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Citation for final published version:
McGrath, Deirdre, Ravikumar, Nishant, Beltrachini, Leandro, Wilkinson, Iain, Frangi, Alejandro
and Taylor, Zeike 2016. Evaluation of wave delivery methodology for brain MRE: Insights from
computational simulations. Magnetic Resonance in Medicine 78 (1) , pp. 341-356.
10.1002/mrm.26333 filefile
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1
EvaluationofWaveDeliveryMethodologyforBrainMRE:Insightsfrom
ComputationalSimulations
DeirdreM.McGrath1,2*
NishantRavikumar1
LeandroBeltrachini1
IainD.Wilkinson2
AlejandroF.Frangi1
ZeikeA.Taylor1
1CISTIBCentreforComputationalImaging&SimulationTechnologiesinBiomedicine,
INSIGNEOInstituteforinsilicoMedicine,
TheUniversityofSheffield,
Sheffield,
UK
2AcademicUnitofRadiology,
FacultyofMedicine,Dentistry&Health,
TheUniversityofSheffield,
Sheffield,
UK
*Addressofauthorforcorrespondence:
DeirdreMcGrath,
CentreforComputationalImaging&SimulationTechnologiesinBiomedicine(CISTIB),
DepartmentofElectronicandElectricalEngineering,
TheUniversityofSheffield,
2
C14,PamLiversidgeBuilding,
MappinStreet,
Sheffield,
S13JD
Tel:0114-2225398
d.mcgrath@sheffield.ac.uk
Note:A.F.FrangiandZ.A.Taylorarejointseniorauthors
Wordcount:
Running head: Simulation of cranial vibration during brain MRE to evaluate wave delivery
methodologies
3
ABSTRACT
Purpose:Magneticresonanceelastography(MRE)of thebrain isbeingexploredasabiomarkerof
neurodegenerativediseasesuchasdementia.However,MREmeasuresforhealthybrainhavevaried
widely.Differingwavedeliverymethodologiesmayhaveinfluencedthis,hencefiniteelement-based
simulationswerecarriedouttoexplorethispossibility.
Methods:Thenaturalfrequenciesofaseriesofcranialmodelswerecalculated,andMRE-associated
vibration was simulated for different wave delivery methods at varying frequency. Displacement
fields and the corresponding brain constitutive properties estimated by standard inversion
techniqueswerecomparedacrossdeliverymethodsandfrequencies.
Results: The deliverymethods producedwidely differentMREdisplacement fields and inversions.
Furthermore, resonancesatnatural frequencies influenced thedisplacementpatterns. Twoof the
wavedeliverymethods(head-cradleandacousticpillow)gaverisetolowerinversionerrors,e.g.,at
90Hztheerrorinthestoragemoduluswas11%lessthanforthebite-barmethod.
Conclusion: Wave delivery has an important impact on brain MRE reliability. Assuming small
variations in brain biomechanics, as recently reported to accompany neurodegenerative disease
(e.g., 7% for Alzheimer's disease), the effect of wave delivery is important. Hence, a consensus
should be established on the optimum methodology, to ensure diagnostic and prognostic
consistency.
KEYWORDS:Magnetic resonance elastography; brain; skull; finite element modeling simulation;
naturalfrequencies;dementia
4
INTRODUCTION
Magnetic resonance elastography (MRE) (1) is a non-invasive method for measuring the
biomechanicalpropertiesofbiologicaltissue.Thisisachievedbydeliveryofmechanicalwavestothe
site of interest, and measurement of the resulting displacement field using MRI. Biomechanical
properties, such as stiffness and viscosity, are reconstructed from the displacement field using
inversionalgorithms.MREofthebrainiscurrentlybeingexploredforthediagnosisofneurological
andneurodegenerativediseasesuchasdementia(2-11).Thisevaluation iscomplicatedbythefact
thattheMREmeasuresobtainedsofarforhealthybrainhavevariedwidely(12-15).Inareviewof
healthybrainMREdata(12)theshearmodulusvaluesreportedforwhitemattervariedbetween2.5
and15.2kPa,andforgreymatterbetween2.8and12.9kPa.Additionally,someMREstudieshave
reportedadependencyofbrainelasticity andviscosityonageandgender (16,17).Moreover, the
expectedinfluenceofneurodegenerativediseaseonbrainbiomechanicsislow,e.g.,in(2)onlya7%
decreaseinshearstiffnesswasreportedforAlzheimer'sdiseasecomparedwithhealthycontrols.
WhilethevariationinhealthybrainMREdatamayreflecttrueheterogeneityacrosspopulations,itis
also possible that thesemeasureswere influencedbymethodological variations between studies,
suchasdifferentinversionalgorithmsorotherpost-processingstepssuchasfilters,orthesignalto
noise(SNR)oftheacquisitions.Forinstance,in(18)itwasfoundthatMREmeasuresforbrainwere
strongly dependent on SNR and on the region of interest selected. Another possibility is that
differencesinwavedeliverymethodmethodologyandexcitationfrequencyhadaninfluence..MRE
wavesaretransmittedtothebrainviavibrationoftheskull.However,asyet little isknownabout
themotionofthecraniumduringthisprocess,whichislikelytodependonthemechanismofwave
delivery, the wave frequency, and the specific characteristics and inter-subject variability of the
anatomyoftheskull.Themodeofwavedeliveryhasvariedgreatlybetweenstudies,e.g.,bitebar
(19)mechanicalheadactuator (16),andacousticpillow(2).Thewave frequencyhasalsodiffered;
however, as brain tissue exhibits viscoelastic properties, different viscoelastic moduli values are
expected for different frequencies, and some studies have sought to characterize frequency-
dependenteffects(16,20,21).
Much previous work has been carried out using finite element model (FEM) based analysis to
simulatemotionof thehumanheadduring injury (22). Somestudieshavesimulatedormeasured
the natural frequencies (NFs) of the human skull, to predict the response of the skull to collision
impact (23,24), tomodel the conduction of sound through the skull to aid hearing (25,26), or to
understand skull vibrationduring surgical intervention (27).Recently,ourgroupused steady state
harmonicanalysistomodelMRE-associatedwavepropagationinthehumanbraintoinvestigatethe
5
influence of reflections and heterogeneity across boundaries of anatomical structures, i.e., the
processesoftheduramaterandtheventricles(28). Itwasfoundthatthisanatomyinfluencedthe
displacementfieldsandledtoerrorartifactsintheinversioncalculationofthebrainbiomechanical
properties.Inthisearlierwork,tosimplifythemodelingofwavedeliverytothebrain,theskullwas
notincludedinthemodel,andvibrationdeliverywasmodeledfromthepiamaterofthebrainusing
displacement loading with a uniform direction and magnitude. However, this simplification also
excludesthepossibilityofmodelingtheeffectsofdifferentskullexcitationapproaches.
InthecurrentstudyitwassoughttoextendourFEMsimulationframeworktomodelthevibration
dynamicsoftheskullduringMRE,andtherebytodeterminetheirdependencyonthewavedelivery
approachandfrequency.Moreover,itwassoughttodeterminetheimpactofvaryingwavedelivery
attheskullontheMREdisplacementfieldinthebrain,andonthederivedbiomechanicalproperties.
Thehypothesiswasthatthemethodofwavedeliveryandwavefrequencywould leadtodifferent
vibrationfieldsintheskullandthereforeinthebrain,whichwouldinturninfluencetheestimation
ofthebiomechanicalpropertiesofthebrain.Asapreliminarystep,modalanalysiswascarriedoutto
understand the influenceof the skull’s various anatomical featureson its natural frequencies and
associatedmodesofvibration.Tothebestofourknowledge,this isthefirstreportaddressingthe
modelingofvibrationdynamicsofthewholehumanheadduringMRE.
METHODS
OverviewofFEMsimulations
AllsimulationswerecarriedoutusingAbaqusv6.12(DassaultSystèmesSimuliaCorp,Johnston,RI,
USA),anddetailsarelistedinTable1.
