Medial Axis and Optimal Locations for Min-Max Sphere Packing

We study the following min-max sphere packing problem originated from radiosurgical treatment planning using gamma knife (Bourland and Wu, 1996; Wu, 1996). Given an input (R, V), where R is a 3-dimensional (3D) bounded region and V a positive integer, find a packing of R using the minimum number of spheres (spheres may not be identical) such that the covered volume is at least V, and the number of points on the boundary of R touched by spheres is maximized. Bourland and Wu (1996) and Wu (1996), devised a greedy algorithm to solve the problem based on medial axis analysis. In particular, the algorithm places the center of each sphere on the medial axis of each subsequent region starting from R. While this approach has met with certain success, we show that medial axis does not always provide optimal locations for min-max sphere packing.