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The problem of optimizing beam orientations for irradiations with multiple fixed beams is investigated. It is shown that this is a complex, in mathematical terms 'non-convex', optimization problem, whose solution requires sophisticated techniques. In this work, the optimization is performed with the method of simulated annealing. In order to keep the calculation time within reasonable limits, the problem is expressed in the spatial frequency domain using Parseval's theorem. All calculations are then performed in the frequency domain. The algorithm is described in detail. Various treatment techniques, including intensity modulation, are considered. The results for various exemplary cases are presented. They are based on a simplified dose calculation model. A general conclusion is that the optimum beam configuration for multiple-beam irradiations (with more than three beams) tends to be an even distribution over an angular range of 0 to 2 pi.
Bortfeld et al. (Mon,) studied this question.