Heart-Muscle Fiber Reconstruction from Diffusion Tensor MRI
Leonid Zhukov
Alan H. Barr
Department of Computer Science, California Institute of Technology
Abstract
In this paper we use advanced tensor visualization techniques to
study 3D diffusion tensor MRI data of a heart. We use scalar and
tensor glyph visualization methods to investigate the data and apply
a moving least squares (MLS) fiber tracing method to recover and
visualize the helical structure and the orientation of the heart muscle
fibers.
CR Categories:
I.3.8 [Computing Methodologies]: Computer
Graphics—Applications, I.6 [Computing Methodologies]: Simula-
tion and Modeling, J.3 [Computer Application]: Life and Medical
Sciences;
Keywords:
Diffusion tensors, DT-MRI, fiber tracing, adaptive fil-
tering, moving least squares, streamlines.
1 Introduction
Diffusion tensor magnetic resonance imaging (DT-MRI) [Basser
et al. 1994] is a technique used to measure the anisotropic diffusion
properties of biological tissues as a function of the spatial position
within the sample. Diffusion properties allow the classification of
different types of tissues and can be used for tissue segmentation
and finding preferred directions in the tissue. Previously, DT-MRI
data was successfully used to recover white matter fiber tracts in
the brain [Basser et al. 2000; Poupon et al. 2001; Parker et al. 2001;
Zhukov and Barr 2002]. Modeling of fiber orientation in the ven-
tricular myocardium based on MR diffusion imaging was proposed
by Tseng et al. [1999], Hsu and Henriquez [2001] and Sachse et al.
[2001]. In this paper we apply tensor visualization and fiber recon-
struction methods to DT-MRI data to recover and demonstrate the
3D orientation and structure of the muscle fibers in the entire heart.
In the brain, white matter fibers are detected by examining the
largest eigenvalue and then “growing” the fiber along the principal
eigenvector direction. This approach is effective, due to the mi-
crostructure of the white matter tissue. Heart muscle tissue has
a different microstructure; as a result we have chosen to detect
the fibers by examining the
sum
of the linear and planar tensor
anisotropies (see Eq. 4), and then trace the fiber along the principal
eigenvector direction as we did before.
In this paper we use a moving least squares (MLS) fiber tracing
algorithm from Zhukov and Barr [2002] to recover heart muscle
fibers. To our knowledge this is the first 3D reconstruction of the
heart muscle fibers from DT-MRI data.
2 Method
2.1 Diffusion Tensors
Diffusion tensors describe the diffusion properties of the media, that
is the ability of water molecules to move around. In biological tis-
sues diffusion properties are dictated by the cell structure of the
tissue. Water molecules can easily move inside the cell, but their
motion across the cells is restricted by the cell membrane. Thus
diffusion properties of the tissue reflect the shape and orientation
of the cells. In the case of an elongated cell, the tissue will have a
preferred diffusion direction along the primary axis of the cell.
Diffusion is measured through a diffusion coefficient, which is
represented with a symmetric second order tensor -
3
x
3
matrix:
D
=
(
D
xx
D
xy
D
xz
D
yx
D
yy
D
yz
D
zx
D
zy
D
zz
)
(1)
The
6
independent values (the tensor is symmetric) of the tensor
elements vary continuously with the spatial location in the tissue.
Eigenvalues
λ
i
and eigenvectors
e
i
of a matrix (1) can be found
as a solution to the eigenvalue problem:
De
i
=
λ
i
e
i
(2)
Since the tensor is symmetric, its eigenvalues are always real num-
bers, and the eigenvectors are orthogonal and form a basis. Geo-
metrically, a diffusion tensor can be thought of as an ellipsoid with
its three axes oriented along these eigenvectors, with the three semi-
axis lengths proportional to the square root of the eigenvalues of the
tensor - mean diffusion distances [Basser et al. 1994].
