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Towards quantitative connectivity analysis: reducing tractography biases
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Gabriel Girard a,b, Kevin Whittingstall c,d, Rachid Deriche b, Maxime Descoteaux a,d
a Sherbrooke Connectivity Imaging Lab (SCIL), Computer Science Department, Faculty of Science, Université de Sherbrooke, 2500 Boulevard Université, Sherbrooke, QC, Canada J1K 2R1b Project Team Athena, INRIA Sophia Antipolis Méditerranée, 2004 Route des Lucioles BP 93, 06902 Sophia Antipolis Cedex, Francec Department of Diagnostic Radiology, Faculty of Medicine and Health Science, Université de Sherbrooke, 12e Avenue Nord, Sherbrooke, QC, Canada J1H 5N4d Sherbrooke Molecular Imaging Center, Department of Nuclear Medicine and Radiobiology, Faculty of Medicine and Health Science, Université de Sherbrooke, 12e Avenue Nord, Sherbrooke, QC,Canada J1H 5N4
RDiffusion MRI tractography is often used to estimate structural connections between brain areas and there is afast-growing interest in quantifying these connections based on their position, shape, size and length. However,a portion of the connections reconstructed with tractography is biased by their position, shape, size and length.Thus, connections reconstructed are not equally distributed in all whitematter bundles. Quantitativemeasures ofconnectivity based on the streamline distribution in the brain such as streamline count (density), average lengthand spatial extent (volume) are biased by erroneous streamlines produced by tractography algorithms. In thispaper, solutions are proposed to reduce biases in the streamline distribution. First, we propose to optimizetractography parameters in terms of connectivity. Then, we propose to relax the tractography stopping criterionwith a novel probabilistic stopping criterion and a particle filtering method, both based on tissue partial volumeestimation maps calculated from a T1-weighted image. We show that optimizing tractography parameters,stopping and seeding strategies can reduce the biases in position, shape, size and length of the streamlinedistribution. These tractography biases are quantitatively reported using in-vivo and synthetic data. This is acritical step towards producing tractography results for quantitative structural connectivity analysis.
Diffusion-weighted (DW) magnetic resonance imaging (MRI)tractography is used to reconstruct white matter (WM) pathwaysbetween brain regions. A growing number of connectomics studiesexploit structural properties of these pathways or streamlines to make‘connectivity’ comparisons between groups or individuals (Fornitoet al., 2013; Hagmann et al., 2007; Ng et al., 2013; Sporns, 2010).However, white matter bundles have various position, shape, size andlength making their reconstruction a challenge for tractographyalgorithms (Jbabdi and Johansen-Berg, 2011; Jones, 2010; Jones et al.,2012; Smith et al., 2012). Bundles positioned in partial volume withcerebrospinal fluid (CSF) are harder to completely reconstruct becausestreamline propagation ismore likely to be stopped (e.g. corpus callosum,fornix). Narrow bundles are harder to reconstruct because they are morelikely to be affected by error in the trackingmask, potentially stopping thestreamline propagation (e.g. cingulum, lower part of the corticospinaltracts). Curved bundles are also harder to reconstruct because noise canmake the tracking direction harder to follow in curved regions, especiallybecause discrete steps are taken in the estimated tangent direction (e.g.cingulum, uncinate fasciculus, U-fibers). Lastly, the length ofwhitematter
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bundles raise two opposite effects that bias their reconstruction: i)seeding from the white matter increases the density because there aremore streamlines that are initiated in longer bundles than in shorterbundles, ii) longer bundles are harder to completely recover because ofpremature stops, which decreases the density of streamlines.
Therefore, it is clear that streamline reconstruction is biased bythe seeding strategy, the stopping and masking criterion and thetractography parameters themselves. Hence, quantitative measures ofconnectivity based on the streamline distribution in the brain such asstreamline count (density), average length and spatial extent (volume)are biased by erroneous streamlines produced by tractography algo-rithms (Jbabdi and Johansen-Berg, 2011; Jones, 2010; Jones et al., 2012).Yet these effects are rarely addressed and reported in the literatureeven though they may lead to incorrect connectivity measures betweenareas. It is thus crucial and timely for the DW-MRI community to tackletractography limitations before it can be robustly used in connectomicsstudies.
In themajority of cases, tractography is done inside amask defined bya white matter segmentation of the T1-weighted image or fractionalanisotropy (FA) thresholded mask (Côté et al., 2013; Hagmann et al.,2007; Li et al., 2012; Smith et al., 2012; Tournier et al., 2011, 2012).Starting from an initial point within the mask, the tractography processfollows diffusion orientations in the forward and backward directionsuntil a stopping criterion is reached. Typical stopping criteria are when
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the tracking takes a step outside the tracking mask, when a minimumFA value or a minimum fiber orientation distribution function (ODF)amplitude value is reached, or when a maximum curvature constraintis exceeded (Toosy et al., 2004; Tournier et al., 2011, 2012). The trackingmask selection and the stopping parameters are thus very important astheywill determinewhen the streamline is included in the reconstructedwhite matter pathway. Discrete binary masks derived from thresholdedFA or T1-weighted images result in aggressive stopping criteria whichcan have a strong impact on connectivity results. For example, Côtéet al. (2013) studied streamlines produced by tractography pipelinesusing the Tractometer (tractometer.org) system analysis on the FiberCupdataset (Fillard et al., 2011; Poupon et al., 2008, 2010). Out of alltractography pipelines that found the seven out of seven true bundlesof the FiberCup (6,360 out of 57,096), between 58% and 97% ofstreamlines did not connect gray matter (GM) regions. Although theseobservations are based on a phantom mimicking a coronal slice(Poupon et al., 2008) of the brain, similar observations are seen usingbrain imaging data. For instance, in (Hagmann et al., 2007), the authorsreported that one third to half of streamlines did not reach theWM/GMinterface mask and thus, are excluded from the structural connectivityanalysis.
To overcome the effect of binary masks, one can use tissue partialvolume estimation (PVE) maps obtained from a structural T1-weightedimage (Zhang et al., 2001). The tissue PVE maps have values between 0and 1 in voxels near the boundary between distinct tissues and in voxelsof the subcortical gray matter. The discretization of these voxels set theirvalue to 1 in the tissue mask for the highest PVE value and 0 for othertissues. This can creates holes in the white matter mask if the highestPVE value varies from one tissue to another, ormakes somewhitematterpathways narrower (see Fig. 1). Thismakes streamlines stop prematurelyin these regions. This problem is especially important when trackingcorticospinal fibers or fibers involved in the motor system as shown in(Girard et al., 2012). Recently, Smith et al. (2012) proposed a methodcalled Anatomically-Constrained Tractography (ACT) taking advantage ofthe tissue PVE maps. They proposed relaxing the stopping criterion byusing WM, GM and CSF PVE maps to determine when a streamlinestops and if it is included or excluded in the reconstruction. Therefore,biological tissue properties are used to better determine the trackingmask and stopping criteria. They proposed to threshold interpolatedPVE maps to define stopping criterion. However, they observed thatsubcortical gray matter have low PVE values, leading to streamlinegoing through these regions, connecting other gray matter regions orreaching CSF regions excluding the streamline. The authors suggestedto cut streamlines going through binary segmentation of the subcorticalgraymatter and include only valid segments. However, it is challengingto choose which regions to define in such a binary mask construction.
