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1、SuchaDifficult“Sphere-PackingProblem”!参赛队员：欧阳稳(00数学)卢福良(00数学)邓青青(00计算机)指导教师：段正敏参赛单位：重庆大学参赛时间：2003年2月7∼10日Team#171page1of15SuchaDifficult“Sphere-PackingProblem”!WenOuyang,FuliangLu,QingqingDengAdvisor:ZhengminDuanAbstractThispaperisabouthowtomakethebesttreatmentplanningofGammaKnifewiththemethodsofopt
2、imization.WeconsiderthisProblemBasasphere-packingproblem,whichwetrytoputthedifferentkindsofspheres(shots)intothetargetvolumes(tumors)forthatmaximizingthesumofspherevolumes,andminimizingthenumberofspheres.First,wehavedevelopedamethodtosolvethisproblembydealingwiththetargetvolumeasasphere,andthenwegen
3、eralizethemethodtovarioustargetvolumes.Toaglobaltumorwedevelopfourmodelsandgivefouralgorithms.Thentoellipsoidaltumorwealsodevelopamodelanddesignanalgorithm.Inthefirstmodelwediscussthecasethatthereisasmallsphereatthecenteroftargetvolume.WeuseGreedyAlgorithmtogetthesolutionofthisproblem.Inthesecondmod
4、elwedesignamodifiedversionofalgorithmtoimprovetheinitialalgorithm,bymovingthosesphereswhichhavebeenputintothetargetvolume,soastomaximumtherightspace,thenchangingthesmallspheresattherightsideintolargerones.Inthethirdandfourthmodelwefillinthetargetwithsmallspheresfromthelefttotheright.Wediscussthecase
5、thatonlyusetwosmallspheresontheX-axisinthethirdmodel,discussthecasethatputthreesmallspheresontheX-axisinthefourthmodel.Inthefifthmodelwediscussthecasethatthetargetvolumeisellipsoidalandothercases.AssumptionsThe201beamsfromthe201cobalt-60unitsourcesare201columns.Therearefourtypesofsuchcolumnsaccordin
6、gtofourdifferenthelmets,withthediametersof4,8,14,and18mm.Thegammarayishardlydispersed,especiallyinsuchashortdistance.Sincethediametersofthebeamchannelsonthefourdifferenthelmetsare4,8,14,and18mm,thediametersofthebeamsarethesameasthatofthebeamchannels.Therearefourtypesof“shots”,withthediametersof4,8,1
7、4,and18mm.Accordingtotheassumptionsabove,the201beamscanbeconsideredas201columns,whichareresultinginasphere(approximately).Thissphereisobviouslytangentialtoallofthese201columns,sothediameterofthesphere
