The Role of Dynamics and Objective in Modeling Cancer Chemotherapy
by
Urszula Ledzewicz
Southern Illinois University, IL
Coauthors: Heinz Schaettler, Washington University, St. Louis

We consider optimal control problems of Bolza type which arise in mathematical models for cancer chemotherapy when treatment protocols over a fixed therapy interval are considered. In some of these models the state is represented by the number of cancer cells in various compartments which combine phases of the cell cycle. In other models the state is given by the number of bone marrow cells in both active and dormant stages. The dynamics describes the growth of the tumor and its implications for the human body as well as the influences of the drugs given. The dynamical equations of the system are balance equations which describe the in- and out-flows from these compartments. Dosages of various drugs given in chemotherapy are serving as control functions in these models. These include killing agents (for example, Taxol, or spindle poisons like Vincristine), blocking agents (antibiotics, like Adriamycin, or Hydroxyurea) and recruiting agents (cytokines, like Interleukin-3). Killing agents are cytotoxic drugs used to destroy the cancer cells, but they are equally toxic for normal cells. Blocking agents try to slow down cancer growth by causing brief and invisible inhibition of DNA synthesis and recruiting agents are used to lower the large residuum of dormant cells which are not sensitive to most cytotoxic agents.

The goal of chemotherapy typically is to minimize the number of cancer cells at the end of the therapy interval while keeping the toxicity to the normal tissues acceptable or, in other models, keep the number of bone marrow cells at the highest possible level. While a lot of research has been done analyzing the dynamics of these problems, much less attention has been given to the mathematical formulation of the objective. Some approaches focus on monitoring the number of cancer or bone marrow cells during the entire period of the chemotherapy while others focus on the final outcome of the therapy. The toxicity of the drug can be measured in the L¹- or L²-norms on the control which also results in a different mathematical analysis and outcomes.

In this talk we discuss the roles of the objective in optimal control formulations for various specific models in cancer chemotherapy and show how different types of optimal solutions arise depending on what choices are made. Some numerical simulations will be presented to support the theoretical results.

Date received: March 27, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caky-76.