Using the ellipsoidal interpretation, one can classify the diffu-
sion properties of a tissue according to the shape of the ellipsoids,
with extended ellipsoids corresponding to regions with strong lin-
ear diffusion (long, thin cells), flat ellipsoids to planar diffusion,
and spherical ellipsoids to regions of isotropic media (such as fluid-
filled regions like the ventricles). The quantitative classification can
be done through the coefficients
c
,c
p
,c
s
[Westin et al. 1997] cor-
responding to linear, planar and spherical diffusion.
c
=
λ
1
−
λ
2
λ
1
+
λ
2
+
λ
3
c
p
=
2(
λ
2
−
λ
3
)
λ
1
+
λ
2
+
λ
3
(3)
c
s
=
3
λ
3
λ
1
+
λ
2
+
λ
3
These coefficients are normalized to the range of
[0
..
1]
. Values of
c
that are close to
1
selects regions with strong linear
(
λ
1
>>
λ
2
≈
λ
3
)
diffusion. Values of
c
p
and
c
s
will be small in those
regions. Large
c
p
values correspond to planar diffusion, large
c
s
values corresponds to isotropic media.
Due to the structure of the heart muscle, we will use the sum of
the
c
and
c
p
coefficients to characterize the tissue, thus looking for
regions with high combined directional and planar anisotropies
c
muscle
=
c
+
c
p
=1
−
c
s
.
(4)
This approach retains all of the directional information from
c
l
and
c
p
and discards all nondirectional
c
s
.
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Figure 1: Grayscale mapped value of diffusion tensor anisotropy
measure
c
p
+
c
l
for the “short axes” and “long axes” views of the
heart. The data is thresholded and all voxels with the signal strength
below the threshold value are zero out.
2.2 Algorithm
We first use standard visualization techniques to investigate the
range of parameters in the data and and locate the starting region
for fiber tracing procedure. We then proceed with moving least
squares (MLS) fiber tracing method developed in Zhukov and Barr
[2002].
The basic steps of the MLS algorithm are:
1. Interpolate given 3D tensor data component-wise in the entire
volume and create a continuous tensor field.
2. Choose the seed region for the fibers (can be the entire vol-
ume).
3. Trace fibers individually using Moving Least Square (MLS)
method for dynamic regularization. Fibers will follow the
regularized principal direction (direction of the eigenvector
with largest eigenvalue) The tracing will stop when the fiber
reaches the areas with below the threshold value of chosen
anisotropy parameter.
4. Render individual fibers as streamlines or stream-tubes.
2.3 Data
The 3D DT-MRI data was collected from an intact fixed ca-
nine heart using reduced encoding and constrained reconstruction
methodology [Hsu and Henriquez 2001] and interpolated via zero
filling to
256
x
256
x
256
voxels, with a voxel side of
∼
0
.
4
mm. The
data was thresholded and all voxels with the signal strength below
the threshold value were zero out. Figure Fig. (1) shows grayscale
mapped value of anisotropy measure
c
p
+
c
l
for the “short axes”
and “long axes” views of the heart.
2.4 Computations
Tracing fibers requires computation of weighted averages of tensor
fields within the filter radius (1-3 voxel sizes) and finding tensor
eigenvalues and eigenvectors on every step. These are the most
expensive computations in the algorithm and are performed us-
ing SVD decomposition and LU factorization using routines from
[Press et al. 1992]. We employ standard Euler and Runge-Kutta
forward integration scheme [Press et al. 1992] to evaluate the line
integrals in tracing procedure.
2.5 Visualization
We use two major techniques to visualize tensor data – glyph based
visualization and fiber tracing. We notice [Worth et al. 1998] that
three-dimensional boxes used as glyphs in the complicated tensor
fields convey information better than ellipsoids. We apply box-
based visualization to the short axis (axial) slice of the data. The
results are shown in Fig. (2), where boxes are scaled according
to
(
λ
1
,λ
2
,λ
3
)
eigenvalues and oriented according to correspond-
ing eigenvectors. As seen in Fig.(2), left and middle images, boxes
have almost equal sides, i.e. in our data first eigenvalue is only
slightly larger than the second and third (see Discussion Section).
In order to visually enhance the directionality of the field we use
(
λ
1
−
λ
3
,λ
2
−
λ
3
,λ
3
/
10)
for scaling factors in further visualiza-
tion. (see Figs. (2-3)).
