Optimal control problems with time delays: Two case studies in biomedicine.

There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem.

Optimal control of a tuberculosis model with state and control delays.

We introduce delays in a tuberculosis (TB) model, representing the time delay on the diagnosis and commencement of treatment of individuals with active TB infection. The stability of the disease free and endemic equilibriums is investigated for any time delay. Corresponding optimal control problems, with time delays in both state and control variables, are formulated and studied.

Local controllability and optimal control for a model of combined anticancer therapy with control delays.

We study some control properties of a class of two-compartmental models of response to anticancer treatment which combines anti-angiogenic and cytotoxic drugs and take into account multiple control delays. We formulate sufficient local controllability conditions for semilinear systems resulting from these models. The control delays are related to PK/PD effects and some clinical recommendations, e.

Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy.

We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given for four versions of this problem: the case when only anti-angiogenic agents are administered, combination treatment with a cytotoxic agent, and when a standard linear pharmacokinetic equation for the anti-angiogenic agent is added to each of these models. It is shown that the solutions to the more complex models naturally build upon the simplified versions.

Realizable protocols for optimal administration of drugs in mathematical models for anti-angiogenic treatment.

Two mathematical models for tumour anti-angiogenesis, one originally formulated by Hahnfeldt et al. (1999, Tumor development under angiogenic signaling: a dynamical theory of tumor growth, treatment response, and postvascular dormancy. Cancer Res.

On optimal delivery of combination therapy for tumors.

A mathematical model for the scheduling of angiogenic inhibitors in combination with a chemotherapeutic agent is formulated. Conditions are given that allow tumor eradication under constant infusion therapies. Then the optimal scheduling of a vessel disruptive agent in combination with a cytotoxic drug is considered as an optimal control problem.