integral equation formulations and related …...biomedical applications, computational models,...

DOI: 10.4018/IJEHMC.2018010105

International Journal of E-Health and Medical CommunicationsVolume 9 • Issue 1 • January-March 2018

Copyright©2018,IGIGlobal.CopyingordistributinginprintorelectronicformswithoutwrittenpermissionofIGIGlobalisprohibited.

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Integral Equation Formulations and Related Numerical Solution Methods in Some Biomedical Applications of Electromagnetic Fields:Transcranial Magnetic Stimulation (TMS), Nerve Fiber StimulationDragan Poljak, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture (FESB), University of Split, Split, Croatia

Mario Cvetković, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture (FESB), University of Split, Split, Croatia

Vicko Dorić, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture (FESB), University of Split, Split, Croatia

Ivana Zulim, Faculty of Electrical Engineering, Mechanical Engineering and Naval Architecture (FESB), University of Split, Split, Croatia

Zoran Đogaš, School of Medicine, University of Split, Split, Croatia

Maja Rogić Vidaković, School of Medicine, University of Split, Split, Croatia

Jens Haueisen, Institute of Biomedical Engineering and Informatics, Technical University Ilmenau, Ilmenau, Germany

Khalil El Khamlichi Drissi, Blaise Pascal University, Clermont-Ferrand, France

ABSTRACT

Thepaperreviewscertainintegralequationapproachesandrelatednumericalmethodsusedinstudiesofbiomedicalapplicationsofelectromagneticfieldspertainingtotranscranialmagneticstimulation(TMS)andnervefiberstimulation.TMSisanalyzedbysolvingthesetofcoupledsurfaceintegralequations(SIEs),whilethenumericalsolutionofgoverningequationsiscarriedoutviaMethodofMoments(MoM)scheme.Amyelinatednervefiber,stimulatedbyacurrentsource,isrepresentedbyastraightthinwireantenna.ThemodelisbasedonthecorrespondinghomogeneousPocklingtonintegro-differentialequationsolvedbymeansoftheGalerkinBubnovIndirectBoundaryElementMethod(GB-IBEM).SomeillustrativenumericalresultsfortheTMSinducedfieldsandintracellularcurrentdistributionalongthemyelinatednervefiber(activeandpassive),respectively,arepresentedinthepaper.

KEywoRdSBiomedical Applications, Computational Models, Induced Fields, Intracellular Current, Nerve Fiber Stimulation, Transcranial Magnetic Stimulation


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1. INTRodUCTIoN

Whilehumanexposuretoradiationfromdifferentelectromagneticinterference(EMI)sourceshasinitiatedanumberofquestionsregardingpotentialadverseeffects,particularlyforthebrainandeyeexposuretohighfrequency(HF)radiation,somebiomedicalapplicationsofelectromagneticfieldsareofparticularimportance,aswell.Thus,theelectromagneticfieldsareappliedinmedicaldiagnosticandfortherapypurposesfeaturingtheuseoftechniquessuchas:transcranialmagneticstimulation(TMS)(Cvetković&etal.,2015;Garvey&Mall,2008;Rajapakse&Kirton,2013,Yamamoto&etal.,2016),percutaneouselectricalnervestimulation(PENS), transcutaneousnervestimulation(TENS)(Frijnas&tenKate,1994;Rattay,1999;Zulim&etal.,2015),orintraoperationalmethodssuchastranscranialelectricalstimulation(TES)anddirectcorticalstimulation(DCS).Transcranialmagneticstimulation(TMS)isanoninvasiveandpainlesstechniqueforexcitationorinhibitionofbrainregions,andinlastfewdecadesisanimportanttoolinpreoperativeneurosurgicaldiagnostic/evaluationofpatients(Picht&etal.,2013;Deletis&etal.,2014;Rogić&etal.,2014).TMSisalsoused in therapeuticpurposes(i.e.depression),and issubjectof interest inneurophysiologicresearch.VariousefficiencyaspectsofTMSstimulationhavebeenprimarilystressedoutprimarilyduetodifferencesinrelevantstimulationparameterssuchaspulsewaveform,frequency,intensityoftreatment,etc.ThechoiceofoptimalstimulationintensityisstillinvestigatedinmanyTMSstudies.

Thecoilorientationandpositioningappreciablyinfluencethemisalignmentfromthetargetedbrainregion, thusreducingtheTMSefficiency,althoughbyusingnavigatedTMS, thisproblemcouldbe somewhat alleviated. In addition to the stimulationparameters,beingadjustable to therequirementsoftheTMSoperator,toacertainextent,thedifferenceinindividualbrainmorphology,duetoage,genderorhealthstatusandthebiologicaltissueparametersappreciablyinfluencethedistributionoftheinducedfieldsinthebrain.Mostoftheparametersareobtainedunderdifferentmeasurementonexvivoanimalandhumantissues,andusuallyexhibitlargevariationsfromtheiraveragevalues.Thelevelofuncertaintyinthevaluesofthebrainconductivityandpermittivity,isevenmorepronouncedatlowfrequencies.

ModelingandcomputersimulationofTMSphenomenacouldberatherusefulindeterminingtheexactlocationofstimulation,intheinterpretationofexperimentalresultsaswellasindesigningsome more efficient stimulation setups. Realistic TMS models can also provide a more reliablepredictionofthedistributionofinternalfieldsandcurrentsbytakingintoaccountthevariabilityofthevariousinputparameters.

