Processing and visualization for diffusion tensor MRI

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1 Medical Image Analysis 6 (00) locate/ media Processing and visualization for diffusion tensor MRI * C.-F. Westin, S.E. Maier, H. Mamata, A. Nabavi, F.A. Jolesz, R. Kiinis Brigham & Women s Hospital, Harvard Medical School, Department of Radiology, 75Francis Street, Boston, MA 05, USA Received 0 December 999; received in revised form 5 July 00; accepted 0 July 00 Abstract his paper presents processing and visualization techniques for Diffusion ensor Magnetic Resonance Imaging (D-MRI). In D-MRI, each voxel is assigned a tensor that describes local water diffusion. he geometric nature of diffusion tensors enables us to quantitatively characterize the local structure in tissues such as bone, muscle, and white matter of the brain. his maes D-MRI an interesting modality for image analysis. In this paper we present a novel analytical solution to the Stejsal anner diffusion equation system whereby a dual tensor basis, derived from the diffusion sensitizing gradient configuration, eliminates the need to solve this equation for each voxel. We further describe decomposition of the diffusion tensor based on its symmetrical properties, which in turn describe the geometry of the diffusion ellipsoid. A simple anisotropy measure follows naturally from this analysis. We describe how the geometry or shape of the tensor can be visualized using a coloring scheme based on the derived shape measures. In addition, we demonstrate that human brain tensor data when filtered can effectively describe macrostructural diffusion, which is important in the assessment of fiber-tract organization. We also describe how white matter pathways can be monitored with the methods introduced in this paper. D-MRI tractography is useful for demonstrating neural connectivity (in vivo) in healthy and diseased brain tissue. 00 Elsevier Science B.V. All rights reserved. Keywords: Diffusion ensor MRI; ractography; Dual basis; Geometrical diffusion measures; Visualization. Introduction perpendicular to a direction is proportional to the concentration gradient. hus, the phenomenon of diffusion Diffusion is the process by which matter is transported was described scientifically before any systematic developfrom one part of a system to another owing to random ment of thermodynamics. his phenomenon, nown as molecular motions. he transfer of heat by conduction is Brownian Motion, is named after the botanist, Robert also due to random molecular motion. he analogous Brown, who observed the movement of plant spores nature of the two processes was first recognized by Fic floating in water in 87. he first satisfactory theoretical (855), who described diffusion quantitatively by adopting treatment of Brownian Motion, however, was not made the mathematical equation of heat conduction derived until much later by Albert Einstein (905) and provided some years earlier by Fourier (8). Fic s law states that strong circumstantial evidence for the existence of molelocal differences in solute concentration will give rise to a cules. net flux of solute molecules from high concentration Anisotropic media such as crystals, textile fibers, and regions to low concentration regions. he net amount of polymer films have different diffusion properties dependmaterial diffusing across a unit cross-section that is ing on direction. Anisotropic diffusion is best described by an ellipsoid where the radius defines the diffusion in a particular direction. he widely accepted analogy between symmetric tensors and ellipsoids maes such tensors *Corresponding author. el.: ; fax: natural descriptors for diffusion. Moreover, the geometric 60. nature of the diffusion tensors can quantitatively character- address: westin@bwh.harvard.edu (C.-F. Westin). ize the local structure in tissues such as bone, muscle, and 6-845/ 0/ $ see front matter 00 Elsevier Science B.V. All rights reserved. PII: S6-845(0)0005-

2 94 C.-F. Westin et al. / Medical Image Analysis 6 (00) 9 08 white matter of the brain. Within white matter, the mobility. Diffusion tensor MRI of the water is restricted by the axons that are oriented along the fiber tracts. his anisotropic diffusion is due to D-MRI is a relatively recent MR imaging modality tightly paced multiple myelin membranes encompassing used for relating image intensities to the relative mobility the axon. Although myelination is not essential for diffu- of endogenous tissue water molecules. In D-MRI, a sion anisotropy of nerves [as shown in studies of non- tensor describing local water diffusion is calculated for myelinated garfish olfactory nerves (Beaulieu and Allen, each voxel from measurements of diffusion in several 994); and in studies where anisotropy exists in brains of directions. o measure diffusion, the Stejsal anner neonates before the histological appearance of myelin imaging sequence is used (Stejsal and anner, 965). (Wimberger et al., 995)], myelin is generally assumed to his sequence uses two strong gradient pulses, symmetribe the major barrier to diffusion in myelinated fiber tracts. cally positioned around a 808 refocusing pulse, allowing Using conventional MRI, we can easily identify the for controlled diffusion weighting (Fig. ). he first functional centers of the brain (cortex and nuclei). How- gradient pulse induces a phase shift for all spins; the ever, with conventional proton magnetic resonance imag- second gradient pulse will invert this phase shift, thus ing (MRI) techniques, the white matter of the brain canceling the phase shift for static spins. Spins having appears to be homogeneous without any suggestion of the completed a change of location due to Brownian Motion complex arrangement of fiber tracts. Hence, the demonstra- during the time period (D in Fig. ) will experience tion of anisotropic diffusion in the brain by magnetic different phase shifts by the two gradient pulses, which resonance has paved the way for non-invasive exploration means they are not completely refocused and consequently of the structural anatomy of the white matter in vivo will result in a signal loss. (Moseley et al., 990; Chenevert et al., 990; Basser, o eliminate the dependence of spin density,, and 995; Pierpaoli et al., 996). we must tae at least two measurements of diffusion- he paper is organized as follows. First, we review the weighted images that are differently sensitized to diffusion basics of D-MRI (Section ). Section. presents a new but remain identical in all other respects. By using, for method for calculating the diffusion tensors from the instance, a measurement without diffusion weighting and diffusion gradient raw data. he method is based on an one with diffusion weighting, diffusion can be calculated analytic solution of the Stejsal anner formula eliminat- using the following equation (Stejsal and anner, 965): ing the need to solve the Stejsal anner equation system bd S 5 S e, for each voxel of the data set. In Section 4, we expand on 0 () our previous quantitative characterization of the geometric where b is the diffusion weighting factor, introduced and nature of the diffusion tensors (Westin et al., 997, 999) defined by LeBihan et al. (986), by capturing macrostructural diffusion properties, and depicting detailed in vivo images of human white matter d b 5 g dsd] Dugu, tracts. Using these methods, we can identify the orientation () and distribution of most of the nown major fiber tracts. In where g is the proton gyromagnetic ratio (4 MHz/esla), Section 5, we discuss visualization methods for tensor ugu is the strength of the diffusion sensitizing gradient diffusion data and describe a method relating to the pulses, d is the duration of the diffusion gradient pulses, barycentric tensor shape descriptors from Section 4. We and D is the time between diffusion gradient RF pulses conclude by describing a novel white matter tractography (Fig. ). he diffusion values, D, are also nown as method (expanded from Westin et al., 999) and show apparent diffusion components (or ADC values), which some tractography results from axial D-MRI data of the emphasizes the fact that the diffusion values generated brain (Section 6). from this procedure depend on the experimental conditions Fig.. he Stejsal anner imaging sequence.

