Voxel-wise comparisons of the morphology of diffusion tensors across groups of experimental subjects

Available online at www.sciencedirect.com

aging 156 (2007) 225–245www.elsevier.com/locate/psychresns

Psychiatry Research: Neuroim

Voxel-wise comparisons of the morphology of diffusion tensorsacross groups of experimental subjects

Ravi Bansala,b,⁎, Lawrence H. Staibc, Kerstin J. Plessenb,d, Dongrong Xua,b,Jason Royalb, Bradley S. Petersona,b

aNew York State Psychiatric Institute, New York, NY 10032, United StatesbDepartment of Psychiatry, Columbia University, New York, NY 10032, United States

cDepartments of Electrical Engineering and Diagnostic Radiology, Yale University, New Haven, CT 06512, United StatesdCenter for Child and Adolescent Mental Health, University of Bergen, Norway

Received 27 June 2006; received in revised form 18 November 2006; accepted 26 December 2006

Abstract

Water molecules in the brain diffuse preferentially along the fiber tracts within white matter that form the anatomicalconnections across spatially distant brain regions. A diffusion tensor (DT) is a probabilistic ellipsoid composed of three orthogonalvectors, each having a direction and an associated scalar magnitude, that represent the probability of water molecules diffusing ineach of those directions. The 3D morphologies of DTs can be compared across groups of subjects to reveal disruptions in structuralorganization and neuroanatomical connectivity of the brains of persons with various neuropsychiatric illnesses. Comparisons oftensor morphology across groups have typically been performed on scalar measures of diffusivity, such as Fractional Anisotropy(FA) rather than directly on the complex 3D morphologies of DTs. Scalar measures, however, are related in nonlinear ways to theeigenvalues and eigenvectors that create the 3D morphologies of DTs. We present a mathematical framework that permits the directcomparison across groups of mean eigenvalues and eigenvectors of individual DTs. We show that group-mean eigenvalues andeigenvectors are multivariate Gaussian distributed, and we use the Delta method to compute their approximate covariance matrices.Our results show that the theoretically computed mean tensor (MT) eigenvectors and eigenvalues match well with their respectivetrue values. Furthermore, a comparison of synthetically generated groups of DTs highlights the limitations of using FA to detectgroup differences. Finally, analyses of in vivo DT data using our method reveal significant between-group differences in diffusivityalong fiber tracts within white matter, whereas analyses based on FA values failed to detect some of these differences.© 2007 Elsevier Ireland Ltd. All rights reserved.

Keywords: Central Limit Theorem; Delta method; Fisher F-distribution; Fractional Anisotropy; Multivariate Gaussian distribution; Magneticresonance imaging

⁎Corresponding author. Room #2410, Unit 74, New York StatePsychiatric Institute, 1051 Riverside Dr., New York, NY 10032,United States. Tel.: +1 212 543 6145.

E-mail address: [email protected] (R. Bansal).

0925-4927/$ - see front matter © 2007 Elsevier Ireland Ltd. All rights resedoi:10.1016/j.pscychresns.2006.12.015

1. Introduction

Diffusion Tensor Imaging (DTI) can detect subtleabnormalities in white matter by assessing the degree towhich fiber tracts within that tissue have lost theirdirectional organization (Basser et al., 1994; Basser,1995). The myelin sheath and cell membranes of axons in

rved.

mailto:[email protected]

http://dx.doi.org/10.1016/j.pscychresns.2006.12.015

226 R. Bansal et al. / Psychiatry Research: Neuroimaging 156 (2007) 225–245

white matter fibers of the brain restrict water moleculesfrom diffusing in a direction perpendicular to the fibertracts. Water molecules therefore diffuse preferentiallyalong the length of axons (Pierpaoli et al., 1996).Disruption of the myelin sheath, loss of the integrity ofmicrostructures with the neurons of the fiber tracts, or anincrease in fluid volume can cause water molecules todiffuse more equally in all directions in the areas ofdisruption. Thus, by quantifying the diffusivity of watermolecules, DTI provides an indirect in vivo measureof the integrity of fiber tracts in white matter. Statisticalanalyses of differences in diffusivity between groups haverevealed associations of possible disruption or disorgani-zation in neuroanatomical connections with specificneuropsychiatric disorders.

DT data representing the diffusivity of water areencoded as 3×3 symmetrical diffusion tensors, and scalarmeasures derived from these DTs, such as FractionalAnisotropy (FA), Apparent Diffusion Coefficient (ADC)(Pierpaoli et al., 1996), Voxel Scale Connectivity (VSC)(Parker et al., 2002), and the coherence measure(Klingberg et al., 2000), have been used extensively tocompare diffusivity across individuals and thereby toidentify disturbances in neuroanatomical connectivitywithin specific diagnostic groups of subjects (Lim andHelpern, 2002). FA and the coherence measure, forexample, have been used to document disruptions in themicroarchitecture of white matter in the brains of infantsborn prematurely (Nagy et al., 2003); an in vivo study ofthe white matter fibers that form temporal–frontalconnections showed that patients with schizophrenialack the left-greater-than-right asymmetry in FA valuesfound in normal controls (Kubicki et al., 2002); and ADCvalues in infants with overt white matter pathology havebeen reported to be significantly higher than ADC valuesin healthy infants (Counsell et al., 2003).

These scalar measures, however, are not ideal fordetecting group differences. FA values, for example, areuseful as a measure of the degree of anisotropy, but theyare related nonlinearly to the eigenvalues of a diffusiontensor (Fig. 1). Therefore, even when sets of eigenvaluesdiffer, the FA values calculated for the sets may besimilar (i.e., the mapping of eigenvalues to FA values isnon-unique) and thus may conceal between-group dif-ferences. Moreover, the sample average FA value cal-culated from the FA values of individual tensors willnot necessarily equal the FA value calculated from themean of those individual tensors. Similarly, the ADC isa linear combination of eigenvalues that provides in-formation about overall diffusivity, but it too is limitedin its ability to detect group differences. Furthermore,the FA and ADC each quantify the magnitude of dif-

fusion but provide little information about the directionof diffusion, and thus comparisons of these scalar indicesprovide little insight into differences in local fiber tra-jectory across groups. The coherence measure, in con-trast, depends only upon the eigenvectors of diffusiontensors and ignores all differences in their associatedeigenvalues; it therefore provides little insight intodifferences in the rates of diffusion across groups. Eachof these measures therefore is limited in its ability toreveal group differences in tensor morphology.

Recently, a number of advanced methods have beenproposed to compare tensors across groups of individuals.Between-group differences, for example, have beenstudied by comparing diffusion along the principaldirections (PDs) of tensors (Schwartzman et al., 2005).The accuracy of these methods, however, requires theunambiguous determination of the principal direction ofeach tensor (the eigenvector of the tensor having thelargest associated eigenvalue), which the presence ofnoise in DTI datasets renders difficult. When estimatingtensors, rise in DTI datasets is commonly assumed tofollow a Gaussian distribution (Pajevic and Basser, 2003;Basser and Pajevic, 2003); the distribution of noise,however, has finite weight where tensors are not positive-definite, because positive-definite tensors are only asubspace of Euclidean space. To ensure that tensors arepositive-definite, some investigators have suggestedusing logarithmic transformations to first convert tensorsto symmetric matrices, then computing statistics for thesesymmetric matrices and using exponential transforma-tions to generate positive-definite tensors (Arsigny et al.,2005; Schwartzman, 2006). These methods, however,depend upon the estimation of only a single parameter tocapture variations in all three eigenvalues. The use ofRiemannian geometry to describe a positive-definitetensor subspace within R6 Euclidean space has allowedcomputation of an intrinsic mean tensor and an index ofgeodesic anisotropy (Fletcher and Joshi, 2004; Batcheloret al., 2005). However, the probability distributions andstatistical properties of the eigenvectors and eigenvaluesof the mean tensor (MT) cannot be computed using thesetechniques because the MT is computed using nonlinearoptimization methods. Furthermore, other methods thatcompare mean eigenvalues (Martin et al., 1999) oreigenvectors (Schwartzman et al., 2005) across indivi-duals are prone to errors in determining the correspondingeigenvalues and eigenvectors across tensors in thepresence of noise.

We have developed a mathematical framework fordirectly comparing the eigenvalues and eigenvectors ofthe MTs across groups of subjects. We first use theCentral Limit Theorem (CLT) to compute the statistical

Fig. 1. Graph of Fractional Anisotropy (FA) values for increasingeigenvalues. Two eigenvalues of the tensor were set to 20 and 30, andthe third was varied from 0 to 200. Note differing eigenvalues can yieldthe same FA value and that small changes in eigenvalues can causelarge changes in the FAs. The FA values were computed using theformula in Basser and Pierpaoli (1996). Both FA and the eigenvaluesare dimensionless, continuous variables.