Furthermore, techniques suchasPENS, electro-acupuncture,orTENSarewidelyused in atreatmentofneurologicaldisorders.Basically,therearetwotypesofelectricpotentialoccurringinthestimulatednervecells;theelectrotonicpotentialandtheactionpotential.Theelectrotonicpotentialexistingdue to the localchanges in the ionconductivitydecaysalong the fiber, and thepassivemembranethenshowslinearnaturewhichsatisfiesOhm’slaw.Theactionpotentialisinitiatedwhenthethresholdpotentialtowhichthemembranepotentialmustbedepolarized,isachieved.Studiesonelectricalexcitationofnerves,amongotheraspectsinvolve:nerveexcitationusingstimulatingelectrodes,nerveconductionvelocitytestsornon-invasivestimulationofnerves.Theexceptionalnervoussystemcomplexity,particularlynervecellsasitsbasis,andwidespreadneurologicaldisordersandaneedforbetterinsightintocomplexfunctioningofthenervoussystemhavebeencontinuouslymotivatingresearchers(Golomolzina&etal.,2014)tocarryoutanefficientandaccuratenervefibermodeling.Thus,suchcomputationalmodelsofnervefiberscouldprovideastudyofthenervefiberresponsetodifferentstimuluswaveforms,oftenusedwithinvariouselectrotherapytechniques,particularlyelectro-acupunctureandPENS.Suchamodelcanbebeneficialandversatiletoolforinterpretingexperimentalresultsandanalyzingvariablesforwhichtherecouldbedifficultiesinthelaboratoryimplementation.

Thepresentworkextendsthereview(Poljak&etal.,2016.a)ofintegralequationmethodapproachtostudyTMSinducedfieldsandnervefiberexcitationusedintherapeuticproceduressuchaselectro-


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acupunctureand/orPENSreportedin(Poljak&etal.2016.b)byprovidingmoremathematicaldetailsandpresentingsomeadditionalcomputationalexamples.Thepresentwork,togetherwith(Poljak&etal.2016.a)and(Poljak&etal.2016.b)hasbeenundertakenthroughtasksandactivitieswithintheframeworkofIEEE/ICESTC95SC6EMFDosimetryModelingandCOSTActionBM1309.

2. ModELING oF TMS INdUCEd FIELdS

ThelossydielectricmodelofthebrainexposedtoTMSelectromagneticfieldisbasedonthesetofcoupledsurfaceintegralequations(SIEs)whichcouldbederivedfromtheequivalencetheoremandbyforcingthecorrespondinginterfaceconditionsfortheelectricand/ormagneticfield,(Cvetković&etal.,2015)asindicatedinFigure1.

Thelossyhomogeneousobjectrepresentingthebrainisilluminatedbytheelectromagneticwavecharacterizedbyincidentelectricfield

�Einc andincidentmagneticfield

�H inc .

Performing somemathematicalmanipulations, the following set of coupled surface integralequationsisobtained(Cvetković&etal.,2015):

j J r G r r dSj

es J r G r r

n S nn

S nωµ

ω

�� ' , ' ' ' ' , '( ) ( ) − ∇ ⋅ ( )∇ (∫∫ ∫∫ ))

+ ( )×∇ ( ) = ==

∫∫

dS

M r G r r dS E n

nS n

inc

'

' ' , ' ',

�� 1

0 2

(1)

where�J and

�M representequivalentelectricandmagneticcurrentdensity,respectively,andGnis

theinterior/exteriorGreenfunctiongivenby(Cvetković&etal.,2015):

G r re

RR r r

n

jk Rn� � � �, ' ; '( ) = = −

−

4π (2)

andRisthedistancefromthesourcetoobservationpoint,respectively,whilekndenotesthewavenumberofaconsideredmediumn.

ThebrainmodelisgeneratedfromafreelyavailableGoogleSketchuprenderingofthehumanbrain,asshownonFigure2.

Figure 1. The lossy homogeneous dielectric brain model


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The set of integral Equation (1) is solved by means of an efficient MoM scheme reportedelsewhere,e.g.in(Cvetković&etal.,2015).Forthesakeofcompleteness,theMoMprocedureisoutlinedinthissection,aswell.

Asafirststep,theequivalentelectricandmagneticcurrents�J and

�M in(1)areexpressedby

meansofalinearcombinationofbasisfunctions�fn

and �gn

,respectively:

J r J f rn n

n

N�� ( ) = ( )

=∑1

(3)

M r M g rn n

n

N�� ( ) = ( )

=∑1

(4)

whereJnandMnareunknowncoefficients,whileNisthetotalnumberoftriangularelements.Applyingtheweightedresidualapproach,i.e.multiplying(1)bythesetofatestfunctions

�fm

andintegratingoverthesurfaceS,aftersomemathematicalmanipulations,itfollows:

j J f r f r G dS dSj

eJ s f

i n mS

n

N

n ii

nn

N

Smωµ

ω

� � � � �( ) ⋅ ( ) + ∇ ⋅∫∫∑ ∑ ∫∫

= =1 1

' ' rr s

f r G dS dS M f r n g r

SS

n i n m n

�

� � � ��

( ) ∇ ⋅

( ) +± ( ) ⋅ × ( )

∫∫∫∫ '

' ' '

''

+ ( ) ⋅ ( )×∇ =

∫∫∑ ∑ ∫∫∫∫= =

Sn

N

nn

N

m nSS

i

dS M f r g r

G dS dS

1 1

� � � �

'

' 'ff r E i

i

m

inc

S

� � ��( ) ⋅ =

=

∫∫ ,

,

1

0 2

(5)

wheresubscriptidenotestheindexofthemedium.Thedetailsofthenumericalsolutionprocedurecouldbefoundelsewhere,e.g.in(Poljak&etal.,2016.a).

