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., 59, 4770-4775) and a modification of this model by Ergun et al. (2003, Optimal scheduling of radiotherapy and angiogenic inhibitors. Bull. Math. Biol., 65, 407-424) are considered as optimal control problem with the aim of maximizing the tumour reduction achievable with an a priori given amount of angiogenic agents. For both models, depending on the initial conditions, optimal controls may contain a segment along which the dosage follows a so-called singular control, a time-varying feedback control. In this paper, for these cases, the efficiency of piecewise constant protocols with a small number of switchings is investigated. Through comparison with the theoretically optimal solutions, it will be shown that these protocols provide generally excellent suboptimal strategies that for many initial conditions come within a fraction of 1% of the theoretically optimal values. When the duration of the dosages are a priori restricted to a daily or semi-daily regimen, still very good approximations of the theoretically optimal solution can be achieved.