Modalanalysistodeterminenaturalfrequencies
FEM-basednaturalfrequency(modal)analysiswascarriedoutonskull-onlymodelsandafullhead
modelderivedfromtheXCATphantom(adatasetbasedonimaging,defininganatomicalstructures
of the human body at high resolution) (29). The purpose of this investigation was to gain
understanding of the influence of different anatomical components on cranial vibration. Varying
materialpropertiesanddifferentboundaryconditionswerealsocompared.Furthermore,forinter-
subjectcomparison,modalanalysiswascarriedoutonaskull-onlymodelderivedfromCTdata.
HarmonicanalysistopredictMREwavepropagationpatterns
6
MREwavepropagationwassimulatedusingharmonicanalysiswiththeXCATandCTderivedmodels
toinvestigatehowskullvibrationchangeswithwavedeliverymethodandfrequency.Comparisonof
theskull-onlyandfullheadXCATmodelsgivesinsightintotheinfluenceofthesofttissuesoncranial
vibration. Moreover, the full head model allows simulation of the complete propagation of
vibrationsfromtheskulltothebrain,asdesired,andexplorationoftheinfluenceofwavedelivery
modesandfrequencyonbraindisplacementfieldsand,thereby,recoveredtissueproperties.
ModelsderivedfromtheXCATphantom(XM)
AsetofskullmodelsandafullheadmodelwerederivedfromtheXCATphantom(29)(XM1-XM6,
Fig. 1). Surface meshes from the phantom were first interpolated onto a regular grid to create
individual segmentations for the included anatomical structures. For the full head model, the
segmentations were assigned different labels and merged to create a single multi-label
segmentation. Next, a volumetric (tetrahedral) finite element mesh was generated in Matlab
(R2012a,Mathworks Inc.,Natick,MA)using the ISO2MESHsoftwarepackage (30). Thevolumetric
mesh generation algorithm used within ISO2MESH is based on the CGAL library (CGAL,
Computational Geometry Algorithms Library, cgal.org). The primary advantage of generating a
volumetric mesh from a multi-label segmentation in this manner is the automatic generation of
shared nodes between adjacent structures. Additionally, the ISO2MESH package provides user
control over the tetrahedral element size to be applied to each region in the mesh, for
computationalefficiency.
The XM models included various combinations of anatomical components, to evaluate their
respectiveinfluences(Fig.1).XM1:upperskull,excludingjawandneck;XM2:upperskull including
jaw, but excluding neck; XM3: upper skull, plus jaw and neck. The upper skull contained a cavity
correspondingtothesinuses.FormodelsXM1-XM3allbonewasassignedpropertiesasforcortical
bone(seesection:"Tissuematerialproperties").XM4:afurtherrefinedmodelwasgeneratedfrom
XM3,inwhichextrastructuresweredefinedattheconnectionpointsofthejawwiththeskull,and
assignedmaterialpropertiesasforcartilage(twoversions:Cartilage#1andCartilage#2inTable2).
XM5:thismodelwasarefinementofXM4inwhichtheupperskullandjawincludedinnerregionsof
cancellousbone.ThecancellousbonewasaddedusingISO2MESHbyerodingtheskullvolume,while
avoidingintersectionswiththesinuses.Theerosionwasperformeduntilarealisticgeometryforthe
cancellous bone was obtained, as assessed by visual comparison with the Colin 27 atlas
segmentation(31).
XM6wasa refinementofXM5 inwhichadditional regionswereadded:1)brain in the inner skull
cavity;2)alayerofcerebrospinalfluid(CSF)surroundingthebraintoapproximatethemeninges;3)
7
avolumetodefinetheventriclesinsidethebrain,filledwithCSF;4)theprocessesoftheduramater,
thefalxcerebriandthetentoriumandfalxcerebellimembranes(denoted"FTM"),whichliebetween
the hemispheres of the cortex, between the cortex and the cerebellum, and between the
hemispheres of the cerebellum, respectively; 5) a single volume for the tissues (skin,muscle, fat,
etc.)surroundingtheskullandneck;and6)thesectionofspinalcordontheinsideoftheneck.The
meningesandFTMwerenotincludedintheoriginalXCATphantomdatabutwereaddedbasedon
estimations informed by manual segmentations of other anatomical MRI data. The approximate
volume of the finite elements of the brain, CSF and FTMmeshesmatched that of the equivalent
meshesemployed in (28) (i.e.,£ 2mm3),while, for computational efficiency, theother structures
weremodeledwitha lowerelementdensity(i.e.,elementvolume£4mm3).AsthebrainandCSF
regions were modeled as near incompressible material, they were meshed with hybrid (linear-
pressure) elements, which discretize and solve for the pressure field independently of the
displacements,toavoidvolumetriclocking.
SkullmodelderivedfromCTdata(CTM)
ToexplorethegeneralityofthefindingsfortheXCATskull,asecondskullmodelwasprepared.This
model (CTM, Fig 2a) was generated from a probabilistic atlas derived from the computed
tomography(CT)imagesofpatients(n=33),providedinapublicdomaindatabaseforcomputational
anatomy (imagenglab.com/pddca_18.html). The probabilistic atlas was used as it represents the
averageskullshapeforapopulation,andhencedescribesamoregeneralanatomy.Additionally,in
comparison to the rawpatientCT images, theatlas is lessnoisyandhenceeasier toprocess. The
skull was segmented semi-automatically from the atlas using ITK-SNAP (32). The segmentation
processinvolvedacombinationofintensitythresholdingandgeodesicactivecontourpropagationto
segment the skull (including the jaw) and the first three vertebrae. A volumetric mesh was
subsequentlygeneratedfromtheskullsegmentation,similarlytothepreparationoftheXMmodels.
Themodelincludedtheskull,jawandneckandagapforthesinuses,andthematerialpropertiesof
elastic cortical bone were employed for the whole skull (i.e., no cartilage or cancellous bone
included).
Tissuematerialproperties
AlltissueconstitutivepropertiesaresummarizedinTable2.Corticalboneandcancellousbonewere
modeledinitiallyaslinearelasticsolidswithpropertiesasdefinedin(33).Viscousdampingwaslater
addedtoboth, inaccordancewith(34).Brainwasmodeledasasofthomogeneousisotropiclinear
viscoelasticnear-incompressiblematerial,withstorage(G')and loss(G'')modulivaluestakenfrom
MREmeasurementsinhealthybrainat25-90Hz(16,19)andthedensitywasapproximatedtothat
8
of water (1,000 kg/m3) (16). The Poisson's ratio was set to 0.499999; estimated using the
approximatespeedofsoundinthebrain(1,550ms-1).Cerebrospinalfluid(CSF)inthemeningesand
ventricles wasmodeled as a soft viscoelastic solid (33). All other tissues weremodeled as linear
elastic solids, with parameters taken from the following sources: cartilage Young's modulus and
density were estimated from (35), and two different Poisson's ratio values (0.5 and 0.1) were
assumedtoexploretheeffectofvaryingthecartilageproperties,andtherebyinfluencingtherange
of relative movement between the jaw and skull; tissues surrounding the skull and neck (skin,
muscle,fat,etc.)weremodeledasauniformvolumewithpropertiesofthescalpusedin(33);spinal
cord was modeled as a linear elastic solid, with the elastic modulus taken from (36), with an
approximatedPoisson's ratioanddensity;and theprocessesof theFTMwereassignedproperties
from(24).
Boundaryconditions
Differentboundary conditions (BC)wereapplied for thevarious simulations (Table1, Fig.1).BC1:
free boundaries; BC2: formodels XM1 and XM2, inwhich the neckwas excluded, nodes near to
wheretheneckwouldattachtotheskullweretethered(x,yandzdisplacementssettozero);BC3:
formodelsincludingtheneck(XM3-XM6),asetofnodesatthebaseoftheneckweretethered;BC4:
forXM6thenodesattheendoftheoutertissueoftheneckwerealsotetheredtoapproximatethe
connectionof theneck to the restof thebody,and to reduce the reflectionofwaveenergyback
fromtheendsurfaceofthenecktissue.
