An Optimal Treatment Strategy for a Leukemia Immune Model Governed by Reaction-Diffusion Equations

In this paper, the authors discuss an optimal control problem of a reaction-diffusion leukemia immune model that describes the dynamics of leukemia cells, normal cells, and CAR-T cells. In order to overcome the defects of traditional biotherapy for leukemia by injecting CAR-T cells in a large dose at one time, dynamic low-dose injection of CAR-T cells is considered in this paper. To minimize the total amount of leukemia cells and the injection amount of CAR-T cells and to maximize the total amount of normal cells, an optimal control problem is proposed. We first show the existence, uniqueness, and some estimates of the positive strong solution to the controlled system by using semigroup and functional analysis techniques. Then, the existence of the optimal control is proved by employing minimal sequence methods. On this basis, we further give the first-order necessary conditions satisfied by the optimal control strategy by using the convex perturbation and dual methods. Finally, a specific example and its numerical implementation are offered, which further confirm the theoretical results.