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TECHNOLOGY REPORT
published: 06 June 2016
doi: 10.3389/fnins.2016.00247
Frontiers in Neuroscience | www.frontiersin.org 1June 2016 | Volume 10 | Article 247
Edited by:
Xi-Nian Zuo,
Chinese Academy of Sciences, China
Reviewed by:
Lauren Jean O’Donnell,
Harvard Medical School, USA
Hyunjin Park,
Sungkyunkwan University,
South Korea
*Correspondence:
Alessandro Daducci
alessandro.daducci@epfl.ch
Specialty section:
This article was submitted to
Brain Imaging Methods,
a section of the journal
Frontiers in Neuroscience
Received: 24 February 2016
Accepted: 19 May 2016
Published: 06 June 2016
Citation:
Daducci A, Dal Palú A, Descoteaux M
and Thiran J-P (2016) Microstructure
Informed Tractography: Pitfalls and
Open Challenges.
Front. Neurosci. 10:247.
doi: 10.3389/fnins.2016.00247
Microstructure Informed
Tractography: Pitfalls and Open
Challenges
Alessandro Daducci 1, 2, 3*, Alessandro Dal Palú 4, Maxime Descoteaux 3and
Jean-Philippe Thiran 1, 2
1Signal Processing Lab, Electrical Engineering, École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland,
2Radiology Department, University Hospital Center, Lausanne, Switzerland, 3Sherbrooke Connectivity Imaging Lab,
Computer Science, Université de Sherbrooke, Sherbrooke, QC, Canada, 4Mathematics and Computer Science Department,
University of Parma, Parma, Italy
One of the major limitations of diffusion MRI tractography is that the fiber tracts
recovered by existing algorithms are not truly quantitative. Local techniques for estimating
more quantitative features of the tissue microstructure exist, but their combination with
tractography has always been considered intractable. Recent advances in local and
global modeling made it possible to fill this gap and a number of promising techniques for
microstructure informed tractography have been suggested, opening new and exciting
perspectives for the quantification of brain connectivity. The ease-of-use of the proposed
solutions made it very attractive for researchers to include such advanced methods
in their analyses; however, this apparent simplicity should not hide some critical open
questions raised by the complexity of these very high-dimensional problems, otherwise
some fundamental issues may be pushed into the background. The aim of this article
is to raise awareness in the diffusion MRI community, notably researchers working on
brain connectivity, about some potential pitfalls and modeling choices that make the
interpretation of the outcomes from these novel techniques rather cumbersome. Through
a series of experiments on synthetic and real data, we illustrate practical situations where
erroneous and severely biased conclusions may be drawn about the connectivity if these
pitfalls are overlooked, like the presence of partial/missing/duplicate fibers or the critical
importance of the diffusion model adopted. Microstructure informed tractography is
a young but very promising technology, and by acknowledging its current limitations
as done in this paper, we hope our observations will trigger further research in this
direction and new ideas for truly quantitative and biologically meaningful analyses of the
connectivity.
Keywords: diffusion MRI, tractography, microstructure imaging, interpretation, pitfalls, open challenges
1. INTRODUCTION
It is commonly acknowledged that the human brain is the most complex system in nature.
The presence of pathological conditions in its intricate structure may lead to a wide variety
of neurological disorders and, thus, the availability of tools to investigate its organization is of
paramount importance. Diffusion Magnetic Resonance Imaging (dMRI) is one of such tools that
allows in-vivo and non-invasive investigation of brain connectivity. In biological tissues, the natural
Daducci et al. Microstructure Informed Tractography
motion of water molecules is highly influenced by the
microstructural environment and, in the white matter, the
anisotropy of the resulting random process can be exploited to
probe important features of the neuronal tissue (Le Bihan et al.,
1986; Beaulieu, 2002).
To help the general reader who is unfamiliar with the field,
the metaphor illustrated in Figure 1 might be convenient. One
can imagine brain imaging as a tool to assess the health condition
of the water supply network of a big city, in which the treated
water (i.e., the information) is distributed to the consumers (i.e.,
gray matter nuclei) through a very intricate pipe network (i.e.,
white matter nerves). To analyze the state of the system, the
plumber has at his disposal a powerful toolbox (i.e., dMRI)
which allows him to perform two complementary evaluations.
On one hand, the topology of the network can be assessed
using tractography; for an overview, see Mangin et al. (2013)
and references therein. However, in spite of the high number
of algorithms developed, none of the existing techniques can
actually measure the capacity of the pipes (i.e., the amount of
water that can flow through them). To obtain this information,
the plumber can use another tool called microstructure imaging;
for a review, see Panagiotaki et al. (2012) and references therein.
These methods can characterize the morphology of the pipes in
each district but, on the other hand, they cannot establish the
origin or the consumers of the water passing through them.
Microstructure informed tractography is a relatively new area
of research that, translated into the previous figure of speech,
aims at combining these two pieces of information using global
optimization techniques in order to draw a quantitative map of
the pipe network which, today, is not available. In fact, several
orders of magnitude separate the resolution achievable with
dMRI from the actual size of the axons and each reconstructed
trajectory has to be considered as representative of a coherent
set of real anatomical fibers, the amount of which is not
easy to assess. As a consequence, nowadays the structural
FIGURE 1 | Plumbing metaphor picturing microstructure informed tractography as a tool to evaluate the health condition of the water supply network
of a big city.
connectivity between brain regions is quantified by counting
the number of recovered tracts or averaging a scalar map
along them, e.g., Fractional Anisotropy (FA); either way, these
quantities are only indirectly related to the actual underlying
neuronal connectivity (Jones et al., 2012). In the past few
years, this limitation has received a fast-growing interest in
the field and a number of interesting solutions have been
proposed. A first class of methods (Kreher et al., 2008; Fillard
et al., 2009; Reisert et al., 2011, 2014; Christiaens et al., 2015;
Girard et al., 2015) reconstruct the full tractogram, i.e., set of
fiber tracts, from the measured data in a bottom-up fashion.
