Bifurcation analysis of a model for hormonal regulation of the menstrual cycle

A model for hormonal control of the menstrual cycle with 13 ordinary differential equations and 41 parameters is presented. Important changes in model behavior result from variations in two of the most sensitive parameters. One parameter represents the level of estradiol sufficient for significant synthesis of luteinizing hormone, which causes ovulation. By studying bifurcation diagrams in this parameter, an interval of parameter values is observed for which a unique stable periodic solution exists and it represent an ovulatory cycle. The other parameter measures mass transfer between the first two stages of ovarian development and is indicative of healthy follicular growth. Changes in this parameter affect the uniqueness interval defined with respect to the first parameter. Hopf, saddle-node and transcritical bifurcations are examined. To attain a normal ovulatory menstrual cycle in this model, a balance must be maintained between healthy development of the follicles and flexibility in estradiol levels needed to produce the surge in luteinizing hormone.     

A delay differential equation model of follicle waves in women

This article presents a mathematical model for hormonal regulation of the menstrual cycle which predicts the occurrence of follicle waves in normally cycling women. Several follicles of ovulatory size that develop sequentially during one menstrual cycle are referred to as follicle waves. The model consists of 13 nonlinear, delay differential equations with 51 parameters. Model simulations exhibit a unique stable periodic cycle and this menstrual cycle accurately approximates blood levels of ovarian and pituitary hormones found in the biological literature. Numerical experiments illustrate that the number of follicle waves corresponds to the number of rises in pituitary follicle stimulating hormone. Modifications of the model equations result in simulations which predict the possibility of two ovulations at different times during the same menstrual cycle and, hence, the occurrence of dizygotic twins via a phenomenon referred to as superfecundation. Sensitive parameters are identified and bifurcations in model behaviour with respect to parameter changes are discussed. Studying follicle waves may be helpful for improving female fertility and for understanding some aspects of female reproductive ageing.     

A model for hormonal control of the menstrual cycle: Structural consistency but sensitivity with regard to data

This study presents a 13-dimensional system of delayed differential equations which predicts serum concentrations of five hormones important for regulation of the menstrual cycle. Parameters for the system are fit to two different data sets for normally cycling women. For these best fit parameter sets, model simulations agree well with the two different data sets but one model also has an abnormal stable periodic solution, which may represent polycystic ovarian syndrome. This abnormal cycle occurs for the model in which the normal cycle has estradiol levels at the high end of the normal range. Differences in model behavior are explained by studying hysteresis curves in bifurcation diagrams with respect to sensitive model parameters. For instance, one sensitive parameter is indicative of the estradiol concentration that promotes pituitary synthesis of a large amount of luteinizing hormone, which is required for ovulation. Also, it is observed that models with greater early follicular growth rates may have a greater risk of cycling abnormally.     

The time interval effect on two consecutive LHRH-TRH tests on adult female baboons

To Clarify the relationship between the time interval and pituitary Luteinizing hormone (LH), Follicular stimulation hormone (FSH) and prolactin (PRL) secretion function under LHRH-TRH stimulation, 4 mature female baboons were used. Two consecutive LHRH (100 micrograms)-TRH (250 micrograms) stimulations with a 60 min interval between them was carried out in the early follicular phase, late follicular phase and mid luteal phase in the same baboon in the first menstrual cycle, then carried out with a 120 min interval between tests in the third menstrual cycle. The LH, FSH and PRL were measured by specific radioimmunoassay. The PRL maximum response to the first bolus of TRH was higher than maximum response to the second bolus of TRH. The PRL maximum response to the second TRH at a 120 min interval was higher than the maximum response to the second TRH at a 60 min interval. It seems that the TRH had the dominant effect on PRL releasing but not on PRL Priming. The maximum LH response to the second bolus of LHRH was higher than the maximum response to the first bolus of LHRH. The LH maximum response to the second bolus of LHRH at a 60 min interval was greater than the maximum response at a 120 min interval in the follicular phase but it was the reverse in the luteal phase. The FSH response to the second LHRH was different from the LH response.(ABSTRACT TRUNCATED AT 250 WORDS)     

