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 preexisting ones, a process which is regulated by proteins that include fibroblast growth factor 2 (FGF2). In this paper we develop a system of timedependent 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.
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References
Beenken A, Mohammadi M (2009) The FGF family: biology, pathophysiology and therapy. Nat Rev 8:235–253
Bernardo M, Budd C, Champneys A, Kowalczyk P (2008) Piecewisesmooth dynamical systems: theory and applications. Springer, Berlin
Boer H, Stotzel C, Roblitz S, Deuflhard P, Veerkamp R, Woelders H (2011) A simple mathematical model of the bovine estrous cycle: follicle development and endocrine interactions. J Theor Biol 278:20–31
Colombo A (2009) A bifurcation analysis of discontinuous systems: theory and applications. Dissertation, Politecnico di Milano, Italy
Filippov A (1982) Differential equations with discontinuous righthand sides, vol 56. Kluwer Academic Publishers, Dordrecht, pp 281–292
Fraser H, Lunn S (2001) Regularization and manipulation of angiogenesis in the primate corpus luteum. Reproduction 121:355–362
Hunter R (2003) Physiology of the graafian follicle and ovulation. Cambridge University press, Cambridge
Juengel J, Niswender G (1999) Molecular regulation of luteal progesterone synthesis in domestic ruminants. J Reprod Fert Supp 54:193–205
Kuznetsov Y, Rinaldi S, Gragnani A (2003) Oneparameter bifurcations in planar filippov systems. IJBC 13:2157–2188
Laird M (2010) Angiogenesis during luteal development in the cow. Dissertation, University of Nottingham, UK
Laird M, Woad K, Hunter M, Mann G, Robinson R (2013) Fibroblast growth factor 2 induces the precocious development of endothelial cell networks in bovine luteinising follicular cells. Reprod Fertil Dev 25:372–386
Lincoln D, Whitney R, Smith J (1982) In vitro proliferation and lifespan of bovine aorta endothelial cells: response to conditioned media. J Cell Sci 56:281–292
Lucy M (2001) Reproductive loss in highproducing dairy cattle: Where will it end? J Dairy Sci 84:1277–1293
Mann G (2009) Corpus luteum size and plasma progesterone concentration in cows. Anim Reprod Sci 115:296–299
Mann G, Lamming G (2001) Relationship between maternal endocrine environment, early embryo development and inhibition of the luteolytic mechanism in cows. Reproduction 121:175–180
Meier S, Roche J, Kolver E, Boston R (2009) A compartmental model describing changes in progesterone concentrations during the oestrous cycle. J Dairy Res 76:249–256
Owen R, Stamper I, Muthana M, Richardson G, Dobson J, Lewis C, Byrne H (2011) Mathematical modeling predicts synergistic antitumor effects of combining a macrophagebased, hypoxiatargeted gene therapy with chemotherapy. Cancer Res 71:2826–2837
Perry G, Smith M, Lucy M, Green J, Parks T, MacNeil M, Roberts A, Geary T (2005) Relationship between follicle size at insemination and pregnancy success. PNAS 102:5268–5273
Piiroinen P, Kuznetsov Y (2008) An eventdriven method to simulate filippov systems with accurate computing of sliding motions. ACM Trans Math Softw 34(3)
Pring S, Owen M, King J, Sinclair K, Webb R, Flint A, Garnsworthy P (2012) A mathematical model of the bovine oestrous cycle: simulating outcomes of dietary and pharmacological interventions. J Theor Biol 313:115–126
Redmer D, Doraiswamy V, Bortnem B, Fisher K, JablonkaShariff A, GrazulBilska A, Reynolds L (2001) Evidence for a role of capillary pericytes in vascular growth of the developing ovine corpus luteum. Biol Reprod 65(3):879–889
