Fast Superposition Algorithm Using Spectral Kernel Decomposition for Photon Beam Dose Calculation In Heterogeneous Media
Abstract
Purpose
Convolution/superposition is a widely-used method for radiotherapy dose calculation, in which energy deposited at the first interaction point (TERMA) is convolved with a scatter kernel to obtain the dose. To account for tissue heterogeneities, the kernel can be scaled radially by the path-averaged relative electron density between the source and calculation points. However, this scaling renders the kernel spatially-dependent, precluding fast Fourier transform (FFT)-based acceleration. Several algorithms have been developed to circumvent this increased computational cost, including collapsed cone convolution (CCC). Here, an alternative approach is presented that decomposes the heterogeneous dose kernel into a linear combination of spatially invariant basis kernels, restoring FFT-based acceleration.
Methods
The analytical kernel is adopted from CCC, i.e., a sum of exponentials with inverse-square decay. A quadrature grid is defined in kernel space using Nr Gauss-Laguerre radial nodes, L Gauss-Legendre polar nodes, and 2L uniformly-spaced azimuthal nodes. Each node corresponds to a fixed displacement in TERMA (source) space, allowing the kernel to be factored into a coefficient array in source space and a spatially-invariant Laguerre×Legendre basis kernel. The coefficient field includes the heterogeneity scaling factor, evaluated for each node-defined displacement in source space. The final dose is then computed as the weighted sum over all nodes of FFT-based convolutions, with asymptotic complexity O(NrL2N3log(N)).
Results
The algorithm was tested for a 6 MV, 10×10cm2 beam incident on a mediastinum-like water-and-cork phantom. Dose distributions computed with Nr=4-12 and L=4-12 converged quickly to agreement with the Geant4 Monte Carlo estimate, with gamma passing rates consistently exceeding 95% at 3%/2mm tolerance and 10% threshold for Nr≥5 and L≥10.
Conclusion
Spectral kernel decomposition represents a new framework for fast kernel-based dose calculation in heterogeneous media, enabling FFT-based evaluation without restricting heterogeneity dependence to a finite number of directions. Next steps will include algorithmic optimization and comparison with CCC.