Technical note: Determining equivalent squares of high-energetic photon fields.

Purpose: We developed a method based on a physical pencil beam model for accurate equivalent square calculations for rectangular and irregular fields, for different definitions of equivalent squares, for beams with and without flattening filter, different photon energies, and depths in water.

Methods: We considered two equivalent square definitions: equal dose at a point on the beam axis and equal depth dose, measured as tissue phantom ratio at 20 and 10 cm depth ( ). As dose engine, we used an analytical pencil beam model. By integrating the pencil beam kernels, we assigned square fields to rectangular fields minimizing the dose, respectively, the difference. The results were compared with measurements at 100 mm depth for nominal beam energies of 6 and 18 MV, the Sterling equation, the geometric mean, and data from BJR Suppl 25 (British Institute of Radiology, 1996).

Results: Pencil beam results were closest to the measurements. An energy dependence of several millimeters for small field dimensions and depth dependencies for very elongated fields were observed. For the assignment of WFF square to FFF rectangular fields, using the equal- definition, our method agrees with previously published results. For circular fields approximated by leaves, we found deviations to the data from BJR Suppl. 25 below 1 mm for diameters smaller than 200 mm.

Conclusions: Our study shows that the validity range for geometric mean and Sterling equation is limited. Ergo, instead of specifying specific validity ranges, we suggest using the pencil beam method, valid for all aspect ratios, including elongated fields in the primary dose dominated regime. We published our method as python library and graphical user interface on GitHub. Users can choose between two definitions of equivalent square and between WFF and FFF fields. The implemented pencil beam method for irregular fields is also usable for quality assurance such as monitor unit checks.