RUI: Collaborative Research: Optimal Control of Multi-Input Mathematical Models for Tumor Dynamics under Combination Therapies

将化疗、放疗等传统治疗方法与抗血管生成治疗或免疫治疗等新方法相结合，以期达到协同效应，是当前癌症研究积极追求的方向之一。这些新方法的潜在生物学机制尚不完全清楚，一些重要的问题，包括如何随着时间的推移最好地安排这些治疗，仍然需要回答。当涉及几种治疗药物时，调度方面变得更加困难和复杂。对于这些联合疗法，目前还没有医学指南，而数学建模和分析能够为建立稳健有效的治疗方案提供有价值的见解。联合治疗的数学模型相当复杂，而且由于所采用的治疗方法多种多样，需要用多输入控制系统来描述。在这个项目中，当化疗或放疗与抗血管生成治疗相结合时，将应用和发展现代最优控制的几何方法来分析这些系统。从简化的，但经过生物学验证的模型开始，越来越多的现实医学特征，如药物的药代动力学和肿瘤免疫系统的相互作用将被纳入。对于这些模型，非常规的数学结构出现了，这些结构以前没有在生物医学应用的背景下进行分析，并且值得根据其自身的优点进行研究。我们的分析使用的工具远远超出了标准必要条件的应用，旨在全面综合最优控制，即对任意初始数据的问题的完整解决方案。这些解决方案将为其他更简单和实际可实现的协议设定理论基准。最终目标是为联合治疗设计稳健、有效、可实现的方案。由于其应用和跨学科的特点，该项目包含了来自数学、生物和工程等各个领域的学生感兴趣的大量教育组成部分。目前吸引妇女和少数民族参加项目的努力将继续进行。

One of the directions actively pursued in current cancer research is to combine traditional treatments like chemotherapy and radiotherapy with novel approaches such as anti-angiogenic treatment or immunotherapy in the hope of achieving synergistic effects. The underlying biological mechanisms of these novel approaches are not fully understood and several important questions including how to best schedule these therapies over time still need to be answered. The scheduling aspect becomes even more difficult and complex when several therapeutic agents are involved. For these combination therapies no medical guidelines are in place yet and mathematical modeling and analysis are able to give valuable insights into establishing robust and effective treatment protocols. Mathematical models for combination therapies are quite complex and, due to the various therapies pursued, are described by multi-input control systems. In this project, geometric methods from modern optimal control will be applied and developed as needed to analyze these systems when chemotherapy or radiotherapy are combined with anti-angiogenic treatments. Starting with simplified, but biologically validated models, increasingly more realistic medical features such as the pharmacokinetics of the agents and tumor immune system interactions will be incorporated. For these models unconventional mathematical structures arise that have not been analyzed in the context of biomedical applications before and are worthwhile to be investigated on their own merit. Our analysis employs tools that go well beyond the application of standard necessary conditions for optimality and aims at a full synthesis of optimal controls, i.e., a complete solution to the problem for arbitrary initial data. These solutions will set theoretical benchmarks to which other - simpler and practically realizable - protocols can be compared. The ultimate goal is to design robust and effective realizable protocols for combination therapies. Due to its applied and interdisciplinary character, the project contains a substantial educational component of interest to students from various fields including Mathematics, Biology and Engineering. Existing efforts to attract women and minorities to the project will be continued.