A free boundary model of epithelial dynamics

Journal article

In this work we analyse a one-dimensional, cell-based model of an epithelial sheet. In the model, cells interact with their nearest neighbouring cells and move deterministically. Cells also proliferate stochastically, with the rate of proliferation specified as a function of the cell length. This mechanical model of cell dynamics gives rise to a free boundary problem. We construct a corresponding continuum-limit description where the variables in the continuum limit description are expanded in powers of the small parameter 1/N, where N is the number of cells in the population. By carefully constructing the continuum limit description we obtain a free boundary partial differential equation description governing the density of the cells within the evolving domain, as well as a free boundary condition that governs the evolution of the domain. We show that care must be taken to arrive at a free boundary condition that conserves mass. By comparing averaged realisations of the cell-based model with the numerical solution of the free boundary partial differential equation, we show that the new mass-conserving boundary condition enables the coarse-grained partial differential equation model to provide very accurate predictions of the behaviour of the cell-based model, including both evolution of the cell density, and the position of the free boundary, across a range of interaction potentials and proliferation functions in the cell based model.

Authors: Baker, R; Parker, A; Simpson, M

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Elsevier
• Journal: Journal of Theoretical Biology
• Volume: 481
• Pages: 61-74
• Publication date: 2018-12-19
• Acceptance date: 2018-12-18
• DOI: 10.1016/j.jtbi.2018.12.025
• EISSN: 1095-8541
• ISSN: 0022-5193
• Pmid: 30576691
• Language: English
• Keywords: moving boundary problem; mechanical model; cell proliferation; cell migration; cell-based model; individual-based model; free boundary problem