Download Fiber Tracking using Magnetic Resonance Diffusion Tensor Imaging

Fiber Tracking using Magnetic Resonance Diffusion Tensor Imaging
and its Applications to Human Brain Development
Richard Watts, Conor Liston, Sumit Niogi and Aziz M. Ulu_
Departments of Radiology, Psychiatry and Neuroscience
Weill Medical College of Cornell University
Address for correspondence:
Richard Watts, D.Phil.
Citigroup Biomedical Imaging Center, Box 234
Weill Medical College of Cornell University
1300 York Avenue, New York, NY 10021
E-mail: [email protected]
Tel: 212-746-5781
Fax: 212-746-6681
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Abstract
Diffusion tensor imaging is unique in its ability to non-invasively visualize white matter
fiber tracts in the human brain in vivo. Diffusion is the incoherent motion of water
molecules on a microscopic scale. This motion is itself dependent on the micro-structural
environment that restricts the movement of the water molecules. In white matter fibers
there is a pronounced directional dependence on diffusion. With white matter fiber
tracking or tractography, projections among brain regions can be detected in the threedimensional diffusion tensor dataset according to the directionality of the fibers.
Examples of developmental changes in diffusion, tracking of major fiber tracts, and
examples of how diffusion tensor tractography and functional magnetic resonance
imaging can be combined are provided. These techniques are complimentary and allow
both the identification of the eloquent areas of the brain involved in specific functional
tasks, and the connections between them. The non-invasive nature of magnetic resonance
imaging will allow these techniques to be used in both longitudinal developmental and
diagnostic studies. An overview of the technique and preliminary applications are
presented, along with its current limitations.
Keywords
Tractography
White matter
Diffusion tensor imaging (DTI)
Magnetic resonance imaging (MRI)
Functional Magnetic Resonance Imaging (fMRI)
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Introduction
Diffusion tensor imaging is unique in its ability to quantify changes in neural tissue
microstructure non-invasively, and in this respect, it is an ideal tool for assessing brain
development from birth to adolescence. It is well known that the human brain undergoes
extensive postnatal development: neurons differentiate and proliferate; neuronal
connectivity is refined as new synapses form and others are pruned away; and the
strength of this connectivity is modulated as neural tracts undergo an on-going process of
myelination that proceeds into adolescence (Dobbing & Sands, 1973). As such, many
groups are beginning to use this technology with developmental populations (e.g.,
Klingberg et al., 1999)
Diffusion in a Uniform Medium
Diffusion is the random motion of particles due to their thermal energy. This motion can
be described as a random walk, in which the particles travel some distance before
colliding with another particle and changing direction (Figure 1). While this process is
random, it may be quantified in terms of probable displacement of particles from their
initial position after a period of time, t (Crank, 1975).
Since the displacement of diffusing particles is equally likely to be positive or negative,
the true average over a large number of particles will be zero and not informative.
Instead, a more useful measure of diffusion is the “root mean-square” (RMS) value of the
displacement. The Einstein equation states that the mean-square displacement is
proportional to the time that the particles diffuse:
r 2 = 6 Dt
Eq. 1
Where r is the displacement, D is the diffusion constant and t is the diffusion time
(Karger et al., 1988). Since diffusion is driven by the thermal energy of the particles, D
is temperature-dependent. The value of 6 is introduced to account for the threedimensional nature of the motion, and would be different if the diffusion were limited to
one- or two-dimensions (Le Bihan, 1995).
For water at 37 ºC, the diffusion constant D is approximately 3.2x10-3 mm2/s. This
allows us to immediately calculate the length scale over which water diffuses over
different periods of time. The RMS displacements over time scales relevant for magnetic
resonance imaging are shown in Table 1.
However, water in the human brain does not exist in such a uniform, isotropic
environment. Instead, its diffusion is restricted due to the presence of structures such as
cellular membranes. Changes in the measured diffusion constant will be observed if the
size of the structures is of the order, or smaller than the distance over which the water
would otherwise diffuse in the measurement time. In the case of MRI, measurement
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times are typically of the order of 10 to 100 ms, making such techniques sensitive to
changes in diffusion caused by structures of approximately 40 µm or smaller.
By monitoring the changes in the diffusion of water molecules, information about the
microstructure which imposes the restriction can be inferred (Basser et al., 1994; 1996).
This methodology can thus give microscopic information from a macroscopic
measurement.
