A mathematical basis for selection of wedge angle and orientation

The treatment plan optimization criterion that dose be homogeneous over an irradiated volume is equivalent to the criterion that the magnitude of the dose gradient be zero throughout that volume. If the dose gradient due to an individual beam is represented by a vector, the dose gradient due to an ensemble of beams is given by the weighted vector sum of the constituent beams' individual gradients. Given a fixed ensemble of beams, the two ways in which the total dose gradient can be modified are (1) by changes in relative beam weights and/or (2) by changes in the direction and/or magnitude of the dose gradient of one or more of the individual beams. Conventional wedges provide a simple mechanism for altering the dose gradient of a single beam. This paper describes a mathematical basis for the selection of wedge angles, wedge orientations, and relative beam weights, with the goal of producing a field of zero gradient over the volume of beam intersection. The approach is based on 3-dimensional vector analysis of dose gradients, and is valuable not only for its formalism, but also for the conceptual basis it provides for discussing and solving general wedge selection problems.