227R. Bansal et al. / Psychiatry Research: Neuroimaging 156 (2007) 225–245

properties of the MT for a group of subjects, and thenwe use the Delta method to derive the probabilitydistribution of the eigenvalues and eigenvectors of this

Fig. 2. Maximum and minimum values of eigenspace components for incre10,000 random tensors for increasing variance and a diagonal covarianceEigenvalue3 are the three eigenvalues of the DTs corresponding to the 5, 25,Blue: Maximum values for eigenspace components; Purple: Minimum valushown in yellow. Therefore, eigenvalues and eigenvectors of the simulated tMST.

MT. Because the space of positive-definite tensors isdense and closed under addition, our computed MT ispositive-definite for each group of subjects being com-pared. Additionally, our method averages the tensorelements together across individuals, thereby avoidingthe need to determine corresponding eigenvalues andeigenvectors across different tensors for a group ofsubjects. Furthermore, in comparing tensors acrossgroups, we are interested in the statistical properties ofthe MT only and not those of the individual tensorsthemselves. Therefore, because the DT space is closedunder addition, and because our estimate of the sampleMT is unbiased, we can apply the CLT to determine thedistribution of the MT. Comparing the MTs across dif-ferent groups of subjects using this method yields thestatistical significance, or P-value, of the differences inaverage tensor morphology across groups at each voxelin the imaging volume. Our method thus allows us tospecify better the morphological origins (in terms of thedirectional orientation of fibers through a voxel element,

asing variance in tensor elements. We generated simulated groups ofmatrix between the tensor elements. Eigenvalue1, Eigenvalue2, andand 50 eigenvalues of the mean specified tensor (MST), respectively.es for eigenspace components. The MST eigenspace components areensors varied in large intervals around the corresponding values of the


or the magnitude of diffusion along fiber tracts) of thestatistically significant between-group differences in ten-sor morphology.

In our framework, we use a multivariate Gaussiandistribution to represent the variability in diffusiontensors across subjects; therefore, 3×3 symmetrictensors are represented in their equivalent 6×1 vectorforms. Furthermore, because correlations among ran-dom variables do not depend upon their representation,a covariance matrix within a multivariate Gaussiandistribution captures the full range of variability in thetensors that compose that distribution. Additionally, theelements of the covariance matrix are uniquely relatedon a one-to-one basis to the elements of the positive-definite and symmetric, fourth-order precision tensor Athat is used to describe diffusion tensors (Basser andPajevic, 2003). Therefore, all information about tensorsrepresented in the precision tensor A can be deducedfrom the covariance matrix. Thus, the covariance matrixand the mean tensor uniquely describe the distributionof the group of tensors, under the assumption thattensors are multivariate Gaussian distributed. Finally,representing tensors as 6×1 vectors allows us to usewell developed tools for multivariate statistics tocompare tensors across groups.

2. Methods

To detect voxel-wise, between-group differences indiffusion tensors, we compute the mean tensor (MT) for agroup of tensors, which in turn is used to calculate theeigenvalues and eigenvectors of the MT, as well as theircovariance matrices. Under the null hypothesis for thiscomparison,MTeigenspace components for the groups areequal at each voxel. We test this hypothesis usingHotelling's T2 statistic to calculate probability values (P-values) at each voxel in the imaging space, therebygenerating a P-value map that depicts where in the volumethe DT measure being compared is likely to differ acrossgroups. By thresholding the P-value map at a specifiedsignificance level, we can then detect statistically signif-icant differences in the diffusivity of water at each voxel.

In the presence of noise in a DT dataset, meaneigenvalues and eigenvectors and their covariance matrixfor a group of subjects cannot be computed from theeigenvalues and eigenvectors of the individual DTs. Forexample, the principal eigenvector of one tensor could beclosely aligned in a second tensor with an eigenvectorother than the principal eigenvector. In this instance, wecannot determine whether to average the eigenvalues thatcorrespond to the PDs of these two tensors, or to averagethe eigenvalues that correspond to the eigenvectors that

are most closely aligned across the two tensors.Computing the average of the principal diffusivity acrossthese two tensors could be erroneous using eitherapproach. Our method explicitly addresses these ambi-guities by averaging elements yi, i=1,…, N across Ndifferent DTs and then computing their mean values μyand covariance matrices Σy, for each group of subjects.According to the Central Limit Theorem (CLT) the meanvalues μy of tensor elements are multivariate Gaussiandistributed. We then infer the probability distribution ofthe eigenvectors and eigenvalues (the eigenspace compo-nents) of theMT from the probability distribution ofμy, asdescribed in the following subsections.

We assume that DT images have been normalized(registered and reoriented) into a common coordinatespace to permit voxel-wise statistical analysis of tensorsacross subjects. Errors in normalizing these images(e.g., from motion, cardiac pulsation, or thermal noise)will increase noise in statistical analyses of the tensorsand thereby reduce our power to detect significantbetween-group differences. Furthermore, using manydegrees of freedom in the spatial transformation oftensor maps could diminish the real differences in tensormorphology across groups. DTI datasets, however, canbe normalized accurately using any one of severalexcellent existing methods (Ashburner and Friston,1999; Alexander et al., 2001; Ruiz-Alzola et al., 2000,2002; Xu et al., 2003; Netsch and Muiswinkel, 2003;Park et al., 2003; Westin and Knutsson, 2003; Wanget al., 2004; Bansal et al., 2005b), thereby avoiding thisinappropriate elimination of real group differences. Weshould note that although the accurate spatial normal-ization of DTI datasets is essential for the validcomparison of tensor morphologies across groups, ourmethod for the statistical comparison of tensor mor-phologies is independent of the method used to nor-malize DT images.

2.1. Distribution of mean tensor elements

Let Di, i=1,…, N be N diffusion tensors at a cor-responding voxel in the spatially normalized andreoriented DT images of N subjects, and let yi=(d

i11

di12 di23 d

i22 d

i23 d

i33)

T∈R6, be a 6×1 random vector ofthe elements of a diffusion tensor Di. Because of thepositive-definite constrain on diffusion tensors, yi isrestricted to only a subspace ofR6. Consider a sequenceof random vectors yn, n=1,…, N, in which yn iscomputed by averaging the first n random vectorsyi : yn ¼ 1

n

Pni¼1 yi. If yi are independent and identically

distributed (i.i.d.) and E||yi||b∞, then by the strong lawof large numbers, yn converges almost surely (a.s.) to


the expected value of the random vector yi, i.e.,ynY

a:s E½yi�. Under the stronger assumption that E||yi||

2b∞, we obtain by the CLT (van der Vaart, 2000)

ffiffiffin

pynQNðE½yi�;RyiÞ;

where Σyi is the covariance matrix of the random vectoryi; that is, the sequence of random vectors yn convergein distribution to a multivariate Gaussian distribution,irrespective of the distributions of the tensors yi.

Because the positive-definite subspace of diffusiontensors inR6 is dense and closed under the mathematicaloperation of addition, the MT for a group of subjects iswell defined and positive-definite. Additionally, thevariance of the MT will be much smaller than is thedistribution of the tensors used to compute the MT itself,with its variance–covariance reducing by a factor of n foran increasing number n of observations. With 20 subjectsin each group, for example, the variance of theMTwill be20 times smaller than the overall variance for the tensorsacross all 20 subjects in the group. Asymptotically, thedistribution of the MT yn is degenerate at E[yi] (i.e. willhave nonzero probability mass at only E[yi]). Therefore,although themultivariateGaussian distribution describingthe mean random vector yn will have nonzero probabilitymass in the portion of Euclidean space R6 where thetensor is not positive-definite, this mass is zero asymp-totically, and we can apply the CLT to determine themultivariate distribution of the MT.

For sufficiently large n, the random vectors yi need notbe identically distributed to apply the CLT. If E(yi)=μiand Cov yi=Σi, and if we define

ly ¼1n

Xnj¼1

lj; and R ¼ 1n

Xnj¼1

Rj;

thenffiffiffin

pynQNðly;RÞas nYl.

2.2. Transformation of tensor elements into eigenspacecomponents

We derive herein a closed form analytic transfor-mation ϕ( ˙ ) between tensor elements and eigenspacecomponents. Because the eigenvectors of a symmetric3×3 matrix form an orthonormal coordinate system(Golub and Loan, 1996), we represent eigenvectors of adiffusion tensor by three angles (α β γ) that rotate theX-, Y-, and Z-coordinate axes into the eigenvectors.We therefore represent eigenspace components by a6×1 random vector x=(λ1 λ2 λ3 α β γ)T whoseelements are three eigenvalues (λ1 λ2 λ3) and threerotation angles (α β γ).

