Incorporating spatial effects to a four-state compartmental model of apoptotic cancerous tissue

Abstract

Cellular and cancer growth dynamics can be captured by a set of ordinary differential equations (ODEs) but limited capacity to incorporate spatial effects. In this study, we extend the usual compartmental cancer model to a four-state system that incorporates apoptotic stage. The spatial effects are incorporated using a probabilistic cellular automata version of the model on a rectangular lattice of size L x L (L = 20) with outer-totalistic Moore neighborhood imposing a closed system boundary condition. A single cancerous cell is initiated at the center of the lattice at time step t = 0. We explored the effects of varying the rates of carcinogenesis and apoptosis described by the model parameters mutation probability (p), apoptosis probability (r), and rate of necrosis from apoptotic state (u). Both models show convergent steady-states. The spatial factor is found to induce oscillatory behavior in the system. An increase in the apoptotic rate (r) is found to modulate cancer growth in the system. The current model can already provide dynamical characteristics of cancer growth in 2D identifying stages that provide optimal control. The model can be extended to a more realistic surface cancer growth and readily to 3D versions of the same.