(Radiology. 1999;212:770-784.)
© RSNA, 1999
Quantitative Diffusion-Tensor Anisotropy Brain MR Imaging: Normative Human Data and Anatomic Analysis1
Joshua S. Shimony, MD, PhD,
Robert C. McKinstry, MD, PhD,
Erbil Akbudak, PhD,
Joseph A. Aronovitz, MD, PhD,
Abraham Z. Snyder, PhD, MD,
Nicolas F. Lori, BS,
Thomas S. Cull, PhD and
Thomas E. Conturo, MD, PhD
1 From the Mallinckrodt Institute of Radiology (J.S.S., R.C.M., E.A., J.A.A., A.Z.S., T.S.C.) and Dept of Physics (N.F.L., T.E.C.), Washington University Medical Center, 510 S Kingshighway Blvd, St Louis, MO 63110. Received Jul 20, 1998; revision requested Aug 19; revision received Nov 10; accepted Jan 15, 1999. Supported in part by grants from Major Grants Program of McDonnell Center for Higher Brain Function, National Multiple Sclerosis Society Pilot Research Award, Charles A. Dana Foundation Consortium on Neuroimaging Leadership Training, and PRAXIS III Fellowship from the Portuguese government. J.S.S. supported by RSNA Research and Education Foundation Research Resident Grant. Address reprint requests to J.S.S. (e-mail: shimonyj@npg.wustl.edu).
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Abstract
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PURPOSE: To obtain normative human cerebral data and evaluate the anatomtomic information in quantitative diffusion anisotropy magnetic resonance (MR) imaging.
MATERIALS AND METHODS: Quantitative diffusion anisotropy MR images were obtained in 13 healthy adults by using single-shot echo-planar MR imaging and a combination of tetrahedral and orthogonal gradient encoding (whole-brain coverage in about 1 minute). White matter (WM) anatomy was assessed at visual inspection, and values were measured in various brain regions. Different anisotropy measures, including total anisotropy (A
), were compared on the basis of information content, rotational invariance, and susceptibility to noise. Partial volume and noise effects were simulated.
RESULTS: Anisotropy MR images depicted WM features not typically seen on conventional MR images (eg, external capsule, thalamic substructures, basal ganglia, occipital WM, thickness of the internal capsule). Statistically significant anisotropy differences occurred across brain regions, which were reproducible within and across subjects. A was highest in commissural WM and progressively lower in projection and association WM. This order paralleled that of known resistance to spread of vasogenic edema, which suggested that anisotropy may be sensitive to WM histologic structure. Gray matter (GM) A data were consistent with zero anisotropy, and partial volume WM-GM effects were approximately linear. A image quality could be effectively improved by means of averaging.
CONCLUSION: Quantitative diffusion anisotropy images can be obtained rapidly and demonstrate subtle WM anatomy. Different histologic types of WM have significant and reproducible anisotropy differences.
Index terms: Brain, diffusion, 10.99 • Brain, gray matter, 10.92 • Brain, MR, 10.121411, 10.121416, 10.12144 • Brain, white matter, 10.92 • Magnetic resonance (MR), diffusion study, 10.12144
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Introduction
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Random diffusive motion of water along the direction of a strong field gradient will cause spin dephasing in magnetic resonance (MR). As shown by Stejskal and Tanner (1), quantitative diffusion coefficients can be measured with MR by encoding diffusion with balanced gradients applied before and after a spin-echo refocusing pulse and comparing this signal loss to a reference signal acquired with no diffusion encoding. The amount of diffusion sensitivity is indicated by the b factor, which is dependent on the timing and strength of the diffusion-encoding gradients.
MR imaging of diffusion effects with a combination of spin-echo imaging and Stejskal-Tanner gradients was used by Le Bihan and colleagues (2,3) and by Taylor and Bushell (4). The technique of echo-planar MR imaging (5) has been particularly useful for acquisition of diffusion information, as demonstrated by Avram and Crooks (6), Turner and colleagues (7,8), and McKinstry et al (9).
In early human diffusion MR imaging studies (10,11), it was realized that water diffusion is anisotropic in many biologic media such as white matter (WM) and muscle (ie, the rate of water diffusion varies with direction). Anisotropic diffusion can, in general, be represented by a symmetric 3 x 3 diffusion tensor D at each position in space (12,13) and can be modeled as ellipsoidal water displacements (12), as was suggested by Basser et al (14) for modeling of MR image data. Anisotropy is a quantitative parameter that represents the degree to which diffusion varies in different directions. The authors of more recent studies (15,16) have demonstrated the benefit of removal of anisotropic diffusion effects by creating a directionally averaged diffusivity. However, anisotropic diffusion effects provide anatomic information in regions of WM (11,17–19) and may provide diagnostic information for certain WM diseases (20), including multiple sclerosis (21–23). Unlike averaged diffusivity, measures of anisotropy in humans have been shown to be different in various regions of the normal adult brain (17,19) and to vary with gestational age in neonates (24) and with postnatal age in the first several months of life (25).
