In this paper we present an individual-based mechanical model that describes the dynamics of two contiguous cell populations with different proliferative and mechanical characteristics. An off-lattice modelling approach is considered whereby: (i) every cell is identified by the position of its centre; (ii) mechanical interactions between cells are described via generic nonlinear force laws; and (iii) cell proliferation is contact inhibited. We formally show that the continuum counterpart of this discrete model is given by a free-boundary problem for the cell densities. The results of the derivation demonstrate how the parameters of continuum mechanical models of multicellular systems can be related to biophysical cell properties. We prove an existence result for the free-boundary problem and construct travelling-wave solutions. Numerical simulations are performed in the case where the cellular interaction forces are described by the celebrated Johnson-Kendall-Roberts model of elastic contact, which has been previously used to model cell-cell interactions. The results obtained indicate excellent agreement between the simulation results for the individual-based model, the numerical solutions of the corresponding free-boundary problem and the travelling-wave analysis.

中文翻译：

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从基于个体的多细胞系统力学模型到自由边界问题

在本文中，我们提出了一个基于个体的力学模型，该模型描述了具有不同增殖和力学特征的两个连续细胞群的动力学。考虑了一种非晶格建模方法，其中： (i) 每个单元都由其中心的位置识别；(ii) 细胞之间的机械相互作用是通过通用的非线性力定律来描述的；(iii) 细胞增殖受到接触抑制。我们正式表明，这个离散模型的连续体对应物是由单元密度的自由边界问题给出的。推导结果证明了多细胞系统的连续机械模型的参数如何与生物物理细胞特性相关。我们证明了自由边界问题的存在性结果并构造了行波解。数值模拟是在细胞相互作用力由著名的 Johnson-Kendall-Roberts 弹性接触模型描述的情况下进行的，该模型以前曾用于模拟细胞-细胞相互作用。获得的结果表明基于个体的模型的模拟结果、相应自由边界问题的数值解和行波分析之间具有极好的一致性。