Composite Waves for a Cell Population System Modeling Tumor Growth and Invasion

Abstract

In the recent biomechanical theory of cancer growth, solid tumors are considered as liquid-like materials comprising elastic components. In this fluid mechanical view, the expansion ability of a solid tumor into a host tissue is mainly driven by either the cell diffusion constant or the cell division rate, with the latter depending on the local cell density (contact inhibition) or/and on the mechanical stress in the tumor.

For the two by two degenerate parabolic/elliptic reaction-diffusion system that results from this modeling, the authors prove that there are always traveling waves above a minimal speed, and analyse their shapes. They appear to be complex with composite shapes and discontinuities. Several small parameters allow for analytical solutions, and in particular, the incompressible cells limit is very singular and related to the Hele-Shaw equation. These singular traveling waves are recovered numerically.

Project supported by the ANR grant PhysiCancer and the BMBF grant LungSys.

Appendix: Derivation of the Cuboid State Equation

Cells are modelled as cuboidal elastic bodies of dimensions at rest L ₀×l ₀×h ₀ in x,y,z directions aligned in a row in x direction. At rest, the lineic mass density of the row of cells, in contact but not deformed, is $\rho_{0} = \frac{M_{\rm cell}}{L_{0}}$. We consider the case that the cells are confined in a tube of section l ₀×h ₀, where the only possible deformation is along the x axis. This situation can be tested in a direct in-vitro experiment. Moreover, this limit would be expected in case a tumor composed of elastic cells is sufficiently large, such that for the ratio of the cell size L and the radius of curvature R, $\frac{L}{R}\ll 1$ holds, and the cell division is mainly oriented in radial direction as well as the cell-cell tangential friction is sufficiently small, such that a fingering or buckling instability does not occur.

When cells are deformed, we assume that stress and deformation are uniformly distributed, and that the displacements are small. Let L be the size of the cells. The lineic mass density is $\rho = \frac{\rho_{0}L_{0}}{L}$. For ρ<ρ ₀, the cells are not in contact and Σ(ρ)=0; for ρ≥ρ ₀, a variation dL of the size L of the cell corresponds to an infinitesimal strain $\mathrm{d}u = \frac{\mathrm{d}L}{L}$. Therefore, the strain for a cell of size L is $u = \ln(\frac{L}{L_{0}})$. Assuming that a cell is a linear elastic body with Young modulus E and Poisson ratio ν, one finds that the component σ _xx of the stress tensor can be written as

$$\sigma_{xx} = -\frac{1-\nu}{(1-2\nu)(1+\nu)}E\ln \biggl(\frac{\rho}{\rho_0} \biggr). $$

The state equation is given by

$$\varSigma(\rho) = \left\lbrace \begin{array}{l@{\quad}l} 0, & \text{if}\ \rho \leq \rho_0,\\ {\frac{1-\nu}{(1-2\nu)(1+\nu)}E\ln(\frac{\rho}{\rho_0})}, & \text{otherwise.} \end{array} \right. $$

Here, Σ(ρ)=−σ _xx is the pressure. Let $\overline{\rho} = \frac{\rho}{\rho_{0}}$, $\overline{\varSigma} = \frac{\varSigma}{E_{0}}$ and $\overline{E}=\frac{E}{E_{0}}$ be the dimensionless density, pressure and Young modulus respectively, with E ₀ a reference Young modulus. Then the state equation can be written as

$$\overline{\varSigma}(\overline{\rho}) = \left\lbrace \begin{array}{l@{\quad}l} 0, &\text{if}\ \overline{\rho} \leq 1,\\ C_\nu\ln(\overline{\rho}),&\text{otherwise}, \end{array} \right. $$

where $C_{\nu}= {\frac{\overline{E}(1-\nu)}{(1-2\nu)(1+\nu)}}$. In the article, equations are written in the dimensionless form, and the bars above dimensionless quantities are removed.