A number of anisotropy measures have been proposed in the literature (16,26–29). The theoretic basis and advantages of several of these will be discussed. In our study, quantitative anisotropy images were computed from echo-planar MR imaging data (30) that sampled the diffusion tensor along four tetrahedral directions (26) and three coordinate axes. Anatomic findings and anisotropy values are presented from different regions of interest (ROIs) in the normal human brain. The purpose of our study was to obtain normative human cerebral data and evaluate the anatomic information in quantitative diffusion anisotropy MR imaging.
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MATERIALS AND METHODS
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Theory
In the general case of homogeneous anisotropic media, the diffusion tensor can be represented by a 3 x 3 matrix. For Gaussian diffusion, the matrix is symmetric (12) and is completely characterized by six scalars: three diagonal elements, Dxx, Dyy, Dzz, and three off-diagonal elements, Dxy, Dxz, Dyz. An ellipsoid, whose surface represents the root mean square diffusive displacement, provides a convenient pictorial representation of anisotropic diffusion in a voxel (14,26,27). Measurement of the diffusion tensor is then equivalent to sampling of enough discrete points on the ellipsoid surface so that the ellipsoid can be uniquely defined (31). Individual elements of the diffusion tensor are not rotationally invariant and, as markers of brain pathologic conditions, have been shown to be misleading in comparison with invariant measures (15,29). In contrast, the trace of the diffusion tensor is invariant under rotations of the coordinate system and is the basis for the directionally averaged diffusivity 
= (Dxx + Dyy + Dzz)/3.
Each voxel has a unique principal ellipsoid coordinate system (x', y', z') that, in general, is rotated with respect to the laboratory coordinate system (27). This principal coordinate system lies along the major and minor axes of the diffusion ellipsoid, and, in this rotated system, the diffusion tensor is diagonal. The diagonal matrix elements are the principal diffusivities of the rotated coordinate system 
x', y', and z' and are the eigenvalues of the diffusion tensor. In the laboratory, the six scalars that represent the diffusion tensor can be expressed in terms of the three eigenvalues (x', y', and z') that are rotationally invariant and the three Euler angles (
, 
, 
) that describe the relative rotation between the principal and laboratory coordinates (26).
Multiple measures of tissue anisotropy have been proposed (16,26–29). An ideal measure of anisotropy should be quantitative and rotationally invariant. It is desirable that it be a function of the directly measured diffusion coefficients, because any diagonalization of the diffusion tensor and evaluation of its eigenvalues may introduce additional noise. Furthermore, it is advantageous to use anisotropy calculations that are not dependent on sorting of the eigenvalues according to size, because this can introduce additional uncertainty (29), especially in regions with low anisotropy or a low signal-to-noise ratio (SNR). Finally, to facilitate comparison across sequences, institutions, and clinical and basic science disciplines, it would be preferable if anisotropy measures were on an absolute anisotropy scale.
The natural choices for anisotropy measures are Amajor and Aminor (26), which describe the variation in ellipsoid shape along the major and minor axes, respectively. These are obtained by decomposing the diffusion tensor into isotropic and anisotropic components in the x', y', z' ellipsoid coordinate system (26):
and
where 
is the average of the eigenvalues (equivalent to averaged diffusivity ), z' is the axis of greatest symmetry, and x' is greater than or equal to y'. The parameter Amajor ranges from -0.5 to 1.0, where the sign contains shape information. An Amajor value that is greater than 0 indicates a prolate ("cigar-shaped") diffusion ellipsoid where z' is the largest eigenvalue. An Amajor value that is less than 0 indicates an oblate ("pancake-shaped") diffusion ellipsoid where z' is the smallest eigenvalue. These measures may be useful but require that one solve for and sort the eigenvalues.
An alternative anisotropy measure is a coefficient of variation determined on the basis of the second moment (variance) of D (26):
with as already given. A represents total anisotropy, irrespective of ellipsoid shape; it is on an absolute anisotropy scale and has a value between 0 and 1. A is an invariant quantitative measure of anisotropy that does not require diagonalization of the diffusion tensor and is independent of the order of the eigenvalues. A was derived to correspond in range and meaning to the spectroscopic definition of anisotropy used in the physical sciences (32). This is most easily seen in the axisymmetric approximation (26), where Aminor is 0, A is the absolute value of Amajor and
with D|| = z' and D
= x' = y'. The parameter A is similar to the relative anisotropy defined by Basser and Pierpaoli (28,29) except for a scaling factor of 2-1/2 contained in Equation 3 (also see factor of 2-1/2 in Eq [A1] in the Appendix) that places A on an absolute anisotropy scale. A also can be written in terms of Amajor and Aminor (see Appendix, Eq [A2]).
The diffusion tensor is completely determined by measuring its six scalar components. This can also be viewed as sampling of sufficient points on the surface of the diffusion ellipsoid to constrain the ellipsoid size (ie, averaged diffusivity), shape (ie, Amajor, Aminor), and orientation (ie, , , ). To reduce the effects of noise on the measurements, the diffusion ellipsoid should be sampled with strong gradients at points widely spaced over its surface while avoiding sampling of points related by inversion symmetry.
