRH=gonadotropin releasing hormone; FSH=follicle stimulating hormone; LH=luteinizing hormone; E2=estrogen; P4=progesterone; MLP=mid-luteal phase. Rasgon NL, Pumphrey L, Prolo P, Elman S, Negrao AB, Licinio J, Garfinkel A. CNS Spectr. Vol 8, No 11. 2003.
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of factors have not been been incorporated into the model due to incomplete knowledge of their actions. Neurohumoral elements, as well as hormones, such as inhibin have not been included in our study. Our findings support the importance of both positive and negative feedbacks within the system, and suggest that the HPG system is self-organizing and it is capable of temporal structure without outside “drivers.” The described low-frequency LH pulses can be distinguished from known gonadotropin pulses in women (eg, diurnal and circhoral).29 Mechanisms sustaining this ultradian periodicity are not known, but they are in part explained by time-delayed negative feedback by one or more hormones (ie, progesterone, estrogen). Deconvolution studies34 of pituitary responses to injections of synthetic GnRH have indicated that the A
differential response of LH-secretagogues to the GnRH stimulus depends in part on the changing amplitude of spontaneous LH pulses in the late follicular and midluteal phases of the menstrual cycle. It has been postulated that this is due to the feedback of gonadal steroids, such as estrogen, progesterone, and inhibin on the hypothalamus and anterior pituitary.25 Our findings of ultradian periodicity in secretion of both estrogen and progesterone (Figures 4D and 4E) in the MLP support the view that steroid feedback on the GnRH pulse generator modulates endocrine rhythms. The rhythms of aforementioned hormones are in fact chaotic, but the function of chaos in the endocrine system is not known (even whether it is functional or dysfunctional). Nonlinear modeling of the endocrine sytem may help to explain human phenomena, such as the human menstrual cycle, as presented in this paper. Chaos theory has already been successfully used to explain observed phenomena, such as the response of cardiac and neural tissue to pacing stimuli32,35 and electroencephalographic activity in epilepsy.36-38 Nonlinear modeling has also been proposed to explain major mental illnesses (ie, schizophrenia,39 bipolar disorder,40,41 and depression42). Further development of models of the endocrine system may also lead to elucidation of interactions between this system and other phenomena, such as affective disorders. Reproductive hormones could exert effect on multiple systems: neurotransmitter, neuroendocrine, or circadian. For example, an investigation of the possible chaotic nature of mood oscillations in seven women with rapid-cycling bipolar disorder yielded only occasional brief episodes of period-like behavior.40 It is possible that the rapidcycling course of bipolar disorder may be affected by reproductive hormones. Oscillators can generate chaotic dynamics when interacting with other oscillatory systems.43 Therefore, the interaction between abnormal mood regulation and the menstrual cycle, for example, could lead to the “chaotic” behavior seen in rapid-cycling bipolar disorder. This hypothesis may help to explain the higher prevalence of rapidcycling bipolar disorder in women despite an approximately equal prevalence of other types of bipolar disorder among men and women.42 Further detailing of this model may also aid in predicting course of illness in relation to menstrual cyclicity among women with bipolar disorder.19 Nonlinear modeling may provide support for hypotheses, which relate the psychoneuroendocrinology of biological and psychological variables of psychiatric disorders. Deeper understanding of these relationships may lead to an integral model of brain function.
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FIGURE 5. Twenty-four hour record of estrogen levels from six human subjects, in comparison to the model (inserts). Both records were smoothed to the same sampling rate of 7 minutes. Note the similar small peaks (Q1 hour) and less frequent larger peaks. Rasgon NL, Pumphrey L, Prolo P, Elman S, Negrao AB, Licinio J, Garfinkel A. CNS Spectr. Vol 8, No 11. 2003.
