Magnetic Resonance in Medicine 53:944 –953 (2005)
Mathematical Framework for Simulating Diffusion
Tensor MR Neural Fiber Bundles
A. Leemans,1* J. Sijbers,1 M. Verhoye,1,2 A. Van der Linden,2 and D. Van Dyck1
White matter (WM) fiber tractography (i.e., the reconstruction of
the 3D architecture of WM fiber pathways) is known to be an
important application of diffusion tensor magnetic resonance
imaging (DT-MRI). For the quantitative evaluation of several
fiber-tracking properties, such as accuracy, noise sensitivity,
and robustness, synthetic ground-truth DT-MRI data are required. Moreover, an accurate simulated phantom is also required for optimization of the user-defined tractography parameters, and objective comparisons between fiber-tracking algorithms. Therefore, in this study a mathematical framework for
simulating DT-MRI data, based on the physical properties of
WM fiber bundles, is presented. We obtained a model of a WM
fiber bundle by parameterizing the various features that characterize this bundle. We then evaluated three different synthetic
DT-MRI models using experimental data in order to test the
proposed methodology, and to determine the optimum model
and parameter settings for constructing a realistic simulated
DT-MRI phantom. Several examples of how the mathematical
framework can be applied to compare fiber-tracking algorithms
are presented. Magn Reson Med 53:944 –953, 2005. © 2005
Wiley-Liss, Inc.
Key words: diffusion tensor MRI; anisotropy; fiber tracking;
simulated phantom; synthetic data; model evaluation
Diffusion tensor magnetic resonance imaging (DT-MRI) is
currently the only method available to obtain quantitative
information about the three-dimensional (3D) anisotropic
diffusion of water molecules in biological tissue (1,2). This
DT anisotropy reflects the presence of spatially oriented
microstructures (e.g., neural fibers in the central nervous
system), where the mobility of the diffusing particles is
mainly determined by the fiber pathway (3). On the basis
of this intrinsic property, which assumes that the orientation of the DT field matches the orientation of the corresponding underlying fiber system, DT-MRI has been applied in several studies to infer microstructural characteristics and obtain valuable diagnostic information
regarding various neuropathological conditions. Excellent
reviews on white matter (WM) and neuropsychiatric diseases can be found in Refs. 4 and 5.
It is known that ambiguous results are obtained when
DT-MRI is used to study regions in which WM fibers cross
1
Vision Laboratory, Department of Physics, University of Antwerp, Antwerp,
Belgium.
2
Bio-Imaging Laboratory, Department of Biomedical Sciences, University of
Antwerp, Antwerp, Belgium.
Grant sponsors: Institute for the Promotion of Innovation Through Science
and Technology; Fund for Scientific Research.
*Correspondence to: Alexander Leemans, Vision Laboratory, Dept. of Physics, University of Antwerp (CMI), 171 Groenenborgerlaan, U314, B-2020 Antwerpen, Belgium. E-mail: [email protected]
Received 15 July 2004; revised 2 November 2004; accepted 5 November
2004.
DOI 10.1002/mrm.20418
Published online in Wiley InterScience (www.interscience.wiley.com).
© 2005 Wiley-Liss, Inc.
or multiple fibers merge (e.g., Ref. 6). In such regions, the
second-rank DT model is incapable of describing multiple
fiber orientations within an individual voxel. To overcome
this problem, a number of new techniques have been proposed, such as high-angular-resolution diffusion-weighted
imaging (HARDI) (7), Q-ball imaging (QBI) (8), diffusion
spectrum imaging (DSI) (9), persistent angular structure
(PAS) reconstruction (10), and generalized DT imaging
(GDTI) (11). Although these recently developed techniques can provide more accurate and unambiguous results, for simplicity the focus in this paper is confined to
classic DT-MRI.
An important application of DT-MRI is the reconstruction of the 3D WM fiber network, which is referred to as DT
tracking (DTT) or fiber tractography. This technique is
based on the assumption that it can accurately retrieve the
spatial information of the underlying fiber network, using
the available diffusion information of the corresponding
tensor field. DTT provides exciting new opportunities to
study the central nervous system (CNS) anatomy, and has
generated much enthusiasm, resulting in the development
of a large number of fiber-tracking algorithms (12–26).
