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Brain and Behavior 2016; e00588 wileyonlinelibrary.com/journal/brb3
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1
© 2016 The Authors. Brain and Behavior
published by Wiley Periodicals, Inc.
Received: 3 May 2016
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Revised: 17 August 2016
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Accepted: 23 August 2016
DOI: 10.1002/brb3.588
Abstract
Background and Purpose: Diusion MRI tractography enables to invesgate white mat-
ter pathways noninvasively by reconstrucng esmated ber pathways. However, such
tractograms remain biased and nonquantave. Several techniques have been proposed
to reestablish the link between tractography and ssue microstructure by modeling the
diusion signal or ber orientaon distribuon (FOD) with the given tractogram and
opmizing each ber or compartment contribuon according to the diusion signal or
FOD. Nevertheless, deriving a reliable quancaon of connecvity strength between
dierent brain areas is sll a challenge. Moreover, evaluang the quality of a tractogram
and measuring the possible error sources contained in a specic reconstructed ber bun-
dle also remains dicult. Lastly, all of these opmizaon techniques fail if specic ber
populaons within a tractogram are underrepresented, for example, due to algorithmic
constraints, anatomical properes, ber geometry or seeding paerns.
Methods: In this work, we propose an approach which enables the inspecon of the
quality of a tractogram opmizaon by evaluang the residual error signal and its FOD
representaon. The automated ber quancaon (AFQ) is applied, whereby the
framework is extended to reect not only scalar diusion metrics along a ber bundle,
but also direconally dependent FOD amplitudes along and perpendicular to the ber
direcon. Furthermore, we also present an up- sampling procedure to increase the
number of streamlines of a given ber populaon. The introduced error metrics and
ber up- sampling method are tested and evaluated on single- shell diusion data sets
of 16 healthy volunteers.
Results and Conclusion: Analyzing the introduced error measures on specic ber
bundles shows a considerable improvement in applying the up- sampling method.
Addionally, the error metrics provide a useful tool to spot and idenfy potenal error
sources in tractograms.
KEYWORDS
diusion, error FA, error maps, ber up-sampling ber opmizaon, tractography
1Instute for Biomedical Engineering,
University and ETH Zurich, Zurich,
Switzerland
2MR-Center of the Psychiatric Hospital and
the Department of Child and Adolescent
Psychiatry, University of Zurich, Zurich,
Switzerland
3Department of Psychiatry, Psychotherapy
and Psychosomacs, Hospital of
Psychiatry, University of Zurich, Zurich,
Switzerland
Correspondence
Stefan Sommer, Department of Psychiatry,
Psychotherapy and Psychosomacs,
Hospital of Psychiatry, University of Zurich,
Zurich, Switzerland.
Email: sommer@biomed.ee.ethz.ch
ORIGINAL RESEARCH
Fiber up- sampling and quality assessment of
tractograms – towards quantave brain connecvity
Stefan Sommer1,2 | Sebasan Kozerke1 | Erich Seifritz3 | Philipp Staempi2,3
1 | INTRODUCTION
Diusion magnec resonance imaging (Le Bihan et al., 1986) is a com-
pelling tool for probing microscopic ssue properes and diusion
tensor imaging (DTI) has become a popular model to inspect white
maer architecture.
Tractography algorithms are able to reveal global fiber struc-
tures by estimating continuous streamline connections based
This is an open access arcle under the terms of the Creave Commons Aribuon License, which permits use, distribuon and reproducon in any medium,
provided the original work is properly cited.
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SOMMER ET AL.
on the local diffusion information throughout the brain (Basser,
Mattiello, & LeBihan, 1994a,b). The performance of tracking al-
gorithms has significantly improved by considering the infor-
mation contained in orientation distribution functions (ODF) or
fiber orientation distribution (FOD), especially in regions with
complex fiber configurations (Behrens, Berg, Jbabdi, Rushworth,
& Woolrich, 2007; Fillard et al., 2011; Tournier, Mori, & Leemans,
2011). However, tractograms remain biased by algorithmic- specific
parameters, that is, stopping criteria, curvature thresholds, seed
point distribution, and the choice of the tracking algorithm itself,
as well as partial volume effects of different fiber populations or
various tissue types within the acquired data voxels. This compli-
cates the estimation of reliable tractograms and thus the extraction
of biologically meaningful connectivity measures between brain
areas which are a crucial requirement for an accurate, quantitative
connectome across different populations (Jbabdi & Johansen- Berg,
2011; Jones, 2010; Jones, Knösche, & Turner, 2012). Lastly, be-
sides validation of diffusion pipelines with dedicated phantom data
mainly focusing on geometrical metrics of fiber tracts (Côté et al.,
2013), there is currently no objective way to inspect the quality of
tractograms in vivo, especially with respect to accurate quantifica-
tion of tracking errors.