The tetrahedral gradient-encoding method (26) uses maximal simultaneous application of all three orthogonal gradients to construct each of the four noncolinear tetrahedral gradient vectors. This method has an inherently high SNR for measurements of averaged diffusivity (18,30), because the tetrahedral gradients have a strength that is 31/2 times larger than the orthogonal gradient-encoding method. Accordingly, the tetrahedral method also is expected to have favorable noise properties for A measurement, particularly because of the wide uniform sampling of coordinate space (31). Linear combinations of the tetrahedral diffusion measurements D1, D2, D3, and D4 are used to evaluate the off-diagonal elements of the diffusion tensor (26). In cases of axisymmetric diffusion (Aminor = 0), these measurements describe the entire diffusion tensor. In more general cases, the individual diagonal elements of the diffusion tensor must be determined by means of additional sampling along other directions, such as the orthogonal x, y, and z magnet coordinate system. Combined tetrahedral-orthogonal encoding was chosen because the tetrahedral and orthogonal gradient vectors are complementary in terms of sampling of the diffusion tensor, since each set of vectors bisects the others to achieve wide coverage of the ellipsoid surface.
The diagonal element information is contained in the orthogonal gradient-encoded data, whereas the off-diagonal information is contained in the tetrahedral gradient-encoded data. Accordingly, A as expressed in Equation (3) can be decomposed into two partial, noninvariant, anisotropy-weighted measures AWortho and AWtet, which are dependent, respectively, on the orthogonal and tetrahedral measurements (see Appendix, Eqq [A4,A5]). The fractional amount of anisotropy contained in each of these measures can be expressed as ftet = AWtet/A and fortho = AWortho/A (see Appendix, Eqq [A7,A8]).
Data Acquisition
Approval for this study was obtained from the human studies committee at our institution. Quantitative diffusion-tensor MR imaging was performed in 13 neurologically healthy adults (11 men, two women; mean age, 31 years; age range, 23–47 years) recruited from the population at our institution. Informed consent was obtained from all subjects after the nature of the experiment was fully explained.
All examinations were performed with a 1.5-T system (Magnetom Vision; Siemens Medical Systems, Erlangen, Germany) equipped with a standard, circularly polarized clinical radio-frequency head coil. Custom single-shot spin-echo echo-planar MR pulse sequences with Stejskal-Tanner gradients were used. In each subject, diffusion-tensor information was collected by using a combination of tetrahedral (26) and orthogonal gradient-encoded diffusion-weighted images, together with reference signal intensity data. This tetrahedral-orthogonal configuration of diffusion-encoding directions was chosen because it was found to provide a high SNR for both averaged diffusivity and A (simulations not shown). This configuration contains tetrahedral gradient vectors that are of maximal strength and spatial separation and three orthogonal vectors that bisect the tetrahedral vectors to provide sufficient measurements for overdetermined estimation of the full tensor, with wide angular coverage. Both angular coverage and gradient strength are important in anisotropy measurement (31). Other configurations for tensor measurement have been used, such as a three-dimensional hexagonal array (17,33,34), where all diffusion-weighted images can have the same diffusion-encoding strength (b factor) and echo time but where gradient strengths are weaker than those of tetrahedral encoding.
Quantitative diffusion-tensor data were acquired first under conditions optimized for accuracy for reporting of normative values. For this purpose, a peripherally gated single-section echo-planar MR imaging sequence (sequence A) was implemented with a nonselective 180° pulse and no cross terms. Axial MR images were acquired at the level of the basal ganglia in a subset of the subjects (nine men, two women; mean age 31 years; age range, 23–47 years). Each acquisition was repeated once under the same conditions to provide two identical data sets for noise calculation (35) and statistical testing (described later). Sequence A was used for measurement of all normative values reported herein.
To assess the performance of quantitative diffusion-tensor and anisotropy MR imaging under more clinically practical conditions, axial MR images were then acquired in six of the subjects (five men, one woman; mean age, 31 years; age range, 28–39 years) by using sequence B, a nongated multisection version of sequence A. Four of the six subjects (three men, one woman; mean age, 29 years; age range, 23–39 years) in the sequence B group underwent imaging with tetrahedral-only encoding and were also in the sequence A group. These data were used for statistical comparison between the single-section (sequence A) and multisection (sequence B) results. For the remaining two subjects (two men, aged 32 and 39 years) in the sequence B group, combined tetrahedral-orthogonal, whole-brain, multisection encoding was used. In one of these subjects, sequence B was used with additional repetitions to assess the effect of averaging of repeated image acquisitions.
For both sequences, the compensating lobes of the readout and phase-encoding gradients were applied after the diffusion-encoding gradients, and the section-selective gradient for the 90° radio-frequency pulse was refocused immediately to prevent cross terms between the diffusion and imaging gradients (31).
For sequence A, cross terms between the diffusion gradients and the imaging section–selective gradients were reduced further by using nonselective refocusing with a composite (one-two-one) 180° radio-frequency pulse. Thus, for sequence A, all phase shifts induced by imaging gradients were zero when the diffusion gradients were applied, and all cross terms with the diffusion gradients were zero. The composite nonselective 180° pulse was chosen to increase the uniformity of the 180° radio-frequency pulse, which potentially reduces the effect of radio-frequency inhomogeneities on diffusion encoding.