2.1. Numerical ResultsFigures3and4shownumericalresultsfortheequivalentelectricandmagneticcurrentdensitiesfor thecaseofa)circularcoilandb)figure-8coil.Theactualcoil insulationandcasingarenotconsidered.Eachcoil isdiscretizedinto80linearsegments.ThecoilsaredrivenbyasinusoidalcurrentofamplitudeI=2843A,f=2.44kHz.Theradiusofcircularcoilandfigure-8coilis4.5cmand3.5cm,respectively,whilethenumberofwindingsforcircularis14andforfigure-8is15turns.

Figure 2. The human brain model for SIE formulation: (a) Detailed 3-D model from Google Sketchup; (b) Final model discretized using the triangular elements


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Figures5and6showthenumericalresultsforthesagittalandtransversalcross-sectionsoftheinternalandexternalfield,respectively,duetocircularcoilandfigure-8coil.Inbothcases,thecoilsarepositionedtangentiallytothebrainsurface,1cmovertheprimarymotorcortexarea,i.e.thedistancebetweenthecoilgeometriccenterandbrainsurfaceis1cm.

TheresultsshowninFigures5and6confirmthatthemaximumelectricfieldoffigure-8coilisinduceddirectlyunderthecoilgeometriccenter,whileforthecaseofcircularcoil,maximumfieldisinducedunderthecoilwindings.Inadditiontotheinducedelectricfieldvalue,itisimportanttoconsidertheorientationofthefieldvector.TheresultsshowtheobviousimportanceofthecorrectTMScoilplacementwithrespecttothebrainsurface.

3. ModELING oF NERVE FIBER EXCITATIoN

Figure7showsthemyelinatednervefiberwithanarbitrarynumberofRanvier’snodesrepresentedbyastraightthinwireantenna(Zulim&etal.,2015;Zulim&etal.2016).

Figure 3. Equivalent electric current density for: a) Circular coil; b) Figure-8 coil

Figure 4. Equivalent magnetic current density for: a) Circular coil; b) Figure-8 coil


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Figure 5. Sagittal cross-section of the electric field induced in the brain model by: a) Circular coil; b) Figure-8 coil. Coils not shown.

Figure 6. Transversal cross-section of the electric field induced in the brain model by: a) Circular coil; b) Figure-8 coil. Coils not shown.

Figure 7. Thin wire antenna model of the myelinated nerve fiber


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NotethatLdenotesthefiberlength,aistheinneraxonradiusandbistheradiusincludingthemyelinsheath.ElectrodenervefiberstimulationistakenintoaccountbymeansoftheequivalentcurrentsourceIglocatedatthefiberbeginning.ThecurrentgeneratorIgrepresentsthenervefiberstimulationusedinelectro-acupunctureorpercutaneouselectricalnervestimulation(PENS),whichbothmakeuseofthinneedlesinjectedthroughtheskin.Thefiber,orientedalongthexaxis,isassumedtobeimmersedinanunboundedhomogenouslossymedium.

TheformulationisbasedonthehomogeneousPocklingtonintegro-differentialequationfortheunknownintracellularcurrentIaalongthenervefiber(Zulim&etal.,2015;Zulim&etal.2016):

−∂

∂−

( ) =∫

1

40

2

2

2

0j x

g x x I x dxeff

a

L

πωεγ , ' ( ') ' (6)

whileg(x,x’)isthelossymediumGreen’sfunctiondefinedas:

g x xe

R

R

, '( ) =−γ

(7)

whereRdenotesadistancefromthesourcetotheobservationpoint,respectively,γisthecomplexpropagationconstantgivenby:

γ ωµσ ω µε ε= −jr

20

(8)

andεeffisthecomplexpermittivityofalossymediumgivenby:

ε ε εσωeff rj= −

0 (9)

withσ, εrandωbeingtheconductivityandrelativepermittivityofthemedium,respectively,andωistheangularfrequency.

ThecurrentgeneratorIgistakenintoaccountviatheconditionsatthenervefiberends:

I I I La g a( ) , ( )0 0= = (10)

It is worth mentioning that the properties of the lossy medium, nerve fiber membrane andmyelinsheathareconsideredviatheconductivityandrelativepermittivitywhichareobtainedfromthecableequationandthetransmissionlineequation,respectively,asdescribedin(Zulim&etal.,2015;Zulim&etal.2016).

TheHelmholtzequationforthecorrespondingtransmembranevoltageisgivenby(Zulim&etal.,2015;Zulim&etal.,2016):

∂

∂− =

2

2

2 0V

xVmγ (11)


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where:

γωτλm

j=

+12

(12)

andVisatransmembranevoltage,whileλisthelengthconstantdefinedby(Zulim&etal.2016):

λρ

=r am

a2

(13)

andτisthetimeconstantgivenby(Picht&etal.,2013):

τ = r cm m

(14)

ρaistheresistivityofaxoplasm,rmthemyelinlayerresistanceforunitareaandcmcapacitanceoftheRanvier’snodemembraneperunitarea.

