Basser DTI Theory (1)

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Received 16 November 2001; Revised 8 January 2002; Accepted 9 January 2002

ABSTRACT: This article treats the theoretical underpinnings of diffusion-tensor magnetic resonance imaging (DT-MRI), as well as experimental design and data analysis issues. We review the mathematical model underlying DT-MRI, discuss the quantitative parameters that are derived from the measured effective diffusion tensor, and describeartifacts thet arise in typical DT-MRI acquisitions. We also discuss difficulties in identifying appropriate models todescribe water diffusion in heterogeneous tissues, as well as in interpreting experimental data obtained in such issues.Finally, we describe new statistical methods that have been developed to analyse DT-MRI data, and their potentialuses in clinical and multi-site studies. Copyright 2002 John Wiley & Sons, Ltd.

KEYWORDS: diffusion; tensor; MRI; methods

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It is now well established that the MR measurement of aneffective diffusion tensor of water in tissues can provideunique biologically and clinically relevant informationthat is not available from other imaging modalities. Thisinformation includes parameters that help characterizetissue composition, the physical properties of tissueconstituents, tissue microstructure and its architecturalorganization. Moreover, this assessment is obtained non-invasively, without requiring exogenous contrast agents.

This article describes methodological issues related tothe estimation of the effective diffusion tensor of water intissue. It also tries to review and showcase new methodsthat have been proposed to advance the state of the art inthis burgeoning field. We focus primarily upon diffusiontensor MRI (DT-MRI) data acquisition, experimentaldesign, artifacts and post-processing issues. The defini-tion and physical interpretation of useful MR parametersderived from the effective diffusion tensor, such as theTrace as well as measures of diffusion anisotropy, havebeen reviewed elsewhere.1 Moreover, several reviewarticles and book chapters have covered many different

aspects of diffusion tensor MRI.25 This article also triesto identify unresolved issues in DT-MRI data acquisitionand analysis, with the hope of interesting readers topursue and address some of these problems.

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In tissues, such as brain gray matter, where the measuredapparent diffusivity is largely independent of theorientation of the tissue (i.e. isotropic), it is usuallysufficient to characterize the diffusion characteristicswith a single (scalar) apparent diffusion coefficient(ADC). However, in anisotropic media, such as skeletaland cardiac muscle68 and in white matter,911 where themeasured diffusivity is known to depend upon theorientation of the tissue, no single ADC can characterizethe orientation-dependent water mobility in these tissues.The next most complex model of diffusion that candescribe anisotropic diffusion is to replace the scalardiffusion coefficient with a symmetric effective orapparent diffusion tensor of water, D.12

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Torrey13 first incorporated anisotropic translational diffu-sion in the Bloch (magnetization transport) equations,14

NMR IN BIOMEDICINENMR Biomed. 2002;15:456467Published online in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002nbm.783

*Correspondence to: P. J. Basser, Section on Tissue Biophysics andBiomimetics, NICHD, National Institutes of Health, Bethesda, Mary-land, 20892, USA.Abbreviations used: ADC, apparent diffusion coefficient; CSF,cerebrospinal fluid; DT-MRI, diffusion tensor MRI; DI, diffusionimaging; DWI, diffusion-weighted imaging; EPI, echo-planar ima-ging; FA, fractional anisotropy; RA, relative anisotropy; RMS, rootmean square; SNR, signal to noise ratio.

Copyright 2002 John Wiley & Sons, Ltd. NMR Biomed. 2002;15:456467

which could lead to additional attenuation of the NMRsignal. Analytical solutions to this equation followed forfreely diffusing species during a spin echo experiment15and, later, for diffusion in restricted geometries.1618About a decade after its introduction, Stejskal and Tannersolved the BlochTorrey equation19 for the case of free,anisotropic diffusion in the principal frame of reference.However, the StejskalTanner formula is not generallyuseable to measure an effective diffusion tensor usingNMR or MRI methods for several reasons. First, thisformula relates a time-dependent diffusion tensor, to theNMR signal, so one must establish a relationship betweenthe time-dependent diffusion tensor and an effectivediffusion tensor. Second, in the pre-MRI era in whichthe StejskalTanner formula was derived, it was alwaystacitly assumed that a homogeneous anisotropic samplecould be physically reoriented within the magnet so that itsprincipal axes could be aligned with the laboratorycoordinate system. After the development of MRI,however, this assumption was no longer tenable. Materialsunder study were often heterogeneous media whose fiberor principal axes were generally not known a priori andcould vary from place to place within the sample. Thus, ageneral scheme had to be developed to measure the entirediffusion tensor (both its diagonal and off-diagonalelements) in the laboratory frame of reference.20