For sequence B, the only nonzero cross term was between the section-selective and the diffusion-encoding gradients, and this cross term (
12) contributed a diffusion coefficient error of less than 0.5% (31). For both sequences, navigator echo gradient pulses were present for all acquisitions, but the navigator echo of the reference image was used to correct all the diffusion-encoded acquisitions, because this procedure was found to reduce the severity of artifacts. The effect from the navigator echo gradient pulses and the readout and phase-encoding gradient pulses occurring during the echo-planar readout can contribute additional self and cross terms, but these are negligible in clinical echo-planar MR imaging (36).
When possible, diffusion-weighted MR images were acquired with a b value of approximately 1,000 sec/mm2 in each encoding direction in conjunction with image data with a b value of approximately 0 sec/mm2, to provide a high SNR per unit time (31,37). For the sequence A tetrahedral acquisitions, the input Cartesian gradients were 20.0 mT/m, and the echo time was 106 msec with a b value of 1,022 sec/mm2 (diffusion-encoding durations of 17.0 msec and a time 
between gradient onsets of 46.85 msec). For the orthogonal acquisitions, the echo time and the diffusion-encoding timings and were lengthened to an echo time of 121 msec, of 27 msec, and of 56.85 msec, to yield a b value of 999 sec/mm2.
Data for each of the seven encoding directions were acquired with a separate pulse sequence, each with its own reference acquisition. Each reference acquisition had weak diffusion encoding (b = 12 sec/mm2) along the same direction as the strong diffusion encoding, to spoil residual spurious free induction decay signals arising from imperfections in the nonselective 180° radio-frequency pulse. For sequence B, all diffusion-weighted images in the tetrahedral-only (four subjects) and tetrahedral-orthogonal (two subjects) cases were acquired with a combined pulse sequence that included one reference acquisition (b = 0 sec/mm2). Weak diffusion encoding in the reference acquisition was found to be unnecessary in sequence B, because spurious free induction decay signals were reduced by means of the volume selectivity of the 180° radio-frequency pulse and the spoiling caused by the 180° section-selective gradient. For the tetrahedral-only sequence B case, the echo time, input gradient strength, and diffusion-weighted b factors were the same as those for sequence A. For the tetrahedral-orthogonal sequence B case, the input diffusion-encoding gradient strength was increased to 22.0 mT/m, and the echo time was reduced to 97 msec, which yielded a b value of 1,003.3 sec/mm2 for tetrahedral encoding ( = 15.75 msec, = 44.80 msec). For the orthogonal encodings in this sequence, echo time and diffusion timings were the same as those in the tetrahedral encodings, with a b value of 334.3 sec/mm2. The same timings were chosen for pulse sequence simplicity. Although this sequence has different diffusion-weighted b factors for tetrahedral and orthogonal encoding, this was not expected to contribute systematic error, on the basis of results from an initial study (not shown) that established that the same averaged diffusivity was measured with orthogonal encoding at b values of approximately 300 and 1,000 sec/mm2.
In addition, for typical b factors and diffusion times used in clinical imaging, different groups (38,39) have found a lack of multiexponential dependence of brain signal intensity on the b factor, as well as a lack of explicit dependence of the brain diffusion coefficient on diffusion time at a constant b factor. Although multiple intravoxel tissue components can potentially lead to multiexponential dependence on the b factor, these results suggest that this effect is small, due possibly to the similarity in averaged diffusivity between gray matter (GM) and WM. Intravoxel averaging of multiple anisotropic components could lead to complex effects on diffusion-weighted images that are not completely described with one tensor. Such partial volume effects were considered in simulations to be described subsequently.
To limit the effects of physiologic motion (eg, cerebrospinal fluid pulsatility), sequence A was gated so that the 90° pulse was applied 500 msec after the peak signal from the peripheral pulse oximeter, which corresponds to brain diastole (30,40). To reduce effects from heart rate variability in the presence of partial saturation of cerebrospinal fluid, the pulse sequence repetition time was set to a minimum of 8,000 msec. One warm-up step was programmed to ensure that cerebrospinal fluid signals were at steady state. The overall acquisition time was thus approximately 26 seconds to encode each direction with a small and a large b factor (approximately 3 minutes for one acquisition of all seven encoding directions).
Sequence B was performed with a repetition time of 3,050–3,500 msec without gating and with three warm-up steps. A 15-section axial set of tetrahedral-orthogonal brain data, including warm-up steps, was acquired in 33 seconds. This sequence was performed twice with gaps that were 100% of the section thickness, with the second acquisition offset by a section thickness. Thus, contiguous whole brain data were collected during 66 seconds, for an overall acquisition time of approximately 1
minutes, including setup and tuning.
For qualitative assessment of the effectiveness of averaging acquisitions, three interleaved multisection data sets were acquired with 200% gaps in one subject, and these acquisitions were repeated to produce a total of 10 thin-section whole-brain data sets for averaging (total acquisition time, 16 minutes 40 seconds). For both sequences A and B, a sinusoidal readout gradient, a constant phase-encoding gradient, and linear time sampling were used to yield a raw data matrix of 96 x 200, which was interpolated to a rectilinear k-space matrix of 96 x 128 by using a gridding procedure. The field of view ranged from 180 x 240 mm to 210 x 280 mm (in-plane voxel size range, 1.88 x 1.88 mm to 2.19 x 2.19 mm). The section thickness was 5.0 mm in all cases except the triple-interleaved acquisition for sequence B, where the section thickness was 3.3 mm.
