Apparent Diffusion Coefficients from High Angular Resolution Diffusion Images

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Apparent Diffusion Coefficients from High Angular
Resolution Diffusion Images: Estimation and
Applications
Maxime Descoteaux, Elaine Angelino, Shaun Fitzgibbons, Rachid Deriche
To cite this version:
Maxime Descoteaux, Elaine Angelino, Shaun Fitzgibbons, Rachid Deriche. Apparent Diffusion Co-
efficients from High Angular Resolution Diffusion Images: Estimation and Applications. [Research
Report] RR-5681, INRIA. 2006, pp.44. ￿inria-00070332￿
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INSTITUTNATIONALDERECHERCHEENINFORMATIQUEETENAUTOMATIQUE
Apparent Diffusion Coefficients from High Angular
Resolution Diffusion Images: Estimation and
Applications
MaximeDescoteaux —Elaine Angelino — ShaunFitzgibbons — RachidDeriche
N° 5681
Septembre2005
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Apparent Diffusion Coefficients from High Angular Resolution
Diffusion Images: Estimation and Applications
MaximeDescoteaux(cid:3) ,ElaineAngelinoy ,ShaunFitzgibbonsz ,RachidDerichex
ThèmeBIO—Systèmesbiologiques
ProjetOdyssée
Rapportderecherche n°5681 —Septembre2005— 44pages
Abstract: Highangularresolutiondiffusionimaging(HARDI)hasrecentlybeenofgreatinterest
incharacterizingnon-Gaussiandiffusionprocesses. Inthewhitematterofthebrain,non-Gaussian
diffusionoccurswhenfiberbundlescross,kissordivergewithinthesamevoxel.Oneimportantgoal
incurrentresearchistoobtainmoreaccuratefitsoftheapparentdiffusionprocessesinthesemulti-
plefiberregions,thusovercomingthelimitationsofclassicaldiffusiontensorimaging(DTI).This
paperpresentsanextensivestudyofhighordermodelsforapparentdiffusioncoefficientestimation
andillustratessomeoftheirapplications. Inparticular,wefirstdeveloptheappropriatemathemat-
ical tools to work on noisy HARDI data. Using a meaningful modified spherical harmonics basis
to capture the physical constraints of the problem, we propose a new regularization algorithm to
estimatea diffusivityprofilesmootherandclosertothe truediffusivitieswithoutnoise. We define
asmoothingtermbasedontheLaplace-Beltramioperatorforfunctionsdefinedontheunitsphere.
Thepropertiesofthesphericalharmonicsarethenexploitedtoderiveaclosedformimplementation
ofthistermintothefittingprocedure. Wenextderivethegenerallineartransformationbetweenthe
coefficientsof a spherical harmonics series of order ‘ and the independent elements of the rank-‘
highorderdiffusiontensor.Anadditionalcontributionofthepaperisthecarefulstudyofthestateof
theartanisotropymeasuresforhighorderformulationmodelscomputedfromsphericalharmonics
ortensorcoefficients. Theirabilitytocharacterizetheunderlyingdiffusionprocessisanalyzed. We
are able to reproduce published results and also able to recovervoxels with isotropic, single fiber
anisotropic and multiple fiber anisotropic diffusion. We test and validate the different approaches
onapparentdiffusioncoefficientsfromsyntheticdata,fromabiologicalphantomandfromahuman
braindataset.