Reynolds L, Redmer D (1996) Angiogenesis in the ovary. Rev Reprod 1:182–192
Reynolds L, Redmer D (1998) Expression of the angiogenic factors, bFGF and VEGF, in the ovary. J Anim Sci 76:1671–1681
Reynolds L, Redmer D (1999) Growth and development of the corpus luteum. J Reprod Fert Supp 54:181–191
Rieck P, Oliver L, Engelmann K, Fuhrmann G, Hartmann C, Courtois Y (1995) The role of exogenous/endogenous basic fibroblast growth factor (FGF2) and transforming growth factor \(\beta \,(\text{ TGF }\beta 1)\) on human corneal endothelial cells proliferation in vitro. Exp Cell Res 220:36–46
Robinson R, Hammond A, Hunter M, Mann G (2008) A novel physiological culture system that mimics luteal angiogenesis. Reprod 9:433–457
Robinson R, Hammond A, Nicklin L, Schams D, Mann G, Hunter M (2006) Endocrine and cellular characteristics of corpora lutea from cows with a delayed postovulatory progesterone rise. Domest Anim Endocrinol 31:154–172
Robinson R, Nicklin L, Hammond A, Schams D, Hunter M, Mann G (2007) Fibroblast growth factor 2 is more dynamic than vascular endothelial growth factor A during the follicle–luteal transition in the cow. Biol Reprod 77:28–36
Robinson R, Woad K, Hammond A, Laird M, Hunter M, Mann G (2009) Angiogenesis and vascular function in the ovary. Reproduction 138:869–881
Royal M, Mann G, Flint A (2000) Strategies for reversing the trend towards subfertility in dairy cattle. Vet J 160:53–60
Schams D, Berisha B, Steffl M, Amselgruber W (2006) Changes in FGF2 and its receptors in bovine follicles before and after GnRH application and after ovulation. Reproduction 131:319–329
Wiltbank M (1994) Cell types and hormonal mechanisms associated with midcycle corpus luteum function. J Anim Sci 72:1873–1883
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. KUKC101304, made by King Abdullah University of Science and Technology (KAUST).
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Appendices
Appendix A: Estimation of parameter values
Guided by estimates of the maximum proliferation rate of bovine aorta ECs cultured in vitro, we suppose initially that the maximum growth rate of all cell types are identical so that \(k_{E}=k_{L}=k_{R} \sim 1\) day(\(d\))\(^{1}\) (Lincoln et al. 1982), and later explore the consequences of relaxing this assumption.
Regarding the decay rate of FGF2, we fixed \(d_{F}=3\,d^{1}\) since its halflife is \(\sim \)8 h (Beenken and Mohammadi 2009). Concerning \(F_{h}\), the FGF2 concentration at which the EC proliferation rate is halfmaximal, we chose it to be a typical late level of FGF2 (\(F^*\); see Fig. 2a), so that \(F_{h}=F^* \simeq 50\) \(\mathrm{ng}\,(\mathrm{cm}^3)^{1}\). We remark that in Fig. 2a FGF2 has units \(\mathrm{ng/g}\) tissue, but since the majority of tissues consist predominantly of water, we convert from \(\mathrm{ng/g}\) to \(\mathrm{ng/cm}^3\) by assuming that 1 \(g\) occupies approximately 1 \(\mathrm{cm}^3\).
A value of \(\overline{V}\) was estimated for the CL volume, above which cells compete for space. From Fig. 2b, the steady value for the CL diameter is approximately 2.5 cm and, therefore, \(\overline{V} \simeq \frac{4}{3}\pi R^3 \simeq \) 8.2 \(\mathrm{cm}^{3}\). In midcycle, LCs comprise approximately 68 \(\%\) of the bovine CL volume and the ECs approximately 13 \(\%\) (Wiltbank 1994). Therefore, we assume that \(E_{h1}\) and \(E_{h2}\), the volume of ECs at which the swelling rate of LCs and stromal cells is halfmaximal, are identical \(\sim 13\,\% \overline{V}=1.07\,\mathrm{cm}^{3}\).
