We develop a continuum model for the aggregation of cells cultured in a nutrient-rich medium in a culture well. We consider a 2D geometry, representing a vertical slice through the culture well, and assume that the cell layer depth is small compared with the typical lengthscale of the culture well. We adopt a continuum mechanics approach, treating the cells and culture medium as a two-phase mixture. Specifically, the cells and culture medium are treated as fluids. Additionally, the cell phase can generate forces in response to environmental cues, which include the concentration of a chemoattractant that is produced by the cells within the culture medium. The model leads to a system of coupled nonlinear partial differential equations for the volume fraction and velocity of the cell phase, the culture medium pressure and the chemoattractant concentration, which must be solved subject to appropriate boundary and initial conditions. To gain insight into the system, we consider two model reductions, appropriate when the cell layer depth is thin compared to the typical length scale of the culture well: a (simple) 1D and a (more involved) thin-film extensional flow reduction. By investigating the resulting systems of equations analytically and numerically, we identify conditions under which small amplitude perturbations to a homogeneous steady state (corresponding to a spatially uniform cell distribution) can lead to a spatially varying steady state (pattern formation). Our analysis reveals that the simpler 1D reduction has the same qualitative features as the thin-film extensional flow reduction in the linear and weakly nonlinear regimes, motivating the use of the simpler 1D modelling approach when a qualitative understanding of the system is required. However, the thin-film extensional flow reduction may be more appropriate when detailed quantitative agreement between modelling predictions and experimental data is desired. Furthermore, full numerical simulations of the two model reductions in regions of parameter space when the system is not close to marginal stability reveal significant differences in the evolution of the volume fraction and velocity of the cell phase, and chemoattractant concentration.

中文翻译：

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趋化细胞聚集的多相模型中的模式形成。

我们开发了一个连续模型，用于在培养孔中的营养丰富的培养基中培养的细胞聚集。我们考虑一个二维几何图形，它代表通过培养孔的垂直切片，并假设与培养孔的典型长度尺度相比，细胞层深度较小。我们采用连续力学方法，将细胞和培养基视为两相混合物。具体而言，将细胞和培养基视为流体。另外，细胞相可以响应于环境提示而产生力，该环境提示包括由培养基中的细胞产生的化学吸引剂的浓度。该模型导致了一个耦合非线性偏微分方程组的系统，该方程组包含细胞相的体积分数和速度 培养基压力和化学吸引剂浓度，必须在适当的边界和初始条件下解决。为了深入了解系统，我们考虑了两种模型缩减，当与培养孔的典型长度尺度相比，细胞层深度较薄时比较合适：（简单的）1D和（更复杂的）薄膜扩展流量缩减。通过分析和数值研究所得的方程组，我们确定了条件，在这种条件下，对均质稳态（对应于空间均匀的单元分布）的小幅度扰动会导致空间变化的稳态（图案形成）。我们的分析表明，在线性和弱非线性条件下，较简单的1D缩减与薄膜拉伸流动减少具有相同的定性特征，从而在需要对系统进行定性理解时激发了使用较简单的1D建模方法。但是，当需要模型预测和实验数据之间的详细定量一致时，减少薄膜拉伸流动可能更合适。此外，当系统未接近边际稳定性时，参数空间区域中两个模型缩减的完整数值模拟显示，细胞相的体积分数和速度以及趋化因子浓度的变化存在显着差异。当需要对系统进行定性了解时，鼓励使用更简单的一维建模方法。但是，当需要模型预测和实验数据之间的详细定量一致时，减少薄膜拉伸流动可能更合适。此外，当系统未接近边际稳定性时，参数空间区域中两个模型缩减的完整数值模拟显示，细胞相的体积分数和速度以及趋化因子浓度的变化存在显着差异。当需要对系统进行定性了解时，鼓励使用更简单的一维建模方法。但是，当需要模型预测和实验数据之间的详细定量一致时，减少薄膜拉伸流动可能更合适。此外，当系统未接近边际稳定性时，参数空间区域中两个模型缩减的完整数值模拟显示，细胞相的体积分数和速度以及趋化因子浓度的变化存在显着差异。