3 C.-F. Westin et al. / Medical Image Analysis 6 (00) such as the direction of the sensitizing gradient and other tion to the baseline image data S 0. hus for each slice in sequence parameters (d and D). the data set, seven images need to be collected with different diffusion weightings and gradient directions. Fig... Imaging parameters shows an example of such data with corresponding gradient directions, where hs 0,S,...,S6j represent the In this wor we applied a version of the Line Scan signal intensities in the presence of gradients g. S0 is the Diffusion Imaging (LSDI) technique (Gudbjartsson et al., signal intensity in the absence of a diffusion-sensitizing 996; Maier and Gudbjartsson, 998; Maier et al., 998). field gradient (ug0u 5 0), giving the baseline to which the his method, lie the commonly used diffusion-sensitized, remaining measurements can be related. Inserting the ultrafast, echo-planar imaging (EPI) technique (urner et gradients g and the signals hsj into the equation for the al., 990), is relatively insensitive to bul motion and loss in signal intensity (Eq. ()) results in physiologic pulsations of vascular origin. bgˆ Dgˆ Our data were acquired at the Brigham and Women s S 5 S0 e, (5) Hospital on a GE Signa.5 esla Horizon Echospeed 5.6 system with standard. Gauss/ cm field gradients. he a system of six equations from which the tensor can be time required for acquisition of the diffusion tensor data calculated, for one slice was min; no averaging was performed. Imaging parameters were: effective R5.4 s, E565 ) 5 ln(s 0) bgˆ Dg ˆ, ms, bhigh s/mm, blow 5 5 s/mm, field of view ln(s ) 5 ln(s 0) bgˆ Dg ˆ, cm, effective voxel size mm, 4 Hz readout ) 5 ln(s 0) bgˆ Dg ˆ, bandwidth, acquisition matrix 88. (6) ln(s 4) 5 ln(s 0) bgˆ 4Dg ˆ4,.. Calculation of diffusion tensors ln(s 5) 5 ln(s 0) bgˆ 5Dg ˆ5, For the case of anisotropic diffusion, Eq. () has to be written in a more general form, g d [D(d /)]g Dg S 5 S e. () 0 his formula reverts to the isotropic case above with D 5 DI, where I is the identity tensor. By inserting normalized gradient vectors, g5g/ ˆ ugu, we can then write Eq. () using LeBihan s b-factor (Eq. ()), bgˆ Dgˆ ln(s ) 5 ln(s ) bgˆ Dg ˆ By solving this equation system for each voxel in the data set, we will arrive at the final diffusion tensor field.. ensor form of Stejsal anner equations.. Dual bases and diffusion measurements S 5 S0 e. (4) In this section we will derive a compact analytic In the typical case, the symmetric diffusion tensor D solution to the Stejsal anner equation system (Eq. (6)) has six degrees of freedom (number of independent using concepts from tensor analysis. ensor analysis is a coefficients in a matrix representation). o estimate the multi-linear extension of traditional linear algebra and a tensor, then, at least six measurements (taen from differ- generalization of the notions from vector analysis. Central ent non-collinear gradient directions) are needed, in addi- concepts in tensor analysis are dual spaces and con- Fig.. Examples of sagittal diffusion measurements with corresponding magnetic field gradients used for diffusion weighting.

