A robust optimization method with successive linear programming for intensity-modulated radiation therapy

Intensity-modulated radiation therapy for cancer is considered to be effective when dealing with complicated tumour shapes because the dose distribution for each irradiation can be modulated. Fluence map optimization is often formulated as an optimization problem with dose volume constraints (DVCs). A linear programming (LP) method that approximated DVCs was proposed, and it was modified to the successive LP method (SLPM) to find a feasible treatment plan in a wider region. In the present paper, we propose a numerical method called SLPM-R (the SLPM with robustness) that enhances the SLPM using a robust optimization approach. We mathematically prove that the proposed method with extended LP problems has the favourable properties of the SLPM, even taking uncertainty in the influence matrix into consideration. In particular, when the optimal value of the LP problem is non-positive, the proposed SLPM-R guarantees that the output solution can satisfy all DVCs. Through numerical experiments, we observed that the proposed method found a feasible plan that the SLPM could not find. In addition, for a test case that even the SLPM-R failed, the largest deviations of 5.65 Gray in the SLPM was reduced to 3.15 Gray by the SLPM-R.