Models for finding treatment plans for intensity modulated radiation therapy are usually based on a number of structure-based treatment plan evaluation criteria, which are often conflicting. Rather than formulating a model that a priori quantifies the trade-offs between these criteria, we consider a multi-criteria optimization approach that aims at finding the so-called undominated treatment plans. We present a unifying framework for studying multi-criteria optimization problems for treatment planning that establishes conditions under which treatment plan evaluation criteria can be transformed into convex criteria while preserving the set of undominated treatment plans. Such transformations are identified for many of the criteria that have been proposed to date, establishing equivalences between these criteria. In addition, it is shown that the use of a nonconvex criterion can often be avoided by transformation to an equivalent convex criterion. In particular, we show that models employing criteria such as tumour control probability, normal tissue complication probability, probability of uncomplicated tumour control, as well as sigmoidal transformations of (generalized) equivalent uniform dose are equivalent to models formulated in terms of separable voxel-based criteria that penalize dose in individual voxels.
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