Optimal two- stage treatment protocol for a blood cancer model

A two-stage combined treatment of blood cancer (leukemia, lymphoma) is considered for a given time interval. At the first stage, the patient is given a "hard" therapy (for example, chemotherapy) to achieve the normal functioning of the body; at the second stage, the patient is prescribed a "soft", maintenance therapy (for example, targeted therapy) to consolidate the achieved remission. The transition time from one therapy to another is not known in advance and depends on the patient's condition. Such treatment is mathematically described by a two-dimensional controlled Lotka-Volterra competition model, the variables of which are the concentrations of healthy and cancerous cells, and two bounded control functions reflect the applied methods of treatment. For such a controlled model, the aim is to minimize the objective function, which is the sum of the total weighted difference in the concentrations of cancer and healthy cells over the entire treatment interval, as well as these weighted differences taken both at the time of changing the therapy used and at the end of the treatment protocol. It is assumed that the moment of changing the type of therapy is not predetermined; it is found as a result of solving the stated minimization problem. Optimal solutions are obtained numerically using the BOCOP-2.0.5 environment and then discussed in detail. Conclusions are drawn about the effectiveness of two-stage combined treatment and the possibility of finding the optimal treatment protocol for a real cancer patient.