A complex mathematical model of the human menstrual cycle

Despite the fact that more than 100 million women worldwide use birth control pills and that half of the world's population is concerned, the menstrual cycle has so far received comparatively little attention in the field of mathematical modeling. The term menstrual cycle comprises the processes of the control system in the female body that, under healthy circumstances, lead to ovulation at regular intervals, thus making reproduction possible. If this is not the case or ovulation is not desired, the question arises how this control system can be influenced, for example, by hormonal treatments. In order to be able to cover a vast range of external manipulations, the mathematical model must comprise the main components where the processes belonging to the menstrual cycle occur, as well as their interrelations. A system of differential equations serves as the mathematical model, describing the dynamics of hormones, enzymes, receptors, and follicular phases. Since the processes take place in different parts of the body and influence each other with a certain delay, passing over to delay differential equations is deemed a reasonable step. The pulsatile release of the gonadotropin-releasing hormone (GnRH) is controlled by a complex neural network. We choose to model the pulse time points of this GnRH pulse generator by a stochastic process. Focus in this paper is on the model development. This rather elaborate mathematical model is the basis for a detailed analysis and could be helpful for possible drug design.     

抑卵激素对家蝇卵巢周期性发育的调控

抑卵激素是调控家蝇Musca dorncstica vicina卵巢周期性发育的关键因子之一。在家蝇中,当第一个周期的卵母细胞处于卵黄发生期或卵黄发生后期时,其第二个周期的卵母细胞的发育不进入卵黄发生期。本文建立了家蝇抑卵激素的生物测定方法,即用一对卵巢提取物注射1头羽化后12h家蝇,并在羽化后60h观察卵母细胞的发育及卵黄蛋白的沉积情况。抑卵激素的作用首先是延缓了卵母细胞在卵黄发生前期的发育;其次,抑卵激素抑制脂肪体中卵黄蛋白的合成,导致血淋巴中卵黄蛋白含量的下降,从而抑制了卵母细胞的发育。抑卵激素并不抑制卵母细胞对卵黄原蛋白的摄取。卵发育神经激素可以颉抗抑卵激素的抑制作用。抑卵激素无种属特异性。     

Oestrogen receptor content of normal breast cells and breast carcinomas throughout the menstrual cycle

To examine the presence and distribution of oestrogen receptors in the normal breast during the menstrual cycle cytological samples obtained by fine needle aspiration from 69 premenopausal women with normal breasts were analysed immunocyto-chemically with a monoclonal antibody to oestrogen receptor; samples from 15 postmenopausal women were also analysed. The receptor content of breast cancers from 83 premenopausal women was also determined in relation to when during the menstrual cycle excision was performed. In the normal premenopausal women oestrogen receptors were detected in the nuclei of epithelial cells in 21 out of 68 (31%) assessable samples. All 21 of these samples were obtained from the 35 women who were studied during the first half of their menstrual cycle (days 28 to 14). None of the 33 samples obtained during the second half of the cycle contained oestrogen receptors. Samples were assessable in eight of the postmenopausal women, six giving a positive result for oestrogen receptor. Fifty one of the 83 carcinomas were positive for oestrogen receptor, 24 having been excised during the first half of the cycle and 27 during the second half.Production of oestrogen receptor protein is suppressed at the time of ovulation in the normal breast epithelium of premenopausal women. In contrast, breast carcinoma cells either synthesise this protein continuously throughout the cycle or fail to express it despite fluctuations of serum hormone concentrations.     

The Impact of Time Delays on the Robustness of Biological Oscillators and the Effect of Bifurcations on the Inverse Problem

Differential equation models for biological oscillators are often not robust with respect to parameter variations. They are based on chemical reaction kinetics, and solutions typically converge to a fixed point. This behavior is in contrast to real biological oscillators, which work reliably under varying conditions. Moreover, it complicates network inference from time series data. This paper investigates differential equation models for biological oscillators from two perspectives. First, we investigate the effect of time delays on the robustness of these oscillator models. In particular, we provide sufficient conditions for a time delay to cause oscillations by destabilizing a fixed point in two-dimensional systems. Moreover, we show that the inclusion of a time delay also stabilizes oscillating behavior in this way in larger networks. The second part focuses on the inverse problem of estimating model parameters from time series data. Bifurcations are related to nonsmoothness and multiple local minima of the objective function.     

On the modeling of breath-by-breath oxygen uptake kinetics at the onset of high-intensity exercises: simulated annealing vs. GRG2 method.