The NMR measurement of the effective diffusiontensor20 and the analysis, and display of the information itcontains in each voxel, is called diffusion tensor MRI(DT-MRI).21 The effective diffusion tensor, D, (orfunctions of it) is estimated from a series of diffusion-weighted images (DWI) using a relationship between themeasured echo attenuation in each voxel and the appliedmagnetic field gradient sequence.20 Just as in diffusionimaging (DI) where a scalar b-factor is calculated for eachDWI, in DT-MRI a symmetric b-matrix is calculated foreach DWI.22 Whereas the b-value summarizes theattenuating effect on the MR signal of all diffusion andimaging gradients in one direction,23 the b-matrixsummarizes the attenuating effect of all gradient wave-forms applied in all three directions, x, y and z.22,24,25

In DI26 one uses a set of DWIs and their correspondingscalar b-factors to estimate an ADC along a particulardirection using linear regression. In DT-MRI, we firstdefine an effective diffusion tensor (by analogy toTanners definition of an apparent diffusion coeffi-cient27), from which a formula relating the effectivediffusion tensor to the measured echo can be derived:

ln AbAb 0

3i1

3j1

bijDij

bxxDxx 2bxyDxy 2bxzDxz byyDyy 2byzDyz bzzDzz TracebD 1

where A (b) and A(b = 0) are the echo magnitudes of the

diffusion weighted and non-diffusion weighted signalsrespectively, and bij is a component of the symmetricb-matrix, b. In DT-MRI a symmetric b-matrix iscalculated for each DWI.22 The b-matrix summarizesthe attenuating effect of all gradient waveforms appliedin all three directions, x, y and z.22,24,25 We then useeach DWI and its corresponding b-matrix to estimate Dusing multivariate linear regression* of eqn (1) as inBasser et al.20

There are two important distinctions between DI andDT-MRI. First, DI is inherently a one-dimensionaltechnique, i.e. it is used to measure the projection of allmolecular displacements along one direction at a time.Therefore, it is sufficient to apply diffusion gradientsalong only one direction. DT-MRI is inherently three-dimensional; one must apply diffusion gradients along atleast six noncollinear, non-coplanar directions in order toprovide enough information to estimate the six indepen-dent elements of the diffusion tensor from eqn (1).20

Second, the b-matrix formalism forces us to expand thenotion of cross-terms between imaging and diffusiongradients, to account for possible interactions betweenimaging and diffusion gradients that are applied inorthogonal directions, and even between imaging gra-dients that are applied in orthogonal directions.22,24,25 Inisotropic media, gradients applied in orthogonal direc-tions do not result in cross-terms; in anisotropic media,however, they can.

Finally, it is easy to see that DT-MRI subsumes DI. Ifthe medium is isotropic, then Dxx = Dyy = Dzz = D, andDxy = Dxz = Dyz = 0. Then, eqn (1) reduces to a model ofisotropic diffusion with b = bxx byy bzz = Trace(b).

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Quantitative parameters provided by diffusion-tensorMRI can be obtained and explained using a geometricapproach. Intrinsic quantities can be found that char-acterize different unique features, for example, describ-ing the size, shape and orientation of the root mean square(rms) displacement profiles within an imaging volume,which can be represented as diffusion ellipsoids. Scalarparameters, functionally related to the diagonal and off-diagonal elements of D(x, y, z), can also be displayed asan image, revealing ways in which the tensor field variesfrom place to place within the imaging volume.28 Thesequantities are rotationally invariant, i.e. independent ofthe orientation of the tissue structures, the patients bodywithin the MR magnet, the applied diffusion sensitizinggradients, and the choice of the laboratory coordinatesystem in which the components of the diffusion tensor

*Multivariate linear regression is just one of a number of techniques,including non-linear regression and singular-value decomposition,that could be used to estimate D from the echo data.


