A Fourth Order Tensor Statistical Model for Diffusion Weighted MRI
Application to Population Comparison
Theodosios Gkamas
1
, F
´
elix Renard
2
, Christian Heinrich
1
and St
´
ephane Kremer
1
1
ICube UMR 7357, University of Strasbourg – CNRS, F
´
ed
´
eration de M
´
edecine Translationnelle de Strasbourg (FMTS),
300 bd S
´
ebastien Brant – CS 10413, 67412 Illkirch cedex, France
2
Gipsa-lab, 11 rue des Math
´
ematiques, Grenoble Campus BP 46, 38402 St Martin d’H
`
eres cedex, France
Keywords:
Diffusion-weighted MRI, Fourth Order Tensor, Non Euclidean Metric, Nonlinear Dimension Reduction,
Permutation Testing.
Abstract:
In this communication, we propose an original statistical model for diffusion-weighted magnetic resonance
imaging, in order to determine new biomarkers. Second order tensor (T2) modeling of Orientation Distribution
Functions (ODFs) is popular and has benefited of specific statistical models, incorporating appropriate metrics.
Nevertheless, the shortcomings of T2s, for example for the modeling of crossing fibers, are well identified. We
consider here fourth order tensor (T4) models for ODFs, thus alleviating the T2 shortcomings. We propose
an original metric in the T4 parameter space. This metric is incorporated in a nonlinear dimension reduction
procedure. In the resulting reduced space, we represent the probability density of the two populations, normal
and abnormal, by kernel density estimation with a Gaussian kernel, and propose a permutation test for the
comparison of the two populations. Application of the proposed model on synthetic and real data is achieved.
The relevance of the approach is shown.
1 INTRODUCTION
Diffusion-Weighted Magnetic Resonance Imaging
(DW-MRI) is a unique way to probe the diffusion of
water molecules in the human brain in vivo. The cor-
responding data provide information about the under-
lying white matter fiber tracts. Diseases such as mul-
tiple sclerosis, Alzheimer’s disease, and stroke can
be monitored using DW-MRI (Horsfield and Jones,
2002).
Data stemming from DW-MRI require mathemat-
ical modeling. Diffusion Tensor Imaging (DTI) is a
popular description tool for such data (Jones, 2011).
The Orientation Distribution Function (ODF) of a
given voxel in white matter is accounted for by a six
parameter model, also known as second order tensor
(T2) model. The major limitation of the T2 model is
its inability to account for crossing fibers. Neverthe-
less, accounting for crossing fibers is important since
about half of the brain voxels are hosting crossings.
To this end, the T2 ODF model has been generalized
to a fourth order tensor (T4) model, comprising 15 pa-
rameters. This model encapsulates the T2 model as a
particular case, and is able to account for fiber cross-
ings (Weldeselassie et al., 2012). This versatility has
led us to rely on T4 modeling (Ozarslan and Mareci,
2003; Weldeselassie et al., 2012; Tuch, 2004).
The goal of the present study is to propose an orig-
inal statistical model related to T4 parameterization,
in order to provide more efficient diagnosis and fol-
low up tools for pathologies of interest. In particular,
we aim at early diagnosis and at the determination of
new biomarkers. Many statistical description models
have been devised in the T2 framework (see e.g. (Ar-
signy et al., 2006)). The main concern is to account
for the particular geometry of the underlying space in
order to propose powerful tools. A popular model is
the log-Euclidean one (Arsigny et al., 2006). The con-
tribution of the present work is to propose a dedicated
metric and corresponding statistics in the T4 frame-
work.
A T4 ODF is parameterized by a vector in R
15
.
This working space will be equipped with a suited
metric. Besides, a 15-dimension space will be
sparsely filled with data: for a given voxel of interest,
we have a number of points corresponding to the num-
ber of patients (abnormal data) and controls (normal
data), which is orders of magnitude below the number
of points required to fill R
15
. Reduction of dimen-
sion of this space will improve the robustness of the
subsequent statistical tests. We resort to a nonlinear
dimension reduction procedure (Isomap (Tenenbaum
277
Gkamas T., Renard F., Heinrich C. and Kremer S..
A Fourth Order Tensor Statistical Model for Diffusion Weighted MRI - Application to Population Comparison.
