Modelling of Cancer Stem Cell Driven Solid Tumors

Cancer stem cells (CSCs) are pluripotent cancer cells, which are less sensitive to treatments, and which can generate new tumors once transplanted into a healthy tissue environment. They have been identified in many cancers and they are the driving force of cancer growth and metastasis. Mathematical modelling of CSCs has contributed to our increased understanding of CSC interactions. Here we will consider CSC feedback mechanisms, treatments with radiation therapy, differentiation promoters, and dedifferentiation inhibitors. In addition, we will consider spatially explicit models. The dominating effect in these models is the tumor growth paradox. It says that a tumor under treatment might grow larger than a similar tumor without treatment. Using geometric singular perturbation methods, we prove mathematically that such a tumor growth paradox can arise. In the spatial context, it leads to a tumor invasion paradox, as we will explain. Many of the models presented here lead to interesting mathematical questions and to the development of new mathematical techniques. We end our chapter with a collection of open problems, which we think, are amenable to further mathematical analysis.