2.2.1. Transformation of eigenvaluesEigenvalues are estimated by setting the determinant

of the matrix (D−λI3) equal to zero, where I3 is anidentity 3×3 matrix. Thus, we have

det ðD−kI3Þ ¼ 0 ¼ −k3 þ c2k2−c1kþ c0;

where (Schneider and Eberly, 2002)

c0 ¼ d11d22d33 þ 2d12d13d23−d11d223−d22d213−d33d212;c1 ¼ d11d22−d212 þ d11d33−d213 þ d22d33−d223; andc2 ¼ d11 þ d22 þ d33:

Note that c0 is the determinant of the matrix D,and c2 is its trace. To evaluate the transformationfor eigenvalues, define a ¼ ð3c1−c22Þ=3; b ¼ −2c32þ9c1c2−27c0

27

� �and Q ¼ b2

4 þ a3

27

� �; then eigenvalues depend upon the

value of Q.Case I: QN0 This is possible only if the tensor D

is a multiple of the identity matrix (Schneider and Eberly,2002), i.e.,D=λI3. In this case, all eigenvalues are equal.

Case II: Q=0 Here, only two distinct eigenvaluesexist:

k1 ¼ c23þ b

2

� �1=3

; k2 ¼ k1; k ¼ c23−2

b2

� �1=3

: ð1Þ

Case III: Qb0 In this instance, each of the threeeigenvalues is distinct. If we define

h ¼ arctan ðffiffiffiffiffiffiffi−Q

p; −b=2Þ; q ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðb=2Þ2−Q

q;

then the eigenvalues are computed as (Hasan et al.,2001; Shen et al., 2006)

k1 ¼ c23þ 2q1=3cos ðh=3Þ

k2 ¼ c23−q1=3½cos ðh=3Þ þ

ffiffiffi3

psin ðh=3Þ�

k3 ¼ c23−q1=3½cos ðh=3Þ−

ffiffiffi3

psin ðh=3Þ�:

ð2Þ

2.2.2. Transformation of eigenvectorsEigenvectors are specified by the columns of the

rotation matrix R that rotates the X-, Y-, and Z-coordinateaxes. If α, β, and γ denote rotations about the X-, Y-,and Z-axes, respectively, then the rotation matrix R is

R ¼cos bcos g cos bsin g −sin bð−cos asin gþ sin asin bcos gÞ ðcos acos gþ sin asin bsin gÞ sin acos bðsin asin gþ cos asin bcos gÞ ð−sin acos gþ cos asin bsin gÞ cos acos b

0@

1A

T

:

Eigenvectors, vi; i=1, 2, 3, are estimated by solvingthe equations (D−λiI3)vi=0. Because eigenvectors of asymmetric matrix are orthogonal, columns of therotation matrix R represent the eigenvectors themselves.


Thus, to estimate eigenvectors, we estimate the rotationangles α, β, and γ. We discuss the degenerate case in thenext subsection.

Let M=(D−λI3)= [mij]. Then we solve the equation

0 ¼ M

cos bcos g

cos bsin g

−sin b

0B@

1CA ¼

m11 m12 m13

m12 m22 m23

m13 m23 m33

0B@

1CA

�cos bcos g

cos bsin g

−sin b

0B@

1CA:

Assuming that cos β≠0, that is, the rotations are lessthan 90° − k

2 bbbk2

� �, we obtain

tan g ¼ m12m33−m13m23

m223−m22m33

; and

tan b ¼ cos gm13 þ m23tan g

m33

� �: ð3Þ

To estimate α, consider a different eigenvalue andlet the eigenvector that corresponds to this eigenvaluebe ((− cos α sin γ+sin α sin β cos γ) (cos α cos γ+sin α sin β sin γ) sin α cos β)T. Solving for α andassuming that cos α≠0 and cos γ≠0, we obtain

tan a ¼ m11m23−m13m12

sin btan gðm13m12−m11m23Þ þ ðm213−m11m33Þ cos b

cos g

" #:

ð4ÞThese relations are used to compute the Jacobian of

the transformation ϕ( ˙ ) in the analysis below. Tocompute the Jacobian using Eqs. (3) and (4) requires acorrespondence between eigenvalues and eigenvectors;we determine this correspondence using numericalmethods (Press et al., 1992) that compute eigenvaluesand eigenvectors of matrices.

2.3. Special case mean tensors

For an isotropic MT, Eqs. (3) and (4) cannot be used todetermine the eigenvectors. Because the eigenvectors of anisotropic MT can be rotated to match those of the othergroup, in this case we compare only the eigenvalues of theMT across the two groups. When the MT is either prolate(i.e. the smaller two eigenvalues are equal) or oblate (i.e.the larger two eigenvalues are equal), we statisticallycompare only the eigenvalues and the one angle that can becomputed uniquely. However, exactly isotropic, prolate, oroblate MTs are rare because eigenvalues are continuous-valued random variables, and the precise equality ofeigenvalues within a tensor is unlikely. Therefore, we donot discuss these cases further in the article.

2.4. Linearized distribution of mean eigenspacecomponents

We estimate the linearized probability distribution ofthe mean values x of the eigenspace components for agroup of subjects and show that it is multivariateGaussian distributed. Eqs. (2), (3), and (4) define thetransformation ϕ( ˙ ) between y and x. Because eigen-space components x=ϕ(y) and tensor elements y arerelated through a nonlinear transformation ϕ( ˙ ), thechange of variables method cannot yield the covariancematrix of the eigenspace components. From thecontinuous-mapping theorem (van der Vaart, 2000;Carew et al., 2006; Anderson, 2001), if the sequence yn

converges in probability to μy, and ϕ( ˙ ) is continuous atμy, then ϕ(yn) converges in probability to ϕ(μy). Toobtain the asymptotic distribution of ϕ(yn), we thereforeuse the Delta method (van der Vaart, 2000) from thetheory of large samples to linearize ϕ( ˙ ). The Deltamethod uses Taylor's expansion of ϕ( ˙ ) around μy tolinearize the transformation, and then estimates anasymptotic distribution of μx. The theorem states that ifyn is a random 6×1 vector with an asymptotic distri-bution

ffiffiffin

pynQNðly;RÞ, and if ϕ(yn) is a differentia-

ble function, then the asymptotic distribution of ϕ(yn) isffiffiffin

p/ðynÞQNð/ðlyÞ;GðlyÞRGðlyÞT Þ; ð5Þ

where the Jacobian G is evaluated as GðlyÞ ¼ A/ð ynÞA yTn jyn¼ly

.

In the Delta method, the linear term (yn−μy)T ˙ G(μy)determines the order at which (ϕ(yn)−ϕ(μy)) convergesto zero (van der Vaart, 2000). The quadratic termdetermines the asymptotic behavior of (ϕ(yn)−ϕ(μy))only when G(μy)=0. Furthermore, calculation of thequadratic term is analytically more challenging andcomputationally unstable in the presence of noise. In ourexperiments, the linear term was nonzero and thus weused only the linear term to determine the limitingbehavior of the sequence

ffiffiffin

p/ðynÞ.

2.5. Hypothesis testing for group differences

The null hypothesis for comparing tensors across twogroups of subjects is that the mean values μ1

x and μ2x of

the eigenspace components will be equal each voxel ofthe imaging space. Thus, we test the null hypothesis

H0 : l1x ¼ l2x;H1 : l

1xpl

2x;with

X1x

¼X2x

:

To test this hypothesis, we compute sample meanvalues μx

1 =ϕ(μy1) and μx

2 =ϕ(μy2). We calculate sample


covariance matrices, Σx1 and Σx

2, for each group ofsubjects using Eq. (5) and then compute the pooledcovariance matrix Spl ¼ ðN1−1Þ R

1x þ ðN2−1Þ R

2x

N1 þ N2−2. Using the sam-

ple mean values and the pooled covariance matrix, wecompute Hotelling's T2 statistic (Stevens, 2002) asT2 ¼ N1N2

N1 þ N2ð l1x− l2xÞTS−1pl ð l1x− l2xÞ. From Hotelling's T2 statis-

tic, we calculate the F-statistic (Stevens, 2002)

Fð6;N1þN2−6−1Þ ¼N1 þ N2−6−1

6T2

N1 þ N2−2; ð6Þ

which follows the Fisher F-distribution with 6 and N1+N2−6−1 degrees of freedom.We also computeFα(6,N1+N2−6−1) for a specified significance level, α. If thecomputed statistic is greater than this value, then the nullhypothesis can be rejected, and we conclude that theaverage tensor morphologies of the two groups do in factdiffer significantly at that particular voxel. We thenperform a post hoc analysis to determine which particularelements of themean tensors contributemost to the overallmultivariate significance by conducting univariate t-testsfor each eigenspace component at the α/6 level ofsignificance. The Bonferroni inequality ensures that theoverall type I error rate for this set of t-tests is less than α(Stevens, 2002). (Equality of covariancematrices could betested using likelihood ratio statistic (Gupta and Tang,1984). If the covariance matrices of the two populationsare not equal, i.e. in the generalized Behrens–Fisherproblem, other advanced procedures must be used to testfor equality of the mean vectors (Kim, 1992; Bennett,1951; Yao, 1965; Anderson, 1984). However, these ad-vanced procedures are unnecessary when comparingtensors across groups containing a sufficiently largenumber of subjects (Anderson, 1984).)