In the fiber visualization, fibers are grown along the “largest
eigenvector” – the eigenvector corresponding to the largest eigen-
value of the regularized continuous tensor field. The integration
step size is taken to be
0
.
1
of the data voxel size and the moving
filter covers
1
-
3
of neighboring voxels. The seeding regions for
the fibers are chosen to be a thick (
10
voxels) vertical slab going
through the center of the dataset and with width and height cov-
ering the entire data volume. The seeding points are located on a
regular grid
{
20
x
20
x
5
}
within the slab, but the lines start only
from the points with
c
p
+
c
l
>
, with
=0
.
1
for this dataset.
For more details and explanation of the fiber tracing technique see
Zhukov and Barr [2002].
We employ several distinct color-mapping schemes for visual-
ization. We use direct RGB
→
XYZ mapping for all glyph based
visualization and some of fiber tracking results. In this scheme, the
elements oriented along the X axis are colored red, along the Y axis
- green and Z axis - blue (see Figs. (2-4)).
For heart muscle visualization, it is important to be able to distin-
guish between clockwise and counter-clockwise orientation of spi-
raling fibers corresponding to left and right handed spirals. To em-
phasize that difference, we developed two color-mapping schemes.
The first one classifies fibers according to the chirality (right or left
handedness) without taking into account the value of the pitch. All
fibers with the same chirality are colored the same way. Such color-
mapping is useful for classification purposes, but creates abrupt
changes in color (singularities) on the boundaries (see Fig. (5)).
The second method adds the pitch angle to the scheme and al-
lows the mapping to smoothly blend colors from the most saturated,
corresponding to the pitch angle of
45
degrees, to neutral for hori-
zontal and vertical fibers (see Fig. (6)).
Both color-mapping schemes rely on the same idea: after choos-
ing an axis of rotation for the spiral one can determine its chirality
by comparing the z component of the cross product between the ra-
dius vector and tangent vector to the fiber at the chosen point and
the z component of the tangent vector itself. Mathematically, if
r
the radius vector pointing to the element and
τ
is a tangent vector
at the point, then
g
=
τ
z
·
(
r
×
τ
)
z
(5)
If
g>
0
, it is a clockwise (right handed) spiral, otherwise it is
counter-clockwise (left handed).
2.6 Rendering
Visualization software is written using OpenGL and GLUT. DT-
MRI anisotropy for the volume is shown using 2D texture mapping
on the three orthogonal planes. Fibers are rendered using illumi-
nated streamline techniques described in Banks [1994] and Hege
[1996] with diffuse and specular terms. Stream tubes are rendered
as a longs sequence of cylinder primitives with axes following the
fiber.
For glyph based rendering, we choose to form rotation and scal-
ing matrices from eigenvectors and eigenvalues for every voxel of
the original data and then apply them to cubes positioned in the
center of those voxels.
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The direction of eigenvectors and magnitude of corresponding
eigenvalues together with anisotropy measures are pre-computed to
allow interactive frame rate during visualization.
All renderings are performed using NVIDIA GeForce4 Ti4200
video card on Pentium4 2.80GHz PC with 1GB of RAM.
3 Discussion
The mammalian heart primarily consists of four chambers: two
ventricles at the bottom of the heart, and two atria above [Streeter
1979]. The heart also contains other structures, such as the valves
between the chambers and the papillary muscles controlling the
valves.
The heart pumps the blood along a multi-part pathway between
the atria and ventricles. Since a muscle fiber can contract only in
one direction, the heart structure is complex, to succeed at pumping
the blood. Anatomically, to achieve this, the muscle walls of the
ventricles and atria are composed of a single helically folded mus-
cular structure, whose fibers continuously connect from one part of
the heart to another [Streeter et al. 1978]. A contraction wave passes
through this muscular helix, which creates the complex pumping
cycle of the heart. The details in the geometric aspects of the he-
lical structure determines the proper functioning of the heart. It is
proposed that a better understanding of this helical structure can
be used to more effectively correct impaired heart function with
surgery [Buckberg et al. 2001].