Diffusion-Weighted Magnetic Resonance Imaging (DW-MRI)
To measure diffusion using magnetic resonance imaging, the data acquisition must be
made sensitive to the diffusion-induced motion. With a typical MR spin-echo image
acquisition scheme, diffusion weighting may be produced by adding magnetic field
gradients of equal magnitude and duration prior to and after the 180º refocusing pulse, as
shown in Figure 2. In the absence of incoherent motion, these diffusion sensitizing
gradients would not affect the MR signal – the first gradient dephases the MR signal
while the second rephases it. However, if the water molecules move between one
gradient pulse and the other, then the rephasing of the MR signal is incomplete and the
measured signal is decreased (Karger et al., 1988; Basser et al., 1994).
The MR signal loss due to diffusion-induced dephasing can be written as follows:
S = S 0 e - bD
Eq. 2
where S is the measured signal, S0 is the signal that would be obtained with no diffusion
weighting gradients, D is the diffusion constant, and b is the diffusion weighting caused
by the additional gradients. This b-value is a measure of the diffusion weighting and
related to the strength and duration of the gradients. For the commonly used trapezoidal
gradients, the b-value can be calculated as:
(
b = g 2 g 2d 2 D - d
3
)
Eq. 3
where _ is the gyro-magnetic ratio for the hydrogen nucleus, g is the gradient strength, _
is the duration of the diffusion weighting gradients, and _ is the separation in time of the
two gradient pulses (the diffusion time) (Le Bihan, 1995). With such experiments in
which the b-value is varied, the diffusion constant can be accurately and quantitatively
determined for each volume element (voxel) in an entire acquisition volume.
Magnetic Resonance Diffusion Tensor Imaging (MR-DTI)
The preceding discussion of diffusion weighted imaging assumed that diffusion could be
represented as a single quantity. However, in the presence of highly oriented biological
structures, such as white matter, this is not the case. In such cases, the diffusion is
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anisotropic – directionally dependent (Figure 3). A more complex diffusion model is
required to account for this anisotropic behavior. The diffusion tensor model represents
the diffusion as a 3x3 symmetric tensor:
Ê D xx
Á
D = Á D xy
ÁD
Ë xz
D xy
D yy
D yz
D xz ˆ
˜
D yz ˜
D zz ˜¯
Eq. 4
The measured MR signal depends on both the direction and magnitude of the diffusion
weighting gradient, g . Through combinations of the x, y, and z gradients, the MR signal
can be sensitized to the component of diffusion in any arbitrary direction.
With this tensor model (Ulu_ & van Zijl, 1999),
S = S 0 e - b0 g
where
T
Dg
(
b0 = g 2d 2 D - d
Eq. 5
3
)
Eq. 6
for trapezoidal gradients, as previously.
In the case of isotropic diffusion (Dxx=Dyy=Dzz, and Dxy=Dxz=Dyz=0), this becomes
identical to the previous equations for the isotropic case.
To calculate the diffusion tensor, measurements must be made with the diffusion
weighting gradients in at least 6 non-collinear directions (since the symmetric tensor
itself has 6 independent components) as well as with no diffusion weighting (Figure 4).
In practice, many more directions are normally measured, and a fitting procedure used to
calculate the six tensor components for each voxel.
The diffusion tensor may be conveniently visualized by a diffusion ellipsoid (Figure 5).
The surface of the ellipsoid represents how an imaginary blob of ink would spread out if
it were placed in that voxel within the brain. The three principal axes of the ellipsoid
correspond directly to the three eigenvalues and eigenvectors of the diffusion tensor. In
large white matter fiber tracts such as the cortico-spinal tract and the corpus callosum, the
diffusion constant parallel to the direction of the fibers can be more than three times
larger than that perpendicular.
Once the diffusion tensor has been calculated, a range of quantities can be easily derived
(Figure 6):
Average diffusion constant. The arithmetic average of the diagonal elements of the
diffusion tensor is often used,
D xx + D yy + D zz
Dav =
Eq. 7
3
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The summation of the diagonal tensor components is also know as the “trace” of the
diffusion tensor, i.e. Trace = Dxx+Dyy+Dzz = 3Dav.
Anisotropy indices. Many different measures of anisotropy have been proposed,
including relative anisotropy, fractional anisotropy and ultimate anisotropy (Ulu_ & van
Zijl, 1999; Pierpaoli & Basser, 1996). While these measures all attempt to quantify
anisotropy, they differ in their response curves and noise characteristics.
