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888 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MIM’-23, NO. 11, NOVEMBER 1975 Computation of the Electromagnetic Fields and Induced Temperatures within a Model of the Human Eye SENIOR MEMBER, IEEE Microwave-h-radiated ALLEN TAFLOVE AND MORRIS E. BRODWIN, Absfract—The electromagnetic fields within a detailed model of the human eye and its surrounding bony orbit are calculated for two different frequencies of plane-wave irradiation: 75o MHz and 1.5 GHz. The comp
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  888 IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,VOL.MIM’-23,NO.11,NOVEMBER1975 ComputationoftheElectromagneticFieldsandInducedTemperatures within aMicrowave-h-radiatedModeloftheHumanEye ALLENTAFLOVEANDMORRISE.BRODWIN,SENIORMEMBER,IEEE Absfract—Theelectromagneticfieldswithinadetailedmodelofthehumaneyeanditssurroundingbonyorbitarecalculatedfortwodifferentfrequenciesofplane-waveirradiation:75oMHzand1.5GHz.Thecomputationisperformedwithafinite-dtierencealgorithmforthetime-dependentMaxwell’sequations,carriedouttothesinusoidalsteadystate.Theheatingpotential,derivedfromthesquareoftheelectricfield,isusedtocalculatethetemperaturesinducedwithintheeyeballofthemodel.Thiscomputationisper-formedwiththeimplicitalternating-direction(IAD)algorithmfortheheatconductionequation. Usinganorder-of-magnitudees@nateoftheheat-sinkingcapacityoftheretinalbloodsupply,itisdeterminedthatahotspotexceeding40.4°Coccursatthecenterofthemodeleyeballatanincidentpowerlevelof100mW/cmzat1.5GHz.I.INTRODUCTION A TPRESENT,littletheoreticalworkhasbeendoneinsolvingforthetemperaturedistributioninducedbymicrowaveradiationincomplicatedbiologicalstruc-tures.Theemphasishasbeenonexperimentalinvestiga-tion.Thishasbeenbroughtaboutinpartbytheexpensiveandtime-consumingnumericalmethodsrequiredtocom-putetheelectromagneticfieldswithinarbitrarydielectricscatterers.Indeed,inhomogeneoustissuesofgreatcomp-lexitymayrequiresomuchdirectcomputerstoragewithwell-knowntechniquesthatsolutionisvirtuallyimpos-sible.Theeyehasbeenofspecialexperimentalinterestbe-causeofevidenceofmicrowave-inducedcataractsinhumans[1],[2].Typicalexperimentsinvolvedtheex-posureofrabbitstohighlevelsofmicrowaveradiationoverashorttimeintervalandtheobservationofthein-ducedlensopacificationoveraperiodofseveralweeks[3],[4].Therelevanceofsuchstudieswasbaseduponthesimilarityoftheanatomyofhumanandrabbiteyes.Theseexperimentsestablishedtime-powerdensitythres-holdlevelsforcataractformation.Resultsindicatedthatthedoseofmicrowaveradiationrequiredforlensinjuryisbaseduponaverageratherthanpeakpower.Themech-anismofmicrowavecataractformationisthereforemost