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Fig. 1. TheWM binary mask (first row) and the WM PVE maps (second row). Red circleshighlight differences between the WM binary mask and the WM PVE map.
Please cite this article as: Girard, G., et al., Towards quantitative connectivdx.doi.org/10.1016/j.neuroimage.2014.04.074
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Aside from stopping criterion, the seeding strategy can also bias thestreamline distribution and change estimation of the brain connectivity.The seedingmask defines all the potential voxels where streamlines areinitiated. It can either be the trackingmask (Centuro et al., 1999; Huanget al., 2004; Tournier et al., 2011, 2012), a WM/GM interface mask (Liet al., 2012; Smith et al., 2012) or a region of interest (Behrens et al.,2007; Huang et al., 2004; Parker and Alexander, 2005; Toosy et al.,2004; Tournier et al., 2011, 2012). Seeding from a whole white mattermask biases the number of reconstructed streamlines in bundles withvarious lengths because streamlines are more likely to be initialized inlonger white matter bundles, covering a larger part of the white mattermask (Jones, 2010; Smith et al., 2013). For example, if two linearbundles have the same size but one twice the length of the other, thenumber of streamlines in the longer bundle will be approximatelydoubled. This increase of density is not related to the connectivity ofthe bundle, it is a seeding bias of the tractography. Recently, Smithet al. (2013) proposed amethod called Spherical-deconvolution informedfiltering (SIFT) to reduce local bias in the streamline density. Themethodfilters the tractography results to improve the fit between the streamlinedistribution in each voxel and the fiber ODF estimated from DW-MRI.SIFT produces streamlines that better represent the measured diffusioninformation. In particular, SIFT reduces density bias resulting from theseeding strategy. However, streamlines are still affected by the choice ofmasking and stopping criterion, and tractography parameters used bythe tractography algorithm.
In this work, we show that careful selection of tractography param-eters and optimal seeding, masking and stopping criterion choicessignificantly reduces the biases in position, shape, size and length ofthe streamline distribution. Firstly, inspired by the work of Smith et al.(2012) which uses anatomical information from a T1-weighted imagefor tractography, we propose novel probabilistic stopping criteriabased on tissue PVE maps. Secondly, we propose a particle filteringmethodusing anatomical information for tractography to enforce stream-lines connecting graymatter regions. Thirdly,wemake recommendationson themost important tractography parameters and optimize the param-eter selection in terms of global connectivity using the Tractometer (Côtéet al., 2013) evaluation strategy on in-vivo and synthetic data. Our overallcontribution is a new tractography framework optimized in terms ofquantitative connectivity, which reduce tractography biases in position,shape, size and length of white matter bundles.
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Method
Streamline tractography
In this work, we relax the tractography stopping criterion usingtissue partial volume estimation (PVE)maps. Since our proposed strategydoes not represent a new tractography algorithm as such, we comparedand applied this relaxation to previously published state-of-the-art fiberODF deterministic and probabilistic algorithms (Descoteaux et al.,2009; Tournier et al., 2011, 2012). In-house implementations of thesetractography algorithms are used, which have been validated againstMRtrix (Tournier et al., 2012) by the Tractometer (Côté et al., 2013). Inour implementation, the spherical harmonics of the fiber ODFs areprojected on a discrete evenly distributed symmetric sphere of 724vertices (Daducci et al., 2013). Propagation directions are always a vectorof orientation corresponding to one vertex of the sphere and of lengthΔs = 0.2 mm (Tournier et al., 2012). A propagation direction is valid ifits corresponding value is greater than a fraction of the maximum valueof the fiber ODF τ, and form an angle smaller than θ with the previouspropagation direction (Behrens et al., 2007; Tournier et al., 2011, 2012).If there is no valid propagation direction, algorithms assume an error inthe fiber ODF and continue in the previous propagation direction. Thisis done for a maximum distance of δundeviated. Implementation detailsare given in Appendix A.
In the current study, we propose a novel approach that takes advan-tage of the completeWM, GM and CSF partial volume estimation (PVE)maps to change theway tractography stopping events are triggered.Wecall our novel strategy ContinuousMapCriterion (CMC). It uses PVEmapsto define the probability of stopping the tracking process. This providessmooth boundaries between tissues as ACT (Smith et al., 2012) andadditionally encodes a stopping behavior in subcortical gray matter.Streamlines reaching the cortex and going through large regions oflow GM PVE such as the subcortical gray matter are proportionalitylikely to be stopped. Using CMC, streamlines can propagate close tosubcortical gray matter without having to define binary segmentationblocking some of the propagation pathways.
CMC uses an inclusion map (Mapin) and exclusion map (Mapex) tostop the streamline propagation. An example of Mapin and Mapex
based on GM and CSF PVE maps is shown in Fig. 2. We hypothesizethat the amount of streamlines stopping in a voxel and included shouldbe proportional toMapin. Similarly, the amount of streamlines stoppingat a voxel and rejected should be proportional toMapex. Using CMC, theprobability that a streamline continues its propagation at position p isgiven by
Pcontinuep ¼ 1− Mapinp þMapexp
� �� �Δs=ρ; ð1Þ
with ρ the maps voxel size (ρ=1 for voxel size of 1 × 1 × 1mm3) andΔs the step size.Δs/ρ allows the probability of stopping to be stablewithrespect to the step size Δs. Otherwise, since the tracking probability isevaluated at each tracking step, using a step size Δs b ρ will increasethe probability of stopping the tractography and decrease the probabilitywhen Δs N ρ. Alternatively, Pcontinue can be computed and adjusted to thestep size following Eq. (1) for each voxels and useddirectly. If the trackingprocess stops, the streamline is included (added to the estimated set ofstreamlines) with a probability given by
Pincludedp ¼ Mapinp = Mapinp þMapexp
� �; ð2Þ
otherwise the streamline is excluded (rejected from estimated set ofstreamlines). Trilinear interpolation is done over Mapin andMapex to getthe probability of continuing the propagation (Eq. (1)) and theprobabilityof including the streamline (Eq. (2)).
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Please cite this article as: Girard, G., et al., Towards quantitative connectivdx.doi.org/10.1016/j.neuroimage.2014.04.074
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Particle filtering tractography – PFT
In addition to CMC, we propose using amodular add-on to streamlinetractography algorithm, called Particle Filtering Tractography (PFT), toreduce the number of streamlines that prematurely stop in the whitematter or in the cerebrospinal fluid, and do not connect the gray matter.Streamline tractography can be modeled as a state system evolvingover time using noisy measurements, where states are the trackingposition, the propagation direction and the tracking status (e.g. ’in theWM’ or ’stopped in the GM’), and are connected over time by a Markovchain. The particle filtering algorithm is described in Appendix B.