2.6. Testing our formulation

2.6.1. DefinitionsWe first define notations and terms used subsequent-

ly in this paper. The mean tensor used to generate a set ofrandom tensors will be called the mean specified tensor(MST). The eigenspace components of the MSTwill becalled the MST eigenspace components and theeigenvalues MST eigenvalues. From the set of tensorsgenerated randomly, we compute the mean tensor (MT)by calculating the average value of each tensor element.The eigenvalue, eigenspace components, FA, and PD ofthe MT will be called the MT eigenvalues, MT eigen-space components,Mean FA (MT-FA), andMT principaldiffusivity, respectively.

In addition to calculating these values from the MT,we also calculate average FA and diffusivity along the

PD for each tensor in a set of sampled tensors. Thesewill be referred to as the sample average FA (SA-FA)and the SA principal diffusivity. In the syntheticallygenerated sets of tensors, where we know correspon-dence across eigenspace components, we also computeaverage eigenspace components, which are referred toas SA eigenspace components and SA eigenvalues.

2.6.2. Generating random tensorsTo generate random tensors, we first specified

elements of the MST y =(23.99, −3.75, 0.0, 11.004,0.0, 50.0), that represented a tensor with eigenvalues50.0, 25.0, and 5.0, and rotation angles α=0.0, β=0.0,and γ=15°. The FA equals 0.7 for the MST, which isrepresentative of the average FA values of white matterin the brain of an adult human (Madden et al., 2004).Furthermore, the MST can always be aligned with thecoordinate axes. However, we elected γ=15° to studythe effect of nonzero angle on the generated randomtensors. Note that even if MST α and β equal zero, theseangles will differ significantly from zero in tensorsgenerated randomly. Groups of tensors were generatedassuming a multivariate Gaussian distribution for thetensor elements. We considered two different covariancematrices for the tensor elements: (1) a diagonal matrixdiag(σ2, σ2, σ2, σ2, σ2, σ2); and (2) an upper block-diagonal matrix (Basser and Pajevic, 2003; Batcheloret al., 2003) of the form

0:75r2 0:25r2 0:25r2 0 0 00:25r2 0:75r2 0:25r2 0 0 00:25r2 0:25r2 0:75r2 0 0 0

0 0 0 r2 0 00 0 0 0 r2 00 0 0 0 0 r2

0BBBBBB@

1CCCCCCA:

The diagonal form of the covariance matrix capturesthe situation in which the tensor elements varyindependently in the presence of noise. However, theblock-diagonal covariance matrix is richer and capturesmore realistic variations in tensor elements (Basser andPajevic, 2003; Batchelor et al., 2003). We increased σ2

from 1.0 to 10.0 in increments of 1.0. For the specifiedmean values and their covariance matrix, we used theCholesky method of decomposition (Press et al., 1992)to factor the correlation matrix and to generate 10,000positive-definite, random tensors from independentlygenerated zero-mean, Gaussian random variables bykeeping only those tensors that were positive-definite.For each value of variance σ2, we computed values ofthe MT eigenspace components and their covariancematrices, both theoretically and empirically. To test the


validity of our formulation, we plotted graphs of the MTeigenspace components and SA eigenspace componentsfor varying σ. We expected that the theoretical andempirical values of the covariance matrices would beequal, and that the MT eigenspace components wouldremain constant.

2.6.3. Single group analysisWe tested the validity of our formulation for

calculating descriptive statistics for tensors in a groupof individuals by generating random diffusion tensorswhose elements were multivariate Gaussian distributedwith a known covariance matrix. From these randomlygenerated tensors, we computed mean values y of thetensor elements and then calculated the MT eigenspacecomponents x using the transformation ϕ( ). We alsoempirically computed the covariance matrix of therandomly generated tensor elements; this covariancematrix, along with the Jacobian G of the transformationϕ( ˙ ), was used to compute the covariance matrix of theMT eigenspace components.

In addition, we computed eigenvalues and eigenvec-tors for each randomly generated tensor. Because thetensors were generated synthetically, we knew whicheigenvalues corresponded across groups. This allowedus to compute empirically the values of the SAeigenspace components and their covariance matrix.We then compared the values of the SA eigenspacecomponents with the MT eigenspace components, aswell as their covariance matrices to assess the accuracyof our proposed formulation.

We computed the MT for a group of subjects byaveraging the tensor elements across each tensor in thegroup. Intuitively, averaging different tensors shouldreduce the MT-FA. We therefore plotted a graph to trackthe effect on MT-FA values of varying amounts of noisein the randomly generated tensors and with a varyingnumber of subjects. Additionally, we plotted graphs ofSA-FA for varying amounts of noise and numbers ofsubjects. Under ideal conditions, the values of MT-FAand SA-FA should equal the value of FA for the MST.

2.6.4. Between-group analysisTo evaluate the performance of our method in

comparing tensor morphologies across groups, wegenerated different sets of 100 diffusion tensors for fivediffering cases of interest (see below). We compared theresults of between-group analyses using our method withresults from the method that uses values of SA-FAs. Thenull hypotheses to be tested for between-group compar-isons were that the values of the various statistics(computed using the MT eigenspace components and

SA-FA) would be equal for the tensors from the twogroups of subjects. We should note that a failure to rejectthe null hypothesis for a given statistic does not imply thatthe values of the statistic were equal for the two groups oftensors.

For the first group of tensors, the MST y =(6.0598,−5.075, 0.0, 23.6402, 0.0, 50.0) represented eigenva-lues 50.0, 25.0, 4.7, and rotation angles α=0.0, β=0.0,and γ=15°. The FA of the MSTwas 0.7. For the secondgroup of tensors, we specified MSTs uniquely for thefive differing cases of interest. In each case, we assumeda diagonal covariance matrix between the MSTelements, with a variance that ranged from 1.0 to 10.0.

Case I: Equal MSTs. In this case, the MSTs for thetwo groups are equal. Therefore, neither the analysesbased on our method nor those based on SA-FA shouldreject the null hypotheses.

Case II: Differentmean specified rotations. In this case,the second set of tensors was generated for MST y =(4.7,0.0, 0.0, 25.0, 0.0, 50.0) with eigenvalues 50.0, 25.0, 4.7,and rotation angles α=0.0, β=0.0, and γ=0°. Therefore,the two groups have equal MST eigenvalues but differingvalues of γ. We expect that our method will reject the nullhypothesis, i.e., that the MT eigenspace components willbe equal for the groups and will localize the origin of thedifference to the angle γ. We expect that themethod basedon SA-FAwill not reject the null hypothesis.

Case III: Differing MST eigenvalues and MST-FAvalues. We generated the second set of tensors with MSTy =(18.189, −1.825, 0.0, 24.511, 0.0, 50.0) to represent atensor having eigenvalues 50.0, 25.0, 17.7, rotation anglesα=0.0, β=0.0, and γ=15°, and FA of 0.5. The MSTeigenspace components and an FA differed between thetwo groups of tensors. Therefore, both ourmethod and themethod based on SA-FA should reject the null hypothesis.

Case IV: Small differences in MST eigenvalues. Herewe generated the second set of diffusion tensors withMST y =(6.8063, −4.875, 0.0, 23.6937, 0.0, 50.0) thatrepresented a tensor having eigenvalues 50.0, 25.0, and5.5, rotation angles α=0.0, β=0.0, and γ=15°, and anFA of 0.687.

We provide this example to illustrate that a relativelysmall change in the eigenvalue (from 4.7 to 5.5) leads toa relatively large change in FAvalues (from 0.7 to 0.687,respectively). Small differences in eigenvalues can becaused by noise in the DT images, which in turnproduces large perturbations in FA. In this instance, it ispreferable that the null hypothesis should not berejected. In the presence of large noise values, weexpect that our method will not reject the null hy-pothesis, whereas the analysis based on SA-FAwill tendto reject it.


Case V: Differing MST eigenvalues but equal MST-FAs. Tensors with differing values for their tensorelements could have equal FA values; therefore, even ifthe MSTs in two groups differ, their difference cannot bedetected using SA-FAs. To illustrate this, we generated asecond set of diffusion tensors with MST y =(134.442429.325 0.0 32.8576 0.0 50.0), representing a tensor witheigenvalues 50.0, 25.0, 142.2, rotation angles α=0.0,β=0.0, and γ=15°, and an FA of 0.7. Whereas ourmethod should reject the null hypothesis and localize theorigin of the between-group difference to the eigenva-lues 142.2 and 4.7, we expect that the method based onSA-FA will not be able to reject the null hypothesis.