The quancaon of white maer properes based on diusion
data also remains challenging. Fiber- specic metrics are quaned by
the generally unreliable ber- count (Jones et al., 2012) or ROI- based
approaches. The evaluaon of diusion metrics along segmented trac-
tography bundles was introduced by (Colby et al., 2012) and (Yeatman,
Dougherty, Myall, Wandell, & Feldman, 2012). The Automated Fiber
Quancaon (AFQ) framework allows the automac idencaon
and segmentaon of major white maer tracts and evaluates scalar
diusion measures such as fraconal anisotropy (FA) along these
trajectories to quanfy changes within the tract diusion proles
among dierent subjects or groups (Yeatman et al., 2012). A rst at-
tempt to correct for tractography biases by esmang an actual
contribuon for each tract was introduced by Sherbondy et al. using
a stochasc algorithm on a supercomputer architecture (Sherbondy,
Dougherty, Ananthanarayanan, Modha, & Wandell, 2009; Sherbondy,
Rowe, & Alexander, 2010). Another method introduced by Smith et al.
is based on a nonlinear gradient descent method called spherical-
deconvoluon informed ltering of tractograms (SIFT). This approach
removes bers of an inially large ber populaon to improve the t
between the streamline distribuon in each voxel and the ber ODF
(Smith, Tournier, Calamante, & Connelly, 2013). Thereby, a cost func-
on describing the deviaon between ber densies and FOD lobe
integrals is minimized by iteravely removing bers. Fiber densies are
calculated by incorporang the length and tangent of reconstructed
bers within a voxel and compared to the corresponding ber ODF
lobes. However, the SIFT approach requires a large amount of inial
bers to determine an opmized subset of included and excluded ber
tracts.
Its successor, SIFT 2 (Smith, Tournier, Calamante, & Connelly,
2015) reduces this requirement, as it determines an eecve cross-
seconal area for each streamline, represented by a oang- point
weighng factor for each ber, instead of a binary keeping or removing
of bers in comparison to the inial SIFT.
Peslli, Yeatman, Rokem, Kay, & Wandell (2014) introduced a sim-
ilar method, that is, linear fascicle evaluaon (LiFE), which is based on
the diusion signal, predicted from the connectome, instead of the
FOD. The default forward model is a degenerated tensor represent-
ing a sck with zero radial diusivity. To deal with isotropic com-
partments, the signal mean is subtracted in each voxel prior to the
opmizaon. Daducci, Dal Palu, Lemkaddem, & Thiran (2015) pursued
a similar approach introducing the Convex Opmizaon Modeling
for Microstructure Informed Tractography (COMMIT) framework,
though using a more complex forward model by describing both the
intracellular sck model, and the extracellular compartment by a ten-
sor. Furthermore, gray maer and cerebrospinal uid (CSF) are also
represented with two disnct isotropic components. It is tempng
to interpret the resulng ber weights as quantave connecvity
measures between brain regions, however, the described opmizaon
methods have their own pialls. For example, in voxels with poor or
incorrect ber representaons due to tracking errors, noise or paral
volume contaminaons, compartments are typically overcompensated
by increasing the weights of the few present bers, isotropic or ex-
tracellular compartments in order to decrease the global t error. An
overview of pialls and open challenges is given in (Daducci, Dal Palu,
Descoteaux, & Thiran, 2016).
Here, we propose a novel approach which enables the inspecon
of the quality and validaon of a tractogram opmizaon such as
COMMIT by evaluang FOD characteriscs of the error signal along
and perpendicular to ber bundles by ulizing the AFQ framework.
The quality metrics proposed allow for a beer understanding of the
accuracy and error sources of tractograms and help idenfying regions
with poorly ed data. We further show that these metrics, combined
with a newly introduced error FA, allow a beer interpretaon of the
direconal error distribuon. These are important steps toward in-
terpreng ber weights from a tractogram opmizaon in a quan-
tave way to, for example, construct a more meaningful connecvity
measure in a connectome. Furthermore, we also present a ber up-
sampling procedure: It allows to increase the number of streamlines
of a given ber bundle, in case of, for example, underrepresentaon
of a certain structure due to anatomical properes, ber geometry,
seeding paern or algorithmic constraints. Analyzing the introduced
error measures on specic ber bundles shows the benet of using
up- sampled ber bundles.
2 | MATERIALS AND METHODS
The major steps of a typical connectome generaon process is shown
in a simplied form in Figure 1. It is crucial to perform the opmiza-
on aer the segmentaon and up- sampling steps in order to avoid
the paral ber problemac discussed in (Daducci et al., 2016). In this
work, in contrast to a connectome pipeline, the segmentaon step
is not based on corcal parcellaon, but performed using the AFQ
framework (AFQ: RRID:SCR_014546). This choice was movated by
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SOMMER ET AL.
the ability of the AFQ framework to reliably quanfy measures along
tracts.
The method secon is organized as follows. First, the acquision
protocol, preprocessing steps and tractography algorithm is described.
