Solution to treatment planning problems using coordinate transformations

The majority of radiation treatment planning problems are relatively straightforward, involving only specified gantry angles in a treatment plane which is perpendicular to the patient longitudinal axis. In addition, there are a number of more complex three‐dimensional problems which require combined rotation of the gantry, collimator, and turntable for their solutions. These include, for example, the use of non‐coplanar fields and oblique treatment planes, the matching of field edges in three dimensions, the treatment of the breast with opposing tangential fields, and the treatment of inclined elongated lesions. Unfortunately, there is no general systematic approach to the solution of these more complex problems. One may attempt an analytic solution, but this approach is often too cumbersome and tedious. On the other hand, one may resort to a “trial and error” session with the simulator. This paper, therefore, presents a mathematical method which is easily applied and applicable to a wide variety of complex three‐dimensional treatment planning problems. The method considers the gantry, collimator, and turntable as coordinate systems. These coordinate systems are derivable from each other by specified coordinate transformations, which contain the rotation angles of the gantry, collimator, and turntable. Within this mathematical framework, the treatment planning problems are found to reduce to two general types, of which various clinical examples are then given. Key words: treatment planning, coordinate systems, coordinate transformations, three‐dimensional treatment planning, rotation operators