Stochastic and Deterministic Modelling of Cell Migration

Mathematical models are vital interpretive and predictive tools used to assist in the understanding of cell migration. There are typically two approaches to modelling cell migration: either micro-scale, discrete or macro-scale, 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 modelled 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 behaviour 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 behaviours (such as cell proliferation and cell-cell interaction) appropriately and are amenable to direct parameterisation using experimental data. It is essential, therefore, to establish direct connections between stochastic micro-scale behaviours and deterministic macro-scale dynamics.
In this Chapter we describe how, in some situations, these two distinct modelling 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.