Theoretical Analysis of the Equivalent Square Field by Means of the Pencil Beam Model

Abstract

The supportive instrument of the equivalent square field associated with a given rectangular radiation field is still frequently applied in photon beam treatment planning. The well-known empirical approximation formula q = 2ab/(a+b), where 2a, 2b und 2q are the side lengths of the given rectangular cross section respectively of the square field with approximately the same central axis depth dose values, is here derived as a first order approximation from the previously published triple Gaussian pencil beam model. The deviation of this formula from the theoretical model and from experimental data has throughout positive sign and usually, e. g. for a 10 × 20 cm2 field, varies from 2 % to 4 %. For extreme elongations, e. g. for a 1 × 40 cm2 field, the error of q caused by this formula however exceeds 20 %. With the help of the pencil beam model, a more accurate approximation formula for q is now derived, whose deviation from the correct equivalent square field never exceeds 1 %. Values of q derived from the pencil beam model are also given in tabular form, and the size of the equivalent square field is studied in dependence upon depth in water and for various photon spectra. For practical applications of the equivalent square concept, the present study is thought as an incentive to establish exact tabulations of the equivalent square field size by dedicated measurements.