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

侵袭组织并形成转移（继发性肿瘤）的能力是癌症如此危险的原因。侵袭过程中发生的关键生物学过程有基质降解酶的分泌、细胞增殖、细胞-细胞黏附丧失和细胞-基质黏附增强以及主动迁移。更好地了解生物化学（细胞内）和细胞过程对细胞和基质的组织尺度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一个非常有趣的研究课题。在这个项目中，我们专注于癌症侵袭建模的两个关键方面：细胞-细胞和细胞-基质粘附（组织水平建模）和参与癌症侵袭的表面结合反应。建模将在组织尺度上完成，这允许有效的模拟。本文将分析模型的正性、图灵型不稳定性以及实验尚未很好表征的参数对模型输出的影响。我们将开发一个模拟框架，根据原始模型测试这些模型。最后，这些结果将被转移到肿瘤中的淋巴管生成模型中。

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.