Thecorrespondingrelationsfortheconductivityandrelativepermittivityareobtainedbymeansof theprocedurereportedin(Zulim&etal.,2015).HomogeneousPocklingtonintegro-differentialEquation(6)issolvedviatheGalerkinBubnovIndirectBoundaryElementMethod(GB-IBEM).Thedetailscouldbefoundelsewhere,e.g.in(Zulim&etal.2016).Amoregeneraltreatmentofathin-wireembeddedinalossymediumcanbefoundin(Poljak&Doric,2006.a,Poljak&Doric,2006.b).Forthesakeofcompleteness,theprocedureisoutlinedinthissection,aswell.

Thus,usingtheboundaryelementformalism,(6)istransformedintothesetoflinearequations,whichcanbewrittenusingthefollowingmatrixnotation(Zulim&etal.2016;Poljak,2007):

Z j nji

e

i

e

i

n { } = =

=∑ α 0 1 2

1

, , ,... (15)

wheren is thenumberofwiresegments, Zji

e is themutual impedancematrix representing the

interactionoftheobservationsegmentjwiththesourcesegmentiand α{ }i

eisthesolutionvector,

expressedinglobalnodes.Themutualimpedancematrixis:

Zj

D D g x x f fji

e

effj i

T

llj i

ij

= − { } { } − { } { }∫∫

1

42

πωεγ' ( , ') '

∆∆

TT

ll

g x x dxdxij

( , ') '∆∆∫∫

(16)

wherematrices{f}and{f’}containlinearshapefunctionsandmatrices{D}and{D’}containtheirderivatives.ΔljandΔliaretheobservationandsourcesegmentlengths,respectively.Thenervefiberstimulation,intheformofanequivalentcurrentgenerator,isincorporatedintomatrixsystemviaboundaryconditions(10).


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3.1. Passive Nerve FiberWhenmodelingthepassivenervefiber,theioniccurrentineachRanvier’snodeistobeconsidered.Thus,theadditionalcurrentsourcesinthenon-activatedRanvier’snodesareimpressed.Figure8showsthepassivenervefibermodelhaving3Ranvier’snodesand4internodes.

AnalyzingtheCRRSS(Chiu,Ritchie,Rogert,StaggandSweeney)model(Rattay,1999;Chiu&etal.,1979;Sweeney&etal.,1987)itfollowsthattheioniccurrentineachnon-activatedRanvier’snodeisapproximatelyequalto-0.6Ia,whereIaistheintracellularcurrentflowingintotheobservedRanvier’snode.IntracellularcurrentIa2whichleavesthe1stRanvier’snodeisequalto0.4Ia1.Byanalogy,theintracellularcurrentIa3flowingoutofthesecondRanvier’snodeisequalto0.4Ia2andtheintracellularcurrentIa4thatleavesthethirdRanvier’snodeis0.4 Ia3.

3.2. Numerical Results for Passive Nerve FiberTheintracellularcurrentiscalculatedforthenervefiberof2cm,with9Ranvier’snodesand10internodes.Thefiberisstimulatedatitsbeginning,usingasubthresholdrectangularcurrentpulse,plottedinFigure9.ThetimedomainresultsareobtainedfromthefrequencydomainresultsbymeansofInverseFastFourierTransform(IFFT).

Ranvier’snodesarelocatedatx = 2, 4, 6, 8, 10, 12, 14, 16, 18mmalongthefiber,respectively.TheintracellularcurrentdistributionalongthepassivenervefiberisplottedinFigure10.TheresultiscomparedtotheCRRSSmodel(Rattay,1999;Chiu&etal.,1979;Sweeney&etal.,1987),andcalculatedforthetimeinstantt =1ms.

Figures11and12showthesubthresholdintracellularcurrentversustimeintheRanvier’snodes2,and6,respectively.

Theresultsfortheintracellularcurrentobtainedbytheantennamodel,seemtobeinarathersatisfactoryagreementwiththeresultsobtainedbyusingtheCRRSSmodel.NotethattheintracellularcurrentwaveforminRanvier’snodesfollowstheexcitationpulseform,asduetothesubthresholdcurrentstimulation,thevoltagegatedchannelsarenotactivated.

3.3. Active Nerve FiberModelofanactivenervefiberispresentedonamyelinatednervefiberwith3activeRanvier’snodesand4internodes,asshowninFigure13.

EachnodeisrepresentedbyawirejunctionoftwothinwiresrepresentinganactiveRanvier’snode.Thus,3additionalcurrentsourcesattheactiveRanviernodesrepresentanioniccurrentIioftheactivatednode.TheioniccurrentisdeterminedbyanalyzingtheCRRSSmodelandtheanalyticalexpressionfortheioniccurrentisobtainedbycurvefittingprocedure(Poljak&etal.,2016.b):

Figure 8. Passive nerve fiber model


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Figure 9. Rectangular subthreshold current pulse

Figure 10. Intracellular current along the passive nerve fiber, t = 1ms, L=2cm


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Figure 11. Intracellular current in the passive Ranvier’s node 2, L = 2 cm

Figure 12. Intracellular current in the passive Ranvier’s node 6, L = 2 cm


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I t Au t t e e Eu t t ei

B t t D t t G t t( ) = − −( ) −

+ −( )− −( ) − −( ) − −( )

1 11 1 2

22

3

2

+ − −( )HeK t t (17)

wheret1 =0.031ms,t2 =0.252msA,A =0.02075mA/ms,B=199.9(ms)-1,D =46.91(ms)-1,E=0.000664mA/ms,G =60(ms)-2.TheioniccurrentisdepictedinFigure14.

