Optimal switching time control of the hyperbaric oxygen therapy for a chronic wound

Chronic wounds, defined as those wounds which fail to heal through the normally orderly process of stages and remain in a chronic inflammatory state, are a significant socioeconomic problem. This paper considers an optimal switching time control problem of the hyperbaric oxygen therapy for a chronic wound. First, we model the spatiotemporal evolution of a chronic wound by introducing oxygen, neutrophils, invasive bacteria, and chemoattractant. Then, we apply the method of lines to reduce the partial differential equations (PDEs) into ordinary differential equations (ODEs), which lead to an ODE optimization problem with the changed time switching points. The time-scaling transformation approach is applied to further transform the control problem with changed switching time into another new problem with fixed switching time. The gradient formulas of the cost functional corresponding to the time intervals are derived based on the sensitivity analysis. Finally, computational numerical analysis demonstrates the effectiveness of the proposed control strategy to inhibit the growth of bacterial concentration.