The equivalence of “biological” and “physical” fluence map optimization models for IMRT

Abstract

We present and explain some recent theoretical finding regarding the equivalence of many intensity modulated radiation therapy (IMRT) fluence map optimization (FMO) models based on both “biological” and “physical” treatment plan criteria. We will present and discuss the clinical and algorithmic significance of this otherwise highly abstract and mathematical work. Our work is best described as a mathematical framework for studying treatment plan evaluation criteria. The framework consists of many mathematical theorems and proofs relating to multi-criteria optimization. We concisely capture the essence of our work with a central theorem that states: applying an increasing function to treatment plan evaluation criteria does not affect the corresponding set of Pareto efficient treatment plans: Theorem 1. Let G1(x),...,GL(x) denote a set of L convex treatment plan evaluation criteria (e.g., structure based EUD criteria), viewed as functions of the vector of beamlet intensities x, for FMO. Moreover, let h1,...,hL be increasing functions. Then the multi-criteria optimization formulations of FMO based on G1(x),...,GL(x) has the same set of efficient treatment plans as the multi-criteria optimization formulations of FMO based on h1(G1(x)),...,hL(GL(x)). We applied the framework to many convex and non-convex structure-based treatment plan evaluation criteria which have previously been published and are under development, including TCP, NTCP, P+, EUD, gEUD, and sigmoidal transformations of EUD and gEUD. In particular, we demonstrated that models employing these criteria are equivalent to two models: dose power functions and exponential dose functions formulated in terms of voxel-based criteria that penalize dose in individual voxels, i.e. traditional voxel-based penalty functions. By equivalent we mean to say that the set of efficient solutions in a multi-critera model are identical. Theorem 1 also implies that the potential additional complexity introduced by applying the increasing transformations, say from penalty function to gEUD to NTCP, does not change the trade-offs that are being considered in the FMO problem. Put differently, this theorem states that if a suitable decomposition of a given nonconvex criterion into a convex criterion and an increasing function can be found, the nonconvex criterion may be replaced by an equivalent convex criterion. As is well know, convex models are efficiently solvable while nonconvex models must be dealt with using heuristic approaches to optimization. Casting FMO models as multi-criteria models has shown that many sets of treatment plan evaluation criteria that have been proposed in the literature are in fact equivalent with respect to the corresponding set of efficient treatment plans, i.e. any efficient plan found with one approach is accessible by any other. In addition, we have established conditions under which nonconvex criteria can be transformed into convex criteria without changing the set of efficient treatment plans. Thus, the presented framework identifies which of the proposed treatment plan evaluation criteria are truly different, which enables researchers to focus their efforts on comparing nonequivalent models. In particular, we have shown that among the “biological” criteria of EUD, gEUD, TCP, and NTCP, as well as two sigmoidal transformations of EUD, only two truly distinct criteria exist. This result should alleviate concerns on the pros and cons of using biological vs. physical criteria that was recently debated in the literature, as they are in essence equivalent. To avoid duplication of efforts, in the pursuit of new treatment plan evaluation criteria one should consider the unifying framework to determine whether potentially new criteria indeed yield different sets of efficient treatment plans and therefore truly new models