ManuscriptreceivedMarch21,1975;revisedMay20,1975.A.TaflovewaswiththeDepartmentofElectricalEngineering,TechnologicalInstitute,NorthwesternUniversity,Evanston,Ill.60201.HeisnowwiththeResearchInstitute,IllinoisInstituteofTechnology,Chicago,Ill.60616.M.E.BrodwiniswiththeDepartmentofElectricalEngineering,NorthwesternUniversity,Evanston,Ill.60201. likelythermal[5].Astandardforhumanexposure,baseduponsuchanimalexperimentation,hasbeenpublished[6].Theuseofanimalexperimentationtoestablishahumanexposurestandardformicrowaveradiationimpliesthattheanatomy,physiology,andelectromagneticenviron-mentofthetestanimalscanberelatedtothatofhumans.However,severalelementsofthisrelationremainunclear.Inparticular,theroleoftissuestructureindeterminingmicrowaveabsorptionmaybesignificant.Weknowthatelectromagneticwaveabsorptioninalossydielectricscattererisafunctionofitsshapeanddimensions.Itisquitepossiblethattheeye–scattererofmandevelopsheatingpotentialcontoursdifferentinlocationandmag-nitudefromthoseoftherabbitbecauseofthedimen-sionalandstructuraldifferencesintissueanatomy.Thispossibilitymustbeexploredifamorepreciseexposurestandardformanistobeestablished.AsGuy[7]hasstated:“Ahighpriorityneedinthisareaisacompletethermodynamicstudyoftheeyeundermicrowaveex-posure.”Directexperimentationwiththelivinghumaneyeusingeithercataractogenicexposuresstillartothoseof[3],[4],orusipgimplantedthermocoupletechniques[8]isimpermissiblebecauseofthetissuedamagecausedbytheexperimentalprocedure.Therefore,themicrowaveheat-ingofthehumaneyemustbestudiedusingmodelsoftheactualorgan.Atheoreticalapproachwouldattempttosolveforthefieldsandtemperaturesusingsomeanalyticalornumericalmethod.Suchamodelwouldpostulateanear-erfar-fieldirradiation,andsimulatethetissuegeometryandthermodynamicstothemaximumpossibleextent.Tb.isistheapproachtakeninthispaper.Earlytheoreticalworkintheareaofthebiologicaleffectsofelectromagneticradiationcenteredontheir-radiationofmodelsoftheentirehumanbody[9].How-ever,becauseexperimentalworkindicatedthatharmfullocaltissuetemperaturerisescouldoccur,interestinpartialbodyirradiationwasstimulated.CookproposedthesolutionofMaxwell’sequationscoupledwiththeheatconductionequationtosolvetheproblemoflocalmicro-waveheating[10].Hedevelopedatheoryfortheheatingofatissuehalf-spacecomposedoflayersofskin,fat,andmuscle,byincidentplanewaves.Thetemperaturedis-tributionpredictedbythetheorywasverifiedinhisre-portedexperimentalprocedure.Shapiro etal. modeledthe  TAFLOVEANDBRODWIN:MICROWAVE-IRRADIATEDEYEMODEL-S&k- plane-waveirradiationofacranialstructure[11].TheabsorbedpowerdensitywithinconcentricsphericalshellswascalculatedusinganapproachparalleltothatofStratton[12].Itwasconcludedthatcalculationofmicro-waveheatingusingsemi-infiniteslabmodelsisnotac-curatefortissuegeometrieswheretheratioofthelocalradiusofcurvaturetothewavelengthisbetween0.05and5.Ineffect,hotspotsmaydevelopdeepwithin10SSY,curvedtissuescatterersofttissize.TheworkofShapiro etal. maybescaledtomodeltheeyeasaIossysphereinfreespace.Suchadirectscalingsuggeststhepresenceofhotspotswithinthemodeleye-ballbecausetheimportantratioparameterisoftheorderofoneatmicrowavefrequencies.Yetsuchascalingisfaultyinthatitneglectsthewavereflectioneffectsofthetissuesofthebonyorbitwhichsurroundtheeyeball.Unfortunately,anyeyemodelthataccountsfortheseeffectsismuchmoredifficulttosolve.Inclusionofthebonyorbiteliminatesthepossibilityofananalyticsolu-tionforthefieldsbecausethegeometryisnolongeramen-abletoclassicalseparation-of-variablestechniques.Theimprovedeyemodelcompelstheuseofsomenumericalmethodtosolvefortheelectromagneticfields.Severalcomputertechniquesthatappearrelevanttothisproblemhaveappearedintherecentliterature[13]-[16].Eachofthesemethodsderivesasetoflinearequa-tionsforeitherfieldvariablesorforfieldexpansionco-efficients,andthensolvesthelinearsystemwithasuitablematrixinversionscheme.However,itseemsthatnoneofthesemethodshasbeenusedtosolvethemicrowaveir-radiationoftheeye.Inspectionoftheproblemsinvolvedwitheachmethodindicateseitherdifficultyinsettingupthelinearsystem,orinfindingsufficientfast,direct-accesscomputerstoragetoinvertthematrixofthelinearsystem,Inthispaper,wereportacalculationofthemicrowavefieldswithinadetailedmodelofthehumaneyeanditssurroundingbonyorbit.Thefieldsarecomputedusingthefinite-difference,time-domainsolutionofMaxwell’sequationsdiscussedinapreviouspaper[17].Thistech-niquehasbeenfoundtoyieldfirst-orderaccuratesolu-tionsoftheelectromagneticfieldswithinarbitrarydi-electricscatterersoftheorderofonewavelengthindi-ameter,situatedinfreespaceoratthesurfaceofahalf-space.Wealsoreportanumericalsolutionoftheheatconductionequationfortheeyeball,usingthecomputedmicrowaveheatingfunctionasthesourcefunction.Theheatequationissolvedusingtheimplicitalternating-direction(IAD)method[18],[19].Itisdeterminedthatasignificanthotspotdevelopsatthecenteroftheeyeballofthemodelatanincidentfrequencyof1.5GHz.II.THEMICROWAVESCATTERINGMODELInthissectionwediscussthefinite-differencelatticeusedfortheeyeandbonyorbitmicrowavescatteringproblem.Thelatticeboundaryplanesandassumedco-ordinateaxesareshowninFig.1.Thefieldvectorcom-ponentsarepositionedatdistincthalf-intervalpointsinthelattice,asshownin[17,fig.1].ThelatticeofFig.1isa19-by-39-by-19arrayofunit-cellcubes,withafixedunit-celldiameter8equalto1.25mm.Aspacelatticepointisdenotedas(i,j,iiI=(ib,j~,k~),wherei,j,andkareintegers.Theeye–scattererisassumedtohaveevensymmetryaboutlatticeplanesz=19$6andz=198.Thissymmetryallowslargesavingsofcom-puterstoragebypermittingthesolutionofthecompleteeyemodelwiththeprogrammingofonlyonespatialquadrantofthemodel.6ischosenlargeenoughsothat,usingthissymmetry,thelatticecoversa47.5-mm-by-48.75-mm-by-47.5-mmvolume,whichissulticienttoen-closemostofthebonyorbit.Yet8issmallenoughtofulfilltheaccuracyrequirementof0.05wavelengthresolu-tioninalltissueforfrequenciesuptoabout1.5GHz.Theincidentplanewaveisassumedtohavethefieldcom-ponents E, and Hz andpropagateinthe+ydirection.Thiswaveisgeneratedatlatticeplaney=36usingasourcecondition.Atplanesy=O,y=396,x=~8,andz=O,thefieldcomponentsaredeterminedusinglattice-truncationconditions.Thescatterersymmetrycondi-tions,plane-wavesourcecondition,andlatticc-truncationconditions,arediscussedinSectionIIIofthispaper.ThelocationofthelatticeofFig.1relativetotheeye-ballandtothefrontoftheskullisshownschematicallyinFig.2,whichdepictsthelocationofplanesy=Oandz=193.The‘anglesanddimensionsoftheskullcrosssectionarebaseduponavailablecephalometIric data[20].Inordettorealizeanevenlysymmetricirradiationoftheeye–scatterer,theincidentwaveisassumedtopropagateinadirectionparalleltotheaxisofonebonyorbit.Theselectedorbitalaxisisassumedtobetheintersectionof thelatticesymmetryplanesz=19~~andz=196.TheskullsurfaceofFig.2is