4π Radiotherapy Using a Linear Accelerator: A Misnomer in Violation of the Solid Geometric Boundary Conditions in Three-Dimensional Euclidean Space

PURPOSE: The concept of 4π(c) 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 0–2π(c) and a table rotation of [Image: see text]. RESULTS: Total achievable solid angle by cantilever-type linear accelerator (or any teletherapy machine employing a cantilever design) is [Image: see text] , 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 2π(c), 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 1/5(th) (maximum value) of the 4π(c). To obtain a 4π(c) 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 4π(c) solid angle.