# Mathematical analysis of a model for the growth of the bovine corpus luteum

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## Abstract

The corpus luteum (CL) is an ovarian tissue that grows in the wound space created by follicular rupture. It produces the progesterone needed in the uterus to maintain pregnancy. Rapid growth of the CL and progesterone transport to the uterus require angiogenesis, the creation of new blood vessels from pre-existing ones, a process which is regulated by proteins that include fibroblast growth factor 2 (FGF2). In this paper we develop a system of time-dependent ordinary differential equations to model CL growth. The dependent variables represent FGF2, endothelial cells (ECs), luteal cells, and stromal cells (like pericytes), by assuming that the CL volume is a continuum of the three cell types. We assume that if the CL volume exceeds that of the ovulated follicle, then growth is inhibited. This threshold volume partitions the system dynamics into two regimes, so that the model may be classified as a Filippov (piecewise smooth) system. We show that normal CL growth requires an appropriate balance between the growth rates of luteal and stromal cells. We investigate how angiogenesis influences CL growth by considering how the system dynamics depend on the dimensionless EC proliferation rate, \(\rho _5\). We find that weak (low \(\rho _5\)) or strong (high \(\rho _5\)) angiogenesis leads to ‘pathological’ CL growth, since the loss of CL constituents compromises progesterone production or delivery. However, for intermediate values of \(\rho _5\), normal CL growth is predicted. The implications of these results for cow fertility are also discussed. For example, inadequate angiogenesis has been linked to infertility in dairy cows.

## Keywords

Corpus luteum Angiogenesis Piecewise smooth systems Sliding bifurcations## Mathematics Subject Classification

92B05 (General biology and biomathematics)## Notes

### Acknowledgments

SAP acknowledges support from the Schools of Biosciences and Mathematical Sciences at the University of Nottingham in the form of a PhD studentship. MRJ’s research is supported by EPSRC grant EP/J001317/1. This publication was based in part on work supported by Award No. KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST).

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