Invasiveness of Cancer Populations in a Two-dimensional Percolation cluster: a Stochastic Mathematical Approach

Abstract

A framework for the software Unstructured Reaction-Diffusion Master Equation (URDME) was developed. A mitogenic paracrine signaling pathway was introduced phenomenologically to show how cells cooperate with one another. We modeled the emerging Allee effect using low seeding density culture (LSDC) assays to fit the model parameters. Finite time scaling (FTS) was found to be a useful tool for quantifying invasiveness in cancer populations. Through simulation, we analyzed the growth-migration dynamics of BT474 cancer cell populations in-vitro in a 2D percolation cluster and calculated the SPR (successful penetration rate). By analyzing the temporal trajectories of the SPR, we could determine the critical exponents of the critical SPR scaling relation $RT^{-\beta}\sim\widetilde f\left(T^{1/\nu}\left(p-p_c\right)\right) $. Moreover, the SPR transition point defined according to the FTS theory, $p_c $, is negatively correlated with the invasiveness of cancer cell populations. The $p_c $ of the three virtual cancer populations distinctly designated by varying the parameter set of the same model are 0.3408, 0.3675, and 0.4454, respectively. Therefore, the FTS algorithm may be useful in determining invasiveness. Through the simplistic phenomenological paracrine model, inter-cell cooperation and mutual mitogenic boosting are enabled, causing the Allee effect to occur. Such a method could be applied to other circumstances as an example of the quantitatively falsifiable emerging theory.