PFT is initiated before the premature stopping event and weighspropagation pathways based on the PVE maps to enforce the trackingin the white matter, as illustrated in Fig. 3. Propagation pathways arechosen to ensure streamlines not to stop in the CSF and reach the graymatter (Bloy et al., 2012; Smith et al., 2012). A backtracking approachfor probabilistic tractography has been proposed in (Smith et al.,2012) which incrementally truncates and re-tracks the streamlinewhen it reaches a premature stop. It shows an increase of the whitematter bundle coverage and helps the reconstruction of some whitematter bundles. However, higher backtracking distances can bias thestreamline reconstruction, especially in crossing regions. PFT uses abacktracking idea by simultaneously estimating many propagationpathways at a short distance of the premature stopping event.
The proposed Particle Filtering Tractography (PFT) estimates a likelystreamline using Mapin and Mapex (see Section 2.2) whenever thetractography reaches a stopping criterion excluding a streamline, asillustrated in Fig. 3(a). The key idea is to backtrack δb mm and computea valid streamline after K = (δb + δf)/Δs steps, where δb and δf arerespectively the backward and forward distances. If the propagationdistance is less than δb, δb is set to the propagation distance done sofar. The goal is to estimate a likely streamline initialized at δb mmbeforethe stopping criterion is reached, and then go δf mm further to ensurethe local stopping event is solved. That is, the streamline stops correctlyin an including region or the streamline continues its propagation in thewhitematter. If the streamline stops in an including region, the trackingis done. If the streamline is in the white matter, the tractographycontinues normally until another stopping criterion is reached.
PFT uses a set {xk(i), wk(i)}i = 1
N of N discrete samples (referred asparticles) xk(i)with an associatedweightwk
(i) to characterize the estimatedstreamline distribution.Weights are normalized over all particles to have∑ i = 1
N wk(i) = 1. A particle xk(i) = [p, v, status] has a the tracking position
p, a propagation direction v and a status∈{active,inactive} which repre-sents the tracking process propagating (active) or stopped in an includingregion (inactive). At each iteration k, if status = active, the particle posi-tion p and propagation direction v are updated following the probabilistictractography algorithm (see Appendix A). Otherwise, if status= inactive,the tracking reached a valid stopping region and is stopped (p and v arenot updated). The status stays activewith a probability of
Pactivep ¼ 1−Mapinp
� �Δs=ρ;
following the CMC strategy (see Section 2.2). The exclusionmapMappex, is
used to estimate the likelihood of the particle xk(i), which is the likelihoodof a streamline propagating at p. Theweightwk
(i), at time k, of a particle atposition p is calculated following
w ið Þk ¼ w ið Þ
k−1 � 1−Mapexp� �Δs=ρ
:
wk(i) is set to 0 if no valid propagation direction is available for a distance
δundeviated (see Section 2.1).PFT estimates a valid streamline distribution around the stopping
event and iteratively estimates subsequent valid streamline distributionsfrom the previous one. The resulting streamline is drawn from the finalvalid streamline distribution. As shown in Figs. 3 (c–e), this algorithm
Fig. 3. PFT algorithm. (a) A streamline prematurely stops in the CSF (white) and (b) a backtracking step is done. (c, d, e) shows the particles at three iterations of PFT. PFT estimates thedistribution of possible streamlines using probabilistic samples and weighs them using anatomical information. Reddish particles have low weight and greenish have high weight. (f) Apath is drawn from the particles distribution. (g) The propagation process then continues using the principal tractography algorithm (deterministic).
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generates multiple probabilistic streamlines and penalizes particlespropagating in the excluding region (white). Reddish particles have lowweights and greenish have high weights. The output of the PFT is eitheran inactive streamline ending in the including region (Mapin) or an activestreamline continuing its propagation in the white matter. If at anyiteration k theweightswk
(i)= 0 ∀ xk(i), the streamline is excluded because
no valid streamline is found (e.g. Mappex = 1 ∀ xk
(i)). The principaltractography algorithm used (deterministic or probabilistic) is doneuntil the propagation reaches a stopping criterion excluding the stream-line, as determined by the CMC (see Section 2.2). In this case, PFT istriggered to find an alternative valid pathway.
Seeding from the white matter–gray matter interface
In this study, we want tractography algorithms to produce a similardensity for bundles with the similar size but various lengths. To achievethis, we seed from the WM/GM interface as in (Li et al., 2012; Smithet al., 2012). We propose to define the WM/GM interface mask bysegmenting all voxel having a GM PVE N 0.1 and a WM PVE N 0.1. Thisresults in a ribbon of voxels at the boundary between gray matter andwhite matter (see Fig. 4). Most of the voxels of the subcortical graymatter are included in the interface since they are partially segmentedas white matter and gray matter (see Fig. 4). An approach based on adilatation of the gray matter mask could have been used to obtain theWM/GM interface such as (Li et al., 2012; Smith et al., 2012). Furtherinvestigation is required to quantify the effect of the definition of theWM/GM interface on tractography, but is outside the scope of thispaper.
The seedingmask contains a partial volume of graymatter, which canlead to premature stopping the streamline propagation using CMC. Toovercome this, CMC (see Eq. (1)) is only triggered once the streamlinehas reached a position p where Mapp
in = 0 (e.g. in the white matter).Otherwise, propagation stops only when reachingMapp
in =1 (included),Mapp
ex=1 (excluded) or when no valid direction is available for distanceδundeviated (excluded). This allows streamlines to exit the initial regionbefore stopping the propagation (see Section 2.2).
When a voxel is identified to initiate a streamline in it, the seedposition is randomly chosen within the voxel boundary (Tournieret al., 2012). A trilinear interpolation over the spherical harmonic
Fig. 4. The WM/GM interface. All voxels of the interface have a GM PVE N 0.1 and a WMPVE N 0.1.
Please cite this article as: Girard, G., et al., Towards quantitative connectivdx.doi.org/10.1016/j.neuroimage.2014.04.074
OFcoefficients of the fiber ODFs image is done to obtain the fiber ODF at
the seed position. The fiber ODF is then thresholded to a predefinedvalue τinit, a fraction of the maximum value of the fiber ODF. The initialpropagation direction is drawn from the empirical distribution definedby thresholded fiber ODF. τinit aims at starting tractography in a tangentdirection to the bundle.
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Datasets for the experimentsThe simulateddataset produced for the IEEE International Symposium
on Biomedical Imaging (ISBI) 2013 Reconstruction Challenge (Caruyeret al., 2014; Daducci et al., 2013) is used to evaluate quantitatively thequality of tractography algorithms. The synthetic dataset consists of 27simulated known ground truth white matter bundles, mimickingchallenging branching, kissing, crossing structures at angles between30° and 90°, with various curvature, and diameters ranging from 2 mmto 6 mm, as seen in Fig. 5(a). The DWI signal is simulated in each voxelbased on the Numerical Fiber Generator (Close et al., 2009) and somefree-water CSF-like partial volume effects. The simulated signal is obtain-ed using a hindered and restricted diffusion model (Assaf and Basser,2005), and adding Rician noise. In this study, we used 64 uniformlydistributed gradient directions using a b-value of b = 1000 s/mm2 atsignal to noise ratio (SNR) 10, 20 and 30. The dataset has a sphericalshapewith the extremities of the simulatedwhitematter bundles endingon the surface of the sphere. The bundles mask is defined as all voxelhaving awhitematter PVE greater 0.1. The simulated graymatter consistsof the voxels in the three outer layers of the sphere, obtained by threeerosion iterations and intersecting the bundles mask. The white mattermask is composed of all voxels of the bundles mask and not part of thegray matter mask. The simulated CSF is all none gray matter or whitematter voxels (see Fig. 5(a)). The simulated WM/GM interface is thefourth outer layer of the sphere and part of the white matter mask.Mapin and Mapex are defined from the gray matter mask and the CSFmask (see Figs. 5 (b, c)).
