Development of methods for beam angle optimization for IMRT using an accelerated exhaustive search strategy

The purpose of this article is to explore the use of the accelerated exhaustive search strategy for developing and validating methods for optimizing beam orientations for intensity-modulated radiation therapy (IMRT). Combining beam-angle optimization (BAO) with intensity distribution optimization is expected to improve the quality of IMRT treatment plans. However, BAO is one of the most difficult problems to solve adequately because of the huge hyperspace of possible beam configurations (e.g., selecting 7 of 36 uniformly spaced coplanar beams would require the intercomparison of 8,347,680 IMRT plans).

An "influence vector" (IV) approximation technique for high-speed estimation of IMRT dose distributions was used in combination with a fast gradient search algorithm (Newton's method) for IMRT optimization. In the IV approximation, it is assumed that the change in intensity of a ray (or bixel) proportionately changes dose along the ray. Evidence is presented that the IV approximation is valid for BAO. The scatter contribution at points away from the ray is accounted for fully in IMRT optimization after the optimum beam orientation has been determined. IVs for all candidate beam angles are generated before the start of optimization. For all subsets of beams selected from a given pool of beams (e.g., 5 of 24 uniformly spaced beams), the distribution of planning scores for the best and the worst plans, optimum angle distributions, dose distributions, and dose-volume histograms (DVH) were analyzed for one prostate and two lung cancer cases. The results of the exhaustive search technique were used to develop a "multiresolution" search strategy. In this approach, a smaller number of beams (e.g., three) is first used to explore the hyperspace of solutions to determine the most preferred and the least preferred directions. The results of such exploration are then used as a starting point for determining an optimum configuration comprising a larger number of beams (e.g., seven). This two-step process is considerably faster than full exhaustive search. The question to be answered was whether the two methods lead to the same or similar solutions. The results of exhaustive search and multiresolution approaches were also compared with a previously published approach that used beam's-eye-view dosimetrics (BEVD).

The relative ranks of plans optimized by an accurate dose calculation method were highly correlated with those of the plans optimized by the fast calculation method (i.e., using the IV approximation), which suggests that an approximate dose calculation algorithm can be used effectively for ranking of plans during BAO. We found that dose distributions and DVH of many beam configurations within a specified subset from a given pool of beams (e.g., 5 of 18) may be clinically indistinguishable and acceptable. Their optimized IMRT scores fall in a narrow range, although beam configurations and dose distributions may be different. We used the frequency distributions as a function of beam angles for the best 100 and the worst 100 plans to determine the most and the least preferred beam angles. We found that the most and the least preferred angle distributions for 3 of 18 configurations were very similar to those for 5, 6, 7, or 8 of 18 or 24 configurations, but the size of the search space was much smaller for the 3 of 18 case. Using fewer than three beams was discovered to be inadequate. This information was used to select the most preferred angles and eliminate the least preferred ones before searching for the optimum angles for the remaining beams. For the cases we studies, the multiresolution strategy produced very similar results to the full exhaustive search. Based on the observation that the worst plans had at least one parallel-opposed pair of beams and virtually all of the best plans had none, we were able to further reduce the size of the search space dramatically by using a pool of only nonparallel-opposed equispaced beams (i.e., 7 of 19 instead of 7 of 36). Another observation was that the probability of finding an optimum configuration in a smaller beam pool is substantially lower than in a larger pool (e.g., 5 of 18 vs. 5 of 24). The implication of this BAO is not very important when a large number of beams (nine or more) is used and vice versa. Our results showed that the plans with fewer but optimally placed beams could be as good as or better than plans using a larger number of unoptimized or uniformly placed beams.

Exhaustive search with fast IMRT algorithms provides a novel and realistic approach to study the characteristics of IMRT dose distributions as a function of beam angles and to design practical BAO strategies for IMRT planning.