A Successive LP Approach with C-VaR Type Constraints for IMRT   Optimization
  Radiation therapy is considered to be one of important treatment protocols for cancers. Radiation therapy employs several beams of ionizing radiation to kill cancer tumors, but such irradiation also causes damage to normal tissues. Therefore, a treatment plan should satisfy dose-volume constraints (DVCs). Intensity-modulated radiotherapy treatment (IMRT) enables to control the beam intensities and gives more flexibility for the treatment plan to satisfy the DVCs. Romeijn et al. (2003) replaced the DVCs in an IMRT optimization with C-VaR (Conditional Value-at-Risk) type constraints, and proposed a numerical method based on linear programming (LP). Their approach reduced the computation cost of the original DVCs, but the feasible region of their LP problems was much narrow compared to the DVCs, therefore, their approach often failed to find a feasible plan even when the DVCs were not so tight.   In this paper, we propose a successive LP approach with the C-VaR type constraints. We detect outliers form the solution of LP problems, and remove them from the domain of the C-VaR type constraints. This eases the sensitivity of C-VaR type constraints to outliers and we can search feasible plans from wider regions. Furthermore, we can give a mathematical proof that if the optimal value of the LP problem in the proposed approach is non-positive, the corresponding optimal solution satisfies all the DVCs. From numerical experiments on test data sets, we observed that our approach found feasible solutions more appropriately than existing LP approaches. In addition, our approach required fewer LP problems, and this led to a short computation time. 
Shogo Kishimoto, and Makoto Yamashita
math.OC