On assessing quality of therapy in non-linear distributed mathematical models for brain tumor growth dynamics

In this paper a mathematical model for glioma therapy based on the Gompertzian law of cell growth is presented. In the common case the model is considered with non-linear spatially varying diffusion depending on a parameter. The case of the linear spatially-varying diffusion arose as a special case for a particular value of the parameter. Effectiveness of the medicine is described in terms of a therapy function. At any given moment the amount of the applied chemotherapeutic agent 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. The main goal of the work is to compare the quality of the optimal strategy of treatment with the quality of another one, proposed by the authors and called the alternative strategy. As the criterion of the quality of the treatment, the amount of the cancer cells at the end of the therapy is chosen. The authors concentrate their efforts on finding a good estimate for the lower bound of the cost-function. Thus it becomes possible to compare the quality of the optimal treatment strategy with the quality of the alternative treatment strategy without explicitly finding the optimal control function.