Optimal therapy protocols in the mathematical model of acute leukemia with several phase constraints

A mathematical model for leukemia therapy based on the Gompertzian law of cell growth is investigated. The effect of the therapy is described in terms of therapy functions. Two types of therapy functions are considered: monotone and non‐monotone. In the first case, the effect of the chemotherapy on the normal cells directly depends on the quantity of the chemotherapeutic agent. In the second case, we consider a therapy function with a threshold effect, which influences the leukemia cells. This effect can describe resistance to the therapy. The optimal control problem has the purpose of minimizing the number of the leukemia cells subject to the constraints of the normal cells, whose number must not fall below a prescribed critical limit and of the amount of the chemotherapeutic agents in the body of the patient (i.e. constraint on the toxicity). Furthermore, a third phase constraint on the cumulative amount of the medicine over the whole therapy process is considered, which can be interpreted as an economic factor of the therapy. This optimization problem is solved analytically with the help of the Pontryagin's Maximum Principle, extension of penalty functions and additional analysis.