A mathematical study of the area over perimeter rule using the sector-integration equation

The equivalence between rectangular photon fields and square fields is currently used to simplify tabulation and handling of data as well as to reduce measurement time. Widely used for routine calculation in the case of rectangular fields of moderate elongation, the so-called "area over perimeter rule" (A/P rule) has a remarkable accuracy. Several approaches have been developed to determine the physical and mathematical grounds of this rule, yet the statement that it is independent of depth and energy was not fully clarified. By means of the Clarkson sector-integration equation and Taylor expansion, this work demonstrates that the A/P rule is a first-order approximation on the elongation variable for all cases, i.e., for moderate rectangular field elongation presents a quadratic deviation. To appreciate the degree of approximation of this rule, a model scatter air ratio for Co60 y-rays at 10.0 cm depth was used to compute rectangular radiation fields and the results were compared to those given by the A/P rule. The model scatter air ratio was the proposed by Day and Aird [Br. J. Radiol. 25, 138-151 (1996)].