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

在正在进行的研究的延续中，将根据需要应用和开发现代最优控制理论中的几何方法，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。工具，超越了最优性的必要条件的应用将被利用，旨在全面综合的最佳控制，以获得定性的见解，这些新的治疗方案的最佳协议的结构。基于最优解的知识，将给出更简单且可能更实用的次优协议的定量评估。将这些新的治疗方法与传统的治疗方法（如化疗）相结合，也将得到解决，希望利用协同效应。在建模和分析方面都出现了挑战，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，一般模型将更加现实，包括这些功能。数学的复杂性和生物医学的相关性给这项研究带来了双重好处：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针对重要应用的新技术，为最优控制理论做出了贡献。传统癌症治疗（如化疗）的一个主要限制因素是耐药性。因此，在癌症研究中存在强有力的努力，以找到不容易产生耐药性的治疗方法。目前在实验阶段和临床试验中正在积极追求的两个突出的新方向是抗血管生成治疗和免疫治疗。由于潜在的医学问题的巨大复杂性，在临床试验中，药物的调度通常是在科学指导下的详尽试错法中进行的。但更复杂的方案相对困难，如果不是不可能的话，或者至少在实验室环境中测试非常昂贵，特别是如果涉及一种以上的药物。在这个项目中，这些新兴疗法的数学模型将与现代最优控制的工具进行分析，以揭示理论上最优协议的结构。虽然这些可能还没有在医学上实现与当前的技术，这种分析提供了理论基准，可实现的协议可以进行比较，从而有助于更有效的次优治疗方案的设计。这对于尚未建立具体指南的新型疗法尤其重要，并且与传统方法如放疗或化疗相结合更是如此，这些传统方法正在试图利用协同效应。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。

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.