4π radiotherapy using a linear accelerator: A misnomer in violation of the solid geometric boundary conditions in three-dimensional Euclidean space

Abstract

Purpose:The concept of 4πc radiotherapy is a radiotherapy planning technique receiving much attention in recent times. The aim of this article is to disprove the feasibility of the 4π radiotherapy using a cantilever-type linear accelerator or any other external-beam delivery machines.Materials and Methods:A surface integral-based mathematical derivation for the maximum achievable solid angle for a linear accelerator was carried out respecting the rotational boundary conditions for gantry and couch in three-dimensional Euclidean space. The allowed movements include a gantry rotation of 0–2πc and a table rotation of .Results:Total achievable solid angle by cantilever-type linear accelerator (or any teletherapy machine employing a cantilever design) is , which is applicable only for the foot and brain radiotherapy where the allowed table rotation is 90°–0°–270°. For other sites such as pelvis, thorax, or abdomen, achievable solid angle as the couch rotation comes down significantly. Practically, only suitable couch angle is 0° by avoiding gantry–couch–patient collision.Conclusions:Present cantilever design of linear accelerator prevents achieving a 4π radian solid angle at any point in the patient. Even the most modern therapy machines like CyberKnife which has a robotic arm also cannot achieve 4π geometry. Maximum achievable solid angle under the highest allowable boundary condition(s) cannot exceed 2πc, which is restricted for only extremities such as foot and brain radiotherapy. For other parts of the body such as pelvis, thorax, and abdomen, the solid angle is reduced to 1/5th (maximum value) of the 4πc. To obtain a 4πc solid angle in a three-dimensional Euclidean space, the patient has to be a zero-dimensional point and X-ray head of the linear accelerator has a freedom to rotate in every point of a hypothetical sphere of radius 1 m. This article establishes geometrically why it is not possible to achieve a 4πc solid angle.

In regards to Pokhrel et al. Clinical validation of ring‐mounted halcyon linac for lung SBRT: Comparison to SBRT‐dedicated C‐arm linac treatments. JACMP 2021 Jan;22(1):261‐70

“…Several groups have tried multiple non-coplanar beam arrangements for SBRT on different sites like lung or liver. 4 This technique is often associated with a mathematically incorrect terminology, namely "4π geometry"which in reality uses a limiting 2π [ lim Ω!2π Ω where Ω is the solid angle] geometry beam-arrangement approach to obtain a larger solid angle at the target [4][5][6] which results in a better dose distribution compared to that of a conventional coplanar beam geometry. 4 Even though the differences in delivery parameters like unflatten beam, higher MLC speed, higher gantry speed, and higher dose rate are in its favor, the Halcyon planning technique is seriously limited by the lack of non-coplanar beam geometry which results in a larger low-dose volume.…”