A flocking based method for brain tractography Ramon Aranda a,⇑ , Mariano Rivera a , Alonso Ramirez-Manzanares b a Department of Computer Science, Centro de Investigacion en Matematicas (CIMAT), A.C., Guanajuato, Gto 36240, Mexico b Department of Mathematics, University of Guanajuato, Guanajuato, Gto 36000, Mexico article info Article history: Received 25 November 2012 Received in revised form 5 November 2013 Accepted 25 January 2014 Available online 10 February 2014 Keywords: Tractography Diffusion tensor Stochastic walks Anatomical brain connectivity Flocking abstract We propose a new method to estimate axonal fiber pathways from Multiple Intra-Voxel Diffusion Orien- tations. Our method uses the multiple local orientation information for leading stochastic walks of par- ticles. These stochastic particles are modeled with mass and thus they are subject to gravitational and inertial forces. As result, we obtain smooth, filtered and compact trajectory bundles. This gravitational interaction can be seen as a flocking behavior among particles that promotes better and robust axon fiber estimations because they use collective information to move. However, the stochastic walks may gener- ate paths with low support (outliers), generally associated to incorrect brain connections. In order to eliminate the outlier pathways, we propose a filtering procedure based on principal component analysis and spectral clustering. The performance of the proposal is evaluated on Multiple Intra-Voxel Diffusion Orientations from two realistic numeric diffusion phantoms and a physical diffusion phantom. Addition- ally, we qualitatively demonstrate the performance on in vivo human brain data. Ó 2014 Elsevier B.V. All rights reserved. 1. Introduction White matter tractography is a technique that estimates neuro- nal tract pathways to determine both the structure and the con- nectivity of the brain. The existence of the neural tracts had only been demonstrated by histochemical and biological techniques in post-mortem specimens. Many white matter structures have been well documented in anatomic studies, for instance see Williams et al. (1997). However, the brain tracts are not well identified by direct examination of computed tomography scans or regular Mag- netic Resonance Imaging (MRI). This explains the poorly under- stood white matter functions and the inaccurate description of the neuro-anatomical atlas based on those technologies. The neural-tract structure estimation in vivo is one of the most important goals in neuroimaging, as it provides information on normal human brain development and about some neurological diseases; e.g., strokes, multiple sclerosis, epilepsy, neurodegenera- tive diseases and spinal cord disorders (Clark et al., 2000; Mukher- jee et al., 2000; Sotak, 2002; Ciccarelli et al., 2008). It can also be used to investigate the spectrum of neuropsychiatric disorders; e.g., language problems and reading disability (Klingberg et al., 2000; Catani et al., 2005). Recently, brain planning surgery procedures take into account the white matter tracts that are at risk in an intervention (Romano et al., 2009; Castellano et al., 2012). Neural tracts for in vivo human brain can indirectly be esti- mated by using the Diffusion-Weighted (DW) modality of MRI. These images provide information about the local brain structure by measuring the molecular water diffusion at local points of the brain. The most popular parametric model for representing and analyzing DW-MR signals is the Diffusion-Tensor MRI [DT-MRI, Basser et al. (1994)]. However, the DT-MRI provides only one Prin- cipal Diffusion Direction (PDD) per voxel and this estimation is insufficient to explain the brain structure at fiber crossings or bifurcations. For this reason, more sophisticated models for esti- mating Multiple Intra-Voxel Diffusion Orientations (MIVDO) have been proposed; see for instance the works of: Tuch et al. (2002) (Diffusion Multi-Tensor), Behrens et al. (2003) (Ball-and-Stick), Tuch (2004) (Q-Ball), Wedeen et al. (2005) (Diffusion Spectrum Imaging), Ramirez-Manzanares et al. (2007) (Diffusion Basis Functions), Tournier et al. (2007) (Spherical Deconvolution), Kaden et al. (2007) (Parametric Spherical Deconvolution), Scherrer and Warfield (2012) (CUSP-MFM), Zhang et al. (2012) (NODDI), Sotiropoulos et al. (2012) (Ball and Rackets), among others. Once the intra-voxel structure is estimated at each voxel of a DW- MRI volume, one can try to estimate the neural tract pathways and to determine connectivity information. The accurate computation of the brain connectivity relies completely on the accurate estimation of the cerebral tract pathways. For this reason, in this paper we propose a http://dx.doi.org/10.1016/j.media.2014.01.009 1361-8415/Ó 2014 Elsevier B.V. All rights reserved. ⇑ Corresponding author. Address: Department of Computer Science, Centro de Investigacion en Matematicas (CIMAT), A.C., Jalisco S/N, Col. Valenciana, Guanaju- ato, Gto 36240, Mexico. Tel.: +52 473 732 7155/735 0800x730; fax: +52 473 732 5749. E-mail address: [email protected](R. Aranda). Medical Image Analysis 18 (2014) 515–530 Contents lists available at ScienceDirect Medical Image Analysis journal homepage: www.elsevier.com/locate/media
Transcript
Medical Image Analysis 18 (2014) 515–530
Contents lists available at ScienceDirect
Medical Image Analysis
journal homepage: www.elsevier .com/locate /media
A flocking based method for brain tractography
http://dx.doi.org/10.1016/j.media.2014.01.0091361-8415/� 2014 Elsevier B.V. All rights reserved.