The tracts are formed starting from a collection of short
segments, whose signal contribution in each voxel is defined
using tissue models, that are encouraged to interact and form
long chains using global energy minimization. Conversely, top-
down approaches start from a collection of tracts constructed
using standard tractography methods, and attempt to assess
their actual contribution, or some other features of interest like
the average axon diameter, resorting to various optimization
techniques such as stochastic algorithms (Sherbondy et al.,
2009, 2010), non-linear gradient descent (Smith et al., 2013,
2015), random-walk simulations (Lemkaddem et al., 2014)
or, more recently, convex optimization (Daducci et al., 2013,
2014a; Pestilli et al., 2014). Despite using quite different
strategies, all previous solutions share the same goal: combine
tractography with tissue microstructural models in the pursuit of
more quantitative and biologically oriented estimation of brain
connectivity.
Recent advances in software development1turned the use of
such complex models into a rather straightforward operation
but, on the other hand, this ease-of-use also pushed some
1A number of open-source software packages are publicly available, e.g., https://
github.com/MRtrix3/, https://github.com/daducci/COMMIT/ and https://github.
com/francopestilli/life/. These tools are well documented and also contain
demos/tutorials to guide users from the raw data to the processed results.
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Daducci et al. Microstructure Informed Tractography
fundamental issues into the background, e.g., the critical
importance of the diffusion model adopted or the impact on the
results of partial/missing/duplicate fibers. The interpretation of
the outcomes from such high-dimensional problems represents a
nontrivial task and it is subject to a number of potential pitfalls
that might be not so obvious at first glance. If overlooked and
these tools are used as black boxes, erroneous inferences might
be drawn from the data, leading to very deceptive conclusions
about the connectivity in the brain. The purpose of this article is to
raise awareness of the reader about some subtle pitfalls that may
be hidden by the apparent simplicity of these novel techniques,
but that can severely impact on the interpretation of the results.
Most of the issues we discuss are not just concerns in the context
of microstructure informed tractography but, in general, are
relevant to most of the existing tractography algorithms, and can
potentially bias any connectivity analysis with them. Advantages
and limitations of classical tractography have been extensively
reviewed and discussed in several previous studies, for example
Jones (2010), Jbabdi and Johansen-Berg (2011), Fillard et al.
(2011), Côté et al. (2013), Thomas et al. (2014), and Jbabdi et al.
(2015). Therefore we hope this article will serve as a complement
to the existing literature and will help potential users of these
novel techniques to correctly interpret their results and, also,
give ideas to the methods developers about the current open
challenges that still need to be solved, hence triggering further
research in this direction.
The manuscript is organized as follows. First, we call
attention to the close affinity that exists between microstructure
informed tractography methods, notably the linear formulation
recently introduced by Daducci et al. (2013, 2014a), and
classical dictionary-based techniques for recovering the local
fiber structure in a voxel. This analogy will provide us with a
solid mathematical framework to describe strengths and pitfalls
of global reconstruction methods that are inherited from their
local counterpart, while keeping the presentation simple and easy
to follow. In the remainder of the article, we perform a series
of experiments on synthetic and real data to illustrate some of
the situations where the interpretation of the outcomes might
go terribly wrong if these pitfalls are overlooked, describing the
causes and providing simple but clear explanations. Please note
that although existing methods vary considerably in terms of
approach and assumptions, e.g., bottom-up vs. top-down, most
of the issues covered in this article are common to most of them
or can be easily generalized.
2. A PARALLEL WITH LOCAL
RECONSTRUCTION AND INHERITED
ISSUES
With local reconstruction we refer to the branch of dMRI that
deals with the estimation of the intra-voxel fiber structure from
the acquired MR data. As known in the field, many features
of interest, such as the fiber orientation distribution function
(ODF) (Ramirez-Manzanares et al., 2007; Tournier et al., 2007) or
more detailed microstructural properties of the tissue (Alexander
et al., 2010; Daducci et al., 2015), can be expressed as linear
combinations of given basis-functions, also called atoms, as
follows:
y=Ax +η , (1)
where y∈RNd
+is the vector containing the dMRI signal acquired
in the voxel, ηaccounts for the acquisition noise, A= {aij} ∈
RNd×Nkis the linear operator, or dictionary, that explicitly maps
the feature of interest to the measurements through the Nkbasis
functions and x∈RNk
+are the corresponding contributions.
These latter can be efficiently estimated, for instance, by solving
the following general regularized least-squares problem:
argmin
x≥0
kAx −yk2
2
|{z }
data fitting
+λ 8(x)
|{z }
regularization
,(2)
where k · k2is the standard ℓ2-norm in Rn,8(·) represents a
generic function and λ≥0 controls the relative strength of the
regularization (Descoteaux et al., 2007).
