Analysis of Optimal Controls for Biomedical Models of Cancer and HIV

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0205093LedzewiczIn the project, optimal control problems will be investigated which arise as mathematical models for biomedical systems in the chemotherapy of diseases which have a strong cell proliferation aspect such as cancer or AIDS. Three directions of research will be pursued: the analysis of models in cancer chemotherapy, an analysis of models for HIV-infection and anti-viral treatment of AIDS, and the design and analysis of a general mathematical framework which combines common features of both cancer and HIV models. The investigators will use their previous experience working on cancer chemotherapy models and apply new tools based on the method of characteristics and high-order conditions for optimality. In compartmental models for cancer chemotherapy the cell-cycle is controlled by clustering its phases into compartments and using phase specific cytostatic killing and blocking agents. The goal is to maximize the number of cancer cells that the cytostatic agent in the drug kills while keeping the toxicity to the normal tissue acceptable. In this project a synthesis of optimal controls for these models will be constructed. Furthermore, more complex mathematical models that take into account additional aspects such as evolving drug resistance or bone marrow destruction will be analyzed. Due to the complexity of HIV infection, mathematical models for AIDS have only recently been developed and still are constantly being revised and updated taking new medical data into account. With the main attention so far given to the form of the dynamics that best models the interactions between the virus and the human immune system under drug treatment, many of these models still have not been formulated as optimal control problems. The investigators will analyze a variety of these models in the framework of optimal control and compare the solutions aiming at a design of optimal treatment protocols.In the project, the investigators will consider mathematical models in the chemotherapy of diseases that have a strong cell proliferation aspect such a cancer or AIDS. These models will be analyzed as optimal control problems with the drug dosage serving as control parameter with the goal of maximizing the number of cancer cells killed by the drug in the case of cancer (respectively maximizing the number of uninfected cells in the case of chemotherapy for AIDS) while minimizing the harmful side effects and cost of the chemotherapy. Although individually these problems are very different in their specifics, they also have many aspects in common and can be put into one general mathematical model that encompasses them all. Thus, while on one side there is a need to consider these problems separately to understand the implications for the underlying disease, on the other side there are simplifications and insights to be gained by looking at the general properties common to these models. This project will address both directions. A biomedical interpretation of the conditions that optimal controls satisfy will be given in terms understandable by biomedical personnel and the conditions will be related to the underlying biological situation. It is expected that the outcomes of this project will be of interest to the medical community by giving some insights into the analysis of existing chemotherapy protocols for cancer or HIV and possibly aid in the design of new improved protocols. These results are particularly important for HIV treatments which so far do not cure the disease, but only provide a long-term maintenance program.