Key-words: restoration and deblurring, high angular resolution diffusionimaging (HARDI), ap-
parentdiffusioncoefficient(ADC),sphericalharmonics(SH),diffusiontensor,anisotropymeasure
(cid:3)[email protected]
[email protected]
zfi[email protected]
[email protected]
UnitéderechercheINRIASophiaAntipolis
2004,routedesLucioles,BP93,06902SophiaAntipolisCedex(France)
Téléphone:[phone]—Télécopie:[phone]
Coefficients de Diffusion Apparents à Partir d’IRM de Diffusion
à Haute Résolution Angulaire: Estimation et Applications
Résumé : L’IRM de diffusion à haute résolution angulaire (HARDI) est maintenant un outil es-
sentiel pour décrire les phénomènes de diffusion non-Gaussiens des faisceaux de fibres de la ma-
tière blanche. Ceux-ci se produient lorsque plusieurs fibres se croisent. Dans ce cas, le tenseur
de diffusion classique (DTI) est limité et insuffisant. Ce rapport fait le point sur les techniques
d’approximationsdescoefficientsdediffusionapparentsàpartirdemodèlesàordressupérieurset
présente aussi leur application dans la définition de mesures d’anisotropies. En particulier, nous
développonsles outils mathématiques adéquats pour traiter et estimer les coefficientsde diffusion
bruitésprovenantdes données HARDI. Àpartird’une base modifiée d’harmoniquessphériques et
desespropriétés,nousproposonsunenouvelleméthodederégularizationobtenantdescoefficients
dediffusionpluslisses. Nousvalidonsl’approchesurdesdonnéessynthétiques,surunfantômebio-
logiqueetsuruncerveauhumain. Deplus,nousétudionsl’étatdel’artdesmesuresd’anisotropies
calculéesàpartirdemodèlesàordressupérieursetnousévaluonsleurhabilitéàdécrireleprocessus
dediffusion.
Mots-clés : restoration et débruitage, IRM de diffusion à haute résolution angulaire (HARDI),
coefficient de diffusion apparent (ADC), harmoniques sphériques, tenseur de diffusion, mesure
d’anisotropie
ApparentdiffusioncoefficientsfromHARDI:estimationandapplications 3
Contents
1 Introduction 4
2 HighAngularResolutionDiffusionImaging(HARDI) 5
3 ApparentDiffusionCoefficient (ADC)Profile Estimation 7
3.1 EstimatingtheADCProfilewiththeSphericalHarmonics(SH). . . . . . . . . . . . 7
3.1.1 SphericalHarmonics(SH) . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.1.2 MethodsforFittingSphericalDatawithSHSeries . . . . . . . . . . . . . . 8
3.2 FittingtheADCProfilewithaHighOrderDiffusionTensor(HODT) . . . . . . . . . 10
4 FittingHODTstoHARDIDataUsingSphericalHarmonics 12
4.1 ARegularizationAlgorithmforADCProfileEstimation . . . . . . . . . . . . . . . 12
4.2 FromSHCoefficientstoHODTCoefficients . . . . . . . . . . . . . . . . . . . . . . 15
5 SyntheticDataExperiments 19
5.1 QuantitativeComparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.1.1 Effectofthe(cid:21)-RegularizationWeight . . . . . . . . . . . . . . . . . . . . . 21
5.2 AnisotropyMeasures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2.1 FrankandChenetalMeasures . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2.2 AlexanderetalMeasure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2.3 ResultsofAnisotropyMeasuresfromSHSeries . . . . . . . . . . . . . . . 26
5.2.4 GeneralizedAnisotropyMeasure . . . . . . . . . . . . . . . . . . . . . . . 27
5.2.5 CumulativeResidualEntropy(CRE) . . . . . . . . . . . . . . . . . . . . . . 31
6 RealDataExperiment 33
6.1 ABiologicalPhantom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.2 HumanBrainHARDIData . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7 ConclusionandDiscussion 38
A RelationofSphericalHarmonicstotheHODT 43
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4 M.Descoteaux,E.Angelino,S.Fitzgibbons,R.Deriche
2fibers trueADCprofile ADCprofilefromDTI
Figure1: ADCprofileestimatefromDTIfailstorecovermultiplefiberorientation. Themaximaof
theADCprofiledonotagreewiththingreenlinescorrespondingtothetruesyntheticfiberdirections.