Although qualitative results may be very valuable, the lack
of a gold standard still precludes an objective quantitative
evaluation of these fiber-tracking algorithms with respect
to precision, accuracy, reproducibility, etc. Although histology has been used to identify major WM fiber bundles,
and can provide complementary anatomical information
for DT-MRI (27–29), technical difficulties related to tissue
preparation impede a quantitative validation of the 3D
WM fiber tract reconstruction.
To address this lack of a gold standard, an accurate
simulated DT-MRI phantom is necessary to evaluate the
numerous criteria that characterize a fiber-tracking algorithm. Only with such a phantom can a comparison between different fiber-tracking algorithms yield decisive answers regarding accuracy, precision, robustness, reproducibility, etc. In addition, with such a phantom one could
study the effect of DT-MRI data processing prior to DT and
fiber tract computing (e.g., image co-registration, noise filtering, and correction of motion artifacts) quantitatively.
For example, fiber tracking requires geometrically registered diffusion-weighted (DW) images. With the use of a
simulated DT-MRI phantom, the sensitivity of fiber-tracking algorithms with respect to the misaligned DW images
can be studied.
Although the need for an accurate simulated DT-MRI
phantom has been emphasized in the literature (e.g., Refs.
13 and 30), only a few tractography-related articles have
described a technique for computing a simulated DT-MRI
phantom. In Ref. 23, a continuous diffusion vector field is
used to describe the tracts, omitting information that is
contained in the remaining degrees of freedom (DOF) of
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Simulating DT-MR Neural Fiber Bundles
the diffusion tensor field. Other methods use the twodimensional (2D) continuous DT field, where the tracts are
represented by rings with a fixed width (12,31). More
advanced techniques employ a cylindrical tube in a 3D
continuous DT field that has a curved trajectory given by a
circular helix (16,32,33). Within this tube, the diffusion
anisotropy measure that characterizes the WM fiber pathway is set to a predefined threshold. However, none of
these DT-MRI tract phantoms simulate a smooth transition
between the actual WM pathway and the surrounding
tissue, although the DT field itself, which is used as “background tissue,” is continuous. Also, nonconstant curvature and torsion within a single tract, derived from experimental data, should be employed to give a more realistic
representation of the WM fiber pathway.
To our knowledge, this paper describes the first mathematical framework for simulating a DT-MRI phantom that
represents the physical properties of a WM fiber pathway
within its surrounding tissue and models the cross-sectional dependency of the fiber density, based on the corresponding fractional anisotropy (FA) and mean diffusivity (MD) (2). The objective of this mathematical framework
is to simulate ground-truth DT-MRI data of a corresponding WM fiber system in order to 1) quantitatively evaluate
the numerous criteria that characterize a fiber-tracking algorithm, 2) optimize the user-defined tractography parameters, and 3) objectively compare different fiber-tracking
algorithms. The analytical form of the tract allows the
simulation of more complex configurations in an elegant
way (e.g., branching and crossing of WM fiber bundles).
Finally, different synthetic DT-MRI models are evaluated
using experimental DT-MRI data, and several applications
to WM fiber tracking are described.
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a set of spatial coordinates {ri} (with i ⫽ 1,. . .,N), where the
direction of the vector line ⌬i ⫽ ri⫹1 – ri smoothly varies
for consecutive points ri. The vector lines ⌬i describe a
piecewise differential 3D space curve, which can be represented by
冘冕
N⫺1
t共r) ⫽
i⫽1
1
␦关r ⫺ (ri ⫹ ␣⌬i )]d␣ ,
where ␦ denotes the Dirac-delta distribution, and ␣ is a
parameterization variable. Basically, t(r) ⫽ 1 for the positions r that define the backbone, and t(r) ⫽ 0 elsewhere. In
the following text, and without loss of generality, it will be
assumed that the length of the vectors ⌬i are constant, i.e.,
@i: ||⌬i|| ⫽ ⌬.
Physical Thickness of the Fiber Bundle
After the coordinates that represent the center of the WM
fiber bundle have been defined, a parameter that models
the physical thickness should be incorporated to refine the
physical properties of the fiber tract. This thickness represents the cross-sectional dependency of the fiber density,
which is assumed here to be nonconstant. It can be introduced in a natural way by convolving the fiber trajectory
t(r), defined in Eq. [1], with a proper kernel k(r). The
resulting convolution function T(r) is calculated as:
冘冕
N⫺1
T共r) ⫽ t共r)*k共r) ⫽
i⫽1
1
k关r ⫺ (ri ⫹ ␣⌬i 兲兴d␣
0
.