4 96 C.-F. Westin et al. / Medical Image Analysis 6 (00) 9 08 travariant vectors. Details on this topic can be found in practical one. As evidenced below, the diffusion basis textboos on vector analysis and differential geometry, functions arising from MR scanner measurements are nonincluding (Stoer, 989; Young, 978; Kendall, 977). orthogonal; thus its dual elements have practical implica- ensor algebra states that for finite dimensional spaces tions. Diffusion tensors belong to the class of symmetric every element can decomposed on a basis tensors and have six degrees of freedom, meaning that six different basis tensors are required to span this D 5O a G, (7) space. hus we may write where a are the coordinates of D in the basis G. If the D 5O G,DlG, (0) basis is orthonormal, the coordinates a are the inner 5 products of the element with the basis elements. If the where the weights a are the coordinates of D in the basis is not orthonormal, the a are the inner products of (tensor) basis hg,g,...,g6j. the element with the dual basis elements, here denoted G. Reverting to the Stejsal anner equation (Eq. ()), and o be more precise, the coordinates are always the inner noting that the double projection of the gradient vectors products of the element with the dual basis elements, can be rewritten using an inner product (contraction) however for an orthonormal basis, the dual basis coincides between the diffusion tensor and the outer product of the with the original basis: G 5G gradient directions, D 5O G,DlG. (8) ˆ ˆ 6 g Dg5 G,Dl, () he elements can also be expressed in the dual basis, and where by symmetry the coordinates are obtained by projecting G 5 gg ˆˆ, () onto the dual of the dual basis, which is the original basis itself, and inserting this result gives the following expression of the Stejsal anner equations: D 5O G,DlG. (9) ln(s ) 5 ln(s ) bg,dl, () 0 Fig. shows an example of how to graphically construct a which give closed expressions for the inner products, dual basis from a basis by drawing parallel lines along the vectors and lines touching the end of the vectors. From the G,Dl 5 ](ln(s ) ln(s )). b 0 figure, it can be seen that if the original basis were (4) orthonormal, the dual basis vectors and the basis vectors Recalling the structure of the coordinates and basis elewould coincide, and the difference between the basis sets ments for the diffusion tensor in Eq. (9) we see that Eq. would become more of a theoretical issue rather than a (4) describes the dual coordinates of the diffusion tensor Fig.. Graphical construction of a dual basis in D. wo lines are drawn along the basis vectors b and b, and two lines are drawn parallel to these touching the end of the vectors. he parallel lines represent a dual basis, which also can be graphically illustrated in a vector form: two vectors that are perpendicular to the lines, b and b.

5 C.-F. Westin et al. / Medical Image Analysis 6 (00) expressed by the diffusion data from the scanner S and the u u??? u b-factor. By identifying the dual basis to G, the outer G G??? G6 product of the gradient directions, Eq. (9) gives that the 5 u u??? u diffusion tensor can be directly calculated by a simple 0 0 linear sum, D 5O ](ln(s ) ln(s ))G. (5) b 0 5 ] 0 (0) he set of six outer product gradient tensors hgj, [ ,...,6, defines a (non-orthogonal) tensor basis and the corresponding dual tensor basis hg j is uniquely defined 0 0 by the following bi-orthogonal relation: G i,gl 5 d i, (6) ] where Kronecer s symbol dij is used with the usual meaning, d 5 ifi5j and 0 if i ± j. See, for example, (Westin, 994) for more details about non-orthogonal () bases. Inserting this result verifies the bi-orthonormality requirement between the basis and dual basis elements:.. Calculation of dual tensor basis u u??? u u u??? u In this section we will derive the dual tensor basis GG5 G G??? G corresponding to the specific gradient configuration in our 6 G G??? G6 D-MRI protocol. he gradient directions used in our u u??? u u u??? u experiments are () 0 g 5, g 5, g 5 0, Œ] SD Œ] SD Œ] SD ] g 5, g 5, g 5 0, ] S D 5 ] S D 6 ] S D Œ Œ Œ #%%%%%%%%%%%%"!%%%%%%%%%%%%$ (7) dual basis 0 giving the following outer products of these gradients: G 5] 0, G 5] 0, S D S D ] () G 5] S0 0 0 D, (8) #%%%%%%%"!%%%%%%%$ 0 0 basis S D S D G45] 0, G55] 0, ] 5 5I. (4) G 5 S D 6. (9) A convenient way of finding the dual basis to G is by stacing the coordinates of each tensor basis element as he rows of the pseudo inverse (Eq. ()) contain the column vectors in a matrix, and calculating the pseudo dual basis elements. Reshaping them into tensors inverse of this matrix G5G, gives