However, these parameters can easily be swapped with other proto-
cols or tractography algorithms. Thereaer, the AFQ segmentaon,
ber up- sampling, COMMIT opmizaon and error quancaons,
including the introduced error measures are described in more detail.
2.1 | In- vivo diusion data acquision
Diusion MRI data were acquired on a Philips Achieva 3T TX system
(Philips Healthcare, Best, the Netherlands), equipped with 80 mT/m
gradients and a 32- element receive head coil array, using a diusion-
weighted single- shot spin echo EPI sequence. The study was ap-
proved by the local ethics commiee and meets the guidelines of the
declaraon of Helsinki. Wrien informed consent was obtained from
all subjects.
Data sets from 16 healthy volunteers (age: 31.6 ± 8.6, gender: 12
male, 4 female) were acquired with the following diusion scan parame-
ters: TR: 11.85 s, TE: 66 ms, FOV: 220 × 220 mm2, with 40 conguous
slices, slice thickness: 2.3 mm, acquision and reconstrucon matrix:
96 × 96, SENSE factor: 2, paral Fourier encoding: 60%. Diusion-
weighted images were acquired along 64 direcons distributed uni-
formly on a half- sphere with a b- value of 3000 s/mm2 in addion to
a b = 0 s/mm2 scan, resulng in a scan me of approximately 13 min.
Addionally, 1 mm isotropic T1-weighted structural images were re-
corded with a 3D MP- RAGE sequence (FOV: 240 × 240 × 160 mm3,
sagial orientaon, 1 × 1 × 1 mm3 voxel size, TR: 8.14 ms, TE: 3.7 ms,
ip angle: 8°).
2.2 | Preprocessing and tractography
For each data set, the diusion data was corrected for eddy- currents
and subject moon by FSL: RRID:SCR_002823 (EDDY) (Jenkinson,
Beckmann, Behrens, Woolrich, & Smith, 2012). The white maer
mask was esmated from the T1- weighted data set using the ssue
segmentaon in SPM8: RRID:SCR_007037 (www.l.ion.ucl.ac.uk/
spm) and transformed back to diusion space using SPMs coregister
funcon based on normalized mutual informaon. A Fiber Assignment
by Connuous Tracking (FACT) inspired determinisc algorithm gen-
eralized to the Orientaon Distribuon Funcon (ODF) was used in
the tractography step. The ODF was reconstructed using the FRACT
method (Haldar & Leahy, 2013). The tracking direcon was selected
according to the local diusion maximum of the ODF. Ten seeds were
started in each white maer voxel, resulng in approximately 700,000
bers per subject. The esmated white maer mask was only used
for seeding purposes and was not ulized as a tractography stopping
criterion.
2.3 | Fiber segmentaon and up- sampling
The segmentaon of the tractograms was performed using the AFQ
framework (Yeatman et al., 2012), which is based on a waypoint ROI
procedure as described in (Wakana et al., 2007). Addionally, a re-
nement step was applied, which compares each candidate ber to
tract probability maps (Hua et al., 2008). To avoid conicng start and
endpoints of bers running through the two ROIs of the target ber
structure, a ip was performed on all tracts which rst passed through
the second ROI, resulng in consistent ber alignment in each bundle.
These segmentaon steps resulted in the selecon of 20 major white
maer ber tracts (Yeatman et al., 2012) out of all white maer b-
ers contained in the whole- brain tractogram (18 bundles as described
in (Yeatman et al., 2012), and two addional tracts as dened in the
online version: hps://github.com/jyeatman/AFQ).
Next, to increase the number of bers of potenally underrepre-
sented ber populaons in the dierent AFQ segmented bundles, for
example, due to tractography algorithm biases, the following method
was applied: The segmented bers were equidistantly resampled using
80 interpolaon points per ber and principal component analysis
(PCA) was applied to all classied and resampled bers (Parker et al.,
2013). The space was truncated to the rst 80 dimensions (from the
240 point descriptors), whereby more than 99% of the explained vari-
ance was sll captured. In the PCA space, for each bundle separately,
new bers were randomly generated according to the point distribu-
on of the transformed bers, assuming a bundle- specic mulvariate
Gaussian distribuon. The newly generated bers were transformed
back by inverng the linear PCA transformaon.
In a further step, potenal outliers were idened based on the
calculaon of a populaon- mean ber, that is, the mean value of all
corresponding resampled points of the inial bers within one ber
bundle. The distance of each randomly generated ber to the original
populaon- mean ber was derived by summing up the distances to
the nearest points on the mean ber. New bers were only accepted if
FIGURE1 A schematic connectome
pipeline is depicted including the positions
for proposed up- sampling and validation
steps
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SOMMER ET AL.
the distance- threshold to the inial populaon was met. This thresh-
old was set to the maximum ber distance of all bers within the inial
populaon relave to its mean ber. Newly generated tracts leaving
the white- maer mask were also rejected. Based on these ber pop-
ulaon up- sampling steps, addional 10,000 bers per bundle were
generated for each data set.