Athinwirejunctionmodeloftheactivenode,canberepresentedbytwoseparatethinwires,asshowninFigure15.

TheKirchhoff’slawhastobesatisfiedatthejunction(Poljak&etal.,2016.b):

I Ii i j i j= ++ +0 4 0 61

1 2. .

, , (18)

Figure 13. Active nerve fiber model

Figure 14. Ionic current of activated node of Ranvier


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whereIi,j+1representstheioniccurrentvalueflowingoutofthejunctioninonedirectionandIi,j+2iscurrentflowingoutofthejunctionintheoppositedirection.

First,thenervefiberisstimulatedbyacurrentgeneratorIgatafiberbeginning,asinthecaseofthepassivenervefiber.Furthermore,IfthestimulatingcurrentexceedsthethresholdinthecorrespondingRanvier’snode,thesecondcurrentsource,representingtheioniccurrent,activates.Theintracellularcurrentfortheactivatedfiberisasummationofthenodeactivationcurrentandtheioniccurrentoftheactivatednode.Theresultingcurrent,flowingoutoftheactivatednode,thenactivatesthenextnode.Thesameprocedureisrepeatedforeachnode.

Thethresholdforthenervefiberactivationdependsonthestrengthanddurationofthestimulus(Geddes&Bourland,1985;Boinagrov&etal.,2010;Ashley&etal.,2005).

3.4. Numerical Results for Active Nerve FiberThe intracellular current for the active nerve fiber is calculated with 9 Ranvier’s nodes and 10internodes.Rectangular currentpulseused todepolarize thenerve fibermembrane isplotted inFigure16.

ThecurrentpulseshowninFigure16stimulatesthenervefiberatthefiberbeginning.Oncea thresholdcurrentof1.5µAreaches the firstRanvier’snode, theothercurrent sourceactivates.Summationoftheintracellularcurrentactivatingthenodeandtheioniccurrentofthesecondcurrentsourceyieldstheresultingintracellularcurrentinthe1stactivatedRanvier’snode.Theresultingcurrentthenactivatesthe2ndRanvier’snode.TherelatedintracellularcurrentplotisshowninFigure17.

The results for the intracellularcurrentobtainedvia theantennaandCRRSSmodel showarathersatisfactoryagreementtoacertainextent.Someminordiscrepanciesappearduetotheioniccurrentapproximation.Intracellularcurrentpartpertainingtotherectangularcurrentpulseisinaverygoodagreement.

Finally,Figure18showstheintracellularcurrentforthe4thRanvier’snode.Obviously,therectangularcurrentpulsedecaysalongthefiber,whileintracellularcurrentpart

keepsalmostthesameamplitudeattheactivatednode.Thesignaloftheactivatednervepropagatesalongthefiberalmostwithoutattenuation.

Finally,Figure19showstheintracellularcurrentdistributionalongthenervefiber,plottedforthetimeinstantof0.2msobtainedbytheantennaandCRRSSmodel,respectively.

Thereisasatisfactoryagreementbetweentheresults,withsomevisiblediscrepanciesinamplitudeof the intracellular current travelling along the fiber, mainly due to the instability of numericalprocedureintheantennamodel.

Figure 15. Two wire junction representation of the active node


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Figure 16. Rectangular superthreshold current pulse

Figure 17. Intracellular current in the active Ranvier’s node 2, L = 2 cm


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Figure 18. Intracellular current in the active Ranvier’s node 6, L = 2 cm

Figure 19. Intracellular current along the active nerve fiber, t = 0.2 ms, L = 2cm


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4. CoNCLUSIoN

Thepaper reviews theuseof integralequationmethods in studiesofbiomedicalapplicationsofelectromagneticfieldspertainingtoTMSandnervefiberstimulation.TMSmodelingisbasedonthesetofcoupledsurfaceintegralequationswhilerelatednumericalsolutionprocedurebasedontheMethodofMoments(MoM)approach.

The integralequationbasedformulationpresentsamore rigorous,andhencemoreaccuratedescriptionoftheTMS,bynothavingtotakeintoaccounttheinductiveandcapacitiveeffects,oftenbeingneglectedwhenquasi-staticapproximationisused.

Themyelinatednerve fiber ismodeledby the thinwireantenna inpassiveandactivestate,respectively,stimulatedbyarectangularcurrentpulseatthefiberbeginning.ThemodelisbasedonthecorrespondingPocklingtonintegro-differentialequation,handledviatheGB-IBEM.

Themyelinatednervefiberbasedonantennatheorypresentsthecompletelynewconceptinthenervefibermodeling.Interestedreadercanfindthedetailedsurveyofotherconceptsrelatedtonumericalelectrostimulationmodelsin(Reilly2016).