Daducci et al. (2013) showed that also the microstructure
informed tractography problem can be recast in terms of the
same linear formulation given in Equations (1) and (2); this
framework was later generalized (Daducci et al., 2014a) to
allow the combination of tractography with any microstructural
tissue-model (Panagiotaki et al., 2012). For the aims of this
study, the possibility to use the same framework to express
both local and global problems allows us to describe known
issues of local formulations (that have been extensively studied
in the literature) and to readily extrapolate them to global
approaches. This analogy is depicted in Figure 2. In the case
of local reconstruction (top row), the estimation of the fiber
configuration in each voxel is usually performed on a voxel-by-
voxel basis or considering small neighborhoods. The dictionary
reflects the specific features of interest and the data to fit; for
instance, it may consist of Gaussian profiles as in Ramirez-
Manzanares et al. (2007) and the estimated coefficients x
correspond then to the contributions of the fiber populations
present in the voxel along any direction. In contrast, rather
than fitting one voxel at the time, microstructure informed
tractography methods (bottom row) consider simultaneously all
the voxels of the brain using global optimization techniques. The
dictionary consists in this case of a combination of the fiber tracts
present in the tractogram with tissue forward-models to assess
their contribution to the dMRI image along their trajectories; the
interested reader is referred to Daducci et al. (2014a) for more
details.
A variety of modeling approaches has been used by different
optimization strategies to predict the contribution of the tracts in
each imaging voxel, from the classical mixture of tensors (Reisert
et al., 2011; Pestilli et al., 2014) to more sophisticated models that
directly relate to features of the tissue microstructure (Sherbondy
et al., 2010; Daducci et al., 2014a). For the scope of our analysis
and sake of simplicity, in our experiments we adopted the Stick-
Zeppelin-Ball model (Panagiotaki et al., 2012): axons represented
as cylinders with zero radius, extra-cellular space modeled with
anisotropic tensors and isotropic free diffusion. The estimated
coefficients xcorrespond to the actual weight (or volume) of each
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Daducci et al. Microstructure Informed Tractography
FIGURE 2 | Parallel between local and global reconstruction. Local reconstruction methods usually recover the fiber configuration in a voxel by expressing the
quantity of interest, e.g., ODF, as a linear combination of a given set of response functions (shown here in 2D for simplicity). Daducci et al. (2013, 2014a) showed that
also microstructure informed tractography can be expressed using the same formulation where, instead of a single voxel, the dictionary models the whole dMRI image
as a superimposition of the signal arising from all the fibers in a tractogram.
fiber, possibly in addition to other signal contaminations from
non-fiber tissues, e.g., cerebrospinal fluid (CSF). Please note that
more realistic biophysical models could be easily considered in
this framework, e.g., to account for bundles composed of axon
populations with different radii as in Sherbondy et al. (2010)
and Daducci et al. (2014a), but their use would only complicate
the exposition without affecting the observations discussed in
this work. In Section 3, though, we will examine the impact on
the recovered fiber parameters of small variations to this tissue
model.
2.1. Data and Experiments
To highlight the hidden dangers mentioned before in
interpreting the outcomes, we performed a series of
experiments on both synthetic and real data aimed at
evaluating microstructure informed tractography methods
in different conditions. In particular, we first illustrate some
well-known issues of local methods with the help of a simple
synthetic example, for which the expected behavior is easily
explained; then we exploit the parallel between local and global
reconstruction to extrapolate these observations to global
approaches. The phantom consists of a single voxel with two
fiber populations crossing at 90◦(Figure 2, top-left). For the
sake of illustration, both the signal corresponding to this
configuration and the response functions of the dictionary have
been simulated using the classical multi-tensor model (Tuch
et al., 2002); all the observations in this article remain the same
if more complex generative models are used (Soderman and
Jonsson, 1995; Assaf and Basser, 2005). Also, although 2D ODF
are shown in the plots, the actual experiments were performed in
signal space on the sphere. Global reconstruction was tested on
real data that is publicly-available; specifically, we used the same
two-shell dataset from Daducci et al. (2014a), i.e., 24 images at
b=700s/mm2and 48 at b=2000s/mm2, as well as 10 datasets
from the Human Connectome Project (HCP) (Van Essen et al.,
2012), i.e., 90 measurements at b=2000s/mm2as done in
Pestilli et al. (2014). If not otherwise specified, the two-shell
dataset was used in all real-data experiments.
For the scope of this paper, tractography was performed with
the probabilistic iFOD2 algorithm (Tournier et al., 2012) and the
GIBBS tracker (Reisert et al., 2011), using default parameters.
The corresponding dictionaries were created by combining the
streamlines in each tractogram with the Stick-Zeppelin-Ball
model, implemented in the DIPY library (Garyfallidis et al.,
2014), and assuming diffusivities values typical for in-vivo human
data, as in Alexander et al. (2010),Zhang et al. (2012), and
Daducci et al. (2014a): longitudinal dk=1.7 ×10−3mm2/s,
perpendicular d⊥=0.5 ×10−3mm2/s and same dkin the
extra-cellular space, and two isotropic compartments with d∈
{1.7,3.0} × 10−3mm2/s. For more details on the construction
of the dictionary, please refer to Daducci et al. (2014a). All
experiments were carried out using the publicly-available2
COMMIT framework, adopting the Douglas-Rachford algorithm
(Combettes and Pesquet, 2007) with no regularization (λ=0)
to solve Equation (2) for both local and global problems. This
2https://github.com/daducci/COMMIT.
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Daducci et al. Microstructure Informed Tractography
data and experimental setup was used consistently throughout
the manuscript.
