BioDDFT: A hybrid continuum-discrete mechanical collective cell model
John Lowengrub, UCI


 The regulation of cell division, cell sizes and cell arrangements is central to tissue morphogenesis. To study these processes, we develop a
mechanistic hybrid continuum-discrete mathematical model of cell dynamics that has advantages over previous approaches. This model borrows ideas from
statistical physics, materials science and applied mathematics and follows the framework of dynamic density functional theory. This approach provides
a strategy for coarse-graining systems of stochastically interacting particles. By appropriately accounting for cell size and shape variability,
we obtain a system of continuum equations that are able to capture plastic, viscoelastic and elastic deformations in the clusters while providing
single-cell resolution. The discrete component of the model implements cell division and thus influences cell sizes and shapes that couple to the
continuum equations. We present efficient numerical methods for solving the coupled systems of equations and validate the model using recent in vitro
studies of epithelial cell colonies using Madin-Darby canine kidney cells. In good agreement with the experiments, we find that mechanical
interactions and constraints on the local expansion of cell size cause inhibition of cell motion and reductive cell division. This leads to
successively smaller cells and a transition from exponential to quadratic growth of the colony that is associated with the emergence of short-range
ordering and a constant-thickness proliferating rim of cells at the cluster edge. We then discuss extensions of the model to account for different cell
types, including stem, progenitor and terminally differentiated cells, that are needed to model stratified epithelial tissues. Finally, we describe how
this framework can provide a rational approach for upscaling from the subcell and cell scales to tissue scales to obtain new, tissue-scale
governing equations.