Heinz S., Urszula L. Optimal Control for Mathematical Models of Cancer Therapies. An application of Geometric Methods

Editors: S.S. Altman, L. Greengard, and P. Holmes. — Springer, 2015. — 496 p. — (Interdisciplinary Applied Mathematics 42). — ISBN: 978-1-4939-2971-9.In this text, we shall apply the tools and methods from optimal control to analyze various minimally parameterized models that describe the dynamics of populations of cancer cells and elements of the tumor microenvironment under different anticancer therapies. In spite of their simplicity, the analysis of these models that capture the essence of the underlying biology sheds light on more general scenarios and, in many cases, leads to conclusions that confirm experimental studies and clinical data. For example, a treatment strategy for the application of chemotherapy known as “chemo-switch” in the medical literature corresponds to optimal controls that are of the type “bang-singular,” and these are shown to be optimal in many of the models considered here. While the underlying topic of this text is medical, the tools and techniques that will be used are mathematical with some elementary dynamical systems theory and more advanced methods from optimal control theory at the core of the reasoning. Our text is intended for a dual audience. On one hand, we illustrate the applicability of the tools and techniques of optimal control theory to a wide class of problems coming from one particular field. As such, we hope it will be of interest to researchers in mathematics and engineering to whom it introduces a fascinating area of potential applications. On the other hand, this text is also aimed at students and researchers in the applied sciences and the widely understood field of mathematical modeling of cancer treatment. An effort has been made to write the text in a self-contained way to make it accessible to these two possibly disjoint sets of target audiences. Hence, as much as feasible, we included the biomedical background to make the models intelligible to the nonexpert in the field. At the same time, we do not dwell on the theoretical basis for optimal control, but focus on the application of results.