2.6.5. Effects of calculation of the MT from errors inreorienting tensors

Errors in registering and reorienting subject DTimages into a common coordinate space will causeerrors in the computed MTs, and therefore errors in thebetween-group analyses. To study the effects of error inreorienting tensors on calculation of MTs, we consideredan MSTwith eigenvalues 10, 1, and 1, and eigenvectorsaligned along the X, Y, and Z axes. The principaleigenvector was aligned along the X-axis. We thengenerated sets of simulated tensors by rotating the MSTabout the Z-axis by varying degrees that were sampleduniformly at random from a specified range of angle. Thesimulated tensors therefore were randomly rotated buthad the same eigenvalues as the MST. We generatedseven sets of 50 simulated tensors for a differing range ofangle: ±1°, ±2°, ±4°, ±8°, ±16°, ±32°, and ±64°. Thelast set of tensors, generated for the rotation angle γsampled uniformly ±64°, represented the situation inwhich the DT images were poorly normalized. (Note thata range of ±90° in fact covers the entire 360° becausetensors are symmetric along the eigenvectors. In this casethe MT cannot be determined.) To study the effect oferrors in registering and reorienting large number oftensors on the computed MT, we generated seven othersets of 5000 simulated tensors for the same ranges ofangle. For each of these sets, we computed the MTeigenvalues, eigenvectors and MT-FA and comparedthese values with those of the MST.We expected that theMT-FA would decrease for an increasing range ofrotation angles. Furthermore, for an increasing numberof subjects, we expected that the MT-FAwould decreasebut that the MT eigenvectors would align better with theX-, Y-, and Z-axes.

2.6.6. A real-world exampleWe used our method to compare DT images from a

group of 24 healthy subjects and a group of 22

individuals with Tourette's Syndrome (TS). The DTimages were acquired along 6 directions (8 averagedmeasurements per slice) on a Siemens 1.5-Tesla MRIscanner, with Repetition Time (TR)=4000 ms, EchoTime (TE)=96 ms, b=1000 s/mm2, an image matrix ofsize 128×128, and a Field of View (FOV) of 240 mm.Nineteen slices, each 4 mm thick, were acquired in thesagittal orientation such that the 10th slice was located atthe interhemispheric fissure in the brain. Therefore, theDTI data covered only the midportion of the brain ineach subject. Furthermore, to ensure that the recon-structed tensors were positive-definite, we masked outthe background in the images and reconstructed tensorsonly in the brain region. Rarely, a reconstructed tensorwithin the brain was not positive-definite. For thesetensors, we imposed the positivity constraint byinverting the sign of the negative eigenvalue and thenrecomputing the tensor. Such an approach reducedcomputation time without compromising the imagingdata or findings. More advanced methods, includingconstrained nonlinear least-squares estimation (Koayet al., 2006), could be used to ensure that the recon-structed tensors are positive-definite; however, theseapproaches are computationally expensive.

To compare DT images across groups of subjects,images were coregistered and reoriented into a commoncoordinate space. We defined this common coordinatespace using an anatomical MR image of a normal subjectas the reference image. DT images from every othersubject were then coregistered to this reference image,and their tensors were reoriented into this commoncoordinate space using a two-step procedure (Bansalet al., 2005b; Xu et al., 2005). First, we used a similaritytransformation to register the FA image of each subject tothe corresponding anatomical MR image, such that themutual information across images wasmaximized (Violaand Wells, 1997). Because DTI data spanned only thesagittal and parasagittal portions of the brain of eachsubject, we cropped the anatomical MR image in thereference space to match the total extent of the FA image.Then, using a fluid dynamical model of spatial trans-formation, we warped the FA image from each subject tothe corresponding cropped anatomical MR image forthat person, such that mutual information across the FAand anatomical images was maximized (Christensenet al., 1994; Bansal et al., 2005a). Second, the uncroppedanatomical image of each subject was warped to thereference anatomical image, such that the differencesin pixel intensities between images were minimized(Christensen et al., 1994).

Thus, each step estimated a deformation field thatsatisfied the smoothness and topology constraints


imposed by fluid dynamics. We then concatenated thesetwo deformation fields and used Procrustean estimation(Xu et al., 2003) to coregister and reorient the DTimages of each subject into the common referencespace.

After coregistering and reorienting the DT images ofeach subject to the template space, we computed theMTs and their covariance matrices for the two groups ofsubjects, as detailed above. We used these MTs andcovariance matrices to compare DTs across diagnosticgroups. To validate visually the accuracy of thecomputed MTs, we computed and compared two FAimages: (1) an image of values of MT-FA at each voxeland (2) an image of the values of SA-FA for the group ofhealthy subjects. Additionally, we compared numerical-ly the values of MT-FA and SA-FA at six correspondinglocations selected manually (two locations were selectedwithin the Corpus Callosum (CC) region and fourlocations were selected from the surface of white matterin the reference image).

To compare tensor morphologies statistically acrossgroups of subjects, we computed the P-values forbetween-group differences in the values of the variousDTI statistics. P-values were computed at each voxelin the common coordinate space without correcting forcorrelations between tensors at neighboring voxels.First, we computed the values of all the MT eigenspacecomponents and their covariance matrices for bothgroups of subjects. The vectors corresponding to theMT eigenvalues (which were arranged in order ofdecreasing magnitude) and their covariance matricesfor the two groups were used to compute Hotelling'sT2 statistics and their corresponding P-values. Second,we statistically compared the MT principal diffusivitiesfor the two groups using Student's t-statistics and P-values. Third, we compared the SA principal diffusiv-ity between-groups using Student's t-statistics. Weexpected that the MT principal diffusivities and SAprincipal diffusivities for the two groups would matchwell, and that therefore P-values of between-groupdifferences would also match well. Computation of SAprincipal diffusivity assumes, however, that the largesteigenvalues always correspond with the principaldirections of a tensor, but the presence of noise inthe DT dataset can invalidate this assumption.Therefore, P-values based on these two statistics maydiffer. Finally, we computed at each voxel the values ofSA-FA and their variances for the two groups; thesevalues were used to compute the t-statistic and P-valuefor between-group differences (Rosner, 1995). P-values were color encoded and displayed on thereference image.

3. Results

3.1. Generation of random tensors

Fig. 2 shows the range of values for the eigenspacecomponents of the 10,000 DTs that were generated atrandom (for a diagonal covariance matrix betweentensor elements). The range of values for eachcomponent increased with increasing variance in thecovariance matrix. While DTs were generated randomly,values of the MT eigenspace components converged totheir respective MST eigenspace components (seeSection 3.2 below).

3.2. Single group analyses

Single group analyses show that the values for theMT eigenspace components and their covariancematrices match well with the corresponding values forthe SA eigenspace components. We present results fortwo sets of 10,000 diffusion tensors generated randomly.The first set of diffusion tensors was generated assuminga diagonal covariance matrix between the elements ofthe MST. The second set was generated assuming ablock-diagonal covariance matrix.

3.2.1. Diagonal covariance matrixThe graphs in Fig. 3 show that the theoretically and

empirically computed values of the covariance matrix ofeigenspace components match closely, even for a largecovariance in the tensor elements. (We did not plot thevalues of the off-diagonal elements in the covariancematrix because they were small.)

The graphs in Fig. 4 show that the values for the SAeigenspace components depend substantially on variabil-ity in the tensor elements and diverge away from theirrespective MST eigenspace components. On the otherhand, the values for the MT eigenspace componentsremain close to their respective MST eigenspacecomponents.

3.2.2. Upper block-diagonal covariance matrixThe graphs of the values for the SA eigenspace and

MT eigenspace components match well for a block-diagonal covariance matrix (Fig. 5). Additionally, thegraphs in Fig. 6 show that the MT eigenspacecomponents are closer to the corresponding MSTeigenspace components than to the SA eigenspacecomponents. A close match of the MT and SAeigenspace components, together with their respectivecovariance matrices (for both the diagonal and theblock-diagonal covariance matrices) shows the accuracy

Fig. 3. Theoretical versus sample average (SA) values of the covariance matrix between eigenspace components. When generating the groups of10,000 random diffusion tensors (DTs), we assumed the presence of a diagonal covariance matrix between the tensor elements with specified meanvalues of y=(6.0598, −5.075, 0.0, 23.6402, 0.0, 50.0). We increased σ2 from 1.0 to 10.0 in the diagonal covariance matrix. Ideally (purple), the SAvalues should equal the theoretical values that were calculated using our method. Theoretical values computed using our method were close (blue)to the SA values, even for large amounts of noise in the tensors. Covar(1,1)=variance in Eigenvalue1; Covar(2,2)=variance in Eigenvalue2;Covar(3,3)=variance in Eigenvalue3; Covar(4,4)=variance in Alpha; Covar(5,5)=variance in Beta; Covar(6,6)=variance in Gamma.


of our mathematical formulation. Because our methodlinearizes the relation between the tensor elements andthe eigenspace components, the theoretical values of thecovariance matrix will not match exactly with thosecomputed empirically.