1 Introduction
For the past decade, there has been a growing interest in diffusion magnetic resonance imaging
(MRI)tounderstandfunctionalcouplingbetweencorticalregionsofthebrain,forcharacterizationof
neuro-degenerativediseases,forsurgicalplanningandforothermedicalapplications.DiffusionMRI
istheonlynon-invasivetooltoobtaininformationabouttheneuralarchitectureinvivo. Itisbased
on the Brownian motion of water molecules in normal tissues and the observationthat molecules
tendtodiffusealongfiberswhencontainedinfiberbundles[19,4]. Usingclassicaldiffusiontensor
imaging(DTI),severalmethodshavebeendevelopedtosegmentandtrackwhitematterfibersinthe
human brain [39, 45, 47, 8, 23, 24]. The common way to analyze the data is to fit it to a second
ordertensor,whichcorrespondstotheprobabilitydistributionofagivenwatermoleculemovingby
acertainamountduringsomefixedelapsedtime. Bydiagonalization,thesurfacecorrespondingto
thediffusiontensorisanellipsoidwithitslongaxisalignedwiththefiberorientation. However,the
theoretical basis for this model assumes that the underlying diffusion process is Gaussian. While
thisapproximationisadequateforvoxelsinwhichthereisonlyasinglefiberorientation(ornone),
itbreaksdownforvoxelsinwhichthereismorecomplicatedinternalstructure,asseeninFig.1,an
exampleoftwofiberscrossing. Thisisanimportantlimitation,sinceresolutionofDTIacquisition
isbetween1mm3 and3mm3 whilethe physicaldiameteroffibers canbeless than1(cid:22)mandupto
30(cid:22)m[30]. Fromanisotropymeasuremaps,weknowthatmanyvoxelsindiffusionMRIvolumes
potentiallyhavemultiplefiberswithcrossing,kissingordivergingconfigurations.
To date, thisisa reasonwhyclinicians andneurosurgeonshavebeenskepticalof trackingand
segmentation methods developed on DTI data. They have doubts on the principal directions ex-
tracted and followed from the diffusion tensor to track fiber bundles. In the presence of multiple
fibers,thediffusionprofileisoblateorplanarandthereisnouniqueprincipaldirection(Fig.1). Ad-
ditionally,notethat maxima ofthe apparentdiffusioncoefficient(ADC) profiledo notcorrespond
to true fiber orientation (thin green lines). In current clinical applications, people instead choose
to use simple anisotropymapscomputed from the ADCprofile [13] to infer white matter connec-
tivity information. These measures are fast and easy to interpret with regions of anisotropy that
clearlystandout. ManyanisotropymeasuresexistandthemostcommonlyusedareFA(fractional
INRIA
ApparentdiffusioncoefficientsfromHARDI:estimationandapplications 5
anisotropy)andRA(relativeanisotropy)[5]butagain,thesemeasuresarelimitedinnon-Gaussian
diffusionareaswhencomputedfromDTIdata. ThisiswellillustratedinOzarslanetal[29]where
theanisotropymeasureinafibercrossingregionisinthesamerangeasvoxelswithnostructure. As
such,recentresearchhasbeendonetogeneralizetheexistingdiffusionmodelwithnewhigherreso-
lutionacquisitiontechniquessuchashighangularresolutiondiffusionimaging(HARDI)[40]. One
naturalgeneralizationistomodeltheapparentdiffusioncoefficient(ADC)withhigherorderdiffu-
siontensors(HODT)[28]. Thismodeldoesnotassumeanyaprioriknowledgeaboutthediffusivity
profileandhasthepotentialtodescribenon-Gaussiandiffusion.
Inthisarticle,westudytheestimateoftheADCprofilefromHARDIdataanditsabilitytode-
scribecomplextissuearchitecture. ContrarytomostrecentpapersonHARDIdataprocessing,we
donotfocusonfindingtheorientationofunderlyingfibersbutwanttodesigntheappropriatetools
to describe noisy HARDI data and explore scalar anisotropy measures computed from high order
formulation. In particular, the paper addresses the problem of fitting HARDI data with a higher
order tensor. One proposed possibility by Ozarslan et al [28] is to use a direct linear regression
by least-squares fitting. This can be effective but its robustness to noise is questionable as there
doesnotappeartobeanystraightforwardwaytoimposeaviablesmoothnessmaximizingcriteria.
