Claudia Stötzel

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.

A mathematical model of the human menstrual cycle for the administration of GnRH analogues

The paper presents a differential equation model for the feedback mechanisms between gonadotropin-releasing hormone (GnRH), follicle-stimulating hormone (FSH), luteinizing hormone (LH), development of follicles and corpus luteum, and the production of estradiol (E2), progesterone (P4), inhibin A (IhA), and inhibin B (IhB) during the female menstrual cycle. Compared to earlier human cycle models, there are three important differences: The model presented here (a) does not involve any delay equations, (b) is based on a deterministic modeling of the GnRH pulse pattern, and (c) contains less differential equations and less parameters. These differences allow for a faster simulation and parameter identification. The focus is on modeling GnRH-receptor binding, in particular, by inclusion of a pharmacokinetic/pharmacodynamic (PK/PD) model for a GnRH agonist, Nafarelin, and a GnRH antagonist, Cetrorelix, into the menstrual cycle model. The final mathematical model describes the hormone profiles (LH, FSH, P4, E2) throughout the menstrual cycle of 12 healthy women. It correctly predicts hormonal changes following single and multiple dose administration of Nafarelin or Cetrorelix at different stages in the cycle.

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.

Advances in modeling of the bovine estrous cycle: Synchronization with PGF2α

Our model of the bovine estrous cycle is a set of ordinary differential equations which generates hormone profiles of successive estrous cycles with several follicular waves per cycle. It describes the growth and decay of the follicles and the corpus luteum, as well as the change of the key reproductive hormones, enzymes and processes over time. In this work we describe recent developments of this model towards the administration of prostaglandin F2α. We validate our model by showing that the simulations agree with observations from synchronization studies and with measured progesterone data after single dose administrations of synthetic prostaglandin F2α.

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.

Complementing ODE-Based System Analysis Using Boolean Networks Derived from an Euler-Like Transformation

In this paper, we present a systematic transition scheme for a large class of ordinary differential equations (ODEs) into Boolean networks. Our transition scheme can be applied to any system of ODEs whose right hand sides can be written as sums and products of monotone functions. It performs an Euler-like step which uses the signs of the right hand sides to obtain the Boolean update functions for every variable of the corresponding discrete model. The discrete model can, on one hand, be considered as another representation of the biological system or, alternatively, it can be used to further the analysis of the original ODE model. Since the generic transformation method does not guarantee any property conservation, a subsequent validation step is required. Depending on the purpose of the model this step can be based on experimental data or ODE simulations and characteristics. Analysis of the resulting Boolean model, both on its own and in comparison with the ODE model, then allows to investigate system properties not accessible in a purely continuous setting. The method is exemplarily applied to a previously published model of the bovine estrous cycle, which leads to new insights regarding the regulation among the components, and also indicates strongly that the system is tailored to generate stable oscillations.

Integrating nutritional and reproductive models to improve reproductive efficiency in dairy cattle

Successful reproduction requires coordination among neural, endocrine and nutritional systems leading to ovulation, insemination and a uterine environment that allows embryonic growth and attachment. These processes are a function of genetics, dietary nutrient composition and intake, and housing and climate. We lack a systems biology approach to study and define the control of reproduction in the dairy cow. Pregnancy rates in the best managed herds only approach 25 to 30%, and may be due to the multifactorial nature of reproductive processes (McNamara, 2010). Lower fertility increases the cost for insemination because of low reproduction performance and remains a primary reason for culling cows in the first three weeks of lactation (Chagas et al., 2007). Additional days the cow is not pregnant beyond the optimal time post-calving are costly to the dairy producer. Low fertility has often been blamed on high rates of milk production, but this is not the only factor affecting reproduction (Sangsritavong et al., 2002). There are three major systems of reproduction that can be affected by genetics, nutrition and management: the hypothalamic-pituitary axis controlling initialization of cycling, the follicular development in the ovary leading to ovulation; and the successful fertilization and growth of an embryo in the uterine environment. Thus, the goal is to improve reproduction efficiency while decreasing any environmental impacts. Better nutritional management, genetic selection for fertility, and more attention to success can improve reproductive efficiency. In order to do all this efficiently, biosystems models can be of great efficacy.

Mathematical model of the bovine oestrous cycle

Mathematical model of the bovine oestrous cycle HMT Boer, C Stötzel, S Röblitz, P Deuflhard, RF Veerkamp, H Woelders Animal Breeding and Genomics Centre, Wageningen UR Livestock Research, Lelystad, Netherlands, Adaptation Physiology Group, Department of Animal Sciences, Wageningen University, Wageningen, Netherlands, Zuse Institute Berlin, Department of Numerical Analysis and Modeling, Research Group Computational Drug Design, Berlin, Germany Email: marike.boer@wur.nl

A systems biology approach to bovine fertility and metabolism: Introduction of a glucose insulin model

To counteract the antagonistic relationship between milk yield and fertility in dairy cow, a deeper understanding of the underlying biological mechanisms is required. For this purpose, we study physiological networks related to reproduction and metabolism in dairy cows. We interactively develop dynamic, mechanistic models by fitting the models to experimental data and mechanistic knowledge. We have already developed models for potassium balance and hormonal regulation of fertility in the dairy cow, which will briefly be reviewed here. The main focus of this article is a glucose-insulin model currently developed by us. This model links the bovine hormonal cycle and the potassium balance to glucose and thus to energy metabolism. The models can be applied in scientific research, education, experimental planning, drug development and production on farms.

Exploration of different wave patterns in a model of the bovine estrous cycle by Fourier analysis

Cows typically have different numbers of follicular waves during their hormonal cycle. Understanding the underlying regulations leads to insights into the reasons for declined fertility, a phenomenon that has been observed during the last decades. We present a systematic approach based on Fourier analysis to examine how parameter changes in a model of the bovine estrous cycle lead to different wave patterns. Even without any biological considerations, this allows to detect the responsible model parameters that control the type of periodicity of the solution, thus supporting experimental planning of animal scientists.