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Comparison of 3 Diffusion Models to Track the Hand Motor Fibers within the Corticospinal Tract Using Functional, Anatomical and Diffusion MRI

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  • Boston Children's Hospital, Harvard Medical School

Abstract

In this paper, we propose to compare three diffusion mod-els to track the portion of the corticospinal tract dedicated to the hand motor function (called hand motor fibers hereafter), using diffusion, func-tional and anatomical MRI. The clinical diffusion data have few gradient directions and low b-values. In this context, we show that a newly intro-duced model, called diffusion directions imaging (DDI) outperforms both the DTI and the ODF models. This new model allows to capture several diffusion directions within a voxel, with only a low number of parame-ters. Two important results are that i) the DDI model is the only one allowing consistent tracking from the mesencephalon to the most lateral part of the cortical motor hand area, and that ii) the DDI model is the only model able to show that the number of hand motor fibers in the left hemisphere is larger than in the contralateral hemisphere for right-handed subjects; the DDI model, as the other two models, fails to find such a difference for left-handed subjects. To the best of our knowledge, this is the first time such results are reported, at least on clinical data.
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Comparison of 3 Diusion Models to Track the Hand
Motor Fibers within the Corticospinal Tract Using
Functional, Anatomical and Diusion MRI
Nicolas Wiest-Daesslé, Olivier Commowick, Aymeric Stamm, Patrick Pérez,
Christian Barillot, Romuald Seizeur, Sylvain Prima
To cite this version:
Nicolas Wiest-Daesslé, Olivier Commowick, Aymeric Stamm, Patrick Pérez, Christian Barillot, et
al.. Comparison of 3 Diusion Models to Track the Hand Motor Fibers within the Corticospinal
Tract Using Functional, Anatomical and Diusion MRI. MICCAI 2011 Workshop on Computational
Diusion MRI (CDMRI’11), Sep 2011, Toronto, Canada. pp.150-157, 2011. <inserm-00628264>
Comparison of 3 diffusion models to track the
hand motor fibers within the corticospinal tract
using functional, anatomical and diffusion MRI
Nicolas Wiest-Daessl´e1,2, Olivier Commowick1, Aymeric Stamm1, Patrick
erez3, Christian Barillot1, Romuald Seizeur1,4, and Sylvain Prima1
1VISAGES: INSERM U746 - CNRS UMR6074 - INRIA - Univ. of Rennes I, France
2Department of Neurology, CHU Rennes, France
3Technicolor, Rennes, France
4Department of Neurosurgery, CHU Brest, France
Abstract. In this paper, we propose to compare three diffusion mod-
els to track the portion of the corticospinal tract dedicated to the hand
motor function (called hand motor fibers hereafter), using diffusion, func-
tional and anatomical MRI. The clinical diffusion data have few gradient
directions and low b-values. In this context, we show that a newly intro-
duced model, called diffusion directions imaging (DDI) outperforms both
the DTI and the ODF models. This new model allows to capture several
diffusion directions within a voxel, with only a low number of parame-
ters. Two important results are that i) the DDI model is the only one
allowing consistent tracking from the mesencephalon to the most lateral
part of the cortical motor hand area, and that ii) the DDI model is the
only model able to show that the number of hand motor fibers in the
left hemisphere is larger than in the contralateral hemisphere for right-
handed subjects; the DDI model, as the other two models, fails to find
such a difference for left-handed subjects. To the best of our knowledge,
this is the first time such results are reported, at least on clinical data.
1 Introduction
Diffusion MRI (dMRI) [11] allows in vivo and non-invasive imaging of tissue
structure. It is based on the facts that i) the diffusion of water molecules is
constrained by the micro-structure of the tissues (such as, typically, the white
matter fibers in the brain), and that ii) MRI can be made sensitive to this diffu-
sion, using specific MR pulse sequences. Diffusion models can then be devised,
and their parameters can be estimated for further study and analysis of tissue
architecture. The simplest model is that of a Gaussian diffusion function, which
amounts to characterise the diffusion with a tensor (i.e. a 3×3 symmetric definite
positive matrix), giving its name to diffusion-tensor imaging (DTI) [2].
