A Mathematical Model of the Growth of Uterine Myomas

Abstract

Uterine myomas or fibroids are common, benign smooth muscle tumours that can grow to 10  cm or more in diameter and are routinely removed surgically. They are typically slow- growing, well-vascularised, spherical tumours that, on a macro-scale, are a structurally uniform, hard elastic material. We present a multi-phase mathematical model of a fully vascularised myoma growing within a surrounding elastic tissue. Adopting a continuum approach, the model assumes the conservation of mass and momentum of four phases, namely cells/collagen, extracellular fluid, arterial and venous phases. The cell/collagen phase is treated as a poro-elastic material, based on a linear stress–strain relationship, and Darcy’s law is applied to describe flow in the extracellular fluid and the two vascular phases. The supply of extracellular fluid is dependent on the capillary flow rate and mean capillary pressure expressed in terms of the arterial and venous pressures. Cell growth and division is limited to the myoma domain and dependent on the local stress in the material. The resulting model consists of a system of nonlinear partial differential equations with two moving boundaries. Numerical solutions of the model successfully reproduce qualitatively the clinically observed three-phase “fast–slow–fast” growth profile that is typical for myomas. The results suggest that this growth profile requires stress-induced resistance to growth by the surrounding tissue and a switch-like cell growth response to stress. Analysis of large-time solutions reveal that while there is a functioning vasculature throughout the myoma, exponential growth results, otherwise power-law growth is predicted. An extensive survey of the effect of parameters on model solutions is also presented, and in particular, the enhanced growth caused by factors such as oestrogen is predicted by the model.

Appendices

Appendix 1: Vascular Versus Passive-Influx in Myomas

By setting $S=1$, the bifurcation plots in Fig. 8 are generated by solving

$$\begin{aligned} \frac{1}{r^2}\frac{\partial }{\partial r}\left( r^2 \frac{\partial p_w}{\partial r}\right) - \frac{\alpha _1}{\phi _w\,k_w} \,\frac{R_\infty \,\sinh (\delta \,r)}{r\,\sinh (\delta \,R_\infty )}\,p_w = \left\{ \begin{array}{ccc} \displaystyle {\frac{1}{k_w}\left( \frac{1}{\phi _w}+\frac{1}{\phi _s}\right) } &{} ~~&{} r \!<\! R,\\ 0 &{}&{} R\!<\!r\!<\!R_\infty , \end{array}\right. \end{aligned}$$

using (37) (with $\psi (p_a,p_v)=0$ using the values in Table 2) and (42) for $p_a-p_v$, and the boundary conditions

$$\begin{aligned} \begin{array}{lll} r \,=\, 0 &{}~~~&{}\displaystyle \frac{\partial p_w^{(1)}}{\partial r} ~=~ 0, ~~~~~\qquad p_w^{(1)} = -\frac{\sinh (\delta \,R_\infty )}{\alpha _1\,\delta \,R_\infty },\\ r\,=\, R &{} &{}\quad \displaystyle p_w^{(1)} \,=\, p_w^{(2)}, ~~~~ \frac{\partial p_w^{(1)}}{\partial r} \,=\, \frac{\partial p_w^{(2)}}{\partial r}, \\ r\,=\,R_\infty &{}&{}\quad p_w^{(2)}\,=\,0, \end{array} \end{aligned}$$

where the second condition at $r=0$ results from $p_w=1/(\alpha _1(p_a-p_v))$ and (42) in the limit $r\rightarrow 0$ and the second condition at $r=R$ follows from (30) and (38) with uniform $\phi _w$ and $k_w$. This is a fourth-order ODE system with five boundary conditions, and hence, there is a free parameter to be determined as part of the solution. This system was solved using the MATLAB boundary value solver bvp4c and the Newton-Raphson method was used to iteratively determine the free parameter. A simple continuation procedure was employed to complete the curves shown in Fig. 8.

Appendix 2: Large-Time Analysis of Intermediate Growth Phase

The numerical results show that up to $t=25$, there are three apparently distinct phases of growth, an initial accelerating phase in which the small myoma grows with little resistance from the outer tissue, an intermediate phase of near linear growth in which the stiffness of outer material slows growth, and a final (transitory) acceleration phase in which the outer tissue becomes thin (due to volume conservation) and resistance to growth becomes negligible. Throughout these phases, the vascular pressure difference, $p_a-p_v$, is $O(1)$; when the myoma becomes large, $p_a-p_v \rightarrow 0$ over most of the myoma causing growth to eventually slow down.