The above estimates imply that typical proportion of ECs and LCs is ECs:LCs = 13:68. Therefore, an estimate for the parameter \(R_{EL}\) (as in Eq. (1)), the value of \(E/L\) at which FGF2 production rate (from LCs) is halfmaximal was obtained. That is, \(R_{EL}=0.19\).
The methods for estimating the parameters \(a_1\) and \(a_2\), and the parameter \(k\) are enumerated below as methods (Ia) and (Ib), respectively.

(Ia)
To estimate \(a_1\) and \(a_2\), we assume a steady state \(\dot{F}\)=0, and solve for \(a_1\) when there are no LCs and solve for \(a_2\) when there are no ECs. Then, for \(a_{1}\), \(\dot{F}=a_{1}E  d_{F}F\), gives \(a_{1}=d_{F}\frac{F}{E} \simeq d_{F}\frac{F^{*}}{E^{*}} \simeq d_{F}\frac{F^{*}}{0.13\,\overline{V}} \simeq \frac{3*50}{1.1} \simeq 136\). For \(a_{2}\), \(\dot{F}=a_{2}\frac{L}{R_{EL}}  d_{F}F\), gives \(a_{2} \simeq d_{F}R_{EL}\frac{F}{L} \simeq d_{F}R_{EL}\frac{F^{*}}{L^{*}} \simeq d_{F}R_{EL}\frac{F^{*}}{0.68\,\overline{V}} = \frac{3*0.19*50}{5.6} \simeq 5.1\).

(Ib)
To estimate the parameter \(k\), the strength of the tissue constraint to the cell growth or proliferation, we assume a steady state \(\dot{E}\)=0 by taking the limiting case where FGF2 is sufficiently high (\(F^* \gg F_h\)). Therefore, \(\dot{E}=0 \Rightarrow k_{E}E=kEV \Rightarrow k=\frac{k_E}{V} \simeq \frac{k_{E}}{\overline{V}}\simeq \) 0.12.
Dimensional estimates of the model parameters in Eqs. (2), (5)–(7) were estimated as accurately as possible. Since there is less data with which to determine \(a_1\), \(a_2\), and \(k\), than the other parameters, we give here an alternative method to estimate them, as a test of robustness. These alternative methods are enumerated below as (IIa) and (IIb):

(IIa)
Here we estimate \(a_1\) and \(a_2\) as follows. In Fig. 3, a schematic is presented which illustrates the high FGF2 production by LCs during the first two days, while ECs are productive all over the cell cycle. Based on that, we assume that the FGF2 production from LCs after the first two days is minimal. That is, at steady state (\(\dot{F}=0\)) \(\frac{a_2 L}{R_{EL} + \frac{E}{L}} \rightarrow 0\), or equivalently,
$$\begin{aligned} a_1 E^* \gg \frac{a_2 L^*}{R_{EL} + \frac{E^*}{L^*}}. \end{aligned}$$(49)In addition, at steady state,
$$\begin{aligned} \dot{F}=0 \Leftrightarrow a_1 E^*\left( 1 + \frac{1}{\gamma }\right) =d_F F^*, \end{aligned}$$(50)where,
$$\begin{aligned} \gamma =\frac{a_1 E^*}{\frac{a_2 L^*}{R_{EL} + \frac{E^*}{L^*}}} = 0.07 \frac{a_1}{a_2}, \end{aligned}$$(51)is the ratio of FGF2 production by ECs to that by LCs. If \(\gamma =10\), then
$$\begin{aligned} a_1 \simeq 143\,a_2. \end{aligned}$$(52)Now Eq. (50) supplies \(a_1 = \frac{d_F F^*}{E^*(1 + \frac{1}{\gamma })}\simeq 124\), in which case \(a_2 \simeq 0.9\).