seentobeapproximatelyparalleltothezaxisofthelattice,withinabout1cmoftherightedgeoftheorbit.Similarly,itmaybeshownthatimmediatelyaboveandbelowtherimoftheorbittheskullsurfaceisalmostparalleltothezaxis.Tocon-structamodeloftheeye–scattererwiththeassumedsym-metry,welettheskullsurfaceofthemodelextendtoinfinityparalleltothex–zplane.Thusthemodelorbitisanindentationcenteredat(19~,j,19)inaninfinite,planar,bonylayer.Thespacetotherearofthislayerisassumedtocontainonlybraintissue.Themajormicro-wavereflectioneffectsofthemodelresultfromtheeye-ball-orbitjuxtapositionalone,Usingavailableanatomicaldata[20],[21],wesketch ‘819+8,398,198+.<*,3981$8t,:,X* A8.39K5-i;K&;9&o.“”” Fig.1.Boundaryplanesofthefinite-differencelatl,iceusedforthemicrowavescatteringproblem.  --866’” IEEETRANSACTIONSONMICROWAVETHEORYANDTECHNIQUES,NOVEMBER1975z=198 LATTICEPI.ANC y=o . Fig.2.LocationofthelatticeofFig.Irelativetotheeyeballandthefrontofthehumanskull. topandsideviewsofthemajortissuesoftheorbitinFig.3(a)and(b).ThelatticeofFig.1containsanidealized,stepped-surfacerepresentationofthetissuesofonequadrantoftheorbit.Thisrepresentationisde-pictedinFig.4(a)and(b)atthetwolatticesymmetryplanes.ComparingFigs.3and4weseethatthemodeleyesimulatesallrectusmusclesassociatedwiththeeye-ball.Thesymmetryassumptions,however,preventsimu-lationoftheobliquemuscles.InFig.4nobone–fatinter-faceisshownbecauseofthesimilardielectricparametersofthesetwotissues.Usingpublisheddataonthecomposi-tionoftheeyeballmedia[21]andtheelectricalparam-etersoftheothertissuesfoundintheorbit[11],[22],thefollowingvaluesofeanduareassignedtoeachtissue. E, at:a(mhos/meter)at:TissueType750MHz1.5GHz750MHz1.5GHzSkin,muscle,lens52491.541.77Fat,bone5.65.60.090412Eyeballhumors80801.901.90Brain,nerves49461.201.40 Inconstructingamodeloftheeyewiththeassumedsymmetry,severalsimplificationsoftheactualtissuegeometryarenecessary.Thesymmetricmodelcannottakeintoaccountthecurvatureoftheskullatthesidesoftheorbit,thesinuscavitiesoftheskull,theexistenceoftheobliquemuscles,andtheexactshapeandthicknessoftheorbitalwall.Theaccuracyofthemodelmaybeincreasedbyomittingtheassumedsymmetryandusingalatticetocovertheentireeye–scatterer.Further,themaximumfrequencyofirradiationmaybeincreasedbydecreasingthelatticespacing.Theseelaborationsarede- - NERVEMEO~AuL&BITALFATSUPERIOROBLIQUELATE~;:C~:CTUSMUSCLESUPERIORRECTUSMUSCLE (a) SUPERIOROBLIQUE—MUSCLEBSUPERIORRECTUSMUSCLEORBITALFATLATERALRECTUS~--MuSCLEOPTICNERVE..INFERIORRECTUS+-JMUSCLEINFERIOROBLIQUEMUSCLE (b) Fig.3.Themajortissuesoftheorbit.(a)Topview.(b)Sideview. MEDIALRECTUSiBRAINMUSCLE39 I[ 353025ORSITALFATANDBONE u 0SKINANO‘5 MUSCLEOF‘v=,‘n 10 r LA ““’””Pd W,W : HUMOR 1-1 (a) EYESALLEYEBALLAOUEOUSVITREOUSHUMORHUMORORBITALFATiAIROPTICNERVEls-- INCIOEF4T 15-P1.,llE WAVE5-%I,5SKINANOMUSCLEORBITALFATOFEYELIOANDBONE(b) Fig.4.Crosssectionsoftheorbitaltissuesofthemodelatthelatticesymmetryplanes.(a)Planez=198.(b)Planez=19*8.  