Finally, the up- sampled bers were again segmented using the
AFQ framework to apply the same classicaon criteria to the newly
generated bers as to the inial tractogram. Around 75% of the up-
sampled bers were successfully classied and therefore kept for the
further analysis. With the procedure described above, a total of four
tractography sets were generated:
AFQ Classied AFQ bers based on the inial tractogram
AFQUP AFQ set combined with the up- sampled AFQ bers
WB Inial whole- brain tractogram
WBUP WB combined with the up- sampled AFQ bers
2.4 | Fiber opmizaon, opmized tractogram
The opmizaon of the dierent tractogram sets was performed
using the COMMIT framework (Daducci et al., 2015) by apply-
ing the Sck- Zeppelin- Ball model (Panagiotaki et al., 2012) for
modeling the ber signal. The intracellular sck model was gener-
ated with a longitudinal diusivity of d∥ = 1.7 × 10−3 mm2/s. In addi-
on, in each voxel, a hindered contribuon was included for every
unique FOD peak using the Zeppelin model assuming a perpen-
dicular diusivity d⊥ = 0.5 × 10−3 mm2/s and longitudinal diusivity
d∥ = 1.7 × 10−3 mm2/s. Lastly, two isotropic compartments account-
ing for paral volume with gray maer and cerebrospinal uid were
modeled with diusivity
d
∈
{1.7,3.0}
×
10−3mm2
∕
s
. The nondiusion
weighted b = 0 image was used to normalize the diusion data. The
convex opmizaon problem of the following form
where y is the vector containing the normalized diusion signal, A is
the linear operator or diconary and x is the vector of the contribu-
ons, was solved using a forward- backward, fast iterave shrinkage-
threshold algorithm (hps://github.com/daducci/COMMIT), resulng
in a soluon
x
. Stopping criteria for the opmizaon were either a
maximum number of 500 iteraons or a minimum relave change of
the objecve funcon of 1e- 4.
2.5 | Error quancaon
In addion to the normalized root mean square error (NRMSE) of the
opmizaon t, an actual signal esmator
s
was calculated using
A
x
, by reverng the b = 0 normalizaon. To further examine the dier-
ences and similaries between this signal esmator
s
and the acquired
diusion data s, a direconal error FOD of the signal esmator
s
and
the original diusion data s was calculated. Remaining signal contribu-
ons from under- or overrepresented bers are assumed to remain
in the error signal. The FOD for the diusion signal esmator was
reconstructed by applying the constrained spherical deconvoluon
(Tournier, Calamante, & Connelly, 2007) to the error signal, which is
dened by the element wise dierence between the measured and
esmated diusion signals:
In order to use a meaningful deconvoluon kernel and to be com-
parable to the FOD derived from the measured signal s, the response
funcon was not re- esmated on the error signal; instead the ber
response from s was used. A maximum spherical harmonics order of
lmax = 8 was used. Furthermore, a tradional tensor t of the signal
error serr was derived in order to calculate the fraconal anisotropy
(FA) of serr.
To quanfy the dierent error measures along the segmented and
opmized AFQ ber bundles, we extended the tract prole genera-
on of the AFQ framework. In (Yeatman et al., 2012), the locaons
of the used waypoint ROIs from the segmentaon step (2.3) isolate
the central trajectories of the fascicles. Next, dierent scalar diu-
sion measures (FA, RD, etc.) are evaluated along the central poron
of the ber bundle by clipping and resampling each ber according
to the main segment between the ROIs. Bundle properes are then
summarized at each node by taking a weighted average according
to the Mahalanobis distance of each ber tract core as described in
(Yeatman et al., 2012).
In this work, instead of invesgang tradional scalar diusion
quanes as proposed in the AFQ framework, we examined scalar
measures such as the t NRMSE and the introduced error FA along
the segmented AFQ tracts. Furthermore, the three- dimensional error
FOD was also evaluated by calculang longitudinal and perpendicu-
lar error FOD amplitudes for each segmented AFQ ber. These mea-
sures depend on the ber direconality and are not scalar maps. The
maximum peak- amplitude along a ber tract is dened by the maxi-
mum FOD amplitude in a cone around the ber orientaon with an
opening angle of π/6. The maximum peak- amplitude perpendicular
to the ber is the maximum of all sampling points outside this cone
(Figure 2).
For every tractogram set (n = 4), following parameters were an-
alyzed along each of the 20 segmented fiber bundles: NRMSE, error
argmin
x≥0
∥Ax−y∥
2
2
s
i
err
=
√
(si−
si)2
FIGURE2 Schematics showing the fiber orientation distribution
(FOD) evaluation along a fiber tract: longitudinal maxima are marked
by stars (within the cone), perpendicular maxima are marked with
circles (outside of the cone)
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SOMMER ET AL.
FOD along, error FOD perpendicular, and error FA. These measures
were tested for statistical significance between the initial and up-
sampled tractogram sets and were corrected for multiple com-
parison, using the nonparametric permutation test implemented
in FSL (Winkler, Ridgway, Webster, Smith, & Nichols, 2014). The
number of permutations were set to 5000 with a significance level
of p < .05.