Fiber tracking, or tractography, has been developed to “reconstruct” or “dis-
sect” the fiber tracts in vivo, and then infer brain anatomy [18]. Many of the
association (e.g. uncinate fasciculus, cingulum) and commissural (e.g. transverse
fibers of the corpus callosum) fiber tracts have been successfully reconstructed
2 Wiest-Daessl´e et al.
using i) clinical dMRI sequences (with few gradient directions and low b-values),
ii) the simple Gaussian diffusion model and iii) simple deterministic streamline
fiber tracking methods [17]. On the contrary, it has proved much more difficult to
reconstruct projection fibers, and especially the motor fibers of the corticospinal
tract (CST) using such standard protocols. On one side, the anatomy of these
fibers between the spinal cord and the internal capsule has been well-studied us-
ing DTI [10, 8]. On the other side, the study of these fibers between the internal
capsule and the cortex, and in particular of those dedicated to a specific mo-
tor function, is much more challenging using DTI, mostly due to the numerous
crossings/kissing/merging/diverging fibers in the corona radiata. A particularly
difficult fiber bundle to track within the CST is the portion corresponding to
the motor hand area, because it is located laterally on the motor cortex [30] (as
shown by the homunculus of Penfield & Rasmussen), compared to the leg or the
trunk areas for instance. In the following, we call this fiber bundle the HMFs,
as “hand motor fibers”.
The HMFs are a crucial fiber bundle to investigate, in the context of normal
anatomy, within the more general study of handedness, cerebral dominance, and
brain asymmetry [27]. More generally, the development of diffusion models and
tractography methods for the CST in general, and the HMFs in particular, which
could be used in clinical routine, is key for a better understanding of pathologies
of the CST such as, typically, amyotrophic lateral sclerosis [9], Wallerian degen-
eration of the CST after ischemic stroke [31], motor dysfunctions in infants [14]
or in patients with relapsing-remitting multiple sclerosis [13].
New types of image acquisition schemes (e.g. HARDI sequences), diffusion
models (e.g. multiple tensors, ODF, etc.) [12] and tractography methods [18–
20] have been introduced to account for intricate fiber configurations, but these
techniques i) have been reported to often miss entirely the lateral portions of
the CST, and thus the HMFs [3] and ii) are not applicable at hand in a clinical
setting, mostly due to long acquisition times.
In this paper, we propose to investigate the usefulness of a new diffusion
model, termed Diffusion Directions Imaging (DDI), to track the HMFs on clinical
data, using a deterministic streamline tractography algorithm. This model allows
to capture several diffusion directions within a voxel, with a low number of
parameters [26, 4].
The two goals of this paper are: 1) to evaluate the ability of this new diffusion
model and of two other standard models (DTI/ODF) to track the left and right
HMFs, using a common tractography algorithm, in a clinical setting, i.e. with
few diffusion gradients (typically, less than 15) and low b-values (typically,
less that 1000s/mm2), and 2) to study whether the number of tracked HMFs
in one hemisphere is different from that in the contralateral hemisphere, for
right-handed and left-handed subjects. Note that we do not test multi-tensor
models, as these have been shown to be unable to provide a unique solution in
the context of single-shell (one unique b-value) acquisitions [24], as is the case
here and in most standard clinical protocols.
Comparison of 3 diffusion models to track the hand motor fibers 3
In Section 2.1, we present the three tested models, and we outline our im-
plementation thereof. In Section 2.2, we describe the common multiple fibers
deterministic streamline tractography algorithm we use for these three models.
The same algorithm is used to make sure that the subsequently reported results
can be interpreted as differences in models, rather than differences in dMRI
sequences or tractography algorithms. The data are described in Section 2.3,
and we perform statistical tests and numerical evaluation in Section 3, before
discussing these results, concluding and giving some perspectives in Section 4.
2 Material & Methods
2.1 Diffusion models
DTI model: The Gaussian model assumes that the diffusion process can
be captured by a tensor (6 parameters), which is proportional to the covariance
matrix of the unknown Gaussian pdf. The tensor Tis parametrised as T=
exp(M), where Mis an unknown 3 ×3 symmetric matrix, and its estimation
is done using a least-squares (LS) fitting on the raw DWI intensities [7]. The LS
criterion is optimised numerically using the NEWUOA algorithm [23]. Within
a given voxel, the single putative fiber direction is considered to be aligned with
the direction of the eigenvectors associated to the largest eigenvalue of the tensor.
