Packing of Unequal Spheres and Automated Radiosurgical Treatment Planning

We study an optimization problem of packing unequal spheres into a three-dimensional (3D) bounded region in connection with radiosurgical treatment planning. Given an input (R, V, S, L), where R is a 3D bounded region, V a positive integer, S a multiset of spheres, and L a location constraint on spheres, we want to find a packing of R using the minimum number of spheres in S such that the covered volume is at least V; the location constraint L is satisfied; and the number of points on the boundary of R that are touched by spheres is maximized. Such a packing arrangement corresponds to an optimal radiosurgical treatment planning. Finding an optimal solution to the problem, however, is computationally intractable. In particular, we show that this optimization problem and several related problems are NP-hard. Hence, some form of approximations is needed. One approach is to consider a simplified problem under the assumption that spheres of arbitrary (integral) diameters are available with unlimited supply, and there are no location constraints. This approach has met with certain success in medical applications using a dynamic programming algorithm (Bourland and Wu, 1996; Wu, 1996). We propose in this paper an improvement to the algorithm that can greatly reduce its computation cost.

[1]  Rangasami L. Kashyap,et al.  Building Skeleton Models via 3-D Medial Surface/Axis Thinning Algorithms , 1994, CVGIP Graph. Model. Image Process..

[2]  J. D. Bourland,et al.  Use of Shape for Automated, Optimized 3D Radiosurgical Treatment Planning , 1996, VBC.

[3]  N. J. A. Sloane,et al.  Sphere Packings, Lattices and Groups , 1987, Grundlehren der mathematischen Wissenschaften.

[4]  C K Bomford,et al.  The Sheffield stereotactic radiosurgery unit: physical characteristics and principles of operation. , 1987, The British journal of radiology.

[5]  Petros Maragos,et al.  Morphological skeleton representation and coding of binary images , 1984, IEEE Trans. Acoust. Speech Signal Process..

[6]  Gabriella Sanniti di Baja,et al.  Ridge points in Euclidean distance maps , 1992, Pattern Recognit. Lett..

[7]  Thomas C. Hales.  Sphere packings, I , 1997, Discret. Comput. Geom..

[8]  Roland T. Chin,et al.  Analysis of Thinning Algorithms Using Mathematical Morphology , 1990, IEEE Trans. Pattern Anal. Mach. Intell..

[9]  C. A. Rogers,et al.  Packing and Covering , 1964 .

[10]  H. Blum.  Biological shape and visual science (part I) , 1973 .

[11]  J. H. Lindsey,et al.  Sphere packing in R 3 , 1986 .

[12]  Ugo Montanari,et al.  A Method for Obtaining Skeletons Using a Quasi-Euclidean Distance , 1968, J. ACM.

[13]  King-Sun Fu,et al.  A parallel thinning algorithm for 3-D pictures , 1981 .

[14]  Gabriella Sanniti di Baja,et al.  Finding local maxima in a pseudo-Euclidian distance transform , 1988, Comput. Vis. Graph. Image Process..

[15]  James C. Mullikin,et al.  The vector distance transform in two and three dimensions , 1992, CVGIP Graph. Model. Image Process..

[16]  A. Wu,et al.  Physics and dosimetry of the gamma knife. , 1992, Neurosurgery clinics of North America.