Comment on `Linear energy transfer incorporated intensity modulated proton therapy optimization'
aa r X i v : . [ phy s i c s . m e d - ph ] J a n Comment on ‘Linear energy transfer incorporatedintensity modulated proton therapy optimization’
Bram L. Gorissen
Department of Radiation Oncology, Massachusetts General Hospital and Harvard Medical SchoolBoston, Massachusetts
Abstract
Cao et al (2018) published an article on inverse planning based on dose-averagedlinear energy transfer (LET). Their claim that the problem can be cast as a linearoptimization model relies on an incorrect application of the Charnes-Cooper trans-formation. In this comment we show that their linear model is simlar to one wheredose-averaged LET is multiplied with dose, explaining why their model was nonethelessable to improve the LET distribution.
This is an author-created, un-copyedited version of an article published in Physics in Medicine and BiologyDOI:10.1088/1361-6560/aaffa6.
With great interest we read the recently published article by Cao et al. (2018) on linear en-ergy transfer (LET) based inverse planning for intensity-modulated proton therapy (IMPT).The biological effectiveness of IMPT is at least correlated with LET (Peeler et al., 2016),and therefore, it is desirable to take LET into account when designing a treatment plan. Caoet al. proposed a linear optimization model that minimizes LET in organs at risk (OARs)and maximizes LET in the target. However, the derivation of their model is erroneous dueto which the solutions are suboptimal with respect to dose and LET.The authors claim to optimize a plan based on the following objective function: f ( w ) = λ + T | T | X i ∈ T max { , d i − d pr i } + λ − T | T | X i ∈ T max { , d pr i − d i } (1)+ λ O | O | X i ∈ O max { , d i − d max i } + λ N | N | X i ∈ N d i (2) − θ T | T | X i ∈ T l i + θ O | O | X i ∈ O l i , (3)where the dose in voxel i is given by d i = P j d ij w j with w j the weight of pencil beam j ,and l i is the dose-averaged LET P j D ij L ij w j / P j D ij w j . The first two lines of this objectivefunction are well known in inverse planning: the terms in (1) incur a penalty when the dosein target T is below or above the prescribed dose d pr i , while the terms in (2) penalize dose1bove the maximum prescribed dose d max i in OARs O and total dose in normal tissue N . Thenovel part are the terms in (3) that maximize dose averaged LET in the target and minimizedose averaged LET in the OARs. The weights λ and θ control the trade-off between thedifferent objectives, and were tuned for each patient to yield similar physical dose as previousclinical plans.It is well known that the terms in (1) and (2) can be linearized and that it is easy to finda global optimum for just these terms. It is considerably harder to optimize (3), which is asum of fractions of linear functions after substituting the formula for l i . Cao et al. (2018)claim that the variable transformation proposed by Charnes and Cooper (1962) results inan equivalent formulation that is linear. This cannot be true, because this transformationcan be applied to a single fraction only , not to a sum of fractions unless all denominators arethe same. Although it is hard to prove that a linear reformulation of (3) does not exist, itis clear that Cao et al. (2018) have used a method that is not suitable.What we could ask ourselves is what Cao et al. have actually optimized, and if there isperhaps an explanation as to why their optimization problem finds treatment plans with animproved LET distribution. It is impossible to answer this question with certainty based onthe information reported in the paper. The authors mention a substitution t = 1 /d i , butsince the left hand side does not depend on i , it is not clear which voxel was used to define t . If we ignore t , the final objective used by Cao et al. is: g ( x ) = λ + T | T | X i ∈ T max { , d i − d pr i } + λ − T | T | X i ∈ T max { , d pr i − d i } + λ O | O | X i ∈ O max { , d i − d max i } + λ N | N | X i ∈ N d i − θ T | T | X i ∈ T ( ld ) i + θ O | O | X i ∈ O ( ld ) i , (4)where ( ld ) i is the dose-averaged LET multiplied with dose: P j D ij L ij x j . Optimization prob-lems with ( ld ) i have been proposed as a first approximation to the additional biological dosedue to high LET (Unkelbach et al., 2016). Cao et al. seemingly missed this interpretationof (4), and did not directly use the pencil beam weights x that are optimal to the objectivefunction (4). Instead, they took the optimal x , and a posteriori scaled it with t to ensureconstraint satisfaction. This step is rather unnecessary since the constraints (omitted herefor brevity) could have been formulated in a way that an a posteriori scaling is not needed.Our expectation is that t was sufficiently close to 1 for all patients, so that the presentedscaled treatment plans were near-optimal to (4).Everything considered, the results by Cao et al. are not as groundbreaking as they mayseem. In contrast to what the authors claim, they have not directly optimized dose averagedLET, but instead used a model where dose averaged LET was multiplied with dose. Modelswith such terms are well known and have a meaningful interpretation, which the authorsseemingly missed. Due to the added scaling factor t , the interpretability and optimality oftheir solution is impaired. 2 cknowledgment Supported in part by NIH U19 Grant 5U19CA021239-38.
References
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