To investigate the feasibility and accuracy of proton dose calculations with artificial neural networks (ANNs) in challenging three-dimensional (3D) anatomies. Methods: A novel proton dose calculation approach was designed based on the application of a long short-term memory (LSTM) network. It processes the 3D geometry as a sequence of two-dimensional (2D) computed tomography slices and outputs a corresponding sequence of 2D slices that forms the 3D dose distribution. The general accuracy of the approach is investigated in comparison to Monte Carlo reference simulations and pencil beam dose calculations. We consider both artificial phantom geometries and clinically realistic lung cases for three different pencil beam energies. Results: For artificial phantom cases, the trained LSTM model achieved a 98.57% γ-index pass rate ([1%, 3 mm]) in comparison to MC simulations for a pencil beam with initial energy 104.25 MeV. For a lung patient case, we observe pass rates of 98.56%, 97.74%, and 94.51% for an initial energy of 67.85, 104.25, and 134.68 MeV, respectively. Applying the LSTM dose calculation on patient cases that were fully excluded from the training process yields an average γ-index pass rate of 97.85%. Conclusions: LSTM networks are well suited for proton dose calculation tasks. Further research, especially regarding model generalization and computational performance in comparison to established dose calculation methods, is warranted.

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