H. Edwin Romeijn, Ravindra K. Ahuja, James F. Dempsey, Arvind Kumar
A column generation approach to radiation therapy treatment planning using aperture modulation

This paper considers the problem of radiation therapy treatment planning for cancer patients. During radiation therapy, beams of radiation pass through a patient. This radiation kills both cancerous and normal cells, so the radiation therapy must be carefully planned to deliver a clinically prescribed dose to certain targets while sparing nearby organs and tissues. Currently, a technique called intensity modulated radiation therapy (IMRT) is considered to be the most effective radiation therapy for many forms of cancer. In IMRT, the patient is irradiated from several different directions. From each direction, one or more irregularly shaped radiation beams of uniform intensity are used to deliver the treatment. This paper deals with the problem of designing a treatment plan for IMRT that determines an optimal set of such shapes (called apertures) and their corresponding intensities. This is in contrast with established two-stage approaches where, in the first phase, each radiation beam is viewed as consisting of a set of individual beamlets, each with its own intensity. A second phase is then needed to approximate and decompose the optimal intensity profile into a set of apertures with corresponding intensities. The problem is formulated as a large-scale convex programming problem, and a column generation approach to deal with its dimensionality is developed. The associated pricing problem determines, in each iteration, one or more apertures to be added to our problem. Several variants of this pricing problem are discussed, each corresponding to a particular set of constraints that the apertures must satisfy in one or more of the currently available commercial IMRT equipment. Polynomial-time algorithms for solving each of these variants of the pricing problem to optimality are presented. Finally, the effectiveness of our approach is demonstrated on clinical data.