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\section{Introduction}  \subsection{Restatement Stereotactic radiosurgery delivers a single high dose of ionizing radiation to a radiographically well-defined, small intracranial 3D brain tumor without delivering any significant fraction  of the Problem}  An optimal treatment plan for prescribed dose to the surrounding brain tissue. Three modalities are commonly used in this area; they are  the gamma knife must be formulated as a sphere-packing problem. This plan must based upon various constraints unit, heavy charged particle beams,  and requirements detailed below. external high-energy photon beams from linear accelerators.\cite{aoyama2006stereotactic}  \subsection{Background}  \subsubsection{Gamma Ray Knife}  Stereotactic radiosurgery delivers a single high dose of ionizing radiation to a small intracranial 3D brain tumor without delivering any significant fraction of the prescribed dose to the healthy brain tissue surrounding the tumor. The gamma knife unit  delivers a single high dose of ionizing radiation emanating from 201 cobalt-60 unit sources through a heavy  helmet. Each irradiation is known as All 201 beams simultaneously intersect at the isocenter, resulting in  a "shot", which resembles spherical (approximately) dose distribution at the effective dose levels. Irradiating the isocenter to deliver dose is termed  a circle. Multiple shots “shot.” Shots  can be used if individual shots do not cover the target. Helmets have represented as different spheres. Four interchangeable outer collimator helmets with  beam channel diameters of 4, 8, 14 \& 14, and  18 mm.\cite{aoyama2006stereotactic} mm are available for irradiating different size volumes. For a target volume larger than one shot, multiple shots can be used to cover the entire target. In practice, most target volumes are treated with 1 to 15 shots. The target volume is a bounded, three-dimensional digital image that usually consists of millions of points.  \subsubsection{Circle Packing}  In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that all circles touch another. The associated packing density of an arrangement is the proportion of the surface covered by the circles. In two dimensional Euclidean space, the optimal lattice arrangement of identically-sized circles with the highest density is the hexagonal packing arrangement, a result that was proven by Lagrange.\cite{chang2010simple} Generalizations can also be made of higher dimensions – this is called "sphere packing," which usually deals only with identical spheres. We dealt only with circles for our first model for simplicity.