Lorenzo Arsini ^1,2 Barbara Caccia ³ Andrea Ciardiello ^1,2 Angelica De Gregorio ^2,4 Gaia Franciosini ⁴ Stefano Giagu ^1,2 Susanna Guatelli ^5,6 Annalisa Muscato ^2,4 Francesca Nicolanti ^1,2* Jason Paino ^5,7 Angelo Schiavi ⁴ Carlo Mancini-Terracciano ^1,2

• ¹ Department of Physics, Faculty of Mathematics, Physics, and Natural Sciences, Sapienza University of Rome, Rome, Lazio, Italy
• ² National Institute of Nuclear Physics of Rome, Rome, Lazio, Italy
• ³ National Institute of Health (ISS), Rome, Lazio, Italy
• ⁴ Department of Basic and Applied Sciences for Engineering, Faculty of Civil and Industrial Engineering, Sapienza University of Rome, Rome, Lazio, Italy
• ⁵ Illawarra Health and Medical Research Institute, University of Wollongong, Wollongong, New South Wales, Australia
• ⁶ School of Computing and Information Technology, Faculty of Engineering and Information Sciences, University of Wollongong, Wollongong, New South Wales, Australia
• ⁷ Centre for Medical Radiation Physics, Faculty of Engineering and Information Sciences, University of Wollongong, Wollongong, New South Wales, Australia

External beam radiotherapy (RT) is a one of the most used treatments against cancer, with photon-based RT and particle therapy (PT) being commonly employed modalities. Very highenergy electrons (VHEE) have emerged as promising candidates for novel treatments, particularly in exploiting the FLASH effect, offering potential advantages over traditional modalities. This paper introduces a Deep Learning (DL) model based on Graph Convolutional Networks to compute dose distributions of therapeutic VHEE beams in patient tissues. The model emulates Monte Carlo (MC) simulated dose within a cylindrical volume around the beam, enabling high spatial resolution dose calculation along the beamline while managing memory constraints. Trained on diverse beam orientations and energies, the model exhibits strong generalization to unseen configurations, achieving high accuracy metrics including a δ-index 3% passing rate of 99.8% and average relative error < 1% in integrated dose profiles compared to MC simulations. Notably, the model offers 3 to 6 orders of magnitude speedup over full MC simulations and fast MC codes, generating dose distributions in milliseconds on a single GPU. This speed could enable direct integration into treatment planning optimization algorithms, also leveraging the model's differentiability for exact gradient computation.