Furthermore, the up- sampling method was also compared with
an increase of seed points during the tractography step. Therefore,
the number of seed points was increased incrementally up to a
factor of eight in a single subject. The resulng tractogram sets
were segmented using the AFQ framework and either opmized
or up- sampled and opmized for the comparison. The up- sampled
tractogram sets were also segmented a second me prior to the
opmizaon.
3 | RESULTS
In Figure 3, the mean NRMSE of all four tractogram sets are shown
for every subject (N = 16) aer the opmizaon with the COMMIT
framework. The error in the up- sampled populaons (AFQUP and
WBUP) is decreased compared to the inial sets (AFQ and WB) for
each subject, and comparison at the group level shows a highly sig-
nicant decrease in the mean NRMSE between AFQ and AFQUP and
between WB and WBUP (paired samples, p < .001). Furthermore, the
whole- brain tractograms (WB and WBUP) also showed lower errors
compared to the AFQ and AFQUP.
The dierent segmented AFQ ber bundles that are discussed
in further detail in the following secons are illustrated in Figure 4.
Figures 5–8 show the tract prole of the NRMSE, error FA, longitudinal
and perpendicular FOD error in selected bundles to illustrate dierent
distribuons of the error signal and performance of the up- sampling
method.
Figure 5 shows the NRMSE along three major bundles (le and
right hemisphere) in the four tractograms sets (AFQ, AFQUP, WB,
WBUP). The colored secon of the depicted bundles describe the
core of the bundle, whereas the x- axis in the subplots shows the 100
parameterized points between ROI 1 and ROI 2. In Figures 5–8, the
ROIs are marked with 1 and 2 to emphasize the start and end region
of the parameterizaon.
The lower error in the up- sampled tractograms (AFQUP, red
line, WBUP, black line) compared to AFQ and WB (blue, green line)
achieved a beer t compared to the inial sets (AFQ, WB). In most
parts, the t error signicantly decreased (p < .05) aer mulple com-
parison correcon using FSL’s randomize. Regions of stascal signif-
icance are highlighted with a transparent overlay in the color of the
tractogram set with a higher value (e.g., blue for AFQ).
The FA of the error signal gives further insight into the opmiza-
on results. In Figure 6, three dierent types of error FA behavior are
shown as an example. The Corcospinal Tract showed a stascally
signicant reducon of the error FA in the up- sampled populaons
(AFQUP vs. AFQ and WBUP vs. WB), which is desirable in order to
reduce a direconal bias in the residual diusion signal. Nevertheless,
structural tendencies along the bundle are sll visible, especially in the
second quarter of the bundle, where the error FA is clearly increased
in all of the tractogram sets. The error FA in the Callosum Forceps
Major could not be reduced by applying the up- sampling method, and
especially in the middle part of the bundle, direconal biases in the
residual diusion signal remain clearly visible. In contrast, the Inferior
Longitudinal Fasciculus (ILF) revealed a relavely isotropic error signal,
expressed by low FA values, and no disnct structure in the error FA,
FIGURE3 Optimization results
showing the mean normalized root mean
square error (NRMSE) for each subject
between (a) automated fiber quantification
(AFQ) and AFQUP, and (b) WB and WBUP;
(c) group average for the four tractogram
sets, the error bars depict one standard
error
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SOMMER ET AL.
that is, no direconal bias in the residual diusion signal along the
bundle was observed.
In Figure 7, the longitudinal FOD error is evaluated along the
disnct ber bundles. The Corcospinal Tract showed a signicantly
(p < .05) reduced longitudinal error in both up- sampled sets compared
to the inial tractograms. In the Superior Longitudinal Fasciculus (SLF),
the up- sampling reduced the error in the AFQ populaon (AFQ vs.
AFQUP,). The longitudinal error was already low in the WB tracto-
gram set for the SLF, and could not be further reduced in a stascally
signicant manner by up- sampling the bundle (WBUP). The Arcuate
Fasciculus showed a similar behavior, whereas the up- sampling signi-
cantly reduced the longitudinal error in the AFQ cases. Addionally,
in the WB sets, the up- sampling sll signicantly reduced the longitu-
dinal error in the temporal part of the bundle (WBUP) but the overall
dierence is drascally reduced.
Figure 8 depicts the perpendicular FOD error in the segmented
ber bundles of the right Thalamic Radiaon, Callosum Forceps Minor
and the le Arcuate Fasciculus. For the AFQ case, the up- sampled sets
showed a signicantly higher error in the Thalamic Radiaon and the
Arcuate Fasciculus in some parts, even though the overall mean t
error (NRMSE) was reduced. If all the bers are taken into account
(WB, WBUP), the up- sampled populaon (WBUP) does not show a
signicant increase of the perpendicular error anymore.