(IIb)
Here we estimate \(k\) as follows. We assume a steady state \(\dot{E}=0\) and FGF2 steady value being smaller than for (Ib), e.g. \(F^*=F_h\), and solving for \(k\) implies: \(k \sim \frac{F^*}{F_h + F^*}\frac{k_E}{\overline{V}} \simeq \frac{1}{2}\frac{k_E}{\overline{V}}=0.06\).
Appendix B: Classification of steady state in Filippov systems
The types of steady states that Filippov systems exhibit are summarised below.
Definition 2
A point \(\underline{x} \in D\) is termed an admissible steady state of (17) if
a point \(\underline{x} \in G^{\pm }\) is termed a virtual steady state of (17) if
Definition 3
A point \(\underline{x}\in D\) is termed a pseudo steady state if
As for Definition 2, there may exist solutions to \(\underline{g}(\underline{x})=0\) which are invalid because \(x\in \varSigma \backslash \hat{\varSigma }\). We distinguish such solutions as follows:
Definition 4
A pseudo steady state is termed admissible if \(0<\lambda <1\) and virtual if \(\lambda < 0\) or \(\lambda >1\), with \(\lambda \) as defined in (28).
For some values of the system parameters, a steady state may lie on the discontinuity boundary. Since \(\underline{f}^+\) or \(\underline{f}^\) vanishes there, we find that \(\underline{g}\) also vanishes by (29), so that a steady state on \(\varSigma \) always coincides with a pseudo steady state. Furthermore this occurs on the sliding boundary where \(\lambda =0\) or 1. We classify such points asfollows.
Definition 5
A point \(\underline{x} \in D\) is termed a boundary steady state of (17) if
The admissibility conditions for the steady states in \(G^+\) and \(\hat{\varSigma }\) are given below.
Conditions for \(A^+_1\)
From Eq. (38) we deduce that \(E_{1}\ge 0\) if either
where \(\frac{\rho _8}{\rho _6}\) is the ratio of the maximal growth rate of the stromal cells to that of the luteal cells, and \(\frac{\rho _9}{\rho _7}\) is the ratio of the halfmaximal EC value of the stromal cells to that of the luteal cells. Similarly, the concentration of \(FGF2\), \(F=E_1\phi (E_1)\), is physically realistic if \(\phi (E_1)\ge 0\) where, since \(\rho _{6}>0\),
since (57) guarantees that \(E_{1}>0\). Thus (58) places a lower bound on \(\rho _{5}\).
The value of \(L\) is physically realistic if \(\psi (E_{1})\ge 0\) (\(\Leftrightarrow \omega (E_{1})\ge 0\)), which implies
an upper bound to \(\rho _{5}\) (since \(E_{1}\ge 0\)).
The value of \(R\) is physically realistic if \(\nu (E_1)\psi (E_1) \ge 1\). Substituting in the expression for \(\psi \) (Eq. (36)) in terms of \(\omega \), implies
The admissibility condition for \(A^+_1\) is simply that the volume \(V=V^+_1\) satisfies
Conditions for \(A^+_2\)
The components of \(A^+_2\) are nonnegative if
The first inequality supplies the following mutually exclusive set of inequalities:
The second inequality (\(\eta (E_{2})\ge 1\)) yields:
The admissibility condition for \(A^+_2\) is that the volume \(V=V^+_2\) satisfies
noting that \(V^+_2\ge 0\) is guaranteed by both cases (i) and (ii) above.
Conditions for \(A^+_3\)
The steady state \(A^+_3\) is physically realistic if \(E_3\), \(\phi (E_3)\), and \((\nu (E_3)1)\) are positive. The solution for \(E_3\) as a root of the cubic polynomial (39) is unilluminating so we do not present it here. We note, however, that (39) has real coefficients, hence it always has at least one real root, but the root may not be positive for all values of the parameters.
The admissibility condition can be written as
Numerical solutions of these equations for the parameter values in (13), with \(\rho _5\) allowed to vary, reveal only one physically realistic steady state solution, \(A^+_3\). It becomes unphysical outside the left bound as \(L=\nu (E_3)1\) becomes negative, and inadmissible outside the right bound as \(E{\varOmega }=V^+_3=E_3\nu (E_3)\) becomes smaller than unity.