[2]
⬅ Ti 共r)
THEORY
This section presents the mathematical framework for simulating DT-MRI data based on the physical properties of
WM fiber bundles. These properties can be summarized as
follows: A realistic fiber trajectory should exhibit a nonconstant curvature and torsion, and have a certain extent.
Also, a smooth transition between the DT field of the WM
fiber bundle and the DT field of the surrounding tissue
should be incorporated.
The first step in the modeling process is to generate a set
of points that define the position and curvature of the WM
fiber pathway. A piecewise continuous 3D space curve can
be associated with these points, which represents the skeleton of the fiber bundle trajectory. This space curve is then
convolved with a model kernel to obtain the physical
extent that defines the cross-sectional dependency of the
fiber density. The principal diffusion direction (PDD) field
is assessed by a weighted sum of the vector lines that
define the backbone of the WM fiber bundle. Finally, the
DT field is calculated using the PDD field and the eigenvalues obtained from the model. Throughout this paper,
the dimensions of the synthetic DT-MRI data are considered to be in units of voxel size.
[1]
0
Consequently, the continuously varying quantitative measures FA and MD can be associated with every point
within the WM fiber bundle, as follows:
FA(r) ⫽
FAM T共r)
;
TM
MD共r兲 ⫽
MDM T共r兲
,
TM
[3]
where FAM and MDM are predefined values that control the
maximum FA and maximum MD, respectively, and TM ⫽
maxrT(r). In Eq. [3], T(r)/TM can be considered as the
normalized cross-sectional dependency of the fiber density, confining FA(r) and MD(r) to their predefined value
range. Depending on the kernel k(r), different diffusion
characteristics within the fiber can be generated by defining another model to describe the physical thickness,
which is equivalent to choosing a different model kernel.
Since FA and MD are independent, different model kernels k(r) can be associated with these diffusion measures.
To illustrate this concept, three specific models (solid,
Gaussian, and saturated) are elaborated.
Solid Model
Backbone of the Fiber Bundle
The first step in the modeling process is the parameterization of a WM fiber. The fiber backbone is constructed from
The techniques that are currently used to simulate a 3D
WM fiber phantom (e.g., Ref. 32) can be obtained by using
a rectangle function ⌸, which is defined as:
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Leemans et al.
k r(r) ⫽ ⌸
冉 冊 再
储r储
⫽
w
0 : 储r储 ⬎ w
1 : 储r储 ⱕ w ,
[4]
where w is the parameter that describes the width of the
fiber bundle (Fig. 1a and d). After some calculations (see
Appendix), the convolution of the fiber tract t(r) from Eq.
[1] with kr(r) is given by:
冘 冉
N⫺1
T r(r) ⫽
i⫽1
冊
⫹
i
冉
k g(r) ⫽ lim ks(r) and kr(r) ⫽ limks(r) .
冊
⫹ H共1 ⫺ ␣i⫺ 兲 ⫺ H共1 ⫺ ␣i⫹ 兲兩 ,
[5]
The PDD field of a WM fiber bundle is characterized by the
first eigenvector field e1(r). To reflect the width and continuity of the WM fiber bundle, e1(r) is calculated as a
weighted sum of the vector lines ⌬i:
where H represents the Heaviside step function and
冘
冐冘
N⫺1
冦
冑
w2
⫺ 储di 共r兲储2
4
⌬ˆ i 䡠 共r ⫺ ri 兲 ⫾
␣ i⫾ ⬅ ␣ i⫾共r兲 ⫽ ᑬ
⌬
with
再
冧
e 1(r) ⫽
i⫽1
[6]
Here ᑬ{ 䡠 } denotes the real part. Also note that ||di(r)||
indicates the distance between the position r and the line
defined by ⌬i, and that Tr(r) ⫽ 0 for ||di(r)|| ⱖ w/2.
Therefore, Tr(r) ⫽ FAM is finally taken for ||di(r)|| ⬍ w/2.
This model assumes no continuous cross-sectional dependency of the microstructural fiber organization, resulting
in a cylindrical tube with constant FA and MD.
