A free boundary model of epithelial dynamics. (21st November 2019)

• Record Type: Journal Article
• Authors: Baker, Ruth E; Parker, Andrew; Simpson, Matthew J
• Is Part Of: Journal of theoretical biology. Volume 481(2019)
• Page Start: 61
• Page End: 74
• Publication Date: 2019-11-21
• Subjects: Cell-based model -- Individual-based model -- Mechanical model -- Cell migration -- Cell proliferation -- Free boundary problem -- Moving boundary problem; Biology -- Periodicals; Biological Science Disciplines -- Periodicals; Biology -- Periodicals; Biologie -- Périodiques; Theoretische biologie; Biology; Periodicals; 571.05
• DOI: 10.1016/j.jtbi.2018.12.025
• Languages: English
• ISSNs: 0022-5193

Abstract

Highlights: We analyse a one-dimensional, cell-based model of an epithelial sheet that includes both cell-cell mechanical interactions and proliferation. 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 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. Abstract: 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.