Multi-Input Control Systems in Biomedical Applications

摘要

Cancer research has made tremendous progress in the past decades and new treatments are emerging. Especially promising are combinations of traditional treatments like chemotherapy or radiotherapy with novels ones such as tumor anti-angiogenesis, an indirect cancer treatment approach that targets the vasculature of the tumor, or immunotherapy, which gives a boost to the immune system, in the hope of achieving synergistic effects. With novel approaches the underlying biological mechanisms are often not fully understood and several important questions such as how to best schedule these therapies over time still need to be answered. In clinical trials, because of the great complexity of the underlying medical problem, the scheduling of drugs is pursued in expensive, exhaustive, medically guided trial-and-error approaches. But these difficult scheduling questions are far from being answered, especially when more than one treatment is involved. Hence there exists a strong opportunity for mathematical modeling and analysis to be useful here. The dynamics of the tumor growth and the interactions between cancer cells, healthy cells, immune system and the vasculature of the tumor can all be described as nonlinear systems with the actions of various drugs providing control inputs. In this talk, we shall present several systems describing these dynamical interactions between various types of cells under combination therapies for cancer. Starting with cell-cycle specific bilinear systems which allow the modeling of multiple drug treatments in chemotherapy, we then shall focus on dynamics given by nonlinear systems appearing in modeling of combinations of novel treatment with traditional treatments. The dimension of the systems increases if the pharmacokinetics of the therapeutic agents is taken into account. Analyzing these systems as optimal control problems with the objective of minimizing the tumor volume, allows us to determine the form of the optimal controls representing drug protocols for these treatments. Both analytical and numerical results about the structures of optimal controls and corresponding responses of the system will be presented. Open questions and challenges will be discussed.