Modalanalysis
Thenaturalfrequencies(oreigenfrequencies)ofvibrationwerecalculatedinAbaqusbyeigenvalue
extraction using the Lanczos eigensolver. This analysis was carried out for each XM model and
materialcombinationwithvaryingBCs(seeTable1),andforCTMwithcorticalboneonlyandBC3.
ThefirstsixNFswerecomparedforthemodelsXM1-XM6andCTM.
MREwavepropagationsimulation
MRE-associated mechanical vibration at specific frequencies was simulated in Abaqus using the
direct-solutionsteady-statedynamicanalysis (hereafter referredtoasharmonicanalysis).This isa
perturbation procedure in which the response of a model to an applied harmonic vibration is
calculatedaboutabasestate,toproducefrequency-spacesteady-statenodaldisplacementsu:
u(x, �) = u(x)exp(���) (1)
9
wherewistheangularfrequency,andxand�arespatialandtemporalcoordinates,respectively.
UsingmodelsXM5,XM6andCTM,MREsimulationwascarriedoutforfrequenciesat5Hzintervals
in the range 5-150 Hz, and additionally at 37.5 and 62.5 Hz, to correspond with the frequencies
includedinthebrainmaterialspecification(Table2).HumanbrainMREisusuallycarriedoutat<100
Hz, as brain exhibits viscoelastic behavior and strongly attenuates the MRE waves at higher
frequencies,resultinginlowdisplacementamplitudesandpoordataquality.Theupperlimitof150
Hzwaschosenheretoinvestigateeffectsinthevicinityof100Hz.
Displacement loading with an amplitude of 10 μm (chosen to approximate wave amplitudes
observed in brainMRE)was delivered to sets of nodes at different positions on the skull surface
correspondingtothedifferentbrainMREwavedeliverymethods(Fig.2):L1:"head-cradle",temples
vibratedinthehead-footdirection;L2:templesvibratedleft-rightinoppositedirections;L3:temples
vibrated left-right in the same direction; L4: "acoustic pillow" (2), nodes at the back of the skull
vibratedinanterior-posteriordirection;L5:"bitebar"(19),nodesonupperandlowerjawvibratedin
left-right direction. For consistency of wave delivery between XM5 and XM6, the loading was
delivered to the skull surface inXM6 rather than theouter skin surface. In the first instance, 100
nodes were selected on the skull for each loading location. To determine the sensitivity to the
numberofloadingnodes,for50and90Hzthenumberofloadingnodeswasvariedto50and200
foreachloadingoption.
The vibration fields in the skull and brain were compared for the different loading options. The
viscoelasticmoduli (G' andG'')were reconstructedusingdirect inversion (28). This algorithmwas
implementedinMatlabthroughderivativecalculationusingafinitedifferencemethodona"virtual
imagingvoxel"grid,whichwasinterpolatedat3mmintervalsfromtheFEnodaldisplacements.To
evaluate the inversion accuracy, themean absolutepercentagedifference (MAPD)was calculated
forthetotalbrainvolume:
���� =011
2
3456378
345
29 (2)
whereNisthetotalnumberofvoxels,nisthevoxelnumber,�;<thegroundtruthvalue(ofG'orG'')
and�=9theinversionvalue.Forselectionofthevolumecorrespondingtothefullbrain,a3Dmask
wascreated.Thevoxelsattheedgeofthebrainareaffectedbyvarioussourcesoferror,including
averagingwiththesurroundingtissues frominterpolation,derivativecalculation,smoothingofthe
curlvector fieldduring inversion,errors inthedirect inversioncausedbytissueheterogeneityand
interference patterns resulting fromwave reflections at tissue boundaries (28). Hence theMAPD
10
was also calculated using a mask eroded by a margin of 3 voxels. By excluding this margin,
understanding can be gained of the specific influence of the errors at the brain tissue edges. To
explorethepossiblebenefitofcombiningtheresultsofdifferentactuationmethods,theG'andG''
voxeldatawasaveragedfor the fivedifferent loadingmethods,andtheMAPDscalculatedfor the
fullanderodedbrainvolumes.
RESULTS
NaturalfrequenciesoftheXCATskullmodels
Thefirstsix(non-zero)NFsoftheXCATskullmodelsare listedinTable3.(ForallmodelswithBC1
thefirstsixvibrationmodeswillalways,trivially,berigidbodymodes,withtheoreticalfrequencies
of0Hz;onlyfrequenciesfornon-rigidmodes,i.e.,non-zerofrequencies,wereincludedinTable3).
For simulations #1-#6 (models XM1-XM3with varying BCs) all of the first six non-zero NFs differ
widelybetweensimulations,indicatingtheinfluenceofthevariousanatomicalcomponentsandthe
boundary conditions. For simulation#6 (XM3,BC3) the first fourNFs are54, 82, 124and281Hz.
Visualizationof theassociateddisplacement fields revealed that the first threeNFsareassociated
withthedirectionsofrotationoftheheadabouttheneck,whilethefourthNFwasassociatedwith
motionofthejaw(Fig.2g-j).Forsimulation#7(XM4withCartilage#1,BC3)thefirstthreeNFsare
unchanged,whileNF#4andsubsequentNFswerealtered.Forsimulation#8(XM4,Cartilage#2,BC3)
thefirstthreeNFswereagainunchanged,whileNF#4andsubsequentNFswereagainaltered.For
XM5,withtheadditionofcancellousbone(simulation#9,BC3),alltheNFsareslightlyaltered,while
the first fourarestillassociatedwiththesamemodesofvibration (Fig.2g-j).Withtheadditionof
viscositytothecorticalandcancellousbone(simulation#10)thefirstsixNFsareunaltered.Hence
the inclusion or exclusion of the jaw and neck had amajor impact on skull vibration, as did the
boundaryconditionoftetheringatthebaseoftheneck.Basedontheresultsofthisanalysis,forthe
MREsimulationintheskullandwholeheadmodelsitwasdeemednecessarytoincludethejawand
neckandtetheringatthebaseoftheneck.
FortheXM6fullheadmodel(simulation#11,BC3+BC4),theNFcalculationwasstronglyinfluenced
bythesoftbraintissue,andthiswasconfirmedbyvisualizationof theassociatedeigenmodes, for
whichvariousresonancepatternsoccurredinthesoftbraintissue.Resonancesoccurredatintervals
ofapproximately1Hz,fromtheminimumNF15.4Hz.IntheMREsimulationswithXM6itwasnoted
that resonantpeaksoccurredatparticular frequenciesassociatedwith rotationof theheadabout
theneck (Fig.2g-i) (see later section:Effectofwavedeliveryand frequencyonMREdisplacement
fieldsandinversions inXM6). ItwasalsoobservedthatforNFsofXM6closetoorattheresonant
11
frequencies for theMRE simulations of XM6, the associated eigenmodes demonstrated a strong
influenceofaparticulardirectionofwholeheadmotionabouttheneck,andhencetheoverallskull
andbraindisplacementswerelargerattheseNFs.
