Reducing the tongue-and-groove underdosage in MLC (multileaf collimator) shape matrix decomposition

We present an algorithm for optimal step-and-shoot multileaf collimator field segmentation minimizing tongue-and-groove effects. Adapting the concepts of an earlier paper, we characterize the minimal decomposition time as the maximal weight of a path in a properly constructed weighted digraph. We also show that this decomposition time can be realized by a unidirectional plan, thus proving that the algorithm from a second earlier paper is monitor unit optimal in general and not only for unidirectional leaf movement. Our characterization of the minimal decomposition time has the advantage that it can be used to derive a heuristic for the reduction of the number of shape matrices.