Fig. 5. Synthetic dataset. (a) Sphere of CSF (black)withWM(white), connecting GMat theextremity of the WM (green), (b)Mapin, (c)Mapex.
Healthy brain datasetDWI was acquired on a single volunteer along 64 uniformly distrib-
uted directions using a b-value of b=1000 s/mm2 and a single b=0 s/mm2 image using the single-shot echo-planar imaging (EPI) sequenceon a 1.5 Tesla SIEMENS Magnetom (128 × 128 matrix, 2 mm isotropicresolution, TR/TE 11000/98 ms and GRAPPA factor 2). An anatomicalT1-weighted 1 mm isotropic MPRAGE (TR/TE 6.57/2.52 ms) imagewas also acquired. Diffusion data were upsampled to 1 mm isotropicresolution using a trilinear interpolation (Dyrby et al., 2011; Girardet al., 2012; Smith et al., 2012; Tournier et al., 2012). The T1-weightedimage was registered to a 1 mm isotropic DWI using FSL/FLIRT(Jenkinson and Smith, 2001). Quality control was done to make surethe registrationwas done robustly bymanual inspection. The FractionalAnisotropy (FA) map and color-FA were overlaid on the T1-weightedimage to make sure optimal alignment between images. The BrainExtraction tool (FSL/BET (Smith, 2002)) and FSL/FAST (Zhang et al.,2001) were also used to extract both binary and PVE maps of the WM,GM and CSF. Mapin and Mapex are respectively set as the gray matterPVE and CSF PVE maps. Additionally, all voxels not in the brain are setto 1 in Mapin to keep streamlines exiting the brain mask. Diffusiontensors, FA, fiber ODFs reconstruction and all tractography algorithmsare carefully detailed in Appendix A. White matter bundles have beenmanually segmented using the Fibernavigator software (scilus.github.io/fibernavigator/) and using FreeSurfer T1-weighted image white matterand gray matter segmentations (Fischl et al., 2004). Streamlines arecolored by their orientation (the vector connecting their extremities)using the standard red-green-blue convention (red: left-right, green:anterior-posterior, blue: inferior-superior) (Calamante et al., 2012;Pajevic and Pierpaoli, 1999).
Quantitative connectivity evaluation
To compare and evaluate reconstructed streamlines, we use theTractometer (Côté et al., 2013) connectivity analysis. We computed thefour metrics of the Tractometer, namely the Valid Connections (VC),the Invalid Connections (IC), the No Connections (NC) and the AverageBundle Coverage (ABC). The definitions of these connectivitymetrics arelisted in Appendix C. The Tractometer identifies two types of erroneousstreamlines: streamlines either connecting unexpected regions (IC) ornot connecting any regions (NC). For in-vivo data, NC can be identifiedand removed, which is not the case for IC. Thus, we defined two newglobal connectivity metrics to quantify reconstructed streamlines:
• The Valid Connection to Connection Ratio (VCCR): relation betweenvalid connections and all connections estimated VCCR= VC/(VC+ IC).
• The Connection to Seed Ratio (CSR): relation between the number ofconnections estimated and the number of seeds S used by thetractography algorithm, i.e. CSR = (VC + IC)/S. If all seeds produceda streamline, CSR= (VC + IC)/(VC+ IC + NC).
Thus, VCCR is a measure of the precision of the estimated connectionsand CSR is an indicator of the performance of the tractography. Allmetricsare reported in percentages (%). An optimal tractography algorithmshould produce global connectivitymetricswith the followingproperties:i) all seeds should lead to streamlines connecting gray matter regions(high connection to seed ratio CSR), ii) all connected gray matter regionsshouldmatch the ground truth (high valid connection to connection ratioVCCR) and iii) streamlines should cover all voxels within a bundle (highaverage bundle coverage ABC). VCCR cannot be computed on in-vivodata because the ground truth is not known. Nevertheless, CSR can beestimated on in-vivo data by defining a connection as any streamlineconnecting two gray matter regions and having a minimum lengthδmin = 10 mm and a maximum length δmax = 300 mm (Tournieret al., 2012). However, an increase in CSR could be due to an increasein invalid connections and must be interpreted carefully.
Hence, parameters are chosen to increase the valid connection toconnection ratio VCCRs on synthetic data and the connection to seed
Please cite this article as: Girard, G., et al., Towards quantitative connectivdx.doi.org/10.1016/j.neuroimage.2014.04.074
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ratios CSRs and CSRb, on both synthetic and in-vivo data. Here, thesubscript b is used to indicate that the metric is computed on in-vivobrain data, as opposed to the subscript s for synthetic data. CSRs andVCCRs are computed seeding from all white matter voxels of thesynthetic dataset and CSRb is estimated seeding randomly in the whitematter mask of the in-vivo dataset.
To evaluate the optimal value for each parameter, we fixed allparameters to a default value: θ = 0°, τ = 0, τinit = 0, δundeviated =0 mm, δf = 0 mm, δb = 0 mm. Iteratively, starting with θ, parametersare fixed to a value within the range of value maximizing VCCRs, CSRsand CSRb. The order of the parameters optimizationwas chosen startingwith parameters producing the most changes of metrics value forin-house algorithms. First, we optimized parameters for in-housetractography algorithms (θ, τ, τinit, δundeviated). Then, using previouslyfound optimal parameters, we optimized parameters for the PFTalgorithm (δf, δb).
Results
Choosing optimal tractography parameters
Maximum deviation angle θdet = 45°, θprob = 20°Fig. 6(a) shows valid connection to connection ratio VCCRs, and
connection to seed ratios CSRs and CSRb for deterministic and probabilistictractography varying themaximumdeviation angle θ. Higher θ decreasesthe stopping issue in the white matter for both algorithms, by allowingthe propagation to curve more rapidly. This can lead to an increase ininvalid connections IC if the direction associated to more than onewhite matter bundle is used to propagate the streamline. Thus, lower θis preferred to reduce this effect. Using deterministic tractography, themetrics on synthetic data tend to stabilize using θdet N 45°. Using in-vivodata, they stabilize for θdet N 60°. Probabilistic tractography showshigher values for VCCRs using angle in range 15° b θ b 30°. CSRs reachesmaximum values using 20° b θ b 40° (Fig. 6(a)). Based on these results,we fix θdet = 45° and θprob = 20°, which provide high VCCRs, CSRs andCSRb.