Effectofwavedeliveryandfrequencyondisplacementfieldsinskull-onlymodels
InFigure3themeandisplacementcomponents inthex,yandzdirections(seeFig.1fortheaxes
orientations) and the displacement vector magnitude are compared for the different MRE wave
delivery methods (loading, L) and frequencies for XM5 and CTM. The plots demonstrate that
resonances occur at the NFs of the models. However, for the various loading options, different
resonancepeaksarepresentorabsent,dependingonthedirectionofmotionoftheskullprescribed
and controlled by the loading. For example, for XM5 with L1 (Fig. 3(a)), a peak occurs for the
displacementsinthexandydirectionsaround125Hz,whichcorrespondstoNF#3at127Hz,while
peaks forNF#1andNF#2are absent. ForCTMwith L1 (Fig. 3(b)), no resonancepeaks are visible;
however,thereappearstobeagradual increasetowardsapeak,whichwouldoccuratthehigher
frequencyof230Hz forNF#3.ForXM5withL2 (Fig.3(c)),a resonancepeakoccurs for theyandz
components at 55 Hz, corresponding to NF#1 of 55 Hz, while for CTM (Fig. 3(d)), resonance is
apparentaround115Hz,correspondingtotheNF#1ofCTM.Therearesimilarpatternsofparticular
resonances occurring for the other wave delivery options (L3-L5). Furthermore (far from the
resonance peaks) for each wave delivery method, displacement is predominantly in a single
direction(x,yorz)correspondingtothedirectionofloadingtotheskull.Itisalsoofimportanceto
notethatthedifferentloadingmethodsachievedifferentdisplacementamplitudes(x,yandz)and
magnitudes. For example at 37.5 Hz (far from resonance) themean displacementmagnitudes of
XM5forL1-L5are:20μm;7μm;9μm;9μm;7μm.ForCTMat37.5Hzsimilarmeandisplacements
magnitudeswereobservedforL1-L5:17μm;5μm;8μm;10μm;7μm.
Figure4presentsthedisplacementmagnitudesplottedontheskullsurfaceforXM5forthevarious
loadingmethods. Ineachcase,vibration is shownat resonance,and foranexamplenon-resonant
frequency.Thedisplacementfieldsdifferbetweenloadingmethods.Additionally,foreachmethod,
the displacement field alters greatly at resonance, when it resembles that of the corresponding
eigenmode(Fig.2g-i).Owingtothe largedisparity inthedisplacementsatresonanceandfar from
resonance,differentcolorscalesareemployedinFig.4forthefrequenciesfarfromresonance,and
theresonantfrequencies.Thisallowscomparisonbetweentheloadingmethodsforresonantandfar
from resonance frequencies withclear depiction of the displacement patterns, and allows ready
comparisonwiththeeigenmodes(Fig.2g-i).
12
EffectofwavedeliveryandfrequencyonMREdisplacementfieldsandinversionsinXM6
Figure5presentsplotsofthemeandisplacements(x,y,zandmagnitude)intheskullandbrainof
theXM6model forthedifferent loadingmethodsandfrequencies.Fortheskull, thedisplacement
componentsagaindifferinmagnitude,andthepredominantdirectionvarieswiththewavedelivery
direction.Furthermore,resonancepeakswhosefrequenciesliewithinabout10Hzofresonancesfor
theXM5(skull-only)model(Fig.3)arevisible.NotalltheNFsofXM6haveanapparentinfluenceon
theseplots,butonlytheNFswitheigenmodesassociatedwithheadrotationabouttheneck(Fig.2g-
i). These have a stronger influence on the displacement plots in Fig. 5 as the overall head
displacements are higher at these frequencies. The relative proportions of the displacement
componentsintheskullaremirroredinthedisplacementcomponentsofthebrain,andlikewisethe
brain resonancepeaksoccur in the vicinityof the resonances in the skull. Figure5f compares the
displacement magnitudes between the loading methods, revealing that for the same (10 μm)
displacement loadingontheskullsurface,L1achievedthehighestdisplacementamplitudes inthe
skullandbrain,whileL2achievedthe lowest (e.g.,at50Hz, themeandisplacementmagnitude in
thebrainwas26μmforL1and5μmforL2).
Figure6displaystherealcomponentsofthecomplexdisplacementfields,themagnitudeofthecurl
andinversionresultsforacentralaxialbrainsliceofXM6at50and90Hz.Thewavepatternsforthe
different displacement components differ widely between loadingmethods and frequencies. The
frequenciesof50and90Hzwerechosenasmoregenerallyrepresentativeoftheloadingmethods,
as they lie far fromtheresonancepeaks forallmethods.Theyalsoaretwoof the frequencies for
whichG'andG''arespecifiedforthebrainmaterial(Table2).LoadingmethodsL1andL4gaverise
tosimilarpatternsinthex,yandzdisplacementfieldsandintheinversionresults,whichisperhaps
to be expected, as bothmethods result in a similar noddingmotion of the head. Also, there are
somesimilaritiesinthepatternsobservedinthez-componentimagesliceofL1andL4andthe50Hz
real displacement component image in (16), where a device is employed which would result in
noddingmotionoftheheadsimilartoL1andL4.Correspondingly,methodsL3andL5,whichboth
prescribe a left-rightmotion of the head, also resulted in similar displacement field patterns and
inversionresults.ForL5at90Hzsimilarx-,y-andz-displacementpatternswereobservedtothose
presented in (19), where 90 Hz actuation is achieved via a bite-bar similar to L5. Conversely, L2
resultedinverydifferentdisplacementpatternsfromalltheothermethods,thoughthepatternsof
errors in the inversion results are similar to thoseof L3 and L5. Themagnitudeof the curl of the
displacement fieldalsodiffersbetween loadingoptionsand frequencies,withgenerally largercurl
magnitudesatthehigherfrequencyof90Hz,asexpectedforamorerapidlyvaryingwaveform,i.e.,
13
ashorterwavelength.Inthedisplayedslices,L2andL4hadloweroverallcurlmagnitudescompared
to theothermethods forboth frequencies.Certain locationson theslicearea tendtohave larger
curlmagnitudes,suchaseithersideofthefalxcerebri(seelastpanelinFigure1forlocationofthe
falxcerebriinanaxialslicethroughthebrain),especiallyforL3andL5.Thismaybeassociatedwith
wavereflectionoffthefalx,butalsopossiblewavetransmissionfromthefalx.
Figure 7 presents plots ofmeanG' andG'' over the brain volume for the differentwave delivery
methods and frequencies, with separate error bar plots (standard deviation error bars) for the
differentloadingmethodstocomparetheerrorsbetweenmethods,andcombinedplotscomparing
the mean values for all loading methods. TheMAPD of G' and G'' for the full and eroded brain
volumes,forthefivefrequenciesofthegroundtruthdata(Table2)arealsopresentedinseparate
error bar plots and combined plots for the five loadingmethods. At higher frequencies, G' varies
betweenthemethodsby~500Pa(Fig.7a),andG''by~300Pa(Fig.7b).Tointerprettheshapeofthe
plotsinFigs.7aand7bitisnecessarytoknowthegroundtruthmodulithatAbaqusemployedinthe
simulations: for frequency-dependent viscoelastic materials, Abaqus interpolates parameters
linearlywithin the rangeof specified frequencies (Table 2), and caps parameters at the bounding
valuesoutsideofthisrange(i.e.atfrequencies<25Hz,the25Hzmoduliareused,andat>90Hz,the
90Hzmoduli are used). Figure 7a-b demonstrate how the variations inG' andG'' over the brain
volumevarywithactuationmethod,andthestandarddeviationinG'andG''overthebrainvolume
tendstobelowerforL1andL4thanfortheothermethods.ThevariationinG'andG''overthebrain
volumealsoincreaseswithfrequency.
TheMAPDsalsovarybetweenactuationmethods,andoverallL1andL4resultinlowererrorsthan
theothermethods(Fig.7c-d)e.g.,at90Hz,MAPDofG'forL1andL4wasapproximately11%less
thanforL3andL5,whileforG''at37.5Hz,itwasapproximately17%lowerforL1andL4compared
with L3.. For the eroded volume, the MAPDs are vastly reduced, though L1 and L4 still have
predominantlythelowererrorvalues,exceptatthehigherfrequencies(62.5and90Hz).However,
the differences between deliverymethods for the erodedmask are only on the order of 1%.The
MAPDsalsovarywithfrequency.ForthefullbrainvolumetheerrorsonG'tendtobehigherforthe
twoextremesof25and90Hz(andthestandarddeviationoftheerror isalsolarger)comparedto
the other frequencies,while forG'' theopposite holds, i.e., 25 and 90Hz have lower errors (and
standarddeviationsoftheerror).Fortheerodedbrainvolume(Fig.7e-f)alltheerrorsarereduced
substantially comparedwith the full volume,and theerrorsand standarddeviationsof theerrors
arehigherfor25and90HzforbothG'andG''.