⇑ Corresponding author. Address: Department of Computer Science, Centro deInvestigacion en Matematicas (CIMAT), A.C., Jalisco S/N, Col. Valenciana, Guanaju-ato, Gto 36240, Mexico. Tel.: +52 473 732 7155/735 0800x730; fax: +52 473 7325749.
Ramon Aranda a,⇑, Mariano Rivera a, Alonso Ramirez-Manzanares b
a Department of Computer Science, Centro de Investigacion en Matematicas (CIMAT), A.C., Guanajuato, Gto 36240, Mexicob Department of Mathematics, University of Guanajuato, Guanajuato, Gto 36000, Mexico
a r t i c l e i n f o
Article history:Received 25 November 2012Received in revised form 5 November 2013Accepted 25 January 2014Available online 10 February 2014
We propose a new method to estimate axonal fiber pathways from Multiple Intra-Voxel Diffusion Orien-tations. Our method uses the multiple local orientation information for leading stochastic walks of par-ticles. These stochastic particles are modeled with mass and thus they are subject to gravitational andinertial forces. As result, we obtain smooth, filtered and compact trajectory bundles. This gravitationalinteraction can be seen as a flocking behavior among particles that promotes better and robust axon fiberestimations because they use collective information to move. However, the stochastic walks may gener-ate paths with low support (outliers), generally associated to incorrect brain connections. In order toeliminate the outlier pathways, we propose a filtering procedure based on principal component analysisand spectral clustering. The performance of the proposal is evaluated on Multiple Intra-Voxel DiffusionOrientations from two realistic numeric diffusion phantoms and a physical diffusion phantom. Addition-ally, we qualitatively demonstrate the performance on in vivo human brain data.
� 2014 Elsevier B.V. All rights reserved.
1. Introduction
White matter tractography is a technique that estimates neuro-nal tract pathways to determine both the structure and the con-nectivity of the brain. The existence of the neural tracts had onlybeen demonstrated by histochemical and biological techniques inpost-mortem specimens. Many white matter structures have beenwell documented in anatomic studies, for instance see Williamset al. (1997). However, the brain tracts are not well identified bydirect examination of computed tomography scans or regular Mag-netic Resonance Imaging (MRI). This explains the poorly under-stood white matter functions and the inaccurate description ofthe neuro-anatomical atlas based on those technologies.
The neural-tract structure estimation in vivo is one of the mostimportant goals in neuroimaging, as it provides information onnormal human brain development and about some neurologicaldiseases; e.g., strokes, multiple sclerosis, epilepsy, neurodegenera-tive diseases and spinal cord disorders (Clark et al., 2000; Mukher-jee et al., 2000; Sotak, 2002; Ciccarelli et al., 2008). It can also beused to investigate the spectrum of neuropsychiatric disorders;e.g., language problems and reading disability (Klingberg et al.,2000; Catani et al., 2005). Recently, brain planning surgery
procedures take into account the white matter tracts that are atrisk in an intervention (Romano et al., 2009; Castellano et al.,2012).
Neural tracts for in vivo human brain can indirectly be esti-mated by using the Diffusion-Weighted (DW) modality of MRI.These images provide information about the local brain structureby measuring the molecular water diffusion at local points of thebrain. The most popular parametric model for representing andanalyzing DW-MR signals is the Diffusion-Tensor MRI [DT-MRI,Basser et al. (1994)]. However, the DT-MRI provides only one Prin-cipal Diffusion Direction (PDD) per voxel and this estimation isinsufficient to explain the brain structure at fiber crossings orbifurcations. For this reason, more sophisticated models for esti-mating Multiple Intra-Voxel Diffusion Orientations (MIVDO) havebeen proposed; see for instance the works of: Tuch et al. (2002)(Diffusion Multi-Tensor), Behrens et al. (2003) (Ball-and-Stick),Tuch (2004) (Q-Ball), Wedeen et al. (2005) (Diffusion SpectrumImaging), Ramirez-Manzanares et al. (2007) (Diffusion BasisFunctions), Tournier et al. (2007) (Spherical Deconvolution), Kadenet al. (2007) (Parametric Spherical Deconvolution), Scherrer andWarfield (2012) (CUSP-MFM), Zhang et al. (2012) (NODDI),Sotiropoulos et al. (2012) (Ball and Rackets), among others.