The tractography algorithm uses a log-Euclidean interpolation scheme [1].
ODF model: The orientation distribution function (ODF) describes the ori-
entational structure of the diffusion function [28]. The raw DWI intensities are
modelled with a modified basis of spherical harmonics, whose ccoefficients
are estimated using a LS fitting including a Laplace-Beltrami regularisation
term. The number of unknown coefficients depends on the order lof the basis:
c= (l+ 1)(l+ 2)/2. This LS problem has a closed-form solution, from which the
optimal ODF (or to be precise, an approximation thereof) can be computed
analytically using the Funk-Hecke theorem [5]. Then, ODF sharpening is per-
formed using spherical deconvolution to compute the fiber ODF [6]. Within a
given voxel, the putative fiber directions are selected as the local maxima of
the normalised, sharpened ODF for which the ODF function value is above a
user-specified threshold set here to 0.1. These local maxima are computed using
NEWUOA (with starting points homogeneously distributed on the unit sphere),
and they are sorted according to their ODF function value. The tractography
algorithm uses a trilinear interpolation scheme [12].
DDI model: The diffusion function is modelled as a mixture of distributions
with a common parametric form [26]. The number of mixture components is that
of the number mof different fiber directions within the voxel. In essence, the
pdf of each of these distributions is defined as the convolution of a von Mises-
Fisher pdf (which models the direction of the fiber) and of a centered cylin-
drical Gaussian pdf (which models the amplitude of the diffusion along the
fiber). The covariance matrix of the Gaussian distribution is actually a function
4 Wiest-Daessl´e et al.
of the two parameters of the von Mises-Fisher distribution: the mean direction µ
(unit vector) and the concentration parameter κ. A scalar parameter λ, assumed
to be identical for all fiber directions, completes the model. Therefore, to allow
for mfibers, the DDI model requires 3m+ 1 parameters. An anisotropy value ξ,
akin to the fractional anisotropy (FA) (resp. the generalised FA (GFA) [28]) in
the DTI (resp. ODF) model, is also defined. The 3m+ 1 unknown parameters
are estimated using a LS fitting on the raw DWI intensities, and this optimi-
sation is performed using NEWUOA. Within a given voxel, the putative fiber
directions are a natural output of this model, sorted according to their diffusion
function values. Note that, as of now, the number of fibers min this model is set
to 2, and automated estimation thereof will be a topic of future investigation.
The tractography algorithm uses a trilinear interpolation scheme.
2.2 The common multiple fibers tractography algorithm
Our goal is to track the fibers linking two ROIs. Our deterministic streamline
algorithm can be viewed as an extension of the original FACT method [16],
adapted to ODF and DDI models, using a breadth-first-type search. It must be
made clear that for the DTI model, the tractography is led without considering
multiple directions; we omit this important detail below for the sake of clarity.
Starting from one of the two ROIs, we define nstarting points within each
voxel of the ROI. The DTI/ODF/DDI models at these starting points are esti-
mated using the previously described interpolation schemes. For each of these
starting points, we compute the two principal putative directions (defined using
the previously described sorting out procedures), we follow the first direction
with a step size of lmillimeters and we record the second direction for future
use, as it can be indicative of crossing/kissing/merging/diverging fibers. We
then reestimate the DTI/ODF/DDI models at this new spatial position (using
the previously described interpolation schemes), and we compute all the pu-
tative directions for each model. Among these, we follow the one closest (i.e.
with minimal angular difference) to the previously estimated first direction. A
second direction, having the highest ODF/diffusion function value among the
remaining putative fiber directions, is recorded for future use. The tracking of
the main fiber is achieved when i) the angle between two successively estimated
first directions is higher than α, or when ii) FA/GFA/ξis lower than β, or when
iii) the fiber reaches the border of a precomputed brain mask [25]. Once this
main fiber has been tracked, we perform the same tracking from all the possi-
ble crossing/kissing/merging/diverging points that have been recorded along its
path. Importantly, for these trackings, the stepping rule and stopping criteria
are identical as those for the main fiber, but we do not record any possible mixed
fiber configuration along these secondary paths, for which we only follow the first
direction at each step. The same tracking is then led on the second ROI, and
only the tracts linking the two ROIS are kept for further analysis. In practice,
we choose the parameters n= 1, l= 1, α= 60 degrees and β= 0.15.
