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arXiv 0812.4708q-bio.CB
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肿瘤细胞迁移模型：从微观到宏观

Modeling tumor cell migration: from microscopic to macroscopic

Christophe Deroulers, Marine Aubert, Mathilde Badoual, Basil Grammaticos

备注：
Final published version; 14 pages, 7 figures
发表信息：
Physical Review E: Statistical, Nonlinear, and Soft Matter Physics 79, 3 (2009) 031917
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摘要：

实验表明，接触相互作用可能会影响癌细胞的迁移。以前的工作已经建模，这要归功于随机的，离散的模型（细胞自动机）在细胞水平。然而，对于具有数百万个细胞的真实大小肿瘤的生长的研究，最好使用具有细胞密度的偏微分方程（PDE）形式的宏观模型。困难的是在宏观尺度上预测微观尺度上发生的接触相互作用的效果。为了解决这个问题，我们使用多尺度的方法：从一个非常简单的，但实验验证，微观模型与接触相互作用的迁移，我们推导出一个宏观模型。我们表明，扩散方程的出现，因为是经常假设在该领域的胶质瘤建模，但它是非线性的，因为相互作用。我们给出了明确的细胞密度和细胞间相互作用的参数上的扩散系数的依赖关系。我们详细讨论了在推导中使用的近似的有效性条件，我们比较分析结果从我们的PDE数值模拟和一些体外实验。我们注意到，我们开始的微观模型的家庭包括作为特殊情况下，一些动力学约束模型，介绍了玻璃，过冷液体和干扰系统的物理研究。

英文摘要：

It has been shown experimentally that contact interactions may influence the migration of cancer cells. Previous works have modelized this thanks to stochastic, discrete models (cellular automata) at the cell level. However, for the study of the growth of real-size tumors with several millions of cells, it is best to use a macroscopic model having the form of a partial differential equation (PDE) for the density of cells. The difficulty is to predict the effect, at the macroscopic scale, of contact interactions that take place at the microscopic scale. To address this we use a multiscale approach: starting from a very simple, yet experimentally validated, microscopic model of migration with contact interactions, we derive a macroscopic model. We show that a diffusion equation arises, as is often postulated in the field of glioma modeling, but it is nonlinear because of the interactions. We give the explicit dependence of diffusivity on the cell density and on a parameter governing cell-cell interactions. We discuss in details the conditions of validity of the approximations used in the derivation and we compare analytic results from our PDE to numerical simulations and to some in vitro experiments. We notice that the family of microscopic models we started from includes as special cases some kinetically constrained models that were introduced for the study of the physics of glasses, supercooled liquids and jamming systems.

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