Three-dimensional fourier transformation calculation of the radiation dose received from a60Co source

Abstract

The main difficulty of the determination of the absorbed dose distribution in a patient's body consists of calculating the scattered radiation contribution. As shown in [4,9,12], the error of calculation of the scattered radiation dose by simple one-dimensional methods (calculation of effective radiological thickness, Bataut's equation, etc.) in some cases can be rather large (up to 20% and more in case of elaborate heterogenic structures).As shown in [5][6][7], the calculation of the dose absorbed by a three-dimensional object can be reduced using several simplifying assumptions to solving the equation of convolution in three-dimensionai space: D(r) = IIf d~(r')p(r')K(r-r')dv ,V where dp(r ~) is the nonscattered photon flux in the irradiated object; 9(r') is the electron density distribution in the object; K(r -r) is the convolution core determining the contribution of photons scattered in the elementary volume (voxel) dv with the coordinates r' into the dose absorbed in the elementary volume with the coordinates r. The convolution equation can be solved using an efficient method based on Fourier transformation. According to the convolution theorem [13], convolution of two functions is equal to the product of their Fourier transforms: f(x) is the product qb(r)p(r); the convolution core g(x), the scattering core K(r -r).If operating with discrete data, the calculation algorithm can be reduced to the three-dimensional fast Fourier transformation (FFT) of two three-dimensional arrays (the function to be convoluted and the scattering core), multiplication of arrays, and inverse FFT of the resulting array.The method of the absorbed dose calculation described above was proposed in the 1980s [5][6][7]. However, in spite of approximately 5000-fold acceleration of calculation provided by this method as compared with direct integration, it did not find wide application because its implementation requires rather powerful computers. According to our estimates [2], modern PCs allow implementation of this method. The goal of this work was to develop special software for implementing this method, construct a convenient physical model of the scattering core K(r -r'), and determine the program operation rate and the accuracy of the results obtained. A 6°Co source of gamma-radiation was used in the test. Dose Calculation ModelThe dose distribution was calculated for two types of irradiated objects: a cubical tissue-equivalent phantom of side 32 cm irradiated from a distance of 75 cm (irradiation field, from 4 x 4 cm to 20 x 20 cm) and a tissue-equivalent cylinder 12 cm in diameter (irradiation field, 20 x 20 cm; the distance from the radiation source to the cylinder axis, 75 cm). The results of calculation were compared to the data contained in standard dose field atlases and to the results of calculation by the Monte Carlo technique.