6 98 C.-F. Westin et al. / Medical Image Analysis 6 (00) 9 08 S D S D square root of the sum of the squared elements of the G ] 5 0, G5] 0, tensor which equals the square root of the sum of the squared eigenvalues, and the trace is calculated as the sum of the diagonal elements which equals the sum of the G ] 5 S0 0 D, G 5] S 0 4 D eigenvalues), 0 0 ]]]]]]]]]] 0 0 œ(l l ) (l l ) (l l ) G 5] ] RA5 ]]]]]]]]]]]] S 0 D, G 5 S 0 0 D. ] 5 6 Œ (l l l ) Œ ] ud ] trace(d)iu Since the dual tensor basis is defined solely by the gradient 5 ]]]]]] Œ], (7) configuration and is not dependent on the data, the trace(d) diffusion tensors can be described by a linear sum of ]]]]]]]]]] œ(l l ) (l l ) (l l ) constant tensor elements, FA5 ]]]]]]]]]]]] Œ] 6 œl l l D 5O b G, (5) ] Œ ud ] trace(d)iu 5 5 ]]]]]] Œ], (8) udu where the dual coordinates are (Eq. (4)) where I is the identity tensor. he constants are inserted to b 5 ](ln(s ) ln(s )). (6) b 0 i ensure that the measures are in the range from zero to one. In the next section we will present alternatives to these he presented method gives an analytical solution to the measures based on the geometric properties of the diffu- Stesal anner equation system, and eliminates the need sion ellipsoid. to solve this equation system for every data point. In Appendix A, the dual tensor basis corresponding to the gradient configuration in (Basser and Pierpaoli, 996) 4.. Geometrical measures of diffusion is calculated as an additional example of how the method wors. he diffusion tensor can be visualized using an ellipsoid where the principal axes correspond to the directions of the eigenvector system. Using symmetry properties of this 4. Anisotropy and macrostructural measures ellipsoid, the diffusion tensor can be decomposed into basic geometric measures (Westin et al., 997), a concept Since MRI methods, in general, obtain a macroscopic upon which we will elaborate in this section. measure of a microscopic quantity (which necessarily Let l > l > l > 0 be the eigenvalues of the symmetric diffusion tensor D, and let eˆ entails intravoxel averaging), the voxel dimensions ineigenvector corresponding to l i be the normalized fluence the measured diffusion tensor at any given location i. he tensor D can then be in the brain. Factors affecting the shape of the apparent described by diffusion tensor (shape of the diffusion ellipsoid) in the D 5 lee ˆˆ lee ˆˆ lee ˆˆ. (9) white matter include the density of fibers, the degree of myelination, the average fiber diameter and the directional Diffusion can be divided into three basic cases depending similarity of the fibers in the voxel. he geometric nature on the ran of the diffusion tensor, of the measured diffusion tensor within a voxel is thus a Linear case (l4 l. l ): diffusion is mainly in the meaningful measure of fiber tract organization. he advent direction corresponding to the largest eigenvalue, of robust diffusion tensor imaging techniques has promp- ted the development of quantitative measures for describing the diffusion anisotropy. However, to relate the mea- D. ldl 5 lee ˆˆ. (0) sure of diffusion anisotropy to the structural geometry of Planar case (l. l4 l ): diffusion is restricted to a the tissue, a mathematical description of diffusion tensors plane spanned by the two eigenvectors corresponding to and their quantification is necessary (Basser, 995). Several different measures of anisotropy have been proposed in the two largest eigenvalues, D. ldp 5 l (eˆeˆ ee ˆˆ ). () the literature. Among the most popular are two that are based on the normalized variance of the eigenvalues; the relative anisotropy (RA) and the fractional anisotropy (FA) Spherical case ( l. l. l ): isotropic diffusion, (Basser and Pierpaoli, 996). An advantage of these D. lds 5 l (eˆeˆ ee ˆˆ ee ˆˆ ). () measures is that they can be calculated without first explicitly calculating any eigenvalues. Both anisotropy In general, the diffusion tensor D will be a combination of measures can be expressed in terms of the norm and the these cases. Expanding the diffusion tensor using these trace of the diffusion tensor (the norm is calculated as the cases as a basis gives

7 C.-F. Westin et al. / Medical Image Analysis 6 (00) D 5 leˆ eˆ leˆeˆ leˆeˆ Fig. 4 contains coronal brain images depicting these geometrical measures. Alternatively the coordinates can be 5 (l l )eˆ eˆ (l l )(eˆ eˆ ee ˆˆ ) normalized with the norm of the tensor giving l (eˆeˆ ee ˆˆ ee ˆˆ ) l l 5 (l l )D (l l )D l D, cl 5 ]]]]], (7) l p s l l l œ where (l l ), (l l ) and l are the coordinates of D (l l ) in the tensor basis hd,d,d j. A similar tensor shape cp 5 ]]]]], (8) l p s analysis has shown to be useful in a number of computer œl l l vision applications (Westin and Knutsson, 99, 994). l ]]]]] he coordinates of the tensor in our new basis classify cs 5. (9) the diffusion tensor and describe how close the tensor is to œl l l the generic cases of line, plane and sphere; and hence can o ensure that the measures remain in the range from zero be used for classification of the diffusion tensor according to one, and the sum is one, the scaling factors and have to its geometry. Since the coordinates are based on the been inserted for the planar and the spherical case. A eigenvalues of the tensor, they are rotationally invariant geometrical anisotropy measure that has a behavior similar and the values do not depend on the chosen frame of to the FA measure (fractional anisotropy, Eq. (7)) is a reference. o obtain quantitive measures of the anisotropy, measure describing the deviation from the spherical case, the derived coordinates have to be normalized, which, in turn, will lead to geometric shape measures. As in the case ca5 cs5 cl c p, (40) of fractional and relative anisotropy, there are several which is the sum of the linear and planar measure. By (rotationally invariant) choices of normalization. For exnormalizing with the trace of the tensor instead of the ample the maximum diffusity, l, the trace of the tensor, / norm, the measure will be more similar to the RA measure l l l, or the norm of the tensor, (l l l ), (relative anisotropy, Eq. (7)). can be used as normalization factors. he presented measures quantitatively describe the By using the largest eigenvalues of the tensor, the geometrical shape of the diffusion tensor and therefore do following quantitive shape measures are obtained for the not depend on the absolute level of the diffusion present. linear, planar and spherical measures, However, in low signal regions, where the noise level l l dominates these shape measures, they mae little sense. In cl 5 ]], () l practice, all shape measures should be regularized by adding a constant in the denominator of size similar to that l l of the noise level. For example, the l normalized linear cp 5 ]], (4) measure (Eq. ()) would be expressed as follows: l l l l c 5 ], (5) cl 5 ]], (4) s s l s l where all measures lie in the range from zero to one, and where a suitable value for s would be the expected value their sum is equal to one, of l in a low signal region. his expression has similarities to classical Wiener filtering where the noise level s cl cp cs5. (6) has very little influence on signals larger than s, but Fig. 4. Coronal brain images showing the three geometrical measures. Note that most of the major fiber tracts are visible despite the low resolution of the data set (.7.74 mm).

