Tumor boundary instability induced by nutrient consumption and supply

Abstract

We investigate the tumor boundary instability induced by nutrient consumption and supply based on a Hele-Shaw model. The model describes the geometric evolution of the tumor region and is derived from taking the incompressible limit of a cell density model, which leads to the pressure vanishing on the free boundary. We investigate the boundary behaviors under two different nutrient supply regimes, in vitro and in vivo, where in the former case the supply is adequate and in the latter case the supply is deficient. Our main conclusion is that by investigating typical solutions of the tumor-nutrient model with the asymptotic analysis with respect to the domain perturbation, the tumor boundary in the in vitro regime is shown to be stable regardless of the nutrient consumption rate. However, boundary instability occurs when the nutrient consumption rate exceeds a certain threshold in the in vivo regime, and the bifurcation threshold has a monotonic dependence on the frequency of the domain perturbation.

Appendix A. Properties of Bessel functions

Since the solutions of the radial case are presented in terms of the second kind modified Bessel functions $I_n(r)$ and $K_n(r)$ (for $n\in {\mathbb {N}}$), we review some basic properties of them in this section. Firstly, $I_n(r)$ and $K_n(r)$ solve the differential equation:

$$\begin{aligned} r^2\frac{\textrm{d}^2f}{\textrm{d} r^2}+r\frac{\textrm{d} f}{\textrm{d} r}-(r^2+n^2)f=0, \end{aligned}$$

(120)

and are strict positive for any $n\in {\mathbb {N}}$ and $r>0$. For the derivatives, we have $I_0'(r)=I_1(r)>0$ and $K_0'(r)=-K_1(r)<0$, and for $n\geqslant 1$:

$$\begin{aligned} I_n'(r)&=\frac{I_{n-1}(r)+I_{n+1}(r)}{2}>0, \end{aligned}$$

(121a)

$$\begin{aligned} K_n'(r)&=-\frac{K_{n-1}(r)+K_{n+1}(r)}{2}<0. \end{aligned}$$

(121b)

Therefore, $I_j(r)$ are monotone increasing functions and $K_j(r)$ are monotone decreasing functions. And the Bessel function $I_n(r)$ satisfies

$$\begin{aligned} I_n'(r)-\frac{n}{r}I_n(r)=I_{n+1}(r), \end{aligned}$$

(122)

for any $n\in {\mathbb {N}}^+$.

When $r\rightarrow 0$, $I_n(r)$ and $K_n(r)$ possess the following asymptotes:

$$\begin{aligned}&I_n(r)\sim \frac{1}{\Gamma (n+1)}\left( \frac{r}{2}\right) ^n,\qquad \text {for}\quad n\in {\mathbb {N}}, \end{aligned}$$

(123a)

$$\begin{aligned}&K_n(r)\sim \frac{\Gamma (n)}{2}\left( \frac{r}{2}\right) ^{-n},\quad \qquad \text {for}\quad n\in {\mathbb {N}}^+, \end{aligned}$$

(123b)

$$\begin{aligned}&K_0(r)\sim -\ln {r}. \end{aligned}$$

(123c)

While as $r\rightarrow +\infty$, $I_n(r)$ and $K_n(r)$ have the asymptotes:

$$\begin{aligned}&I_n(r)\sim \left( \frac{1}{2\pi r}\right) ^{1/2}e^{r}\left( 1-\frac{4n^2-1}{8r}+\frac{(4n^2-1)(4n^2-9)}{128 n^2}+O(\frac{1}{r^3})\right) , \end{aligned}$$

(124a)

$$\begin{aligned}&K_n(r)\sim \left( \frac{\pi }{2r}\right) ^{1/2}e^{-r}\left( 1+\frac{4n^2-1}{8r}+\frac{(4n^2-1)(4n^2-9)}{128 r^2}+O(\frac{1}{r^3})\right) . \end{aligned}$$

(124b)

By using (121), we can also derive the asymptotes for $I'(r)$ and $K'(r)$ for $r\rightarrow +\infty$:

$$\begin{aligned}&I_n'(r)\sim \left( \frac{1}{2\pi r}\right) ^{1/2}e^{r}\left( 1-\frac{4n^2+3}{8r}+\frac{(4n^2-1)(4n^2+15)}{128 r^2}+O(\frac{1}{r^3})\right) , \end{aligned}$$

(125a)

$$\begin{aligned}&K_n'(r)\sim -\left( \frac{1}{2\pi r}\right) ^{1/2}e^{-r}\left( 1+\frac{4n^2+3}{8r}+\frac{(4n^2-1)(4n^2+15)}{128 r^2}+O(\frac{1}{r^3})\right) . \end{aligned}$$

(125b)

Furthermore, $I_n(r)$ and $K_n(r)$ satisfy the so-called Wronskians cross-product:

$$\begin{aligned} I_n(r)K_{n+1}(r)+I_{n+1}(r)K_n(r)=\frac{1}{r}\quad \text {for}\quad \forall n\in {\mathbb {N}}. \end{aligned}$$

(126)