To analyse the second, intermediate growth phase, we assume that $p_a-p_v\equiv 1$ throughout the domain and we assume that the outer tissue extends to infinity. The latter assumption reflects the relative large volume difference between the relatively small myoma and the outer tissue during this growth phase. With these assumptions, the large-time solutions (equivalently, as $R\rightarrow \infty$) are exponential and we write

$$\begin{aligned} \frac{\dot{R}}{R} ~\sim ~ \Theta , \end{aligned}$$

(55)

as $R\rightarrow \infty$; typically $\Theta \ll 1$ in the biologically relevant case. It will be shown in Appendix Sect. “Outer Region, $\rho = O(1)$” that

$$\begin{aligned} \Theta ~\sim ~ \frac{S(\sigma _{rr}^*)}{3\,\phi _s^{(1)}}, \end{aligned}$$

(56)

where $\sigma _{rr}^*$ is the solution to the implicit Eq. (71). To facilitate the analysis we rescale the variables as follows

$$\begin{aligned} r ~=~ R\,\rho , ~~~~ v_s ~=~ \dot{R}\, \hat{v}_s, ~~~~ v_w ~=~ \dot{R}\, \hat{v}_w, ~~~~ p_w ~=~ \hat{p}_w, ~~~~ \sigma _{rr} ~=~ \hat{\sigma }_{rr}, ~~~~ {\overline{\sigma }} ~=~ \hat{{\overline{\sigma }}}, \end{aligned}$$

so that the equations become

$$\begin{aligned} \frac{1}{\rho ^2}\frac{\partial (\rho ^2\hat{v}_s)}{\partial \rho }&= \frac{\hat{S}}{\Theta \,\phi _s}, \end{aligned}$$

(57)

$$\begin{aligned} \frac{1}{\rho ^2}\frac{\partial (\rho ^2\hat{v}_w)}{\partial \rho }&= -\frac{\hat{S}}{\Theta \,\phi _w} \,-\, \frac{\alpha _1}{\Theta \,\phi _w} \,\hat{p}_w, \end{aligned}$$

(58)

$$\begin{aligned} \hat{v}_w \,-\, \hat{v}_s&= -\frac{k_w}{\Theta \,R^2}\,\frac{\partial \hat{p}_w}{\partial \rho }, \end{aligned}$$

(59)

$$\begin{aligned} \frac{\partial \hat{\sigma }_{rr}}{\partial \rho } \,+\, \frac{2}{\rho }\hat{{\overline{\sigma }}}&= \frac{\phi _w}{\phi _s}\,\frac{\partial \hat{p}_{w}}{\partial \rho },\end{aligned}$$

(60)

$$\begin{aligned} \frac{1}{\Theta }\frac{\partial \hat{{\overline{\sigma }}}}{\partial t} \,+\, (\hat{v}_s-\rho ) \frac{\partial \hat{{\overline{\sigma }}}}{\partial \rho }&= 2\,\mu \, \left( \frac{\partial \hat{v}_s}{\partial \rho }\,-\, \frac{\hat{v}_s}{\rho }\right) , \end{aligned}$$

(61)

where $\hat{S}=S(\hat{\sigma }_{kk}/3)$ and $\hat{\sigma }_{kk}=3\hat{\sigma }_{rr}-2\hat{{\overline{\sigma }}}$. From the numerical simulations, it turns out that within the myoma $\overline{\sigma } \ll 1$ over the entire region except in a boundary layer region at $\rho =1$; in fact this is true when non-zero initial conditions are imposed on $\overline{\sigma }(\rho ,0)$ as long as $\overline{\sigma }(0,0)=0$ (a demonstration of this is given by the linear stability analysis of Appendix Sect. “Linear Stability Analysis of Outer Region Solution”). It is useful to combine Eqs. (57–59) to obtain

$$\begin{aligned} \frac{1}{R^2}\frac{1}{\rho ^2}\frac{\partial }{\partial \rho }\left( \rho ^2 \frac{\partial \hat{p}_w}{\partial \rho }\right) \,-\, \gamma ^2 \,\hat{p}_w ~=~ \frac{\hat{S}}{k_w}\left( \frac{1}{\phi _s}+\frac{1}{\phi _w}\right) , \end{aligned}$$

