Collaborative Research: Optimal Control of Multi-Input Mathematical Models for Tumor Dynamics under Combination Therapies

Washington University, Saint Louis MO

Investigators

Abstract

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.