DOI: 10.5220/0005252602770282
In Proceedings of the International Conference on Pattern Recognition Applications and Methods (ICPRAM-2015), pages 277-282
ISBN: 978-989-758-077-2
Copyright
c
2015 SCITEPRESS (Science and Technology Publications, Lda.)
et al., 2000)) which is now widely used. Isomap has
already been used for group comparison with classi-
cal statistical tests in the T2 framework (Verma et al.,
2007). In the reduced space (the space of features),
we propose here an original statistical discriminative
test between the two populations. This test allows to
detect the voxels of interest, permitting the discrimi-
nation between normal and abnormal and discovering
voxels where the populations are significantly differ-
ent (i.e. biomarker).
This communication is organized as follows. The
proposed model is presented in section 2. We detail
in particular the choice of the metric, the reduction
of dimension, and the statistical test in the reduced
space. Experimental results on synthetic data and on
real data are shown in section 3. Conclusion is drawn
in section 4.
2 PROPOSED STATISTICAL
MODEL
This section presents the considered data parameteri-
zation, the chosen data normalization procedure, and
the proposed statistical model.
2.1 T4 ODF Parameterization
A second order semi-definite Cartesian tensor writes
f (g
g
g) = g
g
g
T
D
D
Dg
g
g, where g
g
g is a unit vector of R
3
cor-
responding to a given direction in space and D
D
D is a
matrix containing the coefficients of the tensor. Such
a model may be used to represent ODFs (T2 ODF
model).
The generalization to fourth order semi-definite
Cartesian tensors writes:
f (g
g
g) =
i+ j+k=4
D
i, j,k
g
i
x
g
j
y
g
k
z
(1)
where (g
x
, g
y
, g
z
) are the components of the unit vec-
tor g
g
g and the D
i, j,k
s are the coefficients of the T4 ten-
sor. Such a model may be used to represent ODFs
(T4 ODF model). The coefficients are found by mini-
mizing a quadratic cost function under constraints that
imposes also the positivity constraint to the estimated
tensor model (see (Weldeselassie et al., 2012) for de-
tails, see Fig. 1 for a T2–T4 comparison).
2.2 Data Normalization
DW-MRI data sets have to be normalized in a com-
mon reference space. Such a spatial normalization is
particularly cumbersome for diffusion data, since care
must be taken of the underlying fibers.
Figure 1: ODF profiles for a given brain area. T2 model
(left), T4 model (right). As expected, the T4 model captures
more detail and variability than the T2 model.
Considering a T2 model, we compute Fractional
Anisotropy (FA) maps which are used to determine
a nonlinear transformation between the source and
the template data. The estimated transformation is
applied on the raw data. The mapped raw data are
then reoriented using the local rotation component
extracted from the transformation. This amounts to
reorientation of the underlying fibers (see (Tao and
Miller, 2006; Duarte-Carvajalino et al., 2013)). In
practice, we used the patch proposed in (Duarte-
Carvajalino et al., 2013), and available in the FSL
toolkit (Jenkinson et al., 2012).
2.3 T4 Parameter Space Metric
We have to define a metric in R
15
, the T4 parameter
space. An Euclidean distance would not be conve-
nient, since it would not take into consideration the
specificity of the corresponding underlying ODF.
A point in R
15
corresponds to an ODF on the unit
sphere. The metric in R
15
should thus correspond to a
distance between two positive valued functions on the
unit sphere. Such a positivity constraint has led to log-
based distances (see e.g. (Tarantola, 2005)), as is for
example the case in the Kullback-Leibler divergence
or in the T2 log-Euclidean framework (Arsigny et al.,
2006).
Following (Tarantola, 2005), we define the dis-
tance between two ODFs d
1
and d
2
as:
D (d
1
, d
2
) =
Z Z
log
d
1
(θ, φ)
d
2
(θ, φ)
sinθ dθ dφ (2)
where θ and φ are the angular parameters on the
sphere. This distance induces a metric in R
15
, with d
1
and d
2
issued from two points in R
15
. This distance
is original in the DW-MRI context, to the best of our
knowledge. Other (i.e. non log-based) distances have
been proposed for T4s (see e.g. (Barmpoutis et al.,
2007; Du et al., 2014)).
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278
The computation is achieved by discretization in
θ and φ of the equation above. In practice, a T4
ODF profile d
i
is a regression model of noisy data.
Moreover, the underlying non noisy data are not nec-
essarily perfectly adequate with a genuine T4 profile.
