Speed Switch in Glioblastoma Growth Rate due to Enhanced Hypoxia-Induced Migration

Abstract

We analyze the wave speed of the Proliferation Invasion Hypoxia Necrosis Angiogenesis (PIHNA) model that was previously created and applied to simulate the growth and spread of glioblastoma (GBM), a particularly aggressive primary brain tumor. We extend the PIHNA model by allowing for different hypoxic and normoxic cell migration rates and study the impact of these differences on the wave-speed dynamics. Through this analysis, we find key variables that drive the outward growth of the simulated GBM. We find a minimum tumor wave-speed for the model; this depends on the migration and proliferation rates of the normoxic cells and is achieved under certain conditions on the migration rates of the normoxic and hypoxic cells. If the hypoxic cell migration rate is greater than the normoxic cell migration rate above a threshold, the wave speed increases above the predicted minimum. This increase in wave speed is explored through an eigenvalue and eigenvector analysis of the linearized PIHNA model, which yields an expression for this threshold. The PIHNA model suggests that an inherently faster-diffusing hypoxic cell population can drive the outward growth of a GBM as a whole, and that this effect is more prominent for faster-proliferating tumors that recover relatively slowly from a hypoxic phenotype. The findings presented here act as a first step in enabling patient-specific calibration of the PIHNA model.

Appendix

As discussed in the main body of this work, we wanted to show that our PIHNA simulations were following different eigenvalues depending on the value of $D_h/D_c$. In the linearized regime ahead of the wave, we expect that ${\hat{c}} = {\bar{c}}\exp (\lambda (x-st))$, such that

$$\begin{aligned} \frac{\partial {(\log (c))}}{\partial {r}} \sim \lambda . \end{aligned}$$

(26)

In Fig. 6, we present simulations at a T2 radius of 30cm. We see for the simulation with $D_h/D_c = 0.1$, $\partial {(\log (c))}/\partial {r}$ follows $\lambda _1$ ahead of the traveling wave, whereas for $D_h/D_c = 10$, $\partial {(\log (c))}/\partial {r}$ follows $\lambda _4$.

Fig. 6

The gradient of the $\log$ of the normoxic cells is plotted for a T2 radius of 30 cm. As described in the main text, the leading edge of this simulated gradient (ignoring boundary effects present close to the edge of the domain) should follow the eigenvalue that controls the dynamics of the PIHNA model. Simulations presented here correspond with those presented in Fig. 3. The simulation with $D_h/D_c = 0.1$ agrees more closely with $\lambda _1$, whereas the simulation with $D_h/D_c = 10$ follows $\lambda _4$. The results of these support the eigenvalue and eigenvector analysis in the main body of this work