Someillustrativecomputationalexamplesfortheinducedfieldinthebrain,duetotheTMStreatmentand intracellular current along thenerve fiberdue to thecurrent sourceexcitationarepresentedinthepaper.

Thefutureworkwillberelatedtothedevelopmentofanatomicallyrealisticmodelofthehumanbrainbasedontheintegralequationapproach.Also,thecouplingofthehumanbrainmodelwiththenervefibermodel,i.e.theinducedelectricfieldresultsastheinputparametersofthenervefibermodelisthesubjectoftheongoingactivities.


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REFERENCES

Ashley,Z.,Sutherland,H.,Lanmuller,H.,Unger,E.,Li,F.,Mayr,F.,&Salmons,S.et al.(2005).Determinationofthechronaxieandrheobaseofdenervatedlimbmusclesinconsciousrabbits.Artificial Organs,29(3),212–215.doi:10.1111/j.1525-1594.2005.29037.xPMID:15725219

Boinagrov, D., Loudin, J., & Palanker, D. (2010). Strength–Duration Relationship for Extracellular NeuralStimulation:NumericalandAnalyticalModels.Journal of Neurophysiology,104(4),2236–2248.doi:10.1152/jn.00343.2010PMID:20702740

Chiu,S.Y.,Ritchie,J.M.,Rogart,R.B.,&Stagg,D.(1979).Aquantitativedescriptionofmembranecurrentsinrabbitmyelinatednerve.The Journal of Physiology,292(1),149–166.doi:10.1113/jphysiol.1979.sp012843PMID:314974

Cvetković,M.,Poljak,D.,&Haueisen,J.(2015).AnalysisofTranscranialMagneticStimulationBasedontheSurface Integral Equation Formulation. IEEE Transactions on Biomedical Engineering, 62(6), 1535–1545.doi:10.1109/TBME.2015.2393557PMID:25608302

Deletis,V.,Rogić,M.,Fernández-Conejero,I.,Gabarrós,A.,&Jerončić,A.(2014).Neurophysiologicmarkersinlaryngealmusclesindicatefunctionalanatomyoflaryngealprimarymotorcortexandpremotorcortexinthecaudalopercularpartofinferiorfrontalgyrus.Clinical Neurophysiology,125(9),1912–1922.doi:10.1016/j.clinph.2014.01.023PMID:24613682

Frijns, J.H.,&tenKate, J.H. (1994).Amodelofmyelinatednervefibres forelectricalprosthesisdesign.Medical & Biological Engineering & Computing,32(4),391–398.doi:10.1007/BF02524690PMID:7967803

Garvey,M.A.,&Mall,V.(2008).TranscranialMagneticStimulationinChildren.Clinical Neurophysiology,119(5),973–984.doi:10.1016/j.clinph.2007.11.048PMID:18221913

Geddes,L.A.,&Bourland, J.D. (1985).TheStrength-DurationCurve. IEEE Transactions on Biomedical Engineering,BME-32(6),458–459.doi:10.1109/TBME.1985.325456PMID:4007910

Golomolzina,D.R.,Gorodnichev,M.A.,Levin,E.A.,Savostyanov,A.N.,Yablokova,E.P.,Tsai,A.C.,&Smirnov,N.V.et al.(2014).AdvancedElectroencephalogramProcessing:AutomaticClusteringofEEGComponents. International Journal of E-Health and Medical Communications, 5(2), 49–69. doi:10.4018/ijehmc.2014040103

Picht,T.,Krieg,S.M.,Sollmann,N.,Rösler,J.,Niraula,B.,Neuvonen,T.,&Ringel,F.et al.(2013).Acomparisonoflanguagemappingbypreoperativenavigatedtranscranialmagneticstimulationanddirectcorticalstimulationduringawakesurgery.Neurosurgery,72(5),808–819.doi:10.1227/NEU.0b013e3182889e01PMID:23385773

Poljak,D.(2007).Advanced Modeling in Computational Electromagnetic Compatibility.NewJersey:Wiley-Interscience.doi:10.1002/0470116889

Poljak,D.,Cvetković,M.,Dorić,V.,Zulim,I.,Đogaš,Z.,Rogić-Vidaković,M.,...ElKhamlichiDrissi,K.(2016b).IntegralEquationModelsinSomeBiomedicalApplicationsofElectromagneticFields:Transcranialmagneticstimulation(TMS),Nervefiberstimulation.InProceedings of the International Multidisciplinary Conference on Computer and Energy Science (SpliTech),Croatia.doi:10.1109/SpliTech.2016.7555936

Poljak,D.,Cvetković,M.,Peratta,A.,Peratta,C.,Dodig,H.,&Hirata,A.(2016a,June).Onsomeintegralequationapproachesinelectromagneticdosimetry.InProceedingsofBioEM2016,GhentBelgium.