2.2. Missing Atoms/Fibers
Figure 3 illustrates the situation where some atoms are missing
in the dictionary. It is straightforward to realize how the
90◦configuration (top-left in Figure 2) cannot be accurately
described in terms of only atoms 1, 2, and 4. The best fit
actually consists in assigning a weight of 0.5 to atoms 1 and
3; however, as this latter is not present in the dictionary
(Figure 3A), the optimization compensates for this deficiency by
assigning a contribution to atoms 2 and 4 instead (Figure 3B).
The consequences are clearly visible in Figure 3C: not only
the vertical bundle corresponding to atom 3 is obviously not
recovered, i.e., false negative, but two spurious fiber populations
might also be incorrectly detected, i.e., false positives.
In the context of global reconstruction, this situation arises
when a tracking algorithm fails to reconstruct some fibers, which
are thus not accounted for in the dictionary. A typical example,
as shown in Reisert et al. (2011) and Mangin et al. (2013), is
the well-known difficulty of classical streamline tractography in
reconstructing the lateral projections of the corpus callosum
(Figure 3D). Besides clearly ignoring these fibers in the analysis,
global methods might also incorrectly assign a contribution to
other bundles and thus infer that two gray matter regions are
actually connected when, perhaps, these fibers may not exist
at all. Therefore, missing fibers, or in the specific context of
this example missing atoms in the dictionary, can be hugely
problematic for the correct assessment of the fiber contributions.
The construction of an adequate set of fiber tracts, either by
top-down or bottom-up approaches, is therefore of utmost
importance and currently an open problem in tractography,
in particular for those methods which attempt to estimate the
contribution of the tracts to the measured data or to a feature
of interest (e.g., ODF).
2.3. Duplicate Atoms/Fibers
The most straightforward solution one can think of for ensuring
the inclusion of all possible fibers is to merge tractograms from
different algorithms. However, if performed without due care, this
operation can generate another subtle issue. For example, the
dictionary in Figure 4A contains all the response functions that
are required to fit correctly the 90◦configuration in the top-left
of Figure 2, but it also includes some duplicates. Depending on
the algorithm used to solve Equation (2), the actual contribution
of a response function could be arbitrarily distributed among its
copies; the higher this number and the more the corresponding
coefficients tend to be small (Figure 4B). In this case, however,
a low weight does not necessarily mean that those atoms are
less important than others in explaining the data. Besides, it is
worth recalling that local methods usually employ a threshold to
determine when a response function can be considered spurious
(Tournier et al., 2007); in presence of duplicates, these atoms
are more likely to be discarded because their coefficients can
fall below this level, and thus false negatives can be generated
(Figure 4C). Actually, the same issue may arise also without such
threshold if a form of regularization is naively applied. As an
example, using 8(x)= kxk1for promoting sparsity (Candès
et al., 2006; Donoho, 2006) in the tractogram, small coefficients
may tend to be suppressed depending on how the specific
AB
C D
FIGURE 3 | Improper dictionary: missing atoms. (A) Schematic illustration of a dictionary with a missing atom (number 3 in Figure 2). Estimated coefficients (B)
and ODF (C) when using it to reconstruct the voxel configuration in Figure 2.(D) Parallel with global reconstruction: the common difficulty of classical tractography
algorithms in tracking the callosal projection fibers (yellow boxes) is a typical scenario that leads to missing atoms (i.e., fibers) in the dictionary.
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Daducci et al. Microstructure Informed Tractography
AB
C D
FIGURE 4 | Improper dictionary: duplicate atoms. (A) Dictionary with replicas of an atom (number 3 in Figure 2). Estimated coefficients (B) and ODF (C) when
using it to reconstruct the voxel configuration in Figure 2.(D) Parallel with global reconstruction: the trajectories of an easy-to-track bundle (yellow line) can be
reconstructed many times in a tractogram and the corresponding atoms can be assigned very small contributions; this could lead to conclude that each duplicate
fiber is spurious.
algorithm for solving Equation (2) handles highly correlated
atoms. Thus, false negatives may be generated also in this case
and different regularization functions may lead to even more
unpredictable results; this is why we adopted classical least
squares in all our experiments.
Major fiber bundles such as the U-shaped callosal radiations,
the corticospinal tract or the inferior fronto-occipital fasciculus
are easily reconstructed by most tractography algorithms and
they are likely overrepresented in a tractogram. Figure 4D shows
a subset of the tracts after merging the tractogram of Figure 3D
with the output of the GIBBS tracker, which now include the
callosal projections that were missed previously. It can be noticed
that the callosal radiations appear denser than before, as indeed
these fibers were easily reconstructed by both algorithms; hence
the corresponding atoms in the dictionary are likely to be
assigned small weights. As a consequence, if the raw estimated
weights are used to discriminate between true and spurious fibers,
these might be incorrectly labeled as spurious simply because
they are so easy to track and many replicas are actually recovered
in the tractogram. Note that, once again, this potential danger
is not an issue only for dictionary-based approaches but it may
arise also in other techniques, both top-down and bottom-up,
because in presence of duplicates there exist infinite solutions
providing the same data fitting but characterized by an arbitrary
distribution of the actual fiber weight among its copies, which
depends on the specific optimization algorithm.
