Planning Curvature and Torsion Constrained Ribbons in 3D with Application to Intracavitary Brachytherapy

Abstract

We present an approach for planning ensembles of channels, ribbons, within 3D printed implants for facilitating radiation therapy treatment of cancer. The ribbons are traced out by sweeping a constant width rigid body (cuboid) along spatial curves. We propose a method for planning multiple disjoint and mutually collision-free ribbons of finite thickness along curvature and torsion constrained curves in 3D space. This is equivalent to planning motions for the cross section of the ribbon along a spatial curve such that the cross section is oriented along the unit binormal to the curve defined according to the Frenet-Serret frame. We propose a two-stage planning approach. In the first stage, a customized sampling-based planner uses rapidly exploring random trees (RRTs) to generate feasible curvature and torsion constrained ribbons. In the second stage, the curvature and torsion along each ribbon is locally optimized using sequential quadratic programming (SQP). We use this approach to design curved radiation delivery channels inside custom 3D printed implants that allow temporary insertion of a high-dose radioactive source that is threaded through the channels using a wire and allowed to dwell for specified times to expose cancerous tumors for intracavitary brachytherapy treatment. Constraints on the curvature and torsion are required for 3D printing (to allow flushing of sacrificial material) and for smooth insertion of radioactive sources. In simulation experiments, this approach achieves an improvement of 46% in tumor coverage compared with a greedy approach that generates channels sequentially.