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APPENDIX A. EARLY FOLLICULAR EQUATIONS
APPENDIX B EARLY FOLLICULAR AND MID-LUTEAL EXPLANATIONS
• E2=.01 x (FSH→E2 positive feedback)–(kE2 x E2) • FSH=2x10-7x[(E2→ FSH negative feedback)+ (GnRH→FSH positive feedbsck)]–(kFSHxFSH) • LH=(.1 x Multiplier x GnRH)–(kLHxLH) • P4=(.045 x LH)–(kP4xP4)
Early Follicular Explanation
• E2 is stimulated by FSH by a positive sigmoidal relationship, as shown by FSH→E2 positive feedback. • FSH is stimulated by GnRH. FSH is secreted episodically following pulses of GnRH as shown by GnRH→FSH positive feedback. FSH is inhibited by E2 in a negative sigmoidal relationship, shown by E2→FSH negative feedback. • LH follows the GnRH pulses. It is also linearly dependent on E2. As E2 decreases (see multiplier variable), LH secretion is enhanced, and vice versa. • P4 passively follows LH pulses in a linear fashion. • All of the above variables (E2, FSH, LH, and P4) are affected by their respective metabolic clearance rates, notes by: (kE2 x E2), using E2 as an example. • GnRH pulses at a rate and amplitude that is dependent on the levels of E2. As E2 increases (between 250 and 400 minutes), the rate slows and the amplitude decreases, and vice versa (see period and amplitude variables).
Variables • kE2=.011 • kLH=.002 • kFSH=.009 • kP4=9.2x10–5 • Multiplier=If (E2800) then .5 else (–.002 x E2+2.26) • Period=If (E2400) then 100 else (.267 x E2)–6.75 • Amplitude=If (E2400) then 3 else (–.013 x E2)+8.2 FSH→E2 positive feedback= 150 x (.15 x FSH)7 +250* 1+ (.15 x FSH)7
( (
) )
E2→FSH negative feedback= 1 + (.0025 x E2)15 5 x (.2 x GnRH)15
*
GnRH→FSH positive feedback= 5 x *(.2 x GnRH)15 +5 * 1 + (.2 x GnRH)3
(
)
Mid-Luteal Explanation
• E2 is stimulated by FSH by a positive sigmoidal relationship, as shown by FSH→E2 positive feedback. • FSH passively follows GnRH pulses, as notated by a linear relationship. E2 is stimulated by FSH by a positive sigmoidal relationship, shown by E2→FSH negative feedback. • LH follows the GnRH pulses, as shown by a linear relationship. LH secretion is inhibited by both E2 and P4 separately by negative relationships. This is shown by E2→LH negative feedback and P4→LH negative feedback. • P4 follows LH pulses, as in the early follicular phase, but by the mid-luteal phase, the P4 levels have increased dramatically. • All of the above variables (E2, FSH, LH, and P4) are affected by their respective metabolic clearance rates, noted by: (kE2 x E2), using E2 as an example. • GnRH pulses at a rate and amplitude that is dependent on the levels of E2. As E2 increases (between 250 and 400 minutes), the rate slows and the amplitude decreases, and vice versa (see period and amplitude variables). Also, there is a GnRH “self-priming” effect, where each pulse of GnRH stimulates another two pulses of GnRH. See the GnRH variable: there are 2 subsequent pulses of GnRH (the first coming 20 minutes after the period, the second coming 30 minutes after the period) after the initial pulse. This occurs throughout the mid-luteal phase.
MID-LUTEAL EQUATIONS • E2=.005 x [(FSH→E2 positive feedback)–(kE2xE2)] • FSH=.05 x [5 x GnRH)+(E2→FSH negative feedback)-(kFSHxFSH)] • LH=(.0014 x [(20 x GnRH)+(E2→LH negative feedback)+(P4→LH negative feedback)–(kLH x LH)] • P4=.08 x [(3 x LH)–(kP4 x P4) • GnRH = PULSE (amplitude,1,period + PULSE(amplitude, 3,period + 20))+ PULSE(amplitude,2,(period+30))
Variables • kE2=.9 • kLH=1.9 • kFSH=.4 • kP4=.0835 • Period=If (E2+P4293.3) then 240 else ((11.111 x (E2+P4)) –3018.88889) • Amplitude=If (E2+P4 310 then 1 else ((-.2571 x (E2+P4))+80.7143) E2→FSH negative feedback = 9.5 + (.36 x E2)50 +0.5 1 + (.36 x E2)50
( ( ( (
)
E2→LH negative feedback = 5 + (.36 x E2)30 +5 1 + (.36 x E2)30
)
FSH→E2 positive feedback = .75 + (.2 x FSH)8 +2.5 1 + (.2 x FSH)8
)
P4→LH negative feedback = 5 + (.00347 x P4)175 +2.5 1 + (.00347 x P4)175
)
* See also Appendix C ** See also Appendix D E2=estradiol; FSH=follicle stimulating hormone; kE2=metabolic clearance for estradiol; GnRH=gonadotropin releasing hormone; kFSH=;metabolic clearance for follicle stimulating hormone LH=luteinizing hormone; kLH=metabolic clearance for luteinizing hormone; P4=progesterone; kP4=metabolic clearance for progesterone.