Once the intra-voxel structure is estimated at each voxel of a DW-MRI volume, one can try to estimate the neural tract pathways and todetermine connectivity information. The accurate computation of thebrain connectivity relies completely on the accurate estimation of thecerebral tract pathways. For this reason, in this paper we propose a
516 R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530
novel approach to estimate axon bundle pathways from MIVDO. Ourmethod is based on stochastic walks of particles which, unlike otherapproaches, are simulated with mass. The particle dynamic is gov-erned by three components: a deterministic inertia force, a determin-istic gravitational force (as a result of the mass of the particles) and astochastic path selection. The gravitational force provides interactionbetween particles starting their walks within the same voxel. Weunderstand this interaction as a flocking behavior, which promotesbetter and robust axon fiber pathway estimations in the presence ofnoise because the particle motion incorporates collective informa-tion. Additionally, the stochastic path selection allows us to exploremultiple fiber bundle orientations, meanwhile the inertia and gravitypromote smoothness and compacity on bifurcations in the pathways.On the other hand, the stochastic walks may generate outlier pathswith low support, generally associated to incorrect brain connections.Thus, to eliminate the outliers we propose a clustering method basedon Principal Component Analysis (PCA) and Spectral Clustering (SC).The performance of the proposal is evaluated on MIVDO fields fromtwo realistic numeric diffusion phantoms and a physical diffusionphantom. Additionally, we qualitatively demonstrate the perfor-mance on in vivo human brain data.
In the following, Section 2 reviews the literature background ofthe tractography methods. The flocking paradigm is described inSection 3. Section 4 presents our proposals: the tractography andthe clustering approach. Section 5 shows the evaluation of experi-mental results, followed by our conclusions in Section 6.
2. Background on tractography methods
White matter tractography estimates fiber paths by using localintra-voxel information. There are different methods for brain trac-tography but, in general, the process starts with the definition of apoint on the white matter, known as seed point, which allocates avirtual particle. The particle is iteratively moved along differentvoxels following the local diffusion orientations until a stopping cri-teria is fulfilled. Hence, the estimation of the axonal fiber pathwayconsists of a point set defined by the particle trajectory. Most of thereported tractography methods can be classified as streamline orglobal optimization based techniques. After the axonal fiber path-ways are estimated, the main trajectories can be computed by clus-tering the pathways in bundles based on their trajectory features.
2.1. Streamline methods
Streamline methods estimate the axon pathway step by stepusing the motion of a virtual particle. This process is described bythe updating formula:
Fig. 1. Graphical representation of the streamline methods (three particles areschematized). At time t, the new fiber-segment estimation is only influenced by thepreviously estimated fiber-segment and the observed MIVDO information.
xtþ1 ¼ xt þ Ddt; ð1Þ
where xt ¼ ðxt;x; xt;y; xt;zÞ is the 3D particle position at step t; D is thestep size and dt is the motion direction. Fig. 1 depicts how thestreamline methods work: to estimate the new fiber-segment, onlythe previous fiber-segment estimation and the MIVDO observed bythe particle are used. The streamline based methods differ in theway they compute the direction dt; it can be deterministic or proba-bilistic. Deterministic streamline methods estimate the same fiberpathway for particles starting at the same seed point (Westinet al., 1999; Basser et al., 2000; Lazar et al., 2003; Fillard et al.,2003; Descoteaux et al., 2009). Probabilistic streamline methods setup several particles at a given seed point, which move indepen-dently through stochastically selected motion directions, measuringthe uncertainty of the local fiber bundle orientation at each voxeland they then produce probabilistic connection maps between pairsof brain regions (Parker et al., 2003; Ramirez-Manzanares and Rive-ra, 2006; Seunarine et al., 2007; Behrens et al., 2007; Savadjievet al., 2008; Descoteaux et al., 2009).
Given that the noise on the MRI images and partial volume ef-fects generate untrustworthy local intra-voxel diffusion orienta-tions, the drawback of streamline methods (deterministic orprobabilistic) is that the estimation error might accumulate alongthe distance followed by the particle.
