Mathematical Challenges in the Treatment of Cancer

Ami Radunskaya

Pomona College

Abstract. What can mathematics tell us about the treatment of cancer? Cancer is a myriad of individual diseases, with the common feature that an individual's own cells have become malignant. It is believed that a healthy individual keeps potentially cancerous cells from developing into a threatening tumor through a complicated network of immune response and mechanisms built into the cell cycle that recognize aberrant cells and control their proliferation. Thus, the treatment of cancer poses great challenges, since a attack must be mounted against cells that are nearly identical to normal cells. Mathematical models that describe tumor growth in tissue, the immune response, and the administration of different therapies can suggest treatment strategies that optimize treatment efficacy and minimize negative side effects. However, the inherent complexity of the immune system and the spatial heterogeneity of human tissue gives rise to mathematical models that pose unique analytical and numerical challenges. These include modeling behavior over vastly different time scales, incorporating delays into the model, optimization in high-dimensional spaces, and fitting large sets of dependent parameters to data. In this talk I will present an overview of work that I have done in this area, with the help of many collaborators, over the last ten years, highlighting the various approaches we have taken to tackle these mathematical challenges