14
Figure 7g presents error bar plots for the MAPD of the voxel-wise average G' and G'' for all
five loading methods, for the full and eroded brain volumes. Averaging was found to lead to
marginal reductions of 1-2 % in the lowest MAPD values for G' for the full brain volumes for
frequencies 25, 37.5, 50 and 62.5 Hz, while it did not achieve lower errors for 90 Hz, i.e., a
29% error compared to the lowest error at 90 Hz of 25% for L4 (Fig.7c). For the eroded brain
volume for each frequency the differences between the lowest MAPDs of G' and the MAPD
of the average G' were generally <1%. However for G'' averaging increased the errors
substantially for the full and eroded brain volumes for all five frequencies, e.g., for the full
brain volume, while the maximum error for the non-averaged G'' was 76% at 37.5 Hz for L3
(Fig.7d) the error at 37.5 Hz for the averaged G'' was 85%. The average error (over the
frequencies) of the eroded volume for the averaged G'' was 52% (the minimum error of 22%
was at 90 Hz, Fig. 7g), while the errors for the non-averaged G'' for the eroded volume were
all < 7 % (Fig.7f). Furthermore, averaging of the G' and G'' voxel values tended to reduce the
standard deviation of the MAPD over the brain volume for the full and eroded masks. With
regardtotestingthesensitivitytothenumberof loadingnodes, itwasfoundthat formostofthe
wave delivery methods that changing the number of nodes only led to minor variations in the
displacementfieldsandtheinversionmapsofG'andG'',withtheexceptionofL2.TheMAPDofG'
andG''generallyvariedbylessthan1%between50,100and200nodesforloadingforL1,L3,L4and
L5. However for L2 theMAPDs variedmore substantially. For L2 at 50 Hzwith 50, 100, and 200
nodes, theMAPDofG'was18±23%,17±21%and16±19%respectively,while for theMAPDofG''
thevalueswere74±110%,70±101%,and66±90%.Hence,theMAPDandthestandarddeviationof
theMAPD decreased with increasing loading nodes, however the reductions in error weremore
pronouncedforG''.For90Hzasimilareffectwasobserved:with50,100,and200nodes,theMAPD
of G'was 35±37%, 30±30% and 28±26% respectively,while for theMAPD of G'' the valueswere
63±67%,62±64%,and62±62%.Howeverfor90HzthereductionsinG'weremorepronouncedthan
thoseofG''.ThevariationsininversionerrorsforL2appearstobemainlyassociatedwithvariations
in thedirections and amplitudesof thedisplacement field, anddifferentwave interactions at the
borders of the brain tissue, leading to different reflection and interference effects.While for the
otherloadingmethodsthedirectionofthewaveswasrelativelyunalteredbythechangingnumber
ofloadingnodes. Thatthevariationsaremainlyatthebraintissuebordersissupportedbythefact
thatfortheerodedbrainvolumetheMAPDsandstandarddeviationoftheMAPDvariedbylessthan
1%forL2.
DISCUSSION
15
NaturalfrequenciesofXCATandCTMskullmodels
TheNFanalysisof the skullmodelsXM1-XM5 revealed important informationon the influenceof
thevariousanatomicalcomponentsonskullvibration.AsthedeliveryofMREwavestothebrainis
mainlyachievedviatransmissionthroughtheskull,itisimportanttodeterminetherelevantNFsof
the skull (i.e., those that lie in the typical frequency range for brain MRE: 20-100 Hz), and to
understand the factors that influence the eigenmodes. The inclusion of the jaw and neck, and
tetheringatthebaseoftheneckstronglyinfluencedthevibrationoftheskullandtheNFs.
TheNFsdifferedbetweenCTMandthematchingXM3model(simulations#6and#12,Table3).The
material specifications were identical for these simulations, however the volume (and therefore
mass)ofCTMwaslowerthanthatofXM3.NFstypicallyscaleinverselywiththesquarerootofthe
mass, and hence the NFs for CTM are higher than those for XM3. However, structural variation
betweenthemodelswillalso influencemodaldynamics.While theNFsvarybetweenmodels, the
modesofvibrationforthefirstfourNFsarethesameforbothmodels(Fig.2g-j).
PreviousinvestigatorshavesoughttomeasureorsimulatethevibrationandNFsofthehumanskull.
Howeverthemethodologyhasvariedwidelybetweenstudies:somesimulationsormeasurements
excludedthejaw(24)ortheneck(27),orboth(23),andwhilesomemeasurementswerecarriedout
in dry skull models (27), others were made in live human subjects (25), or both (37,38), and
therefore the reportedNFs have also variedwidely between studies. In (24) two FE skullmodels
werecompared:oneexcludingthejawandneck,andtheotherexcludingthejawbutincludingthe
neck.Tetheringwasalsoincludedatthebaseoftheskullorneck.In(24)therangeofthefirstfour
NFsforthemodelwithouttheneckwas149.1-860.2Hz,whiletherangeofthefirstfourNFsofthe
nearestcorrespondingmodelinthisstudy(XM1withBC2)was321-1488Hz.Asinthepresentstudy,
in (24) itwasfoundthatwhentheneckwas includedtheNFswerereduced(first four:88.9-399.4
Hz),andfurthermoretherotationalmotionoftheskullforthefirstthreemodeswassimilartothose
observedinthisstudy(Fig.2g-j),whilethefourthwasassociatedwithhead-footmotionoftheskull.
However,in(24)theydidnotincludethejawbone,whereasinthepresentstudyitwasfoundthat
thefourthNFwasassociatedwithjawmotion.Moreover,inthisstudyforBC3,NF#4was>200Hz
forXM3and>400HzforCTM;asbrainMREistypicallycarriedoutat<100Hz,thissuggeststhat
jawmotionmayhaveasmallerinfluenceonthemotionoftheskullduringMRE.
In this study theNFs of the full headmodel (XM6)were very different to those of the skull-only
model (XM5). The XM6modal analysis was strongly influenced by the soft brain tissue, and the
differentanatomicalstructureswithinthecranium,andNFsoccurredatintervalsofapproximately1
16
Hz, from theminimumNF of 15.4 Hz. This differed greatly from reported NFs for in vivo human
head:Hakanssonetal. (25)measuredNFs in the range500-7,500Hz for invivohumanskullsand
found14-19 resonances,with theaverageof the two lowest frequenciesat972Hz.Caietal. (37)
alsomadeinvivomeasurementsintherange2-52kHzandmadeacomparisonwithdryskulls.They
found complex resonances and antiresonances in both the dry skulls and live head, which were
stronglydependentonthetransducerposition,andfoundthatdampingintheliveheadreducedthe
resonancepeaks.
TheeffectofdampingfromsofttissuescouldbeobservedintheMREsimulationwithXM6,asthe
resonancepeakswereshiftedwithrespecttotheXM5skull-onlymodel(Figs.3and5).Furthermore,
theMREsimulationcouldexploretheeffectofdeliveringwaveenergyatdifferentpositions,andthe
associatedvibrationeffectsofeachdeliverymode.Hence,forXM6theMREsimulationsweremore
informativethanthemodalanalysis.
