On optimal protocols for combinations of chemo- and immunotherapy

An optimal control problem for combinations of cancer chemotherapy with immunotherapy in form of a boost to the immune system is considered as a multi-input optimal control problem. In the objective, a weighted average of several quantities that describe the effectiveness of treatment is minimized that includes the number of cancer cells at the terminal time, a measure for the immunocompetent cell densities at the terminal point as a negative term, the overall amount of therapeutic agents given as a measure for the side effects of treatment and a small penalty on the terminal time which is free. In the dynamics, a pharmacokinetic model for the concentrations of the pharmaceutical agents is included. The focus of the paper is on the structure of singular arcs. Their local optimality properties will be investigated analytically and then illustrated through numerically obtained results. The resulting geometric structure of optimal controlled trajectories provides some insights into the design of optimal protocols, especially the sequencing of the drugs in these combination treatments.