Validation of a mathematical model of the bovine estrous cycle for cows with different estrous cycle characteristics.

A recently developed mechanistic mathematical model of the bovine estrous cycle was parameterized to fit empirical data sets collected during one estrous cycle of 31 individual cows, with the main objective to further validate the model. The a priori criteria for validation were (1) the resulting model can simulate the measured data correctly (i.e. goodness of fit), and (2) this is achieved without needing extreme, probably non-physiological parameter values. We used a least squares optimization procedure to identify parameter configurations for the mathematical model to fit the empirical in vivo measurements of follicle and corpus luteum sizes, and the plasma concentrations of progesterone, estradiol, FSH and LH for each cow. The model was capable of accommodating normal variation in estrous cycle characteristics of individual cows. With the parameter sets estimated for the individual cows, the model behavior changed for 21 cows, with improved fit of the simulated output curves for 18 of these 21 cows. Moreover, the number of follicular waves was predicted correctly for 18 of the 25 two-wave and three-wave cows, without extreme parameter value changes. Estimation of specific parameters confirmed results of previous model simulations indicating that parameters involved in luteolytic signaling are very important for regulation of general estrous cycle characteristics, and are likely responsible for differences in estrous cycle characteristics between cows.

Candidate mechanisms underlying atypical progesterone profiles as deduced from parameter perturbations in a mathematical model of the bovine estrous cycle.

The complex interplay of physiological factors that underlies fertility in dairy cows was investigated using a mechanistic mathematical model of the dynamics of the bovine estrous cycle. The model simulates the processes of follicle and corpus luteum development and its relations with key hormones that interact to control these processes. Several factors may perturb the regular oscillatory behavior of a normal estrous cycle, and such perturbations are likely the effect of simultaneous changes in multiple parameters. The objective of this paper was to investigate how multiple parameter perturbation changes the behavior of the estrous cycle model, so as to identify biological mechanisms that could play a role in the development of cystic ovaries. Cystic ovaries are a common reason for reproductive failure in dairy cows, but much about the causes of this disorder remains unknown. We investigated in which region of the parameter space the model predicts a normal cycle, and when a progesterone pattern occurred with delayed ovulation (indicating a cystic follicle) or delayed luteolysis (indicating a persistent corpus luteum). Perturbation of the initial values for all parameters simultaneously showed 2 specific parameter configurations leading to delayed ovulation or delayed luteolysis immediately. The most important parameter changes in these 2 configurations involve the regulation of corpus luteum functioning, luteolytic signals, and GnRH synthesis, suggesting that these mechanisms are likely involved in the development of cystic ovaries. In the multidimensional parameter space, areas exist in which the parameter configurations resulted in normal cycles. These areas may be separated by areas in which irregular cycle patterns occurred. These irregular patterns thus mark the transition from one stable (normal) situation to another. Interestingly, within a series, there were some cycles with delayed ovulation and some with delayed luteolysis in these patterns. This could represent a situation of resumption of normal cyclicity (e.g., after parturition). In conclusion, the method of parameter perturbation used in the present study is an effective tool to find parameter configurations that lead to progesterone profiles associated with delayed ovulation and delayed luteolysis. Thereby, the model helps to generate hypotheses regarding the underlying cause of the development of cystic ovaries, which could be investigated in future experiments.

A differential equation model to investigate the dynamics of the bovine estrous cycle.

To investigate physiological factors affecting fertility of dairy cows, we developed a mechanistic mathematical model of the dynamics of the bovine estrous cycle. The model consists of 12 (delay) differential equations and 54 parameters. It simulates follicle and corpus luteum development and the periodic changes in hormones levels that regulate these processes. The model can be used to determine the level of control exerted by various system components on the functioning of the system. As an example, it was investigated which mechanisms could be candidates for regulation of the number of waves of follicle development per cycle. Important issues in model building and validation of our model were parameter identification, sensitivity analysis, stability, and prediction of model behavior in different scenarios.

Mechanisms regulating follicle wave patterns in the bovine estrous cycle investigated with a mathematical model.

