Exploration of spatial and stochastic effects in a four-state compartmental model of apoptotic cancerous tissue model

Abstract

In this work, we explore the spatial and stochastic effects to a previously established four-state model of cancerous tissue by comparing the results of an equivalent Gillespie stochastic simulation algorithm (SSA) of the system. An exploration of the model parameters for the equivalent probabilistic cellular automata (pCA, also known as stochastic CA) model and the SSA shows ranges of parameters that result in close equivalency between model types. We reduced the parameters from five to two dimensions by normalizing all rates by the apoptotic transition rate u. We used the root mean squared deviation (RMSD) of both the pCA and SSA time series results for each of the four state population fractions to compare them with the ODE counterpart as reference. We identified ranges of model parameters wherein the three modeling approaches diverge due to stochastic and spatial effects. In particular, the stochasticity incorporated in the SSA allows the system to explore unique states of the system that are not possible with the ODE approach. Meanwhile, spatial effects incorporate dynamic boundary effects that isolate cellular states, resulting in non-accessible system states. This work can provide further insights and interests as to which effects can be significant in modeling biological systems such as the one presented herein and can help choose which modeling technique is appropriate for the given dynamical questions.