Mathematical Analysis of Spatial Cancer Models

癌症的成功预防、早期发现和治疗需要了解癌症在空间和时间上演变的潜在机制。肿瘤的不同部位出现不同类型的细胞，肿瘤的空间异质性，可能导致仅针对一种类型的癌细胞的治疗失败。此外，在许多癌症类型中，空间异质性与较差的预后相关，营养物质和氧气的空间分布有助于发展更具侵袭性的肿瘤细胞群。通过对空间肿瘤模型的分析，这项研究将为实体肿瘤中肿瘤内异质性的机制基础以及细胞流动性在形成预后不良的肿瘤表型中的作用提供新的见解。为了确保结果与治疗相关，该项目将与医生和一名癌症生物学家合作进行。研究生参与这项研究将增强他们在数学和生物之间工作的能力。该项目解决了在抗击癌症的斗争中寻求更有效的诊断工具和治疗方法方面的关键挑战。复杂的基于试剂的模型经常被用来研究空间对癌症进化的影响。然而，即使模拟成功地再现了观察到的现象，也并不意味着已经找到了真正的潜在机制。对于这个项目，研究人员计划通过在基本模型上添加新的功能来扩展先前在空间Moran模型上的工作。为了进一步缩小易于分析的计算模型和复杂计算模型之间的差距，将对新系统的行为进行数学分析。该项目要解决的生物学问题是由最近的实验发现推动的，包括研究癌细胞迁移对实体肿瘤生长的影响，以及阐明癌症干细胞在推动癌症进化中的作用。可以包括基因组数据的数学模型将被用于研究结肠癌的系谱和肿瘤内的异质性。描述肿瘤-微环境相互作用的混合模型的严格结果也将被推导出来。更广泛地说，对模型的严格分析将需要开发新的数学技术，这些技术将有助于更深入地理解空间随机系统和在不断扩大的人口中谱系的行为。

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.