Comparison of 3 diffusion models to track the hand motor fibers 5
2.3 Data
The data consist of dMRI, anatomical (aMRI) and functional (fMRI) MRI on 14
right-handed (8 males, 6 females) and 9 left-handed (6 males, 3 females) healthy
volunteers. The mean age was 30.3 (21 to 45). Handedness was determined using
the Oldfield questionnaire [22]. The aMRI, dMRI and fMRI data were acquired
using standard sequences on a Philips Achieva 3T system:
aMRI: T1-w 3D TFE, 184 sagittal slices of size 256×256 (1mm×1mm) and
1mm thickness.
fMRI: gradient echo EPI using BOLD contrast, 24 contiguous axial slices
of size 128×128 (1.8mm×1.8mm) and 4mm thickness. The hand motor task
consisted in opening and closing the hand, and was implemented in a stan-
dard block design. Motion correction, slice-timing and detection of the acti-
vation areas for both right and left hands were performed within SPM5.
dMRI: single shot EPI, 60 contiguous axial slices of size 128×128 (2mm×
2mm) and 2mm thickness. Diffusion gradients were applied in 15 non-col-
linear directions with b= 800s/mm2. Each diffusion-weighted MRI was
corrected for eddy current-induced geometric distortions [21] and denoised
using the Rician non-local means algorithm [29]. Given this low number of
directions, the (modified) spherical harmonics basis of order 4 (15 parame-
ters) was used for ODF estimation, while 6 parameters (resp. 7) were to be
estimated for the DTI (resp. DDI) model.
For each subject, the aMR and fMR images were rigidly registered to the B0
image of the dMRI sequence [15]. A first ROI was manually delineated by an
expert neuroanatomist in an axial slice on the aMRI data through the superior
part of the mesencephalum, both on left and right sides. Tractography was then
performed between these two anatomical ROIs and the two (left and right) cor-
tical functional ROIs to reconstruct the HMFs using the three above-mentioned
diffusion models and the previously described tractography algorithm.
3 Results
3.1 Connections between the ROIs
Our objective here was to evaluate whether the three diffusion models, coupled
with the tractography algorithm, were able to connect fully, partially, or not at
all, the anatomical and functional ROIs. First of all, we split each functional ROI
into a medial and a lateral area, the latter corresponding to the extremity of the
hand representation on the motor homunculus, i.e. the thumb. Then for each
model DTI/ODF/DDI, each hemisphere, and each of the 23 subjects, we com-
puted a discrete score of 0, 1 or 2 depending on the quality/strength (subjectively
based on the number of fibers) of the connection between the anatomical ROI and
the medial part of the functional ROI; in a word, we estimated 3 ×2×23 = 138
scores. Similarly, we computed another set of 138 scores for the connection with
6 Wiest-Daessl´e et al.
the lateral part of the functional ROI. At last, the overall number of fibers com-
posing the reconstructed HMFs, i.e. linking the two (anatomical and functional)
ROIs, was also computed.
The Pearson χ2test is particularly adequate to handle such qualitative, dis-
crete scores. We performed pairwise Pearson χ2tests with a significance level of
0.05, corrected for multiple comparisons (Bonferroni) to compare DTI vs ODF,
ODF vs DDI, and DDI vs DTI for the medial and lateral areas. To compare
the overall number of fibers, we first showed that the data were not Gaussian-
distributed using the Jarque-Bera test, and then we performed pairwise sign
tests (which allows to test for differences in medians) with a significance level of
0.05, corrected for multiple comparisons (Bonferroni).
The p-values are reported in Tab. 1, left, and mainly show that i) the ODF
model was able to track more medial fibers than the DTI model, but as many
lateral fibers, and that ii) the DDI model did not track more medial fibers than
the ODF model, but did track more lateral fibers, at the 0.05 significance level.
These two results are confirmed by the sign test on the overall number of fibers.
