THEORY AND APPLICATIONS OF DIFFUSION MRI

Alexander Leemans

Image Sciences Institute, University Medical Center Utrecht, 3584 CX Utrecht, the Netherlandsk Alexander@isi.uu.nl m http://www.isi.uu.nl/

ABSTRACT

Boosted by the tremendous success of diffusion tensor MRI, whichled to the first in vivo and non-invasive characterization of mi-crostructural tissue properties, many advancements in diffusionMRI have been made during the last decade. With new methodolog-ical developments in data acquisition, modeling, pre-processing,analysis, and visualization, diffusion MRI is rapidly becoming astandard addition to clinical MRI, especially, for investigating brainwhite matter architecture. In this work, an overview of methodsand applications in diffusion MRI is presented. In particular, con-cepts underlying diffusion tensor MRI are discussed, techniques forreconstructing fiber pathways (both deterministically and probabilis-tically) are described, and the “crossing fibers” issue is explained.

Index Terms – Diffusion MRI, Diffusion Tensor Imaging, De-terministic & Probabilistic Tractography, Crossing Fibers

1. THE DIFFUSION MRI SIGNAL

Roughly five years after Le Bihan and Breton introduced diffusion-weighted MRI (DWI) in the clinic as a new MRI contrast to investi-gate microstructural changes in the brain [1], directional dependenceof DWI data was observed in vivo [2]. An example of this anisotropyin both the measured diffusion signal and the corresponding apparentdiffusion coefficient (ADC) is presented in Fig. 1-(a). To understandthe underlying mechanisms that may cause anisotropy in white mat-ter (WM), the reader is referred to the work of Beaulieu et al [3].

2. DIFFUSION TENSOR MRI

The next most complex model to characterize 3D Gaussian diffusionin which anisotropy may be present is the second-rank diffusion ten-sor [4]. In summary, the symmetric diffusion tensor D,

D =

⎡⎣

Dxx Dxy Dxz

Dxy Dyy Dyz

Dxz Dyz Dzz

⎤⎦ , (1)

is calculated by solving the generalized anisotropic form of theStejskal-Tanner relationship [5]:

Sk = S0 e−bgTk ·D·gk , (2)

with Sk(r) the DWI signal, measured along the unit diffusion gra-dient direction gk (k = 1, . . . , K, where K lies typically in therange of 6 ∼ 60 directions) [6], S0(r) the reference image acquiredwithout diffusion weighting, and b the diffusion weighting factor(roughly in the range of 700 s/mm2 ∼ 1500 s/mm2 in a clinical set-ting) [7]. The reader is referred to the work of Koay et al regardingdiffusion tensor estimation procedures (e.g., [8]).

With diffusion tensor MRI (DTI), the estimated displacementprofile of the diffusing particles can be interpreted as an ellipsoidaliso-probability surface, reflecting the distance that molecules dif-fused to from the origin. The radii and principal axes of this ellipsoidare then defined by the square root of the eigenvalues (λ1 ≥ λ2 ≥λ3 ≥ 0) and eigenvectors (e1, e2, e3) of the diffusion tensor D,respectively [4], as obtained from the eigenvalue decomposition (seeFig. 1-(b)):

D = E · Λ · E−1, (3)

where

E =[e1 e2 e3

]and Λ =

⎡⎣

λ1 0 00 λ2 00 0 λ3

⎤⎦ . (4)

The major benefit of the diffusion tensor framework is that itprovides for each voxel (i) a local coordinate system of the predomi-nant diffusion orientations – as defined by the eigenvectors ei – and(ii) rotationally invariant measures of diffusion, such as the ADC(also referred to as mean diffusivity or trace/3), radial (ADC⊥) andaxial (ADC‖) diffusion, fractional anisotropy (FA), and the “Westin-measures” (CL, CP , CS) – as defined by the eigenvalues λi [11, 10](see Fig. 1 (b) and (c)):

ADC =Tr

[D

]3

=Dxx + Dyy + Dzz

3=

λ1 + λ2 + λ3

3, (5)

ADC⊥ =λ2 + λ3

2and ADC‖ = λ1 , (6)

FA =

√3[(λ1 − λ2)2 + (λ2 − λ3)2 + (λ3 − λ1)2

]√

2(λ21 + λ2

2 + λ23)

, (7)

and

CL =λ1 − λ2

3〈λ〉 ; CP =2(λ2 − λ3)

3〈λ〉 ; CS =3λ3

3〈λ〉 . (8)

More detailed information regarding the theoretical underpinningsof DTI can be found in a recent review of Mukherjee et al [12].

