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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 substances over time. In this work we describe recent improvements of this model, including the introduction of new components, and elimination of time delays. We validate our model by showing that the simulations agree with observations from synchronization studies and with measured progesterone data after a single dose administration of synthetic prostaglandin F2alpha.
Potassium is fundamental for cell functioning including signal transduction, acid-base- and water-metabolism.
Since diet of dairy cows is generally rich in potassium, hypokalemia was not in the focus of research for long time. Furthermore, hypokalemia was not frequently diagnosed because blood potassium content is difficult to measure. In recent years, measurement methods have been improved. Nowadays hypokalemia is increasingly diagnosed in cows with disorders such as abomasal displacement, ketosis or down cow syndrome, calling for intensified research on this topic.
In this report we describe the development of a basic mechanistic, dynamic model of potassium balance based on ordinary differential and algebraic equations. Parameter values are obtained from data of a clinical trial in which potassium balance and the influence of therapeutic intervention in glucose and electrolyte metabolism on potassium balance in non-lactating dairy cows were studied. The model is formulated at a high abstraction level and includes information and hypotheses from literature. This work represents a first step towards the understanding and design of effective prophylactic feed additives and treatment strategies.
Exploration of different wave patterns in a model of the bovine estrous cycle by Fourier analysis
(2014)
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.
High-performing dairy cows require a particular composition of nutritional ingredients, adapted to their individual requirements and depending on their production status. The optimal dimensioning of minerals in the diet, one being potassium, is indispensable for the prevention of imbalances. Potassium balance in cows is the result of potassium intake, distribution in the organism, and excretion, and it is closely related to glucose and electrolyte metabolism. In this paper, we present a dynamical model for potassium balance in lactating and nonlactating dairy cows based on ordinary differential equations. Parameter values were obtained from clinical trial data and from the literature. To verify the consistency of the model, we present simulation outcomes for 3 different scenarios: potassium balance in (1) nonlactating cows with varying feed intake, (2) nonlactating cows with varying potassium fraction in the diet, and (3) lactating cows with varying milk production levels. The results give insights into the short- and long-term potassium metabolism, providing an important step toward the understanding of the potassium network, the design of prophylactic feed additives, and possible treatment strategies.
The estrous cycle of mono-ovulatory species such as cows or humans, is known to show two or more waves of follicular growth and decline between two successive ovulations. Within each wave, there is one dominant follicle escorted by subordinate follicles of varying number. Under the surge of the luteinizing hormone a growing dominant follicle ovulates. Rarely the number of ovulating follicles exceeds one. In the biological literature, the change of hormonal concentrations and individually varying numbers of follicular receptors are made responsible for the selection of exactly one dominant follicle, yet a clear cause has not been identified. In this paper, we suggest a synergistic explanation based on competition, formulated by a parsimoniously defined system of ordinary differential equations (ODEs) that quantifies the time evolution of multiple follicles and their competitive interaction during one wave. Not discriminating between follicles, growth and decline are given by fixed rates. Competition is introduced via a growth-suppressing term, equally supported by all follicles. We prove that the number of dominant follicles is determined exclusively by the ratio of follicular growth and competition. This number turns out to be independent of the number of subordinate follicles. The asymptotic behavior of the corresponding dynamical system is investigated rigorously, where we demonstrate that the omega-limit set only contains fixed points. When also including follicular decline, our ODEs perfectly resemble ultrasound data of bovine follicles. Implications for the involved but not explicitly modeled hormones are discussed.
Exploration of different wave patterns in a model of the bovine estrous cycle by Fourier analysis
(2016)
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.
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.
We present a mechanistic pharmacokinetic-pharmacodynamic model to simulate the effect of dexamethasone on the glucose metabolism in dairy cows.
The coupling of the pharmacokinetic model to the pharmacodynamic model
is based on mechanisms underlying homeostasis regulation by dexamethasone.
In particular, the coupling takes into account the predominant role of dexamethasone in stimulating glucagon secretion, glycogenolysis and lipolysis and in
impairing the sensitivity of cells to insulin. Simulating the effect of a single
dose of dexamethasone on the physiological behaviour of the system shows that
the adopted mechanisms are able to induce a temporary hyperglycemia and
hyperinsulinemia, which captures the observed data in non-lactating cows. In
lactating cows, the model simulations show that a single dose of dexamethasone
reduces the lipolytic effect, owing to the reduction of glucose uptake by the
mammary gland.
Nutrition plays a crucial role in regulating reproductive hormones and follicular
development in cattle. This is visible particularly during the time of negative
energy balance at the onset of milk production after calving. Here, elongated
periods of anovulation have been observed, resulting from alterations in luteiniz-
ing hormone concentrations, likely caused by lower glucose and insulin concen-
trations in the blood. The mechanisms that result in a reduced fertility are
not completely understood, although a close relationship to the glucose-insulin
metabolism is widely supported. Following this idea, a mathematical model of
the hormonal network combining reproductive hormones and hormones that are
coupled to the glucose compartments within the body of the cow was developed.