2.2. Global methods
Global tractography methods improve the local fiber-segmentestimation at a given voxel by using the underlying informationon its spatial neighborhood. Fig. 2 depicts how the global methodswork: the information in the neighboring voxels is propagated tostrengthen the fiber-segment estimations. Global approachesestimate the fiber pathways by an energy minimization process(Parker et al., 2002; Prados et al., 2006; Fletcher et al., 2007). Thedisadvantage of these methods is their high computational cost.A sub-family of global methods introduces a stochastic componentin their formulation to reduce the computational cost (Kreher et al.,2008; Reisert et al., 2011). Although these methods reduce thecomputational burden with respect to (w.r.t.) the deterministicglobal approaches, it is still high for practical uses.
Global methods are robust because they integrate neighboringinformation to estimate fiber bundle pathways. On the other hand,the trajectory of a particle by using streamline techniques (determin-istic or probabilistic) is a noisy estimation of the entire axonal fiberpathway because each trajectory is formed by individually walkingparticles. Namely, to the best of our knowledge, no streamline meth-od shares information between the trajectories of the particles. Thus,
Fig. 2. Graphical representation of global tractography methods. A local fiber-segment estimation are influenced by the MIVDO information (continuous arrows)and by the spatial interactions between the rest of local fiber-segment estimations(dashed arrows).
R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530 517
we explore the benefits of introducing a collective information termto the particle’s motion. We combine a streamline technique withthe use of spatial information to estimate the fiber pathway. There-fore, we propose a novel regularization scheme based on a flockingmodel. Different from the classic streamline, our method integratesthe information over all the particles starting at the same seed voxel.Fig. 3 depicts the interactions among neighboring fiber-segment esti-mations in our model, which is similar to the global approaches, butwith a significant reduction of the computational time.
2.3. Fiber clustering
Once the tractography is computed, it is possible to cluster thetrajectories into compact bundles to find the main pathways. Theprincipal problem in fiber clustering is the high dimensionality ofthe data: assuming W steps, a single 3D pathway has dimensionR3�W . Reported fiber clustering methods differ in the way they ana-lyze and model the shape and the features of the trajectories (Brunet al., 2004; O’Donnell et al., 2006; Klein et al., 2008; Maddahet al., 2008; Liang et al., 2009; Wang et al., 2010; Ratnarajah et al.,2012). In this paper, we propose a fiber clustering based shape pro-cedure where the main trajectory features can efficiently be ex-tracted by a PCA. Then, we apply an SC procedure to find thenumber of clusters. Finally, we eliminate the paths with low support(outliers) which are generally associated to nonexistent neuraltracts or incorrect brain connections.
3. The flocking paradigm
Flocking is a collective behavior of a large number of interactingindividuals with a common objective (e.g., going from point a topoint b). This is a common phenomenon in animals such as fish,birds, and butterflies. A flock member is any individual of a groupwith similar characteristics, which observes and acts upon an envi-ronment and directs its activity towards achieving common goals.Over the last few decades, scientists from different disciplines;including animal behavior analysis, computer science, biophysicsand physics; have studied and modeled flocking, swarming andschooling in groups of individuals with local interactions (Okubo,1986; Toner and Tu, 1998; Olfati-Saber, 2006; Luo et al., 2010; Vic-sek and Zafeiris, 2012). All evidence indicates that flock motionmust be merely the aggregate result of the actions of individualanimals, each one acting solely on the basis of its own local percep-tion of the world. Reynolds (1987) introduced three heuristic rulesthat lead the simulation of flocking behavior:
(i) Flock centering: individuals attempt to stay close to nearbyflockmates.
(ii) Collision avoidance: flock members maintain prudent separa-tion from their nearby neighbors.
Fig. 3. Graphical representation of our proposal: flocking behavior to estimate thenew fiber-segment. The estimation of each fiber-segment is influenced by itscorresponding previously estimated fiber-segment at time t, the observed MIVDOinformation (both in continuous arrows), and by the previously estimated fiber-segments of the rest of the neighboring trajectories at time t (dashed arrow).
(iii) Velocity matching: nearby flock members must head inapproximately the same direction and at the same speed.
Other important characteristics about the flocks are:
(a) An independent flock member navigates according to itslocal perception of the dynamic environment.
(b) Solitary flock members and smaller flocks join to becomelarger flocks.
(c) In the presence of external obstacles, larger flocks can splitinto smaller flocks to navigate around such obstacles.Namely, an individual member can stay close to its nearbyneighbors, not caring if the rest of the flock turns away.
(d) The trajectory of each flock member is smooth.
All previous rules and characteristics are subject to a broadinterpretation and implementation.
