Analysis of virotherapy in solid tumor invasion
In this paper, two mathematical models for the dynamics of tumour growth in the presence of oncolytic viruses and immune system cells (cytotoxic T-cells) are developed and analysed. First, the authors present a spatially homogeneous system of ordinary differential equations: \[ \dot x=\sigma-x+\frac{\gamma_1 x z}{\eta_1+z}-\nu xz\quad\text{(cytotoxic T-cells)}, \] \[ \dot y=\alpha_1 y(1-\alpha_2 y)-\frac{\theta_1 zy}{\eta_2+z}\quad\text{(uninfected tumour cells)}, \] \[ \dot z=\beta_1 z(1-\beta_2 z)+\frac{\theta_2 z y}{\eta_2+z}-\mu xz\quad\text{(infected tumour cells)}. \] A linear stability analysis of stationary states is made; in a nutshell, tumour-free steady states are unstable, whereas -- at least in a specific, biologically sensible parameter situation -- a steady state reflecting tumour dormancy is asymptotically stable. Numerical simulations are given. The ODE model is extended to a spatially heterogeneous framework including diffusion of tumour and immune system cells as well as chemotaxis of the latter towards increasing concentrations of a specific signalling substance. It is assumed that cytotoxic T-cells are supplied only in a specific spatial subdomain to invade the cancerous tissue afterwards. The dynamics of the resulting reaction-diffusion system in one spatial dimension are numerically simulated using finite differences. In a simplified setting (e.g. neglecting chemotactic effects), travelling wave solutions are derived analytically.
scientific article; zbMATH DE number 6477283