The proton arc therapy treatment planning problem is NP-Hard

Proton arc therapy (PAT) is an advanced radiotherapy technique using charged particles in which the radiation device rotates continuously around the patient while irradiating the tumor. Compared to conventional, fixed-angle beam delivery mode, proton arc therapy has the potential to further improve the quality of cancer treatment by delivering accurate radiation dose to tumors while minimizing damage to surrounding healthy tissues. However, the computational complexity of treatment planning in PAT raises challenges as to its effective implementation. In this paper, we demonstrate that designing a PAT plan through algorithmic methods is a NP-hard problem (in fact, NP-complete), where the problem size is determined by the number of discrete irradiation angles from which the radiation can be delivered. This finding highlights the inherent complexity of PAT treatment planning and emphasizes the need for efficient algorithms and heuristics to address the challenges associated with optimizing the delivery of radiation doses in this context.