Thereby, the precision in the T4 parameters and in the
corresponding ODF profile is not infinite. In addi-
tion, a small variation on low ODF values may have a
tremendous impact on D, because of a huge variation
of the logarithm. To avoid this harmful effect, we es-
timate the confidence we have on ODF values by way
of a Monte Carlo study, thus yielding a standard de-
viation error for the ODF. This standard deviation is
determined off-line once for all, and each ODF value
at stake in the computation of D by a discrete sum is
corrected by addition of the standard deviation.
2.4 ODF Patches
In practice, it is hazardous to rely on the distance
D between two voxels, in particular because of po-
tential registration errors. This is why we resort to
a distance based on 3 × 3 × 3 voxel patches. The
distance between two patches is defined as the sum
of the 3 × 3 × 3 distances between the enclosed cou-
ples of ODF profiles. Moreover, the matching patches
will be searched among all couples of patches in two
5 × 5 × 5 neighborhoods (one neighborhood for each
dataset). Another (popular) option would have been
to resort to some kind of data smoothing. We wanted
to avoid such a preprocessing, to avoid losing poten-
tial important information. Still another option would
have been to concentrate on skeletons of white mat-
ter fiber bundles (Smith et al., 2006), but still at the
expense of losing information.
2.5 Dimension Reduction
We now have a matrix of interpoint distances attached
to a voxel of interest (i.e. to a 5×5 × 5 ROI). Relying
on T4s, the underlying space would be R
s
, with s =
15 for a mere voxel (no patch), or s = 3 × 3 × 3 ×
15 = 405 for a 27-voxel patch. Given the number of
data points (in the order of 100) in usual studies, the
space is sparsely filled. Reducing the dimension will
provide more robust and reliable results.
We tested several nonlinear dimension reduction
methods, among which Isomap (Tenenbaum et al.,
2000), maximum variance unfolding (Weinberger and
Saul, 2006), and locality preserving projection (He
and Niyogi, 2003; He et al., 2005). There was no sig-
nificant difference between the aforementioned meth-
ods, from a discrimination point of view. We finally
chose Isomap, which enables to reproduce the initial
structure of points in a reduced space, say R
2
or R
3
,
by preserving the geodesic interpoint distances. Scree
plots lead us to work in R
2
, as is the case in (Verma
et al., 2007).
2.6 Statistical Reduced Space Model
Each of the two populations (normal and abnormal)
corresponds to a set of points in the reduced space.
We represent the probability densities p
1
and p
2
cor-
responding to the populations using kernel density es-
timation, with a Gaussian kernel (Hastie et al., 2011).
One Gaussian kernel is attached to each point. There
is one parameter to be determined for each popula-
tion, which is the covariance matrix attached to each
point of this population. These matrices are deter-
mined using Scott’s rule (Scott, 1992).
The discrepancy between densities p
1
and p
2
has
to be quantified. The Kullback-Leibler divergence
and its symmetrized version are popular choices.
Nevertheless, in the case of a mixture of Gaussian
distributions corresponding to our kernel density es-
timation, there is no closed formula available and the
numerical estimation of the Kullback-Leibler diver-
gence is time consuming. We rely on another discrep-
ancy measure, noted P , which provides good results
with a low computation time (Sfikas et al., 2005). The
computation of P is about 150 times faster than the
computation of the symmetrical Kullback-Leibler dis-
tance in the present case. The discrepancy P writes:
P (p
1
, p
2
) =
log
2
Z
p
1
(x
x
x) p
2
(x
x
x) dx
x
x
Z
(p
1
(x
x
x))
2
dx
x
x +
Z
(p
2
(x
x
x))
2
dx
x
x
(3)
Considering the mixtures
p
a
(x
x
x) =
I
i=1
π
(a)
i
N
x
x
x; µ
µ
µ
(a)
i
, Σ
Σ
Σ
(a)
i
p
b
(x
x
x) =
J
j=1
π
(b)
j
N
x
x
x; µ
µ
µ
(b)
j
, Σ
Σ
Σ
(b)
j
(4)
a straightforward computation yields
Z
p
a
(x
x
x) p
b
(x
x
x) dx
x
x =
i, j
π
(a)
i
π
(b)
j
v
u
u
t
exp(k)
|
V
V
V
|
(2π)
N
x
Σ
Σ
Σ
(a)
i
Σ
Σ
Σ
(b)
j
(5)
where N
x
is the dimension of x
x
x and with
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V
V
V =
Σ
Σ
Σ
(a)
i
1
+
Σ
Σ
Σ
(b)
j
1
1
(6)
k = µ
µ
µ
T
V
V
V
1
µ
µ
µ µ
µ
µ
(a)T
i
Σ
Σ
Σ
(a)
i
1
µ
µ
µ
(a)
i
µ
µ
µ
(b)T
j
Σ
Σ
Σ
(b)
j
1
µ
µ
µ
(b)
j
(7)
µ
µ
µ = V
V
V
Σ
Σ
Σ
(a)
i
1
µ
µ
µ
(a)
i
+
Σ
Σ
Σ
(b)
j
1
µ
µ
µ
(b)
j
(8)
and from which we derive P (p
1
, p
2
) (we pick a in
{
1, 2
}
and b in
{
1, 2
}
). Indices i, j, a, b have been
dropped from V
V
V , k, and µ
µ
µ for the sake of simplicity.
