Collaborative Research: Optimal Control of Mathematical Models for Cancer Treatments

在这个项目中，将用现代最优控制理论的方法来研究各种癌症治疗的数学模型。建立了一类基于单杀伤剂或多杀伤剂化疗的耐药进化模型，并对其进行了分析。在单一药物的情况下，这导致了双线性结构，但对于多药物情况，出现了非常规配方，需要应用新的工具。除了涉及耐药性的问题外，还将关注代表血管生成抑制剂(通过剥夺肿瘤新血管的招募来减缓肿瘤生长)治疗的模型。这些模型中的动力学是完全非线性的，在几何最优控制方法框架内的分析有望回答关于这种新疗法的一些公开问题。与化疗、放射或抗血管生成治疗相关的耐药性和其他副作用自然导致需要联合治疗。从数学上讲，这样的治疗不仅在制定复杂的动力学方面提出了挑战，而且在选择适当的目标方面也提出了挑战，这些目标将捕捉到反映这些治疗的各种和多重方面的医学标准。所有这些努力都将服务于找到最佳控制的总体目标，这可能有助于设计改进的治疗方案。传统癌症治疗的目标是直接杀死肿瘤细胞，无论是通过化疗的药物还是通过放射治疗的方法。然而，有许多限制因素，可能是最重要的一个，被称为“肯定是最令人沮丧的因素”，就是耐药性。到今天为止，对于开发有效的癌症治疗方法的这一障碍，还没有医学解决方案。在项目中，将使用现代最优控制理论的工具来分析代表癌症治疗的各种数学模型，重点放在重要的医学耐药问题上。将对捕捉到这些现象的新模型进行分析，以更好地反映潜在的生物学情况和治疗目标。进行的研究将为理解这些重要的生物医学模型提供数学见解，这些模型旨在设计更有效的治疗方案，这些方案可能很难在实验室环境中进行测试，或者至少非常昂贵，如果不是不可能的话。它还将通过开发和采用针对重大应用的新技术来促进最优控制理论。由于其应用性和跨学科的特点，该项目引起了学生的兴趣，因此将包含大量的教育内容。

In this project mathematical models for various cancer treatments will be investigated with the methods of modern optimal control theory. A class of models based on chemotherapy with single or multiple killing agents will be formulated and analyzed under evolving drug resistance. In the case of a single drug this leads to bilinear structures, but for the multiple drug case unconventional formulations arise and new tools need to be applied. Aside from problems involving drug resistance, attention also will be directed to models representing treatments with angiogenic inhibitors (which slow tumor growth by depriving it from recruiting new blood vessels). The dynamics in these models is fully nonlinear and its analysis within the framework of geometric methods of optimal control is expected to answer some open questions about this novel therapy. Drug resistance and other side effects related to chemo-, radio- or antiangiogenic therapy naturally lead to the need for combination treatments. Mathematically such treatments provide a challenge not only in the formulation of the complex dynamics, but also in the choice of proper objectives which would capture the various and multi-fold aspects reflecting the medical criteria by which to judge these therapies. All of these efforts will serve the overall goal of finding the optimal control, which could aid in designing improved therapy protocols.Conventional cancer treatments aim at directly killing tumor cells, be it by means of drugs in chemotherapy or by means of radiation in radiation treatments. However, there exist many limiting factors and probably the single most important, and what has been called "certainly the most frustrating one," is drug resistance. As of today no medical solution exists for this obstacle to developing effective cancer treatments. In the project tools of modern optimal control theory will be employed to analyze a variety of mathematical models representing cancer treatments with the focus on the important medical issue of drug resistance. New models that capture these phenomena will be analyzed in an effort to better reflect the underlying biological situation and goals of the treatment. The research conducted will provide mathematical insights into an understanding of these important biomedical models aimed at designing more effective therapy protocols which may be difficult, or at least very expensive, if not impossible, to test in a laboratory setting. It will also contribute to optimal control theory by developing and employing new techniques aimed at significant applications. Due to its applied and interdisciplinary character the project is of interest to students and consequently will contain a substantial educational component.