Optimal Control of Cancer Chemotherapy with Delays and State Constraints

A mathematical model of cancer chemotherapy is considered as an optimal control problem with the objective of either minimizing a weighted sum of tumor cells and drug dosage or the terminal tumor volume. The control process is subject to three state constraints involving an upper bound on drug toxicity, a lower bound on the white blood cells (WBCs) population, and a constraint to prevent the WBCs count from staying too long below a fixed upper level. The state constraints are imposed for safeguarding the health of patients during treatment. The dynamics of the WBCs population involves delays due to the delay chain of granulocyte development. The control problem is based on a similar model presented in Iliadis and Barbolosi (Comput. Biomed. Res. 33(3):211–226, 2000). However, the authors do not treat necessary conditions which only recently have been presented in the literature. We introduce two variants of the control problem imposing different state constraints. For a basic control problem, we give a thorough discussion of the necessary optimality conditions. Discretization and nonlinear programming methods are employed to determine extremal solutions that precisely satisfy the necessary conditions. It is surprising that in both control problems the time-delayed solution agrees with the non-delayed solution except for the WBCs which are obtained as backward time shifts of the non-delayed WBCs. Since the control variable appears linearly in the Hamiltonian, we find that the control is a combination of bang-bang arcs and boundary arcs of the state constraints. The direct optimization of the switching times and junction times with the boundary defines a finite-dimensional optimization problem for which we can verify second-order sufficient conditions (SSC). To obtain a more practical protocol, we construct an approximative control with fewer control sub-arcs resulting in only a marginal increase in the cost function.