Poljak,D.,&Doric,V.(2006).a).Wireantennamodelfortransientanalysisofsimplegroundingsystems,PartI:Theverticalgroundingelectrode.Progress in Electromagnetics Research,64,149–166.doi:10.2528/PIER06062101

Poljak,D.,&Doric,V.(2006).b).Wireantennamodelfortransientanalysisofsimplegroundingsystems,PartII:Thehorizontalgroundingelectrode.Progress in Electromagnetics Research,64,167–189.doi:10.2528/PIER06062102

Rajapakse,T.,&Kirton,A.(2013).Non-invasivebrainstimulationinchildren:Applicationsandfuturedirections.Transl. Neurosci.,4(2),1–29.doi:10.2478/s13380-013-0116-3PMID:24163755

Rattay,F.(1999).Thebasicmechanismfortheelectricalstimulationofthenervoussystem.Neuroscience,89(2),335–346.doi:10.1016/S0306-4522(98)00330-3PMID:10077317

http://dx.doi.org/10.1111/j.1525-1594.2005.29037.x

http://www.ncbi.nlm.nih.gov/pubmed/15725219

http://dx.doi.org/10.1152/jn.00343.2010

http://dx.doi.org/10.1152/jn.00343.2010


http://dx.doi.org/10.1113/jphysiol.1979.sp012843


http://dx.doi.org/10.1109/TBME.2015.2393557


http://dx.doi.org/10.1016/j.clinph.2014.01.023



http://dx.doi.org/10.1007/BF02524690




http://dx.doi.org/10.1109/TBME.1985.325456


http://dx.doi.org/10.4018/ijehmc.2014040103

http://dx.doi.org/10.4018/ijehmc.2014040103

http://dx.doi.org/10.1227/NEU.0b013e3182889e01


http://dx.doi.org/10.1002/0470116889

http://dx.doi.org/10.1109/SpliTech.2016.7555936

http://dx.doi.org/10.2528/PIER06062101

http://dx.doi.org/10.2528/PIER06062102

http://dx.doi.org/10.2478/s13380-013-0116-3


http://dx.doi.org/10.1016/S0306-4522(98)00330-3



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Rogić,M.,Deletis,V.,&Fernández-Conejero,I.(2014).Inducingtransientlanguagedisruptionsbymappingof Brocas area with modified patterned repetitive transcranial magnetic stimulation protocol. Journal of Neurosurgery,120(5),1033–1041.doi:10.3171/2013.11.JNS13952PMID:24405070

Sweeney,J.D.,Mortimer,J.T.,&Durand,D.(1987).Modelingofmammalianmyelinatednerveforfunctionalneuromuscularelectrostimulation.InProc 9th Ann Conf IEEE EMBS(pp.1577–1578).

Yamamoto,K.,Takiyama,Y.,Saitoh,Y.,&Sekino,M.(2016).Numericalanalysesoftranscranialmagneticstimulation based on individual brain models by using a scalar-potential finite-difference method. IEEE Transactions on Magnetics,52(7),1–4.

Zulim,I.,Dorić,V.,&Poljak,D.(2016).Myelinatednervefiberantennamodelactivation.InProceedings of the 2016 24th International Conference on Software, Telecommunications and Computer Networks (SoftCOM).IEEE.doi:10.1109/SOFTCOM.2016.7772159

Zulim,I.,Dorić,V.,Poljak,D.,&ElKhamlichiDrissi,K.(2015).Antennamodelforpassivemyelinatednervefiber.InProceedings of the 2015 23rd International Conference on Software, Telecommunications and Computer Networks (SoftCOM).IEEE.doi:10.1109/SOFTCOM.2015.7314069

http://dx.doi.org/10.3171/2013.11.JNS13952


http://dx.doi.org/10.1109/SOFTCOM.2016.7772159

http://dx.doi.org/10.1109/SOFTCOM.2015.7314069


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Dragan Poljak was born on 10 October 1965. He received his BSc in 1990, his MSc in 1994 and PhD in electrical engineering in 1996 from the University of Split, Croatia. He is the Full Professor at Department of Electronics, Faculty of electrical engineering, mechanical engineering and naval architecture at the University of Split, and he is also Adjunct Professor at Wessex Institute of Technology. His research interests include frequency and time domain computational methods in electromagnetics, particularly in the numerical modelling of wire antenna structures, and numerical modelling applied to environmental aspects of electromagnetic fields. To date professor Poljak has published nearly 200 journal and conference papers in the area of computational electromagnetics, seven authored books and one edited book, by WIT Press, Southampton-Boston., and one book by Wiley, New Jersey. Professor Poljak is a member of IEEE, a member of the Editorial Board of the journal Engineering Analysis with Boundary Elements, and co-chairman of many WIT International Conferences. He is also editor of the WIT Press Series Advances in Electrical Engineering and Electromagnetics. In June 2004, professor Poljak was awarded by the National Prize for Science. In 2013, he was awarded by the Nikola Tesla Prize for achievements in Technical Sciences. From 2011 to 2015 professor Poljak was the Vice-dean for research at the Faculty of electrical engineering, mechanical engineering and naval architecture. In 2011 professor Poljak became a member of WIT Bord of Directors. In June 2013 professor Poljak became a member of the board of the Croatian Science Foundation.

