Dynamics and Pattern Formation in Invasive Tumor Growth

We study the in vitro dynamics of the malignant brain tumor glioblastoma multiforme. The growing tumor consists of a dense proliferating zone and an outer less dense invasive region. Experiments with different types of cells show qualitatively different behavior: one cell line invades in a spherically symmetric manner, but another gives rise to branches. We formulate a model for this sort of growth using two coupled reaction-diffusion equations for the cell and nutrient concentrations. When the ratio of the nutrient and cell diffusion coefficients exceeds some critical value, the plane propagating front becomes unstable with respect to transversal perturbations. The instability threshold and the full phase-plane diagram in the parameter space are determined. The results are in a qualitative agreement with experimental findings for the two types of cells.

Figure 1

Growing tumor spheroids from in vitro experiments [5] in collagen gel for the wild-type (a) and mutant (b) cells. These in vitro cell clusters consist of an inner proliferation zone with a very high density of cells and an outer invasive zone, where the cell density is smaller. The structure in vivo is believed to be similar. The radius of the inner zone here is about $250\mu$. (a) For wild-type cells a spherically symmetric pattern is observed. (b) Mutant cells are organized in branches. Note also that the invasive region for the mutant-type cells grows slower than for wild-type cells.

Figure 2

Density profiles of cells and nutrient from Eq. (3). Cell density $u=u(\xi )$, solid curve. Inset: nutrient concentration, $c=c(\xi )$, dashed curve. Parameters are $m=1$, $\delta =100$.

Figure 3

Dispersion curve $\gamma (k)$. The instability occurs if the ratio of diffusion coefficients $\delta >{\delta }_{\mathrm{cr}}\approx 2.300$. The growth rate $\gamma$ is positive for small $k$; for larger $k$, cell diffusion in the transverse direction stabilizes the instability. Parameters are: $m=0.1$, $\delta =20$, $p=1$ (the dashed line), and $p=0.3$ (the solid line). Inset: dependence of the largest unstable wave number $k_{*}$ on $ϵ=(\delta -{\delta }_{\mathrm{cr}})/{\delta }_{\mathrm{cr}}$ for $p=1$. Numerical simulations are the asterisks, the asymptote $k_{*}=(0.36/m){ϵ}^{1/2}$ is the dotted line.

Figure 4

Gray scale representation of the cells density for isotropic ($p=1$, the right panel) and anisotropic ($p=0.3$, the left and center panels) diffusion, computed numerically from Eq. (2). Dark color corresponds to smaller density, and light to larger density. The initial conditions are in the form of a radial step function with large amplitude random azimuthal perturbations. On the left $\delta =20$, $m=0.25$, $t=1.2$, and radial growth is stable. The central and right panels correspond to $\delta =200$, $m=0.05$, which we expect to be in the unstable regime. The anisotropic case (the central panel, $t=1.2$) is more unstable and the developing branches are much thinner than for isotropic case (the right panel, $t=1.95$).

Figure 5

Phase plane ($\delta ,m$) with two families of curves: $v=\mathrm{const}$ and $k_{*}=\mathrm{const}$, calculated from Eqs. (3, 4) for an isotropic case $p=1$. The two curves marked by ○’s are $v=\mathrm{const}$; $v=0.1$ (left), and $v=0.02$ (right). The two curves marked by *’s are $k_{*}=\mathrm{const}$. The left-hand curve corresponds to a stronger instability, $k_{*}=10$, while the right one corresponds to a weaker instability, $k_{*}=1$. Also shown are the large $\delta$ asymptotes: $\delta =\mathrm{const}/m$ for $v=\mathrm{const}$ curves, and $m=\mathrm{const}$ for $k_{*}=\mathrm{const}$ curves, see text. The instability threshold for an infinite stripe, $\delta \approx 2.300$, is the dotted line. The large-$\delta$ and small-$m$ region corresponds to wild-type cells, while mutant cells are in the region of smaller $\delta$ and larger $m$. This suggests that wild-type cells have a larger $D_u$ and a smaller $\alpha$ compared to mutant cells.