Data Analysis
Images were sinc interpolated to a 192 x 256 image matrix (0.94 x 0.94 x 5.00 mm pixels in sequence A) to facilitate selection of ROIs. In some cases, the rectilinear matrix produced after k-space gridding was zero filled prior to reconstruction; in the other cases, the interpolation was performed after reconstruction.
All images were realigned in two dimensions to the reference images to correct for two-dimensional displacements and linear distortions (ie, stretch and shear) caused by eddy currents. These linear distortions have been found to constitute the major eddy-current effects in echo-planar diffusion MR imaging (41), which also was supported by cine visual inspection of the realigned images in the present study. The realignment algorithm used a combination of intramodality (42) and cross-modality affine realignment procedures; the cross-modality procedure was used for realignment of image pairs with substantial contrast differences. We routinely use these intra- and cross-modality realignment procedures for functional MR imaging (43). Because all realignments were to the reference echo-planar MR image, the realignment was specifically targeted to low-order eddy-current–induced distortions, which must be corrected to bring the reference and diffusion-weighted images into register for tensor computation (described subsequently). Higher-order distortions also were typically present on the echo-planar MR images (in comparison with those on anatomic spin-echo MR images), but these need not necessarily be corrected for the diffusion computation because the distortions are constant on all diffusion-weighted and reference images. Eddy-current effects also were specifically targeted by performing the realignments in two dimensions, because the principal eddy-current effects are in plane. The realignment procedure additionally corrects for in-plane subject motion, although typical in-plane and through-plane motion effects are estimated to be very small in comparison with the eddy-current effects at these acquisition times.
Tetrahedral and orthogonal diffusion-weighted images were obtained. The intensities are functions of the b factors and the elements of the diffusion tensor, weighted by the direction of the diffusion gradient used (27,31). The logarithm of these intensities was computed to create a set of linear equations that were solved by using standard weighted least-squares techniques (13,14,44). The six diffusion tensor elements were determined with global analysis (see Appendix) of all eight tetrahedral-orthogonal log intensities, and averaged diffusivity and A were then computed directly from the diffusion-tensor elements according to Equation (3) (see Appendix, Eq [A1]). Images were computed on a pixel-by-pixel basis, without spatial filtering.
From the estimated diffusion tensor, the principal axes and eigenvalues were then computed by using standard matrix procedures. The eigenvalues were sorted according to size and symmetry (z' > x' > y' for the prolate case, x' > y' > z' for the oblate case), and Amajor and Aminor were calculated by using Equations (1,2). In addition, by using only the tetrahedral or orthogonal data, anisotropy-weighted images were calculated (see Appendix, Eqq [A4,A5]) by means of direct computation of D1, D2, D3, and D4, or Dxx, Dyy, and Dzz, from log intensity ratios. In the one sequence B case where the acquisition was repeated multiple times for the purpose of averaging, the diffusion-weighted images and reference image were arithmetically averaged prior to the log intensity computation. Averaging was performed at this stage, as opposed to averaging of computed A images, to minimize anisotropy bias caused by input noise (discussed subsequently). The fraction of total anisotropy contained in the anisotropy-weighted measures also was computed for arbitrary orientations and was displayed with MATLAB software (MathWorks, Natick, Mass) by using Equations (A7,A8) in the Appendix.
Elliptical ROIs were positioned on each subject data set by using interactive software (ANALYZE; Mayo Foundation, Rochester, Minn) for evaluation of anisotropy in different anatomic areas of the brain. The ROI calculations were performed by using a custom program that partially weights edge voxels according to the fraction of the voxel that lies inside the ROI, as determined with analytic ellipse expressions. This procedure was performed instead of more standard routines that sample only voxels that are completely inside the ROI. This procedure is equivalent to obtaining ROI measurements on highly interpolated images (but without large memory or disk requirements).
Normative data were measured from ROIs of 12 brain regions in the 11 subjects examined with sequence A. The ROIs were placed in equivalent structures in each hemisphere for averaging and analysis of hemispheric differences. In each subject, the ROIs were initially located by using the T2-weighted reference images (echo time = 106 msec, repetition time > 8,000 msec), and ROI sizes and positions were refined by means of overlay onto the A images. Mean ROI values were measured for all 12 regions in each hemisphere, for each of the two acquisitions, and for each subject. Statistical variations across these factors were assessed with the two-tailed Student t test of the mean ROI values. The random (thermal) noise properties of images also were estimated in individual subjects by subtracting the two images acquired under identical experimental conditions (35). The thermal noise of a single acquisition was taken as 2-1/2 times the single-pixel SD measured from the subtraction image by using the ROIs. The SNR was estimated as the mean of the two acquisitions divided by the thermal noise value. The subtraction images were also inspected visually for artifacts, especially in the locations of the ROIs. No ROI positions were changed, and all data were included in the measurements and statistical analyses.
Computer Simulations
Computer simulations were performed to evaluate the effects of noise and partial volume averaging on the measurement of anisotropy. In a noise-free isotropic region, the true A is 0. In the presence of noise, the measured A is greater than 0 even in isotropic regions, because, according to Equation (3), noise causes variance in the diffusion measurements that elevates A. This low-level anisotropy is expected to be strongly dependent on the SNR of the measurement.
