Multiscale Modeling &Simulation, Volume 18, Issue 2, Page 824-850, January 2020.
The ability to locally degrade the extracellular matrix (ECM) and interact with the tumor microenvironment is a key process distinguishing cancer cells from normal cells, and is a critical step in the metastatic spread of the tumor. The invasion of the surrounding tissue involves the coordinated action of the cancer cells, the ECM, the matrix degrading enzymes, and the epithelial-to-mesenchymal transition. In this paper, we present a mathematical model which describes the transition from an epithelial invasion strategy of the epithelial-like cells (ECs) to an individual invasion strategy for the mesenchymal-like cells (MCs). We achieve this by formulating a genuinely multiscale and hybrid system consisting of partial and stochastic differential equations that describe the evolution of the ECs and the MCs while accounting for the transitions between them. This approach allows one to reproduce, in a very natural way, fundamental qualitative features of the current biomedical understanding of cancer invasion that are not easily captured by classical modelling approaches, for example, the invasion of the ECM by self-generated gradients, and the formation of EC invasion islands outside of the main body of the tumor.

中文翻译：

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癌细胞侵袭细胞外基质的混合多尺度模型。

多尺度建模与仿真，第18卷，第2期，第824-850页，2020年1月。
局部降解细胞外基质（ECM）并与肿瘤微环境相互作用的能力是区分癌细胞与正常细胞的关键过程，并且是肿瘤转移扩散的关键步骤。周围组织的浸润涉及癌细胞，ECM，基质降解酶和上皮-间质转化的协同作用。在本文中，我们提供了一个数学模型，该模型描述了从上皮样细胞（ECs）的上皮浸润策略过渡到间充质样细胞（MCs）的个体浸润策略的过程。我们通过制定一个真正的多尺度混合系统来实现这一目标，该系统由偏微分方程和随机微分方程组成，这些方程描述了EC和MC的演化，同时考虑了它们之间的转换。这种方法允许人们以非常自然的方式重现当前对癌症侵袭的生物医学理解的基本定性特征，而经典建模方法不容易捕获这些特征，例如，自生梯度对ECM的侵袭，以及在肿瘤主体之外形成EC侵入岛。