A Huygens’ surface approach to rapid characterization of peripheral nerve stimulation 2021 Mathias Davids.pdf

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Received:24December2020|Revised:18June2021|Accepted:22July2021
DOI:
FULLPAPER
AHuygens’surfaceapproachtorapidcharacterizationof
peripheralnervestimulation
Mathias Davids1,2,3Bastien Guerin1,2LawrenceL. Wald1,2,4
||
,DepartmentofRadiology,MassachusettsGeneralHospital,Charlestown,Massachusetts,USA
2HarvardMedicalSchool,Boston,Massachusetts,USA
3ComputerAssistedClinicalMedicine,MedicalFacultyMannheim,HeidelbergUniversity,Heidelberg,Germany
4Harvard-MITDivisionofHealthSciencesTechnology,Cambridge,Massachusetts,USA
Correspondence
MathiasDavids,:Peripheralnervestimulation(PNS)modelinghasapotentialroleinde-
forBiomedicalImaging,DepartmentofsigningandoperatingMRIgradientcoilsbutrequirescomputationallydemanding
Radiology,MassachusettsGeneralHospital,-
14913thStreet,,Boston,MA
02129,
Email:mathias.******@(P-matrix)definedonanintermediaryHuygens’surfacetoallowfastPNS
characterizationofarbitrarycoilgeometriesandbodypositions.
Fundinginformation
NationalInstituteofBiomedicalImagingMethods:TheHuygens’surfaceapproachdividesPNSpredictionintoanextensive
andBioengineering(NIBIB);Nationalpre-computationphaseoftheelectromagneticandneurodynamicresponses,which
InstituteofMentalHealth(NIMH),
Grant/AwardNumber:R01EB028250,isindependentofcoilgeometryandpatientposition,andafastcoil-specificlinear
R24MH106053,U01EB025121,
U01EB025162andU01EB026996theHuygens’approachbyperformingPNScharacterizationsfor21bodyandhead
gradientsandcomparingthemwithfullelectromagnetic-neurodynamicmodeling.
WedemonstratethevalueofHuygens’surface-basedPNSmodelingbycharacter-
izingPNS-optimizedcoilwindingsforawiderangeofpatientpositionsandposes
intwobodymodels.
Results:ThePNSpredictionusingtheHuygens’P-matrixtakeslessthanaminute
(insteadofhourstodays)withoutcompromisingnumericalaccuracy(error≤ %)
,wedemonstratethatcoilsoptimized
forPNSatthebrainlandmarkusingamalemodelcanalsoimprovePNSforother
imagingapplications(cardiac,abdominal,pelvic,andkneeimaging)inbothmale
andfemalemodels.
Conclusion:RepresentingPNSinformationonaHuygens’surfaceextendedtheap-
proach’sabilitytoassessPNSacrossbodypositionsandmodelsandtesttherobust-
nessofPNSoptimizationingradientdesign.
ThisisanopenaccessarticleunderthetermsoftheCreativeCommonsAttributionLicense,whichpermitsuse,distributionandreproductioninanymedium,providedtheoriginal
workisproperlycited.
©
;00:1–|1:.
2|DAVIDSetAl.
KEYWORDS
electromagneticfieldsimulation,gradientcoildesign,magneto-stimulationthresholds,MRIsafety,
neurodynamicnervemodel,peripheralnervestimulation
1|INTRODUCTIONPNSthresholdisdeterminedbythelowestcoilcurrentampli-
tudethatgeneratesanactionpotentialinanynerve.
TheabilitytoquicklyassessperipheralnervestimulationThePNSoraclereplacesthecomplextitrationofanonlin-
(PNS)isbecomingincreasinglyimportantforassessing,opti-earneurodynamicmodelwithaquicklycalculablelinearmet-
mizing,andmonitoringthesafetyofgradientcoilsinMRI1,2ricthatcanbeincorporatedinthenumericgradientwinding
aswellasdrivecoilsinmagneticparticleimaging3,4orstim--35ThePNSoracleiscomputeddirectlyfrom
,assigninga
ofwhetheracandidatecoilandpatientpositionwillinducevaluetoeachsegmentthatcorrespondstothereciprocalPNS