Fiber ODF threshold τ = 0.1 and τinit = 0.5Fig. 6(b) shows connection to seed ratio CSRs, valid connection to
connection ratio VCCRs and CSRb varying the fiber ODF threshold τ.The objective of the τ parameter is to remove noisy directions fromthe fiber ODF, but keep true and small white matter volume fractioncontributions in the fiber ODF. We see that τ∈[0.0–0.2] has little effecton synthetic data for deterministic tractography. τ N 0.4 tends to reduceCSRs, removing some of the maxima. Increasing τ parameter has apositive effect on VCCRs using probabilistic tractography, but tends todecrease CSRs using τ N 0.5. CSRb decreases with the increase of τ.Removing some of the propagation directions increases the stoppingissue in the white matter, especially when tracking in lowwhite matterpartial volume fraction regions, where the fiber ODF has lower values.We thus fix τ = 0.1 since it does not reduce the score of metrics onsynthetic data. However, CSRb is reduced by 15% for probabilistictractography. Fig. 7 shows examples of streamline varying τ seeding1,000 streamlines using probabilistic tractography from a single voxelof the synthetic dataset. Thresholding the fiber ODF helps reducingerroneous streamlines. Then, we varied τinit (using τ= 0.1) to increasethe initial propagation direction tangency to thewhitematter bundle. Ithas little effect on CSRs, VCCRs, and CSRb when τinit ≥ 0.5 (see Fig. 6(c)).Lower value of τinit reduces CSRs, especially using low SNR. Based on thisobservation, we fixed τinit = 0.5.
Undeviated propagation distance δundeviated = 1 mmFig. 6(d) shows how varying the value of the undeviated propagation
distance δundeviated affects tractography results. Increasing δundeviateddecreases the number of streamlines stopping in the white matter byallowing the tracking to propagate through regions where propagation
Fig. 6. Valid connection to connection ratio (VCCRs), and connection to seed ratio (CSRs) obtained from synthetic dataset and CSRb obtained from in-vivo data. (a) Themaximumdeviationangle θ (°), (b) thefiber ODF threshold τ, (c) the initial fiber ODF threshold τinit, (d) themaximumundeviated propagation distance δundeviated (mm), (e) the PFT forward tracking distanceδf (mm), (f) the PFT backward tracking distance δb (mm). The vertical dashed line indicates the chosen value for each parameter.
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directions are missing. δundeviated has little effect on synthetic data,meaning that the propagation rarely stops in the white matter. This isnot the case on in-vivo data, where δundeviated increases connection toseed ratio CSRb, especially using deterministic tractography (seeFig. 6(d), black curve). δundeviated allows streamlines to propagatethrough voxels where propagation directions are missing. We observedthat bigger value of δundeviated produces more erroneous streamlinesexiting the white matter bundles. This parameter has a similar effects asincreasing the step size when no valid direction is available and thus,
Please cite this article as: Girard, G., et al., Towards quantitative connectivdx.doi.org/10.1016/j.neuroimage.2014.04.074
produces similar behavior as using a bigger step size. For this reason,we set δundeviated to a maximum distance of 1 mm, half the size of thein-vivo diffusion space voxel size.
Forward and backward distances δb = 2 mm, δf = 1 mmFigs. 6 (e, f) show the effect of the forward δf and the backward δb
tracking distances using PFT on connection to seed ratios CSRs andCSRb, valid connection to connection ratio VCCRs. Fig. 6(e) shows anincrease of the connection to seed ratios with δf ≥ 0.2 and a small
Fig. 7. Streamlines estimated varying the fiber ODF threshold τ parameter using probabi-listic tractography. A thousand streamlines were initiated at the seed voxel indicated bythe arrow. Increasing τ reduce the number of erroneous streamlines.
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decrease of VCCRs, especially for deterministic tractography. Resultsvarying the parameter δb are shown in Fig. 6(f). We can observe anincrease of the connection to seed ratios until δb ≥ 2 mm, where ittends to stabilize. Based on these observations, we set δb = 2 mm andδf = 1 mm, which increase both CSRs and CSRb, and keep the trackingprocess near the stopping issue.
Number of particles N =25The number of particles N should be large enough to produce a
good approximation of the distribution and small enough to keep tocomputation requirement low (Arulampalam et al., 2002; Doucet et al.,2001). In our experiments, we observe (results not shown) that allmetrics are stable using N ≥ 15. In this work, results are obtained usingN = 25.
Connectivity analysis on synthetic data
Table 1 shows the average bundle coverage ABCs, the connection toseed ratio CSRs and the valid connection to connection ratio VCCRs forour in-house probabilistic and deterministic tractography algorithmsused with and without the Particle Filtering Tractography (PFT) (in-housePFT). Results are shown on synthetic data at SNR 10, 20 and 30.PFT increases the connection to seed ratio CSRs by 37.1% on averageusing deterministic tractography and by 51.8% on average usingprobabilistic tractography. Out of the 6 experiments shown in Table 1,in-housePFT algorithms have on average 89.0% of connection to seedratio CSRs, against 44.6% for in-house algorithms. This means that onaverage, the tracking connects gray matter regions from a seed positiontwice as often using particle filtering approach. However, in-housePFTshows a decrease in valid connection to connection ratio VCCRs by 8.3%on average using deterministic tractography and 2.5% on average usingprobabilistic tractography. VCCRs is always lower for probabilisticalgorithms than for deterministic algorithms (see Table 1). Thus, thedecrease of VCCRs is expected for in-housePFT deterministic tractographysince a probabilistic algorithm is used with PFT. The average bundlecoverage ABCs is always higher for in-housePFT, which suggests that thatstreamlines recoveredbyPFTpropagate in regionspreviously not coveredby streamlines produced by the default algorithms.
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Table 1Comparison between deterministic and probabilistic tractography algorithms on synthetic dat
In-house: tracking within a binary mask, In-housePFT: in-house tracking using CMC and PFT. A
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Next, in Table 2, we study the effect of Particle Filtering Tractography(PFT) on the connection to seed ratio CSRs on the individual bundles ofthe synthetic dataset, seeding from its WM/GM interface using SNR =20 dataset. We grouped bundles by their diameter size, and computedthe average (μ) and the standard deviation (σ) of CSRs and VCCRs. Twoof the 27 bundles have been omitted because they share a commonending region, making CSRs, VCCRs not practicable over these bundlesindependently. Similar to what is shown in Tables 1 and 2 shows adecrease in valid connection to connection ratio VCCRs using in-housePFT(deterministic: 7.1%, probabilistic: 3.6%) but the standard deviation σ ofVCCRs is reduced by 7.9% for deterministic tractography and 8.6% forprobabilistic tractography. This decrease is higher for small white matterbundles. This means that, on average, the valid connection to connectionratio VCCRs is less biased by various bundle sizes and shapes. Mostimportantly, in-housePFT shows a clear increase of connection to seedratio CSRs using both probabilistic and deterministic tractography (seeTable 2). It reflects that more alternative connections are found due tothe relaxation of the stopping criterion of in-housePFT. CSRs showsincreases from 45.8% ± 23.1% to 91.9% ± 5.3% for deterministictractography and from 32.8% ± 21.9% to 90.8% ± 5.2% for probabilistictractography. Thismeans that streamlines aremore uniformly distributedamong white matter bundles having various shapes and sizes. This isobserved by a higher increase of CSRs on white matter bundle havingsmall diameter and a lower standard deviation σ of CSRs in individualbundle reconstruction.
