Using weighted power mean for equivalent square estimation

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 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 (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.