3.2 Asymmetry of the hand motor fibers
Our objective here was to evaluate whether the three diffusion models, coupled
with the tractography algorithm, were able to show significant differences (in
terms of number of fibers) between the left and right reconstructed HMFs, in
right-handed (14) and left-handed (9) subjects. We pooled males and females
for increased statistical power. After showing that the data were not Gaussian-
distributed using the Jarque-Bera test, we performed pairwise sign tests with a
significance level of 0.05, corrected for multiple comparisons (Bonferroni).
The p-values are reported in Tab. 1, right, and mainly show that i) none
of the models is able to show differences in left-handed subjects, but that ii)
the DDI model is the only model allowing to show that the bundle of HMFs is
larger (in terms of number of fibers) in the left hemisphere than in the right for
right-handed subjects.
Medial Lateral # Fibers
DTI vs ODF 2.6×1093.4×1013.3×107
ODF vs DDI 3.7×1027.6×1012 3.1×107
DDI vs DTI 1.3×1014 5.5×1014 2.8×1014
RH LH
DTI 0.0923 1
ODF 0.5811 0.1797
DDI 0.0189 0.5078
Table 1. p-values of the statistical tests. Left table: “Is there a significant difference
(level=0.05) between the 3 models in recovering tracts reaching the medial and lateral
areas of the functional ROI, in each hemisphere, for the 23 subjects? And in the overall
number of tracked fibers?”. Right table: “Are the 3 models able to show significant
differences (level=0.05), in terms of number of fibers, between the left and right tracked
HMFs, for the 14 right-handed (RH) and the 9 left-handed (LH) subjects?”.
Comparison of 3 diffusion models to track the hand motor fibers 7
Fig. 1. Tractography of the HMFs in the left and right hemispheres. From
left to right: DTI, ODF and DDI. Note that we use the neurological convention, i.e.
the left (resp. right) hemisphere is displayed on the left (resp. right). The DDI model
is the only one able to consistently reach the lateral area within the functional ROI.
4 Discussion, Conclusion & Perspectives
In this paper, we showed that i) the DDI model outperforms both the DTI and
the ODF models to track the HMFs (see Fig. 1), and, maybe more importantly,
that ii) the DDI model is the only model able to show that the number of
HMFs in the left hemisphere is larger than in the contralateral hemisphere for
right-handed subjects, which seems to fit the intuitive idea that the hand motor
fibers in the hemisphere contralateral to the dominant hand should be some-
what more developed than those in the other hemisphere. Interestingly, the DDI
model, as the other two models, failed to find such a difference for left-handed
subjects, which may suggest that left-handedness is not simply a mirrored right-
handedness. This is already known from e.g. the notoriously differing patterns of
hemispheric dominance for language between left and right-handed subjects, but
to our knowledge, this is the first time such results are reported on white matter
fibers, at least on clinical data. These results must now be further investigated
in light of the huge literature on brain asymmetry and cerebral dominance [27].
In particular, recruiting more males/females and right/left-handed subjects for
increased statistical power and population-specific analysis would be necessary
to confirm these first results, of potentially important anatomical significance.
As expected, the DTI model performs very poorly when tracking the HMFs.
Importantly, we stress here that we do not state that the DDI model outper-
forms the ODF model in general, but only in this particular experimental setting.
Low angular resolution of the order-4 ODF model can explain why it is outper-
formed here. It would be of high interest to try to replicate our experiments on
HARDI data using higher-order ODF models and improved (e.g. probabilistic)
tractography methods, to further support our first findings.
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... The angular and spatial resolutions of the data are very low: 20 or 30 diffusion sensitizing gradient (DSG) encoding directions distributed on a single shell at b = 1000s/mm 2 and slices of 5mm thickness, raising the demand for new appropriate diffusion models and/or tractography algorithms that may manage that type of data and propose an accurate reconstruction of the CST. We proposed a diffusion model coined Diffusion Directions Imaging (DDI) [8– 10] that meets these requirements even when using a simple deterministic streamline algorithm [11, 12] . We also recently design a probabilistic tractography algorithm based on an adaptive multi-modal particle filter [13] that is able to tract multiple fascicles over extended volumes. ...
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