Mathematical Analysis of Spatial Cancer Models

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Successful prevention, early detection, and treatment of cancer requires an understanding of the underlying mechanisms by which cancer evolves spatially and temporally. The occurrence of different cell types in different parts of a tumor, the spatial heterogeneity of the tumor, can lead to failure of therapy that is directed only against one type of cancer cell. Furthermore, spatial heterogeneity is associated with a worse prognosis in many cancer types, and the spatial distribution of nutrients and oxygen can favor the development of more aggressive tumor cell populations. By analyzing spatial cancer models, this research will provide new insights into the mechanistic underpinnings of intra-tumor heterogeneity in solid tumors, and into the role of cellular mobility in shaping tumor phenotypes with poor prognosis. To guarantee that the results are relevant to treatment, the project will be carried out in collaboration with physicians and a cancer biologist. The involvement of graduate students in this research will enhance their ability to work at the interface between mathematics and biology. This project addresses critical challenges in the quest for more effective diagnostic tools and therapeutic approaches in the fight against cancer. Complex agent-based models have often been employed to investigate the impact of space on cancer evolution. However, even if a simulation successfully reproduces observed phenomena, it does not mean that the true underlying mechanism has been found. For this project the investigators plan to extend prior work on spatial Moran models by adding new features to the basic model. To further reduce the gap between analytically tractable and complex computational models, the behavior of the new systems will be analyzed mathematically. The biological questions to be addressed by the project are motivated by recent experimental findings, including studies of the impact of cancer cell migration on solid tumor growth and elucidation of the role of cancer stem cells in driving cancer evolution. Mathematical models that can include genomic data will be used to study genealogies and intra-tumor heterogeneity in colon cancer. Rigorous results for hybrid models that describe tumor-microenvironment interactions will also be derived. More generally, the rigorous analyses of the models will require the development of novel mathematical techniques that will contribute to a deeper understanding of spatial stochastic systems and the behavior of genealogies in an expanding population.