2.4. Partial fibers
Despite the fact that axons connect neurons located in the gray
matter, a number of factors actually cause part of the tracts
reconstructed with tractography to stop prematurely in the white
matter; it was shown in Côté et al. (2013) and Girard et al. (2014)
that up to 70% of the streamlines produced with state-of-the-art
tracking algorithms actually do not reach the gray matter. This is
a very well-known problem in tractography which, in turn, can
severely bias any subsequent connectivity analysis. Probabilistic
algorithms (Behrens et al., 2003; Parker et al., 2003) have been
proposed to deal with this uncertainty, but the interpretation
of the generated probabilistic maps as connection strength is
a controversial matter (Jones et al., 2012). In this section we
show that, if these “partial fibers” are not properly considered,
also microstructure informed tractography techniques can result
ineffective in fixing this issue with estimating the connectivity.
As an example, consider the simplistic scan-rescan analysis in
Figure 5 and assume it corresponds to a healthy subject, i.e., no
significant change is expected. If the connection strength between
the four regions R1, . . . , R4is quantified by the fiber count (Jones
et al., 2012), then a reduction in R1↔R2connectivity is observed
between the two sessions (from 2 to 1), and an increase in R3↔
R4(from 1 to 3), as the dashed tracts do not connect and so do
not contribute to the count. Hence, one might erroneously infer
that some sort of “disruption” took place in the former case and
“axonal remodeling” in the latter.
Microstructure informed tractography methods may help
obtaining unbiased connectivity estimates, but only in case
these partial fibers are removed from the tractogram before
optimization. In fact, their presence can potentially lead to the
same problem discussed in the previous section, i.e., contribution
of a bundle distributed among similar tracts; besides, although
partial fibers do not connect, they still contribute to the dMRI
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Daducci et al. Microstructure Informed Tractography
image during optimization. As a consequence, they “steal” in
practice part of the actual weight of a bundle and bias the final
connectivity estimation. It is interesting to notice that, since the
tracts are almost the same between scan and re-scan sessions,
the two tractograms actually fit the data equally well and thus, in
such situations, it is very difficult to spot possible troubles from
the fitting errors. In contrast, if global optimization is run using
only the tracts that actually connect, the total contribution of each
bundle can be recovered consistently between the two sessions, as
the trajectories provide a valid support for the optimization; this
FIGURE 5 | Toy problem to illustrate some issues of partial fibers on
connectivity analysis. Tracts connecting the four gray-matter regions
R1, . . . , R4are marked as solid lines and those stopping prematurely in the
white matter as dashed. It is well-known that the (random) number of partial
fibers recovered with tractography can significantly bias the estimation;
unfortunately, if these partial fibers are not properly handled, also
microstructure informed tractography techniques can result ineffective.
is true also for bottom-up approaches. In the example shown in
Figure 5, the weight of the connection R1↔R2will be estimated
to be 1 in both cases: in the first scan it is distributed over the two
tracts (but with sum equal to 1, as the total contribution has to
fit the measured data) whereas, in the re-scan, the sole fiber that
reached the gray matter is assigned a weight of 1. Similarly for
R3↔R4.
3. IMPORTANCE OF THE FORWARD
MODEL
The tissue forward-model is a very important ingredient for
investigating the evidence underpinning tractograms. Of course,
it is well-known that simpler models usually result in higher
fitting errors, but it is also easy to realize that when a model does
not explain adequately the measured signal, then the estimated
parameters are most likely unreliable and biased inferences may
be drawn about the connectivity. Although the identification of
the most appropriate model is an ongoing quest in the field, and
beyond the scope of this work, in this section we wish to draw the
attention of the reader to the repercussions of small differences in
the tissue model on the recovered parameters about the tractogram
and, consequently, on the estimation of the connectivity.
3.1. Fit Accuracy
Consider the toy problem in Figure 6A sketching a typical
situation observed in the brain (Figure 6D): the corticospinal
tract (CST) and the callosal projections of the corpus callosum
(CC) are two major fascicles consisting of tightly-packed axons
FIGURE 6 | Importance of the forward-model. The toy problem in (A) is used to analyze the behavior of two different forward-models. The model in (B) assumes
that the signal in each voxel originates exclusively from the tracts of the two fascicles F1 and F2 passing through it, whereas (C) considers all the possible water pools
that can contribute to the dMRI signal, in particular the extra-cellular space (EC) around the axons. (D–F) Analogous situation in the brain using the HCP datasets. See
description in the text for details.
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Daducci et al. Microstructure Informed Tractography
(yellow circles) that progressively fan out and eventually cross
(green circle). As shown by several independent studies (Assaf
and Basser, 2005; Alexander et al., 2010; Zhang et al., 2012),
differences in the packing density are compensated by variations
in the space around the axons themselves; note that this
consideration is implicitly or explicitly assumed in all state-
of-the-art microstructure techniques, and histological studies
corroborate this hypothesis (Jespersen et al., 2010).
Let us first analyze a model in which the signal in a voxel is
assumed to arise exclusively from the tracts passing through it,
as e.g., in Reisert et al. (2011), Smith et al. (2013), and Pestilli
et al. (2014). Therefore, given the tractogram in Figure 6A with
the blue and magenta ideal fascicles of fibers F1 and F2, their
contribution to the image can be controlled via the unknowns
xF1
1,...,4and xF2
1,...,4. An assignment where P4
i=1xF1
i=P4
i=1xF2
i=
1 (Figure 6B, left) would fit correctly the four corner voxels
but in the central one the sum of the contributions would be
over-estimated, i.e., PxF1
i+PxF2
i=2, violating the physical
constraint that the volume fractions in a voxel intrinsically sum
to one (Ramirez-Manzanares et al., 2007; Daducci et al., 2014b).