Figure 9 shows a coronal cross secon through the Corona Radiata
of a single subject. The reconstructed FODs from the measured diu-
sion signal are depicted in gray, with the colored error FODs derived
from the WB set shown on top. Most voxels exhibit a small error FOD
compared to the signal FOD, implicang a good agreement between
the signal esmator from the opmizaon and the measured signal.
Nevertheless, in some voxels, the error FOD is relavely large compared
to the signal FOD. Three of those voxels are highlighted in a, b and c.
Figure 10 depicts the comparison between increasing the num-
ber of seed points during the tractography and up- sampling the
segmented ber bundles in a single subject. Each tractogram set is
ploed with the number of bers on the x- axis in order to compare
the same number of bers. The up- sampling method clearly outper-
forms the increase in seed points, whereby the largest improvement is
achieved by the rst up- sampling step.
4 | DISCUSSION
We have introduced a tool to invesgate the quality of a tractogram
by further inspecng the direconally dependent error signal be-
tween the signal predicon and the measured diusion signal along
reconstructed ber bundles. Addionally, we presented a method to
up- sample a given ber populaon in order to achieve beer opmi-
zaon results, that is, a decreased t error.
The overall mean t error averaged over all the white- maer
voxels and all subjects showed only small, but nevertheless signi-
cant changes comparing the inial (AFQ, WB) with the up- sampled
(AFQUP, WBUP) ber tractograms. These small changes at the group
level could be aributed to large intersubject variability; however, the
up- sampled sets achieved a reduced t error in each single subject
(Figure 3a and b). The improved signal t achieved by the up- sampling
method was highly stascally signicant for both the AFQ vs. AFQUP
and WB vs. WBUP tractogram sets. Further inspecon of the NRMSE
along the major segmented ber bundles showed high similarity be-
tween the matched le and right structures. However, dierences
were found between various structures, for example, the superior part
of the Corcospinal Tract was highly improved by the up- sampling
method, whereas the frontal part of the Thalamic Radiaon was mostly
unaected by the up- sampling procedure. Variable performance of the
up- sampling method across structures might be caused by the qual-
ity of the inial bundle representaon and also by voxels surround-
ing these bundles. Systemic errors were expected and observed in
FIGURE4 A selection of the discussed
segmented fiber bundles of a single
representative subject are shown in
different colors. In the sagittal view, the
right Corticospinal Tract, right Arcuate
Fasciculus, right Inferior Longitudinal
Fasciculus (ILF), the Callosum Forceps
Minor, and the right Uncinate Fasciculus
are illustrated. The axial slice depicts the
left and right Thalamic Radiation, left and
right Superior Longitudinal Fasciculus (SLF)
and the Callosum Forceps Major
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SOMMER ET AL.
FIGURE5 Mean normalized root mean square error (NRMSE) of the optimization over all subjects for the left and right Corticospinal Tract,
Thalamic Radiation, and Arcuate Fasciculus. Subplots 1–6 shows the mean NRMSE of the automated fiber quantification (AFQ) fiber set (blue,)
and of the up- sampled tractogram set AFQUP (red). The dashed red and blue lines indicate one standard error. Subplots 7–12 shows the mean
NRMSE of the WB fiber set (green), and of the WBUP- tractogram set (black). The dashed green and black lines indicate one standard error.
Areas with a significant error reduction (according to FSL’s randomize, p < .05) in AFQUP compared to AFQ are overlaid in transparent blue
(AFQ > AFQUP) or green (WB > WBUP), respectively
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the AFQ tractogram sets (AFQ, AFQUP) due to the fact that many
bers are not covered by the 20 major bundles and therefore excluded
from the opmizaon. The signal of crossing, nonsegmented struc-
tures are missing in regions with high NRMSE in the AFQ and the
AFQUP tractograms. While the AFQUP set showed a reduced error
in almost all structures compared to the AFQ set, a compensaon of
nonsegmented crossing structures remains unachievable by merely
up- sampling the segmented bundles without the introducon of miss-
ing crossing structures. Segmenng the AFQ bundles introduces an
addional source of error due to predened ROIs and registraon
steps during the AFQ bundle classicaon. Fibers, which pass through
the disnct bundles but, for example, not through the two ROIs are
consequently unclassied, and therefore missed in the opmizaon.
In the whole- brain sets (WB and WBUP, Figure 5) no ber populaons
were purposely omied, and therefore a much more homogeneous
NRMSE distribuon was found in the brain.
