Using weighted power mean for equivalent square estimation.

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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.The proposed WPM formula is ESWPMa,b=waα+1-wbα1/α for 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 (Scp ) from the Varian TrueBeam Representative Beam Data (Med Phys. 2012;39:6981-7018), and published Scp data for Varian TrueBeam Edge (J Appl Clin Med Phys. 2015;16:125-148).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.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.





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Zhou, Sumin, Qiuwen Wu, Xiaobo Li, Rongtao Ma, Dandan Zheng, Shuo Wang, Mutian Zhang, Sicong Li, et al. (2017). Using weighted power mean for equivalent square estimation. Journal of applied clinical medical physics, 18(6). pp. 194–199. 10.1002/acm2.12201 Retrieved from

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Qiuwen Wu

Professor of Radiation Oncology

My research interests include intensity-modulated radiation therapy (IMRT), volumetric-modulated arc therapy (VMAT), Dynamic Electron Arc Radiotherapy (DEAR), and image-guided radiation therapy (IGRT). For IMRT, my work includes the development of the research platform, fast and accurate dose calculations, optimization based on physical and biological objectives such as generalized equivalent uniform dose (gEUD), and delivery with a dynamic multi-leaf collimator (DMLC). For VMAT, I am interested in optimization, quality assurance, and novel applications. For DEAR, I'm interested in the treatment planning and delivery verifications. For IGRT, my work includes the development of the infrastructure of the online and offline image guidance, characterization of patient anatomic changes and treatment uncertainties, margin calculations, and adaptive treatment planning. My recent research interests also include the use of AI in treatment planning, Brachytherapy dose calculation and plan optimization.

My clinical interests include prostate cancer, head and neck cancer, total body irradiation (TBI), and total skin irradiation (TSI)

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