As mentioned in Section 2, some tractography methods consistof walks of particles that start their journey at the same seed pointand have a common objective: to reach a pathway termination.Then, inspired by Reynolds’ rules of flocking behavior, we proposea novel flocking-based tractography method. In our case, the flockmembers are particles walking in the white matter for estimatingbrain structure and connectivity.
Our proposal introduces a simple term of communicationbetween pairs of particles (local interaction) that results in acollaborative behavior of the particle group (global interaction).This interaction is modeled by the gravitational force between theparticles. In this context flock obstacles are produced by partialvolume effects, fiber bundle crossings and bifurcations. The pro-posal is described in detail in the next section.
4. Methods
In this section we explain the main components of ourapproach: the flocking model for brain tractography and the fiberclustering. In the first subsection, we introduce the procedure toestimate the fiber tracts by using stochastic walks of massive par-ticles with a flock behavior. In addition, we explain how the nextposition for each particle is computed by taking into account theinertia, the gravitational influence of the surrounding particles(the flocking interaction) and the medium (the MIVDO informa-tion). The second subsection details our procedure for clusteringthe random walks into trajectory bundles in order to find the mainpathways.
4.1. Flocking-based tractography
The tractography associated to a set of particles started at thesame seed voxel can be divided into two independently computedwalks: one set moving forward of a selected diffusion direction andthe other one moving in the opposite direction. Now, let N be thenumber of walking particles at a seed point and xt;m be the positionof the mth particle at step t. Then, based on Eq. (1), we compute thenew position of the particle at time t þ 1 with
xtþ1;m ¼ xt;m þ Ddtþ1;m; ð2Þ
where D is the a step size and the motion direction is denoted bythe unitary vector dtþ1;m. In our approach, such a motion directionis computed with the three-term formula:
where the positive scalar parameters, ci, weigh the contributions ofthe directional vectors: dð1Þ; dð2Þ and dð3Þ, respectively. The firstdirection, dð1Þ, is associated with the medium where the particles
518 R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530
are moving: it codifies the information of the local fiber bundle ori-entation. Second direction, dð2Þ, introduces an inertial component inthe particle trajectory. The last direction, dð3Þ, is based on theflocking interaction that results from the gravitational effect ofthe surrounding particles and promotes that each particle has asimilar trajectory to the rest of the set. Fig. 3 describes how our ap-proach integrates the collective behavior in the particle set: eachparticle is influenced by (a) its corresponding position and motiondirection at time t, (b) the MIVDO information observed at its posi-tion, and (c) by the positions of the rest of the particles at time t. Inthis way, our proposal is based on the streamline model, but whenwe integrate a collective intelligence term we obtain advantagescomparable to the global approaches with a significant lower com-putational cost.
4.1.1. MIVDO representationMIVDO information in each voxel can be represented as a set
V ¼ fqj; bjgj¼1...;J; ð4Þ
where qj 2 R3 is jth PDD associated with the size compartment(contribution value) bj 2 R and J indicates the number of compo-nents estimated for a voxel (J can be different for each voxel). Theset V can be computed, for instance, by methods such as thosepresented in Tuch et al. (2002) (Diffusion Multi-Tensor), Behrenset al. (2003) (Ball-and-Stick), Tuch (2004) (Q-Ball), Wedeen et al.(2005) (Diffusion Spectrum Imaging), Ramirez-Manzanares et al.(2007) (Diffusion Basis Functions), Tournier et al. (2007) (SphericalDeconvolution), Kaden et al. (2007) (Parametric Spherical Deconvo-lution), Scherrer and Warfield (2012) (CUSP-MFM), Zhang et al.(2012) (NODDI), Sotiropoulos et al. (2012) (Ball and Rackets),among others.
4.1.2. Initialization of the particlesGiven a seed point, N particles at time t ¼ 0 are randomly posi-
tioned with a uniform distribution within the voxel that containsthe seed point; i.e., at sub-voxel positions. Then, the initial motiondirection for the N particles is set equal to a jth PDD, qj, of the intra-voxel information. In this paper, we select the qj associated withthe largest contribution value bj.
4.1.3. Medium dependent direction: dð1Þ
The most important component of the particle dynamics in (3)is the effect of the medium; i.e., the local tissue structure codifiedin the MIVDO. The calculation of this direction is independent foreach particle: each particle navigates according to its localperception.