DIFFUSION-TENSOR MRI 457

and magnet field gradients are measured.21,29 Someexamples are given below.

1 7

The first moment of the diffusion tensor field, or theorientationally averaged value of the diffusion tensorfield can be calculated at each point within an imagingvolume:

D TraceD3 Dxx Dyy Dzz3 1 2 33

2

where 1, 2 and 3 are the three eigenvalues and their mean.

Physically, an estimate of D can be obtained bytaking the arithmetic average of ADCs acquired in allpossible directions.30 By integrating over all directionsuniformly, we obtain an intrinsic property of the tissue,which is independent of fiber orientation, gradientdirections, etc. Recently, terms like trace-ADC, meantrace, trace mean, etc. have been used to signify D,however these terms are not meaningful. We suggest, asan alternative, the term bulk mean diffusivity.

Several interesting issues remain unresolved about thedistribution of Trace(D) within tissue. For example, whyis it so uniform within normal adult brain parenchyma? Inparticular, why is its value so similar in normal white andgray matter,31 even though these tissues are so differenthistologically? This spatial uniformity has contributed tothe increasing clinical utility of Trace(D) in diseaseassessment and monitoring since it makes diseasedregions more conspicuous when juxtaposed against thehomogeneous background of normal parenchyma. Asecond reason that makes Trace(D) useful is that itappears so similar between and among normal humansubjects. In fact, it appears to be quite similar across arange of normal mammalian brains including mice, rats,cats,32,33 monkeys34 and humans.31,35 It is worthconsidering whether mammalian brains are designedto force Trace(D) to lie within such a narrow range ofvalues and, if so, what these optimal design criteria are.

As an aside, trace-weighted or isotropicallyweighted DWIs have become a popular means ofdepicting regions in which the diffusivity has changed(particularly dropped) with respect to the surroundingtissue.3638 Some isotropically weighted sequences havealready been implemented commercially. In a Trace-weighted DWI, image intensity is brighter in regions oflow diffusivity, making them more conspicuous. Oneway to construct a Trace-weighted image is to take ageometric mean of N DWIs, which we designate by A(bi),so that the trace-weighted intensity (TWI) becomes:

TWI Ni1

AbiN 3

If the DWI signal attenuation is given by:

Abi A0ebixxDxx2bixyDxy2bixzDxzbiyyDyy2biyzDyzbizzDzz4

then the conditions for producing a Trace-weighted DWIare:

Ni1

bixx Ni1

biyy Ni1

bizz N 0 5

i.e. that the total diffusion weighting along the x, y and zdirections is the same, and

Ni1

bixy Ni1

biyz Ni1

bixz 0 6

i.e. that the sum of each of the off-diagonal elements ofthe b-matrix is zero. In this way

TWI A0 e TraceD 7which results in an image whose intensity is weightedby Trace(D).

- 7

The second and higher moments of D have been proposedfor use as diffusion anisotropy measures because theycharacterize different ways in which the diffusion tensorfield deviates from being isotropic. This has resulted in anumber of diffusion anisotropy measures based upon thesecond moment of the distribution of the eigenvalues ofD: (1 )2 (2 )2 (3 )2, such as therelative anisotropy (RA), and the fractional anisotropy(FA),1 which characterize the eccentricity of the diffusionellipsoid. The RA is just the coefficient of variation of theeigenvalues, which has been previously used in crystal-lography as an aspherism coefficient.39 Anisotropymeasures based upon the higher moments of the diffusiontensor or the distribution of eigenvalues of D, such as theskewness or kurtosis, could potentially be used tocharacterize diffusion anisotropy more completely, butthe effects of noise make such measures unreliable (seebelow).