Fig. 4. Group-average values of eigenspace components computed using di(blue); sample average (SA) eigenspace components (purple); mean specifiedrandom tensors were generated for increasing variance and assuming a diagcomponents matched well with those of the MST, thereby validating our me

Fig. 7 shows that theMT-FA decreases with increasingvariance in the tensor elements, even though the MTeigenspace components are similar to their respectivespecified values (Fig. 6). Therefore, even though the MTeigenvalues differ only slightly from their respectiveMST

ffering methods. Values of mean tensor (MT) eigenspace componentstensor (MST) eigenspace components (yellow). The groups of 10,000onal covariance matrix between the tensor elements. MT eigenspacethod of computing the MT for a group of subjects.

Fig. 5. Theoretical versus sample average (SA) values of the diagonal elements in the covariance matrix for increasing variance. Groups of 10,000random tensors were generated using upper block-diagonal covariance matrix across tensor elements. The σ2 of the covariance matrix was increasedfrom 1.0 to 10.0. Ideally (purple), the SAvalues equal the theoretical values. SAvalues (blue) closely matched the theoretical values computed usingour method. The off-diagonal covariance values are not plotted because they were all small. Theoretically computed covariance matrix match wellwith the covariance matrix computed from simulated sample of tensors, in which correspondences across eigenvectors were known. Therefore, theDelta method computed the covariance matrix of the eigenspace components with high accuracy. Definitions for Covar(x,x) are as indicated in thelegend for Fig. 3.


eigenvalues (Fig. 6), the cumulative effect of thedifference is to decrease the anisotropy of the MT,consistent with findings from an earlier report (Joneset al., 2002). Figs. 6 and 7 suggest that the increased

Fig. 6. Group-average values of eigenspace components computed by variousample average (SA) eigenspace components (purple), and MST eigenspacegroup of 10,000 random tensors generated for increasing σ2 in the block-diag

values of the smallest eigenvalue (labeled as Eigenvalue1in Fig. 6) decrease the MT-FA.

Decreasing values of the MT-FA with increasingnoise is of concern when comparing DT images across

s methods. Values of mean tensor (MT) eigenspace components (blue),components (yellow). We computed the group-average values for eachonal covariance matrix between mean specified tensor (MST) elements.

Fig. 7. Group-average Fractional Anisotropy (FA) values computed for increasing variance and increasing numbers of subjects. Mean tensor FA (MT-FA) values (blue); sample average FA (SA-FA) values (purple). The top row of Figs. (a, b, c) plots the graphs for varying number of subjects at threediffering levels of noise in the generated tensor elements. The bottom row of Figs. (d, e, f) plots the graphs for varying amounts of noise for threediffering numbers of subjects. The FA of the mean specified tensor (MST) was 0.7.


groups of subjects because errors in coregistration andreorientation of the images of each subject will increasenoise in the corresponding DTs. Our results, however,suggest that reduced values of MT-FA may notnecessarily imply that the values of the MT eigenspacecomponents differ greatly from their true (thoughunknown in a real-world setting) values.

3.3. Between-group analyses

When the MSTs were equal for the two groups oftensors, the null hypothesis could not be rejected for anystatistic; therefore, as expected, the two groups oftensors did not differ significantly in any statistic at thesignificance level of 0.05. When only one rotation anglediffered between the MSTs of the two groups (Fig. 8,Case II), our method determined that the null hypothesisfor the Hotelling's T2 statistic could be rejected;therefore, our method detected that the averagemorphology of the two groups of tensors differedsignificantly. Additionally, we were able to localize thedifferences between the MSTs to the difference in therotation angle γ based on its associated P-value. TheSA-FA method could not reject the null hypothesisbecause the MST eigenvalues were equal for the twogroups. This case example demonstrates the superiorityof our approach in detecting differences between-groups

that would otherwise be overlooked by the methodbased on SA-FA.

When the MST eigenvalues differ between-groups(Fig. 8, Case III), the null hypothesis can be rejectedbased both by our method and by the values of SA-FA.Additionally, however, our method localizes the causeof the between-group difference to the correct eigen-value (Eigenvalue1 in this case). When comparing twosets of diffusion tensors that differed only slightly in theMST eigenvalues (Fig. 8, Case IV), our method did notreject the null hypotheses for small amounts of variancesin the tensor elements; however, the method based onthe values of SA-FA rejected the null hypothesis even inthe presence of large variances. A small between-groupdifference in eigenvalues can cause large differences inthe SA-FAs (Fig. 1), leading to the conclusion that whenusing this statistic group differences are statisticallysignificant. This example demonstrates that an analysesbased on SA-FA may erroneously detect differencesbetween-groups when in fact none exist, therebyincreasing the rate of false-positives in the presence ofnoise. Finally, when the MST eigenvalues of the twogroups differed but their FA values were equal (Fig. 8,Case V), our method was able correctly to reject the nullhypothesis and to localize the origin of between-groupdifferences to the correct eigenvalues; the method basedon SA-FA, however, could not reject the null hypothesis

Fig. 8. Graphs of the P-values for differences when comparing two groups of synthetically generated diffusion tensors (DTs). The MST elements forthe first group of tensors were y=(6.0598, −5.075, 0.0, 23.6402, 0.0, 50.0). TheMSTelements for the second group of tensors were varied for each ofthe four differing cases. P-values of Fractional Anisotropy (FA) and various statistics that were significant are plotted. In each case, our methodcorrectly identified the eigenspace component that differed significantly across the two groups. The graphs and their labels were color encoded: Ht =Hotelling's T2 statistics (dark blue); FA = FA values (blue); E1, E2, E3 = Three eigenvalues (E1 = purple); A = Alpha; B = Beta; G = Gamma (green).Case I: Because the MSTs were equal for the two groups in this case, between-group differences in tensor morphology were not statisticallysignificant using any statistic. Therefore, a graph for this case is not shown. Case II: Only γ differed between the two groups. Our method correctlyidentified γ as the source of the between-group difference in tensor morphology. Case III: Only one eigenvalue (E1) differed between the two groups.Both our method and the method using FA determined that the two groups differed significantly in tensor morphology. Our method, in addition,correctly identified differences in morphology due to E1. Case IV: E1 differed only minimally (0.8) between the two groups. Case V: Although themean specified tensor FAs (MST-FAs) were equal for the two groups, their E1s differed. Our method correctly detected that the between-groupdifference derived from difference across groups in E1. However, the method using FA values failed to detect significant difference between the twogroups in tensor morphology.


even though the eigenvalues differed significantlybetween-groups.

These cases show that the method using SA-FA willsometimes erroneously detect nonexistent group differ-ences (Fig. 8, Case IV) or fail to detect true groupdifferences (Fig. 8, Case V). In contrast, our methodcorrectly detected between-group differences in bothexamples. Note that the P-values of the statistics, such asFA, changed wildly when the null hypothesis could not berejected. This is because under the null hypothesis, thedistributions of these indices are equal for the two groupsof subjects, and therefore the between-group differencesin the index and its associatedP-valuewill vary randomly.

3.4. The effects of errors in the reorienting of tensors

TheMT-FA declined significantly for the sets of tensorsgenerated with rotation angle γ sampled uniformly in therange ±32° or greater (Fig. 9). However, for smaller rangesof γ, MT eigenvalues and eigenvectors, and MT-FA,

matched closely those of the MST (Fig. 9). In contrast,even though a larger number of subjects further reducedthe MT-FA only slightly, the MT eigenvectors were betteraligned with those of the MST (Fig. 9). Therefore, thearithmetical averaging of tensors yielded an MT thatapproximated closely the MST. In the presence of largenoise levels, however, the MT differed substantially fromthe MST, indicating that precise registration and reorien-tation of DT images during spatial normalization isessential for detecting between-group differences. Fur-thermore, in the absence of systematic bias in theregistration of subjects in a group to the reference brain,errors in the registration will contribute to Type II(false negative) error by increasing variance in our DTImeasures.