8 00 C.-F. Westin et al. / Medical Image Analysis 6 (00) 9 08 penalizes signals that are smaller. When the normalization seen as an external measure since it is based on the tensors is done using the trace or norm, s should have the in a neighborhood, is to use an internal voxel based expected value of the trace or norm respectively in low measure (RA, FA, c l, p,s) onafiltered version of the signal regions. diffusion tensor field. For example, local averaging of the When applied to white matter, the linear measure, c l, tensor field with a spatial mas a (normalized so the reflects the uniformity of tract direction within a voxel. In coefficients sum to one), other words, it will be high only if the diffusion is restricted in two orthogonal directions. he anisotropy Da5O ad, (47) measure, c a, indicates the relative restriction of the diffu- sion in the most restricted direction and will emphasize describes the local average diffusion where the ran of the white matter tracts, which, within a voxel, will exhibit at average tensor Da describes the complexity of the macro- least one direction of relatively restricted diffusion. scopic diffusion structure. If the ran is close to one, the structure is highly linear, which will be the case in regions 4.. Macrostructural tensor and diffusive measures of bundles of fibers having the same direction. If the ran is two, fibers are crossing in a plane, or the underlying In the previous section, we characterized the diffusion diffusivity is planar. Applying the above geometrical linear isotropy and anisotropy within a voxel. Here, we will measure, c l (Eq. ()), to this tensor gives a measure that discuss methods for examining the pattern or distribution is high in regions with coherent tensors. of diffusion within a local image neighborhood. Basser and Fig. 5 compares the three original geometrical measures Pierpaoli proposed a scalar measure for macrostructural (top) and the same measures applied to the same tensor diffusive similarity based on tensor inner products between data averaged by a Gaussian mas (bottom). Major white the center voxel tensor and its neighbors (Basser and matter tracts such as the corpus callosum show high Pierpaoli, 996; Pierpaoli and Basser, 996) (as in the linearity in the averaged data set, indicating high macrovector case, the inner product between two tensors meaa diffusion tensor field and then deriving a scalar measure structural organization. It should be noted that averaging of sures their degree of similarity). his intervoxel scalar measure is nown as a lattice index and is defined by from the averaged field is not the same as averaging the ] ]] scalar measure derived from the original field. Œ œ,l,l LI 5O a ]]]] ]]]]]], (4) he relatively simple approach of averaging is useful SŒ] 4 D 8 œd,dl œd,dlœd,dl because the ran of the the tensors increases when lower ran, non-collinear tensors are summed. his effect is where a is a spatial mas, for example, voxels, with illustrated in Fig. 6 and compared to adding vectors which sum of coefficients equal to one, and is the anisotropic does not have the freedom to change ran. Adding the two part of the diffusion tensor D. he anisotropic part of the vectors (a) and (b) results in a new vector (c), which is of tensor has trace zero and can be written as the same order of complexity as the original vectors. 5 D ] trace(d)i, (4) However, adding two ran tensors (d) and (e), e.g. diffusion tensors from two differently oriented white 5 D ] D,Il, (44) matter tracts, results in a ran tensor (f), i.e. the output has more degrees of freedom than the input tensors and D,Il 5 D ]], (45) describes the plane in which diffusion is present. In this I,Il sense, averaging of tensors is different from averaging a vector field. he average of a set of vectors gives the where I is the identity tensor. By rewriting Eq. (4) it mean event, while the average of a set of tensors gives becomes clear that all the tensor inner products are the mean event and the range of the present events. individually normalized, Fig. 7 shows a D example illustrating the effect of ] ]] ]] ]]] ] ]], l, l, l Gaussian filtering of a diffusion tensor field. he filtered ] ]] ]]] LI 5O as D. (46) œ8 D,D l 4 D,Dl D,D l areas that contain inconsistent data give a result of almost œ œ œ round ellipses (upper right half of the image). Moreover, Since the components in the sum are normalized, small and Gaussian filtering results in more stable estimates of the large diffusion tensor components will have equal weight field directionality in the areas where there is a clear bias in determining the lattice index. Unfortunately the smaller in one direction (lower left). tensors are more influenced by noise and hence affect the he macrostructural measure achieved (by averaging the index more than is desirable. tensor field using an isotropic mas) is essentially a feature An alternative measure to the lattice index, which can be extraction method rather than a restoration method, where the latter aims at reducing the noise level in the data. Although our method does remove noise, the incorporation Corrected formula, Pierpaoli, personal communication, 997. of more advanced regularization methods (Poupon et al.,

9 C.-F. Westin et al. / Medical Image Analysis 6 (00) Fig. 5. Axial brain images showing the three geometrical measures and corresponding macrostructural diffusion measures. op: shows the geometrical measures derived from the original data. Bottom: shows the corresponding macrostructural diffusion measures: the geometrical measures derived from the tensors averaged with a 99 Gaussian ernel. 000; Parer et al., 000) should be explored, if noise al measure would limit its purpose, the description of the reduction is the main target. Anisotropic filter mass are organization within an area. If the signal is changing due to preferable since they reduce the ris of blurring edges. edges inside the local area of interest, this should be However, using an anisotropic mas for the macrostructur- reflected in the measure. Fig. 6. Vector and tensor summation. wo vectors, (a) and (b), and their sum (c). wo diffusion tensors, (d) and (e), of ran close to visualized as ellipsoids with eigenvectors forming principal axes. he summation of the two tensors gives a ran tensor (f).

