搜索
*,WangYueke2,LinXin1,FuFeng1,ShiXuetao11Dep.ofBiomedicalEng.2Sch.ofMechatronicalEng.&AutomationFourthMil.MedicalUniv.Nat.Univ.ofDef.Technol.Xi’an,China,710032Changsha,China,410073AbstractTherelationshipbetweentheresistivecomponentandthereactivecomponentofthemeasureddataofbioelectricalimpedanceisstudiedinthispaper.TheoreticalproofbasedontheimpedancemodelandCauchyintegralformulaispresentedindetail.Emulationisperformedonvariousexperimentaldatatoverifyandillustratetheproposition.Resultsoftheemulationagreewiththeprooftoagreatextentinfrequencydomain.TherelationshipbetweentherealpartandtheimaginarypartofthebioelectricalimpedancemeasurementisjustifiedanditshowsthattherealpartandtheimaginarypartcanrepresenteachotherbyHilberttransformKeywords:electricalimpedancetomography,Cole-Cole,Hilberttransform1IntroductionElectricalimpedancetomography(EIT)isanonlinearnondestructivetestingtechnology.Multi-frequencyelectricalimpedancetomography(MFEIT)andelectricalimpedanceparametertomography(EIPT)differfromtheconventionalEITinthattheformertwopaymoreattentiontotheelectricalfrequencycharacteristicoftheobject.InMFEITandEIPT,impedanceimagesarereconstructedonthebasisofboundarydatacomposingofseveralfrequencies.Generally,therealpartandtheimaginarypartoftheboundarydataaremeasuredandanalyzedseverallyinordertoacquiremoreinformationoftheinterestedobject.Thisarticleaimstostudythebioelectricalimpedancefrequencycharacteristicintheoryandrevealtherelationshipbetweentheresistivecomponentandthereactivecomponentofthemeasureddata,theoreticproposalforMFEITandEIPTisyieldedintheend.1SupportbyNationalNaturalScienceFoundationofChina(No.50337020).(a)aisoutsideG;3(b)2Principle2.1ModelofthebioelectricalimpedanceBioelectricalimpedanceofthetissuevarieswithfrequency.AccordingtotheCole-Coletheory[1],athree-elementmodelcanmodelthebiologicalimpedance.ThemodeliscomposedoftheextrcellularresistanceRe,intracellularresistanceRiandcellmembraneCm.Figure1illustratethemodel.Additionally,differenttissueshavedifferenttimeconstants,givingadispersionoftimeconstants.Theempiricalformulaforthebioelectricalimpedanceofatissueis[1]:])/(1/[)()(0acfjfRRRfZ+-+=¥¥(1)whereReR=0isthelow-frequencyimpedance;eRRiR//=¥isthehigh-frequencyimpedance;micCRfp2/1=istherelaxationfrequencyforthetissue;aistheconstantthatcharacterizestheColedistributionfunction(0a1)Forformula(1),ifwesubstitutejfwith)(vsj+thatisjf==s)(vsj+(2a)andextendsthedomainofdefinitiontothecomplexplane,theformulacanberewritedasfollow:])/(1/[)()(0acfsRRRsZ+-+=¥¥(2b)2.2AnalysisinthecomplexplaneInthecomplexplane,Z(s)hasapoleinthelefthalfplane,whileZ(s)isanalyticbothintherighthalfplaneandontheboundary.ConsidertheCauchyintegralofZ(s)ontheimaginaryaxisandinthefunctionrighthalfplane,wheretheCauchyintegraltheorem[2]isapplicableas:îíì=-òG0)(2)(aifdsassZpGisapiecewisesmoothclosedcontourinanopendomain.TogetaresultwhenaliesonG,anewcontour/eGiscreatedasshowninFigure2,andwehave)(2)(/aizdsassZpe=-òG(4)RiReCmCmcellmembraneRiintracellularresistanceReextrcellularresistanceFigure1:EquivalentModelforTissueImpedance],[RiRiR-+=Gg-RiFigure2aFigure2bFigure2:APiecewiseSmoothClosedContouronAnOpenDomainAccordingtoLemma1[3],thecontributionfromthesemicircleegtothewholeintegralalong/eGapproaches)(2aizpwhentheradiusofegtendstozero.Lemma1.Ifghasasimplepoleatas=andrgisthecirculararcdefinedbyrg:qireas+=(1q£q£2q)(5)then)()()(Relim120sgidssgsasrr=+-ò-=gqq(6)FromthedefinitionoftheCauchyprincipalvalue[2],wehave)(2)()()(limlim00/aizdsasszdsasszPdsasszpeeeeg=-+-=-òòòG-G-(7)TheintegraloftheCauchyprincipalvalueis)()()(lim0aizdsasszdsasszPpee=-º-òòGG-(8)whereeGisanonclosedcontourwithouttheindentationeg.Figure3.GIsAPiecewiseSmoothClosedContourConsideracontourGasinFigure3thatisclosedbyasemicircleinthelefthalfplaneandtheimaginaryy-axis.[3],theintegraloverRgwilldisappearwhen¥®Rif|||)(|sCsZ,(9)foranypositiveconstantC.Jordan'slemmaIfm0andP/QisthequotientoftwopolynomialssuchthatdegreeQ1+degreeP,(10)Thenò=¥-CpimsdsesQsP0)()(limr(11)WhereCpistherighthalf-circlewithradiusr.Accordingtoformula(2),itisobviousthatassZ-)(complywiththecriticalconditionthatdegreeQ=1+degreeP(12)sotheintegraloverRgwilldisappearwhen¥®Randformula(8)canberewrittenasòòG¥--º-RdsasszRdsasszPg)()(lim+ò¥+¥--dsasszP)(=ò¥+¥--dsasszP)()(aizp=(13)IfZ(s)isputasZ(s)=R(s)+jX(s)(13)Onbothsidesof(13)withargumentsontheimaginaryy-axisandequatingrealandimaginarypartsthenweobtainfortherealpartR(s)=)()(1sHXdsXP=-ò¥+¥-hhhp(14)and)()(1)(sHRdsRPsX-=--=ò¥+¥-hhhp(15)whereHistheHilberttransformoperator.Accordingtoformula(2a),let0=s,wecanrewriteformula(14)andformula(15)as)()(fHXfR=)()(fHRfX-=(16)