Connectivity analysis on in-vivo data
Table 3 shows the distribution of included and excluded streamlineson in-vivo data. Streamlines are obtained by seeding from the WM/GMinterface. The included streamlines are those ending in the gray matterand having a length in [δmin= 10mm, δmax= 300mm]. The distributionof included and excluded streamlines in the brain (see Table 3) showsthat for deterministic tractography, 50.0% of the seeds produced includedstreamlines and 21.2% of the seeds produced excluded streamlines eitherending in the CSF or in thewhitematter (62.9% and 13.6% for probabilistictractography respectively). Using PFT, the streamlines previouslyincluded (not using PFT) are exactly the same, but additionally 19.0%of the excluded streamlines are recovered by the particle filteringapproach using deterministic tractography, and 10.7% using probabilistictractography (indicated in the ExtraPFT row). These additional streamlinesdo not share the same length distribution as the previously includedstreamlines. This can be observed in Table 3 by the higher average lengthof the recovered streamlines (50.6 mm and 54.6 mm) than averagelength of the other included streamlines (32.5 mm and 37.2 mm), usingdeterministic and probabilistic algorithms respectively.
Finally, Table 4 shows the streamline count and their average lengthfor seven in-vivo data bundles using deterministic and probabilistictractography. Each experiment reports 100,000 included streamlinesusing the in-house and the in-housePFT algorithms, seeding from theWM/GM interface. For comparison, 100,000 streamlines with defaultMRtrix parameters are reported. We also randomly select 100,000streamlines included using the particle filter (ExtraPFT). In-housePFT
t2:11 In-house: tracking within a binary mask, In-housePFT: in-house tracking using CMC and PFT. All metrics are reported in %.
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streamlines can be seen as a fraction of in-house streamlines plus afraction of ExtraPFT streamlines (previously excluded streamlines). Weobserve in Table 4 that shorter bundles (Uncinate Fasciculus (UF), andshort association fibers U1 and U2) have a higher streamline countusing in-house and in-housePFT algorithms than using MRtrix (e.g. U1:619, 639 and 105 streamlines using in-house, in-housePFT andMRtrix de-terministic tractography respectively). Longer bundles (the corticospinaltract (CST), the corpus callosum (CC), the superior longitudinal fasciculus(SLF) and the inferior longitudinal fasciculus (ILF)) are overrepresentedseeding from the white matter (e.g. CST: 159, 289 and 584 streamlinesusing in-house, in-housePFT and MRtrix deterministic tractographyrespectively). However, the streamline count is generally higher forin-housePFT in long white matter bundles (e.g. CC: 3,139 and 4,078streamlines using in-house and in-housePFT deterministic tractographyrespectively). This can be observed in Figs. 8, 9, 10, and 11. Longerwhite matter bundles are well reconstructed usingMRtrix, seeding fromthe white matter mask, because there are more seeds that are initiatedin these bundles than in shorter bundles. However, seeding from theWM/GM interface with in-housePFT algorithms provide a less biasedreconstruction of bundles with respect to their length. Finally, MRtrixreconstructed theUFwith the lowest streamline density. It is likely causedby both the seeding strategy and the use of a binarywhitematter trackingmask.
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Optimal parameters
We used the Tractometer strategy (Côté et al., 2013) to investigatethe influence of tractography parameters on in-vivo and syntheticdatasets. Optimal tractography parameters were chosen using twonew global connectivity metrics: the valid connection to connection
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Table 3Streamline distribution and average length, seeding from WM/GM interface. Includedstreamlines end in the GM. Extra PFT shows the percentage of streamlines included usingPFT. Excluded streamlines either stop in the CSF or in the WM, or end in the GM buthave a length not in [δmin = 10 mm, δmax = 300 mm].
Streamlines Algorithms
Deterministic Probabilistic
In-house In-housePFT In-house In-housePFT
Include 50.0% 50.0% 62.9% 62.9%32.5 mm 32.5 mm 37.2 mm 37.2 mm
ExtraPFT 19.0% 10.7%50.6 mm 54.6 mm
Exclude CSF 2.6% 0.1% 3.4% 0.1%42.8 mm 57.2 mm 48.9 mm 56.7 mm
WM 18.6% 3.4% 10.2% 3.2%29.7 mm 34.5 mm 34.5 mm 38.9 mm
δmin, δmax 28.8% 27.6% 23.5% 23.2%7.2 mm 6.7 mm 8.3 mm 7.3 mm
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OFratio VCCR and the connection to seed ratio CSR. The proposed metrics
provide information on the precision (VCCR) and on the performance(CSR) of the reconstructions. However, we observed interactionsbetween individual tractography parameters. For instance, an increaseof both θ and τ for probabilistic tractography can provide similar CSRand VCCR results. The rationale was to find the optimal deviationangle θ considering all the directional information of the fiber ODF.Then, to find theoptimal threshold τ, improving the results by removingnoisy directions in the fiber ODF. In a similar fashion, we successivelyfixed parameters to a value in the range maximizing both CSR andVCCR. Parameter values are not strongly set because we have VCCRonly on synthetic data and an increase in CSRb could be due to anincrease in invalid connections IC and must be interpreted carefully.Nonetheless, we can get a general tendency. Hence, using the determin-istic and probabilistic tractography algorithms described in 2.1, werecommend θdet ∈ [45, 60]°, θprob ∈ [20, 30]°, τ ∈ [0.1, 0.2], τinit∈ [0.2, 0.5] and δundeviated set to a maximum of half the diffusion acquisi-tion voxel size. Taken together, setting these tractography parametersaccordingly is the first step towards reducing position, shape, size andlength biases.
Deterministic versus probabilistic tractography
We observed from Tables 1 and 2 that deterministic tractographyalways shows better performance in terms of valid connection toconnection ratio VCCRs and similar or better performance in terms ofconnection to seed ratio CSRs. Probabilistic tractography shows anaverage bundle coverage ABCs always higher with both in-house andin-housePFT algorithms. Deterministic tractography tends to reducethe proportion of invalid connection IC in comparison to probabilistictractography, but decreases the average bundle coverage ABC. Thus,the tractography algorithm (deterministic or probabilistic) must bechosen to be the most suitable for the streamline analysis (less IC ormore ABC), which will be application-driven.
New seeding, stopping and masking strategies
Through our novel Continuous Map Criterion (CMC) and ParticleFiltering Tractography (PFT), we have shown that injecting anatomicalprior information into tractography seeding, stopping and maskingreduces biases in the streamline distribution. CMC determines wherethe valid and invalid stopping regions are. PFT gives a more uniformstreamline distribution, finding alternative valid pathways for stream-lines stopping in invalid regions, such as in the white matter or in theCSF. It uses the partial volume estimation (PVE) maps to reduce thebiases in long and curved bundles. It increases the average bundlecoverage ABC (see Table 1) and the connection to seed ratio CSR (seeTable 2). Qualitatively, PFT provides a better coverage of known whitematter pathways of the brain and helps reduce bias in the streamlinedistribution. We showed that this relaxation of the stopping criterionenhances the density of complex streamline bundles (e.g. high curvature
54.9 mm 142.5 mm 96.3 mm 131.8 mm 81.9 mm 71.8 mm 38.0 mm 47.4 mmt4:20
t4:21 ExtraPFT shows streamlines additionally included using PFT. The streamline count and the average streamline length are shown for each bundle. From left to right: All streamlines, thet4:22 corticospinal tract (CST), the Corpus Callosum (CC), the Superior Longitudinal Fasciculus (SLF), the Inferior Longitudinal Fasciculus (ILF), the Uncinate Fasciculus (UF), the associationt4:23 fibers between the precentral gyrus and postcentral gyrus (U1) and the association fibers between the superior frontal gyrus and middle frontal gyrus (U2).