Direction of Greatest Diffusion. It is usually assumed that the white matter fibers are
oriented along the direction of greatest diffusion (Conturo et al., 1999; Mori et al., 1999;
Poupon et al., 2000; Jones et al., 1999). Fiber tracking follows this direction in three
dimensions through the imaging volume. Once the diffusion tensor is calculated, the
direction of greatest diffusion can be determined as the direction of the principle
eigenvector by diagonalizing the tensor and obtaining the eigenvectors using standard
mathematical methods (Appendix 1).
Display of Diffusion Tensor Data
While the quantities derived from the diffusion tensor may be plotted individually as
grayscale images, such as Figure 6, it may be difficult to visually interpret how these
images combine together. The direction of greatest diffusion is a three-dimensional
vector quantity. While the three components of the vector may be viewed separately, it is
much easier to interpret a single image in which all three components have been encoded
in some way.
In Figure 7, lines are used to represent the fiber tract orientation. The degree of
anisotropy is represented by the length of the line (the eigenvector itself is by definition
unit length) with lines omitted for anisotropies lower than some threshold value. Since
the page is only two-dimensional, only the projection of each line onto the axial plane is
shown, while color is used to represent the z-value (superior-inferior) of the eigenvector.
Red indicates fibers in the axial plane, while green represents fibers perpendicular to the
plane.
Figure 8 shows an alternative representation of fiber direction and degree of anisotropy.
In this case the hue of each voxel is determined by the direction and the brightness by the
anisotropy. For example, voxels that appear bright red have high anisotropy (bright) and
contain fibers that run in the right-left direction (red). Similarly, green and blue fibers
run anterior-posterior and superior-inferior respectively. Low anisotropy voxels appear
dark. It is fortuitous that we have three primary colors to match the three spatial
dimensions that we need to represent.
Developmental Changes in Diffusion and Anisotropy
Diffusion tensor imaging is an ideal tool for assessing brain development because it is
unique in its ability to quantify changes in neural tissue microstructure non-invasively.
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The human brain undergoes extensive postnatal development: neurons differentiate and
proliferate; neuronal connectivity is refined as new synapses form and others are pruned
away; and the strength of this connectivity is modulated as neural tracts undergo an ongoing process of myelination that proceeds into adolescence (Dobbing & Sands, 1973).
These changes in the strength and directionality of neuronal tracts can be indexed by
diffusion and anisotropy measures.
Developmental differences in average diffusion are clear in Figure 9, which depicts
whole brain diffusion coefficient (Dav) histograms for six children (red; three males,
three females, ages 7-10) and six adults (blue; three males, three females, ages 22-31). A
shift toward lower Dav values from childhood into adulthood is shown. To quantify this
difference, voxels were classified as belonging to one of three compartments: a lowdiffusion compartment representing brain tissue; a high-diffusion compartment
containing primarily cerebrospinal fluid (CSF) and non-brain tissue; and a mediumdiffusion compartment representing brain tissue mixed with CSF as described in Ulu_
(2002). The mean Dav value for the tissue compartment was used as an approximation of
the mean whole brain diffusion constant. The 7.2% decrease in mean whole brain
diffusion depicted in Figure 9 was significant (t = 11.40, p < .001), indicating that
developmental changes in myelination were detectable using this method. Moreover,
these findings are in accord with other work (Ulu_, 2002), which predicts mean Dav
values within 0.5% of those reported here for subjects of this age, suggesting that this
methodology is highly reliable. Recent DTI studies have begun to elucidate how
myelination proceeds within specific areas of the brain (Klingberg, 1999). The use of
higher field strength magnets will greatly facilitate this process and may aid in the
diagnosis of pediatric disease states (e.g. Filippi et al, 1999). Moreover, combining this
method with functional MRI data will enhance efforts to assess how changes in
myelination and connectivity underpin normal cognitive and behavioral development.
This is discussed in more detail below.
Fiber Tracking
The method used for fiber tracking is shown schematically in Figure 10. Starting from a
user-selected seed point, a line in the direction of maximum diffusion is followed until
the edge of the current voxel is encountered. The line then abruptly changes direction to
follow the maximum diffusion in the new voxel. The process is repeated until some
criterion for the end of the track is met. The criteria used in the studies presented here is
that the relative anisotropy falls below some threshold value, indicating that the track is
no longer in white matter, and that the uncertainty in the direction of maximum diffusion
is large.