Modeling in the time domain, the non-steady-state O2 uptake on-kinetics of high-intensity exercises with empirical models is commonly performed with gradient-descent-based methods. However, these procedures may impair the confidence of the parameter estimation when the modeling functions are not continuously differentiable and when the estimation corresponds to an ill-posed problem. To cope with these problems, an implementation of simulated annealing (SA) methods was compared with the GRG2 algorithm (a gradient-descent method known for its robustness). Forty simulated Vo2 on-responses were generated to mimic the real time course for transitions from light- to high-intensity exercises, with a signal-to-noise ratio equal to 20 dB. They were modeled twice with a discontinuous double-exponential function using both estimation methods. GRG2 significantly biased two estimated kinetic parameters of the first exponential (the time delay td1 and the time constant tau1) and impaired the precision (i.e., standard deviation) of the baseline A0, td1, and tau1 compared with SA. SA significantly improved the precision of the three parameters of the second exponential (the asymptotic increment A2, the time delay td2, and the time constant tau2). Nevertheless, td2 was significantly biased by both procedures, and the large confidence intervals of the whole second component parameters limit their interpretation. To compare both algorithms on experimental data, 26 subjects each performed two transitions from 80 W to 80% maximal O2 uptake on a cycle ergometer and O2 uptake was measured breath by breath. More than 88% of the kinetic parameter estimations done with the SA algorithm produced the lowest residual sum of squares between the experimental data points and the model. Repeatability coefficients were better with GRG2 for A1 although better with SA for A2 and tau2. Our results demonstrate that the implementation of SA improves significantly the estimation of most of these kinetic parameters, but a large inaccuracy remains in estimating the parameter values of the second exponential.     

A structured erythropoiesis model with nonlinear cell maturation velocity and hormone decay rate

We develop a quasilinear structured model that describes the regulation of erythropoiesis, the process in which red blood cells are developed. In our model, the maturation velocity of precursor cells is assumed to be a function of the erythropoietin hormone, and the decay rate of this hormone is assumed to be a function of the number of precursor cells, unlike other models which assume these parameters to be constants. Existence-uniqueness results are established and convergence of a finite difference approximation to the unique solution of the model is obtained. The finite difference scheme is then used to investigate the effects of these nonlinear parameters on the model dynamics. Our results show that a velocity of precursor cells maturation rate which is an increasing function of the hormone level and a decay rate of the hormone which is an increasing function of the number of precursor cells have a stabilizing effect on the dynamics of the model. While assuming that one parameter is a function and letting the other be a constant stabilizes the oscillations in the mature cells level, the effect is more significant when both parameters are taken to be functions. A study of robustness with respect to the forms of these functions and parameter sensitivity is also carried out.     

Multiple stable periodic solutions in a model for hormonal control of the menstrual cycle

This study presents a nonlinear system of delay differential equations to model the concentrations of five hormones important for regulation and maintenance of the menstrual cycle. Linear model components for the ovaries and pituitary were previously analyzed and reported separately. Results for the integrated model are now presented here. This model predicts serum levels of ovarian and pituitary hormones which agree with data in the literature for normally cycling women. In addition, the model indicates the existence and stability of an abnormal cycle. Hence, the model may be used to simulate the effects of external hormone therapies on abnormally cycling women as well as the effects of exogenous compounds on normally cycling women. Such simulations may be helpful in understanding the role of xenobiotics in fertility problems, in predicting successful hormone therapies, and for testing hormonal methods of birth control.     

Evaluating the Use of Global Sensitivity Analysis in Dynamic MFA

Dynamic material flow analysis (MFA) provides information about material usage over time and consequent changes in material stocks and flows. In order to understand the effect of limited data quality and model assumptions on MFA results, the use of sensitivity analysis methods in dynamic MFA studies has been on the increase. So far, sensitivity analysis in dynamic MFA has been conducted by means of a one‐at‐a‐time method, which tests parameter perturbations individually and observes the outcomes on output. In contrast to that, variance‐based global sensitivity analysis decomposes the variance of the model output into fractions caused by the uncertainty or variability of input parameters. The present study investigates interaction and time‐delay effects of uncertain parameters on the output of an archetypal input‐driven dynamic material flow model using variance‐based global sensitivity analysis. The results show that determining the main (first‐order) effects of parameter variations is often sufficient in dynamic MFA because substantial effects attributed to the simultaneous variation of several parameters (higher‐order effects) do not appear for classical setups of dynamic material flow models. For models with time‐varying parameters, time‐delay effects of parameter variation on model outputs need to be considered, potentially boosting the computational cost of global sensitivity analysis. Finally, the implications of exploring the sensitivities of model outputs with respect to parameter variations in the archetypical model are used to derive model‐ and goal‐specific recommendations on choosing appropriate sensitivity analysis methods in dynamic MFA.     