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Novel anisotropy measures have been proposed that arebased upon a barycentric representation of the diffusiontensor, in which it is decomposed into line-like, plane-like, and sphere-like tensors corresponding to diffusionellipsoids that are prolate, oblate and spherical, respec-tively.40,41 The information provided by this interestingapproach should be compared systematically with the


458 P. J. BASSER AND D. K. JONES

information contained in the first three moments of D, themean, variance and skewness. One issue that should beexamined is the sensitivity of the barycentric representa-tion to the order in which the eigenvalues of D are sorted.Whereas the moments of D given above are insensitive tothe order of the eigenvalues, dependence on their orderrenders quantities susceptible to a statistical bias causedwhen these eigenvalues are sorted.34

Another novel anisotropy measure has recently beenproposed by Frank42 to treat cases in which two or moredistinct fiber populations may occupy a voxel. When thisoccurs, the diffusion tensor measured using the singletensor model represents only a powder average of theunderlying tensors. This always results in a reduction inthe measured diffusion anisotropy. Franks method is tomeasure ADCs in many non-collinear directions, and tocalculate the variance of these ADC measurements abouttheir mean value, which he defines as a new anisotropymeasure. The sensitivity of this measure to the SNR ofthe acquisition, the degree of diffusion sensitization, andother features of the experimental design should befurther investigated.

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Another important development in DT-MRI is theintroduction of quantities that reveal architecturalfeatures of anisotropic structures, such as nerve fibertracts in brain. Useful information can be gleaned fromthe directional pattern of diffusion ellipsoids within animaging volume. Early on, it was proposed that, inordered fibrous tissues, the eigenvector associated withthe largest eigenvalue within a voxel is parallel to thelocal fiber orientation.21 Imaging methods that apply thisidea include direction field mapping (in which the localfiber direction is displayed as a vector in each voxel) andfiber-tract color mapping (in which a color, assigned to avoxel containing anisotropic tissue, is used to signify thelocal fiber-tract direction4346).

) 79

DT-MRI fiber tractography21,4760 is a method forfollowing fiber-tract trajectories within the brain andother fibrous tissues. Here, fiber-tract trajectories aregenerated from the fiber-tract direction field in much thesame way that fluid streamlines are generated from a fluidvelocity field. Many unexpected artifacts in DT-MRIfiber tractography can arise when discrete, coarselysampled, noisy, voxel-averaged direction field data48 areused, or when one attempts to follow incoherentlyorganized nerve pathways.61,62 These artifacts couldsuggest phantom connections between different brainregions that do not exist anatomically. Therefore, great

care must be exercised both in obtaining and ininterpreting such connectivity data.47,53

9

A less intuitive, but powerful method of motivating anddeveloping quantitative imaging parameters from DT-MRI data is by considering the differential geometry andalgebraic properties of the diffusion tensor field itself,whose local features are sampled discretely in a DT-MRIexperiment. Until recently, this approach was only ofacademic interest since there was no practical method toobtain a continuous representation of a diffusion tensorfield from the noisy, voxel-averaged, discrete diffusiontensor data. However, this situation has changed with theadvent of methods to construct such tensor fieldrepresentations.57,63,64 For instance, this approach haslead to new applications such as DT-MRI tractography,hyperstreamline and hyperstreamsurface imaging,65 con-nectivity analysis,66 and should lead to other innovationsthat were not previously feasible.

For example, in structurally complex anisotropicmedia, such as the heart, which has a laminar architec-ture, one can also attempt to describe the deformation(curving, twisting, and bending) of the normal, rectifying,and oscillating sheets formed by muscle and connectivetissue. To do this, we can construct surfaces from thediffusion tensor field, which can be parameterized by twovariables. Concepts of the differential geometry ofsurfaces67 can then be used to determine additionalgeometric features of sheet shape that can be calculatedand displayed as intrinsic MRI parameters. These includethe First and Second Fundamental Forms, I and II, and thenormal, gaussian and mean curvatures.67 These par-ameters are intrinsic because they characterize differentfeatures of the local shape of the lamina, independent ofthe coordinate frame of reference, and constitute newparameters.

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Subject motion during data acquisition can causeghosting or artifactual redistribution of signal intensitieswithin DWIs. Artifacts resulting from rigid body motionare the easiest to correct for, since this involves applyinga uniform phase correction to an entire image. Thisproblem has been only partially addressed by incorporat-ing navigator echoes in the DWI pulse sequence.68,69However, artifacts due to other physiological motion, forexample, eye movements or pulsation of cerebrospinalfluid, are more difficult to correct. While these artifactsare mitigated by the use of fast echo-planar DWIsequences and cardiac gating, no general theoretical



approach has been developed to model and correct forthem.

.

