Epsilon-Rotation Invariance using Approximate Euclidean Spheres Packing Algorithm for Cancer Treatment Planning

Abstract

Cancer treatment planning using SRS (Stereotactic Radio Surgery) uses approximate sphere packing algorithms by guiding multiple beams to treat a set of spherical cancerous regions. Usually volume data from CT/MRI scans is used to identify the cancerous region as set of voxels. Computationally optimal Sphere Packing is proven NP-Complete. So usually approximate sphere packing algorithms are used to find a set of non-intersecting spheres inside the region of interest (ROI). We implemented a greedy strategy where largest Euclidean spheres are found using distance transformation algorithm. The voxels inside of the largest Euclidean sphere are then subtracted from the ROI, and the next Euclidean sphere is found again from the subtracted volume. The process continues iteratively until we find the desired coverage. In this paper, our goal is to analyze the rotational invariance properties of resulting sphere-packing when the shape of the ROI is rotated. If our sphere packing algorithm generate spheres of identical radius before and after the rotation, then our algorithm could also be used for matching and tracking similar shapes across data sets of multiple patients. In this paper, we describe unique shape descriptors to show that our sphere packing algorithm has high degree of rotation invariance within ±epsilon. We estimate the value of epsilon in the data set for 30 patients by implementing these ideas using Slicer3D™ platform.