Accurate and efficient discretisations for stochastic models of cell migration and cell proliferation with crowding

Understanding how cells proliferate, migrate, and die in various environments is essential in determining how organisms develop and repair themselves. Continuum mathematical models, such as the logistic equation and the Fisher-Kolmogorov equation, can describe the global characteristics observed in commonly-used cell biology assays, such as proliferation and scratch assays. However, these continuum models do not account for single-cell-level mechanics observed in high-throughput experiments. Mathematical modelling frameworks that represent individual cells, often called agent-based models, can successfully describe key single-cell-level features of these assays, but are computationally infeasible when dealing with large populations. In this work, we propose an agent-based model with crowding effects that is computationally efficient and matches the logistic and Fisher-Kolmogorov equations in parameter regimes relevant to proliferation and scratch assays, respectively. This stochastic agent-based model allows multiple agents to be contained within compartments on an underlying lattice, thereby reducing the computational storage compared to existing agent-based models that allow one agent per site only. We propose a systematic method to determine a suitable compartment size. Implementing this compartment-based model with this compartment size provides a balance between computational storage, local resolution of agent behaviour, and agreement with classical continuum descriptions.