To evaluate the baseline isotropic level of A at different noise levels, a Monte Carlo program was written that generated voxel intensity data for an idealized tissue having the same averaged diffusivity as the brain ROI under consideration but with the assumption of zero anisotropy (A = 0). The signal intensities that would be observed in isotropic tissues in a tetrahedral-orthogonal diffusion experiment were simulated by calculating the ideal signal intensity and adding Gaussian random noise. The ideal signal intensity was calculated by assuming (a) that b equalled 1,000 sec/mm2 for all seven diffusion-weighted images and (b) that all diffusion-weighted images and the reference T2-weighted image had the same echo time. Gaussian noise was added to real and imaginary channels at various reference image SNR levels. A constant level of uncorrelated noise was assumed for the diffusion-weighted and reference images. The A value was calculated from these simulated intensities by using Equation (3), and A statistics were determined from 16,000 sets of simulated tetrahedral-orthogonal diffusion experiments. The mean and SD of the computed A values were compared with the experimentally determined, average ROI A value within subjects.
The second simulation was performed to evaluate the effect of partial volume averaging on the measurement of A. This is especially important in transition zones between regions of high anisotropy and those of low anisotropy, such as the borders between GM and WM and in areas of WM with fibers that cross in different directions. Anisotropy measurements in voxels located at GM-WM interfaces were simulated by assuming a mixture of idealized WM and GM tissue components on the basis of ROI measurements in homogeneous areas of GM and WM. The WM tissue component was modeled to have the same and A values as the splenium of the corpus callosum ( = 0.72 x 10-3 mm2/sec, A = 0.5), but with zero minor-axis anisotropy (Aminor = 0) to represent an idealized axisymmetric case imaged at high spatial resolution. The GM component was chosen to be representative of the putamen ( = 0.72 x 10-3 mm2/sec), but with zero anisotropy (A = 0). Noise-free signal intensities were calculated separately for GM and WM and were combined at various partial volume weightings, from which A was calculated by using Equation (3).
In the third simulation, we evaluated the case of WM regions with crossing fibers. Anisotropy measurements were simulated for WM tracts that crossed at variable angular separations within the plane of a transverse image. Signal intensities were calculated for two tissue components, each modeled as for the idealized WM tissue in the second simulation and each containing fibers oriented in the plane ( = 90°). The signals were added at equal weightings for varying in-plane angles (one component with = 0°, the other with -90° < 
+90°). The A value of the combined voxel was calculated with Equation (3) and compared with the two component values.
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RESULTS
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Figure 1 presents a complete set of characteristic single-section diffusion-weighted MR images obtained from one of the subjects (sequence A): four tetrahedral diffusion-weighted images (Fig 1, a–d), three orthogonal diffusion-weighted images (Fig 1, e–g), and the baseline T2-weighted reference image (Fig 1, h). Different levels of brightness in the diffusion-weighted images are due to the anisotropy of WM fiber tracts that were sampled along different directions.
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Figure 1. Characteristic axial single-section MR images (echo time = 106 msec, repetition time > 8,000 msec) obtained from a single acquisition of combined tetrahedral-orthogonal gradient encoding. a-d, Diffusion-weighted MR images encoded along tetrahedral directions one (a), two (b), three (c), and four (d) (26) by using a b value of 1,022 sec/mm2. e-g, Diffusion-weighted MR images encoded along orthogonal directions x (e), y (f), and z (g) by using a b value of 999 sec/mm2. h, One of the seven T2-weighted reference images acquired with very weak diffusion encoding (b = 12 sec/mm2) for each encoding direction. Voxel sizes were 2.11 x 2.11 x 5.00 mm, and the imaging time was approximately 26 seconds per encoding direction (approximately 3 minutes total). All diffusion-weighted images are from a single acquisition and are displayed with the same window width and center, which are one-third the window width and center of h. Note the different WM contrasts in a-g, which encode anisotropic diffusion.
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Figure 2 presents the anisotropy images computed from the data in Figure 1. Images of the various anisotropy measures have a wide range of intensities in the brain, in contrast to the typically uniform appearance of the adult brain on averaged diffusivity images (17,18). Areas of high A in Figure 2, a, are seen in the corpus callosum, internal capsule, and optic radiations. The external capsule also is well depicted. The ROIs were located on the reference image and were confirmed with A images, as shown in Figure 3. Comparison of A values between the matched left and right ROIs demonstrated no statistically significant differences on the basis of measured within-subject thermal noise levels and the results of a two-tailed pairwise Student t test at the 95% confidence level. Comparison of A ROIs between the first and second acquisition in each volunteer similarly demonstrated no significant differences from thermal noise at the 95% confidence level.
