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.
Mathematical methods and tools from a field called geometric optimal control were applied to problems in biomedicine. The analysis concerned mathematical models describing the dynamics of cancer growth under various treatments. The goal was to find an optimal way of administering these protocols with the criterion of minimizing the tumor volume, possibly maximizing the immune system response, and minimizing the overall side effects. Emphasis was put on the study of models for combination therapies where two treatments, usually a traditional one and a more novel one were combined to lead to synergistic effects. In this project, a mathematical model for the novel therapy of antiangiogenic inhibitors (a treatment which targets the vasculature of the tumor) was combined with traditional chemo- and radiotherapy. This led to mathematical models with multiple controls that represent the actions of each of the treatments. Another model sharing similar characteristics described tumor-immune interactions under treatment that combined chemo- and immunotherapy. Analysis of these type of problems is challenging mathematically and under this project some interesting results were obtained that shed some light on how these combination treatments should be administered. The outcome of our mathematical analysis was that the antiangiogenic agent in most cases should be given originally at full dose and then switch to properly calibrated lower doses, whereas the cytotoxic agent should be administered in one session at maximum dose rates. This structure of optimal solutions has biological backing in some Phase II medical trials on this topic that affirm the beneficial effects of a regularization of the tumor vasculature prior to the delivery of chemotherapy. Our results show that there exists an optimal relationship between tumor volume and its vasculature and that the best possible tumor reductions can be achieved if this relationship is maintained. For combinations of antiangiogenic treatment with radiotherapy, we analyzed mathematical models of various dimensions, depending on whether radiation effects on the tumor, its vasculature, and healthy cells are taken into account. We showed that so-called totally singular controls, i.e., controls where both the antiangiogenic dose-rate and the radiation fractionization schedule follow time varying doses play a role in optimal protocols. These theoretical results are in agreement with some results available in the literature on the topic, called dose intensification in radiology. In the research on tumor immune system interactions, we have formulated a novel approach where the objective is set up so that optimal controls move an initial condition that lies in a region of uncontrolled growth (malignant region) into a region of attraction of a tumor free or microscopic (benign) equilibrium. Our results show that chemotherapy remains the dominant method to reduce high tumor volumes, but as the tumor burden shrinks, chemotherapy switches from the maximum tolerated dose bang-bang controls to singular controls at lower dose rates. Overall, our results raise questions as to the appropriateness of an MTD (maximum tolerated dose) approach when other components of the tumor's microenvironment, such as the immune system or vasculature of the tumor, are brought into the modeling. Our research has intrinsic intellectual merit in illustrating how mathematical tools from optimal control theory , some established and some developed for this purpose, can be applied to the analysis of timely and relevant problems in biomedicine. This research both enriches the mathematical toolbox in this field and the results themselves are of interest to the medical community for whom the question is not just what drugs to give, but also how to apply them. In the case of combination treatments, the questions are not only about the dosage and the timing, but also the sequencing is a relevant aspect that can significantly affect the outcome. These combinations include novel therapies which are in the stage of clinical trials and no established protocols or practical guidelines are available. The PIs’ theoretical results could be useful in establishing some benchmarks for medically realizable protocols. All our results have been published in journals oriented towards optimal control and biomedical applications, including interdisciplinary ones. They were also disseminated through invited (including plenary) and contributed talks at a large number of conferences in these fields. Because of its interdisciplinary character, the project was of a great interest to students and had significant impact on education. Students (including one woman and one minority student) were actively involved co-authoring papers and presenting talks and posters at conferences. During the duration of this grant our graduate level textbook on methods of optimal control, "Geometric Optimal Control - Theory, Methods and Examples" was published by Springer in 2012. In this text, the procedures that we are using in our research on the topics under this grant are presented to a broader audience. This book provides a comprehensive study of tools and methods of geometric optimal control not available currently in the literature in such a form.