Conditions for \(A^+_4\)
The steady state solution \(A^+_4\) is physically realistic if \(E_4\ge 0\), implying \(\frac{\rho _5}{\rho _6}\frac{\rho _4}{\rho _1}\ge 0\) or, as a condition on \(\rho _5\),
The admissibility condition is
Notice that the ratio \(\frac{\rho _{4}}{\rho _{1}}\) is involved in determining whether both of the steady states \(A_{2}^{+}\) and \(A_{4}^{+}\) are physically realistic.
We now establish conditions under which the steady states \(A^s_n\) are physically realistic and satisfy the conditions for \(\hat{\varSigma }\) in Table 2.
Conditions for \(A^s_1\)
Because the first three components of \(A^s_1\) are the same as \(A^+_1\), the conditions (57)–(59) ensure that \(E_1\), \(\phi (E_1)\), and \(\psi (E_1)\) are positive. By contrast, the condition for \(R\) to be physically realistic becomes \(\frac{1}{E_1}\psi (E_1)1\ge 0\). Substituting in the expression for \(\psi \) in terms of \(\omega \), a little manipulation gives the condition
Now consider the admissibility condition \(0\le E{\varOmega }\le 1\). The third (\(L\)) component of \(g(A^s_1)\) from (30), which vanishes because \(A^s_1\) is a steady state, gives
Substituting in \(E_1\) from (38), after a little rearranging the admissibility condition becomes
Conditions for \(A^s_2\)
Physical values of \(A^s_2\) require \(E_2\ge 0\) and \(\frac{1}{E_2}1\ge 0\), implying \(0\le E_2\le 1\), which gives
As for \(A^+_2\) there are two cases to consider,

(i)
\(\rho _5\ge \rho _8\) implies \(0\le \rho _4\rho _8 \rho _1\rho _5\rho _{9}\le \rho _1(\rho _5\rho _8)\),

(ii)
\(\rho _5\le \rho _8\) implies \(0\ge \rho _4\rho _8 \rho _1\rho _5\rho _{9}\ge \rho _1(\rho _5\rho _8)\).
which can be rearranged to give conditions on \(\rho _5\),

(i)
\(\rho _5\ge \rho _8\) implies \(\rho _5\ge \frac{\rho _8}{\rho _1}\frac{\rho _1+\rho _4}{1+\rho _{9}}\) and \(\rho _5 \le \frac{\rho _4\rho _8}{\rho _1\rho _9}\),

(ii)
\(\rho _5\le \rho _8\) implies \(\rho _5\le \frac{\rho _8}{\rho _1}\frac{\rho _1+\rho _4}{1+\rho _{9}}\) and \(\rho _5 \ge \frac{\rho _4\rho _8}{\rho _1\rho _9}\).
To evaluate the admissibility condition on \(E{\varOmega }\), consider the fourth (\(R\)) component of \(g(A^s_2)\) from (30), which vanishes because \(A^s_1\) is a steady state, and therefore gives
Substituting in \(E_2\) from (38) and rearranging gives \(E_2{\varOmega }=\frac{\rho _4\rho _8\rho _1\rho _5\rho _{9}}{\rho _4\rho _1\rho _{9}}\), hence the admissibility condition becomes
Equation (70) consists of two cases:

(a)
\(\rho _4< \rho _1\rho _{9}\) implies \(\rho _5 \ge \frac{\rho _4\rho _8}{\rho _1\rho _9}  \frac{\rho _4}{\rho _1\rho _9} + 1\),

(b)
\(\rho _4> \rho _1\rho _{9}\) implies \(\frac{\rho _4\rho _8}{\rho _1\rho _9}  \frac{\rho _4}{\rho _1\rho _9} + 1 \le \rho _5 < \frac{\rho _4\rho _8}{\rho _1\rho _9}\), and this can only happen in case (i).
