On the non-uniqueness of optimal radiation treatment plans

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Abstract
The possibility of multiple locally optimal dose distributions in radiation treatment planning has been discussed and documented in the literature. Here we study a different question related to uniqueness: Is it possible for different treatment plans to generate the same dose distribution? For greatly simplified two-dimensional model problems, we show that the answer is `yes' in regions where two or more beams intersect. In realistic problems, those are of course not the only regions of interest. However, as a result of cancellations in regions of intersection, substantial perturbations of beam profiles in certain directions may still have only small effects on the dose distribution. This is interesting because it offers an opportunity to optimize some other useful property, for instance simplicity, among all treatment plans generating a desired dose distribution with sufficient accuracy. We take a first step beyond our model problems by proving the stability of our results with respect to small perturbations of problem parameters. Since realistic problems differ from our model problems by much more than small perturbations, we plan to present a numerical study of more realistic examples in a sequel to this article.

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