Patterns of estrogen and progesterone receptors in monkey endometrium during the normal menstrual cycle

Total concentrations of estradiol-17β (E₂) and progesterone (P) receptors (R) were measured in the endometrium of rhesus monkeys ($\text{Macaca mulatta}$) during the normal menstrual cycle. The endometrium was collected at abdominal fundal hysterotomy on days 8, 12, 15, 18 and 24 of the menstrual cycle. Visual inspection of the ovaries and measurement of E₂, P, follicle stimulating hormone (FSH) and luteinizing hormone (LH) provided assuredness of normal ovarian function. Exchange procedures were used in order to measure the total concentrations of E₂R and PR in nuclear and cytosol fractions. The pattern of estrogen receptor showed a slight increase in the cytosol and nuclear concentrations at the preovulatory interval. Later, the total E₂R concentration was decreased when P increased during the luteal phase. Cytosol PR synthesis was parallel to the serum E₂ increase during the late follicular phase. Secretion of P by the corpus luteum was accompanied by a rapid nuclear translocation and concomitant decrease in cytoplasmic PR. Thereafter the total PR concentration declined during the second half of the luteal phase. These findings in monkey endometrium are similar to those reported for human endometrium during the normal menstrual cycle and further establish the utility of these surrogate primates in investigations indicative of human endometrial function.     

Influence of the hypothalamic-pituitary-adrenal axis on the menstrual cycle and the pituitary responsiveness to estradiol in the female rhesus monkey (Macaca mulatta)

In order to examine the effect of glucocorticoids on the menstrual cycle of rhesus monkeys, cortisol was injected twice daily during the follicular phase. This cortisol treatment did not alter basal gonadotropin secretion but blocked the normal follicular rise of estrogens, the gonadotropin surge and the luteal rise of progesterone, and delayed the onset of the next cycle. In a second study, estradiol benzoate (E2B) was injected on the sixth day following the start of menstrual bleeding either with or without concurrent adrenocorticotropic hormone (ACTH) treatment. E2B injection was able to stimulate surges of luteinizing hormone (LH) and follicle-stimulating hormone (FSH) whether or not the animals had been treated with ACTH. These data suggest that, the action of cortisol, the final mediating step in the hypothalamic-pituitary-adrenal axis, occurs at the level of the gonads versus the pituitary in the rhesus monkey. While the pituitary response to endogenous gonadotropin-releasing hormone or exogenous E2B stimulation appears to remain unaffected, normal folliculogenesis is disrupted, preventing the follicular secretion of estrogens and the subsequent gonadotropin surges. The effects of corticosteroids are temporary, with normal cycling returning when plasma corticosteroids return to basal concentrations, albeit after a delay.     

Novel hyperprolactinemia and hyperprolactinemic anovulation model using the cynomolgus monkey (Macaca fascicularis)

We investigated the relationship between the menstrual cycle and hormone levels in cynomolgus monkeys, and developed a sulpiride-induced hyperprolactinemic anovulation model. On this study, we demonstrated the usefulness of the commercial human prolactin immunoradiometric assay kit for the measurement of cynomolgus monkey serum samples. In the normal menstrual cycle of the cynomolgus monkey, serum prolactin concentrations were not significantly different between luteal and follicular phases. However, the serum prolactin concentration tended to elevate at the ovulation stage. And serum progesterone began to increase after an estradiol surge, and then declined before the ensuing preovulatory rise in estradiol. During the luteal phase, the serum concentration of progesterone was elevated. Moreover, we aimed to develop an anovulation model, using sulpiride-induced hyperprolactinemia in the cynomolgus monkey. The serum prolactin level gradually increased during the twice-daily administration of sulpiride, and the drug produced as big a response at 5 mg/kg. In this study, the length of the menstrual cycle was approximately 29 days in normal cynomolgus monkeys. When treatment with sulpiride had been continued for more than one month, serum progesterone and estradiol levels fell to within the range seen in the follicular phase of the normal cycle, and the absence of ovulation was recognized by laparoscopy. Moreover, in this period we found that amenorrhea or anovulatory menstruation in the experimental animals. We could produce an anovulatory model induced by sulpiride repeatedly administered over a long time period. Our findings suggest that the cynomolgus monkey is useful as a endocrinological model that uses prolactin as a parameter and as an anovulatory model; thus, it could be a useful model for the hyperprolactinemic amenorrhea and/or anovulation seen in humans.     