(62)

where $\gamma ^2 = \alpha _1/k_w \phi _w$. For simplicity, we assume $\phi _w^{(1)}=\phi _w^{(2)}=\phi _w$ as was adopted in the simulations, and we subject these equations to the following

$$\begin{aligned} \left. \begin{array}{ccl} \rho = 0 &{}~~~&{} \hat{v}_s^{(1)} = 0, ~~ \hat{v}_w^{(1)} = 0,\\ \rho = 1 &{}&{} \hat{v}_s^{(1)} = \hat{v}_s^{(2)} ~=~ 1, ~~ \hat{v}_w^{(1)} = \hat{v}_w^{(2)},~~ \hat{p}_w^{(1)} = \hat{p}_w^{(2)}, ~~ \phi _s^{(1)}\hat{\sigma }_{rr}^{(1)} = \phi _s^{(2)}\hat{\sigma }_{rr}^{(2)},\nonumber \\ \rho = \infty &{}&{} \hat{p}_w^{(2)} = 0, ~~ \hat{\sigma }_{rr}^{(2)} =0. \end{array}~~~~ \right\} \\ \end{aligned}$$

(63)

The analysis for $\phi _w^{(1)}\ne \phi _w^{(2)}$ follows the same lines, but the water pressure $p_w$ scales with $R$, as opposed to being $O(1)$ when $\phi _w^{(1)}= \phi _w^{(2)}$, where it can be shown that $p_w(1,t) \sim R(t) \Theta (\phi _w^{(1)}-\phi _w^{(2)})/k_w\gamma (\phi _w^{(1)}+\phi _w^{(2)})$ as $R(t)\rightarrow \infty$.

1.1 $\rho >1$

With $\hat{S}=0$, the equations can be solved for the tissue region, giving

$$\begin{aligned} \hat{v}_s^{(2)}&= \frac{1}{\rho ^2},\\ \hat{v}_w^{(2)}&= \frac{1}{\rho ^2} \,+\, \frac{P_0\,k_w\,(\gamma R \rho +1)}{R^2\,\Theta \,\rho ^2} e^{-\gamma R(\rho -1)},\\ \hat{p}_w^{(2)}&= \frac{P_0}{\rho } e^{-\gamma R(\rho -1)},\\ \hat{\sigma }_{rr}^{(2)}&= -\frac{4\mu ^{(2)}}{3}\,\text{ dilog } \left( \frac{1}{\rho ^3}-\frac{1}{R^3\rho ^3}\right) \,+\, \frac{P_0\,\phi _w}{\rho \,\phi _s^{(2)}} e^{-\gamma R(\rho -1)}, \end{aligned}$$

and $\hat{{\overline{\sigma }}}=2\mu \ln \left( 1\!-\!1/\rho ^3\!+ \!1/(\rho ^3R^3)\right)$ and recalling $\text{ dilog }(x)=-\int _0^x \ln (1-w)/w\,dw$, and, in particular,

$$\begin{aligned} \left. \begin{array}{ccl} \rho = 1 &{}~~~~&{} \displaystyle \hat{v}_s^{(2)} = 1, ~~ \hat{v}_w = 1+\frac{P_0\,k_w\,(\gamma R +1)}{R^2 \, \Theta },~~ \hat{p}_w^{(2)} = P_0,\\ &{}&{} \displaystyle \hat{\sigma }_{rr}^{(2)}=-\frac{4\mu ^{(2)}}{3}\,\text{ dilog } \left( 1-\frac{1}{R^3}\right) \,+\, \frac{P_0\,\phi _w}{\phi _s^{(2)}}, \end{array} \right\} \end{aligned}$$

(64)

where $P_0$ is a constant of integration.

1.2 $\rho <1$

There are two layers in the limit $R\rightarrow \infty$, an outer layer where $\rho =O(1)$ and an inner layer where $1-\rho =O(1/R)$.

