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Packing of Unequal Spheres and Automated Radiosurgical Treatment Planning

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  • Jie Wang

    (University of North Carolina at Greensboro)

Abstract

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.

Suggested Citation

  • Jie Wang, 1999. "Packing of Unequal Spheres and Automated Radiosurgical Treatment Planning," Journal of Combinatorial Optimization, Springer, vol. 3(4), pages 453-463, December.
    • Handle: RePEc:spr:jcomop:v:3:y:1999:i:4:d:10.1023_a:1009831621621
    DOI: 10.1023/A:1009831621621
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    Citations

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    Cited by:

    1. Andreas Fischer & Igor Litvinchev & Tetyana Romanova & Petro Stetsyuk & Georgiy Yaskov, 2023. "Quasi-Packing Different Spheres with Ratio Conditions in a Spherical Container," Mathematics, MDPI, vol. 11(9), pages 1-19, April.
    2. Bortfeldt, Andreas & Wäscher, Gerhard, 2013. "Constraints in container loading – A state-of-the-art review," European Journal of Operational Research, Elsevier, vol. 229(1), pages 1-20.
    3. Evgueniia Doudareva & Kimia Ghobadi & Dionne Aleman & Mark Ruschin & David Jaffray, 2015. "Skeletonization for isocentre selection in Gamma Knife® Perfexion™," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 23(2), pages 369-385, July.
    4. Hifi, Mhand & Yousef, Labib, 2019. "A local search-based method for sphere packing problems," European Journal of Operational Research, Elsevier, vol. 274(2), pages 482-500.
    5. A. Sutou & Y. Dai, 2002. "Global Optimization Approach to Unequal Global Optimization Approach to Unequal Sphere Packing Problems in 3D," Journal of Optimization Theory and Applications, Springer, vol. 114(3), pages 671-694, September.
    6. Jie Wang, 2000. "Medial Axis and Optimal Locations for Min-Max Sphere Packing," Journal of Combinatorial Optimization, Springer, vol. 4(4), pages 487-503, December.
    7. Stoyan, Yu. & Chugay, A., 2009. "Packing cylinders and rectangular parallelepipeds with distances between them into a given region," European Journal of Operational Research, Elsevier, vol. 197(2), pages 446-455, September.
    8. Li, S.P & Ng, Ka-Lok, 2003. "Monte Carlo study of the sphere packing problem," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 321(1), pages 359-363.

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