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

Biplab Sarkar; Tharmarnadar Ganesh; Anusheel Munshi; Arjunan Manikandan; Bidhu Kalyan Mohanti

doi:10.4103/jmp.JMP_2_19

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

J Med Phys. 2019 Oct-Dec;44(4):283-286. doi: 10.4103/jmp.JMP_2_19. Epub 2019 Dec 11.

Authors

Biplab Sarkar¹, Tharmarnadar Ganesh¹, Anusheel Munshi¹, Arjunan Manikandan², Bidhu Kalyan Mohanti¹

Affiliations

¹ Department of Radiation Oncology, Manipal Hospitals, Dwarka, New Delhi, India.
² Department of Medical Physics, Apollo Proton Cancer Center, Chennai, Tamil Nadu, India.

Abstract

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 .

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 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.

Keywords: 4π; 4π radiotherapy; Euclidean space; linear accelerator; solid angle.