1.2.1 Inner Region, $1-\rho =O(1/R)$

Writing $\rho = 1 - \hat{\rho }/R$, where $\hat{\rho } = O(1)$ as $R\rightarrow \infty$, the inner equations become

$$\begin{aligned} R\,\frac{\partial \hat{v}_s}{\partial \hat{\rho }} \,-\, \frac{2\,\hat{v}_s}{1\!-\!\hat{\rho }/R}&= -\frac{\hat{S}}{\phi _s^{(1)}}, \end{aligned}$$

(65)

$$\begin{aligned} \hat{v}_w\,-\, \hat{v}_s&= \frac{k_w}{\Theta \,R}\,\frac{\partial p_w}{\partial \hat{\rho }}, \end{aligned}$$

(66)

$$\begin{aligned} \frac{\partial ^{2} \hat{p}_w}{\partial \hat{\rho }^2}\,-\,\frac{1}{R}\frac{2}{(1\!-\!\hat{\rho }/R)}\, \frac{\partial p_w}{\partial \hat{\rho }} \,-\, \gamma ^2 \,\hat{p}_w&= \frac{\hat{S}}{k_w}\left( \frac{1}{\phi _s^{(1)}}+ \frac{1}{\phi _w}\right) ,\end{aligned}$$

(67)

$$\begin{aligned} R\,\frac{\partial \hat{\sigma }_{rr}}{\partial \hat{\rho }} \,-\, \frac{2}{1\!-\!\hat{\rho }/R}\hat{\overline{\sigma }}&= R\,\frac{\phi _w}{\phi _s^{(1)}}\,\frac{\partial \hat{p}_{w}}{\partial \hat{\rho }}, \end{aligned}$$

(68)

and we adopt the following expansions

$$\begin{aligned} ~~ \hat{v}_s^{(1)} \sim \hat{v}_{s_0}^{[i]}, ~~ \hat{v}_w^{(1)} \sim \hat{v}_{w_0}^{[i]}, ~~ \hat{p}_w^{(1)} \sim \hat{p}_{w_0}^{[i]}, ~~ \hat{\sigma }_{rr}^{(1)} \sim \hat{\sigma }_{rr_0}^{[i]}, ~~ \hat{\overline{\sigma }} \sim \hat{\overline{\sigma }}_0^{[i]}, ~~ \Theta \sim \Theta _0. \end{aligned}$$

Imposing the boundary conditions (64) we obtain $\hat{v}_{s_0}^{[i]}=1$, $\hat{v}_{w_0}^{[i]}=1$ and

$$\begin{aligned} \hat{\sigma }_{rr_0}^{[i]} = -\frac{2\, \phi _s^{(2)} \,\mu ^{(2)} \,\pi ^2}{9\, \phi _s^{(1)}} \,+\, \frac{\phi _w}{\phi _s^{(1)}}\,\hat{p}_{w_0}^{[i]}, \end{aligned}$$

(69)

using the fact $\text{ dilog }(1)=\pi ^2/6$, where

$$\begin{aligned} \frac{\partial ^{2} \hat{p}_{w_0}^{[i]}}{\partial \hat{\rho }^2}\,-\, \gamma ^2 \,\hat{p}_{w_0}^{[i]}&= \frac{S\left( \hat{\sigma }_{kk_0}^{[i]}/3\right) }{k_w}\left( \frac{1}{\phi _s^{(1)}}+\frac{1}{\phi _w}\right) , \end{aligned}$$

which cannot be solved analytically for general $S(.)$, but we can deduce that

$$\begin{aligned} \hat{p}_{w_0}^{[i]} ~\sim ~-\,\frac{S\left( \hat{\sigma }_{kk_0}^{[i]}/3\right) }{\gamma ^2\,k_w}\left( \frac{1}{\phi _s^{(1)}}+\frac{1}{\phi _w}\right) , \end{aligned}$$

as $\hat{\rho }\rightarrow \infty$, where $\hat{\sigma }_{kk_0}^{[i]} = 3\hat{\sigma }_{rr_0}^{[i]}-2\hat{\overline{\sigma }}_0^{[i]}$.