Inter-subjectdifferencesinskullNFsandpossibleimplicationsforMRE
ThedifferentNFsoftheXM3andCTMmodelsindicatethattheNFswillchangebetweenindividuals
dependingonthesizeandshapeoftheskull.ThedifferentresonanceeffectsintheMREsimulations
ofXM5andXM6also indicatehowNFswill shiftdue to thedampingeffectsof the tissues in the
head, and this is likely to vary between individuals. According to the in vivo measurements of
(25,37),theNFsoftheinvivohumanheadarelikelytooccurat>500Hz,whichiswelloutsidethe
typicalfrequenciesemployedforbrainMRE,i.e.,20-100Hz.However,thesimulationsinthisstudy
have demonstrated that when resonances do occur at or in the vicinity of the MRE excitation
frequency they can have a major impact on the wave fields in the brain. Hence, it is the
recommendation of this study that further exploration should be carried out with volunteers to
determinetheresonancesofthehumanheadandtheimpactoftheseontheMREmeasurements.If
actuation is carried out at a resonance frequency, there is the possibility of persistent nodes and
anti-nodes occurring in the brain tissue, whichmay lead to errors in the inversions. As theMRE
simulationsofthisstudycalculatedtheharmonicsteadystatedisplacements,itwasnotpossibleto
observetheoccurrenceofpersistentnodes,and,moreover,theerrorsintheinversionsofthisstudy
appeared mainly related to the effects at the boundaries of brain tissue with other anatomy.
However in the natural frequency simulations with the full head model XM6, symmetric
displacement patterns were observed in the brain tissue that would suggest the likelihood of
persistent nodes occurring in real acquisitions. This is therefore an important avenue for future
investigationviaMREacquisition.
17
ImplicationsofthechoiceofwavedeliverymethodandfrequencyinbrainMRE
Theresultsof this studyhaveproventhehypothesis that, in thecontextof simulation,MREwave
deliverymethodologyandfrequencyaffect thedisplacement fields intheskullandbrain,andalso
the inversion accuracy. Different displacement components were dominant for the different
methods,whilesomemethodshadsimilarpatternsofdisplacementandinversionerror,i.e.,L1was
similar to L4, and L3 was similar to L5. Furthermore, if the NFs lie at or close to theMRE wave
frequency, a resonance peak can occur in the MRE displacement fields. Also, only particular NF
resonancesoccur for thedifferent loadingmethods,andthepeakscanaccentuatethedifferences
betweendisplacementcomponents.Foraccurateinversionitisimportanttohavebalancebetween
thedisplacementcomponentsinordertoachievefullrankinthesystemofequationssolvedinthe
direct inversion (39). Hence, large disparities between displacement components caused by a
particular wave delivery or resonancesmay lead to inaccurate inversion. However, as brainMRE
studieshavenotreportedsucha largedisparitybetweendisplacementcomponents, thispotential
effectofresonance,orindeedtheparticulardirectionofwavedelivery,appearsunlikelytooccurin
reality,andthemanyapproximations involved in thesesimulationsmayaccount for theseeffects.
However it is a recommendation of this study that future MRE studies should investigate the
possibleinfluencesofresonancesorthepreferentialdirectionofwavesduetothedeliverymethod,
andverifythebalanceofdisplacementcomponents.
OverallL1(head-cradle)andL4(acousticpillow)producedthelowesterrorsintheinversions.InFig.
6,theinversionerrorsappeartobemainlyassociatedwithinteractionofthewavefieldwiththefalx
cerebrimembrane, as large inversionerrorsoccur at either sideof this structure. Formethods L1
andL4thedisplacementfieldismovingpredominantlyinadirectionparalleltothefalx(y-direction),
whilefortheothermethodsthedominantmotionisleft-right(x-direction),andlesserartifactsoccur
for L1 and L4 (especially at 50Hz, Fig. 6). In our previous brainMRE simulationwork (28) itwas
found that inversion artifacts occurred close to interfaces between brain tissue and the FTM and
ventricles.Theconclusionofthatearlierstudywasthaterrorsattheboundarieswerecausedbya
combinationoffactors:1)reflection,refractionandscatteringattissueboundariesleadingtowave
interference, which results in inversion artifacts at larger sampling steps (3 mm); 2) material
heterogeneity bringing about errors in the direct inversion algorithm (which assumes local
homogeneity (39)); 3) averaging across the tissue boundaries due to interpolation, derivative
calculationsandsmoothingofthecurlvectorfieldduringtheinversion(28).However,thefindingsof
this present study emphasize the importance of wave reflection and the resulting interference
patterns, as the different wave delivery methods produce different predominant directions of
18
motionaccompaniedbydifferentmagnitudesofinversionerror.Somesensitivitytothenumberof
loadingnodeswasalsoobserved,andparticularlyforL2.Basedontheseobservations,inrealMRE
acquisitionsitwouldbeimportanttodeterminethesensitivitytotheareaofcontactandpositioning
oftheactuator,andperhapscarryoutoptimizationofthesefactors.
AveragingoftheG'andG''dataoverthefiveactuationmethodsdemonstratedonlyaminorbenefit
totheaccuracyofG',andwasdetrimentaltotheaccuracyofG''.Howeverasthesimulationsofthis
studyweresimplifiedinmanyaspects,averagingovermultipleactuationmethodsmayprovemore
beneficial in real MRE acquisitions, and comparison of different actuation methods will inform
assessmentofthestabilityandreliabilityofmeasurements.Infactrecentworkhasaddressedthisin
the comparison of themethod of (16)with a new remote actuationmethod for brainMRE (40).
Acquisition via multiple actuation methods may be facilitated by newer faster MRE sequences
(41,42).
Limitationsofthecurrentstudyandfuturework
Themajor limitationofthisstudywasthesimplicityofthemodelsemployed, intermsofanatomy
andmaterial specifications.Theapproximations involved in theFEmodelingandsimulationswere
furtherlimitations.Forinstance,theanatomicalmodelsemployedwerebasedontheanatomyofa
single individual (XM) and on the average model of a small cohort (CTM), and therefore do not
captureallthevariabilityofanatomyacrossthepopulation.Skullshapeislikelytovarywithfactors
suchgender,ageandrace.Futureworkwillinvestigatethevariabilityofresonantfrequenciesacross
thepopulationbymeansofstatisticalshapemodelingof theskullbasedonawiderpopulationof
data.Themodelswerealsosimplified intermsofthestructures includedandthematerialmodels
used,suchasasoftviscoelasticsolidforCSFasopposedtoafluid.Furthermore,themeningeshave
inrealityacomplexstructure:theduramater(attachedtotheskull)isconnectedtothepiamater
(attached to the brain) via filaments called trabeculae running through the subarachnoid space,
whichispermeatedwithCSF.Thebrainisalsotetheredtotheskullatthebrainstemandviaother
vascularandneuralconnections.Furthermore,inrealityslippageoccursbetweentheskullandbrain,
andthemeningesarelikelytohavenon-linearmaterialproperties,andthesefactorshavenotbeen
accounted for in themodels. Given the findings ofMRI spin taggingmotion tracking of the brain
duringimpact(43),themotionofthebrainwithintheskullduringMREislikelytobeverycomplex,
with displacement, deformation and rotational motion occurring to varying degrees at different
locationsat thebrain-skull boundary. Another studymeasuringMREwave transmission from the
skulltobrain(44)concludedthatthemeningesstronglyattenuateMREwaves.Furthermore,other
anatomicalfeaturesintheheadthatwerenotincludedarelikelytocausewaveattenuationthrough
19
viscosityandscatteringattissueinterfaces,andindeedbraintissueitselfisinrealityheterogeneous
(20),meaningwavesarelikelytobescatteredattheinterfacesofdifferentbrainregions(45).Based
on the degree of wave and motion damping measured in (43,44) the degree of damping of the
resonancepeaksfortheXM6model(Fig.5)islikelytobeunderestimated.Furthermore,braintissue
isanisotropic,asthewhitematterisfibrous(46),andthiswouldinfluenceMREdisplacementfields.
Brain tissue is likely to be under anisotropic pre-stress, which will affect estimates of material
parametersobtainedusingthedifferentwavedeliverylocationsanddirections.Indeedrecentwork
has found that theMREmeasures for brain whitematter differed depending on whether waves
were delivered in the anterior-posterior direction or the left-right direction (47). Therefore the
interactionofwaveswith theanisotropicstructuresof thebrainmayaccount for thevariability in
measures obtained from different brain MRE studies employing varying actuation methods, as
opposed to differences in the overall displacement fields and wave interactions at the brain
boundaries, or differences in inversionmethods employed or other post-processing steps. Future
studieswillexplorethesensitivityofthefindingsofthisworktovariationsinmaterialpropertiesof
thedifferentanatomicalstructures.
