Solving the dose distribution problem in the radiotherapy planning applying the interior point methods

The dose distribution problem in the Radiation Therapy planning consists in optimize the total radiation dosage delivered into the patient. Its main aim is to attack the tumor delivering as radiation as possible, avoiding severe damage the nearby tissues and organs, for that we use a linear programming model to formulate this problem and the Intensity-modulated radiation therapy (IMRT) technique that allows us to deliver non-uniform radiation flow obtaining a best adjustment of radiation doses to the tumor cells. However, in practice, the oncologist can vary the dose according to his/her expertise making it difficult to guarantee an exact quantity of radiation. Therefore in this paper, the doses are considered as triangular fuzzy numbers and we use the surprise function, that can be understood as a penalty for the violation of each constraint, translate these fuzzy constraints in a non-linear function that must be minimized. Thus we present a mathematical model with non-linear and convex objective function with linear constraints for bounding the radiation flow. We propose an algorithm that was designed using a specially tailored Primal-Dual Interior-Point Method to solve this problem. Our algorithm is implemented in MATLAB and tested in real world large-scale problems in which the tumor is localized on the Head and Neck area. The numerical experiments show the amount of radiation delivered in each region of patient and, in general, the developed approach provides favorable solutions for the dose distribution problem.