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A Diffuse Interface Framework for Modeling the Evolution of Multi-cell Aggregates as a Soft Packing Problem Driven by the Growth and Division of Cells

  • Special Issue: Multiscale Modeling of Tissue Growth and Shape
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Abstract

We present a model for cell growth, division and packing under soft constraints that arise from the deformability of the cells as well as of a membrane that encloses them. Our treatment falls within the framework of diffuse interface methods, under which each cell is represented by a scalar phase field and the zero level set of the phase field represents the cell membrane. One crucial element in the treatment is the definition of a free energy density function that penalizes cell overlap, thus giving rise to a simple model of cell–cell contact. In order to properly represent cell packing and the associated free energy, we include a simplified representation of the anisotropic mechanical response of the underlying cytoskeleton and cell membrane through penalization of the cell shape change. Numerical examples demonstrate the evolution of multi-cell clusters and of the total free energy of the clusters as a consequence of growth, division and packing.

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Notes

  1. In some simulations, a buffer zone, \(\Omega ^{'}\), is needed around the simulation domain to inhibit unrealistic cell shapes resulting from the enforcement of \(\kappa {\varvec{\nabla }} c_k \cdot {\varvec{n}}= 0\) on \(\partial \Omega \). In addition, this buffer zone also acts as a membrane around the cell cluster. In the buffer zone, an additional term of the form \(\sum _{k=1}^{N}\lambda c_k^2\) is added to the free energy density to penalize the movement of any cells from the active simulation domain (\(\Omega \)) to the buffer zone (\(\Omega ^{'}\)).

  2. In this initial boundary value problem, \(\partial \Omega \) is composed of only a Neumann (flux) boundary (\(\partial \Omega ^{h} = \partial \Omega \)) and hence the Dirichlet boundary is a null set (\(\partial \Omega ^{g} = \varnothing \)).

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Authors and Affiliations

  1. Mechanical Engineering, University of Michigan, Ann Arbor, USA

    J. Jiang

  2. Mechanical Engineering and Mathematics, University of Michigan, Ann Arbor, USA

    K. Garikipati

  3. Mechanical Engineering, University of Wisconsin-Madison, Madison, USA

    S. Rudraraju

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  1. J. Jiang
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  2. K. Garikipati
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  3. S. Rudraraju
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Correspondence to K. Garikipati.

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Jiang, J., Garikipati, K. & Rudraraju, S. A Diffuse Interface Framework for Modeling the Evolution of Multi-cell Aggregates as a Soft Packing Problem Driven by the Growth and Division of Cells. Bull Math Biol 81, 3282–3300 (2019). https://doi.org/10.1007/s11538-019-00577-1

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