1.2.2 Outer Region, $\rho =O(1)$

We assume the following expansions

$$\begin{aligned} \hat{v}_s^{(1)} \sim \hat{v}_{s_0}^{[o]}, ~~ \hat{v}_w^{(1)} \sim \hat{v}_{w_0}^{[o]}, ~~ \hat{p}_w^{(1)} \sim \hat{p}_{w_0}^{[o]}, ~~ \hat{\sigma }_{rr}^{(1)} \sim \hat{\sigma }_{rr_0}^{[o]}, ~~ \Theta \sim \Theta _0, \end{aligned}$$

and adopting the assumption that $\hat{\overline{\sigma }}\ll 1$ (in fact $\overline{\sigma }=O(1/R)$ is sufficient, see Appendix Sect. “Linear Stability Analysis of Outer Region Solution”), then $\hat{\sigma }_{kk_0}^{[o]}/3=\hat{\sigma }_{rr_0}^{[o]}$. At leading order (62) leads to

$$\begin{aligned} \hat{p}_{w_0}^{[o]} ~\sim ~-\,\frac{\phi _w\,S(\hat{\sigma }_{rr_0}^{[o]})}{\alpha _1}\left( \frac{1}{\phi _s^{(1)}}+\frac{1}{\phi _w}\right) , \end{aligned}$$

(70)

whereby matching implies $\hat{\sigma }_{rr_0}^{[o]}=\hat{\sigma }_{kk_0}^{[i]}/3$ as $\rho \rightarrow 1^-$ and $\hat{\rho }\rightarrow \infty$, respectively. Integrating (60), with $\hat{\overline{\sigma }}\ll 1$, and using (69) and (70) leads to the fixed point problem

$$\begin{aligned} \sigma _{rr}^* ~=~ -\frac{2\, \phi _s^{(2)} \,\mu ^{(2)} \,\pi ^2}{9\, \phi _s^{(1)}}~-~\frac{S(\sigma _{rr}^*)\,\phi _w}{\alpha _1\,\phi _s^{(1)}} \left( 1+\frac{\phi _w}{\phi _s^{(1)}}\right) , \end{aligned}$$

(71)

where we have defined $\sigma _{rr}^* = \hat{\sigma }_{rr_0}^{[o]}$ which is a constant; note that since $\hat{S}$ is a monotonically decreasing, continuous function, then this constant is unique. With $\hat{S}$ being constant to leading order, then integrating (57) and from (59) we have $\hat{v}_{s_0}^{[o]}=\hat{v}_{w_0}^{[o]}=\hat{S}(\sigma _{rr}^*)\rho /3\phi _s^{(1)}\Theta _0$, where in order to match with $\hat{v}_{s_0}^{[i]}=\hat{v}_{w_0}^{[i]}=1$ as $\rho \rightarrow 1$ we determine the leading order growth constant as $\Theta _0=\hat{S}(\sigma _{rr}^*)/3\phi _s^{(1)}$, confirming the approximation (56).

A complete analysis requires additional correction terms in powers of $1/R$ and explicit consideration of $\overline{\sigma }$.

1.3 Linear Stability Analysis of Outer Region Solution

To strengthen the claim $\overline{\sigma } \ll 1$ in the outer region, we undertake a linear stability analysis of a reduced system relevant to the problem of Appendix Sect. “Outer Region, $\rho = O(1)$”, under a perturbation of $\overline{\sigma }$ of the form

$$\begin{aligned} \overline{\sigma } ~\sim ~ \varepsilon \,\overline{\Sigma }(\rho )\,e^{\,\omega \,t}, \end{aligned}$$

where $\varepsilon = ||\overline{\sigma }(\rho ,0)||_\infty \,\ll \,1$ (such that $1/R \ll \varepsilon$ is assumed), $\overline{\Sigma }(\rho )$ is a differentiable, initial distribution function (with $\overline{\Sigma }(0)=0$ assumed) and exponent $\omega$ is such that, in general, stability of the unperturbed state requires $\mathfrak {R}(\omega )<0$ for all its solutions. The reduced system is given by (57–62) with $R\rightarrow \infty$, and we expand the other variables as follows

