# Evaluating equivalent square field size definitions for rectangular small fields

## TL;DR

This study compares different methods to calculate equivalent square field sizes for rectangular small fields in radiation therapy to improve accuracy.

## Contribution

The study identifies the Fogliata model as the most accurate for calculating equivalent square field sizes for rectangular small fields.

## Key findings

- The Fogliata model provided the best accuracy for equivalent square field size calculations.
- The Sterling-Partial Superellipse model also showed good accuracy without using fitting parameters.
- The Cranmer-Sargison and Superellipse models had the largest discrepancies in results.

## Abstract

In the IAEA TRS‐483 Code of Practice (CoP), rectangular small field sizes are approximated to equivalent square small fields using the definition proposed by Cranmer‐Sargison et al. However, the CoP estimates the uncertainties associated with this formula only for rectangular fields with dimensions within the range 0.7<X/Y<1.4.

The objective of the present study was to compare the accuracy of the Cranmer‐Sargison definition with other formulas for equivalent square small fields in the context of measuring field output factors (FOFs) for rectangular small fields, both within and outside the range covered by the CoP.

Measurements were conducted using Gafchromic EBT4 radiochromic films. The models compared included Cranmer‐Sargison, Sterling, Superellipse, Sterling‐Partial Superellipse, Sterling‐Superellipse, Vadash and Bjärngard, and Fogliata. The most accurate definition of equivalent square field size was identified as the one yielding the lowest discrepancy between measured and analytical values, with the log‐likelihood of the measurements selected as the metric. Analytical values were derived using the function introduced by Sauer and Wilbert, which relates FOFs to equivalent square field sizes.

The best results were achieved with the Fogliata model, followed in terms of accuracy by the Sterling‐Partial Superellipse model. The Sterling‐Superellipse and Vadash and Bjärngard models came next. It should be noted that the Sterling‐Partial Superellipse and Sterling‐Superellipse models rely solely on the geometric shape of the irradiation field size, whereas the Fogliata and Vadash and Bjärngard models incorporate a fitting parameter. The Sterling definition, while less accurate than these models, improved upon the Cranmer‐Sargison definition and retained computational simplicity. Finally, the Cranmer‐Sargison and Superellipse models exhibited the largest discrepancies.

This study identified several definitions of equivalent square small field size that could refine the IAEA TRS‐483 CoP by improving the accuracy of field output correction factors for rectangular small fields. Among these definitions, the Fogliata model obtained the best results.