We approachthe problemwith a sphericalharmonicsseriesapproximation[15,2, 10]. An impor-
tantcontributionofourworkistoproposeageneralizationofthestandardleast-squaresevaluation
methodtoincludearegularizationcriterion. Fromthisresult,wecomputethelineartransformation
taking the coefficientsof the spherical harmonic series to the independent elements of the HODT
usingtherelationpresentedin[28]. Therefore,ourapproachaswellasanytechniquedevelopedfor
sphericalharmonicformulationcanbequicklyandeasilyappliedtothehighorderdiffusiontensor
formulation and vice versa. This bridge is very useful for comparison purposes between state of
the art anisotropymeasures for high order models computed from spherical harmonics and tensor
coefficients. Published results are reproduced accurately and it is also possible to recover voxels
withisotropic,singlefiberanisotropicandmultiplefiberanisotropicdiffusion.
Thepaper isoutlinedas follows. InSection 2, we reviewthebasic principleofdiffusionMRI
and the differences between DTI and HARDI data. In Section 3, we review the existing state of
the art techniques to estimatethe ADC profilefrom noisy HARDIdata. In Section 4, we propose
a new regularization method which recovers a smoother ADC that is closer to the ADC without
noise.InSections5and6,weevaluateouralgorithmagainststateoftheartmethodsandreviewthe
different anisotropy measures and algorithms proposed using spherical harmonic coefficients and
independent elements of the HODT. We show the potential and usefulness of the generalized GA
measureforHARDIproposedbyOzarslanetal[29]. Weconcludewithadiscussionoftheresults
andourcontributionsinSection7.
2 High Angular Resolution Diffusion Imaging (HARDI)
Diffusion magnetic resonance imaging, introduced in the mid 1980s by Le Bihan et al [19], has
become intensely used for the past ten years due to important image acquisition improvement. It
is the unique non-invasive technique capable of quantifying the anisotropic diffusion of water in
biologicaltissuessuchasmuscleandbrainwhitematter. Shortlyafterthefirstacquisitionofimages
RR n°5681
6 M.Descoteaux,E.Angelino,S.Fitzgibbons,R.Deriche
162points 642points
Figure2:Discretesamplingsofthespherefordifferentnumbersofgradientdirectionscorresponding
toorder3andorder4tessellationofthesphererespectively.
characterizingtheanisotropicdiffusionofwatermoleculesin1990[26,27],Basseretal. proposed
the diffusion tensor model [4, 3]. DTI computes the apparent diffusion coefficients (ADC) based
ontheassumptionthatthediffusionorBrownianmotionofwatermoleculescanbedescribedbya
zero-meanGaussiandistribution
P(r)= 1 12 exp rTD(cid:0)1r : (1)
(4(cid:25)(cid:28))3jDj 4(cid:28)
(cid:18) (cid:19) (cid:18) (cid:19)
P(r) istheprobability thata water moleculestartingata givenposition in avoxelwillhavebeen
displacedbysomeradialvectorrintime(cid:28). Thediffusionprocessisthenfullydescribedbyarank-
2 diffusion tensor D which is a positive-definite 3x3 symmetric matrix. Using a minimum of six
differentencodedgradientdirectionsg,thediffusiontensorcanbeconstructedateachvoxelinthe
volume. TheresultingsignalattenuationisgivenbytheStejskal-Tannerequation[37],
S(g)=S exp (cid:0)bgTDg (2)
0
where b is the diffusion weighting factor depending(cid:0)on scann(cid:1)er parameters such as the length and
strength of the diffusion gradient and time between diffusion gradient pulses and S is the T2-
0
weighted signal acquired without any diffusion gradients. Numerous methods for estimating and
regularizingthediffusiontensorhavebeenproposed[39,45,47,7]. Segmentationandtractography
onknownfiberbundleswithclearanisotropicregionsofsinglefiberbundlesworkwell[20,21,22,
23,24]buttheapproachesareintrinsicallylimitedandunstableinregionsofmultiplefibersdueto
therestrictiveassumptionofthediffusiontensormodel.