TAFLOVEANDBRODWIN:MICROWAVE-IRRADIATEDEYEMODEL 891 pendentupontheacquisitionofsufficientcomputertimeonamachinewithmorefaststoragecapacitythantheNorthwesternUniversityCDC6400whichwasusedforthepresentresearch.However,thesymmetricmodelsimulatesthebasicgeometryoftheeye-scatterer,thatofawater-likesphereencasedinalow-lossdielectriccavity.Thismodelshouldbeaccurateenoughtolocateandde-terminethemagnitudeofanyhighconcentrationsofmicrowaveenergy,withanacceptablelevelofuncertainty.III.ELEMENTSOFTHEMICROWAVESCATTERINGALGORITHM1Inthissectionwediscussthemodificationsoftheal-gorithmof[17]necessaryforthepresentresearch.Thesemodificationsincludeatime-steppingalgorithmwithfewermultiplications,simplersymmetryconditions,adap-tivelattice-truncationconditions,andincreasedstabilityconsiderations.Inthediscussion,anyfunctionofspaceandtimeisdenotedasF“(i,j,k)=F(i8,j8,1c8,n8t).Assumingthatthequantity 6t/P i,j,k)6isconstantforall(i,j,k)ofthelattice,thealgorithmof(6a)–(6f)of[17]requiresninemultiplicationsperunitcellpertimestep.Thenumberofrequiredmultiplicationscanbere-ducedtosixandthealgorithmconsiderablysimplifiedinthefollowingmanner.WedefinetheconstantsR=8t/eo(la)l?.= M’/ IYpl eo (lb)Rb= 8t/’/.u@ (lC)c.(m)=1.0–RIT(n’z)/6,(7n)(id)Cb(m)=Ra/er(?n)(le)wheremisatissue-typeintegerfrom1to5assignedinthefollowingway:1,air;2,skin,muscle,lenstissue;3,fat,bonetissue;4,eyeballhumors;5,brain,nervetissue.Wealsodefinetheproportionalelectric-fieldvectorUsingthedefinitionsof(la)–(le)and(2),werewrite[17,eq.(6a)-(6c)]inamannersimilartothefollowing:Hz”+’/2(i,j++,k++)=Hz”-1/2(i,j+ *,k++)+ -a”(isi++,~+ 1)– E.”(ijj+ *J)+ I?z”(i,j,k+ ;)–17z”(i,j+ l,k++)0(3)ThismodificationeliminatesthethreemultiplicationspreviouslyneededintheHpartofthealgorithm.Further,werewrite [17, eq. (6d)–(6f)]ina mannersimilarto the following:m=MEDIA(i+~,j,k)(4a) 1Thelistingofthe362cardFortranIVsourcedeckisavailablefromtheauthors. + C,(m)[H=n+’/2(i+*,j++,k) – Hzn+ll’(i++,j– +,,~) + Hvn+’/2(i+*,j,ii–*) – Hg”+’/z(i+ +,j,lc+~)].(4b)Thismodificationeliminatestheneedforcomputerstorageofseparatecandalattices.Now,onlyaMEDIAlattice,whichspecifiesthetissuetypeateachlatticepoint,needbestored.Inaddition,thee,anduofeachtissuecannowbechangedwithouthavingtorepunchalargedatacarddeck.SuchachangeinvolvesonlytherecalculaticmofthefivevaluesofC.(m)andthefivevaluesofCb(m).Finally,werewrite[17,eq.(10)]asfi.”(i,3,k+~)e--1000R~sin(27rfn M +~i’a -i,3,1c1-~ . (5)Thismodificationisneededtoprovideaplane-wavesourceconditionatlatticeplaney=38thatagrweswiththedefinitionof~,Symmetrycondition[17,eq.(12)]requiresextensionofthelattice0.58beyondtheplaneofsymmetry.Asetofconditionsusefulfortheassumedevensymmetryoftheeye-scatterer,andrequiringnolatticepointsbeyondthesymmetryplanes,isgivenasHv”(19*,j,k++)=If=”(19*,j++,k)=o(6a)flm”(i+*,j,19)=I?.”(i,j+*,19),=O.(6b)Wenextconsiderasetofsimple,approximatelattice-truncationconditionsanalogousto[17,eq[9a)-(9d)].Fromthebasictime-steprelationofthisalgorithm2cfst=a(7)itisseenthatawaverequirestwotimestepstopropagateacrossasingleunitcell,inair.Wedefinetheintegercon-stantl(m)=2X(l)/X(m),x(m):wavelengthintissue-typem(8)asthenumberoftimestepsrequiredforawavetopropa-gateacrossasingleunitcell,intissue-typem.Wealsodefinethestoredfieldvectorsi (i,j,lc)= &z(MEDIA(’s~-@ i,j,k)(9a)Thenthetruncationconditionforlatticeplanex=48isgivenbyHy”(~,j,k+’$)=[fiy(%,j,k–~)+~J$,j,?:++)+l?u(~,j,k+~)]/3(lOa)
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