Physically, muscle consists of many honeycombed “sarcom-
eres,” consisting of interdigitating myosin filaments, in an actin
matrix. Water molecules have difficulty in crossing cell membranes
and the sarcomere boundaries. It is much easier for water molecules
to diffuse to different parts within the same cell and within the same
sarcomere than it is for them to cross into other cells or into other
sarcomeres. Thus, in muscle tissue, water molecules diffuse in a
preferential direction – they are more free to move up and down the
length of the muscle fiber.
Due to these diffusion properties, muscle tissue can be charac-
terized by one somewhat larger eigenvalue of the diffusion tensor,
whose eigenvector is associated with the axial direction of the fiber,
and two somewhat smaller eigenvalues; the eigenvectors for these
should be perpendicular to the direction of the muscle fiber. Since
there is a great deal of structure within the sarcomere, the large
eigenvalue and the smaller eigenvalues are not vastly different from
one another.
Our main results are presented in Figs. (4-6). Heart muscle fibers
in Fig. (4) are colored according to RGB-XYZ color scheme. The
spiraling diagonal orientation of the muscle fibers on the inside and
outside surfaces of the heart reaches
∼
60
degrees angle with re-
spect to the vertical. In Figs. (5- 6) the color scheme is sensitive
to clock/counterclockwise direction of the spiraling muscle fibers.
Clockwise spiral orientation on the inside surface of the heart (en-
docardium) shown in purple , and counterclockwise muscle fiber
orientation on the outside surface of the heart (epicardium) is shown
in blue.
4 Conclusions
In this paper we presented the results of visualization of a DT-MRI
dataset of a canine heart using several visualization techniques:
scalar field rendering, tensor glyph and fiber reconstruction meth-
ods.
Our results clearly show the 3D spiraling diagonal orientation of
the muscle fibers on the heart and systematic smooth variation in
pitch and direction of the fibers from endocardium to epicardium.
5 Acknowledgments
We would like to thank Prof. Edward W. Hsu, Department of Bio-
engineering, Duke University, for providing the dataset, technical
discussions and valuable comments on the paper; Dr. John Wood
for medical and anatomical consultations and visualization sugges-
tions and Prof. David Banks for suggestions on rendering.
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Figure 2: Glyph-based visualization of the the raw data tensors. RGB colors correspond to XYZ components of the largest eigenvector. Boxes
are oriented according to eigenvectors and scaled by eigenvalues. For example, red boxes are oriented along the red (X) axis. Blue boxes
correspond to the vertical muscle fibers in the inside the ventricle. From left to right: axial slice of the data; magnified region from the left
image; the same region with enhanced scaling for the glyph boxes emphasizing principal direction. Note the transition from the red boxes
along the X axis to the green boxes, parallel to Y axis. This transition follows the direction of the heart muscle fibers.
Figure 3: Glyph based visualization of diffusion tensor shown on the background of anisotropy measure: left - the entire dataset; right image
- zoomed in view. Notice the vertical (blue) component of the boxes and the “flatter” orientation of the red boxes.
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Figure 4: Reconstruction of heart muscle fibers using MLS algorithm. This image uses the RGB-XYZ color scheme. Notice spiraling
diagonal orientation of the muscle fibers on the inside and outside surfaces of the heart.
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Figure 5: In this figure we use a color scheme which is sensitive to clock/counterclockwise direction of spiraling muscle fibers. Clockwise
spiral orientation on the inside surface of the heart (endocardium) shown in purple, and counterclockwise muscle fiber orientation on the
outside surface of the heart (epicardium) is shown in blue. This result illustrates the hypothesis that the spiraling fiber orientation in the
heart muscle changes from endocardium to epicardium. The fibers are color-mapped independently on how steep the pitch is, causing abrupt
change of color on the top.
Figure 6: The color scheme used in this figure changes smoothly from clockwise to counterclockwise spiral oriented fibers. Horizontal parts
(very small pitch angle) of the fibers are shown in white. This coloration is consistent with observations of some heart researchers, who have
described a systematic smooth variation in pitch and direction of heart muscle fibers from endocardium to epicardium.
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