Equivalent squares of circular electron fields

Background/Purpose

It is useful to be able to make manual estimates of the output (dose rate) of shaped electron fields. Such estimates can be used to check the treatment planning computed output. For rectangular (or approximately rectangular) fields, the square root rule may be used. Many electron apertures, however, are approximately circular and therefore a method for finding the equivalent square from the radius of circular apertures would be useful.

Methods

A grid of Monte Carlo calculated output values for a variety of square and circular field sizes, applicators, and beam energies of 6 and 15 MeV has been used to find an expression for the equivalent square of circular fields. This equivalence has been tested using a larger data set. The test data set consists of electron energies of 6, 9, 12, and 15 MeV and radii ranging from 1.1 to 5.5 cm, for applicators of size 6 × 6, 10 × 10, 14 × 14, 20 × 20, and 25 × 25 cm². A total of 104 combinations of these parameters have been tested. Some clinical examples are provided to demonstrate how the equivalent square may be used to check treatment planning system MU calculations. Comparisons are made with measured output.

Results/Conclusions

The results show that there is an exceptionally simple and remarkably accurate relationship for the equivalent square of circular electron fields, namely: X = 1.83 R, where R is the radius and X is the side length of the equivalent square. This is approximately midway between the two limiting values predicted by Fermi–Eyges theory ($\sqrt{\pi }\le X/R\le 2$).

For the 104 combinations of the parameters described above, the average ratio of the circular field output to the equivalent square output is 1.000, and the standard deviation is 0.003. In every case, the accuracy is better than 1% and, in most cases, better than 0.5%. Almost all the ratios fall within the ±0.4% accuracy expected based on the statistical uncertainty in the Monte Carlo calculations. It is shown that the equivalent square rule for circles is more accurate than the square root rule for a range of common widths, W, and L/W, where L is the length of the rectangle. For the clinical examples cited, the agreement between estimated and measured output is within a few percent.