Inorderto betterdescribethecomplexityofwatermotion, a clinicallyfeasibleapproach, high
angular resolution diffusion imaging (HARDI), has been proposed by Tuch et al [43, 40]. At the
cost of longer acquisition times, the idea is to sample the sphere in N discrete gradient directions
(Fig.2)andcomputetheapparentdiffusioncoefficient(ADC)profileD(g)alongeachdirection[42].
Hence, at each voxel, we have a discrete spherical function with no a priori assumption about the
natureofthediffusionprocesswithinthevoxel.Therehavebeeninterestingworksdonerecentlyon
alternativewaystoobtaincomplexsub-voxeltissuearchitecture,suchasDSI[46,40],Q-ball[41],
INRIA
ApparentdiffusioncoefficientsfromHARDI:estimationandapplications 7
PASMRI[25]. However,inthispaper,wefocusonHARDIdataandinparticular,thedevelopment
ofnewandefficienttechniquestoprocessnoisysphericaldataobtainedinmultipledirections. Sev-
eralrecentapproacheshaveattemptedtoestimateandinvestigatepropertiesofnoisysphericaldata
obtainedfromHARDI[43,15,2,10,28]tocharacterizetissueswithnon-Gaussiandiffusion. The
problemistorecoverasmoothADC,D(g),closetothetrueADCfromthemeasureddiffusionMRI
noisysignalS(g),
1 S(g)
S(g)=S exp((cid:0)bD(g))=)D(g)=(cid:0) ln : (3)
0
b S
(cid:18) 0 (cid:19)
Westudytheexistingmethods,discusstheirlimitationsandproposeafastandrobustalgorithmfor
ADCprofileestimationinthefollowingsections.
3 Apparent Diffusion Coefficient (ADC) Profile Estimation
AteachvoxelofHARDIdata,wehaveanoisysamplingoftheunderlyingADCprofiledescribing
the diffusion of water molecules within the voxel. In this section, we review tools for analyzing
function defined on the sphere and describe state of the art methods for obtaining the underlying
ADC profile from discrete noisy samplings. There are two classes of algorithms for ADC profile
estimation. Thefirstusesatruncatedsphericalharmonicseriestoapproximatethefunctiononthe
sphere[15,2,10]whereastheotherfitsahighorderdiffusiontensortothedata[28].
3.1 EstimatingtheADCProfile withtheSphericalHarmonics (SH)
Before discussing the fitting of data to a spherical harmonic series, we first define the spherical
harmonicsanddiscussbrieflysomeoftheirproperties.
3.1.1 SphericalHarmonics(SH)
Thesphericalharmonics, normallyindicatedbyYm (‘denotesthe orderandmthe phasefactor),
‘
areabasisforcomplexfunctionsontheunitspheresatisfyingtheSHdifferentialequation
1 @ @F 1 @2F
(sin(cid:18) )+ +‘(‘+1)F =0; ‘2Z (4)
sin(cid:18)@(cid:18) @(cid:18) sin2(cid:18) @(cid:30)2 +
ThefirsttwotermsofthisequationcorrespondtotheLaplacianinsphericalcoordinates,alsocalled
thethreedimensionalLaplace-Beltramioperator4 ,whichisdefinedby
b
1 @ @F 1 @2F
4 F = (sin(cid:18) )+ : (5)
b sin(cid:18)@(cid:18) @(cid:18) sin2(cid:18) @(cid:30)2
TheLaplace-Beltramioperatorisanaturalmeasureofsmoothnessforfunctionsdefinedontheunit
sphere and has been used in many image processing applications [16, 35, 34]. Furthermore, it is
RR n°5681