Gaussian Model
A second, more realistic model to characterize the physical extent of a WM fiber bundle is obtained by convolving
the fiber backbone with a 3D Gaussian kernel:
,
␴
T g(r) ⫽
⌬
冑冘
␲
2
N⫺1
⫺储di 共r)储 2 /2␴ 2
e
i⫽1
再 冋
册
ˆ i 䡠 共r ⫺ ri 兲
⌬
erf
冑2␴
冋
⫹ erf
册冎
冑2␴
k s(r) ⫽
冊 冉
冉冑 冊
w ⫹ 2储r储
2 冑2␴
2 erf
⫹ erf
w
2 2␴
冊
,
␭1 共r兲 ⫹ 2␭2 共r兲
.
3
[12]
␭ 1共r兲 ⫽
3MD共r)关3⫹FA共r) 冑9⫺2FA2 共r)兴
9⫺2FA2 共r)⫹FA共r) 冑9⫺2FA2 共r)
[13]
and
3MD共r)[3⫺FA2 共r)]
9⫺2FA2 共r)⫹FA共r) 冑9⫺2FA2 共r)
,
[14]
Total DT Field
w ⫺ 2储r储
2 冑2␴
⫽
[8]
Finally, a general model can be obtained by combining the
Gaussian and solid models. Basically, the resulting kernel
is defined as the sum of two error functions:
冉
[11]
where FA(r) and MD(r) are now given by Eq. [3].
.
Saturated Model
erf
冐
,
From this equation system, the eigenvalue fields ␭1(r) and
␭2(r) ⫽ ␭3(r) are calculated as:
␭ 2共r兲 ⫽
ˆ i 䡠 共ri⫹1 ⫺ r)
⌬
T 共r兲⌬i
i
冑3关␭1共r兲 ⫺ ␭2共r兲兴
; MD共r兲
冑␭12共r兲 ⫹ 2␭22共r兲
[7]
where the standard deviation ␴ controls the extent of the
fiber bundle (Fig. 1b and e). The convolution of the fiber
tract t(r) with kg(r), denoted as Tg(r), is given by:
Ti 共r)⌬i
where the weighting coefficients T i(r) are derived from the
model kernel, as described in Eq. [2]. The other eigenvector fields e2(r) and e3(r) can easily be found by solving the
equations e1(r) 䡠 e2(r) ⫽ 0 and e1(r) ⫻ e2(r) ⫽ e3(r).
To calculate the eigenvalue fields ␭i(r), several aspects of
the WM fiber bundle with respect to FA and MD should be
considered. The diffusion of a single ideal WM fiber bundle is mainly along the PDD and has an axial symmetric
DT field, i.e. ␭1(r) ⬎ ␭2(r) ⫽ ␭3(r). Therefore, FA(r) and
MD(r) are calculated as:
FA共r兲 ⫽
2 /2␴ 2
i⫽1
N⫺1
ˆ i ⫽ ⌬i /⌬
⌬
.
di 共r) ⫽ ⌬ˆ i ⫻ 共r ⫺ ri 兲
k g(r) ⫽ e⫺储r储
[10]
␴30
w30
PDD and Eigenvalue Fields
1
1
兩␣ ⌸ ␣ ⫺
⫺ ␣i⫺ ⌸ ␣i⫺ ⫺
2
2
⫹
i
where w controls the width of the fiber bundle, and ␴
determines the edge steepness (Fig. 1c and f). Note that the
previous kernels kr(r) and kg(r) are special cases of this
saturation model, i.e.,
[9]
After the eigenvalues and eigenvectors that describe the
WM fiber bundle have been defined, the corresponding DT
field T(r) is calculated as T(r) ⫽ E(r) ⌳(r) E(r)–1, where the
columns of the matrix E(r) define the orthonormal eigenvectors ei(r) and the diagonal matrix ⌳(r) represents the
eigenvalues ␭i(r). Analogously, the DT field of a background tissue with particular anisotropic properties, denoted as Db(r), can be computed. Finally, the total DT field
Dtot(r) of J WM fiber bundles T{j}(r) with j ⫽ 1,. . .,J is
determined as follows:
Simulating DT-MR Neural Fiber Bundles
947
FIG. 1. Different model kernels
that describe the physical extent
of a WM fiber bundle. a– c: 3D
color-coded representations of
the kernels. d–f: 1D projection
(dashed line, with p as the parameterization variable).