Care must also be taken to account for the 180º symmetry of diffusion. The tracking
algorithm follows tracks from the seed point in both the positive and negative going
directions, with the two paths added together. The same directional degeneracy at each
voxel edge is avoided by limiting the maximum angle between the directions for one
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voxel and the next to 90º. The tracking algorithm is then repeated for other seed points.
Only those tracks longer than some minimum value are displayed.
The choice of the position, shape and density of seed points, as well as the threshold
anisotropy must be made with a view to the types of structures that are of interest. If an
overview of the entire brain is required, then a uniform, low density of seed points
throughout the entire brain may be used, as shown in Figure 11. Increasing the density of
seed points leads to large numbers of tracks and increasing computational time required
for both the tracking and display rendering.
To view the corpus callosum, a sagittal plane of seed points through the center of the
brain may be chosen, limiting the fibers displayed to those passing through this plane
(Figure 12). A higher density of seed points may be used because they need only cover a
two-dimensional plane rather than the entire volume of the brain.
For the corticospinal tract, a high density of seed points within small regions of interest in
the internal capsule may be used. Figure 13 shows an example of such tracking in which
each region of interest has been subdivided and color coded according to its seed
position. It is interesting to note that the tracks from different parts of the internal
capsule fan out separately from each other, with no substantial mixing. This indicates
that individual fiber bundles may be uniquely and reliably traced from the cortex to this
level.
Combining Fiber Tracking with Functional Magnetic Resonance Imaging (fMRI)
One of the exciting applications of fiber tracking is in its combination with functional
magnetic resonance imaging (fMRI). fMRI images changes in the blood oxygenation
level as the subject performs a task. This allows in vivo mapping of the functional centers
of the cortex. Fiber tracking provides complementary information, potentially showing
the way in which these activated areas communicate with each other. An example of a
combination of fMRI with MR-DTI is shown in Figure 14, in which regions of the brain
activated during a finger tapping activity (motor strip, red to yellow colors) are overlaid
onto both a T2-weighted image and fiber tracking of the cortico-spinal tract. As
expected, fibers are seen originating close to the activated motor strip that may be
followed all the way to the spinal cord.
This powerful combination of techniques has a wide range of potential applications.
From a basic neuroscience perspective, it provides the unique ability to determine in vivo
both function and connectivity within the brain. Many applications, from the basic
science interest of how brain processes for example visual information (Werring et al.,
1999) to the applied science interest of developmental maturation (Casey, 2002; Ulu_,
2002; Huppi et al., 2001) or aging (Sullivan et al., 2001; Chun et al., 2000) can benefit
from these techniques. The non-invasive nature of MRI allows longitudinal studies to be
done at ease. These techniques can also be used as a diagnostic tool in wide range of
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disorders, ranging from attention deficit in young children, Alzheimer’s, normal pressure
hydrocephalus, stroke recovery, and amyotrophic lateral sclerosis (Ellis et al., 1999).
Limitations of Diffusion Tensor Imaging
The diffusion tensor model is simply a mathematical model of the physical process of
diffusion within each voxel. While the model generally provides a good fit to the
experimental data, there are certain situations in which this model is not sufficient.
An important limitation of the model is that it cannot represent voxels containing
multiple fiber bundles with different orientations. In this case, the principal eigenvector
is determined by an average over the fibers within the voxel. If two fibers are of
comparable size then the combined vector may not represent either direction, causing the
tracking algorithm to follow neither tract. Alternatively, if one track is much larger than
the other then the tracking is likely to incorrectly jump from the minor to the major tract
(Figure 15).
It may also difficult to differentiate between “kissing” and “crossing” fibers, as shown in
Figure 16. In the case of kissing fibers, the two fibers merge into a single voxel before
diverging without crossing over each other. The direction given by the diffusion tensor
in each case may be the same, although the anisotropy should be lower for crossing fibers
compared to kissing fibers. In both of these cases, the eigenvector does not correspond to
the direction of either fiber.
One way to reduce these problems is to simply reduce the voxel size; with smaller voxels
there is less chance that a single voxel will contain more than a single fiber tract, so the
validity of the diffusion tensor model is increased. However, decreasing the voxel size
comes at a cost of increased acquisition time and decreased signal-to-noise ratio, such
that with current MR scanner hardware and clinically realistic acquisition times, it is
difficult to reduce the voxel size to less than a few mm3. Since this is still several orders
of magnitude greater than the fiber dimensions, decreasing the voxel size will reduce but
not eliminate these problems.
