Effect of heterogeneity and spatial correlations on the structure of a tumor invasion front in cellular environments

Analysis of invasion front has been widely used to decipher biological properties, as well as the growth dynamics of the corresponding populations. Likewise, the invasion front of tumors has been investigated, from which insights into the biological mechanisms of tumor growth have been gained. We develop a model to study how tumors' invasion front depends on the relevant properties of a cellular environment. To do so, we develop a model based on a nonlinear reaction-diffusion equation, the Fisher-Kolmogorov-Petrovsky-Piskunov equation, to model tumor growth. Our study aims to understand how heterogeneity in the cellular environment's stiffness, as well as spatial correlations in its morphology, the existence of both of which has been demonstrated by experiments, affects the properties of tumor invasion front. It is demonstrated that three important factors affect the properties of the front, namely the spatial distribution of the local diffusion coefficients, the spatial correlations between them, and the ratio of the cells' duplication rate and their average diffusion coefficient. Analyzing the scaling properties of tumor invasion front computed by solving the governing equation, we show that, contrary to several previous claims, the invasion front of tumors and cancerous cell colonies cannot be described by the well-known models of kinetic growth, such as the Kardar-Parisi-Zhang equation.