Direct spatial convolution for pencil beam dose calculation and feasibility of dij-free IMRT optimization on GPU

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• Picked for Medical Physics (top studies of the week, 4 October 2026): Spatial convolution for pencil‑beam dose calculation

Abstract

PURPOSE: In pencil beam dose calculation, the convolution is usually performed in the frequency domain via FFT. The dose is assembled into a sparse dose-influence matrix (dij) for optimization. We propose to perform the convolution directly in the spatial domain with a deep-learning operator (Conv2d), and investigate the feasibility of a dij-free optimization. METHODS: A 6 MV photon kernel (1 Gaussian penumbra and 3 singular value decomposition (SVD) scatter components, 0.5 mm resolution) was adopted. The Conv2d operator was validated against FFT on head-and-neck (HN), prostate and liver cases, with 7, 5, and 4 beams and corresponding 7503, 2649, and 845 bixels. The cost of building dij was investigated to identify the bottleneck. A one-layer network optimization framework was implemented to investigate the feasibility of the dij-free optimization (workflow A). Workflow A recomputes the dose with Conv2D at each iteration, with no pre-calculated dij. The traditional dij-based workflow (B) was implemented for comparison. RESULTS: Across all 16 beams Conv2d reproduced FFT at machine precision (maximum absolute dose difference 1.35 × 10-13; Pearson correlation 1.0). The computational time of FFT versus Conv2d was 0.995s versus 0.502s (HN), 0.650s versus 0.199s (prostate), and 0.370s versus 0.082s (liver). Dij construction time was 132.9s, 76.4s, and 31.9s, with assembly accounting for ∼89%. Workflow A performs no build but a full forward and backward convolution (2.166s, 0.832s, 0.368s) per-iteration; workflow B performs the one-time dij build (130.2s, 76.0s, 31.5s), then a sparse matrix-vector product (0.25s, 0.18s, 0.15s). The break-even points were 69, 116, and 145 iterations. CONCLUSIONS: Pencil beam kernel convolution can be formulated as Conv2d spatial convolution with machine-precision equivalence to FFT and competitive speed. The spatial-domain formulation enables a differentiable, dij-free optimization workflow that recomputes dose without storing a dose-influence matrix and that, for short optimization runs, is competitive with the traditional FFT-built sparse dij pipeline.

Abstract as published, via PubMed.