An alternative approach is to increase the number of directions in which the diffusion is
measured and then fit this data to a different, higher order model than the tensor. This
high angular resolution diffusion imaging (Tuch et al., 2002) holds the promise to be able
to determine the directions of multiple fibers within a single voxel, but also leads to
increased acquisition times. Indeed, if one uses a sufficiently large number of directions,
the direction of greatest diffusion can be obtained from the data directly without needing
to use a model.
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Currently, the most common acquisition scheme for acquiring diffusion tensor data is that
of echo-planar imaging (EPI). While this technique is fast, acquiring an entire image in
approximately 200ms, it is also susceptible to degradation due to non-uniformities in the
magnetic field strength. Since the magnetic field is altered by the presence of structures
within the human subject, they cannot be eliminated. However, work is underway to
reduce the imaging artifacts using different data acquisition schemes and post processing
of the data.
Conclusions
Magnetic resonance diffusion tensor imaging provides a unique insight into the white
matter fiber connections within the brain in vivo. The non-invasive nature of the
technique allows for systematic studies of both development and disease progression.
While substantial technical difficulties exist, much work is ongoing to address these
limitations. Combining diffusion tensor imaging with functional magnetic resonance
imaging will allow both brain function and connectivity to be probed in a single
examination.
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Appendix 1. Diagonalization of the Diffusion Tensor
The diffusion tensor can be rotated so that the axes of diffusion ellipsoid are aligned with
the axes of the coordinate system. This can be written mathematically as:
Ê D xx D xy D xz ˆ
0
0 ˆ
Ê D11
Á
˜
-1 Á
˜
0 ˜R
Eq. 8
Á D xy D yy D yz ˜ = R Á 0 D22
ÁD
˜
Á
˜
0 D33 ¯
Ë 0
Ë xz D yz D zz ¯
where R is the rotation matrix, of which columns correspond to the eigenvectors of the
diffusion tensor and also to the axes of the diffusion ellipsoid. The diagonal elements
D11, D22 and D33 of the rotated tensor are the corresponding eigenvalues. The largest
(principal) eigenvalue is associated with the major axis of the ellipsoid – the maximum
diffusion constant. The eigenvectors v are defined such that their direction is invariant
under transformation by D .
Ê D xx
Á
Á D xy
ÁD
Ë xz
D xy
D yy
D yz
D xz ˆÊ v1 ˆ
Ê v1 ˆ
˜Á ˜
Á ˜
D yz ˜Á v 2 ˜ = Dii Á v 2 ˜
Áv ˜
D zz ˜¯ÁË v3 ˜¯
Ë 3¯
Eq. 9
Standard mathematical techniques, such as the Jacobi transform, may be used to calculate
the eigenvalues and eigenvectors of the diffusion tensor.
9
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DiffusionTime
1 ms
10 ms
100 ms
1s
RMS Distance
(unrestricted)
4.4 _m
14 _m
44 _m
139 _m
RMS Distance
(brain tissue)
2.1 _m
6.7 _m
21 _m
67 _m
Table 1. Root mean square (RMS) displacement of water molecules for various times,
assuming diffusion constants of 3.2x10-3 mm2/s (unrestricted), and 0.75x10-3 mm2/s
(brain tissue).
13
Figure 1. Simulation of a molecule undergoing a random walk (diffusion). The
molecule travels some random distance between collisions with other molecules, after
which it continues in another random direction.
14
180º
90º
Echo
RF
time
TE
d
g
Gx
Diffusion Gradients
D
Figure 2. Schematic of a diffusion weighted pulse sequence. The two trapezoidal
waveforms on the x gradient (Gx) either side of the 180º RF pulse sensitize the MR signal
to the diffusion in the x direction. The degree of diffusion weighting (b-value) depends
on the timing (_, _) and strength (g) of the diffusion weighting gradients.
15
Figure 3. The effects of surrounding structures on diffusion. Diffusion, represented by
an ellipsoid, is greater along the axis of the white matter fiber tracts (double-headed
arrow, represented as cylinders) compared to perpendicular to them.
16
Figure 4. Echo-planar images with (a) no diffusion weighting, and diffusion weighting
(b=815 s/mm2) in the (b) x, (c) y, and (d) z directions obtained on a 3T clinical MR
scanner. The intensities of the diffusion-weighted images are scaled by a factor of 3 in
comparison to (a). The cerebrospinal fluid (CSF, arrows) is bright on the T2-weighted
image, but strongly suppressed in the diffusion-weighted images due to the high diffusion
constant. More subtle differences between the images with diffusion direction indicate
areas of white matter with anisotropic diffusion.
