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Mathematical Medicine and Biology Advance Access published online on July 17, 2009
Mathematical Medicine and Biology, doi:10.1093/imammb/dqp015
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© The author 2009. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

Continuum approximations of individual-based models for epithelial monolayers

J. A. Fozard{dagger}, H. M. Byrne, O. E. Jensen and J. R. King

School of Mathematical Sciences, University of Nottingham, University Park, Nottingham NG7 2RD, UK

{dagger} Email: john.fozard{at}maths.nottingham.ac.uk

Received on October 16, 2008. Revised on March 31, 2009. Accepted on May 18, 2009.

This work examines a 1D individual-based model (IBM) for a system of tightly adherent cells, such as an epithelial monolayer. Each cell occupies a bounded region, defined by the location of its endpoints, has both elastic and viscous mechanical properties and is subject to drag generated by adhesion to the substrate. Differential-algebraic equations governing the evolution of the system are obtained from energy considerations. This IBM is then approximated by continuum models (systems of partial differential equations) in the limit of a large number of cells, N, when the cell parameters vary slowly in space or are spatially periodic (and so may be heterogeneous, with substantial variation between adjacent cells). For spatially periodic cell properties with significant cell viscosity, the relationship between the mean cell pressure and length for the continuum model is found to be history dependent. Terms involving convective derivatives, not normally included in continuum tissue models, are identified. The specific problem of the expansion of an aggregate of cells through cell growth (but without division) is considered in detail, including the long-time and slow-growth-rate limits. When the parameters of neighbouring cells vary slowly in space, the O(1/N2) error in the continuum approximation enables this approach to be used even for modest values of N. In the spatially periodic case, the neglected terms are found to be O(1/N). The model is also used to examine the acceleration of a wound edge observed in wound-healing assays.

Keywords: multicellular; homogenization; individual based; continuum; epithelial; monolayer


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