In a next step, we further explored the error distribuon across
the diusion direcons in each voxel. Therefore, the error FA was cal-
culated and evaluated to reveal potenal anisotropy in the error sig-
nal (Figure 6). In voxels with a good t, a low anisotropy is expected,
that is, a homogeneous distribuon of the error across all diusion
direcons. In comparison to the NRMSE, the error FA appears to be
a sensive measure for recognizing badly represented regions, even if
all the bers are taken into account (WB, WBUP). A bad ber repre-
sentaon or an inaccurate forward model can cause a high error FA as,
for example, observed in the middle secon of the corpus callosum
FIGURE6 The fractional anisotropy (FA) of the error signal is shown along three selected bundles (Corticospinal Tract, Callosum Forceps
Major, and Inferior Longitudinal Fasciculus). The mean error FA over all subjects derived from the initial optimized, nonup- sampled sets are
depicted in blue (automated fiber quantification [AFQ]) and green (WB), the error FA derived from the up- sampled sets are displayed in red
(AFQUP) and black (WBUP). The dashed lines indicate one standard error
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SOMMER ET AL.
and parts of the Corcospinal Tract. These bundle segments also have
a high signal FA, which might indicate that the chosen forward model
underperforms in high FA voxels. Despite the clear disncon of high
error FA regions in the graphs, it is rather dicult to dene an accurate
baseline for the residual error signal in order to disnguish structurally
related residuals from pure noise. An accurate signal- to- noise rao es-
maon of the diusion signal would be needed which is typically also
spaally varying due to mulple acceleraon methods.
Complex ssue architecture of crossing ber populaons within a
single voxel cannot be fully modeled by tensor based metrics, there-
fore the FOD of the error signal was also evaluated along and perpen-
dicular to the major ber bundles (Figures 7 and 8). By disentangling
the error signal into a perpendicular and longitudinal error, the exact
source of the tractogram error can be observed. Poorly represented
structures can therefore be discriminated from over- or underes-
mated crossing structures. The longitudinal error in the Superior
Longitudinal Fasciculi and in the Arcuate Fasciculi diers strongly in
the inial populaons (AFQ, WB), which is most plausibly caused by
segmentaon dicules. However, in case of the Arcuate Fasciculus,
applying the up- sampling method in the WB set further signicantly
reduced the longitudinal error. By further invesgang the perpendic-
ular error, a signicantly higher error in the AFQUP set was idened
for the rst me. In the WB sets, this eect diminishes and can there-
fore be explained by missing ber populaons not embodied in the
segmented AFQ bundles.
These ber- dependent direconal measures combined with the
error FA enable to detect and disnguish possible error sources,
namely bad ber bundle representaon, missing crossing structures or
a poor forward- model t.
Missing crossing structures in the AFQ sets can be found for ex-
ample, in the lower part of the Corcospinal Tract. The Cerebellar
Peduncles, are passing superior to the rst Corcospinal ROI and might
FIGURE7 Longitudinal error fiber orientation distribution (FOD) peak amplitudes along three representative bundles are shown (Left
Corticospinal, Right Superior Longitudinal Fasciculus and Right Arcuate Fasciculus). The mean across all subjects using the initial tractograms
are shown in blue (automated fiber quantification [AFQ]) and green (WB), the mean longitudinal FOD error in the up- sampled sets are depicted
in red (AFQUP) and black (WBUP). The dashed lines indicate one standard error, statistically significant regions (p < .05) are highlighted with
transparent surfaces (blue: AFQ > AFQUP, green: WB > WBUP)
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SOMMER ET AL.
be the reason for an increased error FA and NRMSE. Another example
is along the middle part of the Arcuate Fasciculus next to the supe-
rior region of the Corona Radiata. Besides The Superior Longitudinal
Fasciculus, which is also included in the major AFQ bundles, the
Posterior Vercal Arcuate and the Vercal Occipital Fasciculus also
populate this area and are not segmented, using the AFQ framework.
The provided tools allow an extensive inspecon of tractograms
and their opmizaon by exploring bundle- specic and direconally
dependent error measures. To the author’s knowledge, this is the rst
study that facilitates a deepened insight into the remaining local er-
rors induced by the tractogram or the opmizaon procedure itself.
This step is crucial in order to get a beer understanding of the actual
goodness of t of the tractogram.
In Figure 9, dierent cases of error FODs are highlighted. In voxel
a), the direconality of the error FOD matches the inial FOD. Most
certainly, some of the passing bers were under or over- esmated in
that parcular voxel. The error FOD in voxel b) shows a completely dif-
ferent characterisc as the inial FOD. The geometry of the recon-
structed bers do not match the measured diusion signal. The third
case (voxel c) is a combinaon of both cases, where the local weighng
deviates from the diusion signal and also the error FOD peaks are
slightly lted. Other voxels show very small error FODs and some spu-
rious peaks do occur, whereby the assumpon of an underlying ber re-
sponse was violated. Nevertheless, the amplitudes of these error FODs
are very small and will not inuence the resulng along- tract analysis.
A possibility to migate the ber assumpon of the kernel func-
on would be to apply a model- free Funk- Radon- Transformaon to
the error diusion signal instead of deconvolving the error signal with
a ber response funcon. The resulng error orientaon distribuon
funcon (ODF) would not suer from spurious peaks.