Mario Cvetković was born in Split, on October 30, 1981. He received the B.S. degree in electrical engineering from the University of Split, Croatia in 2005. In 2009, he obtained master degree in environmental electromagnetic compatibility from the Wessex Institute of Technology, University of Wales, United Kingdom, and in December 2013 he received Ph.D. from University of Split, Croatia, for the thesis entitled “Method for Electromagnetic Thermal Dosimetry of the Human Brain Exposed to High Frequency Fields”. In December 2010, he held a seminar to graduate and postgraduate students at the Technical University of Ilmenau, Germany, and in September 2014 he held a seminar at the Mälardalen University, Västerås, Sweden. To date he has published 28 journal and conference papers and two book chapters. He is currently working as a postdoc at the Faculty of electrical engineering, mechanical engineering and naval architecture (FESB), University of Split. His research interests are numerical modeling including finite element and moment methods, computational bioelectromagnetics and heat transfer related phenomena. He is a recipient of the “Best Student Paper Award”, awarded at the 16th edition of the international conference SoftCOM 2008. Also, in 2012, at the Scientific Novices Seminar held at FESB, he was awarded with the recognition for his previous scientific achievements.

Vicko Doric was born on 27th of April 1974. in Split, Croatia. He received the M.Sc. and PhD. degrees from the University of Split, Split, Croatia, in 2003 and 2009, respectively. He is employed at the Department of Electronics, University of Split, since January 2001. He is currently Assistant Professor and ERASMUS coordinator at same Department. Vicko Doric is a member of IEEE since 2005. He currently serves as a President of Croatian chapter of IEEE EMC society. His previous duties within Chapter include position of Treasurer of Croatian chapter of IEEE EMC society (2007.-2010.) and Vice president of Croatian chapter of IEEE EMC society (2011.-2015.) His research interests include computational methods in electromagnetics, particularly in the numerical modeling of wire antenna structures with strong applications in the field of electromagnetic compatibility. To date, he has published two books, 10 journal papers and nearly 45 conference papers in the area of computational electromagnetics and electromagnetic compatibility.

Ivana Zulim was born on 21 November 1985, in Split, Croatia. She received her M.S. degree in electrical engineering with honors from the University of Split in 2009, and the Ph.D degree in electrical engineering from the University of Split in 2015. Since 2009, she is employed at the Department of Electronics and Computer Science, University of Split, where she is currently a Senior Research Assistant. Dr. Zulim is a recipient of the “Best Student Paper Award”, awarded at the international conference SoftCOM 2009. She is a member of IEEE since 2009. Her primary research interest is the nerve fiber stimulation modeling. She is also dealing with the electromagnetic problems and the numerical simulations.

Maja Rogić Vidaković was born on 23 December 1982. She received her BA in speech and language pathology in 2005 by Faculty of Special Education and Rehabilitation, Department of Speech and Language Pathology, University of Zagreb, Croatia. She received her MSc in 2008 in neuroscience from Graduate School of Neural & Behavioural Sciences, International Max Planck Research School, Eberhard Karly University, Tübingen, Germany. In 2012, she received her PhD in neuroscience from School of Medicine, University of Split under the mentorship of Professor Vedran Deletis. She is a senior research assistant and research associate at the Department of Neuroscience, School of Medicine, University of Split, Croatia. From 2013 she is the Head of the Laboratory for Human and Experimental Neurophysiology (LAHEN) at the Department of Neuroscience. Her research interests are preoperative mappings of higher cognitive functions (particularly speech and language) with special expertise in nTMS, with future desire to apply the research achievements for intraoperative mappings of patients in Croatia. To date dr. Rogić Vidaković published 26 works: 13 scientific publications (9 in Current Contents), 13 conference papers, she has been invited speaker to 4 international conferences. The number of citations was more than 60 in 2015 and ΣIF 30.669 (2014 year). She is a member of Croatian Neuroscience Society, Croatian Society for EEG and Clinical Neurophysiology, Croatian Logopedic Association. She received Rector’s award from the University of Zagreb, and 1st award for best scientific poster for 2015 and 2012 at the International symposium on Navigated Brain Stimulation in Neurosurgery in Berlin, Germany.


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Jens Haueisen received a M.S. and a Ph.D. in electrical engineering from the Technical University Ilmenau, Germany, in 1992 and 1996, respectively. From 1996 to 1998 he worked as a Post-Doc and from 1998 to 2005 as the head of the Biomagnetic Center, Friedrich-Schiller-University, Jena, Germany. In 2003, he received the habilitation (professorial thesis). Since 2005 he is Professor of Biomedical Engineering and directs the Institute of Biomedical Engineering and Informatics at the Technical University Ilmenau, Germany. He has authored and co-authored more than 200 research articles in peer reviewed scientific journals and serves on two editorial boards. From 2002 to 2004 he served as President and from 2004 to 2006 as Secretary General of the International Advisory Board on Biomagnetism. Since 2005, he is chair of the study program development commission and chair of the examination commission of the Bachelor and Master program “Biomedical Engineering”. He is member of the academic senate of the Technical University Ilmenau and full member of the Saxon Academy of Science. His research interests include the investigation of active and passive bioelectric and biomagnetic phenomena and medical technology for ophthalmology.

Khalil El Khamlichi Drissi received his M.Sc., and Ph.D. degrees in Electrical Engineering from Ecole Centrale de Lille and the University of Lille, in 1987 and 1990 respectively. He received the Habilitation in electronics, at the Doctoral School “Sciences Pour l’Ingénieur” of Blaise Pascal University, in 2001. Currently, he is Professor at the Department of Electrical Engineering where he was the Dean in the period from 2007 to 2011. He is also Senior Researcher at Institute Pascal Laboratory and his research interests include EMC in Power Electronics and Power Systems, in particular numerical modeling, EMI reduction and converter control.

integral equation formulations and related …...biomedical applications, computational models,...

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