Accuracy limits of the equivalent field method for irregular photon fields

A mathematical approach is developed to evaluate the accuracy of the equivalent field method using basic clinical photon beam data. This paper presents an analytical calculation of dose errors arising when field equivalencies, calculated at a certain reference depth, are translated to other depths. The phantom scatter summation is expressed as a Riemann-Stieltjes integral and two categories of irregular fields are introduced: uniform and multiform. It is shown that multiform fields produce errors whose magnitudes are nearly twice those corresponding to uniform fields in extreme situations. For uniform field shapes, the maximum, local, relative dose errors, when the equivalencies are calculated at 10 cm depth on the central axis and translated to a depth of 30 cm, are 3.8% and 8.8% for 6 MV and cobalt-60 photon beams, respectively. In terms of maximum dose those errors are within 1-2%. This supports the conclusion that the equivalencies between rectangular fields, which are examples of uniform fields, are applicable to dose ratio functions irrespective of beam energy. However, the magnitude of such errors could be of importance when assessing the exit dose for in vivo monitoring. This work provides a better understanding of the influence of the irregular field shapes on the accuracy of the equivalent field method.