Exact reachability analysis for planning skew-line needle arrangements for automated brachytherapy

When planning skew-line needle arrangements for automated brachytherapy, one objective is to identify a set of candidate needles that enter from a specified entry region, avoid specified organs-at-risk and sufficiently cover the target (tumor) volume. Existing methods use uniform or random sampling to generate a set of candidate needles, which may not adequately cover the target volume. In this paper we present an exact reachability analysis that can be used to guide the selection of candidate needles and to identify which subset of the target volume may not be reachable. Assuming linear needles, convex polyhedral representations of entry zone, organs-at-risk and target volume, we give an exact polynomial time algorithm for checking existence and calculation of the non-reachable set in the target volume. We perform experiments using patient data from 18 brachytherapy cases and found that 11 cases had non-empty occluded volume inside the target ranging from 0.01% to 4.3% of target volume. We also report a sensitivity study showing the change in the occluded volume with dilation of the avoidance volume and entry zone.