A general analytical solution to the geometrical problem of field matching in radiotherapy

Several authors studied the problem of geometrical matching of fields produced by medical linear accelerators. However, a general solution has yet to be published. Currently available solutions are based on parallelism arguments. This study provides a general solution, considering not only parallelism but also field sizes.

A fixed field with arbitrary field size, gantry, collimator, and couch angle is considered, and another field with a fixed gantry angle is matched to it. A single reference system attached to the treatment couch is used, and two approaches are followed. In the first approach, fixed field sizes are assumed and parallelism of the adjacent field-side planes is imposed. In the second approach, fixed isocenter positions are considered and both parallelism and coincidence between field-side planes are required.

When fixed field sizes are assumed, rotation angles are obtained; however, the isocenters may need to be shifted to make side planes coincident and therefore achieve a proper match. When fixed isocenter positions are considered, solutions for all parameters, including the field size, are obtained and an exact geometrical match is achieved.

General expressions to the field-matching problem are found for the two approaches followed, fixed field sizes, and fixed isocenter positions. These results can be applied to any treatment technique and can easily be implemented in modern treatment planning systems.