Collaborative Research: (RUI) Analysis of Optimal and Suboptimal Controls for Mathematical Models Arising in Novel Cancer Therapies

在持续进行的研究中，现代最优控制理论中的几何方法将根据需要应用和发展，以分析新型癌症治疗的新兴数学模型，重点是抗血管生成治疗和免疫治疗。将利用超越最优性必要条件应用的工具，旨在全面综合最优控制，以获得对这些新治疗的最佳方案结构的定性见解。基于最优解的知识，将给出更简单和可能更实用的次优协议的定量评估。这些新治疗方法与传统治疗方法（如化疗）的结合也将得到解决，以期利用协同效应。在建模和分析中都出现了挑战，需要加以解决。在这种情况下，药物的药代动力学和药效学成为一个重要的方面，通常模型将使更现实的包括这些特征。数学复杂性和生物医学相关性给这项研究带来了双重价值：它丰富了对重要生物医学问题的理解，同时通过开发和采用旨在重要应用的新技术，为最优控制理论做出了贡献。化疗等传统癌症治疗的一个主要限制因素是耐药性。因此，在癌症研究方面存在着巨大的努力，以找到不容易产生耐药性的治疗方法。目前，无论是在实验阶段还是临床试验阶段，人们都在积极追求两个突出的新方向：抗血管生成治疗和免疫治疗。由于潜在的医学问题非常复杂，在临床试验中，药物的安排通常采用科学指导的详尽的试错方法。但是，更复杂的方案相对来说比较困难，如果不是不可能的话，或者至少在实验室环境中进行测试非常昂贵，特别是在涉及多种药物的情况下。在这个项目中，这些新出现的治疗方法的数学模型将用现代最优控制的工具进行分析，以阐明理论上最优方案的结构。虽然目前的技术可能还无法在医学上实现这些，但本分析提供了理论基准，可以比较可实现的方案，从而有助于设计更有效的次优治疗方案。这对于尚未制定具体指南的新疗法尤其重要，对于与放疗或化疗等传统方法相结合的疗法更是如此，这些方法正在努力利用协同效应。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。

In a continuation of ongoing research, geometric methods in modern optimal control theory will be applied and developed as needed to analyze emerging mathematical models for novel cancer treatments with a focus on anti-angiogenic treatments and immunotherapy. Tools that go beyond an application of necessary conditions for optimality will be utilized aiming at a full synthesis of optimal controls to gain qualitative insights into the structure of optimal protocols for these novel treatments. Based on the knowledge of optimal solutions, a quantitative assessment of simpler and potentially more practical suboptimal protocols will be given. Combinations of these novel treatment approaches with conventional ones, like chemotherapy, will also be addressed in the hope of harnessing synergistic effects. Here challenges arise both in the modeling and analysis and will need to be resolved. In this context pharmacokinetics and pharmacodynamics of the drugs become an important aspect and generally models will be made more realistic by including these features. Mathematical complexity and biomedical relevance give double merit to this research: it enriches the understanding of important biomedical problems while it at the same time contributes to optimal control theory by developing and employing new techniques aimed at significant applications. A major limiting factor in traditional cancer treatments like chemotherapy is drug resistance. Consequently there exist strong efforts in cancer research to find treatments that would not be prone to drug resistance. Two prominent new directions that are actively being pursued nowadays, both in experimental stages and clinical trials, are anti-angiogenic treatments and immunotherapy. Because of the great complexity of the underlying medical problem, in clinical trials the scheduling of drugs is typically pursued in scientifically guided exhaustive trial-and-error approaches. But more complex protocols are relatively difficult, if not impossible, or at least very expensive to test in a laboratory setting, particularly if more than one drug is involved. In this project mathematical models for these newly emerging therapies will be analyzed with the tools of modern optimal control to shed some light into the structure of theoretically optimal protocols. While these may not yet be medically realizable with current technologies, this analysis provides theoretical benchmarks to which realizable protocols can be compared and thus aids the design of more effective suboptimal therapy protocols. This is of particular importance for novel therapies for which no specific guidelines have been established yet and even more so in combination with traditional approaches like radiotherapy or chemotherapy which are being pursued in an attempt to harness synergistic effects. 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.