The model is built on ordinary differential equations and relies on previously
introduced models on the bovine estrous cycle and the glucose-insulin dynam-
ics. Necessary modifications and coupling mechanisms are thoroughly discussed.
Depending on the composition and the amount of food, in particular the glu-
cose content in the dry matter, the model quantifies reproductive hormones and
follicular development over time. Simulation results for different nutritional
regimes in lactating and non-lactating dairy cows are examined and compared
with experimental studies. Regarding its applicability, this work is an early
attempt towards developing in silico feeding strategies and may eventually help
refining and reducing animal experiments.
Lactating dairy cows require a particular composition of nutritional ingredients depending on their production status.
The optimal supply of energy and minerals in diet, one of them potassium, is indispensable for the prevention of disbalances like hypokalemia or hypoglycaemia.
Potassium balance in cows is the result of potassium intake, distribution in the organism, and excretion, and closely interacts with glucose and electrolyte metabolism, in which postpartum veterinary treatments frequently intervene.
We present a mechanistic, dynamic model for potassium balance together with a glucose insulin model in non-lactating and lactating dairy cows based on ordinary differential equations.
Parameter values were obtained from data of a clinical trial as well as from literature. To verify the mechanistic functioning of the model, we validate the model by comparing simulation outcomes with clinical study findings. Furthermore we perform numerical experiments and compare them with expected behaviour according to mechanistic knowledge. The results give insight into the dynamic behaviour of the network and open the way for further open questions and hypothesis to be tested.
The reproductive cycle of mono-ovulatory species such as cows or humans is known to show two or more waves of follicular growth and decline between two successive ovulations. Within each wave, there is one dominant follicle escorted by subordinate follicles of varying number. Under the surge of the luteinizing hormone a growing dominant follicle ovulates. Rarely the number of ovulating follicles exceeds one. In the biological literature, the change of hormonal concentrations and individually varying numbers of follicular receptors are made responsible for the selection of exactly one dominant follicle, yet a clear cause has not been identified. In this paper, we suggest a synergistic explanation based on competition, formulated by a parsimoniously defined system of ordinary differential equations (ODEs) that quantifies the time evolution of multiple follicles and their competitive interaction during one wave. Not discriminating between follicles, growth and decline are given by fixed rates. Competition is introduced via a growth-suppressing term, equally supported by all follicles. We prove that the number of dominant follicles is determined exclusively by the ratio of follicular growth and competition. This number turns out to be independent of the number of subordinate follicles. The asymptotic behavior of the corresponding dynamical system is investigated rigorously, where we demonstrate that the ω-limit set only contains fixed points. When also including follicular decline, our ODEs perfectly resemble ultrasound data of bovine follicles. Implications for the involved but not explicitly modeled hormones are discussed.
High performing dairy cows require a particular composition of nutritional ingredients, adapted to their individual requirements and depending on their production status. The optimal dimensioning of minerals in the diet, one of them being potassium, is indispensable for the prevention of imbalances. The potassium balance in cows is the result of potassium intake, distribution in the organism, and excretion, it is closely related with the glucose and electrolyte metabolism. In this paper, we present a dynamical model for the potassium balance in lactating and non-lactating dairy cows based on ordinary differential equations. Parameter values are obtained from clinical trial data and from the literature. To verify the consistency of the model, we present simulation outcomes for three different scenarios: potassium balance in (i) non-lactating cows with varying feed intake, (ii) non-lactating cows with varying potassium fraction in the diet, and (iii) lactating cows with varying milk production levels. The results give insights into the short and long term potassium metabolism, providing an important step towards the understanding of the potassium network, the design of prophylactic feed additives, and possible treatment strategies.
Boolean delay equations (BDEs), with their relatively simple and intuitive mode of
modelling, have been used in many research areas including, for example, climate
dynamics and earthquake propagation. Their application to biological systems has been
scarce and limited to the molecular level. Here, we derive and present two BDE models.
One is directly derived from a previously published ordinary differential equation
(ODE) model for the bovine estrous cycle, whereas the second model includes a
modification of a particular biological mechanism. We not only compare the simulation
results from the BDE models with the trajectories of the ODE model, but also validate
the BDE models with two additional numerical experiments. One experiment induces
a switch in the oscillatory pattern upon changes in the model parameters, and the
other simulates the administration of a hormone that is known to shift the estrous
cycle in time. The models presented here are the first BDE models for hormonal
oscillators, and the first BDE models for drug administration. Even though automatic
parameter estimation still remains challenging, our results support the role of BDEs
as a framework for the systematic modelling of complex biological oscillators.
Hypokalemia, characterized by too low serum potassium levels, is a severe mineral disorder which can be life threatening. It is increasingly diagnosed in veterinarian healthcare and a topic of ongoing research. In this paper, we explore the different originating conditions of hypokalemia: reduced potassium intake, increased excretion, acid-base disturbances, or increased insulin, by using a dynamic mathematical model for potassium balance in non-lactating and lactating cows.
Simulations are compared with literature. The results give insights into the network dynamics and point to scenarios on which experimental effort should be focused. Application of mathematical models can assist in experimental planning as well as the reduction, refinement and replacement of animal experiments.