$$\begin{aligned}&\displaystyle \hat{v}_s \sim \rho \,+\, \varepsilon \, V_s(\rho ) \,e^{\omega \,t}, ~~~ \hat{p}_w \sim -\frac{\phi _w S(\sigma _{rr_0}^*)}{\alpha _1}\left( \frac{1}{\phi _s^{(1)}}\!+\!\frac{1}{\phi _w}\right) \,+\,\varepsilon \,P_w(\rho ) \,e^{\omega \,t},&\\&~~ \hat{\sigma }_{rr} \sim \sigma _{rr}^* \,+\, \varepsilon \,\Sigma _{rr}(\rho )\,e^{\omega \,t}, ~~~ \displaystyle \frac{\dot{R}}{R} ~\sim ~ \Theta _0 \,+\, \varepsilon \,\Theta _1 \,e^{\omega \,t},&\end{aligned}$$

with $\hat{v}_w=\hat{v}_s$ to this order and noting that $\hat{\sigma }_{kk}\sim 3\sigma _{rr}^* + \varepsilon \,(3\Sigma _{rr}-2\overline{\Sigma })\,e^{\omega \,t}$. Substituting these expansions into (61) yields on integration at $O(\varepsilon )$,

$$\begin{aligned} V_s(\rho ) ~=~-\rho \,\frac{3\,\omega \,\phi _s^{(1)}}{2\, S(\sigma _{rr}^*)\,\mu ^{(1)}}\,\int _\rho ^1 \frac{\overline{\Sigma }(\hat{\rho })}{\hat{\rho }}\,d\hat{\rho }, \end{aligned}$$

using $V_s(1)=0$ (as $\hat{v}_s(1,t)=1$); we note that the integral is bounded $\forall \rho \in [0,1]$ because $\overline{\Sigma }(0)=0$. The function $\Sigma _{rr}(\rho )$ can now be determined from (57), that leads from (60) and (62) to two formulations for $P_w(\rho )$, which, due to $\overline{\Sigma }(\rho )$ being arbitrary, supplies a consistency condition requiring that $\omega$ takes a unique value, namely

$$\begin{aligned} \omega ~=~-\frac{4\,S'(\sigma _{rr}^*)\,\alpha _1\,\phi _s^{(1)}\, \mu ^{(1)}}{3\,\left( \alpha _1\phi _s^{(1)}\,+\,\phi _w\,S'(\sigma _{rr}^*) (\phi _s^{(1)}+\phi _w)\right) }. \end{aligned}$$

Since $S'(\sigma _{rr}^*)>0$, it follows that $\omega <0$; hence, any small perturbation to $\overline{\sigma }$, satisfying $1/R(t) \ll ||\overline{\sigma }(\rho ,0)||_\infty =O(\varepsilon ) \ll 1$, decays exponentially to at least $O(1/R)$ or smaller.

Appendix 3: Phase 4 Growth Analysis

The leading order growth behaviour during Phase 4 as $t\rightarrow \infty$ can be determined using term balancing arguments. Except for the relatively thin outer rim, growth is approximately described by Eqs. (57–62) with $\alpha _1 = 0$ (i.e. vascular-influx switched off). We suppose $R \sim R_{[4]} \,t^{\chi }$, where $R_{[4]}$ is a constant and $\chi > 0$ is the power-law growth constant to be determined. It immediately follows that $\dot{R}/{R}=\Theta = O(1/t)$ and, since the scalings prescribed in Appendix 2 imply $\hat{v}_s=O(1)$, Eq. (57) implies $S=O(1/t)$ and hence $\sigma _{kk} = O(t^{1/m})$ from (40). Using (59) it follows that $p_w = O(t^{2\chi -1})$ leading to $\sigma _{rr} = O(t^{2\chi -1})$ from (60). Assuming $\sigma _{kk}=O(\sigma _{rr})$, as is supported numerically, then we can deduce $2\chi -1 = 1/m$ and hence,

$$\begin{aligned} R ~\sim ~ R_{[4]}\,t^{(m+1)/2m}, \end{aligned}$$

as $t\rightarrow \infty$.

Using the parameters in Table 2, we expect in the simulations that $R \sim R_{[4]}\,t^{\,61/120}$ in large time. This is indeed the case, but it took $t\approx 50{,}000$ and $R\approx 1{,}300$ for the exponent $(m+1)/2m$ to be within 5 % of the final theoretical value, and $t\approx 90{,}000$ and $R\approx 1{,}550$ to be within 1 %. Faster convergence was observed with larger values of $k_w$.