MDM 共r兲
D tot(r) ⫽
冘
J
j⫽1
冘
J
j⫽1
Tj 共r)
MDj 共r)
M
MDM
M ⫺ MD 共r)
Db(r) ,
⫹
M
MDM
[15]
where MDj(r) represents the MD of the jth WM fiber bundle
and
information (35). The effective DT and the derived FA and
MD maps were calculated in each voxel according to
Ref. 1.
RESULTS
Theoretical Model
Single WM Fiber Bundles
MD 共r) ⫽ max jMDj 共r); MD ⫽ maxrMD 共r) .
M
M
M
M
[16]
The first step in constructing the simulated phantom is to
define a set of points {ri} that represent the fiber backbone
This weighted sum allows one to model more complex
configurations, such as the crossing, merging, kissing, and
branching of WM fiber bundles. The first term in Eq. [15]
ensures that if multiple fibers with their unique corresponding diffusion properties coincide, their global combined mean diffusion is normalized. The second term in
Eq. [15] enables the continuous transition of the diffusion
properties of each fiber separately to the diffusion properties of the surrounding background tissue.
MATERIALS AND METHODS
In vivo DT-MRI of a starling brain was performed on an
MRRS 7 T system. A total of 24 sagittal slices (0.6 mm
thick), covering the whole brain, were obtained. Spin-echo
images incorporating symmetric trapezoidal diffusion gradients were sequentially applied in seven noncollinear
directions. We calculated the components of the b-matrices using analytical expressions, as described in Ref. 34,
incorporating both the diffusion gradients (0 or 70 mT m–1
with ␦ ⫽ 12 ms and ⌬ ⫽ 20 ms) and the image gradients.
Additional acquisition parameters were as follows: BW ⫽
25 kHz, FOV ⫽ 22 mm, TE ⫽ 43 ms, TR ⫽ 2400 ms, ramp
time ⫽ 0.1 ms, acquisition matrix ⫽ (256 ⫻ 128), and
number of averages ⫽ 14. The six DW images were coregistered to the non-DW image by maximization of mutual
.
FIG. 2. A single WM fiber skeleton t(r) is completely defined by the
set of points {ri}. a: Several examples of the convolution function
T(r), according to the different model kernels k(r). b: PDD of the WM
fiber bundle, using the color coding of T(r) to reflect the corresponding model kernel (in this case kg(r))
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Leemans et al.
t(r) (see Eq. [1]). These points may describe a model of a
particular anatomical structure or they may be obtained
from the fiber-tracking results of experimental data. The
width of a WM fiber bundle is obtained by convolving the
fiber skeleton with a predefined model kernel. This physical thickness reflects the natural continuous dependence
of FA and MD values across the fiber pathway, and is
elucidated in Fig. 2a.
The PDD (i.e., the first eigenvector e1(r)) is constructed
using a weighted sum of the vector line segments that
define the WM fiber skeleton (see Eq. [11]). In this way, the
PDD field is continuous and aligned along the fiber backbone. Figure 2b illustrates the first eigenvector field after
the real eigenvalue decomposition is calculated. Note that
only the orientation (and not the direction of e1(r)) is
defined, for diffusion is inherently a center symmetric
phenomenon, i.e., both e1(r) and –e1(r) represent the main
diffusion direction.
As described in Eqs. [13] and [14], the eigenvalues ␭1(r)
and ␭2(r) ⫽ ␭3(r) are defined using the modeled diffusion
maps. Thus, FA(r) and MD(r) can be chosen independently
and with high flexibility (see Fig. 3a). Analogously, the DT
field of the background tissue can be constructed. Figure
3b gives an example of the WM fiber bundle embedded in
its surrounding tissue. Notice the continuous transition of
the diffusion properties between the WM fiber bundle and
the surrounding tissue.
Complex WM Configurations
In DT-MRI tractography, fiber crossing is an important
issue in determining the PDD (36). Lower anisotropy values are observed, and eigenvector directions do not correspond with the direction of both fiber tracts at the crossing.
In a previous study using a geometric analysis of the DT
(37), the predominance of the planar component was observed at fiber crossings.
It is generally known that the rank of DTs increases
when lower-rank, noncollinear tensors are summed (26).
Therefore, calculating the weighted sum of the fiber bundle DT fields (see Eq. [15]) yields a natural representation
of WM fiber crossing, which is consistent with the experimental observations. An example of two crossing fiber
bundles is given in Fig. 4. Also, other configurations (e.g.,
fiber merging, branching, and kissing) can easily be constructed with the use of this mathematical framework.