Large, rapidly switched magnetic field gradients pro-duced by the gradient coils during the diffusion sequenceinduce eddy currents in the electrically conductivestructures of the MRI scanner, which in turn produceadditional unwanted, rapidly and slowly decayingmagnetic fields. This results in two undesirable effects:first, the field gradient at the sample differs from theprescribed field gradient, resulting in a differencebetween the actual and prescribed b-matrix; second, aslowly decaying field during readout of the image causesgeometrical distortion of the DWI. These artifacts canadversely affect diffusion imaging studies because thediffusion coefficient or diffusion tensor is calculated ineach voxel from a multiplicity of DWIs assuming that thegradients actually being applied to the tissue are the sameas the prescribed gradients. Uncompensated imagedistortion can lead to significant, systematic errors inthese estimated diffusion parameters. Figure 1 illustratesa typical artifact arising from uncorrected eddy currentinduced distortionsa rim of high anisotropy along thephase-encode direction.

Unfortunately, single-shot echo-planar image (EPI)acquisitions are quite susceptible to eddy-current artifactsso correction schemes have to be used. Alexander et al.70suggested the use of bipolar diffusion-encoding gradientsand Papadakis et al.71 have also suggested applyingpreemphasis to diffusion-encoding gradients as means ofameliorating the problem at the acquisition stage.

Post-processing strategies generally aim to warp eachDWI to a common template.72,73 An interesting approachis to use a mutual information criterion to determine awarp that maximizes the overlap within a series ofdiffusion-weighted images.57,74 Other post-processingmethods that have been proposed include correcting thephase map using a model of the effects of the eddycurrent on it,75,76 and mapping the eddy current inducedfields directly.77,78

9

Large discontinuities in bulk magnetic susceptibility,such as those occurring at tissueair interfaces, producelocal magnetic field gradients that notoriously degradeand distort DWIs, particularly during echo-planarimaging. In addition to image distortion, susceptibilityvariations within the brain adversely affect DWIsbecause the additional local gradients act like diffusiongradients causing the b-matrix to be spatially varying.This problem is partly compensated for by the use of thelogarithm of the ratio of the diffusion-weighted to non-diffusion-weighted intensity [see eqn (1)], in which casethe effect of these susceptibility-induced gradients, whichare present during both diffusion-weighted and non-diffusion-weighted images, is cancelled.

Susceptibility effects are particularly acute in the brainin regions adjacent to the sinuses.79 As DWIs are beingacquired increasingly on higher field strength magnets(3 T and above), these problems will become moresevere. Figure 2 illustrates a typical example ofsusceptibility-induced artifacts at 3 T using a single-shotEPI sequence. It is clear that, at higher field strengths,

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different strategies to those used for imaging at 1.5 T areneeded. While the distortions in Figure 2 are typical ofdistortions using single-shot EPI at 3T, diffusionweighted SENSE acquisitions appear to mitigate theseartifacts.76

While at low levels of diffusion weighting the logarithmof the signal attenuation decreases linearly with increas-ing b-value, in common with all MRI acquisitions,background noise causes the DWI intensity to approach abaseline noise floor as one progressively increases thedegree of diffusion weighting. Even above this baseline,noise in DWIs can introduce significant bias in theestimates of the eigenvalues, which makes isotropicmedia appear anisotropic, and anisotropic media appearmore anisotropic.34 Pierpaoli et al. have proposed theLattice Anisotropy index which combines pure measuresof anisotropy with measures of inter-voxel coherence ofeigenvectors in neighboring voxels to try to amelioratethis problem.34 This bias often makes higher ordermeasures, such as the skewness or kurtosis, tooinaccurate to be used reliably.

More recently, it was found that RF noise also biasesthe mean and variance of the eigenvectors of D.80Unfortunately, the current understanding of the deleter-ious effects of this artifact is more advanced than ourunderstanding of useful remedies to correct for it.80,81

In the presence of noise, it is possible to estimate

negative eigenvalues of the diffusion tensor using theregression methods described above. Physically, eachof the principal diffusivities of the diffusion tensormust be non-negative (i.e. the tensor is positive semi-definite). To ensure this condition, some groups haveimposed the constraint of positive definiteness on thediffusion tensor explicitly.82 This can be done, forexample, by minimizing the error norm or 2 valuesubject to the constraints that the three principaldiffusivities are positive.