3. FIBER TRACTOGRAPHY

Fiber Tractography (FT), also referred to as fiber tracking or tracttracing, can be defined as the virtual 3D reconstruction of fiber path-ways and was originally developed for DTI data at the end of the

20th century (e.g., [16]). Conceptually, global trajectories of WMtracts are obtained by piecing together the local fiber orientationsof each voxel estimated from the underlying diffusion data (Fig 2(a) and (b)). FT has provided exciting new opportunities to studyseveral architectural characteristics of fibrous tissue in vivo, and hasgenerated much enthusiasm, resulting in the development of a largenumber of deterministic and probabilistic algorithms.

628978-1-4244-4126-6/10/$25.00 ©2010 IEEE ISBI 2010

DWIleft-right

ADCleft-right

DWItop-bottom DWIfront-back

ADCtop-bottom ADCfront-back

(a)

1 1,e

2 2,e

3 3,e min

max

1 2 3

diffu

sivi

ty

(b)

ADC

2 3ADC2

FA DEC

D

CL DL CP DP CS DS

(c)

Fig. 1. (a) DWI maps (top row) and their corresponding ADC maps (bottom row) measured along three orthogonal diffusion gradient direc-tions. The highlighted region clearly shows diffusion anisotropy. (b) The shape of the diffusion ellipsoid (top left) is fully characterized bythe eigenvalues λi (bottom row). Also shown are the ADC and radial (also referred to as transverse or perpendicular) diffusivity (ADC⊥). (c)Top left: fractional anisotropy (FA); top right: directionally encoded color (DEC) map of the FA [9]; bottom row: geometrical decompositionof the diffusion tensor D into its linear (CLDL), planar (CP DP ), and spherical (CSDS) components [10]. Combining the linear (CL – inred) and planar (CP – in green) “Westin-measures” into a single image provides a more specific view of diffusion anisotropy (bottom right).The greenish hue suggests “crossing fibers”.

3.1. Deterministic FT approaches

The first deterministic FT algorithms were based on streamline prop-agation methods, which were already well known from engineeringsciences (e.g., reconstructing fluid streamlines from discretely sam-pled velocity field data [17]), using only the predominant direction ofthe diffusion tensor to compute the trajectories (e.g., [16, 18, 19, 20].These initial FT techniques led to a proliferation of other algorithmswith varying degrees of complexity and emphasizing different as-pects of the diffusion tensor model to extricate the complex WMfiber network (e.g., [21, 22, 23]).

3.2. Probabilistic FT approaches

One of the major limitations of deterministic FT algorithms is thatthey do not provide any indication of the confidence with whichthe fiber trajectories were reconstructed. The uncertainty associatedwith the first eigenvector (Fig 2 (c)), for instance, will have a sig-nificant effect in fiber tractography as it accumulates over long dis-tances [24]. In summary, with probabilistic FT methods, this degreeof confidence can be estimated by drawing the local fiber directionin which to proceed next from a distribution of possible orientations(e.g., [25, 26, 13]) (Fig 2 (d)).

4. “CROSSING FIBERS”

It is well known that the diffusion tensor can not adequately model“crossing fibers” at the scale of a single voxel [27]. In other words,if a voxel contains any form of complex fiber architecture which isdifferent from the one-fiber population configuration (see top row ofFig 2 (e)), the diffusion tensor will be biased. Several approacheshave been proposed as an alternative, more accurate way to charac-terize more than one fiber orientation, such as Q-ball imaging [28]and spherical deconvolution methods (e.g., [15] – see bottom of Fig2 (e)) (see the review by Alexander for a more complete overview of

multiple-fiber reconstruction algorithms [29]). It is clear that DTI-based FT algorithms (Section 3) need to be extended/modified totake “crossing fibers” into account (e.g., [30, 31]).

5. APPLICATIONS

Ranging from the songbird’s brain [32, 33] to the swine’s heart [34]– diffusion MRI has already proven to be a useful tool for a largenumber of research areas as it provides a new way to study fibroustissue (e.g., [35, 36, 37, 38, 39, 40, 41]). A recent review describingseveral applications in brain research is given by Assaf et al [42].

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(a) (b) (c) uncertainty min max

min max tract density (d)

seed point

seed point (e)

projection association

commissural

Fig. 2. (a) Fiber tractography (FT) example: WM pathways reconstructed from 5 seed points. (b) Full brain DTI-based deterministic FTexample. (c) The “cone of uncertainty” and hyperstreamtubes showing the uncertainty associated with e1. (d) Probabilistic DTI-basedFT example (wild-bootstrap approach [13]). (e) Top row: several simulated configurations of complex fiber architecture [14]; bottom:reconstruction of the fiber orientation distribution function using the constrained spherical deconvolution method of Tournier et al [15].

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