Projections onto the Pareto surface in multicriteria 1 radiation therapy optimization

Abstract


INTRODUCTION
1 Introduction 31 Radiation therapy treatment planning is generally guided towards fulfilment of a set 32 of physician-defined plan evaluation criteria. These criteria are sometimes incompat-33 ible, and the treatment planner is therefore asked to find a suitable tradeoff between proved without deteriorating at least one of the others is consider optimal (Pareto 50 optimal), see, e.g., Miettinen [21]. 51 The benefits of navigation are at the cost of that interpolation between precal-52 culated plans introduces an error to Pareto optimality. Algorithms exist that can 53 bound the magnitude of this error [4, 5,27], but the number of plans that are required 54 to maintain a given error bound increases exponentially with the number of objec-55 tives in the worst case (because hypervolume grows exponentially with increasing 56 dimension). To some relief, studies report that the relation between the required 57 number of plans and the number of objectives is more benign for radiation therapy 58 optimization [8,10]. Nevertheless, Craft and Bortfeld [10] and Bokrantz [4] both 59 observed approximation errors above 10 % for representations with less than about 60 20 plans, and that up to about 75 plans are needed to reduce the error below 5 %. 61 These studies considered between five to ten objectives.

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In view of these concerns, we present a technique that eliminates or reduces the 63 error to Pareto optimality through the minimization of a projective distance between 64 the navigated plan and the Pareto surface. We use a formulation of this projection as 65 in Nakayama [23], which attempts to find a plan that is at least as good as or better 66 than the navigated plan with respect to all objectives. We also suggest an augmented 67 formulation where constraints are imposed on maintained dose-volume histogram 68 (DVH) quality. We quantify the dosimetric benefit of the suggested technique by 69 application to planning for step-and-shoot intensity-modulated radiation therapy 70 (ss-IMRT), sliding window intensity-modulated radiation therapy (sw-IMRT), and 71 spot-scanned intensity-modulated proton therapy (IMPT). We formulate treatment planning for radiation therapy as a multicriteria optimiza-75 tion problem with n objective functions f 1 , . . . , f n that are to be minimized with 76 respect to the vector of variables x. The minimization occurs over a feasible set X 77 that represents the physical limitations of the delivery method and, possibly, con-78 straints on the planned dose, according to We understand optimality to this formulation in a Pareto sense, meaning that a Formulation (1) has an infinite number of Pareto optimal solutions in general. To 85 solve this problem from a practical perspective therefore entails to select the single, 86 best preferred, Pareto optimal solution. We perform this selection by Pareto surface 87 navigation, meaning that a representative set of Pareto optimal solutions x 1 , . . . , x m 88 is first calculated and a convex combinationx = m j=1 λ j x j of these solutions then 89 selected, where the components of λ need to be nonnegative and sum to unity.

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The selection is guided by a navigation interface that permits the priorities of each 91 objective to be continuously adjusted using associated slider bar controls. The sliders 92 are coupled to an algorithm that updates λ accordingly, see, e.g., Craft et al. [11] and 93 Monz et al. [22]. For the navigation to be valid, we assume that X is a nonempty 94 and convex set and that all functions f 1 , . . . , f n are convex and bounded on X . The navigated planx is feasible thanks to the assumed convexity of formulation (1).

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It also has objective values that are bounded by Jensen's inequality for convex func- . . , n, as illustrated in Figure 1

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To mitigate that the navigated planx in general is not Pareto optimal, we pro-106 pose to convert this plan into a Pareto optimal plan with at least as good or better 107 performance in all objectives. Specifically, we propose to solve the following opti-approximation error mization problem as a post-processing step to the navigation: where z * is the ideal point, i.e., z * i = min j=1,...,m f i (x j ) for i = 1, . . . , n. This formu-110 lation is a variant of an achievement function suggested by Wierzbicki [31], which

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The navigated planx is a feasible solution to (2) with an objective value of 118 zero. An optimal solution x * to (2) therefore satisfies meaning that x * is as least as good as or better than the navigated plan with respect 120 to all objectives. Further, x * is weakly Pareto optimal because the achievement Pareto optimal but not Pareto optimal are to be avoided, it is possible to augment the 123 objective function of (2) with the term ρ n i=1  the prescription level for targets and at zero for organs at risk (OARs), see Figure 3. 146 These requirements ensure that a DVH criterion that is satisfied by the navigated 147 plan cannot become violated after the projection.

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The DVH requirements are implemented using functions that impose a one-sided 149 penalty on the error between the DVH curves associated with the current dose d where v denotes cumulative volume in percent, d(q) is the dose to some point q ∈ V, and |V| is the volume of V. In other words, D(v; d) is the DVH point associated with the some dose d and cumulative volume v. Let alsod denote the prescription level for targets and zero for OARs,v denote the cumulative volume such thatd = D(v,d), and (·) + denote the positive part operator max{·, 0}. Then, a min reference DVH constraint takes the form The requirements in Figure 3  The projections were evaluated with respect to three delivery techniques: ss-IMRT, e.g., Thieke et al. [29].

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The approximation (3) means that the results in Section 2.2 do not hold rigorously for our numerical implementation. In particular, it is possible that the projection can lead to a mild degradation in objective function value compared to the navigated point for some objectives. Nevertheless, formulation (3) amounts to minimization of a strongly increasing achievement function, and it therefore finds Pareto optimal points [21, Theorem 3.5.4]. If it is critical to maintain objective function values exactly, then an everywhere differentiable epigraph reformulation of (2) according to x ∈ X ,    The projection of a navigated plan onto the Pareto surface led to improved OAR   Table 3: Mean two-sided and one-sided differences along the dose axis between DVH curves of projected plans and corresponding plans generated without performing any projection.  We have presented a method that eliminates or reduces the error to Pareto optimality 307 that arises during Pareto surface navigation. The error is removed through mini-308 mization of a projective distance to the ideal point in the objective function space.

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An augmented form of the projection was also suggested where the DVH distribu-  Tables 4 and 5.
319 Table 4: Optimization formulation for the prostate case. The reference dose level of a function is denotedd. The constraints used during proton and photon therapy planning are indicated by "Pr" and "Ph" in subscript, respectively.