Setting P4
i=1xF1
i=P4
i=1xF2
i=0.5 (Figure 6B, right) fixes this
mismatch but then the signal in the four corners would be under-
estimated. Clearly, there is a tension in the model and not all
the voxels can be explained simultaneously; hence, a suboptimal
and biased solution is always returned which, in turn, might
affect the estimation of connectivity. The problem is that this
model does not take into account that the dMRI signal originates
from different water pools, i.e., intra- and extra-cellular, and that
their contribution is not homogeneous (Assaf and Basser, 2005;
Jespersen et al., 2010). In comparison, Figure 6C demonstrates
that all the voxels can be accurately explained when the signal is
modeled as a mixture of intra-cellular diffusion inside the tracts
and extra-cellular diffusion around them, as e.g., in Sherbondy
et al. (2010), Daducci et al. (2014a), and Reisert et al. (2014); in
this work, this latter is assumed anisotropic and parallel to the
axons, and its contribution in each voxel is controlled by the
variables xEC
1,...,5.
We also investigated how well the two models cope with such
situations in real brain and we computed, for each HCP dataset,
the normalized root-mean-square error (NRMSE) between the
measured dMRI signal in each voxel and the one predicted from
500,000 tracts recovered with the iFOD2 algorithm; Figure 6E
reports the overall mean (solid line) and standard deviation
(dashed lines). Indeed, the histograms reveal rather high fitting
errors (≈62% on average) if the extra-cellular space is not
considered; in contrast, the fitting is significantly more accurate
(≈19% error) when this compartment is included in the model.
Figure 6F shows the results for a representative subject in a slice
corresponding to panel d. In regions with densely-packed axons
(yellow circles), where the extra-cellular space is modest, the fit
is reasonably good (≈35%). However, large deviations from the
measured signal can be observed in voxels with crossings (green
circle, ≈60%) or partial volume with gray matter (≈90%),
where indeed the (missing) extra-cellular compartment appears
fundamental to properly explain all the voxel configurations
present in the brain. The fitting error certainly provides us with
useful information concerning how well a tractogram explains
the measured data, but nothing about its biological plausibility,
i.e., how well the tracts and the estimated contributions are in
agreement with the known brain anatomy. To this aim, in the
next section we will analyze the impact of these local fitting
inaccuracies on the estimation of brain connectivity.
3.2. Biological Plausibility
Besides the fitting errors, Figure 7 inspects as well the fraction of
the intra- (icvf ) and extra-cellular (ecvf ) compartments in each
voxel as predicted with global optimization from the input tracts
and using the two previous tissue-models. Tractography was
performed on the two-shell dataset using both iFOD2 and GIBBS
algorithms, without performing any filtering/pre-processing on
the reconstructed tracts. The NRMSE maps confirm our earlier
observation about the extra-cellular space, but now we can
also evaluate its impact on the estimated parameters. Using the
forward-model without extra-cellular compartment (left images),
the icvf map shows indeed a spatial distribution that does not
follow the expected pattern of neuronal density as found in
previous studies (Assaf and Basser, 2005; Alexander et al., 2010;
Jespersen et al., 2010; Zhang et al., 2012), that is higher density in
the major white-matter bundles, e.g., CC and CST, homogeneous
distribution in crossing regions and reduced values close to gray
matter. In fact, the icvf map appears almost flat, clear sign of
an incorrect assessment of the tract contributions. In contrast,
when the extra-cellular space is considered in the forward-model
(middle column), the estimated icvf and ecvf fractions closely
resemble known anatomy (blue arrows); as expected, the ecvf
map shows the opposite behavior. It is worth noting how this
latter map actually follows the same spatial pattern as the fitting
error of the model without extra-cellular space; we speculate it
reflects, once more, the need to consider in the forward model
all possible water pools that can contribute to the dMRI signal
in order to correctly assess the actual contribution of the tracts.
But this may not be enough. The rightmost maps show in fact
that, despite the model being the same and the fitting accuracy
very similar, distinct tractograms can lead to rather different
spatial distributions of the estimated tract weights, which are not
always consistent with the underlying anatomy (white arrows).
Therefore, it is really important not to rely only on the fitting
errors to compare tractograms or evaluate a model, but one should
always have a look as well at the parameters of interest estimated
with these microstructure informed tractography approaches,
e.g., in this example the voxelwise icvf and ecvf maps computed
from the tracts but, if more sophisticated biophysical models are
used, also more biological indices of tissue microstructure.
4. GLOBAL-SPECIFIC PITFALLS
4.1. Signal Intensity Inhomogeneity
Diffusion MR images can be affected by spatially-varying
modulations of the signal intensity that are caused, besides
actual tissue changes, by external factors such as local magnetic
field variations, gradient nonlinearities or imperfections in the
transmitter/receiver coils (Belaroussi et al., 2006). Although
intensity inhomogeneities are usually not a problem for visual
inspection of the images or voxelwise analyses, they can have
Frontiers in Neuroscience | www.frontiersin.org 8June 2016 | Volume 10 | Article 247
Daducci et al. Microstructure Informed Tractography
n
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FIGURE 7 | Biological plausibility. The fitting error (NRMSE) provides us with useful information about how well a tractogram explains the measured dMRI signal,
but nothing concerning the biological plausibility of the estimated tract contributions. To this end, it can be very useful to inspect, for example, the fractions of the intra-
(icvf) and extra-cellular (ecvf ) compartments in each voxel as predicted by global optimization and compare to the known anatomy. For example, blue arrows highlight
areas where the estimated fractions follow the expected pattern of neuronal density (see text for details), whereas white arrows point to the same regions where these
contributions appear unrealistic in the last column.