E2=estrogen; FSH=follicle stimulating hormone; GnRH=gonadotropin releasing hormone; LH=luteinizing hormone; P4=progesterone. Rasgon NL, Pumphrey L, Prolo P, Elman S, Negrao AB, Licinio J, Garfinkel A. CNS Spectr. Vol 8, No 11. 2003.
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Original Research
Treatment efficacy may vary across the menstrual cycle. Two well-documented cases indicate that serum lithium levels vary in relation to the menstrual cycle in some women with catamenial bipolar symptoms.43 In bipolar women followed longitudinally, serum lithium levels were increased during depression and decreased during mania,44,45 whereas, in asymptomatic women, lithium levels remained constant over the menstrual cycle.46 The mechanism by which bipolar disorder gets entrained to the menstrual cycle remains unknown. Better understanding of the oscillatory behavior of both the menstrual cycle and affective changes may lead to more effective treatment strategies.
modeling of biological and psychological variables contributing to the pathogenesis of complex behavior aberrations may contribute to create an integral model of brain function, predict course of the illness, and eventually, offer novel strategies for therapeutic intervention. Further studies are required to evaluate the extent to which alterations in the ultradian pattern of hormonal release in HPG axis actually contribute to various clinical disorders of menstrual cyclicity, fertility or sexual function. The correlation between simulated and actual data helps to confirm the model, and will allow further development of the model to reflect more subtle changes in hormonal dynamics. Future developments of this model will include hormonal-neuromediator interactions for each particular neuroendocrine process. Neuroendocrine computer modeling may lead to the identification of the causes of endocrine dysregulation, and may be a key element in the development of prevention algorithms. CNS
CONCLUSION Our findings have several implications. The first is that chaos may well be present in nonlinear systems like the HPG axis, in which multiple feedback relationships are present. Such level of chaotic organization would produce emergent rhythms, with different types of duration (from ultradian to diurnal and longer). Secondly nonlinear modeling of the HPG axis may be expanded by introducing additional physiological variables into the system, such as neuromediators or other hormones. Next, nonlinear
REFERENCES 1. Horacek J, Kuzmiakova M, Hoschl C, Andel M, Bahbonh R. The relationship between central serotonergic activity and insulin sensitivity in healthy volunteers. Psychoneuroendocrinology. 1999;24:785-797.
APPENDIX C. EARLY FOLIC ACID SIGMOID PLOTS DEPICTING RELATIONS BETWEEN INDIVIDUAL HORMONES FSH→E2 positive feedback=
E2→negative feedback=
GnRH→FSH positive feedback=
440
10
10.5
8
9.5
390
8.5 6 7.5
340 4 290
240
0
3
6
9
12
15
FSH
6.5
2
5.5
0 250 300 350 400 450 500 550 600 E2
4.5 0
1
2
3 GnRH
4
5
6
FSH=follicle stimulating hormone; E2=estradiol; GnRH=gonadotropin releasing hormone. Rasgon NL, Pumphrey L, Prolo P, Elman S, Negrao AB, Licinio J, Garfinkel A. CNS Spectr. Vol 8, No 11. 2003.
APPENDIX D. MID-LUTEAL SIGMOID PLOTS DEPICTING RELATIONS BETWEEN INDIVIDUAL HORMONES E2→FSH negative feedback= 10
E2→LH 10.5
negative feedback=
FSH→E2 positive feedback=
P4→LH negative feedback=
3.25
10.5
9.5
8 6
8.5
4
7.5
2
6.5
2.65
0
5.5
2.45
2.3
2.5
2.7
2.9 E2
3.1
3.3
9.5
3.05
8.5 2.85 7.5
2.3
2.5
2.7
2.9
3.1
3.3
6.5 1.0
E2
2.8
4.6 6.4 FSH
8.2
10
5.5 280
284
288 292 FSH
296
300
E2=estradiol; FSH=follicle stimulating hormone; LH=leutinizing hormone; P4=progesterone. Rasgon NL, Pumphrey L, Prolo P, Elman S, Negrao AB, Licinio J, Garfinkel A. CNS Spectr. Vol 8, No 11. 2003.
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