10 0 C.-F. Westin et al. / Medical Image Analysis 6 (00) 9 08 Fig. 7. A D diffusion tensor field (left) and the effect of relaxation using a Gaussian filter (right). Note how the tensors in an inconsistent region become rounder, whereas in consistent areas their orientation is stabilized. 5. Visualization of diffusion tensors opacity mapping. hey compare volume renderings using opacity maps based on the indices c l, cp and c a (Eqs. Several methods have been proposed for visualizing the (7) (40)). information contained in D-MRI data. Pierpaoli et al. In Fig. 8 (left) a diffusion tensor field from an axial slice (996) renders ellipsoids to visualize diffusion data in a of the brain is shown (using the visualization method slice. Peled et al. (998) used headless arrows to represent presented in Peled et al., 998) and the filtered tensor field the in-plane component of the principal eigenvector, along (right). Prior to visualization, the tensors have been with a color coded out-of-plane component. Recently, weighted with their linear diffusion measure. Kindlmann and Weinstein (999) applied our geometric he filtered images shows the result of applying the shape indicies (Westin et al., 997) to opacity maps in macrostructural measure presented in Section 4.. First a volume rendering. hey termed this method barycentric 55 Gaussian window, a, with standard deviation Fig. 8. Left: Diffusion tensors, weighted with their linear measure c, from an axial slice of a human brain. Right: Averaged diffusion tensors using a l 55 Gaussian ernel weighted with their linear measure c, resulting in a macrostructural measure of fiber tract organization. l

11 C.-F. Westin et al. / Medical Image Analysis 6 (00) Fig. 9. Comparison of an ellipsoid and a composite shape depicting the same tensor with eigenvalues l 5, l and l equal to mm was applied to the data (Eq. (47)). Since the x 5 x avˆ, (48) out-of-plane resolution is slightly less than half the inplane resolution, there is almost no smoothing performed D 5 F(D(x )), (49) between the slices. he tensors resulting from the Gaussian filtering D a (Eq. (47)) have been weighted with their linear v 5 Dv, (50) diffusion measure, c l (Eq. (7)), respectively. he result illustrates the fact that the filtering increases the ran of the where D(x ) is the tensor in spatial position x tensors in non-structured areas since the linear measures (interpolated from the tensor field D) and F is a function are decreased in those areas. that maps the eigenvalues, As mentioned above, in D, a diffusion tensor can be visualized using an ellipsoid where the principal axes F(D) 5O f (l )eˆˆ e, (5) correspond to the tensor s eigenvector system. However, it 5 is difficult to distinguish between an edge-on, flat ellipsoid where f are scalar functions. Since the effect of the and an oblong one using the surface shading information. function F is only a weighting of the original tensor basis Similar ambiguity exists between a face-on, flat ellipsoid functions ee ˆˆ, only the eigenvalues of the tensor are and a sphere. We propose a technique for the visualization changed, and not its eigenvector system. he functions f of tensor fields that overcomes the problems with ellipcan, for example, serve as thresholding operators, which, soids. Fig. 9 compares the ellipsoidal representation of a in turn, results in decreased tensor ran when the eigentensor (left) with a composite shape of linear, planar and values are smaller than a specified threshold. spherical components (right). he components are here scaled according to the eigenvalues, but can alternatively be scaled according to the shape measures c l, cp and c s. Additionally, coloring based on the shape measures c, l cp and cs can be used for visualization of shape. Fig. 0 shows a coloring scheme where the color is interpolated ellipsoid, since a bias will be introduced by the contribu- tions from the projections on the other two eigenvectors of D. One possible remedy to this is to use a function F that between the blue linear case, the yellow planar case and the red spherical case. 6. White matter tractography When the tensor D in Eq. (50) is anisotropic, the result of the operation Dv is a vector v which is turned towards the largest eigenvector of the tensor relative to the vector v. However, the direction of this vector will not coincide with the principal axis of the diffusion sets the two smaller eigenvalues to zero ( l 5 l 5 0) creating a projection operator that projects any vector onto the orientation of the principal eigenvector, e, D-MRI provides a unique tool for investigating brain F l (D) 5 lee. (5) structures and for assessing axonal fiber connectivity in vivo. Recent wor includes (Conturo et al., 999; Jones et With this function, the iteration formula above becomes al., 999; Poupon et al., 000; Basser et al., 000). In this equivalent to following the direction of the principal section we will expand on the tractography method pre- eigenvector. he ambiguity whether to step in the direction sented in (Westin et al., 999) and present some novel e or e is avoided in this projection operator formularesults. tion. he algorithm for tracing co-linear diffusion tensors is Unfortunately, the strategy to follow the principal based on using the diffusion tensors as projection direction is inherently unstable. he main reason is that if operators. Let x0 be the initial seed point, and v0 be the l5 l, the direction of e is only defined up to a plane, seed direction, e.g. the eigenvector corresponding to the but not which direction in the plane. Fig. illustrates this largest eigenvalue. A tracing sequence hx 0,x,... j can then be obtained by the following iteration formula: problem in a D example. Collinear tensors with varying degrees of isotropy, with maximum isotropy in the middle

12 04 C.-F. Westin et al. / Medical Image Analysis 6 (00) 9 08 Fig. 0. Visualization of diffusion tensors. he tensors are color coded according to the shape: linear case is blue, planar case is yellow, and spherical case is red. he radius of the sphere is the smallest eigenvalue of the diffusion tensor; the radius of the dis is second largest; and the length of the rod is twice the largest eigenvalue. he right image shows a simulated tensor field of three crossing white matter tracts. Due to partial voluming effects, the tensors in the area where the fibers are crossing have spherical shape. where l 5 l, are shown (Fig. (a)). he line inside the ellipses shows the direction of the principal eigenvector. For the middle ellipses, all directions are potential eigenvectors, and thus the calculated direction depends only on the eigenvalue algorithm at hand, and not the data. Using Matlab from Mathwors to calculate the eigenvector results in a vertical vector (Fig. (a)). In D, the same ambiguity arises when the tensor is spherical, since all vectors are eigenvectors. Adding a noise component (Fig. (b)) introduces not only small biases on the directions of the eigenvectors, but introduces sorting errors, maing the previously minor axis becoming the major axis of the ellipse. Since the direction of the eigenvector corresponding to the largest eigenvalue is very unstable when in proximity to the generic cases of planar and spherical diffusion, an

