Year: 2026
Type: article
Abstract: Background and Aims: Forward Automatic Differentiation (FAD) is a generic method for computing exact gradients that, in contrast to its reverse counterpart, has minimal memory overhead and scales well for functions with many outputs.Having proven useful in various computational disciplines, we are currently developing an FAD-compatible proton dose calculator targeting two application areas: efficient robustness quantification as presented here and optimisation of spot positions and/or field directions.Methods: Our FAD dose calculation has been implemented using dual numbers, which are first order Taylor expansions (a+b*x) with the imposition of x 2 =0.This definition allows to work with dual numbers in the same way as with ordinary ones, with the first component representing dose whilst the others represent its potentially multivariate gradient.The resulting algebra can be implemented using operator overloading, which allows performing complex calculations with dual numbers using the same syntax as for ordinary floating point operations.Based on this concept, we have written a proton dose engine from scratch in C++20/CUDA that supports tracking of gradient components in every step, from the user interface to the underlying dose algorithm.As a first proof of principle, we demonstrate here how to infer both dose and dose uncertainty maps from a single FAD-based calculation.Results: We tested the method on eight, non-robustly optimised skull-based tumour plans on an Nvidia RTX 4090 GPU. Figure 1 shows an axial slice of the dose and its derivatives jointly computed using FAD, together with relative calculation times, compared to the standard approach of recalculating dose on multiple shifted scenarios.Dose uncertainty maps resulting from different numbers of shifts (5, 9 and 100) and those using the single-pass FAD technique are shown for an example case in Figure 2, together with quantitative metrics. Conclusions:We have successfully applied FAD to simultaneously calculate both proton dose distributions and their gradients.For robustness analysis, preliminary results suggest a performance gain of 3.1 ± 0.5 compared to the most common conventional method with comparable accuracy.Future directions include connecting to polynomial chaos expansions, Monte Carlo simulation and plan optimisation.
Authors: Maximilian Hofer, Antony Lomax, Ye Zhang
Institutions: Paul Scherrer Institute, ETH Zurich
Language: English
Subfield: Radiation
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $1,700
OpenAlex ID: W7213264534