-

Background gradients can be present if the magnetic fieldis improperly shimmed, leading to additional signalattenuation if not properly compensated for. Thisproblem can often be remedied by measuring thebackground gradients directly83 and incorporating themexplicitly in the b-matrix.

Gradient non-linearity and miscalibration can lead tosmall but significant errors in the calculation of thediffusion coefficient or diffusion tensor elements. Thesignal attenuation and the gradient pulse sequence mustbe known for each DWI. If the gradients are not wellcalibrated, or they are not linear, signal attenuationattributed to the diffusion processes in the sample couldbe miscalculated using eqn (1). If the gradients in the x, yand z directions are coupled to one-another (i.e. there iscross-talk) because of misalignment of the gradient coils,then gradients applied in logical directions may have

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components in other directions, which could be particu-larly problematic in DT-MRI.84,85

Flipping the signs of the diffusion gradients onalternate averages has been suggested as an aid inremoving cross-terms. However, owing to noise, there isa question whether such averaging should be done on thelogarithm of the intensity or whether the geometric meanshould be performed on the measured intensities. Clearly,this method would ameliorate cross-terms arising fromspecific interactions between imaging and diffusiongradients applied in parallel or orthogonal directions,but not cross-terms arising between imaging gradients,which, albeit small in MRI clinical applications, wouldnot be corrected. The advantage of this method is that itcould simplify the analysis of DWI data. In new single-shot acquisition schemes that acquire DWIs withgradients applied in many directions,74,86 this strategyhas the disadvantage that twice as many DWIs wouldhave to be acquired. Moreover, an additional parameter isrequired per voxel as a free parameter in the estimation ofthe diffusion tensor to determine the cross-terms due tothe imaging gradients alone.

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One of the advantages of using the single effectivediffusion tensor formalism in tissue is that it provides agreat deal of new information without making manyexplicit assumptions about the underlying tissue archi-tecture and microstructure. The only explicit assumptionis that the diffusion characteristics can be represented bya single symmetric gaussian displacement distribution.We can appreciate its conceptual economy by consider-ing the great complexity of minimal models one couldpropose to represent the actual known biologicalcompartments with tissues.

Inferring the microstructure and the underlyingarchitectural organization of tissue using diffusionimaging data is complicated by several factors. First,homogeneity within each voxel cannot be assured.Numerous microscopic compartments exist within brainparenchyma. A priori, we must assume that gray matter,white matter and cerebrospinal fluid (CSF) could occupythe same macroscopic voxel. The crudest model onecould pose, then, for these three compartments is:

AA0

f1eTracebDwm f2eTracebDgm f3eTracebDcsf

8where Dwm represents the diffusion tensor for whitematter, and Dgm and Dcsf represent the apparent diffusioncoefficients for gray matter and for CSF, respectively,which are assumed to be isotropic. In this model, thethree compartments are not assumed to be exchanging.We could further simplify this model by requiring that the

f-coefficients sum to one. This would implicitly involveassuming that the T1 and T2 in each compartment werethe same. Even so, this model contains 1 7 2 1= 11 parameters to estimate in each voxel.

If we were to assume that there is an additional whitematter compartment, described by two individual diffu-sion tensors, D1wmand D2wm,such as in regions where whitematter fibers cross,34 the model would be furthercomplicated, involving 18 free parameters:

AA0

f1eTracebD1wm f2eTracebD2wm

f3eTracebDgm f4eTracebDcsf 9If the diffusion processes within the white matter fibercompartments were cylindrically symmetric, eitherdescribed by cylindrically symmetric prolate or oblateellipsoids, then we could impose additional constraints onthe diffusion tensors, D1wmand D2wm,which would reducethe number of free parameters to estimate each tensorfrom 6 to 4.87 This is because only two parameters areneeded to specify the orientation of the ellipsoid in space,and two parameters to specify its principal diffusivities.