3.5. Analyses using a real-world dataset

Visually, fiber pathways in the two FA images (onecomputed from the values of MT-FA and the other from

Fig. 9. The effects of errors in reorientation on mean tensor (MT) morphology. We generated two sets of simulated tensors (one with 50 and the otherwith 5000) by rotating the mean specified tensor (MST) about the Z-axis by randomly varying degrees but sampled uniformly within specified ranges(±1°, ±2°, ±4°, ±8°, ±16°, ±32°, and ±64°) of the angle γ. The MST eigenvalues were E1=10, E2=1, and E3=1, and the eigenvectors were alignedalong the X, Y, and Z axes, respectively. We computed MTeigenvalues and eigenvectors for each set of tensors. In the figure, the horizontal axis plotsthe logarithm of the absolute value of the range in which γ was sampled uniformly. The vertical axis plots the eigenvalues and γ computed from theMT. The eigenvalue E3, and the angles α and β of the MT equaled those of the MST in each experiment. Even for large amounts of errors in correctlyreorienting tensors, the MT eigenvalues and eigenvectors matched closely to those of the MST. Left Column: E1 = purple; E2 = yellow.


the values of SA-FA at each voxel) match well (Fig. 10).However, the values of MT-FA are smaller than those ofSA-FA, and therefore the MTs are slightly less anisotrop-ic; this does not necessarily imply, however, that the MTeigenvalues will differ greatly from their true (althoughunknown) values (see Figs. 6 and 7).

Fig. 11 displays the P-values of the differences acrossgroups in various statistics. The computed P-values forthe entire 3D volume are color-coded and displayed on 4representative sagittal slices from the reference MRimage. Images in Fig. 11 Column (a) show statisticallysignificant between-group differences in the SA-FA ofthe two groups, especially in the Corpus Collosum (CC).However, images in Fig. 11 Column (b) show that the SAprincipal diffusivities in the CC do not differ statisticallyacross groups, implying that the differences in the valuesof SA-FA are a function of differences in the diffusivitiesin directions orthogonal to the PDs.

We compared the vectors of the MT eigenvalues forthe two groups using the Hotelling's T2 statistic. Imagesin Fig. 11 Column (c) show the differences in theHotelling's T2 detected across groups; significantdifferences included the CC, where SA-FA values also

differed statistically. Because Hotelling's T2 statistic(which compares all three eigenvalues simultaneously)differed significantly across groups but the values of SAprincipal diffusivity did not, we conclude that theobserved group differences in SA-FA were attributableto differences in diffusivity across groups in directionsorthogonal to the PDs. Note that while we have beenvery careful in registering and reorienting subject DTIdatasets into a common coordinate space, errors innormalization will effect these analysis by reducing ourstatistical power to detect between-group differences.Therefore, in the absence of grossly obvious morpho-logical deformities in the brain of one group of subjectsas compared to the other, errors in the registration willcontribute to Type II (false negative) error by increasingvariance in our DTI measures.

Images in the Fig. 11 Column (d) depict between-group differences in MT principal diffusivities. Theirgood correspondence with the location of groupdifferences in SA principal diffusivities (Fig. 11Column (b)) further attested to the accuracy of themean values and variances of the principal diffusivitiesthat were computed using our method. In addition, our

Fig. 10. Fractional Anisotropy (FA) images computed for the group of 24 healthy individuals. The top row shows the values of the mean tensor FA(MT-FA); the bottom row shows the values of the sample average FA (SA-FA). The values of FA at 6 locations in the upper row of images were0.7528, 0.75524, 0.4861, 0.5787, 0.4725, and 0.5301; values at the corresponding locations in the lower row of images were 0.7963, 0.7781, 0.5382,0.6044, 0.5323, and 0.5765. We set all three eigenvalues of the tensors in the background (i.e. voxels containing no brain tissue) equal to zero. Then inthe common coordinate space, we computed mean FA values at only those locations where every tensor for each subject has nonzero eigenvalues.


method allows for an unambiguous computation andcomparison of diffusivities along PDs because thesediffusivities were computed from the MTs. Our results

Fig. 11. Comparing tensor morphologies in 24 healthy and 22 ts subjects. Diffuinto a common reference space. The P-values of differences in mean valuecomputed, color encoded, and displayed on 4 parasagittal slices of the referencin (1) sample average FA (SA-FA) values, (2) SA principal diffusivities,diffusivities. For images in the 1st, 2nd, and the 4th columns, red shows that thTS group as compared to its values in healthy individuals, with a P-value ofred shows statistically significant (P-value=0.001), between-group differencvalues of 0.2. See color bar on the right in Fig. 12. A = Anterior; B = Poste

show that the differences in SA-FA are because ofdifferences in diffusion in directions orthogonal to PD(Fig. 12).

sion tensor (DT) image of each subject was coregistered and reorienteds of the four statistics of tensor morphology for the two groups weree brain. Shown in each image are the P-values for the group differences(3) vectors of mean tensor (MT) eigenvalues, and (4) MT principale statistic is significantly greater (violet shows statistic is smaller) in the0.001 or less. See color bar on the right. For images in the 3rd column,es in MT eigenvalues, and violet shows regions of differences with P-rior; CC = Corpus Callosum.

Fig. 12. Comparing the TS and healthy subjects in the axial orientation. The results were resliced, color encoded, and displayed on four slices in theaxial orientation. Shown in each image are the P-values for the between-group differences in (1) sample average FA (SA-FA) values, (2) SA principaldiffusivities, (3) vectors of mean tensor (MT) eigenvalues, and (4) MT principal diffusivities. Color encoding for the P-values of the various statisticsis the same as that in Fig. 11. Note that diffusion tensor data was acquired in only 19 slices around the interhemispheric fissure for each subject. A =Anterior; P = Posterior.


The relatively lower values of SA-FA in the CC of TSsubjects, combined with the absence of significantdifferences between principal diffusivities across groups,suggest that fiber tracts are likely intact in the CCs ofpersons with TS (because principal diffusivities arenormal in this region), but that either the numbers ofnerve fibers or their degree of myelination in the CC maybe reduced compared to normal controls (becausediffusivity was significantly increased in directionsorthogonal to the PDs). This interpretation is consistentwith our prior findings of reduced size of the CC in a largesample of TS children (Plessen et al., 2004).

3.5.1. SummaryAcross these two groups of real subjects, while SA-

FA differed significantly, SA-PD was not significantlydifferent in the CC region: i.e., while eigenvalues weredifferent, the eigenvalue along the principal direction(PD) was not significantly different across groups.Therefore, while the tensors were equally elongatedalong the PD, tensors in one group were either fatter orthinner as compared to those in the other group. Thiswas further confirmed by our method that testeddifferences in the eigenvalues using T2 statistics.Differing eigenvalues (and therefore diffusion of watermolecules) in directions perpendicular to PD maysuggest fewer fibers in individuals with TS.

4. Discussion

We have presented a voxel-wise method for accu-rately detecting between-group differences in the dif-fusivity of water that likely reflect group differences inanatomical connectivity or white matter integrity(Pfefferbaum et al., 2000; Plessen et al., 2004). Becausewhite matter fiber bundles are not visible usinganatomical MRI, DTI is currently the only viablemeans to identify within- and between-group differencesin the diffusion of water when studying the integrity ofwhite matter fibers in vivo. Differences in connectivity orwhite matter integrity typically have been assessed bycomparing mean FAs across diagnostic groups, whichdoes not require identification of corresponding eigen-space components because FA is a simple scalar index.This method may not provide an optimal or completecomparison of tensors across groups, however, becauseFA is related nonlinearly to the underlying eigenspacecomponents and it does not permit comparison of thedirectionality of tensors across groups. Other methodshave therefore been developed that first establish acorrespondence of the various eigenspace componentsand then compares statistically the numerical values ofcorresponding components across tensors from differingsubjects. These approaches are limited by the difficultyof establishing a correct correspondence of eigenspace


components across individuals in the presence of thenoise that are inherent in DTI datasets.

We have developed a method for the comparison oftensors across individuals, or groups of individuals, thatcircumvents the difficulty of correctly identifyingeigenspace components that correspond across indivi-duals. We first compute the MT for each group ofsubjects, and then we use this MT to compute thevarious eigenspace components of the MT. We haveshown that MT eigenspace components are multivariateGaussian distributed, irrespective of the distributionfrom which the tensors at each voxel are sampled. Oncethe MT eigenspace components are calculated, we thenuse the Delta method to compute their covariancematrix. Once the distribution of the MT eigenspacecomponents is determined, standard multivariate statis-tical procedures can be used to detect between-groupdifferences in tensor morphology at each voxel.Analyses of simulated data demonstrated that the MTeigenspace components computed using this approachmatched well with the corresponding MST eigenspacecomponents, whereas the values of SA eigenspacecomponents diverged quickly as the noise increased.Therefore, arithmetic averaging of the tensors yielded avalid MT for a group of subjects, which can be then usedfor detecting between-group differences.