A normal bovine estrous cycle contains 2 or 3 waves of follicle development, and ovulation takes place in the last wave. However, the biological mechanisms that determine whether a cycle has 2 or 3 waves have not been elucidated. In a previous paper, we described a mathematical model of the bovine estrous cycle that generates cyclical fluctuations of hormones, follicles, and corpora lutea in estrous cycles of approximately 21 d for cows with a normal estrous cycle. The parameters in the model represent kinetic properties of the system with regard to synthesis, release, and clearance of hormones and growth and regression of follicles and corpora lutea. The initial model parameterization resulted in estrous cycles with 3 waves of follicular growth. Here, we use this model to explore which physiological mechanisms could affect the number of follicular waves. We hypothesized that some of the parameters related to follicle growth rate or to the time point of corpus luteum regression are likely candidates to affect the number of waves per cycle. We performed simulations with the model in which we varied the values of these parameters. We showed that variation of (combinations of) model parameters regulating follicle growth rate or time point of corpus luteum regression can change the model output from 3 to 2 waves of follicular growth in a cycle. In addition, alternating 2- and 3-wave cycles occurred. Some of the parameter changes seem to represent plausible biological mechanisms that could explain these follicular wave patterns. In conclusion, our simulations indicated likely parameters involved in the mechanisms that regulate the follicular wave pattern, and could thereby help to find causes of declined fertility in dairy cows.

A simple mathematical model of the bovine estrous cycle: follicle development and endocrine interactions.

Bovine fertility is the subject of extensive research in animal sciences, especially because fertility of dairy cows has declined during the last decades. The regulation of estrus is controlled by the complex interplay of various organs and hormones. Mathematical modeling of the bovine estrous cycle could help in understanding the dynamics of this complex biological system. In this paper we present a mechanistic mathematical model of the bovine estrous cycle that includes the processes of follicle and corpus luteum development and the key hormones that interact to control these processes. The model generates successive estrous cycles of 21 days, with three waves of follicle growth per cycle. The model contains 12 differential equations and 54 parameters. Focus in this paper is on development of the model, but also some simulation results are presented, showing that a set of equations and parameters is obtained that describes the system consistent with empirical knowledge. Even though the majority of the mechanisms that are included in the model are based on relations that in the literature have only been described qualitatively (i.e. stimulation and inhibition), the output of the model is surprisingly well in line with empirical data. This model of the bovine estrous cycle could be used as a basis for more elaborate models with the ability to study effects of external manipulations and genetic differences.

Estrous behavior in dairy cows: identification of underlying mechanisms and gene functions.

Selection in dairy cattle for a higher milk yield has coincided with declined fertility. One of the factors is reduced expression of estrous behavior. Changes in systems that regulate the estrous behavior could be manifested by altered gene expression. This literature review describes the current knowledge on mechanisms and genes involved in the regulation of estrous behavior. The endocrinological regulation of the estrous cycle in dairy cows is well described. Estradiol (E2) is assumed to be the key regulator that synchronizes endocrine and behavioral events. Other pivotal hormones are, for example, progesterone, gonadotropin releasing hormone and insulin-like growth factor-1. Interactions between the latter and E2 may play a role in the unfavorable effects of milk yield-related metabolic stress on fertility in high milk-producing dairy cows. However, a clear understanding of how endocrine mechanisms are tied to estrous behavior in cows is only starting to emerge. Recent studies on gene expression and signaling pathways in rodents and other animals contribute to our understanding of genes and mechanisms involved in estrous behavior. Studies in rodents, for example, show that estrogen-induced gene expression in specific brain areas such as the hypothalamus play an important role. Through these estrogen-induced gene expressions, E2 alters the functioning of neuronal networks that underlie estrous behavior, by affecting dendritic connections between cells, receptor populations and neurotransmitter releases. To improve the understanding of complex biological networks, like estrus regulation, and to deal with the increasing amount of genomic information that becomes available, mathematical models can be helpful. Systems biology combines physiological and genomic data with mathematical modeling. Possible applications of systems biology approaches in the field of female fertility and estrous behavior are discussed.