Additionally, intra and extracellular compartmentswithout exchange can further complicate the model.Returning to the model in eqn (8), we now have to addtwo additional compartments for isotropic gray andanisotropic white matter, although obviously not for CSF:

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f1eTracebDintwm f2eTracebDextwm f3eTracebDintgm

f4eTracebDextgm f5eTracebDcsf 10Already, we have accumulated 1 7 6 2 1 1

= 18 free parameters for this model. If we use the rule ofthumb that one would like to obtain at least four times asmany DWIs as the number of free parameters to estimatethem, we require now at least 72 DWIs. Generally,however, the number of these distinct tissue types andtheir distribution within the voxel is unknown. At anultrastructural level, gray and white matter are them-selves generally quite heterogeneous, having a distribu-tion of macromolecular structures of varying size, shape,composition and physical properties (such as T2, D).Thus, even the model in eqn (10) is quite naive.

Several groups have recently tried to address theproblem of multiple white matter compartments in thebrain. Their approach was to use a two-compartmentmodel with non-exchanging spins, in which eachcompartment is characterized by its own diffusiontensor.88,89

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f1eTracebD1 f2eTracebD2 11

Even with this simple model, there are 14 free parametersto estimate. As each white matter compartment is added,



another six parameters are needed to prescribe thediffusion tensor. This leads to a proliferation of freeparameters to estimate. It also taxes experimentalresources and ones ability to design an efficientexperiment. Which diffusion gradient directions andgradient strengths are optimal? How many DWIs areneeded? How can we tell if this model is adequate? Thesebecome complicated multifactorial problems to consider.At present, a general framework for assessing theadequacy of different models to describe water diffusionwithin a voxel has yet to be proposed and implemented.

*

Differences in relaxation parameters can lead to differentrates of echo attenuation in each compartment, making itmore difficult to explain the cause of signal loss within avoxel. There are also irregular boundaries betweenmacromolecular and microscopic-scale compartments.Different macromolecular structures comprising theseboundaries may affect the displacement distribution ofwater molecules differently, necessitating even morecomplex models. Water molecular motion may berestricted or hindered. Even within a compartment, somewater will be associated with certain macromoleculeswhile some will be free to diffuse.

Another unknown is whether there is exchangebetween compartments, which can also affect therelaxation rates of the spin system. How water moveswithin and between compartments is still not wellunderstood. Owing to differences in blood flow andthermal conductivity, temperature cannot be assumed tobe uniform throughout a tissue sample. It is well knownthat temperature affects the measured diffusivity (1.5%per 1 C)9092 and is predicted to have the same effect onall diffusion tensor components.

For all these reasons, the underlying cause of diffusionanisotropy has not been fully elucidated in brainparenchyma, although most investigators ascribe it toordered, heterogeneous structures, such as large orientedextracellular and intracellular macromolecules,supermacromolecular structures, organelles, and mem-branes. In the central nervous system (CNS), diffusionanisotropy is not simply caused by myelin in whitematter, since several studies have shown that, even beforemyelin is deposited, diffusion anisotropy can bemeasured using MRI.9396 Thus, despite the fact thatincreases in myelin are temporally correlated withincreases in diffusion anisotropy, structures other thanthe myelin sheath must be contributing to diffusionanisotropy.97 This is an important point, because there isa common misconception that the degree of diffusionanisotropy can be used as a quantitative measure orstain of myelin content, when in reality no such simplerelationship exists.

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Recently, with the advent of stronger magnetic fieldgradients, several groups have reported multi-exponentialdecay of the MR signal intensity as a function of b-value.98,99 Some have inferred from this data thatproperties of distinct tissue compartments can be mean-ingfully observed using DT-MRI. Clearly, there isinteresting biological information to be gleaned in thenon-linear regime that may help to resolve some of theseissues. High b-value acquisitions are being treated byothers in this issue, so only a few points will be madebelow that pertain to the DT-MRI aspects.

Putting aside the complexities of obtaining stableestimates of discrete exponentials (i.e. diffusion relax-ography), numerous microstructural and architecturalconfigurations could produce the same multi-exponentialrelaxation data. For example, Peled recently showed thata system of impermeable tubes with a distribution ofdiameters consistent with those found in histologicalbrain slices could give rise to multi-exponentialdecay.100 Similar behavior is expected when there is astatistical distribution of any relevant physical propertyor microstructural dimension within a voxel. With theexception of CSF, it is unlikely that a particularexponential can ever meaningfully be assigned to aparticular and distinct tissue compartment. Clearly,without invoking additional a priori information abouttissue structure, tissue composition, the physical proper-

ties of the different compartments and their spatialdistribution, determining tissue microstructural andarchitectural features from the NMR signal is an ill-posed intractable inverse problem.

