Modelling Oscillatory Patterns in the Bovine Estrous Cycle with Boolean Delay Equations

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.

Follicular waves in ontogenesis and female fertility

Antral follicle growth and recruitment are the basis of female reproduction. Follicular wave theory explains the recruitment, growth, and selection of antral follicles. This article is devoted to the follicular wave pattern in female reproduction throughout life. We highlight progress in understanding the rhythmic follicle changes based on clinical studies and studies on animal models. We review the follicular wave pattern before puberty, during pregnancy, and in perimenopause. Several mathematical models are known which quite accurately describe follicular wave dynamics. The follicular waves theory allows the implementation of the new approaches to ovarian stimulation. Stimulation in the luteal phase and double stimulation are used more widely nowadays for fertility preservation in cancer patients and for increasing the chances of IVF programs success in poor responder patients.

Ovarian and uterine arteries blood flow waveform response in the first two cycles following superstimulation in cows

To investigate the ovarian and uterine blood flow responses, hemodynamic, circulating ovarian hormones and nitric oxide (NO) after end of treatment by Folltropin. Holstein Friesian (12) cows previously synchronized with CIDR underwent Doppler ultrasound after administrating of FSH daily for 4 days in eight injections started on day 10 of the second ovulation (day -5). Oestradiol (E2), progesterone (P4) and nitric oxide (NOMs) were measured. During the follicular phase, follicle area and antrum area of the second cycle reached maximum value on the day of ovulation compared with that in the first cycle, while during the luteal phase, both showed a pattern of increase and decrease. The luteal area and total coloured area increased till day 10 in the first and second cycle. The first cycle ipsilateral ovarian artery (Ov.A) had higher pulsatility (PI) (p = .001), resistance (RI) (p = .001), peak velocity (PSV) (p = .009) and lower end-diastolic velocity (EDV) (p = .003) compared with the second cycle. The increased ipsilateral Ov.A PSV (p = .009) was accompanied by lower EDV. The first cycle ipsilateral middle uterine artery (MUA) had higher PI (p = .001) and RI (p = .001), with lower PSV (p = .001) and EDV (p = .001). It was concluded that blood flow of ovarian and middle uterine arteries changed after the end of superstimulation as the increased ipsilateral Ov. A and MUA PSVs accompanied by lower EDV and both Doppler indices that reflect the amount of ovarian and uterine blood flow waveform.

Effect of progesterone administration during growing phase of first dominant follicle on follicular wave pattern in buffalo heifers

In buffaloes, like other domestic mammals, antral follicles develop in a wave-like pattern. Factors predictive of a particular follicular wave pattern are yet to be identified. In this study, we examined the preponderance of 2- versus 3-wave patterns in 46 interovulatory intervals (IOIs) from 36 buffalo heifers, in which a subset of 10 heifers was scanned for 2 consecutive IOIs to record the repeatability of follicular wave pattern. Two-wave pattern was detected in 63.0% and 3-wave follicular pattern in 27.0% IOIs. The dominant follicles (DF) of both wave 1 as well as the ovulatory wave attained a smaller (P < 0.05) maximum diameter in 3-wave cycle as compared to 2-wave cycle. The mean duration of IOI was significantly shorter in 2-wave compared to three-wave cycles (20.5 ± 0.3 vs. 22.3 ± 0.2 days; P < 0.05). Out of 10 buffalo heifers, 7 displayed non-alternating patterns and 3 had alternating follicular wave patterns. We also tested the hypothesis that progesterone administration during early IOI results in increased preponderance of 3-wave pattern and heifers inseminated after ovulation of the third wave DF will have greater fertility. Sixteen heifers subjected to progesterone treatment from D0 (day of ovulation) in a decreasing dose until D5 were compared with control heifers (n = 10). Progesterone treatment significantly reduced the maximum diameter of DF of wave 1 (P < 0.001), whereas the mean duration of IOI remained unchanged (P > 0.05) between the two groups. Progesterone administration during early IOI significantly increased the proportion of 3-wave cycles as compared to control (P < 0.05). The hypothesis that progesterone administration during IOI results in increased preponderance of 3-wave pattern was supported. However, no change in fertility was recorded in progesterone-treated heifers (7 pregnant out of 16; 43.8%) as compared to untreated control heifers (4 out of 10 heifers; 40.0%). In summary, progesterone administration in buffalo heifers during the growing phase of wave 1 resulted in greater preponderance of 3-wave follicular patterns, with no significant effect on fertility.

Dynamics of uterine and ovarian arteries flow velocity waveforms and their relation to follicular and luteal growth and blood flow vascularization during the estrous cycle in Friesian cows