The preceding formulas may be simplified, since Σ
Σ
Σ
(a)
i
does not depend on i and since Σ
Σ
Σ
(b)
j
does not depend
on j.
The discrepancy P is a statistic. We have to esti-
mate the corresponding p-value ν
?
, i.e. the probability
of getting a distance P larger than the reference one
P (p
1
, p
2
) under the hypothesis that both populations
are indiscernible. This will be achieved resorting to
permutation testing. Besides, we want to determine a
confidence interval associated to the estimation of ν
?
.
This interval will depend on the number of permu-
tations (i.e. label shufflings) used. To address these
issues, we will resort to the Bayesian framework.
We shuffle N times the labels of the points in the
reduced space, and for each shuffling we compute
the value of P . Comparison of the n
th
sample of P
with the reference P (p
1
, p
2
) yields a binary value q
n
,
which is a sample of the Bernoulli distribution of (un-
known) parameter ν
?
. The problem is now the es-
timation of ν
?
from binary samples q
1
, . . . , q
N
. Us-
ing a uniform prior for ν, we derive the posterior
p(ν|q
1
, . . . , q
N
). The confidence interval (or credibil-
ity interval) in ν is taken here as being the smallest in-
terval capturing 99 % of the a posteriori mass, which
is known as Highest Probability Density (HPD) inter-
val in the Bayesian framework. We thus obtain a con-
fidence interval for the p-value. If needed, the length
of the confidence interval may be reduced by using
more shufflings.
Finally, the voxels are sorted according to in-
creasing p-values. The most dissimilar voxels (the
biomarkers) appear on top of the list. We do not ad-
dress multiple comparison issues here, since we do
not provide a list of voxels considered as significantly
different between populations. This is left for future
work.
3 EXPERIMENTAL RESULTS
In this section, we present experimental results based
on synthetic data and on real data. The pathology of
(a) (b)
Figure 2: Reduced space for (a) an Euclidean distance and
(b) the proposed distance in presence of an outlier (red
point). The outlier is well isolated in (b), as should be the
case.
interest for real data is Neuromyelitis Optica (NMO)
(see e.g. http://en.wikipedia.org/wiki/Neuromye-
litis optica).
3.1 Material
The proposed statistical model was implemented in
Python. The FSL toolbox (Jenkinson et al., 2012) was
used for the registration step.
Synthetic data were generated using a code writ-
ten by A. Barmpoutis et al (Barmpoutis et al., 2009).
Real data are composed of 22 normal (control)
and 36 abnormal DW-MRI datasets. Brain acquisi-
tion consisted in High Angular Resolution Diffusion
Imaging (HARDI) with 30 non colinear directions of
gradients. The size of the images is 128 × 128 × 41
for a resolution of 1.8 × 1.8 × 3.5 mm
3
. The b-value
is equal to 1000 s/mm
2
.
3.2 Results
Influence of the metric. In Fig. 2, we compare two re-
duced spaces obtained respectively with an Euclidean
distance (L
2
norm) and with the proposed distance,
both on the ODF profiles and for T4 parameteriza-
tion. The outlier is an ODF resulting from a tensor
for which one coefficient was divided by a factor 60.
As expected, this outlier is separated from the rest of
the group in the case of the proposed metric, but not
in the case of the Euclidean metric.
Reduced space model and statistical test. We
represent the reduced space for three voxels in Fig.
3. The corresponding p-value HPD intervals are re-
ported in Table 1. The first voxel is typical of a dis-
similar configuration for both populations, whereas
the second and third voxels are typical of similar con-
figurations for both populations.