The PSO family: deduction, stochastic analysis and comparison

The PSO algorithm can be physically interpreted as a stochastic damped mass-spring system: the so-called PSO continuous model. Furthermore, PSO corresponds to a particular discretization of the PSO continuous model. In this paper, we introduce a delayed version of the PSO continuous model, where the center of attraction might be delayed with respect to the particle trajectories. Based on this mechanical analogy, we derive a family of PSO versions. For each member of this family, we deduce the first and second order stability regions and the corresponding spectral radius. As expected, the PSO-family algorithms approach the PSO continuous model (damped-mass-spring system) as the size of the time step goes to zero. All the family members are linearly isomorphic when they are derived using the same delay parameter. If the delay parameter is different, the algorithms have corresponding stability regions of any order, but they differ in their respective force terms. All the PSO versions perform fairly well in a very broad area of the parameter space (inertia weight and local and global accelerations) that is close to the border of the second order stability regions and also to the median lines of the first order stability regions where no temporal correlation between trajectories exists. In addition, these regions are fairly similar for different benchmark functions. When the number of parameters of the cost function increases, these regions move towards higher inertia weight values (w=1) and lower total mean accelerations where the temporal covariance between trajectories is positive. Finally, the comparison between different PSO versions results in the conclusion that the centered version (CC-PSO) and PSO have the best convergence rates. Conversely, the centered-progressive version (CP-PSO) has the greatest exploratory capabilities. These features have to do with the way these algorithms update the velocities and positions of particles in the swarm. Knowledge of their respective dynamics can be used to propose a family of simple and stable algorithms, including hybrid versions.     

Models for the pressing rate response of rats and pigeons on a certain class of variable interval schedules of behavior

Two mathematical models are proposed for a certain class of variable interval schedules. These schedules are derived from the fixed interval schedule by adding (with probabilityp on any interval) reinforcing stimuli at a fixed point within the interval. The first model is a deterministic one in which the excitation producing the response is treated as a superposition of an excitation generated by the added reinforcements upon that generated by the fixed interval reinforcements. A method for estimating some of the parameters is presented. The model is applied with slight variations to data from performances by pigeons and by normal and brain-operated rats. The second model is a probabilistic one in which the basic entity is the interval between successive responses. This quantity must be represented by a probability distribution which is changing throughout the intervals between reinforcements. A solution is presented for a class of interresponse time distributions. Results from a simulation of a variant of this model are discussed briefly.     

A Mathematical Model of the Human Menstrual Cycle

A mathematical model for the hormonal interactions of the human menstrual cycle is presented. The feedback effects of estrogen on the release of follicle-stimulating hormone (FSH) and luteinizing hormone (LH) are considered, including a mechanism describing the midcycle LH peak. Computer simulation with this model yields results which are periodic and in good agreement with physiological data.     

Significance of the occurrence and distribution of glycogen in cervical cells exfoliated under different physiologic and pathologic conditions

The occurrence and distribution of glycogen was investigated in cervical cells exfoliated under different physiologic and pathologic conditions using cytochemical parameters. The conditions studied included the menstrual cycle, menopause, the reproductive cycle and neoplastic lesions of the cervix in both premenopausal and postmenopausal and postmenopausal situations. The results obtained on the localization of this carbohydrate substance offer interesting information with respect to the absence and presence of glycogen in various degrees in different cell types. The presence of glycogen seems to be linked with cellular maturation; it disappears with the loss of differentiation during neoplasia. The high glycogen content during the menstrual cycle and reproductive cycle and its absence in postmenopause indicates a specific hormonal influence. The discrepancies noted in glycogen distribution in the same or different exfoliated cells obtained from different biologic states assume significance in light of various factors governing the synthesis and utilization of this polymer.