Conditions for \(A^s_3\)
For the state \(A^s_3\) to be physically realistic requires that the three quantities \(E^s_3\), \(\phi (E^s_3)\), and \(\frac{1}{E^s_3}1\), are positive. As for \(A^+_3\), the cubic root solution for \(E^s_3\) is unilluminating, but we note that (44) always has at least one real root which need not be positive for all values of the parameters.
The third (\(L\)) component of \(g(A^s_3)\) from (30), which vanishes because \(A^s_3\) is a steady state, gives
with which the admissibility condition can be written as
in terms of the cubic root \(E^s_3\).
Conditions for \(A^s_4\)
The condition for \(A^s_4\) to be physically realistic is simply \(E_4>0\), which gives (65). For the admissibility condition, note that using the second (\(E\)) component of \(g(A^s_4)\), which vanishes since \(A^s_4\) is a steady state, we can write \(E_4{\varOmega }=\frac{\rho _5F}{1+F}=\frac{\rho _1\rho _5}{\rho _1+\rho _4}\), and therefore admissibility requires \(0\le \frac{\rho _1\rho _5}{\rho _1+\rho _4}\le 1\), which rearranges to
Appendix C: Stability of the plane \(\Pi ^{}\)
The stability of the plane \(\Pi ^{}\) (as defined in Eq. (46) is of some importance being a distributed object in the regions \(G^\) and \(\hat{\varSigma }\). It is also rather more simple to express, requiring the calculation of stability in only two directions orthogonal to each other and to the plane.
To determine whether \(\Pi ^{}\) is an attractor we first take coordinates \(u_{1}=E\), \(u_{2}=F\rho ^*L\), \(u_{3}=\rho ^*F+L\), \(u_{4}=R\), where \(\rho ^*=\frac{\rho _{2}}{\rho _{3}\rho _{4}}\). The \(u_i\) form an orthogonal coordinate system, since \(\nabla u_{i}\cdot \nabla u_{j}=0\) for all \(i\ne j\in (1,2,3,4)\) with \(\nabla =\left( \frac{d}{dF},\frac{d}{dE},\frac{d}{dL},\frac{d}{dR}\right) \). The \(u_1\) and \(u_2\) coordinate axes lie perpendicular to \(\Pi ^{}\) (so \(\Pi ^{}\) is the plane \(u_1=u_2=0\)), while \(u_3\) and \(u_4\) form a coordinate system over the plane \(\Pi ^{}\).
The Jacobian of the \(u_{1}\), \(u_{2}\) system at \(u_{1}=u_{2}=0\) expresses the derivative of the flow through \(\Pi ^{}\). Using \(\dot{u}_{1}=\dot{E}\) and \(\dot{u}_{2}=\dot{F}\rho ^*\dot{L}\), this is given by
with eigenvalues \(\mu _{1}=\rho _{4}\frac{1+3 \rho ^{*2}}{\sqrt{1+\rho ^{*2}}}\) and \(\mu _{2}=\frac{\rho _{5}u_{3}\rho ^{*}}{u_{3}\rho ^{*} + \sqrt{1+\rho ^{*2}}}\). Note that \(\mu _{1}<0\) since \(\rho ^{*}>0\). Also \(u_{3}=\rho ^{*} F+L>0\) given that \(F,L>0\), and therefore \(\mu _{2}>0\).
As a result, the plane \(\Pi ^{}\) is of ‘saddle type’, having one stable and one (orthogonal) unstable direction. \(\varPi ^{}\) is therefore not a global attractor.
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Prokopiou, S.A., Byrne, H.M., Jeffrey, M.R. et al. Mathematical analysis of a model for the growth of the bovine corpus luteum. J. Math. Biol. 69, 1515–1546 (2014). https://doi.org/10.1007/s0028501307222
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DOI: https://doi.org/10.1007/s0028501307222