Doppler ultrasonography enabled understanding of the reproductive system hemodynamics in cyclic and pregnant cattle. To confirm the hypothesis that the ipsilateral ovarian and uterine arterial blood flows to the ovulating ovary are higher than the contralateral one along days and phases (follicular, early luteal, mid-luteal, late luteal) of the estrous cycle, eight cyclic spontaneously ovulating cows were scanned with Doppler ultrasound each other day along three oestrous cycles to monitor the follicular dynamics, the vascularization of the ovulatory follicle (OF), the corpus luteum (CL) developmental dynamics, the ipsilateral and the contralateral ovarian and uterine arterial diameters and their blood flow. Results proved the hypothesis. Both days and phases of the estrus cycle influenced (P = 0.0001) the follicular dynamic, the luteal hemodynamics, the ovarian and uterine hemodynamic. The ovulatory wave and the mid-luteal non-ovulatory wave had expanding numbers and the diameters of small, medium and large follicles. Though area, antral area, vascularization area of the OF ascended from Day -4 to the day of ovulation (Day 0), but the percent of its vascularization area and that of granulose layer increased till Day -3. The CL diameter increase till Day 15, and its vascularization area increased till Day 13, but its% of vascularization area ascended (P = 0.0001) from Days 1-4 and declined from Days 9-13. Both RI and PI of the ipsilateral ovarian artery were lower than the contralateral one; but, both obtained high values during the follicular phase. A linear increase (P = 0.0001) of uterine horns vascularization area and both ovarian and uterine arteries diameters, PSV and EDV from follicular to late luteal phases accompanied a linear decrease of their PI and RI. In conclusion, the ovarian and uterine blood flows vary according to the estrous day, estrous phase, the ovulating ovary, ovulatory follicle growth and corpus luteum developmental stage.

Coupling a reproductive function model to a productive function model to simulate lifetime performance in dairy cows

Reproductive success is a key component of lifetime performance in dairy cows but is difficult to predict due to interactions with productive function. Accordingly, this study introduces a dynamic model to simulate the productive and reproductive performance of a cow during her lifetime. The cow model consists of an existing productive function model (GARUNS) which is coupled to a new reproductive function model (RFM). The GARUNS model simulates the individual productive performance of a dairy cow throughout her lifespan. It provides, with a daily time step, changes in BW and composition, fetal growth, milk yield and composition and food intake. Genetic-scaling parameters are incorporated to scale individual performance and simulate differences within and between breeds. GARUNS responds to the discrete event signals 'conception' and 'death' (of embryo or fetus) generated by RFM. In turn, RFM responds to the GARUNS outputs concerning the cow's energetic status: the daily total processed metabolizable energy per kg BW (TPEW) and the net energy balance (EB). Reproductive function model models the reproductive system as a compartmental system transitioning between nine competence stages: prepubertal (PRPB), anestrous (ANST), anovulatory (ANOV), pre-ovulating (PREO), ovulating (OVUL), post-ovulating (PSTO), luteinizing (LUTZ), luteal (LUTL) and gestating (GEST). The transition from PRPB to ANST represents the start of reproductive activity at puberty. The cyclic path through ANST, PREO, OVUL, PSTO, LUTZ and LUTL forms the regime of ovulatory cycles, whereas ANOV and GEST are transient stages that interrupt this regime. Anovulatory refers explicitly to a stage in which ovulation cannot occur (i.e. interrupted cyclicity), whereas ANST is a pivotal stage within ovulatory cycles. Reproductive function model generates estradiol and progesterone hormonal profiles consistent with reference profiles derived from literature. Cyclicity is impacted by the GARUNS output EB and clearance of estradiol is impacted by TPEW. A farming system model was designed to describe different farm protocols of heat detection, insemination, feeding (amount and energy density), drying-off and culling. Results of model simulation (10 000 simulations of individual cows over 5000 days lifetime period, with randomly drawn genetic-scaling parameters and standard diet) are consistent with literature for reproductive performance. This model allows simulation of deviations in reproductive trajectories along physiological stages of the cow reproductive cycle. It thus provides the basis for evaluation of the relative importance of different factors affecting fertility at individual cow and herd levels across different breeds and management environments.

Estrus behavior, ovarian dynamics, and progesterone secretion in Criollo cattle during estrous cycles with two and three follicular waves

In beef and dairy cattle, the number of follicular waves affects endocrine, ovarian, and behavioral events during a normal estrous cycle. However, in Mexican-native Criollo cattle, a shortly and recently domesticated breed, the association between wave patterns and follicular development has not been studied. The objective of this study was to evaluate the effect of number of follicular waves in an estrous cycle on development of anovulatory and ovulatory follicles, corpus luteum (CL) development and functionality, as well as estrual behavior in Criollo cows. Ovarian follicular activities of 22 cycling multiparous Criollo cows were recorded daily by transrectal ultrasound examinations during a complete estrous cycle. Additionally, blood samples were collected daily to determine serum progesterone concentrations. Only two- (n = 17, 77.3%) and three-wave follicular (n = 5, 22.7%) patterns were observed. Duration of estrus, length of estrous cycle, and length of follicular and luteal phases were similar (P > 0.05) between cycles of two and three waves. Two-wave cows ovulated earlier (P < 0.05) after detection of estrus than three-wave cows. Detected day and maximum diameter of first anovulatory follicle were not affected (P > 0.05) by number of waves. Growth rate of first dominant follicle was higher (P < 0.05) in three-wave cycles. Onset of regression of the first dominant follicle was earlier (P < 0.01) in cycles with three waves than in those with two waves. In two-wave cycles, ovulatory follicles were detected earlier (P < 0.01) and had lower (P < 0.01) growth rate than in three-wave cycles. Development (i.e., maximum diameter and volume) and functionality (minimum and maximum progesterone concentration) of CL were similar (P > 0.05) between two- and three-wave patterns. In conclusion, Criollo cows have two or three follicular waves per estrous cycle, which alters partially ovulatory follicle development and ovulation time after detection of estrus. Length of estrous cycle, as well as CL development and functionality, was not affected by number of follicular waves.

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.

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.