The first voxel examined indicates that the two
ODF populations are different (see Fig. 3, case (a)). If
we resort to smoothing instead of using patches, the
populations do not differ anymore according to this
voxel. Information has been lost in the smoothing,
thus hampering the diagnosis potential.
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(a)
(b)
(c)
Figure 3: Representation of probability densities in the re-
duced space. Green dots: normal cases; red squares: ab-
normal cases. (a) dissimilar populations; (b) and (c) similar
populations. Left column: representation of the probability
density corresponding to the normal population; right col-
umn: representation of the probability density correspond-
ing to the abnormal population (blue: low density; red: high
density. The same color scale is used across all subfigures.).
The densities correspond to kernel density estimation with
a Gaussian kernel.
Table 1: P-value HPD intervals of the cases depicted in
Fig. 3. We used 1000 shuffles. It took approximately 3 min-
utes on a standard computer to process each voxel. Most of
the computation time was devoted to determining the patch
correspondence.
Fig.3 p-value width of
Cases HPD interval interval Decision
(a) [0, 0.0046] 0.0046 Dissimilar
(b) [0.35, 0.43] 0.08 Similar
(c) [0.988, 0.999] 0.011 Similar
T2-T4 comparison on Synthetic Data. We con-
sider a case of two fibers crossing at 90 degrees. Both
fibers have the same amplitude. The two populations
differ by a rotation angle of 5 degrees affecting both
fibers likewise. Noise was added to the synthetic data.
The T4 model is able to discriminate the two popula-
tions, whereas the T2 model is not able to do so (this
fiber crossing is seen by the T2 model as an isotropic
ODF, independently of the rotation angle).
Comparison of the proposed Statistical Test with
the Hotelling Test on Synthetic Data. We consider a
Figure 4: Distribution of p-values for a NMO-related ROI.
As expected, a large number of voxels have low p-values
and are thus identified as abnormal.
case of crossing fibers, where the angle between the
two fibers differs from one population to the other (the
values considered are 90 degrees for the normal pop-
ulation, and 85 degrees for the abnormal population).
The amplitude of the fibers are different as well (the
amplitude is 1 for the fiber common to both popula-
tions; for the fiber with a different orientation in both
populations, the amplitude is 0.35 for the normal pop-
ulation, and 0.45 for the abnormal population).
The proposed statistical test is able to discriminate
both populations. Besides, we perform a Hotelling
test. This test takes place in the reduced space. Each
population is modeled by a Gaussian distribution, and
means are compared, looking for a statistical differ-
ence. The Hotelling test was not able to discriminate
both populations.
T2-T4 comparison on Real Data. We compared
the results (similar/dissimilar) issued from T2 model
and from T4 model. Whenever the results disagreed,
it happened that the T4 model gave the correct answer
and that the T2 model was not able to capture all infor-
mation conveyed by the data. This was validated by
testing the residuals (the dissimilarity information is
contained in the T2 residual). Using a log-Euclidean
distance on second order tensors to reduce the dimen-
sion of the data yields globally the same results as
those issued by the metric we propose, when applied
on T2 profiles.
P-value Distribution on Real Data. On Fig. 4, we
represent the distribution of p-values (one p-value for
each voxel), for a region of interest (ROI). The ROI is
identified as being sensitive to the disease of interest
(NMO). This is confirmed on the plot, where a peak
of low p-values appears.
The proposed statistical T4 model is robust and
sensitive. On the evaluation tests proposed, it showed
better performance than models relying on Euclidean
(L
2
) norms or on T2 ODF profiles. The proposed
model is thus pertinent to extract biomarkers with a
view to early diagnosis in NMO. Other pathologies
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are concerned as well.
4 CONCLUSION
We proposed an original statistical model for fourth
order tensor ODF modeling of DW-MRI. This models
incorporates a suitable metric in the parameter space.
It relies on nonlinear dimension reduction, and on
an original statistic in the reduced space, allowing to
compare two populations and to extract biomarkers.
This approach has shown better ability to discriminate
two populations, as compared to models relying on
other metrics, on T2 ODF profiles, or on the Hotelling
test. It has thus a potential for early diagnosis.
Resorting to T4 ODF profiles will also enable us
to identify more precisely the changes effectively tak-
ing place between the two populations, as compared
to changes identified by T2 models where the profiles
are described less accurately. This is left for future
work.
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