As the particle positions are in real coordinates (sub-voxel posi-tions), a particle in the position xt;m is allocated into a cube formedfrom the eight integer 3D coordinates of the neighboring voxels,
(a)Fig. 4. 2D examples of the factors that influence the choosing of the next PDD for the mthdt;m and v r;m ¼ ½xt;m � yr;m�. (c) Closeness factor between xt;m and yr;m; dr ¼ jjxt;m � yr;mjj.
yr;m (r ¼ 1; . . . ;8). Each of those vertices have associated MIVDOinformation. Hence, given the current position xt;m, its previousmotion direction dt;m, and assuming independence w.r.t. the previ-ous steps t � 1; t � 2; . . . ;1, the probability density function forselecting the new direction from the medium by the mth particleat the time t þ 1 is given by:
where qr;j and br;j are the jth PDD and its respective contribution ofthe MIVDO information on the neighboring voxel yr;m, respectively.Function U promotes to set as dð1Þtþ1;m the qr;j more similar to the pre-vious motion direction:
Uðqr;j;br;j; dt;mÞ ¼ br;j qTr;jdt;m
��� ���w; ð6Þ
where w is a positive scalar parameter that controls the variance ofthe distribution of the function. The value of this function is directlyproportional to the parallelism between qr;j and dt;m and its corre-sponding size comportment br;j, see Panel Fig. 4(a). Function W pro-motes to select the PDDs of the nearest neighbor position, yr;m,which is located along the current particle’s trajectory, dt;m. There-fore, we model these two factors in the function W as:
as a measure of the collinearity between the vectors dt;m and½xt;m � yr;m�, and z as a positive scalar parameter that controls thevariance of the function distribution. A neighboring voxel will havethe largest value if the vector ½xt;m � yr;m� is aligned to dt;m, see Panel4(b). Next term
W2ðyr;m; xt;mÞ ¼ exp � 12r2 kxt;m � yr;mk
2� �
ð9Þ
weighs the Euclidean proximity of the current position to the neigh-boring voxel center yr;m; this is illustrated by Panel 4(c).
Finally, the direction associated to the medium is randomly se-lected among the PDDs, qr;j, corresponding to the neighboring vox-els by the sampling of (5). The sampling is reduced to a randomtournament equivalent to rolling weighted dice, see Appendix A.
4.1.4. Inertial force dð2Þ
The second direction, dð2Þ, is nothing but the previous motiondirection:
dð2Þtþ1;m ¼ dt;m: ð10Þ
(b) (c)particle. (a) Parallelism factor between dt;m and qr;j. (b) Collinearity factor between
R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530 519
It introduces an inertial component in the particle displacementavoiding sudden changes in the pathways, i.e., promoting smoothtrajectories.
4.1.5. Gravitational effect dð3Þ
To implement the flocking interaction, we use an analogy in-spired by Newton’s Universal Law of Gravitation: the force exertedon a particle at position xa by another one at position xb, both withunitary mass, is inversely proportional to the square of the distancebetween them. Thus, given ~um;i as the unit vector addressed fromxt;m to xt;i, we propose to introduce a flocking interaction in our sto-chastic particles with:
~Ft;m ¼ �GXN
i–m
~um;i
jjxt;m�xt;i jj2; if rmin < jjxt;m � xt;ijj2 < rmax
~um;irmin
; if rmin > jjxt;m � xt;ijj2
0 Otherwise;
8>>>>><>>>>>:
ð11Þ
where G is a user-defined gravitational factor; rmin and rmax are theminimum and maximum allowed distances, respectively, such thata particle exerts an influence on another one. The lower limit rmin isused to satisfy the third Reynolds rule (see Section 3) and to keepthe system stable: it avoids that Ft;m !1 when the distance be-tween two particles is going to zero. On the other hand, rmax helpsus to prevent unnecessary computations: when jjxt;m � xt;ijj2 has alarge value the gravitational force is assumed to be zero. Finally,we use (11) as the third component in our dynamic formula in (3):
dð3Þtþ1;m � F!
t;m: ð12Þ
4.1.6. Stopping criteria
Algorithm 1. Compute fibers pathways.
Require: A seed point, step size D, values c1; c2 and c3,number of particles N, gravity factor G, FA threshold s,MIVDO and FA fields.
1: Seeding N particles randomly with uniform distributionwithin the seed voxel;
2: Set d0;1:N equal to the PDD associated with the largestcontribution value bj in the seed voxel;
3: Set t ¼ 0 and X ¼ f1;2; . . . ;Ng;4: while X – ; do5: for 8 m 2 X do6: Compute dtþ1;m from (3) using the particles in X and
the MIVDO;7: Update xtþ1;m ¼ xt;m þ D dtþ1;m;8: if FAðxtþ1;mÞ < s then9: The mth particle is stopped;10: X ¼ X n fmg;11: end if12: t ¼ t þ 1;13: end for14: end while
The particles are kept walking until they reach a no-white-mat-ter voxel (a pathway termination). The detection of a path-end canbe achieved based on a white-matter mask or when the pathreaches a voxel where the corresponding DT has a FractionalAnisotropy (FA) coefficient lower than a given threshold s 2 ½0;1�(Basser and Pierpaoli, 1996). Therefore, our global stopping criteriais reached when all the walking particles are stopped. Algorithm 1summarizes our tractography procedure based on particle randomwalks and flocking interaction.