Experimental Evaluation of the Mathematical Framework
In this section, our method is evaluated with real characteristics of existing fibers, which were derived from experimentally measured DT-MRI data of the starling brain.
More specifically, a particular part of the cerebellum
where WM fiber bundles are clearly visible (see Fig. 5a– c)
was synthesized. A comparison of the different synthetic
DT-MRI data sets with the corresponding experimental
data allows one to determine the optimum model and
parameter settings. The evaluation procedure is performed
as follows:
1. The WM fiber backbone trajectories at a specific region of interest (ROI) are defined by the corresponding fiber tractography results, which in this case are
FIG. 3. Elliptical representations of a WM fiber bundle. a: Without
background tissue: 1 and 2 are calculated using the solid model
kr(r), both with equal MD(r) ⫽ MDM but different FA(r) ⫽ FAM; 3 and
4 are calculated using the Gaussian model kg(r), both with equal
MDM but different FAM; 5 and 6 are combinations of the models kr(r)
and kg(r) (in 5, FA(r) ⫽ FAM; and in 6, MD(r) ⫽ MDM). b: With
background tissue. In this example, a uniform field is given: FA(r) ⫽
FA, MD(r) ⫽ MD and e1(r) ⫽ e1.
obtained by the fiber assignment by continuous tracking (FACT) approach (20).
2. FAM and MDM of the WM fibers, and the characteristics of the background tissue (global MD, FA, and
direction of principal diffusivity) are experimentally
measured.
3. Different model kernels are applied and the corresponding model parameters are optimized (in a
least-squares sense) to fit the experimental DT-MRI
data.
To account for partial-volume effects, the total DT field
was integrated over its local neighborhood, as specified by
the experimental voxel dimensions. The resulting simulated DT-MRI data sets according to the solid, Gaussian,
and saturated models are depicted in Fig. 5f– h. The highest similarity (i.e., the lowest mean squared difference
(MSD) between the DT components of the experimental
and the synthetic data) was obtained from the Gaussian
and saturated models.
Simulating DT-MR Neural Fiber Bundles
949
FIG. 4. Elliptic DT field representation of fiber crossing using the
Gauss model. The ellipses are
color-coded according to their FA
to differentiate planar from spherical diffusion. Note the decreasing
FA at the crossing area of the fiber bundles, where the planar
component is much larger than
the linear component.
Example of a Simulated Phantom From Experimental Data
Synthetic Data
Figure 6a shows the WM fiber network of the starling
cerebellum, which is computed with the FACT algorithm
(20). These fiber pathways are then synthesized according
to their experimentally derived diffusion properties to define the ground-truth DT-MRI data set. As shown in Fig. 6b
and c, different fiber-tracking algorithms can now be compared using the simulated DT-MRI data set. This conceptual illustration already demonstrates the feasibility of
evaluating DTT algorithms.
The simulated DT-MRI data (100 ⫻ 100 ⫻ 100 matrix size)
is constructed by randomly generating 29 WM fiber pathways with step size ⌬ ⫽ 0.5, varying cross-sectional widths
({␴,w} 僆 [0.5, 1.5]), FA values (FAM 僆 [公3/10, 公3]), MD
values (MDM 僆 [0.5, 1.5]), and local curvatures ␬ (␬ 僆 [0,
1]) in an isotropic background with MD equal to one (see
Fig. 7a).
Application to WM Fiber Tractography
The similarity measure S between a pair of fiber pathways
(i.e., the simulated tract ts and the experimentally reconstructed tract te) is here defined as (38):
The main application of the simulated DT-MRI phantom is
to quantitatively test and objectively compare different
fiber-tracking algorithms. Such a study consists of evaluating the numerous criteria that characterize a tractography algorithm, as elaborately described in Ref. 32. Here we
present only a few examples to demonstrate how our
methodology can be applied to evaluate and compare fiber-tracking algorithms.
Fiber Similarity
S共t s ,t e 兲 ⫽ R CS e ⫺ E共t s,t e兲/C
with
RCS ⫽
LCS
,
Ls ⫹ Le ⫺ LCS
[17]
where Ls and Le are the lengths of ts and te, respectively,
and Lcs represents the length of the corresponding fiber
FIG. 5. (a) MD and (b) FA sagittal map of a
starling brain. c: FA map of the cerebellum. The color-coding of the FA maps provides directional information regarding the
local fiber orientation (39). The white lines
indicate the ROI from which the fiber
tracking is started. d: Reconstructed fiber
pathways of a particular part of the cerebellum. e: Experimental FA map of the
specified WM fiber bundles, and corresponding (f) solid, (g) Gaussian, and (h)
saturated modeled synthetic FA map.
