A viscoelastic model of blood capillary extension and regression: derivation, analysis, and simulation

Abstract

This work studies a fundamental problem in blood capillary growth: how the cell proliferation or death induces the stress response and the capillary extension or regression. We develop a one-dimensional viscoelastic model of blood capillary extension/regression under nonlinear friction with surroundings, analyze its solution properties, and simulate various growth patterns in angiogenesis. The mathematical model treats the cell density as the growth pressure eliciting a viscoelastic response from the cells, which again induces extension or regression of the capillary. Nonlinear analysis captures two cases when the biologically meaningful solution exists: (1) the cell density decreases from root to tip, which may occur in vessel regression; (2) the cell density is time-independent and is of small variation along the capillary, which may occur in capillary extension without proliferation. The linear analysis with perturbation in cell density due to proliferation or death predicts the global biological solution exists provided the change in cell density is sufficiently slow in time. Examples with blow-ups are captured by numerical approximations and the global solutions are recovered by slow growth processes, which validate the linear analysis theory. Numerical simulations demonstrate this model can reproduce angiogenesis experiments under several biological conditions including blood vessel extension without proliferation and blood vessel regression.

Appendix

1.1 A Numerical method

We consider solving a more general nonlinear problem

$$\begin{aligned} \left\{ \begin{aligned}&\beta \left(1+\frac{\partial u}{\partial x}\right) \frac{\partial u}{\partial t}= \frac{\partial }{\partial x}\left( \frac{\partial u}{\partial x} + \mu \frac{\partial ^2 {u}}{\partial x\partial t} - (f(x,t)-1)\right), \\&u(0,t) = 0, \quad \frac{\partial u}{\partial x} + \mu \frac{\partial ^2 {u}}{\partial x\partial t } - { (f(x,t)-1)} = g(t), \\&u(x,0) = 0. \end{aligned}\right. \end{aligned}$$

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with a finite element method in the Sobolev space $H^1(0,1)$, that is, the square-integrable functions up to the first order weak derivative.

Choose a uniform time step $\Delta t > 0$ and denote the time points when solutions are sought as $t^k=k\Delta t, k=0, 1, \ldots$. The numerical scheme is to find $u^{k+1}(x) \in H^1(0,1)$ with $u^{k+1}(0)=0$, such that for any test function $\phi \,{\in }\, H^1(0,1)$ with $\phi (0)\,{=}\,0$,

$$\begin{aligned}&\int \limits _0^1 \beta \left(1+\frac{\partial u^{k}}{\partial x}\right) \frac{u^{k+1}-u^k}{\Delta t} \phi = \phi (1) g(t^{k+1})\nonumber \\&\quad - \int \limits _0^1 \left(\frac{\partial u^{k+1}}{\partial x} + \mu \frac{\partial }{\partial x } \frac{u^{k+1}-u^k}{\Delta t} - { (f^{k+1}-1)}\right) \phi _x. \end{aligned}$$

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The spatial domain $[0,1]$ is uniformly discretized into $n$ equal-sized sub-intervals with mesh size $h=\frac{1}{n}$, and mesh points are denoted as $x_i=(i-1)h, i=1, \ldots , n+1$. The space $H^1(0,1)$ is approximated by the continuous piecewise linear finite element space:

$$\begin{aligned} V_h(0,1)&= \{ v_h\in C^0(0,1): v_h \text{ is} \text{ a} \text{ linear} \text{ function} \text{ on} \text{ each} \text{ subinterval} [x_i,x_{i+1}], \\&i=1,\ldots , n \}. \end{aligned}$$

Numerical tests show this scheme is first order accurate in time (data not shown). A Matlab version of the code has been provided in the website http://www.cst.cmich.edu/users/zheng1x/ In all the numerical simulations in this work, we have chosen $n=200$ and $\Delta t=10^{-4}$, and each simulation result is almost identical to that with the more refined choice $n=400$ and $\Delta t=5\times 10^{-5}$.