A novel approach for the definition of small-field sizes using the concept of superellipse

Abstract

In radiotherapy, field sizes are defined in terms of the dimensions of the irradiation area. However, geometric square fields result in irradiation areas with rounded corners, which become almost elliptical for small fields. Superellipses are a family of curves encompassing shapes lying between ellipses and rectangles. The purpose of this work was to analyze the advantages and disadvantages of a novel approach that describes small-field sizes with superellipses. Square fields with nominal side lengths ranging from 0.5 to 10 cm were irradiated with two different linacs using 6 and 10 MV photon beams with and without flattening filters. Field size dimensions and output factors were measured by employing radiochromic films and the Radiochromic.com software. An alternative definition of equivalent square small-field size based on the superellipse (Sse) was introduced. The degree n of the superellipse for 10 cm nominal fields measured between 14.8 ± 1.0 to 27.7 ± 1.9. However, it decreased with the field size, down to between 2.26 ± 0.10 and 2.64 ± 0.15 for 0.5 cm nominal side lengths. A relation between the degree n and the equivalent square small-field size (Sclin) as defined by Cranmer-Sargison et al. [“A methodological approach to reporting corrected small field relative outputs,” Radiotherapy and Oncology 109, 350–355 (2013)] was found. For nominal side lengths of 10 cm, Sse was between 0.34 ± 0.04% and 0.10 ± 0.01% smaller than Sclin, while for 0.5 cm nominal side length Sse was between 9.5 ± 0.6% and 7.4 ± 0.7% smaller than Sclin. There was no significant difference in the goodness of the regression between using Sse or Sclin to fit field output factors with the function proposed by Sauer and Wilbert. Small fields were found to be more accurately characterized with superellipses. The advantages and disadvantages of describing field sizes with superellipses were examined. Field output factors can be derived with equivalent square small-field sizes based on the superellipse approach.