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

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.

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 half-life is $\sim$8 h (Beenken and Mohammadi 2009). Concerning $F_{h}$, the FGF2 concentration at which the EC proliferation rate is half-maximal, 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 mid-cycle, 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 half-maximal, 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 half-maximal 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

$$\begin{aligned} \underline{f}^+(\underline{x})=0\quad and\quad {\varTheta }(\underline{x})>0,\quad or \quad \underline{f}^-(\underline{x})=0\quad and \quad {\varTheta }(\underline{x})<0; \end{aligned}$$

(53)

a point $\underline{x} \in G^{\pm }$ is termed a virtual steady state of (17) if

$$\begin{aligned} \underline{f}^+(\underline{x})=0\,\,\,but\quad {\varTheta }(\underline{x})<0,\quad or\quad \underline{f}^-(\underline{x})=0\quad but\quad {\varTheta }(\underline{x})>0. \end{aligned}$$

(54)

Definition 3

A point $\underline{x}\in D$ is termed a pseudo steady state if

$$\begin{aligned} \underline{g}(\underline{x})=0\quad and\quad {\varTheta }(\underline{x})=0. \end{aligned}$$

(55)

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

$$\begin{aligned} \underline{f}^+(\underline{x})=0\quad and\;\;\;{\varTheta }(\underline{x})=0, \;\;\;or\;\;\;\underline{f}^-(\underline{x})=0\quad and\;\;\;{\varTheta }(\underline{x})=0. \end{aligned}$$

(56)

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

$$\begin{aligned} \frac{\rho _{9}}{\rho _{7}}\le \frac{\rho _{8}}{\rho _{6}}\le 1 \qquad \text{ or }\qquad 1\le \frac{\rho _{8}}{\rho _{6}}\le \frac{\rho _{9}}{\rho _{7}}, \end{aligned}$$

(57)

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 half-maximal 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 $\rho _{6}>0$,

$$\begin{aligned} \phi (E_1)\ge 0 \Leftrightarrow \rho _{5}\rho _{7} + E_{1}(\rho _{5}-\rho _{6})\ge 0 \Leftrightarrow \frac{\rho _{5}}{\rho _{6}} \ge \frac{E_{1}}{\rho _{7} + E_{1}} \Leftrightarrow \rho _{5} \ge \rho _6\frac{\frac{\rho _{9}}{\rho _7} -\frac{\rho _8}{\rho _6}}{\frac{\rho _{9}}{\rho _{7}}-1}\nonumber \\ \end{aligned}$$

(58)

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

$$\begin{aligned} \rho _{5}\le \frac{\rho _{6}(\rho _{4} + \rho _{1}E_{1})}{\rho _{1}(\rho _{7} + E_{1})}, \end{aligned}$$

(59)

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

$$\begin{aligned} \rho _2\left( \nu (E_1)-1\right) ^2-\rho _3\omega (E_1) \left( \nu (E_1)-1\right) \ge \omega (E_1)\ge 0. \end{aligned}$$

(60)

The admissibility condition for $A^+_1$ is simply that the volume $V=V^+_1$ satisfies

$$\begin{aligned} V^+_1=E_1\nu (E_1)=\frac{\rho _6\rho _{9}-\rho _7\rho _8}{\rho _{9}-\rho _7}\ge 1. \end{aligned}$$

(61)

Conditions for $A^+_2$

The components of $A^+_2$ are non-negative if

$$\begin{aligned} E_2=\frac{\rho _{4}\rho _{8}-\rho _{1}\rho _{5}\rho _{9}}{\rho _{1}(\rho _{5}-\rho _{8})}\ge 0,\quad \text{ and } \quad \eta (E_2)=\frac{\rho _{8}}{\rho _{9}+E_2}\ge 1. \end{aligned}$$

(62)

The first inequality supplies the following mutually exclusive set of inequalities:

$$\begin{aligned} (i)\, 1\le \frac{\rho _{5}}{\rho _{8}}\le \frac{\rho _{4}}{\rho _{1}\rho _{9}}, \quad or \quad (ii)\, \frac{\rho _{4}}{\rho _{1}\rho _{9}} \le \frac{\rho _{5}}{\rho _{8}} \le 1. \end{aligned}$$