4.2. Fiber clustering with outlier rejection
After the tractography is performed for a large number of par-ticles, it is convenient to cluster the trajectories into bundles toestimate the main pathways. Given that our tractography methodincorporates a stochastic component and we are dealing withnoisy data, the stochastic walks may generate paths with low sup-port (outliers), generally associated to nonexistent neural tracts orincorrect brain connections. Therefore, in order to improve theestimation of the axonal fiber pathways, we cluster the fiber pathsand reject trajectory outliers. This procedure is independently per-formed to the forward and the backward estimated pathways perseed point.
In general, the clustering of high dimensional data is compli-cated. Due to this reason, the main difference between the meth-ods for computing trajectory clusters is the way they representthe feature set of the pathways. Given that our tractography meth-od computes trajectories from particles seeded at the same voxel,their pathways are correlated, thus, we propose a fiber clusteringshape-based where the main trajectory-features can efficientlybe extracted by a PCA. To apply PCA on the pathway data it is nec-essary to pre-process them because the number of steps for eachparticle can be different. We set M as the maximum number ofsteps on all the particles that start at the same seed voxel:
M ¼maxfMmg; m 2 ð1; . . . ;NÞ; ð13Þ
where Mm is the number of steps of particle m. Then, we interpolateall trajectories to have the length M by using cubic splines (typicallyin our experiments M � 150). In this way, we define the matrixX 2 RN�3M as:
X ¼
X1
X2
..
.
XN
266664
377775; ð14Þ
where the mth row of X:
Xm ¼ ½x0;m; x1;m; . . . ; xM;m� 2 R3M ð15Þ
is the vector make up of the 3D points from the trajectory of the mthparticle [see Eq. (1)]. We preserve the largest principal componentsthat represent 99% of the matrix information. Thus, in the reducefeature space, the mth fiber pathway is represented by:
Algorithm 2. Clustering and Outlier Rejection.
Require The particle trajectories of Algorithm 1, thresholds uand c (typically u ¼ 5% and c ¼ 0:3).
1: Compute the largest trajectory and set M as in (13);2: Compute the matrix X; 3: Apply PCA to X to obtain X0;4: Compute matrix L using (18);5: Set K equal to the number of eigenvalues of L smaller than
c;6: Compute the set of pathway clusters G ¼ fg1; g2; . . . ; gKg
from k-means;7: G ¼ G;8: for k ¼ 1 to K do9: Set nk equal to the number of fibers of cluster gk;10: if nk < u then11: G ¼ G n fgkg;12: end if13: end for14: return G
ge Analysis 18 (2014) 515–530
X0m 2 RQ ; ð16Þ
where, in general, Q � 3M is the number of principal components.In our experiments we observed that Q 6 20. The trajectories areclustered in the principal components space by using an SC method.To this aim, we compute the generalized Adjacent matrix A with:
Aij ¼ exp � 12r2 X0i � X0j
��� ���2� �
; ð17Þ
for i – j and Aii ¼ 0. Then, the Laplacian matrix is defined as:
L ¼ D� A; ð18Þ
where D is the diagonal grade matrix with elements computed asDii ¼
PjAij. Then the number of cluster k is set equal to the number
of eigenvalues equal to zero [for more details, about SC see Luxburg(2007)]. Due to the noise and numerical errors, the detection of thezero eigenvalues is done by using a small threshold, c. Then, we ap-ply k-means algorithm to group the pathways setting K equal to thenumber of eigenvalues of L smaller than c. Once the trajectory clus-ters, gk (for k ¼ 1; . . . ;K), are computed, we define G ¼ fgkg
k¼Kk¼1 as
the set of trajectory clusters. Then, we discard the false pathways(outliers) if their associated cluster, gk, has a number of pathways,nk, lower than a given percentage u of the total number of walksstarting from the same voxel (in our experiments we typically setu ¼ 5%). Finally, we denote G as set of clusters without the outliers.Algorithm 2 summarizes the whole clustering procedure.
