A unified term for directed and undirected motility in collective cell invasion

In this paper, we develop mathematical models for collective cell motility. Initially we develop a model using a linear diffusion-advection type equation and fit the parameters to data from cell motility assays. This approach is helpful in classifying the results of certain cell motility assay experiments. In particular, this model can determine degrees of directed versus undirected collective cell motility. Next we develop a model using a nonlinear diffusion term that is able to capture in a unified way directed and undirected collective cell motility. One goal of this work is to demonstrate that the forms of collective cell motility seen in the scratch assays and possibly other systems of interest need not reference external and more complicated migratory signals such as chemotaxis, but rather could be based on quorum sensing alone, collectively represented as density-dependent diffusivity. As an application we apply the nonlinear diffusion approach to a problem in tumor cell invasion, noting that neither chemotaxis or haptotaxis are present in the system under consideration in this article.