Non-Local Effects and Surface-Bound Reactions in Cancer Invasion Model Development, Analysis and Numerical Simulation

侵袭组织和形成转移(继发性肿瘤)的能力是癌症如此危险的原因。在入侵过程中发生的关键生物学过程是基质降解酶的分泌、细胞增殖、细胞-细胞黏附的丧失和细胞-基质黏附的增强以及活跃的迁移。更好地了解生化(细胞内)和细胞过程对细胞和基质的组织规模重排的影响可能有助于制定治疗策略。因此，癌细胞侵袭的建模和数值模拟是人们非常感兴趣的，也是目前正在进行的研究的主题。在这个项目中，我们集中在癌症侵袭建模中的两个关键方面：细胞-细胞和细胞-基质黏附(组织水平建模)和与癌症侵袭相关的表面结合反应。模型将从正性、图灵型不稳定性和参数对模型输出的影响等方面进行分析，这些参数没有被实验很好地表征。我们将开发一个模拟框架，以针对原始模型测试这些模型。最后，这些结果将被转移到肿瘤中淋巴管生成的模型中。

The ability to invade tissue and form metastases (secondary tumours) is what makes cancer so dangerous. Key biological processes occurring during invasion are the secretion of matrix degrading enzymes, cell proliferation, the loss of cell-cell adhesion on one hand and enhanced cell-matrix adhesion on the other hand as well as active migration. A better understanding of the effect that biochemical (intracellular) and cellular processes have on tissue scale rearrangement of cells and matrix may help to develop treatment strategies. Hence, the modelling and numerical simulation of cancer cell invasion is of great interest and is the subject of ongoing research.In this project, we focus on two key aspects in the modelling of cancer invasion: cell-cell and cell-matrix adhesion (tissue-level modelling) and surface-bound reactions involved in cancer invasion.The modelling will be done on the tissue scale, which allows for an efficient simulation. The model will be analyzed regarding positivity, Turing-type instabilities and the influence of parameters, which are not well-characterized by experiments, on the model output. We will develop a simulation framework to test these models against the original model. Finally, these results will be transferred to a model of lymphangiogenesis in tumors.