case,identifyingthestimulationlocationisalsoimportanttoandguaranteesalinearrelationshipbetweenthePNSoracle
informdesignchoicesthatbalancethepropensitytostimulateandtheelectricpotentials,andthustheE-fieldsandthecoil
,
stimulatestheshouldersbeforethehead,itmightbedesirable(suchasdisretecoilsorcurrentdensitybasiselements)along
toalterthedesigntobalancethetwobodyparts,toincreaseallnervefibersarecombinedinasingleP-matrixthatcanbe
theoverall(worst-case),de-incorporatedduringthecoilwindingoptimizationasalinear
,6Althoughuseful
incorporatedintothenumericalcoiloptimizationframeworkforcharacterizingspecificconfigurations,precalculationof
,6theP-matrixisnumericallyintensive(uptoaweekofcalcu-
Combiningelectromagnetic(EM)modelingindetailedlation)andisspecifictoaparticularcoilformergeometry,
bodymodelsfollowedbyneurodynamiccalculationstopre-bodymodel,andpatientpositionandorientationwithinthe

-newP-matrixforthenewconfiguration.
ingapproachesofRothetal7andothers8-14reliedonsimpleHere,weproposeageneralizationoftheP-matrixformu-
homogeneousgeometriesmimickingthehumanbodyandlationthatallowsasingleprecomputedP-matrixdefinedon
simplifiednervestostudyeffectssuchastheimpactofnerveaHuygens’surfacelyingjustoutsideofthebodymodeltobe
,15Laaksoetal,16quicklytranslatedtoP-matricesspecificforeachofthecoil
Neufeldetal,17,18andothers19-
ofrealisticallyshapednervefibersinmorerealistichetero-theHuygens’P-matrixiscomputationallylaborious(days),
geneousbodymodelstostudyspecificapplications,suchasgenerationofnewP-matricesforspecificcoilgeometriesor
vagusnerve,sciaticnerve,
Thissignificantlyincreasedthecomputationalcomplexityofisnotlimitedtocylinders,allowingtheapproachtobeusedto
,ourgroup22-24andothers18,25explorenoncylindrical,
developedwhole-bodyPNSmodelingframeworks,consist-windingpatternofagradientisestablished,itsfieldscanbe
ingofdetailedEMbodymodelsequippedwithatlasesofquicklymappedtotheHuygens’surfacestoevaluatethePNS
hundredsofnervefibers(∼1900inourmodel).Usingthesethresholdsformultiplepatientpositionswithinthesegeome-
models,’P-matrixby
goodaccuracy(errorsmallerthan15%)inthecontextofPNSperforminganextensivecharacterizationofPNS-optimized
characterizationofMRIgradientcoilsandmagneticparticlegradientcoilsformultiplepatientpositions.
-24
OurmodelingworkflowstartswithsimulatingtheE-
fieldsinducedinadetailedheterogeneousbodymodelbythe2|METHODS
externalcoilandcalculatingtheresultingelectricpotential
-|Basicworkflow
culatethenerveresponsesusingeitheranonlinearelectric
circuit-basedneurodynamicmodel26-28oralinearpredictorTheHuygens’surfaceconsistsofameshenclosingtheen-
builtontheneuralactivatingfunctionandmodifieddrivingtirebodymodelat5cmdistancetotheskin(Figure1).The
function,29-32whichwerefertoasthe“PNSoracle.”24ThesurfaceispopulatedbyanintermediateHuygens’basisset:.
DAVIDSetAl.|3
FIGURE1FemaleandmalebodymodelswithenclosedHuygens’
-fieldmaps(maximumintensityprojectionswithlogarithmicscale)areshownfortwoexemplary
Huygens’basesneartheheartandneartheleftarm
(smallcurrentloops).Huygens’sprincipleandGreen’sthirdthePNSoraclevalues39(reciprocalPNSthreshold)along