The second inequality ($\eta (E_{2})\ge 1$) yields:

$$\begin{aligned} \rho _5 \ge \frac{\rho _4}{\rho _1} + \rho _8 - \rho _9. \end{aligned}$$

The admissibility condition for $A^+_2$ is that the volume $V=V^+_2$ satisfies

$$\begin{aligned} V^+_2=E_2\eta (E_2)=\frac{\rho _4\rho _8-\rho _1\rho _{9}\rho _5}{\rho _6(\rho _4-\rho _1\rho _{9})}\ge 1, \end{aligned}$$

(63)

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

$$\begin{aligned} V^+_3=E_3\nu (E_3)=\frac{\rho _6 E_3}{\rho _7+E_3}\ge 1. \end{aligned}$$

(64)

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 $\rho _5$,

$$\begin{aligned} \rho _5\ge \frac{\rho _4}{\rho _1}. \end{aligned}$$

(65)

The admissibility condition is

$$\begin{aligned} V^+_4=\rho _5-\frac{\rho _4}{\rho _1}\ge 1\quad \Leftrightarrow \quad \rho _5\ge \left( 1+\frac{\rho _4}{\rho _1}\right) . \end{aligned}$$

(66)

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

$$\begin{aligned} \rho _2\left( \frac{1}{E_1}-1\right) ^2-\rho _3\omega (E_1) \left( \frac{1}{E_1}-1\right) \ge \omega (E_1)\ge 0. \end{aligned}$$

(67)

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

$$\begin{aligned} E_1{\varOmega }=\frac{\rho _6E_1}{\rho _7+E_1}. \end{aligned}$$

Substituting in $E_1$ from (38), after a little rearranging the admissibility condition becomes

$$\begin{aligned} 0\le \frac{\rho _6\rho _{9}-\rho _7\rho _8}{\rho _{9}-\rho _7}\le 1. \end{aligned}$$

(68)

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

$$\begin{aligned} 0\le \frac{\rho _{4}\rho _{8}-\rho _{1}\rho _{5}\rho _{9}}{\rho _{1} (\rho _{5}-\rho _{8})}\le 1. \end{aligned}$$

(69)

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

$$\begin{aligned} E_2{\varOmega }=\frac{\rho _8E_2}{\rho _{9}+E_2}. \end{aligned}$$

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

$$\begin{aligned} 0\le \frac{\rho _4\rho _8-\rho _1\rho _5\rho _{9}}{\rho _4-\rho _1\rho _{9}} \le 1. \end{aligned}$$

(70)

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

$$\begin{aligned} E^s_3{\varOmega }=\frac{\rho _6E^s_3}{\rho _7+E^s_3}, \end{aligned}$$

with which the admissibility condition can be written as

$$\begin{aligned} 0\le \frac{\rho _6E^s_3}{\rho _7+E^s_3}\le 1, \end{aligned}$$

(71)

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

$$\begin{aligned} 0\le \rho _5\le 1+\frac{\rho _4}{\rho _1}. \end{aligned}$$

(72)

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

$$\begin{aligned} J_{\Pi ^{-}}= \left( \begin{array}{c@{\quad }c@{\quad }c} \frac{\partial \dot{u}_{1}}{\partial u_{1}} &{} \frac{\partial \dot{u}_{1}}{\partial u_{2}} \\ \frac{\partial \dot{u}_{2}}{\partial u_{1}} &{} \frac{\partial \dot{u}_{2}}{\partial u_{2}} \end{array} \right) _{|_{u_{1}=u_{2}=0}}=\left( \begin{array}{cc} \frac{\rho _{5} u_{3} \rho ^{*}}{u_{3} \rho ^{*} + \sqrt{1+\rho ^{*2}}} &{} \quad 0 \\ \rho _{1}-\rho ^{*} \frac{\rho _{4}}{\rho _{3}} - \frac{\rho _{7} \rho ^{*} u_{3}}{\rho _{8}\sqrt{1+\rho ^{*2}}} &{}\quad -\rho _{4}\frac{1+3 \rho ^{*2}}{\sqrt{1+\rho ^{*2}}} \end{array} \right) , \end{aligned}$$

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.