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or narrow white matter pathways). This is in-line with recent worksthat show that anatomical information and filtering can help reduce
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Fig. 8. Deterministic and probabilistic tractography of the corticospinal tracts (CST) in sagittal awith CMC and PFT, (d) additionally included streamlines using PFT. In-housePFT streamlines ca
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Rtractography biases (Bloy et al., 2012; Li et al., 2012; Smith et al., 2012,2013).
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nd coronal views. (a)MRtrix, (b) in-house algorithms with CMC, (c) in-house algorithmsn be seen as a fraction of in-house streamlines plus a fraction of ExtraPFT.
Fig. 9. Deterministic and probabilistic tractography of the association fibers of between the superior frontal and the middle frontal gyrus (U1) in sagittal view. (a)MRtrix, (b) in-house al-gorithms with CMC, (c) in-house algorithms with CMC and PFT, (d) additionally included streamlines using PFT. In-housePFT streamlines can be seen as a fraction of in-house streamlinesplus a fraction of ExtraPFT.
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Reducing the position, shape and size biases
White matter bundles have various positions, shapes and sizes,making their reconstruction a challenge for tractography algorithms.
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Fig. 10. Deterministic and probabilistic tractography of the Corpus Callosum (CC) in coronal anCMC and PFT, (d) additionally included streamlines using PFT. In-housePFT streamlines can be
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ROBundles positioned in partial volume of CSF are harder to completely
reconstruct because the streamline propagation is more likely to bestopped. Narrow bundles are more likely to be affected by errors inthe tracking mask that could stop the streamline propagation, making
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d axial views. (a)MRtrix, (b) in-house algorithms with CMC, (c) in-house algorithms withseen as a fraction of in-house streamlines plus a fraction of ExtraPFT.
Fig. 11. Deterministic and probabilistic tractography of the association fibers between the precentral and the postcentral gyrus (U1) in coronal view. (a) MRtrix, (b) in-house algorithmswith CMC, (c) in-house algorithms with CMC and PFT, (d) additonally included streamlines using PFT. In-housePFT streamlines can be seen as a fraction of in-house streamlines plus afraction of ExtraPFT.
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their reconstruction harder. Because tractography algorithms followtangent directions of bundles, curved bundles are harder to reconstruct.Noise canmake the tracking direction harder to follow in curved region,especially because discrete steps in the estimated tangent direction aretaken. CMC reduces biases in position and size by making smoothboundaries between distinct tissues. PFT reduces biases in position,shape and size byfinding alternative pathwayswhen errors in the prop-agation lead to premature stops.
Reducing the length bias
There are two opposite effects that bias the streamlinedensity due towhite matter bundles length: i) seeding from the white matter in-creases the density because there aremore streamlines that are initiatedin longer bundles than in shorter bundles, ii) longer bundles are harderto completely recover because of premature stops (in WM or CSF),which decreases the streamline density. Seeding from WM/GMinterface reduces the effect of i) by initiating the propagation at extremi-ties of bundles. Thus, bundles of similar size, but various lengths, have asimilar number of seeds initiated in them. The premature stop bias causedby ii) is reduced using CMC and PFT in the same fashion as the position,size and shape biases. This can be observed in Table 3 where the averagelength of streamlines connecting gray matter regions is increased usingPFT. This means that more seeds initiated in longer bundles are includedin the final result, reducing the effect of ii).
In the end, in-housePFT generates streamlines connecting graymatter regions together with more than 95% of success rate for stream-lines reaching a length of 10mm, for both deterministic and probabilistictractography. PFT improves streamline distribution and can be triggeredin conjunction with any streamline tractography algorithm. Our resultssuggest that that streamlines recovered by PFT propagate in regionspreviously not covered by streamlines. PFT reduces the portion of prema-turely stopping streamlines and can have a positive effect on brainconnectivity studies. However, inaccuracies in the registration of anatom-ical and diffusion images might occur (Glasser et al., 2013), and couldimpact the performance of tractography algorithms using informationfrom anatomical images.
It is worth pointing out that tractography algorithms based on graphmodels or energy minimization method, often referred to as globaltractography algorithms, can also encode anatomical information andenforce connections between gray matter regions. For instance, thegraph-based tractography algorithm proposed in (Iturria-Medinaet al., 2008) penalized pathways going through CSF PVEwhen searchingfor a shortest path.Global tractography techniques have shownpromisingresults in recent years (Mangin et al., 2013) and their development is ofinterest. However, in most cases, anatomical information is not used in
Please cite this article as: Girard, G., et al., Towards quantitative connectivdx.doi.org/10.1016/j.neuroimage.2014.04.074
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guaranteed to connect gray matter regions and make premature stopsbias the structural connectivity analysis. Moreover, interpretation ofconnectivity based on global tractography is challenging, making’classical’ streamline tractography often used in connectomics studies(Fornito et al., 2013).
Finally, Particle Filtering Tractography (PFT) does not address theissue of invalid connections IC. Many included streamlines result fromnoise and errors in the propagation directions and manage to connectgray matter regions. They do not represent anatomical connections assuch. In this sense, one of the next big challenges is to reduce the invalidconnections and to perform better brain structural connectivity estima-tion. We believe PFT, CMC and the proposed tractography parametersare important steps towards tackling this challenge.
Conclusion
We have shown that optimizing tractography parameters, stoppingand seeding strategies can reduce the biases in position, shape, size andlength of the streamline distribution. These tractography biases are quan-titatively reported on both in-vivo and synthetic data. These findings arecritical for future quantitative structural connectivity analysis. We havetherefore proposed a novel framework for tractography. Informationfrom the T1-weighted image must be included in tractography and canno longer be ignored. This represents a paradigm shift in tractographyand strengthens the message that tractography cannot be a DW-MRI-only technique, as also proposed by Smith et al. (2012). Other priorinformation could be included from brain atlases, white matter bundlesprobability maps, blood vessels (Vigneau-Roy et al., in press) map orfunctional connectivity maps. Our novel tractography framework isflexible to these future add-ons and is therefore promising for newdevelopments in quantitative connectomics.
Acknowledgment
The authorswish to thank Emmanuel Caruyer, Ph.D. for the develop-ment and sharing of the synthetic data used in this study, and theTractometer team (Jean-Christophe Houde and Marc-Alexandre Côté,tractometer.org) for the tractography evaluation system.
Appendix A. Streamline tractography
A.1. Local reconstruction technique
Diffusion Tensor estimation and corresponding Fractional Anisotropy(FA)mapgenerationwere doneusingMRtrix (Tournier et al., 2012). From
Fig. C.1. Examples of Valid Connections (VC), Invalid Connections (IC) andNo Connections(NC) on the synthetic dataset (Côté et al., 2013). A thousand streamlines were initiated atthe seed voxel indicated by the arrow.