13 C.-F. Westin et al. / Medical Image Analysis 6 (00) Fig.. Collinear diffusion tensor. alternative is to map the eigenvalues of the tensor so the resulting tensor belongs to one of the generic cases of line, plane, or sphere, or at least mae it closer to those cases. An example of this is to use the shape coordinates, the axial slice shown in Fig.. he four largest connected components were chosen, and different colors (red, blue, yellow and green) were assigned to the seeds. cld l, ifc l. c p,c s, 7. Conclusions D 5 cpd p, ifc p. c l,c s, (5) 5c D, ifc. c,c. We have proposed measures classifying diffusion ten- s s s l p sors into three generic cases based on a tensor basis his operation forces the tensor into the closest of the three expansion. When applied to white matter, the linear index generic shapes: line, plane and sphere. For example, when shows uniformity of tract direction within a voxel, while the tensor is almost spherical, the mapped tensor D in the anisotropic index quantifies the deviation from spatial Eq. (50) then will become the identity matrix (the tensor homogeneity. he non-orthogonal tensor basis chosen is ellipsoid is a sphere), and the result of the operation intuitively appealing since it is based on three simple, yet Dv (Eq. (50)) is v, i.e., the output vector v 5 v. In descriptive, geometrically meaningful cases. our experiments, we have used the mapping in Eq. (5). We have presented a new method for calculating the Although the tracing results are very promising, clearly, diffusion tensors from the diffusion gradient raw data. he the question of how to best map the eigenvalues should be method is based on an analytic solution of the Stejsal investigated further. anner formula eliminating the need to solve the Stejsal Fig. shows the result of tracing three crossing fiber anner equation system for each voxel of the data set. We tracts. A 555 Gaussian operator was used to define the have also described how tensor diffusion data can be macroscopic organization. Each trace line is composed of processed without reverting to the use of only scalar the combined results from both seed directions e and measures of the tensor data. By staying in the tensor e. domain, macroscopical features can be derived using Fig. shows the result of three-dimensional tracto- relatively simple methods such as averaging the tensor graphy of a normal subject showing the anterior (yellow) field and classifying the change of diffusion geometry. We and posterior (blue) part of the corpus callosum as well as discussed the geometric addition of tensors and argued that the left and right (red and green) cortico-spinal tract. he adding tensors and vectors differ in that tensor summation tracts pass through an axial section of the later ventricles. gives more than the mean event due to more degrees of he path was computed using the max-shape mapping freedom. By using the geometric diffusion measures on function (Eq. (5)) in the iteration formula (Eq. (48)). he locally averaged tensors, local directionality consistency seeds for the tractography were automatically determined can be determined (e.g. existence of larger fiber tracts). We by applying a connectivity algorithm on the linear measure are confident this averaging approach can be used to derive c above a threshold. he connectivity was calculated in a tensor field that describes macrostructural features in the l

14 06 C.-F. Westin et al. / Medical Image Analysis 6 (00) 9 08 Fig.. he image shows the result of automatic tracing of collinear tensors in the three crossing tracts data above (trace line in red, yellow and green). he data was first filtered with a 555 Gaussian operator to remove noise, followed by the max shape operator to facilitate the tracing. en points were randomly seeded at one end of each of the three branches. Note how the stream lines tunnel through the area in the center where the information of directionality is uncertain. tensor diffusion data. he linear measure cl derived from Pierpaoli, 996) is presented. hese gradients are the averaged tensor field can, for example, be used for quantitative evaluation of fiber tract organization. One 0 limitation with this method is that if a large mas is g5 ] S 0 D, g5 ] S 0 D, g5 Œ Œ Œ] SD, chosen, information about medium to small fiber bundles 0 will be lost. Care has to be taen when choosing the mas. g45, g 5, g 5 Œ] S D 5 Œ] SD 6 Œ] S D, We also have described how mapped diffusion tensors can 0 0 be used for tracing white matter tracts in the human brain. he mapping alleviates the need to blindly follow the (A.) direction of maximum diffusion which is inherently ungiving the following outer products of these gradients: stable in regions of planar or spherical tensors. 0 0 G5] S0 0 0 D, G5] S D, Acnowledgements his wor was funded, in part, by NIH grants P4- G 5] S0 D, (A.) RR8, R0-RR747, P0-CA6765-0, R0-0 NS95-0A, and the Whitaer Foundation G4 5] S0 D, Appendix A G55] S 0 D, G65] S 0 D. (A.) In Section. we derived a compact solution to the Stejsal anner equation system (Eq. (6)) using concepts A convenient way of finding the dual basis is by from tensor analysis, and presented explicit numbers for stacing the coordinates of each basis element as a column our specific gradient configuration. In this section, the vector in a matrix, and calculating the pseudo inverse of result from using the gradient configuration in (Basser and this matrix G 5 G,

15 C.-F. Westin et al. / Medical Image Analysis 6 (00) Fig.. hree-dimensional tractography of a normal subject showing the anterior (yellow) and posterior (blue) part of the corpus callosum as well as the left and right (green and red) cortico-spinal tract. he tracts pass through an axial section of the later ventricles. op and bottom images present two different views of the tractography result.

