Open AccessUsing weighted power mean for equivalent square estimation
Author(s) -
Zhou Sumin,
Wu Qiuwen,
Li Xiaobo,
Ma Rongtao,
Zheng Dandan,
Wang Shuo,
Zhang Mutian,
Li Sicong,
Lei Yu,
Fan Qiyong,
Hyun Megan,
Diener Tyler,
Enke Charles
Publication year - 2017
Publication title -
journal of applied clinical medical physics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.83
H-Index - 48
ISSN - 1526-9914
DOI - 10.1002/acm2.12201
Subject(s) - truebeam , akaike information criterion , mathematics , standard deviation , residual , statistics , mean squared error , square (algebra) , field (mathematics) , table (database) , nuclear medicine , algorithm , optics , physics , beam (structure) , geometry , medicine , linear particle accelerator , computer science , pure mathematics , data mining
Purpose Equivalent Square ( ES ) enables the calculation of many radiation quantities for rectangular treatment fields, based only on measurements from square fields. While it is widely applied in radiotherapy, its accuracy, especially for extremely elongated fields, still leaves room for improvement. In this study, we introduce a novel explicit ES formula based on Weighted Power Mean ( WPM ) function and compare its performance with the Sterling formula and Vadash/Bjärngard's formula. Methods The proposed WPM formula is E S W P Ma , b = wa α + 1 − wb α1 / αfor a rectangular photon field with sides a and b. The formula performance was evaluated by three methods: standard deviation of model fitting residual error, maximum relative model prediction error, and model's Akaike Information Criterion ( AIC ). Testing datasets included the ES table from British Journal of Radiology ( BJR ), photon output factors ( S cp ) from the Varian TrueBeam Representative Beam Data (Med Phys. 2012;39:6981–7018), and published S cp data for Varian TrueBeam Edge ( J Appl Clin Med Phys . 2015;16:125‐148). Results For the BJR dataset, the best‐fit parameter value α = −1.25 achieved a 20% reduction in standard deviation in ES estimation residual error compared with the two established formulae. For the two Varian datasets, employing WPM reduced the maximum relative error from 3.5% (Sterling) or 2% (Vadash/Bjärngard) to 0.7% for open field sizes ranging from 3 cm to 40 cm, and the reduction was even more prominent for 1 cm field sizes on Edge ( J Appl Clin Med Phys . 2015; 16 :125–148). The AIC value of the WPM formula was consistently lower than its counterparts from the traditional formulae on photon output factors, most prominent on very elongated small fields. Conclusion The WPM formula outperformed the traditional formulae on three testing datasets. With increasing utilization of very elongated, small rectangular fields in modern radiotherapy, improved photon output factor estimation is expected by adopting the WPM formula in treatment planning and secondary MU check.