Stability analysis of radiotherapy cancer treatment model with fractional derivative

Abstract

This paper presents the condition for uniqueness, the stability analysis, and the bifurcation analysis of a mathematical model that simulates a radiotherapy cancer treatment process. The presented model was the previous cancer treatment model integrated with the Caputo fractional derivative and the Linear-Quadratic with the repopulation model. The metric space analysis was used to establish the conditions for the presence of unique fixed points for the model, which indicated the presence of unique solutions. After establishing uniqueness, the model was used to simulate the fractionated treatment process of six cancer patients treated with radiotherapy. The simulations of the cancer treatment process were done in MATLAB with numerical and radiation parameters. The numerical parameters were obtained from previous literature and the radiation parameters were obtained from reported clinical data. The solutions of the simulations represented the final volumes of tumors and normal cells. Subsequently, the initial values of the model were varied with 200 different values for each patient and the corresponding solutions were recorded. The continuity of the solutions was used to investigate the stability of the solutions with respect to initial values. In addition, the value of the Caputo fractional derivative was chosen as the bifurcation parameter. This parameter was varied with 500 different values to determine the bifurcation values. It was concluded that the solutions are unique and stable, hence the model is well-posed. Therefore, it can be used to simulate a cancer treatment process as well as to predict outcomes of other radiation protocols.