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

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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.