16 08 C.-F. Westin et al. / Medical Image Analysis 6 (00) 9 08 u u??? u F., 996. Line scan diffusion imaging. Magn. Reson. Med. 6, 509 G G??? G Jones, D.K., Simmons, A., Williams, S.C.R., Horsfield, M.A., 999. u u??? u Non-invasive assessment of axonal fiber connectivity in the human brain via diffusion tensor MRI. Magn. Reson. Med. 4, 7 4. Kendall, D.E.B.P., 977. Vector Analysis and Cartesian ensors. Van Nostrand Reinhold, UK Kindlmann, G., Weinstein, D., 999. Hue balls and lit-tensors for direct volume rendering of diffusion tensor fields. In: IEEE Visualization ] 0 0 (A.4) 999, VIS999, Salt Lae City, U LeBihan, D., Breton, E., Lallemand, D., Grenier, P., Cabanis, E., Laval Jeantet, M., 986. MR imaging of intravoxel incoherent motions: application to diffusion and perfusion in neurologic disorders. Radiology 6, Maier, S., Gudbjartsson, H., 998. Line scan diffusion imaging. USA patent [5, Maier, S., Gudbjartsson, H., Patz, S., Hsu, L., Lovblad, K.-O., Edelman, Roentgenol R., Warach, S., Jolesz, F., 998. Line scan diffusion imaging: 5 ] characterization in healthy patients and stroe patients. Am. J (), Moseley, M.E., Cohen, Y., Kucharczy, J., Mintorovitch, J., Asgari, H.S., Wendland, M.F., suruda, J., Norman, D., 990. Diffusion-weighted (A.5) MR imaging of anisotropic water diffusion in the central nervous system. Radiology 76, Parer, G.J.M., Schnabel, J.A., Symms, M.R., Werring, D.J., Barer, G.J., he rows of the pseudo inverse (Eq. (A.5)) contain the 000. Nonlinear smoothing for reduction of systematic and random dual basis elements. Reshaping them into tensors errors in diffusion tensor imaging. J. Magn. Reson. Imaging, gives Peled, S., Gudbjartsson, H., Westin, C.-F., Kiinis, R., Jolesz, F., 998. Magnetic resonance imaging shows orientation and asymmetry of G5] S0 0 D, G5] S 0 0 D, white matter tracts. Brain Res. 780 (), Pierpaoli, C., Basser, P.J., 996. oward a quantitative assessment of diffusion anisotropy. Magn. Reson. Med. 6, G ] 5 S 0 D, G45] S 0 D, Pierpaoli, C., Jezzard, P., Basser, P.J., Barnett, A., Chiro, G.D., Diffusion tensor MR imaging of the human brain. Radiology 0, 67. Poupon, C., Clar, C.A., Frouin, F., Regis, J., Bloch, I., Bihan, D.L., 0 0 Bloch, I., Mangin, J.-F., 000. Regularization of diffusion-based G55] S 0 D, G65] S 0 D. direction maps for the tracing of brain white matter fascicles NeuroImage, Stejsal, E.O., anner, J.E., 965. Spin diffusion measurements: spin echoes in the presence of a time-dependent field gradient. J. Chem. References Phys. 4, Stoer, J.J., 989. Differential Geometry. Wiley, New Yor. Basser, P., 995. Inferring microstructural features and the physiological urner, R., le Bihan, D., Maier, J., Vavre, R., Hedges, L.K., Pear, J., state of tissues from diffusion-weighted images. NMR Biomed. 8, 990. Echo planar imaging of intravoxel incoherent motions. Radiolo- 44. gy 77, Basser, P., Pierpaoli, C., 996. Microstructural and physiological features Westin, C.-F., 994. A tensor framewor for multidimensional signal of tissues elucidated by quantitative-diffusion-tensor MRI. J. Magn. processing. Ph.D. thesis, Linoping University, Sweden (dissertation Reson. Ser. B, No. 48). Basser, P., Pajevic, S., Pierpaoli, C., Duda, J., Aldroubi, A., 000. In vivo Westin, C.-F., Knutsson, H., 99. Extraction of local symmetries using fiber tractography using D-MRI data. Magn. Reson. Med. 44, 65 tensor field filtering. In: Proceedings of the nd Singapore Internation- 6. al Conference on Image Processing. IEEE, Singapore. Beaulieu, C., Allen, P., 994. Determinants of anisotropic water diffusion Westin, C.-F., Knutsson, H., 994. Estimation of motion vector fields in nerves. Magn. Reson. Med., using tensor field filtering. In: Proceedings of IEEE International Chenevert,., Brunberg, J., Pipe, J., 990. Anisotropic diffusion in Conference on Image Processing. IEEE, Austin, X, pp human white matter: demonstration with MR techniques in vivo. Westin, C.-F., Maier, S., Khidhir, B., Everett, P., Jolesz, F., Kiinis, R., Radiology 77, Image processing for diffusion tensor magnetic resonance Conturo,.E., Lori, N.F., Cull,.S., Abuda, E., Snyder, A.Z., Shimony, imaging. In: Medical Image Computing and Computer-Assisted Inter- J.S., McKinstry, R.C., Burton, H., Raichle, A.E., 999. racing vention. Lecture Notes in Computer Science, pp neuronal fiber pathways in the living human brain. Neurobiology 96, Westin, C.-F., Peled, S., Gudbjartsson, H., Kiinis, R., Jolesz, F., Geometrical diffusion measures for MRI from tensor basis analysis. Einstein, A., 905. Uber die von der moleularinetischen theorie der In: ISMRM 97, Vancouver, Canada, p. 74. warme geforderte bewegung von in ruhenden flussigeiten suspend- Wimberger, D.M., Roberts,.P., Barovich, A.J., Prayer, L.M., Moseley, ierten teilchen. Ann. Phys. 7, M.E., Kucharczy, J., 995. Identification of premyelination by Fic, A., 855. Uber diffusion. Ann. Phys., diffusion-weighted MRI. J. Comp. Assist. omogr. 9 (), 8. Fourier, J., 8. heorie analytique de la chaleur. Academie des Young, E.C., 978. Vector and ensor Analysis. Deer. Sciences. Gudbjartsson, H., Maier, S., Mulern, R., Morocz, I.A., Patz, S., Jolesz,

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