severe consequences for methods considering all voxels in
the optimization such as segmentation and registration (Vovk
et al., 2007). Figure 8 shows an example when microstructure
informed tractography is performed on raw dMRI images which
are corrupted by such a bias field, whose main component can
be observed in the direction pointed by the arrow. Compare
these results, obtained from GIBBS, with the icvf and ecvf maps
shown in Figure 7 (middle column), which were estimated after
correcting this bias using the N4 algorithm (Tustison et al., 2010),
as done in Daducci et al. (2014a) and Smith et al. (2015).
Without correction, this intensity modulation induces a
scaling in the estimated parameters as well, because the actual
contributions of the compartments are adjusted to fit the (scaled)
measurements. An immediate consequence is that these artificial
variations could be mistakenly interpreted as real alterations in
brain connectivity, e.g., a decreased efficiency of communication
in the frontal lobe in this example. But more subtle issues that
are more difficult to spot can emerge. Current microstructure
informed tractography techniques implicitly assume that the
properties of the tracts, e.g., axonal volume, remain constant
Frontiers in Neuroscience | www.frontiersin.org 9June 2016 | Volume 10 | Article 247
Daducci et al. Microstructure Informed Tractography
FIGURE 8 | Impact of signal inhomogeneities on global analyses. In this example, the artificial modulation of the intensities, e.g., along the arrow direction, leads
to a spatially-varying scaling of the estimated parameter maps that might be erroneously interpreted as a real change in the underlying connectivity.
along their trajectories3. Although this assumption might be
not totally correct, it is commonly accepted as good enough
for current applications. In the presence of such bias, however,
this constraint cannot be satisfied by the long association fibers
connecting the frontal and occipital lobes, for instance; these
tracts are likely to receive very low contributions as they are
inconsistent with the (scaled) data and, as discussed in Section 2,
artificial patterns might be introduced in the connectivity
estimates. Please note that this observation applies to both top-
down or bottom-up approaches. Hence, adequate procedures to
correct/compensate for any signal inhomogeneity in the data
are mandatory and should be employed in all analyses that use
microstructure informed tractography techniques for assessing
the connectivity.
4.2. Fiber Length and Convergence
The approaches recently-proposed in Daducci et al. (2013),
Daducci et al. (2014a), and Pestilli et al. (2014) have enabled
a drastic reduction in the computation time for this class of
problems by resorting to convex optimization. The convergence
rate of classical iterative methods for solving large-scale problems
of the form (2) largely depends on the spectral properties
of the matrix A. Although this is usually not a concern in
local reconstruction, for the reduced size of the problem and
the availability of well-conditioned basis functions (Ramirez-
Manzanares et al., 2007; Tournier et al., 2007), it can definitely
be an issue for these dictionary-based algorithms. In fact, as
illustrated in Figure 9A, axons vary greatly in length and,
evidently, so do the tracts reconstructed with tractography. Their
contribution to the image, and thus their relative importance
in the optimization process, can differ quite considerably; this
disparity can make the problem badly ill-conditioned and actions
must be taken to reach convergence in a reasonable amount of
time.
3One could argue that almost all tractography methods actually use this
assumption. To date the sole exception is represented by MesoFT (Reisert et al.,
2014), which allows for spatially-varying response functions but still enforces the
estimated parameters to be similar in adjacent segments of the tracts.
In Figure 9B, we monitor the progress of global
reconstruction when the dictionary Ais built from the original
tractogram (blue line) and after normalizing its columns by their
corresponding ℓ2-norms (red) so that all response functions
are treated uniformly during optimization; in mathematics this
operation is known as preconditioning. Although both problems
will eventually converge to the same global minimum, in the
preconditioned case the icvf map appears plausible already
after 50 iterations, whereas the original problem is still far
from an acceptable solution after 500. Apart from execution-
time considerations, a slow convergence-rate may result in
a premature termination of the optimization process, either
because the maximum number of iterations is reached or the
relative decrease in the cost function is very small and falls below
a predefined threshold. Hence, the configuration returned at this
stage might be sub-optimal if not arbitrarily flawed.
5. DISCUSSION AND CONCLUSION
In this article, we have discussed the problem of quantifying the
structural connectivity in the brain with diffusion MRI, placing
a particular emphasis on the effectiveness of a novel class of
algorithms known as microstructure informed tractography. In
the spirit of stimulating a constructive discussion in the field, we
have pointed out a series of pitfalls concerning the interpretation
of the results from these methods so that these issues may be
addressed. The intent of this work is not to suggest the idea that
these novel techniques are flawed or to discourage their usage; on
the contrary, we truly believe that the proposed solutions are very
promising and represent a viable direction to further improve the
estimation of brain connectivity. Nevertheless, the critical pitfalls
as presented in this work indicate that, perhaps, these methods
are not yet ready for use in “real-world” applications, due to a
number of serious questions that remain unanswered. If these
questions are overlooked or disregarded, a large potential for
severely biased findings and erroneous conclusions exists.
