| Abstract: |
| \noindent Current targeted treatments for melanoma are based on the continuous use of the maximum tolerated dose by a patient. At the same time, they quickly eliminate drug-sensitive cancer cells. As a result, such treatment changes the competition between drug-sensitive and drug-resistant cancer cells in favor of the latter. Therefore, drug-resistant cancer cells begin to dominate in the patient`s body, and the applied treatment may be ineffective.
\noindent A new direction in the treatment of melanoma is adaptive therapy. It allows a significant number of drug-sensitive cancer cells to survive through the use of minimally effective doses of drugs or temporary interruptions in their intake. As a result, these cells inhibit the proliferation of drug-resistant cancer cells by competing for shared limited resources. For successful results of adaptive therapy, it is extremely important to find the optimal moments of switching from the stage of its active implementation to the stage of its absence (rest intervals for a patient) and vice versa, taking into account the characteristics of the patient.
\noindent In this report, for a given time interval, which is a general period of melanoma treatment, the Lotka-Volterra mathematical model, given by a system of ordinary differential equations, is considered, which describes the competition between drug-sensitive and drug-resistant cancer cells during adaptive therapy. This model also contains a control function of time responsible for the transition from the stage of active implementation of adaptive therapy to the stage of its absence and vice versa. To find the optimal moments of switching between these stages, the task is to minimize the cancer cell load both during the entire general period of melanoma treatment and at its final moment. An analytical study of such a minimization problem is carried out using the Pontryagin maximum principle. The results of the study are confirmed by numerical calculations performed in the BOCOP-2.0.5 environment for the values of the parameters of the original Lotka-Volterra model and its initial conditions taken from real clinical practice data. |
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