17
Figure 5. Diffusion ellipsoids for (a) gray matter (RA = 0.11), and (b) white matter from
the corpus callosum (RA = 0.48). The ellipsoids represent how a blob of ink would
spread out if it were placed in each structure. In the case of the corpus callosum, the
diffusion constant along the fiber is three times greater than the directions perpendicular.
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Figure 6. Selected quantities derived from fitting of the diffusion tensor. (a) Fitted S0
(T2-weighted image), (b) Average diffusion constant (Dav), (c) Relative Anisotropy (RA),
(d) Diffusion constant along x (Dxx), (e) y (Dyy), and (f) z (Dzz). The cortico-spinal tract
(arrows) extends perpendicular to the slice shown, and can be seen as a region of high
anisotropy, with high diffusion along z and low along x and y. Note the contrast for the
diffusion constant images is opposite to that for the diffusion-weighted images shown in
Figure 4. A high diffusion constant leads to a low signal on a diffusion weighted image.
19
Figure 7. Vector plot in which the length and direction of each line represent the relative
anisotropy and the direction of maximum diffusion respectively. The color of the vectors
is used to show the z-component (superior-inferior) of the direction, with red indicating
fibers in the axial plane, and green perpendicular. The vectors are overlaid onto a T2weighted image.
20
Figure 8. Composite color map in which the brightness corresponds to the relative
anisotropy and the hue to the direction of greatest diffusion (red = right-left, green =
anterior-posterior, blue = superior-inferior). The cortico-spinal tract can be seen running
superior-inferior (blue), and the corpus callosum running right-left (red).
Figure 8. Composite color map in which the brightness corresponds to the relative
anisotropy and the hue to the direction of greatest diffusion (red = right-left, green =
anterior-posterior, blue = superior-inferior). The cortico-spinal tract can be seen running
superior-inferior (blue), and the corpus callosum running right-left (red).
21
Figure 9. Whole brain diffusion coefficient (Dav) histograms for six children (red; three
males, three females, ages 7-10) and six adults (blue; three males, three females, ages 2231). A 7.2% decrease (t=11.4, p<0.001) in mean Dav values from childhood to
adulthood is shown, indicating that developmental changes in myelination are detectable
using this method.
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Figure 10. Schematic of the tracking algorithm. Each track follows the direction of
greatest diffusion (double-headed arrows) until it hits the neighboring voxel, at which
point it follows the new direction. In this case, 7 seed positions have been selected.
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Figure 11. Example of fiber tracking of the entire brain (adult normal), with different
view angles. A coronal section from the fitted T2-weighted image is overlaid onto the
fibers (green).
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Figure 12. Fiber tracking of the corpus callosum (adult normal). An array of seed points
in a sagittal plane through the center of the brain is used to isolate the corpus callosum
from the surrounding structures.
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(a)
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(b)
Figure 13. (a) Fiber tracking of the corticospinal tract with different projection angles
from anterior-posterior to posterior-anterior. A coronal section from the fitted T2weighted image is overlaid onto the fibers. (b) Corresponding color map (see Figure 4 for
description), showing placement and color coding of the tracts. Note that fibers starting
from different areas of the internal capsule remain separate as they are tracked superiorly.
INSERT NEW FIGURE HERE CONOR as new FIG 14.
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Figure 14. Combining fiber tracking with functional magnetic resonance imaging
(fMRI) gives a unique opportunity to map out both the areas of cortical activation and the
connections between them. The figure shows areas of activation during a finger tapping
paradigm, along with the cortico-spinal tract traced from the internal capsule.
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Figure 15. Example of crossing fibers in a human brain. Inset: Enlargement showing
motor fibers crossing right-left through the larger longitudinal fasciculus which runs in
the anterior to posterior direction. The major axis of the diffusion ellipsoid in the
crossing voxels is determined primarily by the longitudinal fasciculus such that the
tracking algorithm is unable to connect the motor fibers to cortico-spinal tract. The
crossing regions exhibit reduced anisotropy compared to the rest of the longitudinal
fasciculus. The tracking algorithm is unable to connect fibers (1) and (2) to (3) due to the
presence of (4).
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Figure 16. (a) Crossing and (b) kissing fibers may not be distinguished using the
eigenvectors of the diffusion tensor since the direction of the greatest diffusion (arrow) is
the same in both cases. This illustrates an important limitation of the model.
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