A duality based algorithm for multileaf collimator field segmentation with interleaf collision constraint

Intensity maps are nonnegative matrices describing the intensity modulation of beams in radiotherapy. An important step in the planning process is to determine a segmentation, that is a representation of an intensity map as a positive combination of special matrices corresponding to fixed positions of the multileaf collimator, called segments. We consider the problem of constructing segmentations with small total numbers of monitor units and segments. Generalizing the approach of Engel [Discrete Appl. Math., in press, doi:10.1016/j.dam.2004.10.007] so that it applies to the segmentation problem with interleaf collision constraint, we show that the minimal number of monitor units in this case can be interpreted as the maximal length of a path in a layered digraph. We derive an efficient algorithm that constructs a segmentation with this minimal number of monitor units, and we propose a heuristic approach to the reduction of the number of segments.