To drive the discussion we have focused our analyses on
dictionary-based methods (Daducci et al., 2013, 2014a; Pestilli
Frontiers in Neuroscience | www.frontiersin.org 10 June 2016 | Volume 10 | Article 247
Daducci et al. Microstructure Informed Tractography
A
B
FIGURE 9 | Fiber length and convergence. (A) The fibers in a tractogram vary greatly in length and thus their relative contributions to the dMRI image are quite
different. (B) This imbalance may lead to very slow convergence and to a premature termination of the optimization process.
et al., 2014), mainly because they provided us with (i) a solid
mathematical framework and (ii) a convenient analogy with
well-known local reconstruction techniques that allowed this
discussion to remain concrete and easy to follow. However, it
is worth noting that the concerns raised in this paper are quite
fundamental and are relevant to any other global technique.
For example, no matter what optimization strategy is used, all
filtering approaches are unable to recover the false negative
connections that are not included in the initial set of candidates.
In a bottom-up algorithm, the potential tensions that a particular
forward-model can introduce, as seen in Section 3.1, might
drive the minimization procedure to suppress segments in some
areas and prevent a number of streamlines to be eventually
reconstructed. It is relatively straightforward to extend this list
and extrapolate all the pitfalls previously described to other
tractography techniques; however, considering the variety of
algorithms available in the literature, such description would
be too lengthy for the scope of this work. To summarize our
general observations, in the following we outline a list of the
main take-home messages and open challenges that potential users
must keep in mind when trying to use microstructure informed
tractography techniques in their studies.
•As the methods covered in this work are all based on global
optimization, i.e., all voxels considered simultaneously, it is
important to realize that a problem in a specific location or
fiber bundle may have side effects anywhere else in the brain.
For example, if a fiber bundle is not reconstructed, some other
tracts in the tractogram might be assigned contributions to
compensate and fit the measured data. Conversely, if some
tracts are recovered multiple times, e.g., because easy to track
with tractography, then their weights may be arbitrarily split
among the duplicates. As a consequence, spurious tracts might
be detected as plausible and genuine ones as invalid. Therefore,
to make valid inferences about the data with these tools, the
composition as well as the features of the tractogram are
absolutely critical.
•A large number of models has been developed to estimate
properties of the neuronal tissue in a voxel from the observed
data, but there is still considerable debate about which one
provides the most accurate and useful features. Moving the
fitting from a single voxel to the whole brain does not solve this
dilemma but, instead, it may further amplify the uncertainty.
Different models explain the data more or less well and may
result in rather contrasting feature estimates; inaccuracies
in a voxel, combined with the global nature of the fitting
problem, can lead to unpredictable results occurring in any
other location of the brain or lead to biologically unfeasible
solutions. Carefully ponder the choice of the tissue model used
to assess the contribution of the tracts and consider its impact
in the interpretation of the observed outcomes regarding the
tractogram.
•The fitting error is certainly the first quantity one inspects
to evaluate the goodness-of-fit of a given model. Indeed,
high errors are a good indicator that the model fits the data
poorly and the estimated parameters are most likely unreliable.
However, the other way around is not always true. These
global methods optimize over an extremely high number
of unknowns and the degrees-of-freedom of such problems
are very large; hence, overfitting can easily occur and the
estimated parameters may not reflect the underlying anatomy.
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Daducci et al. Microstructure Informed Tractography
Therefore, it is utterly important to inspect and carefully
examine as well the maps of the obtained parameters in order
to actually evaluate the biological validity of the tractogram at
hand.
•Most existing tractography algorithms do not enforce in
the tracking process the anatomical prior that fibers must
origin and terminate in the gray matter; as a consequence,
a lot of recovered tracts actually stop prematurely in the
white matter. Although this is not an issue when visually
inspecting tractograms, these partial fibers can severely bias
the quantification of the structural connectivity and their
careful handling is essential for unbiased estimations.
•Due to a number of acquisition artifacts, MR images are
usually affected by spatial variations of the signal intensity.
This modulation may introduce artificial patterns in the
estimated parameters which, as previously pointed out, can
occur rather far from the actual location of the artifact in
the image. In the case of analyses with diffusion MRI, these
variations might be mistakenly interpreted as real alterations
in the connectivity; hence, correction strategies to remove
these artifacts from the images before applying these global
techniques are crucial, otherwise unpredictable results might
be obtained.
That being said, we encourage researchers working
on brain connectivity with diffusion MRI to try these
tools and explore the new and exciting possibilities they
offer, but also be aware of the current limitations. By
acknowledging their shortcomings as previously discussed
we hope to stimulate new ideas and research in this
direction, paving the way to enable, in the near future, truly
quantitative and biologically meaningful analyses of brain
connectivity.
AUTHOR CONTRIBUTIONS
Study conception and design: AD, ADP, JT; analysis and
interpretation of experimental results: AD, ADP, MD; drafting of
manuscript: AD, MD; critical revision: all authors.
ACKNOWLEDGMENTS
This work is supported by the Center for Biomedical Imaging
(CIBM) of the Geneva-Lausanne Universities and the EPFL, as
well as the foundations Leenaards and Louis-Jeantet. The authors
wish to thank Dr. Rafael Carrillo (EPFL, Switzerland) for all the
helpful discussions about convex optimization.
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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2016 Daducci, Dal Palú, Descoteaux and Thiran. This is an open-access
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BY). The use, distribution or reproduction in other forums is permitted, provided the
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Frontiers in Neuroscience | www.frontiersin.org 13 June 2016 | Volume 10 | Article 247
