Theoretical developments on fast Fourier transform convolution dose calculations in inhomogeneous media.

A theory is presented on dose calculations in inhomogeneous media that takes advantage of fast Fourier transform (FFT) convolution for practical three-dimensional treatment planning using photon beams. While the initial work of Boyer and Mok [Med. Phys. 13, 503-509 (1986)] provided a theory which is based on first principles, it failed to give satisfactory predictions inside inhomogeneities. Subsequently, Zhu and Boyer [Phys. Med. Biol. 35, 351-368 (1990)] showed that their formulas agreed well with measured data, but these formulas were empirically altered from Boyer and Mok's. In this work, Boyer and Mok's first-order theory is extended to include second-order inhomogeneity effects. A new correction dose formula is derived which corrects the first scattered dose due to the presence of inhomogeneities. This correction dose formula works better than Zhu and Boyer's empirical correction dose formula. Furthermore, the primary dose formula used by Zhu and Boyer, which was empirically modified from Boyer and Mok's, is justified theoretically. Clear statements are made about the assumptions and the approximations that enter into the derivation which in turn uncover the limitations of this FFT convolution dose calculation.