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Leemans et al.
FIG. 6. a: 3D representation of a starling cerebellum with
three orthogonal FA maps (again color-coded to provide
directional information). b: After this fiber network was synthesized, the fiber-tracking results of the FACT approach
(colored in cyan) and the approach developed in Ref. 26
(colored in red) were compared with the ground-truth fiber
pathways (colored in white). c: Subtle differences between
these approaches can be observed (indicated by the arrows), indicating a high sensitivity of the synthetic data.
segment (i.e., the overlapping part of both fiber tracts). E(ts,
te) is defined as the mean point-by-point Euclidean distance between the corresponding segments of ts and te, and
the coefficient C regulates the trade-off between E(ts, te)
and the corresponding segment ratio Rcs. In this study, C
was chosen to be one voxel wide.
Optimal Curvature Threshold
Most DTT algorithms require multiple user-defined tractography termination thresholds, such as the maximum
fiber curvature ␬M, minimum FA (FAm), etc. As shown in
Fig. 7b, the optimal curvature threshold ␬˜ M can now be
determined via the similarity measure S, averaged over all
fiber pathways (Sav), i.e. ␬˜ M ⫽ arg max␬M Sav(␬M) Here ␬˜ M is
observed to be independent of the predefined FAm and ⌬.
Sensitivity to Noise
By adding different levels of Rician distributed noise to the
DW images of the corresponding ground-truth DT-MRI
data set (see Fig. 7c–f), the sensitivity to noise of DTT
algorithms can now be studied. As shown in Fig. 7g, the
FACT approach is more sensitive to noise than the technique developed in Ref. 26.
DISCUSSION
DTT has become a specialized application of DT-MRI,
resulting in the development of a wide variety of algo-
rithms. Basic DTT techniques reconstruct WM fiber pathways by consecutively following the local first eigenvector
of the corresponding DT, which is similar to calculating
fluid streamlines in hydrodynamics (12,15,20,24). To evaluate these approaches, simulated vector fields and elementary DT phantoms suffice, because it is not necessary to
incorporate all DT characteristics to compute the fiber
pathways. Although these basic DTT methods can offer
qualitative and diagnostic information about the global
anatomical connectivity of the brain, it is difficult to obtain
reliable quantitative results (13).
More advanced DTT algorithms take into account the
entire DT information, and use statistical approaches to
explore many potential connections and select appropriate tracts (14,17,18,23). These fiber-tracking techniques are less sensitive to noise and make it possible to
study fiber crossing and diverging in a quantitative way.
To objectively evaluate the DTT algorithms that accomplish these improvements, we developed a mathematical framework to simulate more advanced and realistic
DT-MRI phantoms, which are useful for studying specific architectural configurations (e.g., fiber crossing and
merging) or single WM fiber bundles with varying crosssectional diffusion properties. DTT techniques that utilize all DOF of the DT to compute WM fiber tracts
require synthetic DT-MRI data that also exhibit these
DOF of the DT. Otherwise, an evaluation of these DTT
techniques could yield biased and incomplete results.
Simulating DT-MR Neural Fiber Bundles
951
FIG. 7. a: 3D representation of the simulated WM fiber system (for clarity, only fibers longer than 50⌬ are shown). b: The averaged fiber
similarity Sav as a function of ␬m with FAm ⫽ 公3/10 and ⌬ ⫽ 0.5. c–f: A slice of the synthetic DT-MRI data set with noise levels ␨ ⫽
{0,5,10,20} respectively, where ␨ represents the standard deviation of the Gaussian distribution that underlies the Rician distribution of the
noise. g: The averaged fiber similarity Sav as a function of ␨.
Therefore, these synthetic DT-MRI phantoms will also
play an important role in testing probabilistic fibertracking algorithms (14,18).