Another approach to analyzing tissues with multiplecompartments is to use three-dimensional q-space tech-niques, originally developed by Callaghan.101,102 Mostrecently, this strategy has been suggested by Tuch et al.for use in clinical scanners, using conventional diffusion-weighted images.89,103 In principle, the potential benefitsof using such an approach are numerous: q-space MRIwould provide a probability displacement distributionwithin each voxel in a model-independent way. Thismethod would also allow one to vary the diffusion timeand the length scale probed independently and system-atically.

However, unlike q-space MRI in which high andlow q-values provide an unambiguous physical inter-pretation of the length scale probed and the diffusion timeof the experiment, in clinical DWI acquisitions, neither ofthese experimental quantities is well defined. True three-dimensional q-space MRI requires ultrashort (1 ms)and ultrastrong (500 G cm1) gradient pulses. Produ-cing these using a whole body or even a head gradient setwould clearly cause a severe shock to the patient. If oneattempts to use DWIs with longer duration and weakerdiffusion gradient pulses, one loses the ability to infer theunderlying displacement distribution, the diffusion timeand the length scale probed.

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Since the diffusion tensor data is statistically estimated ineach voxel, we must treat it as a set of random variables.However, the statistical analysis of DT-MRI data iscomplicated by several factors. Although in an ideal DT-MRI experiment D has been shown to be distributedaccording to a multivariate normal distribution,104 andthus Trace (D) has been shown to be normallydistributed,105 the parametric distribution of many otherderived DT-MRI parameters is either unknown or knownnot to be normal. In these cases, we are precluded fromusing statistical hypothesis testing methods that assumean underlying normal distribution to determine whetheran observed difference between different regions ofinterest (ROI) is statistically significant. In such cases,empirical methods like the bootstrapwhich allowsdetermination of the distribution of a statistical parameterempirically without knowledge of the form of itsdistribution a priorishow great promise in diffusionimaging studies.106 This is particularly true now thatmany single-shot DWIs can be acquired during a singlescanning session. Bootstrap methods become morepracticable and accurate as more raw data are available.One promising application of this empirical statisticalmethod is to assure data quality and to test for systematicartifacts that might be present during the acquisition. Inthis way, statistical properties of measured DT-MRIparameters can be studied meaningfully on a voxel byvoxel basis.

+3)2). ( +3)2+" .,) 2)+.2

In performing multi-site or longitudinal diffusion tensorimaging studies, several additional issues arise. The mostbasic is how to compare high-dimensional diffusiontensor data from different subjects or from the samesubject acquired at different time points. Applyingwarping transformations developed for scalar images,can produce nonsensical results when applied to DT-MRIdata without taking appropriate precautions.107 Ourunderstanding of admissible transformations that can beapplied to warp and register diffusion tensor field data isstill limited.

A second issue to consider is the proliferation ofmeasures derived from the diffusion tensor to character-ize different features of the isotropic and anisotropicdiffusion. Consistent definitions of quantities such as theorientationally averaged diffusivity, RA, FA, etc. shouldbe employed. It is advisable to use the same imagingacquisition hardware, reconstruction software and post-processing routines to control for unnecessary variability.All sites should use a well-characterized phantom, even ifit is an isotropic phantom, to ensure that no systematic

artifacts occur, and that the DWI acquisition is stable intime and across platforms.

,*(,3+(0 .&2

DT-MRI provides new means to probe tissue structure atdifferent levels of architectural organization. Whileexperimental diffusion times are associated with watermolecule displacements on the order of microns, thesemolecular motions are ensemble-averaged within avoxel, and then subsequently assembled into multi-sliceor three-dimensional images of tissues and organs. Thus,this imaging modality permits us to study and elucidatecomplex structural features spanning length scalesranging from the macromolecular to the macroscopic,without the use of exogenous contrast agents.

New structural and functional parameters provided byDT-MRI, such as maps of the eigenvalues of the diffusiontensor, its Trace, measures of the degree of diffusionanisotropy and organization and estimates of fiberdirection all advance our understanding of nerve path-ways, fiber continuity and, potentially, functional con-nectivity in the CNS.

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The Wellcome Trust supports DKJ.

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Basser DTI Theory (1)