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this, the single fiber response function was estimated from all FA valuesabove a threshold of 0.7, within the WM binary mask. This single fiberresponse was used as input for spherical deconvolution (Descoteauxet al., 2009; Tournier et al., 2007) to compute the fiber ODFs, with spher-ical harmonic order 8, at every voxel. In this work, we used the efficientimplementation publicly available inMRtrix (Tournier et al., 2012).
A.2. Implementation details
In this study, we used deterministic and probabilistic streamlinetractography algorithms. In our implementation, the spherical harmonicsof thefiberODFs are projected on a discrete evenly distributed symmetricsphere of 724 vertices (Daducci et al., 2013; Garyfallidis et al., 2014).Propagation directions are always a vector of orientation correspondingto one vertexof the sphere andof lengthΔs=0.2 mm. ‘Overshoot’ errorshave been observed when using large Δs in curved structures, and smallΔs increases the computational burden and increase the sensitivity tonoise in diffusion direction (Tournier et al., 2012). No bias was observedusing Δs = 0.2 mm, which is consistent with the work of (Tournieret al., 2012). The single difference betweenprobabilistic anddeterministicalgorithms is the way the propagation direction vi + 1 is chosen. Given aposition pi, a propagation direction vi, the maximum deviation angle θ,and the fiber ODF threshold τ, the discrete set of potential propagationdirections can be estimated: all discrete directions on the sphere withan associated value greater than a fraction of the maximum value of thefiber ODF τ, and within the aperture cone define by θ and vi. The maxi-mum deviation angle θ between two consecutive steps (or a minimumradius of curvature R= Δs/(2 ⋅ sin(θ/2)) (Behrens et al., 2007; Tournieret al., 2012)), limits thehigh angle variations of streamlines and addressesthe smoothness assumption of WM fibers. The fiber ODF threshold τremoves some of the noise directions of the fiber ODF. Given the discreteset of potential propagation directions, vi + 1 is:
• Deterministic: The chosen propagation direction vi + 1 is the closestaligned maximum of the fiber ODF with the previous propagationdirection. Maxima of the fiber ODF are defined as any values greaterthan all its neighbors (6 to 9 vertices) in a cone of an angle of π/16(≈ 11°). No bias was observed due to the use of this strategy. However,other methods exist to extract maxima of the fiber ODF such as thoseproposed in (Bloy and Verma, 2008; Descoteaux et al., 2009; Ghoshet al., 2013; Tournier et al., 2012). Further investigation on the maximaextractionmethod on brain connectivity study is of interest but outsidethe scope of this paper.
• Probabilistic: The chosen propagation direction vi + 1 is drawn fromthe empirical distribution defined by the fiber ODF values of the poten-tial propagation directions (Behrens et al., 2007; Parker and Alexander,2005). The higher the value associated with a direction (vertex) is, thehigher the probability of propagating the streamline in this direction is.
The new trackingposition is pi+ 1= pi+Δs ⋅ vi+ 1. If thediscrete setof potential propagation directions is empty, vi + 1 = vi. Thetractography algorithmassumes an error in the fiber ODF and continuesin the previous propagation direction. This is done for a maximumdistance of δundeviated. This can be seen as allowing the step size Δs toincrease up to the size of δundeviated if there is no propagation directionlocally available.MRtrix andmost other algorithms stop the propagationif no valid direction is available.
From an initial propagation direction, the streamline propagates bymaking discrete steps of size Δs until a stopping criterion is reached.Then, the same is done in the opposite initial direction, creating thestreamline. The seeding position and the initial propagation directionare obtained following the seeding strategy (see Section 2.4). Thenext propagation directions are obtained following the tractographyalgorithm.
Once the tractography is done, streamlines with length within theinterval [δmin mm, δmax mm] are included in the estimated set of stream-lines and excluded otherwise. The minimum length criterion (δmin)
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ensures that connections are between minimally distanced graymatter regions. Themaximum length (δmax) criterion eliminates spuriousstreamlines that loop around or have impossible trajectories.
Appendix B. Particle filtering
The particle filter model has been widely used for localization(Arulampalam et al., 2002; Doucet et al., 2001) using sensor measure-ments to estimate position. Recently, it has also been used for whitematter tractography (Pontabry and Rousseau, 2011; Savadjiev et al.,2010; Zhang et al., 2009). Particle filtering methods aim to estimate asequence of target state variables X0 : t = {Xk, k = 0, …, t} from asequence of observation variables Y0 : t = {Yk, k = 0, …, t}. The goal isto sequentially estimate the posterior distribution p(Xk|Y0 : k). X0 : t is afirst order Markov process such that Xk|Xk − 1 ∼ p(Xk|Xk − 1) with aknown initial distribution p(X0) and Y0 : k are conditionally independentif X0 : k is known. The posterior distribution p(Xk|Y0 : k) is represented bya set of random sampleswith associated weights. It estimates the targetdistribution based on the samples and weights (Arulampalam et al.,2002; Doucet et al., 2001). {xk(i), wk
(i)}i = 1N denotes the set of N discrete
random samples that characterize the posterior distribution, where{xk(i), i = 1, …, N} is the set of random samples with {wk
(i), i = 1, …, N}their associated weights. The weight of a sample xk
(i) at time k corre-sponds to its weight at time k − 1 times the likelihood of the observationyk(i).Weights are then normalized over all particles to have∑ i = 1
N wk(i)=
1. Such a discrete model suffers of degeneracy since the variance of theweights increases over time, leading to a situation where all samplesexcept one have a weight close to zero. To overcome this problem aresamplingmethod is appliedwhen a significant degeneracy is observed.The degeneracy problem can be observed when the number of effectivesamples Neff falls below some threshold NT (Arulampalam et al., 2002).Neff is obtained following
Neff ¼ 1=XN
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The resampling eliminates samples with low weights and concen-trates on samples that have high weights. The resampling generates Nnew samples with equal weights from the current discrete estimationof p(Xk|Y0 : k) (Arulampalam et al., 2002). In this study, the resamplingis done when Neff b NT = N/10 (Arulampalam et al., 2002).
Appendix C. Global connectivity metrics defined in the Tractometer
The Tractometer is a novel tractography evaluation system based onnew global connectivity measures detailed in Côté et al. (2013). Here,we recall them for completeness.
• Valid Connections (VC): streamlines connecting expected regions ofinterest (ROIs) and not exiting the expected bundle mask (Côtéet al., 2013) (see Fig. C.1(a)).
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• Invalid Connections (IC): streamlines connecting unexpected ROIs orstreamlines connecting expected ROIs but exiting the expectedbundle mask. These streamlines are spatially coherent, havemanagedto connect ROIs, but do not agree with the ground truth (Côté et al.,2013) (see Fig. C.1(b)).
• No Connections (NC): streamlines that do not connect two ROIs. Thesestreamlines either stop prematurely due for example to angularconstraints or due to hitting the boundaries of the tracking mask(Côté et al., 2013) (see Fig. C.1(c)).
• Average Bundle Coverage (ABC): average of the number of voxelscrossed by streamlines divided by the total number of voxels in thebundle (Côté et al., 2013). This is the average proportion of bundlescovered by streamlines.
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