Lorentz function convolution kernel of narrow photon beam profiles in homogeneous and inhomogeneous media

Abstract

A successful method for the mathematical description of the lateral dose profiles of narrow photon beams for IMRT treatment is herewith presented. Based on the measurement of dose profiles in homogeneous water equivalent absorbers and on their analysis by deconvolution in Fourier space, the Lorentz function 1/(x2+λ 2) has been identified, in a good approximation, as the convolution kernel of narrow photon beam dose profiles [1,2,3]. Its full half width is 2λ and its lateral tails are more strongly expressed compared to Gaussian kernels. Convolutions of this kernel with rectangular and wedge-shaped photon fluence profiles have been performed analytically, and good agreement of the resulting lateral dose profiles with measured dose profiles has been obtained at accelerators of various manufacturers by adjusting the λ values. Complete knowledge of all measured profiles has been stored in condensed form by storing the λ values, which vary with the size of the effective photon source of the accelerator, with photon energy and with absorber thickness. Lateral dose profiles of photon beams have also been measured in inhomogeneous absorbers, e.g. in a water equivalent phantom with an air layer of variable thickness in between phantom plates. By Fourier analysis, each dose profile measured at the bottom of such air layer has been identified as the sum of two components, one due to the secondary electrons moving away from the upper phantom plate, the other due to backscattered secondary electrons originating from the lower phantom plate. The variation of these components as a function of air layer thickness has been quantitatively evaluated in order to facilitate dose calculations for points at the bottom of an air-filled cavity.