However,thevariabilitythatmightoccurbetweenindividualsandtheapproximationsemployedin
thematerialmodelingdonotnegatetheoverallfindingsofthiswork, i.e.,thatthechoiceofwave
delivery methodology can influence brain MRE data. Rather, studies with wider populations and
varied properties would provide a better estimate of the actual impact of using different wave
deliverymethods.
Althoughfurthersimulationsarewarrantedtoexplorethe limitationsof thefindingsof thisstudy,
ultimatelyinvivoMREstudiesarerequiredtodeterminetheactualimpactonvaryingwavedelivery.
Hence, the main recommendation from this work is that volunteer studies comparing MRE
acquisitions with different wave deliverymethods be undertaken. In fact, in the recent study by
Fehlner at al. (40) athe actuation method of (16) wascompared with a newer remote excitation
method,anditwasfoundthatthemagnitudeandphaseofthecomplexshearmoduluscoulddiffer
byasmuchas6and13%respectivelyinthebrainregionsexamined.Furthersimilarstudiesshould
becarriedouttodetermineaconsensusmethodologyforoptimumaccuracyandstability,although
patientcomfortandthepracticalityofthemethodareotherprimaryconsiderations.
CONCLUSIONS
Throughsimulation,thisstudyhasdemonstratedthatinbrainMREthemethodofwavedeliveryand
wavefrequencystrongly influencethedisplacementfields intheskullandbrain,andconsequently
20
theaccuracyoftheinversionreconstructionsofthebrainbiomechanicalproperties(e.g.,at90Hzan
11%lowerinversionerrorforthehead-cradle(L1)andacoustic-pillow(L4)comparedwiththebite
bar (L5)). However, most of these differences are associated with brain tissue located at the
boundaries with other anatomy, and it is uncertain how much of the inversion error in these
locationsisassociatedwiththeinversionmethodorothersourcesoferrorfromthesimulation,and
futurework should employ other inversionmethods for comparison. But given that the inversion
errorsatthebrainboundariesdifferedstronglybetweentheloadingmethods,itislikelythatthese
errors areassociatedwithdifferent reflection, scatteringand interferenceeffects, anddestructive
interference and standing waves from interference pose very difficult challenges for inversion.
Furthermore,thenaturalfrequenciesofvibrationoftheheadcouldinfluencetheMREdisplacement
fields inthebrainandthereforethe inversionaccuracy,throughtheoccurrenceofstandingwaves
withpersistentnodesandanti-nodes,oranimbalanceindisplacementcomponentsatresonance.
Asthemodelsemployedinthisstudyweregeneratedfromalimitedrepresentationofhumanhead
anatomyandwere simplified in variousaspects, future simulation studies are required toexplore
thelimitationsofthesefindings.Furthermore,itisrecommendedthatinvivoMREstudiesaremade
onvolunteersusing thevariouswavedeliverymethodsandvarying frequencies, todetermine the
stability of the measures of brain tissue biomechanics, and the possible influence of resonant
frequencies.
ACKNOWLEDGEMENTS
ThisstudywasfundedbytheEuropeanUnion’sSeventhFrameworkProgramme(FP7/2007–2013)
aspartoftheprojectVPH-DARE@IT(grantagreementno.601055).Therearenoconflictsofinterest
associatedwiththiswork.
21
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24
Table1Simulationdetails
Simulation
No.
NF/MRE Model No.of
Elements
Element
volume
Boundary
conditions
andloading
Materials
1 NF XM1 823001 £2mm3 BC1 Corticalbone
2 NF XM1 823001 £2mm3 BC2 Corticalbone
3 NF XM2 779408 £2mm3 BC1 Corticalbone
4 NF XM2 779408 £2mm3 BC2 Corticalbone
5 NF XM3 838627 £2mm3 BC1 Corticalbone
6 NF XM3 838627 £2mm3 BC3 Corticalbone
7 NF XM4 838627 £2mm3 BC3 Corticalbone+cartilage#1
8 NF XM4 838627 £2mm3 BC3 Corticalbone+cartilage#2
9 NF XM5 838627 £2mm3 BC3 Cortical+cancellousbone+
cartilage#2
10 NF XM5 838627 £2mm3 BC3 Viscoelasticcorticalviscoelastic
cancellousbone+cartilage#2
11 NF XM6 1420763 £4mm3
(brain,CSF
andFTM£
2mm3)
BC3+BC4 Viscoelasticcorticalandcancellous
bone,cartilage#2,outerhead,
brain,spinalcord,FTM,CSF
12 NF CTM 712434 £2mm3 BC3 Corticalbone
13 MRE(5-
150Hz)
XM5 838627 £2mm3 BC3+L1 Viscoelasticcorticalbone+
viscoelasticcancellousbone+
cartilage#2
14 MRE(5-
150Hz)
XM5 838627 £2mm3 BC3+L2 Viscoelasticcorticalbone+
viscoelasticcancellousbone+
cartilage#2
15 MRE(5-
150Hz)
XM5 838627 £2mm3 BC3+L3 Viscoelasticcorticalbone+
viscoelasticcancellousbone+
cartilage#2
16 MRE(5-
150Hz)
XM5 838627 £2mm3 BC3+L4 Viscoelasticcorticalbone+
viscoelasticcancellousbone+
cartilage#2
25
17 MRE(5-
150Hz)
XM5 838627 £2mm3 BC3+L5 Viscoelasticcorticalbone+
viscoelasticcancellousbone+
cartilage#2
18 MRE(5-
150Hz)
CTM 712434 £2mm3 BC3+L1 Corticalbone
19 MRE(5-
150Hz)
CTM 712434 £2mm3 BC3+L2 Corticalbone
20 MRE(5-
150Hz)
CTM 712434 £2mm3 BC3+L3 Corticalbone
21 MRE(5-
150Hz)
CTM 712434 £2mm3 BC3+L4 Corticalbone
22 MRE(5-
150Hz)
CTM 712434 £2mm3 BC3+L5 Corticalbone
23 MRE(5-
150Hz)
XM6 1420763 £4mm3
(brain,CSF
andFTM£
2mm3)
BC3+BC4+L1 Viscoelasticcorticalandcancellous
bone,cartilage#2,outerhead,
brain,spinalcord,FTM,CSF
24 MRE(5-
150Hz)
XM6 1420763 £4mm3
(brain,CSF
andFTM£
2mm3)
BC3+BC4+L2 Viscoelasticcorticalandcancellous
bone,cartilage#2,outerhead,
brain,spinalcord,FTM,CSF
25 MRE(5-
150Hz)
XM6 1420763 £4mm3
(brain,CSF
andFTM£
2mm3)
BC3+BC4+L3 Viscoelasticcorticalandcancellous
bone,cartilage#2,outerhead,
brain,spinalcord,FTM,CSF
26 MRE(5-
150Hz)
XM6 1420763 £4mm3
(brain,CSF
andFTM£
2mm3)
BC3+BC4+L4 Viscoelasticcorticalandcancellous
bone,cartilage#2,outerhead,
brain,spinalcord,FTM,CSF
27 MRE(5-
150Hz)
XM6 1420763 £4mm3
(brain,CSF
andFTM£
2mm3)
BC3+BC4+L5 Viscoelasticcorticalandcancellous
bone,cartilage#2,outerhead,
brain,spinalcord,FTM,CSF
26
Table2:Constitutiveparametervaluesemployedinsimulations
Tissuetype Parametervalues
Young'smodulus
(MPa)
Poisson'sratio Density(kg/m3)
Corticalbone 15,000 0.21 1,900
Cancellous
bone
4,600 0.05 1,500
Cartilage#1 1 0.5 1,100
Cartilage#2 1 0.1 1,100
Outerhead
tissues
16.7 0.42 1,000
Spinalcord 1.02 0.5 1,000
FTM 31.5 0.45 1,130
Brain Frequency(Hz) G'(Pa) G''(Pa) 1,000
25 1110 480
37.5 1310 570
50 1520 600