fieldswithinthebodymodelgeneratedbyanysourcesituatedunitexcitationofasingleHuygens’
outsidetheHuygens’,37The5cmsurface-bodydis--fieldsimulationand
tanceallowsthebodymodelanditsHuygens’surfacetobePNSoraclecalculationstepsaredescribedfurtherinthe
placedatclinicallyrelevantpatientpositionswithinthecoilSupportingInformation.
(withoutintersectingthecoil),whilemaintainingsufficient
distancetothebodytoallowaccuraterepresentationoftheAfterthisisdoneforeachHuygens’basis,thedatais
EMfieldswiththefinitebasissetdefinedontheHuygens’assembledintomatricesofnHcolumns(wherenHdenotes
(magneticdipoles)thenumberofHuygens’basisfunctions).Weusethesub-
withdiameters1-2 “H”formatricesdefinedfortheHuygens’basisset.

otherbasiselementssuchaselectricdipolescanbeaddedtoB-fieldsandPNSoracles,
theHuygens’basissettoallowformodelingnonconserva-sizenB×nH(nB,numberofB-fieldanalysispoints),whereas
tiveE-×nH(nP,
ForaunitcurrentineachHuygens’basiswecomputed:sectionsinthebody).Multiplicationofthesematricesbythe
vectorofbasiscurrentweightsdescribesthetotalB-fieldand
-fieldcomponentsatanalysispointsequallydis-PNSoracleresponses,respectively.
tributedwithinthebodymodel(10mmstepsize).TheTheHuygens’P-matrixisbothbodymodelandcurrent-
B-fieldscanbesimulatedquicklyusingtheBiot-Savartwaveformspecificbutisindependentofthecoilgeometry
;itrepresentsonlyhow
-fieldcomponentsateachvoxelinthebodymodeltheHuygens’
(asshowninFigure1)usingalow-frequencymagneto-informaspecificcoilgeometryandpatientposition,amap-
quasistaticsolverbasedonthemodularfiniteelementpingmatrixisneededtolinktheHuygens’P-matrixPHto
methods(MFEM)C++-fieldissampledtheP-matrixspecifictoaparticulargradientcoilgeometry,
:.
4|DAVIDSetAl.
mappingmatrixMwithsizenH×nC(nC,numberofexci-plusPNSoracleextraction).Weassessedtheaccuracyofthe
tationsourcesofthegradientcoilgeometry)asfollows:approachforthefemaleandmalemodelinsevenSiemens
gradientcoils(SiemensHealthineers,Erlangen,Germany):
PC≈P̃C=PHM(1)fourwhole-bodygradients(Sonata,Quantum,Prisma,and
Connectome)andthreehead-onlygradients(AC84,AC88,
whereP̃isonlyanestimateofthedesired“ground-truth”PNSandthemorerecent“Impulse”headgradient54).Wecom-
C
,ẼC,andP̃Cwiththeirdirectlycom-
theHuygens’surfaceisneededtoensurethatP̃CissufficientlyputedcounterpartsBC,EC,andPC(computedusingfullEM
,thesamelinearrelationholdsforthesimulationsfromthecoilwindings)asfollows:
B-fields(ie,themagneticfieldBCofasetofexcitationsources
canbeapproximatedbyaweightedsumoftheB-fieldsoftheB−B̃E−ẼP−P̃
B=maxCCE=maxCCP=maxCC.(4)
Huygens’basisset):BCECPC
BC≈B̃C=BHM(2)Notethatweonlyassessedthismetricinregionswhere
theamplitudeofthequantityisabove1%ofitsglobalmaxi-
whereBChasasizeofnB×nC(nB,numberofanalysispointsmum,asthemetricisill-posedinlow-
inthebodymodel;nC,numberofexcitationsourcesofthegra-assessedtheerrorasafunctionofthenumberofbasisfunc-
dientcoilgeometry).BecausetheB-fieldisrelativelysmoothtionsontheHuygens’surfacebyrandomlyremovingbasis
withinthebodymodel,nBcanbesignif