Using weighted power mean for equivalent square estimation.

Area of Science:

• Medical Physics
• Radiotherapy Physics
• Computational Dosimetry

Background:

• Equivalent Square (ES) is crucial for calculating radiation quantities in radiotherapy, but its accuracy for elongated fields needs improvement.
• Existing ES formulas may not adequately address the complexities of modern treatment fields, particularly those that are very elongated and small.

Purpose of the Study:

• Introduce a novel explicit Equivalent Square formula based on the Weighted Power Mean (WPM) function.
• Compare the performance of the WPM formula against established Sterling and Vadash/Bjärngard formulas for accuracy in radiation field calculations.

Main Methods:

• Developed the WPM formula: ESWPMa,b=waα+1-wbα1/α for rectangular photon fields.
• Evaluated formula performance using standard deviation of residual error, maximum relative prediction error, and Akaike Information Criterion (AIC).
• Tested datasets included British Journal of Radiology (BJR) ES data and Varian TrueBeam photon output factors (S_cp).

Main Results:

• The WPM formula achieved a 20% reduction in ES estimation residual error compared to traditional formulas on the BJR dataset.
• Reduced maximum relative error from 3.5% (Sterling) or 2% (Vadash/Bjärngard) to 0.7% for Varian datasets (3-40 cm fields).
• The WPM formula consistently showed lower AIC values, indicating superior performance, especially for very elongated small fields.

Conclusions:

• The Weighted Power Mean (WPM) formula demonstrates superior performance over traditional methods in Equivalent Square calculations.
• Adopting the WPM formula in treatment planning and secondary MU checks is expected to improve photon output factor estimation for modern radiotherapy fields.
• The WPM formula offers enhanced accuracy for increasingly utilized very elongated, small rectangular fields in radiotherapy.