Optimal control applied to competing chemotherapeutic cell-kill strategies

Optimal control techniques are used to develop optimal strategies for chemotherapy. In particular, we investigate the qualitative differences between three different cell-kill models: log-kill hypothesis (cell-kill is proportional to mass); Norton - Simon hypothesis (cell-kill is proportional to growth rate); and, E-max hypothesis (cell-kill is proportional to a saturable function of mass). For each hypothesis, an optimal drug strategy is characterized that minimizes the cancer mass and the cost ( in terms of total amount of drug). The cost of the drug is nonlinearly defined in one objective functional and linearly defined in the other. Existence and uniqueness for the optimal control problems are analyzed. Each of the optimality systems, which consists of the state system coupled with the adjoint system, is characterized. Finally, numerical results show that there are qualitatively different treatment schemes for each model studied. In particular, the log-kill hypothesis requires less drug compared to the Norton - Simon hypothesis to reduce the cancer an equivalent amount over the treatment interval. Therefore, understanding the dynamics of cell-kill for specific treatments is of great importance when developing optimal treatment strategies.