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Figure 2. Axial anisotropy (a-c) and anisotropy-weighted (d, e) MR images computed at the level of the basal ganglia by using the data in Figure 1. A (a), Amajor (b), and Aminor (c) images were obtained from a single acquisition of combined tetrahedral-orthogonal encoding. d, AWtet image was obtained from the tetrahedral encodings. e, AWortho image was obtained from the orthogonal encodings. All images have the same window width and center. Note the high sensitivity to WM depiction in a, with delineation of the external capsule, the peripheral occipital WM projections, the thalamic heterogeneity, and the width of the internal capsule. Note also the structural heterogeneity of WM in a (eg, dark bands between the internal capsule, corpus callosum, and adjacent WM) and the heterogeneity in the thalamus, which is not seen on T2-weighted MR images (see Fig 1, h). Visible differences in anisotropy strength exist between WM classes, where the image intensity in a and b can be ranked, from highest to lowest, as follows: commissural WM, projection WM, and association WM. In comparison with a, the anisotropy-weighted component images in d and e show a loss of intensity in the splenium and genu of the corpus callosum, respectively, and there generally is less intensity in e, as compared with that in d.
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Figure 3a. ROIs used in data analysis are superimposed on (a) axial T2-weighted reference MR image (echo time = 106 msec, repetition time > 8,000 msec) (same image as Fig 1, h) and (b) axial A MR image (same image as Fig 2, a). 1 = frontal WM, 2 = frontal GM, 3 = head of the caudate nucleus, 4 = genu of the internal capsule, 5 = putamen, 6 = external capsule, 7 = posterior limb of internal capsule, 8 = thalamus, 9 = occipital-temporal GM, 10 = splenium of the corpus callosum, 11 = occipital WM, and 12 = optic radiations. The location of the ROI for occipital-temporal GM varied across subjects. ROI sizes and shapes were kept constant across hemispheres but varied across subjects.
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Figure 3b. ROIs used in data analysis are superimposed on (a) axial T2-weighted reference MR image (echo time = 106 msec, repetition time > 8,000 msec) (same image as Fig 1, h) and (b) axial A MR image (same image as Fig 2, a). 1 = frontal WM, 2 = frontal GM, 3 = head of the caudate nucleus, 4 = genu of the internal capsule, 5 = putamen, 6 = external capsule, 7 = posterior limb of internal capsule, 8 = thalamus, 9 = occipital-temporal GM, 10 = splenium of the corpus callosum, 11 = occipital WM, and 12 = optic radiations. The location of the ROI for occipital-temporal GM varied across subjects. ROI sizes and shapes were kept constant across hemispheres but varied across subjects.
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Figure 2, b and c, are images of Amajor and Aminor, respectively. Amajor and A are similar in appearance and, when diffusion is axisymmetric, equal in magnitude. Figure 2, d and e, are images of AWtet and AWortho, respectively. In general, the intensity of AWtet is larger than that of AWortho. Artifactual loss of intensity is seen in AWtet in the midline of the splenium of the corpus callosum and in AWortho in the genu of the corpus callosum owing to fiber direction. The fraction of anisotropy recovered by separate tetrahedral and orthogonal anisotropy-weighted indexes is displayed in Figure 4 for different orientations by using Equations (A7,A8), in the Appendix.
Figure 5a presents A in the 66-second multisection experiment (sequence B). Comparison of ROI A values between single-section and multisection acquisitions at the same level demonstrated no significant differences within subjects on the basis of measured thermal noise and the results of a two-tailed pairwise Student t test at the 95% confidence level. Figure 5b demonstrates the ability to average A. These data were acquired as an average of 10 66-second multisection acquisitions. In Figure 5, high anisotropy is present in WM tracts throughout the brain, such as the centrum semiovale, the subcortical U fibers, and the cerebral and cerebellar peduncles.
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Figure 5a. Axial multisection MR imaging demonstration of A obtained (a) without and (b) with data averaging. Numbers indicate the section numbers. Combined tetrahedral-orthogonal multisection data were acquired with one axial T2-weighted reference MR image obtained with a single sequence (echo time = 97 msec, repetition time = 3,100 msec), with b of 1,003.3 sec/mm2 for tetrahedral data, b of 334.4 sec/mm2 for orthogonal data, and b of 0 sec/mm2 for reference data. In a, only one image was acquired per encoding direction (one shot per image; total acquisition time, 66 seconds), with a voxel size of 1.88 x 1.88 x 5.00 mm. In b, the pulse sequence used in a was repeated a total of 10 times in a different subject (total acquisition time, 16 minutes 40 seconds) but with a section thickness of 3.3 mm. In b, the source image data as depicted in Figure 1 were averaged, followed by computation of the diffusion tensor. In both a and b, A was computed from the estimated diffusion tensor according to Equation (3). Note the sensitive demonstration of WM structures throughout the brain, including subcortical U fibers. In addition to the findings shown in Figure 2, note the detection of other WM structures such as cerebral peduncles (section 18 in a and section 30 in b) and the cerebellar peduncles (sections 38 and 40 in b), as well as heterogeneity in the thalamus, basal ganglia, and occipital WM (eg, in b, note band of anisotropy in the region of the globus pallidus in sections 24 and 26 and ridges of low anisotropy along the optic radiations in sections 20-32). Comparison of a and b indicates that A image quality can be improved by increasing the number of signals acquired and obtaining thinner sections.