Because diffusion of water is nonnegative in anyparticular direction, a tensor must be positive-definite todescribe quantitatively the diffusion of water in the brain.For detecting between-group differences in diffusion ofwater, we computed MT as an arithmetic average of theDTs across individuals in a group. For an increasingnumber of subjects, the distribution of our MT isincreasingly restricted to the subspace portion of R6 inwhich tensors are positive-definite. Another interestingway to ensure that the positive-definite constraint on theMT is satisfied is to first decompose individual tensors intothe lower triangular factor (Li) using Cholesky decompo-sition (Press et al., 1992; Koay et al., 2006), then computethe Mean Lower (ML) factor and its distribution by takingthe arithmetic average of Lis across subjects in a group,and then finally calculate the MT and its distribution fromML. Using this method, the distribution of ML does notneed to be restricted to a particular subspace ofR6 becausethe MT computed from any ML is always positive-definite. However, the comparison of tensor morphologiesacross groups require determination of the distributions ofthe mean eigenvalues and eigenvectors in each group, andin this respect the method of tensor decomposition suffersfrom at least two serious limitations First, the elements ofthe ML are nonlinearly related to those of the MT, andtherefore when the number of subjects is small, estimation

of the distribution of the MT using linear methods willhave nonzero probability mass in subregions of R6 fortensors that are not positive-definite. Second, the nonlinearrelation of the elements ofML to those ofMTwill increasethe errors in linear approximations of the distributions ofeigenvectors and eigenvalues in theDeltamethod. Thus, incomparison, our method satisfies the positivity constraintwhile also producing relatively fewer errors whendetermining the distributions of the MT eigenspacecomponents.

In comparing the performance of our method withthat based on SA-FA, we found that both methods wereequally likely to correctly reject null hypotheses intrivial cases. However, when the MST eigenvaluesdiffered from one another only minimally (Fig. 8, CaseIV), the method using SA-FA rejected the nullhypothesis even in the presence of high noise levels,whereas our method (appropriately) did not reject thenull hypothesis, even in the presence of low noise levels.These results suggest that, in the presence of noise,comparing tensor morphologies across groups using theSA-FA method can detect between-group differenceswhen in fact none exist. Finally, Fig. 8, Case V showedthat real group differences may not be detected using theSA-FA method (i.e., the method is prone to generatingfalse negative findings), whereas our method correctlyidentified real group differences in this particular case.

We evaluated the performances of the two methods ina real-world setting by comparing the DT images of agroup of 24 normal subjects with those of a group of 22subjects with TS. In analyses of SA-FA within theCorpus Callosum, we found that the SA-FA for the TSgroup was significantly lower than that for the healthysubjects. These analyses, however, could not determinewhether SA-FAs differed because of variations in meandiffusivities along the PDs or because of differences indiffusivities in directions orthogonal to PDs. Further-more, SA principal diffusivities did not differ statisticallybetween-groups within the CC, indicating that between-group differences in values of the SA-FA in this regionarose from differences in the diffusivity of water indirections orthogonal to the PD. This interpretation wassupported by findings of significant between-groupdifferences in the remaining MT eigenvalues (Song etal., 2005). These between-group differences in diffusiv-ities could be caused by any of several differences intissue characteristics across groups, including a reduc-tion in the numbers of fibers connecting the left and rightcerebral hemispheres, and reduced myelination of axonswithin the CCs of TS children (Song et al., 2002; Moriand Zhang, 2006). By combining statistical analysesbased on MT principal diffusivity, MT eigenvalues, and


SA-FA, we can therefore describe better the between-group differences in DT images.

Between-group differences could also be detected bycomparing corresponding eigenvalues that were orderedaccording to their magnitude and coherence in localfiber direction in each subject brain (Martin et al., 1999;Basser and Pajevic, 2000). These approaches areattractive because they are simple to implement andbecause when the data are free of noise these methodsshould yield results comparable to those obtained usingour approach. In the ubiquitous presence of noise,corresponding eigenvalues across different tensorscannot be determined uniquely, therefore causing biasin the ordered eigenvalues. Thus, our method will yieldaccurate results even in the presence of noise because itdoes not have to identify the corresponding eigenvectorsacross individuals.

In the presence of noise, the FA of the mean tensorfor a group of subjects in a normalized reference spacemay be lower than the corresponding SA-FA. In oursimulation data, we found that increasing the number oftensors used to compute the MT did not correct thisreduction in anisotropy. Furthermore, in our real subjectdatasets, MT-FA was smaller than the SA-FA for thegroups of 24 healthy subjects. We attribute the loweranisotropy of the MTs to the introduction of errors intensor alignment that accompany errors when normal-izing DT images into the common coordinate space.Better normalization of the DT images will reducedifferences between the values of MT-FA computedusing our method and the values of SA-FA, leading toaccurate detection of between-group differences.

Selection of the appropriate index or measure oftensor morphology to detect between-group differ-ences in tissue organization depends upon thescientific context and aims of a particular study. Thebest index will presumably be the one that is mostsensitive to the particular morphological feature of thetensor that is of interest to the investigator, and usingone that is not sufficiently sensitive will fail to detectsignificant differences in morphology across groups.Whereas a multivariate index, such as the vector ofeigenvalues, is sensitive to differences in a wide rangeof linear combinations of the component eigenvalues,this index will be less sensitive to overall differencesin tensor morphology than will other scalar measures,such as FA, for those tensors in which the index valuechanges rapidly for a small change in the tensor'seigenvalues. Our results (Figs. 11 and 12) suggest thatthe index that is most sensitive in detecting between-group differences in tensor morphology may actuallyvary according to brain region and tissue type. Thus,

assessing group differences in tensor morphologyusing both a scalar index (such as FA) and a mul-tivariate measure (such as the vector of eigenvalues)may yield a more complete understanding of themorphological origin of between-group differences thanwill either measure alone.

Comparing tensor morphologies across groups at eachvoxel within the entire brain entails many statistical testsand a proportionately increased probability of detectingbetween-group differences by chance (i.e., of generatingfalse positive, or type I, errors). Standard multiplecomparison procedures can be used to limit the falsepositive rate to a desired level. The simple method ofBonferroni adjustment to control for false positive rateis generally considered to be excessively conservativebecause test statistics on neighboring voxels are correlat-ed, and therefore the actual number of independentcomparisons is less than the number used for correction.Additional methods, including those based on falsediscovery rate (Benjamini andHochberg, 1995; Genoveseet al., 2002; Schwartzman et al., 2005) and permutationstesting (Bullmore et al., 1999; Golland and Fischl, 2003),can be used to computed corrected P-values of voxels andclusters of voxels. These procedures can be applied to thetest statistic F (Eq. (6)) within our method to control formultiple comparisons and to compute corrected P-value.

We used the Central Limit Theorem to justify ourestimation of the multivariate Gaussian distribution forthe elements of the mean tensor. Whereas the MT itselfwill be positive-definite, the Gaussian distribution inEuclidean space, R6, will have nonzero probabilitymass for vectors that do not form positive-definitetensors. Because the mean tensor of the sample is anunbiased, consistent estimator of the mean, however, themean tensor of the sample asymptotically equals the truemean in probability, and the distribution of the meantensor is degenerate. Therefore, the mass of thedistribution that describes the MT, will equal zeroasymptotically in R6 for tensors that are not positive-definite. Despite this potential limitation, however, ourmethod yields a valid and complete description of themean eigenvalues and eigenvectors by determining theirmultivariate distribution. This multivariate Gaussiandistribution of the MTs can be used to detect between-group differences in tensor morphologies.

We have presented a mathematical framework forcomputing the voxel-wise mean values of the eigen-space components and their covariance matrices fortensors within groups of subjects. Because we computethe mean eigenvalues and eigenvectors from the meantensor, which is computed by the element-wiseaveraging of the tensors across individuals, we need


not define the correct correspondence of eigenvaluesand eigenvectors across those tensors. Furthermore, wehave shown that the mean values are multivariateGaussian distributed. Once the distribution of the meaneigenspace components has been estimated, we cancompare these mean values statistically across groups.This method allows for a comparison of mean DTsacross groups of subjects that is more direct and accuratethan are methods that are based on calculation of scalarindices alone. Our method also permits determination ofthe morphological origins of group differences in tensormorphology, in that it indicates whether between-groupdifferences derive either from differences in diffusivityalong PDs, or from differences in diffusivity in direc-tions perpendicular to the PDs.

Acknowledgment

This work was supported in part by NIMH grantsMH068318, MH36197, and K02-74677, NIDA grantDA017820, a NARSAD Independent InvestigatorAward, the Thomas D. Klingenstein and Nancy D.Perlman Family Fund, and the Suzanne Crosby MurphyEndowment at Columbia University.

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Voxel-wise comparisons of the morphology of diffusion tensors across groups of experimental subjects

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