The evaluation of our methodology for simulating DTMRI data of WM fiber bundles indicates that a simple FA
threshold within the modeled fiber structure (i.e., the
solid model) does not provide the most realistic synthetic data (see Fig. 5). We obtained an improved similarity between the experimental and synthetic data by
defining the WM fiber bundle diffusion properties that
vary cross-sectionally, and incorporating a smooth transition between the WM fiber bundle and its surrounding
tissue (Gaussian and saturated models). These results
suggest that partial-volume effects alone do not provide
the optimum way to model WM fiber structures.
An important aspect of evaluating and comparing different fiber-tracking algorithms is uniformity, i.e., equal
conditions for testing different DTT algorithms. Therefore, a library of simulated DT-MRI phantoms, each with
specific configurations and diffusion properties, should
be developed and disseminated. Such a library can easily be constructed with our framework, since it allows
one to model specific anatomical configurations that are
obtained from actual experimental results. Also, the diffusion properties can be chosen independently and with
high flexibility, or they can be determined experimentally to attain a maximal accuracy.
The applications of this mathematical framework are
not limited to the evaluation, optimization, and comparison of DTT algorithms. Several DT-MRI preprocessing
techniques, such as noise filtering, image co-registration, and regularization, also require ground-truth data
for testing and evaluation, and therefore can make use of
this framework to simulate DT-MRI phantoms.
952
Leemans et al.
CONCLUSIONS
We have developed a general mathematical framework for
simulating DT-MRI data sets of WM fiber bundles based on
the corresponding physical diffusion properties. This
framework allows one to model a smooth transition between the WM fiber system and its surrounding tissue. In
addition, complex configurations of multiple fiber bundles, such as crossing, merging, and kissing of fiber pathways, can also be constructed.
We quantitatively evaluated several models using experimental DT-MRI data, and the results indicate that a higher
correspondence between experimental and synthetic DTMRI data exists when the cross-sectional dependency of
the WM fiber density is modeled as nonconstant. Furthermore, these results suggest that modeling of partial-volume effects alone is not the optimum way to model WM
fiber structures.
We have demonstrated that the proposed mathematical
framework can provide the necessary ground-truth DTMRI data for the quantitative evaluation and optimization
of user-defined tractography termination parameters.
Moreover, the synthetic ground-truth data allow one to
objectively compare different fiber-tracking algorithms. Finally, an example of how this framework can be applied to
study fiber-tracking sensitivity with respect to different
noise levels was presented.
ACKNOWLEDGMENTS
This work was financially supported by the Institute for
the Promotion of Innovation Through Science and Technology, Belgium (I.W.T.), and the Fund for Scientific Research, Belgium (F.W.O.).
APPENDIX
Using Eqs. [1] and [4], the resulting convolution Tr(r) is
calculated as follows:
T r(r) ⫽ t共r)*kr(r)
1
冘冕 冉
N⫺1
⫽
⌸
i⫽1
冊
储r ⫺ ri ⫺ ␣⌬i 储
d␣
w
0
1
冘冕 冉
N⫺1
⫽
H
i⫽1
冊
1 储r ⫺ ri ⫺ ␣⌬i 储2
d␣
⫺
4
w2
0
1
冘冕
N⫺1
⫽
H关a i共r)⫹bi 共r)␣⫹c␣2 兴d␣ ,
[A-1]
i⫽1
0
where H represents the Heaviside step function, and the
coefficients of the second-order polynomial form
fir(␣)⫽ai(r) ⫹ bi 共r)␣⫹c␣2 are given by:
a i共r兲 ⫽
1 储r ⫺ ri 储2
;
⫺
4
w2
bi 共r兲 ⫽
2⌬i 䡠 共r ⫺ ri 兲
;
w2
c⫽ ⫺
⌬2
.
w2
[A-2]
Note that fir(␣) is a concave function, i.e.,
᭙r :
⳵ 2f ir共␣兲
2⌬2
⫽ ⫺ 2 ⬍0.
2
⳵␣
w
[A-3]
Hence, it is clear that
再
H关f ir共␣兲兴 ⫽ 1 for ␣i⫺ ⬍ ␣ ⬍ ␣i⫹
H关f ir共␣兲兴 ⫽ 0 for ␣i⫹ ⬍ ␣ ; ␣ ⬍ ␣i⫺ ,
[A-4]
where ␣i⫹ and ␣i⫺ are the real roots of fir(␣), which are
written out in Eq. [6]. Using Eqs. [A3] and [A4], one can
easily see that the convolution Tr(r) can be calculated as
given in Eq. [5].
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