On strategies on a mathematical model for leukemia therapy

Abstract

A mathematical model for leukemia therapy based on the Gompertzian law of cell growth is studied. It is assumed that the chemotherapeutic agents kill leukemic as well as normal cells. Effectiveness of the medicine is described in terms of a therapy function. Two types of therapy functions are considered: monotonic and non-monotonic. In the former case the level of the effect of the chemotherapy directly depends on the quantity of the chemotherapeutic agent. In the latter case the therapy function achieves its peak at a threshold value and then the effect of the therapy decreases. At any given moment the amount of the applied chemotherapeutic is regulated by a control function with a bounded maximum. Additionally, the total quantity of chemotherapeutic agent which can be used during the treatment process is bounded too. The problem is to find an optimal strategy of treatment to minimize the number of leukemic cells while at the same time retaining as many normal cells as possible. With the help of Pontryagin’s Maximum Principle it was proved that the optimal control function has at most one switch point in both monotonic and non-monotonic cases for most relevant parameter values. A control strategy called alternative is suggested. This strategy involves increasing the amount of the chemotherapeutical medicine up to a certain value within the shortest possible period of time, and holding this level until the end of the treatment. The comparison of the results from the numerical calculation using the Pontryagin’s Maximum Principle with the alternative control strategy shows that the difference between the values of cost functions is negligibly small.