Metaheuristics and Pontryagin's minimum principle for optimal therapeutic protocols in cancer immunotherapy: a case study and methods comparison.

In this paper, the performance appropriateness of population-based metaheuristics for immunotherapy protocols is investigated on a comparative basis while the goal is to stimulate the immune system to defend against cancer. For this purpose, genetic algorithm and particle swarm optimization are employed and compared with modern method of Pontryagin's minimum principle (PMP). To this end, a well-known mathematical model of cell-based cancer immunotherapy is described and examined to formulate the optimal control problem in which the objective is the annihilation of tumour cells by using the minimum amount of cultured immune cells. In this regard, the main aims are: (i) to introduce a single-objective optimization problem and to design the considered metaheuristics in order to appropriately deal with it; (ii) to use the PMP in order to obtain the necessary conditions for optimality, i.e. the governing boundary value problem; (iii) to measure the results obtained by using the proposed metaheuristics against those results obtained by using an indirect approach called forward-backward sweep method; and finally (iv) to produce a set of optimal treatment strategies by formulating the problem in a bi-objective form and demonstrating its advantages over single-objective optimization problem. A set of obtained results conforms the performance capabilities of the considered metaheuristics.