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

当前癌症研究中积极追求的方向之一是将诸如化学疗法和放射疗法的联合收割机传统治疗与诸如抗血管生成治疗或免疫治疗的新方法相结合，以期实现协同效应。这些新方法的潜在生物学机制尚未完全了解，几个重要的问题，包括如何最好地安排这些治疗，随着时间的推移仍然需要回答。当涉及几种治疗剂时，调度方面变得更加困难和复杂。对于这些组合疗法，还没有医学指南，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。用于组合疗法的数学模型是相当复杂的，并且由于所追求的各种疗法，由多输入控制系统描述。在这个项目中，将根据需要应用和开发现代最优控制的几何方法来分析这些系统，当化疗或放疗与抗血管生成治疗相结合时。从简化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之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。我们的分析采用的工具远远超出了最优性的标准必要条件的应用，旨在全面综合最优控制，即，一个完整的解决方案的问题，为任意的初始数据。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。最终的目标是设计强大的和有效的可实现的协议的组合治疗。由于其应用和跨学科的特点，该项目包含了大量的教育组成部分的兴趣，学生从各个领域，包括数学，生物学和工程。将继续努力吸引妇女和少数民族参加该项目。

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.