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Figure 5b. Axial multisection MR imaging demonstration of A obtained (a) without and (b) with data averaging. Numbers indicate the section numbers. Combined tetrahedral-orthogonal multisection data were acquired with one axial T2-weighted reference MR image obtained with a single sequence (echo time = 97 msec, repetition time = 3,100 msec), with b of 1,003.3 sec/mm2 for tetrahedral data, b of 334.4 sec/mm2 for orthogonal data, and b of 0 sec/mm2 for reference data. In a, only one image was acquired per encoding direction (one shot per image; total acquisition time, 66 seconds), with a voxel size of 1.88 x 1.88 x 5.00 mm. In b, the pulse sequence used in a was repeated a total of 10 times in a different subject (total acquisition time, 16 minutes 40 seconds) but with a section thickness of 3.3 mm. In b, the source image data as depicted in Figure 1 were averaged, followed by computation of the diffusion tensor. In both a and b, A was computed from the estimated diffusion tensor according to Equation (3). Note the sensitive demonstration of WM structures throughout the brain, including subcortical U fibers. In addition to the findings shown in Figure 2, note the detection of other WM structures such as cerebral peduncles (section 18 in a and section 30 in b) and the cerebellar peduncles (sections 38 and 40 in b), as well as heterogeneity in the thalamus, basal ganglia, and occipital WM (eg, in b, note band of anisotropy in the region of the globus pallidus in sections 24 and 26 and ridges of low anisotropy along the optic radiations in sections 20-32). Comparison of a and b indicates that A image quality can be improved by increasing the number of signals acquired and obtaining thinner sections.
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Table 1 presents values for several anisotropy measures in the brain ROIs demonstrated in Figure 3. These include A, Amajor, and Aminor, as in Figure 2, a–c. The ROIs are sorted in order of decreasing A value and demonstrate the variability of anisotropy between different types of WM and GM. Additional columns in Table 1 show the comparison of A with its noninvariant components and with AWtet and AWortho, which were obtained separately by using the tetrahedral and orthogonal techniques, respectively. The larger anisotropy contribution contained in the tetrahedral component as compared with that in the orthogonal measure is again demonstrated. Also, our human A results are compared with the relative anisotropy divided by 21/2, as evaluated in monkeys by Pierpaoli and Basser (29). This comparison was made because relative anisotropy is the anisotropy measure mentioned in the MR imaging literature that is most closely related to A. There is general agreement between A and relative anisotropy divided by 21/2. The A SNR averaged across subjects in the reported WM structures ranged from 5.1 to 15.6. The measured thermal noise levels in A, Amajor and Aminor (0.0118, 0.0112, and 0.0121, respectively) were similar to one another.
Table 2 presents characteristic values of sorted eigenvalues obtained from the same ROIs as those in Table 1. Table 3 presents results of the pairwise Student t test comparison of A between different ROIs. The null hypothesis was that there is no significant difference in A between ROIs at the 95% confidence level. Table 3 indicates ROI comparisons for which the null hypothesis was true, that is, ROIs that were not significantly different from each other at the 95% confidence level. If anisotropy were uniform across the brain, the null hypothesis would have been true for all comparisons. If all ROIs differed from one another, only the diagonal elements of Table 3 would have been filled. The actual results in Table 3 resemble a block-diagonal structure, in which A is significantly similar within tissue groups (ie, blocks) and significantly different across groups.
The simulation results for noise and partial volume effects on measurement of A are given in Figures 6 and 7, respectively. In Figure 6a, the measurement bias in the A of isotropic tissues caused by noise is graphed versus the SNR on the reference T2-weighted image for tetrahedral-orthogonal encoding with a b value of 1,000 sec/mm2 for all encodings. The results of this simulation demonstrate that the values measured in GM and the basal ganglia are consistent with zero anisotropy (A = 0) at the reference T2-weighted SNR levels present in the experiment (reference SNR of approximately 40–60).
In Figure 6b, the A measurement bias is graphed versus true anisotropy for anisotropic tissues at various SNR levels for the same tetrahedral-orthogonal encoding. The results of this simulation demonstrate that the anisotropy values measured in the thalamus (A = 0.19) and in other WM areas in Table 1 are consistent with an A value that is greater than 0 at the experimental SNR levels. The results of this simulation also indicate that the noise in A was not stationary but instead increased as A increased (see error bars in Fig 6b).
A simulation of the partial volume effects on A in border regions between GM and WM is presented in Figure 7a. The results of this simulation demonstrate that a combination of signals from anisotropic and isotropic diffusion elements within a voxel generally caused reduction in the measured A proportional to the relative amount of isotropic tissue (given the assumption that the components have similar T1 and T2 weightings). The partial volume effects of averaging between two groups of WM fibers with equal anisotropy is simulated in Figure 7b as a function of angular separation. In this case, the measured A value was reduced by an amount that was dependent on the degree of angular separation between the component tensors.
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DISCUSSION
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Quantitative images of different anisotropy measures were obtained by means of a combination of measurements with tetrahedral and orthogonal gradient encoding (Figs 2, 5). This combination provides wide spatial coverage of the diffusion ellipsoid surface at high gradient strength. The tetrahedral encodings represent the most widely spaced set of four encoding directions (and are at the maximum gradient strength), whereas the orthogonal encodings are in directions that bisect the angular gaps of the tetrahedral encodings. The invariant anisotropy measure A has multiple advantage
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Abstract
Introduction
