We prove the non-existence of ovoids in finite orthogonal spaces of type O 2 n ϩ 1 ( q ) for n у 4 .

At an intermediate stage of radiation treatment planning for IMRT, most commercial treatment planning systems for IMRT generate intensity maps that describe the grid of beamlet intensities for each beam angle. Intensity map segmentation of the matrix of individual beamlet intensities into a set of MLC apertures and corresponding intensities is then required in order to produce an actual radiation delivery plan for clinical use. Mathematically, this is a very difficult combinatorial optimization problem, especially when mechanical limitations of the MLC lead to many constraints on aperture shape, and setup times for apertures make the number of apertures an important factor in overall treatment time. We have developed, implemented and tested on clinical cases a metaheuristic (that is, a method that provides a framework to guide the repeated application of another heuristic) that efficiently generates very high-quality (low aperture number) segmentations. Our computational results demonstrate that the number of beam apertures and monitor units in the treatment plans resulting from our approach is significantly smaller than the corresponding values for treatment plans generated by the heuristics embedded in a widely use commercial system. We also contrast the excellent results of our fast and robust metaheuristic with results from an 'exact' method, branch-and-cut, which attempts to construct optimal solutions, but, within clinically acceptable time limits, generally fails to produce good solutions, especially for intensity maps with more than five intensity levels. Finally, we show that in no instance is there a clinically significant change of quality associated with our more efficient plans.

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