62.5 2010 800
90 3100 2500
CSF G0(Pa) G¥(Pa) b(s-1) K(MPa) 1,000
1,000 900 80 1,050
Viscous
damping
corticalbone
t(s) 10 102 10
3 10
4 10
5
G(t)/G0 0.973 0.95 0.915 0.853 0.773
Viscous
damping
cancellousbone
t(s) 1 2 3 4 5 6 7 8 9 10
G(t)/G0 0.93 0.9 0.888 0.873 0.865 0.875 0.852 0.834 0.74 0.7
27
Table3:Firstsixnon-zeronaturalfrequenciesforsimulations
Simulationdetails NaturalFrequencies(Hz)
No. Model BC MaterialTypes #1 #2 #3 #4 #5 #6
1 XM1 BC1 Corticalbone 2747 3223 3733 3833 3998 4507
2 XM1 BC2 Corticalbone 321 459 817 1488 2348 2628
3 XM2 BC1 Corticalbone 341 846 1352 1837 2482 2726
4 XM2 BC2 Corticalbone 288 348 449 707 875 1356
5 XM3 BC1 Corticalbone 336 792 827 1350 1682 1839
6 XM3 BC3 Corticalbone 54 82 124 281 407 600
7 XM4 BC3 Corticalbone
+cartilage#1
54 82 124 275 402 598
8 XM4 BC3 Corticalbone
+cartilage#2
54 82 124 233 384 586
9 XM5 BC3 Cortical+
cancellousbone+
cartilage#2
55 84 127 233 387 578
10 XM5 BC3 Viscoelasticcorticaland
cancellousbone+cartilage
#2
55 84 127 233 387 578
11 XM6 BC3+BC4 Viscoelasticcorticaland
cancellousbone,cartilage#2,
outerhead,brain,spinal
cord,FTM,CSF
15.4 16.0 16.1 16.4 17.1 17.8
12 CTM BC3 Corticalbone 115 127 230 406 974 1087
28
FIGURELEGENDS
Figure1:ModelsfromXCATphantom(ananatomicaldatasetderivedfromimaging).Themodels
aredenotedXM, and contain varying anatomical components andmaterial properties. XM1:
upperskull,excludingjawandneck;XM2:upperskullincludingjaw,butexcludingneck;XM3:
upper skull, plus jaw and neck. (XM1-XM3 were ascribed uniform material properties of
corticalbone).XM4:a refinementofXM3,withextra structuresat the connectionpointsof
thejawwiththeskull,whichweremodeledascartilage.XM5:arefinementofXM4inwhich
theupperskullandjawincludedinnerregionsofcancellousbone.XM6:arefinementofXM5
adding: 1) brain; 2) cerebrospinal fluid (CSF) surrounding the brain to approximate the
meninges;3)ventricles,filledwithCSF;4)theprocessesoftheduramater,thefalxcerebriand
thetentoriumandfalxcerebellimembranes(denoted"FTM");5)asinglevolumefortheouter
headtissues(skin,muscle,fat,etc.);6)thesectionofspinalcordontheinsideoftheneck.The
nodes selected for the prescribed boundary conditions (BC) are also displayed (BC1: free
vibrationofallnodes).BC2:tetheringofnodesatthebaseoftheskullclosetowheretheneck
wouldconnect;BC3:tetheringofnodesatthebaseoftheneck;BC4:tetheringofnodesatthe
basesurfaceoftheouterheadtissues.
Figure2:a)ThemodelderivedfromthepopulationaverageofCTdatafrompatients(CTM).b)-f)
Loading(L)positionsanddirectionsdisplayedwithXM3model(derivedfromXCATphantomand
consistingofskull, jawandneck,anduniformmaterialpropertiessettocorticalbone)..b)L1or
"headcradle",with templesvibrating inhead-footdirection. c) L2 templesvibrating left-right in
oppositedirections.d)L3templesvibratingleft-rightinsamedirection.e)L4or"acousticpillow",
posteriorofskullvibratinginanterior-posteriordirection.f)L5or"bite-bar",upperandlowerjaw
vibrating in left-right direction. g)-j) Motion of skull associated with the first four natural
frequencies(NF)ofXM4(refinementofXM3withconnectpointsbetweenskullandjawmodeled
ascartilage)withboundaryconditionBC3:g)NF#1;h)NF#2;i)NF#3;j)NF#4.
Figure 3: Comparison of mean x, y and z displacement components and overall displacement
magnitudesagainstfrequencyforskull-onlymodelsfordifferentwavedeliverymethods(loading,
L).XM5isthemodelderivedfromXCATphantomandconsistsofskull, jawandneck,regionsof
cartilageandcancellousbone,whiletheCTMmodelisderivedfromthepopulationaverageofCT
datafrompatientsandincludesonlycorticalbone.a)XM5L1;b)CTML1;c)XM5L2;d)CTML2;e)
XM5L3;f)CTML3;g)XM5L4;h)CTML4;i)XM5L5;j)CTML5.
29
Figure4:MRE simulationnodaldisplacementmagnitudes in theXM5model (derived fromXCAT
phantomand consistsof skull, jawandneck, regionsof cartilageand cancellousbone) far from
resonanceandattheresonancepeaksfordifferentwavedeliverymethods(loading,L,seeFig.2)
foraloadingamplitudeof10μm:a)examplefrequenciesfarfromresonance:L1at50Hz,L2at90
Hz,L3at90Hz,L4,at50Hz,L5,at30Hz;b)exampleresonantfrequencies:L1at125Hz,L2at55
Hz,L3at55Hz,L4at85Hz,andL5at55Hzand115Hz.
Figure5:ForXM6model(fullheadmodelderivedfromXCATphantom),comparisonofskulland
brain mean x, y and z displacement components and overall displacement magnitudes against
frequency for different wave delivery methods (loading, L): a) L1; b) L2; c) L3; d) L4; e) L5; f)
Comparisonofmeandisplacementmagnitudesforwavedeliverymethodsforskullandbrain.
Figure 6: Comparison of the real component of the displacement fields in x, y and z, and the
magnitudeofthecurl,andinversionreconstructionsofG'andG''at50Hzand90Hzfordifferent
wavedeliveryoptions (loading,L).Thedisplacementcomponentsareplottedon thesamecolor
scale of -0.03 to 0.03mm for ready comparison between directions,methods and frequencies.
However for Re(uy) of L1 at 90Hz the displacements are also plotted on themore appropriate
scaleof0to0.07mm.
Figure 7: ComparisonofmeanG' andG'' from inversions ofMRE simulations in XM6 (full head
modelderivedfromXCATphantom),andMeanabsolutepercentagedifferencewithgroundtruth
(MAPD)ofG'andG''forfullanderodedbrainmasksfordifferentwavedeliverymethods(loading,
L),andMAPDsofvoxel-wiseaverageofallfiveloadingmethodsforG'andG''.a)MeanG'forfull
brainmask against frequency, separate error bar plots for each loadingmethodwith standard
deviationerrorbars,andplotcomparingmeanvaluesforloadingmethodswithouterrorbars;b)
MeanG'' forfullbrainmaskagainstfrequency,separateerrorbarplots,andcomparisonplot;c)
MAPDofG'forfullbrainmaskforfivegroundtruthfrequenciesusedinmaterialspecificationfor
brain tissue, error bar plots and comparison plot; d)MAPDofG'' for full brainmask, error bar
plots and comparison plot; e) G'MAPD for eroded brainmask, error bar plots and comparison
plot;f)G''MAPDforerodedbrainmask,errorbarplotsandcomparisonplot;g)Errorbarplotsfor
voxel-wiseaverageoverloadingmethodsofG'andG'',forfullanderodedbrainvolumes.
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