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。在肿瘤的不同部分中出现不同的细胞类型，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，空间异质性与许多癌症类型的预后较差相关，营养物质和氧气的空间分布可能有利于更具侵袭性的肿瘤细胞群的发展。通过分析空间癌症模型，这项研究将为实体瘤中肿瘤内异质性的机制基础提供新的见解，以及细胞移动性在形成预后不良的肿瘤表型中的作用。为了保证结果与治疗相关，该项目将与医生和癌症生物学家合作进行。研究生参与这项研究将提高他们在数学和生物学之间的接口工作的能力。该项目解决了在对抗癌症的斗争中寻求更有效的诊断工具和治疗方法的关键挑战。复杂的基于代理的模型经常被用来研究空间对癌症演变的影响。然而，即使模拟成功地再现了观察到的现象，这并不意味着真正的潜在机制已经找到。对于这个项目，研究人员计划通过在基本模型中添加新的功能来扩展空间Moran模型的先前工作。为了进一步缩小易于分析的模型和复杂的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激发的，包括研究癌细胞迁移对实体瘤生长的影响，以及阐明癌症干细胞在推动癌症演变中的作用。可以包括基因组数据的数学模型将用于研究结肠癌的谱系和肿瘤内异质性。描述肿瘤微环境相互作用的混合模型的严格结果也将得到。更一般地说，模型的严格分析将需要开发新的数学技术，这将有助于更深入地了解空间随机系统和不断扩大的人口中的家谱行为。

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.