FIGURE8 Perpendicular error fiber orientation distribution (FOD) peak amplitudes across the selected automated fiber quantification (AFQ)
bundles (Right Thalamic Radiation, Callosum Forceps Minor, Left Arcuate Fasciculus): the mean over all subjects using the initial tractograms
are shown in blue (AFQ) and green (WB). The perpendicular error FOD peak amplitudes in the up- sampled sets are drawn in red (AFQUP) and
black (WBUP). The dashed lines indicate one standard error statistically significant regions (p < .05) are highlighted with transparent surfaces
(red: AFQUP > AFQ, green: WB > WBUP)
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SOMMER ET AL.
The ulmate goal of using streamline weights to quanfy connec-
vity strength between corcal regions is sll not achievable if the
tractogram or the opmizaon is awed. The ber- dependent error
esmaon could also be used to esmate a condence of the re-
sulng ber weights aiming to verify the integrity of a quantave
connecvity matrix between corcal regions. Addionally, the di-
culty of oming bers due to any segmentaon and its inuence on
to the opmizaon was shown. In the given framework, the segmen-
taon method can easily be exchanged with a segmentaon, ,for ex-
ample, based on a corcal parcellaon. Thereby, the up- sampling can
also be applied to more ber bundles. However, other segmentaon
methods, for example, based on a corcal parcellaon scheme also
suer from unclassied bers which cannot be included in the tracto-
gram opmizaon step.
The introduced ber up- sampling method improved the tractogram
representaon, hence led to a signicantly superior opmizaon ex-
pressed by a reduced NRMSE. Even though, the introduced approach
did not enhance the opmizaon in every single structure, impairment
caused by the ber up- sampling was not observed. Addionally, the
up- sampling was performed per bundle, whereby a bundle segmenta-
on is a necessary prerequisite.
This requirement might be eliminable, if a dierent sampling strat-
egy is applied to randomly draw samples in the PCA space. The as-
sumpon of a mulvariate Gaussian distribuon is no longer valid in
a whole- brain ber populaon and would lead to many implausible
up- sampled bers.
In Figure 10, the ber up- sampling is compared with increasing
the number of seed voxels during tractography. Even an eight- fold in-
crease of bers did not improve the opmizaon result substanally.
However, the up- sampling is not only computaonally less intensive,
FIGURE9 A coronal section through
the Corona Radiata is shown in a
single subject, whereby the signal fiber
orientation distribution (FOD)s derived
from the measured diffusion signal are
depicted in transparent gray, the error
FODs derived from the WB set are
overlaid in color. Three different voxels are
highlighted where the error is significantly
larger compared to the other voxels
FIGURE10 Different number of seed voxels and the resulting
mean normalized root mean square error (NRMSE) for the automated
fiber quantification (AFQ) and AFQUP tractogram sets are shown
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SOMMER ET AL.
it also introduces new bers with dierent features, a larger spaal
extent, and therefore novel trajectories and the up- sampled bers do
dier substanally from the inial populaon. These new bers con-
tribute signicantly to a more opmal tractogram set. Similar eects
were reported by (Takemura, Caiafa, Wandell, & Peslli, 2016) while dif-
ferent algorithm parameters were introduced instead of only increas-
ing the number of seed points. The presented framework also enables
the comparison of dierent tractogram sets from various sources and
allows a more extensive inspecon of each tractogram and its strength
and weaknesses without dening an explicit gold standard.
For a reliable ber quancaon, it is crucial to eliminate error
sources such as a bad t due to wrong choice of the forward model,
poor tractogram representaon caused by the choice of tracking
algorithm and tractography parameters or an overcompensaon
in voxels with a low number of streamlines. As discussed in (Smith
et al., 2015), FOD lobes of voxels containing very lile streamlines
compared to the actual measured ber density will result in assign-
ing high weights to those bers in order to reduce the error in that
parcular voxel, but introducing biases in all the other voxels tra-
versed by those bers. Similarly, the COMMIT model will assign
higher volume fracons to extracellular compartments in order to
compensate for missing streamlines represenng the intracellular
compartment. The presented up- sampling procedure can help to
limit voxels containing a low number of streamlines, especially in
cases where a higher track density cannot be achieved by simply
increasing the number of seed voxels. With respect to global tractog-
raphy algorithms, an increase of bers is typically computaonally
very expensive, whereas the described up- sampling method is very
ecient. The introduced error measures such as the error FA or the
error FOD itself could also be fed back into the tractography process
to inuence the placement of new seed voxels or adjust tractogra-
phy parameters in order to achieve a more representave tracto-
gram. In this study, only single shell diusion data was used for the
opmizaon, whereas the COMMIT framework clearly benets from
mulple shells, for example, in order to disnguish between dierent
isotropic compartments.
CONFLICTS OF INTEREST
None declared.
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How to cite this arcle: Sommer, S., Kozerke, S., Seifritz, E. and
Staempi, P. (2016), Fiber up- sampling and quality assessment of
tractograms – towards quantave brain connecvity. Brain and
Behavior, 00: 1–13. e00588, doi: 10.1002/brb3.588
