Optimal control methods for drug delivery in cancerous tumour by anti-angiogenic therapy and chemotherapy.

There are numerous mathematical models simulating the behaviour of cancer by considering variety of states in different treatment strategies, such as chemotherapy. Among the models, one is developed which is able to consider the blood vessel-production (angiogenesis) in the vicinity of the tumour and the effect of anti-angiogenic therapy. In the mentioned-model, normal cells, cancer cells, endothelial cells, chemotherapy and anti-angiogenic agents are taking into account as state variables, and the rate of injection of the last two are considered as control inputs. Since controlling the cancerous tumour growth is a challenging matter for patient's life, the time schedule design of drug injection is very significant. Two optimal control strategies, an open-loop (calculus of variations) and a closed-loop (state-dependent Riccati equation), are applied on the system in order to find an optimal time scheduling for each drug injection. By defining a proper cost function, an optimal control signal is designed for each one. Both obtained control inputs have reasonable answers, and the system is controlled eventually, but by comparing them, it is concluded that both methods have their own benefits which will be discussed in details in the conclusion section.