Stochastic simulation tools and continuum models for describing two-dimensional collective cell spreading with universal growth functions

Two-dimensional collective cell migration assays are used to study cancer and tissue repair. These assays involve combined cell migration and cell proliferation processes, both of which are modulated by cell-to-cell crowding. Previous discrete models of collective cell migration assays involve a nearest neighbour proliferation mechanism where crowding effects are incorporated by aborting potential proliferation events if the randomly chosen target site is occupied. There are two limitations of this traditional approach: (i) it seems unreasonable to abort a potential proliferation event based on the occupancy of a single, randomly chosen target site; and, (ii) the continuum limit description of this mechanism leads to the standard logistic growth function, but some experimental evidence suggests that cells do not always proliferate logistically. Motivated by these observations, we introduce a generalised proliferation mechanism which allows non-nearest neighbour proliferation events to take place over a template of $r \ge 1$ concentric rings of lattice sites. Further, the decision to abort potential proliferation events is made using a crowding function, $f(C)$, which accounts for the density of agents within a group of sites rather than dealing with the occupancy of a single randomly chosen site. Analysing the continuum limit description of the stochastic model shows that the standard logistic source term, $\lambda C (1 - C)$,where $\lambda$ is the proliferation rate, is generalised to a universal growth function, $\lambda f(C)$.Comparing the solution of the continuum descriptionwith averaged simulation data indicates that the continuum model performs well for many choices of $f(C)$ and $r$. For nonlinear $f(C)$, the quality of the continuum-discrete match increases with $r$.