Stochastic and Deterministic Modeling of Cell Migration

Abstract Mathematical models are vital interpretive and predictive tools used to assist in the understanding of cell migration. There are typically two approaches to modeling cell migration: either microscale, discrete or macroscale, continuum. The discrete approach, using agent-based models (ABMs), is typically stochastic and accounts for properties at the cell-scale. Conversely, the continuum approach, in which cell density is often modeled as a system of deterministic partial differential equations (PDEs), provides a global description of the migration at the population level. Deterministic models have the advantage that they are generally more amenable to mathematical analysis. They can lead to significant insights for situations in which the system comprises a large number of cells, at which point simulating a stochastic ABM becomes computationally expensive. However, finding an appropriate continuum model to describe the collective behavior of a system of individual cells can be a difficult task. Deterministic models are often specified on a phenomenological basis, which reduces their predictive power. Stochastic ABMs have advantages over their deterministic continuum counterparts. In particular, ABMs can represent individual-level behaviors (such as cell proliferation and cell–cell interaction) appropriately and are amenable to direct parameterization using experimental data. It is essential, therefore, to establish direct connections between stochastic microscale behaviors and deterministic macroscale dynamics. In this chapter we describe how, in some situations, these two distinct modeling approaches can be unified into a discrete-continuum equivalence framework. We carry out detailed examinations of a range of fundamental models of cell movement in one dimension. We then extend the discussion to more general models, which focus on incorporating other important factors that affect the migration of cells including cell proliferation and cell–cell interactions. We provide an overview of some of the more recent advances in this field and we point out some of the relevant questions that remain unanswered.