5. Experiments and results
This section presents experimental results of our proposalshowing how the flocking interaction modifies particle behavior.Also, we show how the clustering procedure improves the fibertracking by grouping trajectories in bundles and eliminatingoutliers. To evaluate our approach, three different types of DW-MRI are used: from realistic numerical phantoms, from a physical
520 R. Aranda et al. / Medical Ima
Table 1Compilation of the Empirical Range of Values (ERV) and the Recommended Values (RV). A
Parameter ERV RV Comments
N 128–1024 512 This number is for each seed. It bounds tG 0.0–0.001 0.0001 This value depends on N and it sets the lD 0.1–1.0 0.4 Large values could lead to stepping out o
increase the computational timec1 0.0–1.0 1.0 Small values are recommended for noisyc2 0.0–1.0 0.4 A large value generates smooth trajectorc3 0.0–1.0 1.0 This value is related with G and it is direw and z 0–10 8 and 4 resp. Large values decrease the particle explor
Fig. 5. (a) MIVDO field of the Synthetic crossing data from
diffusion phantom and from an in vivo human brain. For each data-set, we computed the MIVDO fields by using the Diffusion BasisFunctions (DBF) method reported in Ramirez-Manzanares et al.(2007).
5.1. Parameters
To provide a guide for the parameter tuning, Table 1 shows therecommended values for each one. Those values were empiricallychosen according to experimental tests. For all the experimentsin this paper, we use the values shown in the third column ofTable 1. To illustrate the collective intelligence effect, in the exper-iments different values for the gravitational constant G are used, asis indicated in each case.
5.2. Synthetic data
For this experiment, we build a coherent fiber crossing field,such that the DW signal is synthesized with the following features:
� Synthetic data: The DW-MRI signal is synthesized from theGaussian Mixture Model (Tuch et al., 2002). The DT principaleigenvalue is set to 1� 10�3 mm2=s, the second and the thirdtensor eigenvalues are set equal to 2:22� 10�4 mm2=s; thisresults in a FA ¼ 0:74. The above values are taken from a sampleof tensors observed in brain data from a healthy volunteer. Ab-value equal to 1000 is used and Rician noise is added to eachmeasurement to produce a Signal to Noise Ratio (SNR) equal to9. For these data, we acquire 64 measurements. The generatedDBF MIVDO field is shown in Fig. 5.
Fig. 6 depicts the tractography behavior on the synthetic fibercrossing data for different values of the gravitational factor G. Inthis experiment, we used as seed point the position marked in
ll experiments were conducted with the RV.
he time complexity of the proposal as OðN2Þevel of the flocking effectf a bundle. Small values could cause to wrongly bend at fiber crossings and
dataies. Too smooth trajectories are in most cases incorrectctly proportional to the level of interaction between particlesation
the DBF model. (b) Zoom on the crossing area in (a).
Fig. 6. Fiber tracking behavior on a fiber crossing bundle of synthetic data using 512 particles and different G values (rows). (a) Fiber tracking using G ¼ 0, (d) fiber trackingusing G ¼ 1e�4 and (g) fiber tracking using G ¼ 0:003. (b), (e) and (h) are the results after the outlier rejection procedure from (a), (d) and (g), respectively. (c), (f) and (i) showthe average trajectory of each cluster gk 2 G from (b), (e) and (h), respectively.
Fig. 7. (a) Spatial seed point used for the Fiber Cup diffusion phantom. (b) Ground Truth fiber trajectories corresponding to each seed point.
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Panel 5(a) and 512 particles. The seed point was chosen such that atractography method could confuse the fiber crossing with a fiberbranching. One can see that for G ¼ 0 many particles take thewrong pathway because of their incomplete individual perceptionof the environment [see Panel 6(a)]; i.e., they do not behave as a
flock. Panel 6(b) shows the fiber tracking result after the applica-tion of the outlier rejection procedure (trajectories in set G). Onecan see that a wrong fiber branching is recovered, see Panel 6(c).A different result can be obtained if a proper gravitational factorG is used; the problem is corrected and the fiber crossing is
Fig. 8. Fiber tracking on the Fiber Cup diffusion phantom from the seed number 12 [blue mark in Panel (a)]. (a) Estimated fiber pathways. (b) Pathway bundles are shown bycolor (set G). (c) The cluster, gk 2 G, with the largest number of pathways. (d) main trajectory: the average of the trajectories in (c). (For interpretation of the references tocolour in this figure legend, the reader is referred to the web version of this article.)
Fig. 9. DBF MIVDO field. (a) Using the raw diffusion phantom data. (b) Using aneighborhood mean denoising filter over the DW phantom data.
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adequately recovered: the majority of particles take the correctpath and they still explore potential bifurcations [see Panels:6(d–f)]. Finally, for a large G value, the particle trajectories remaintogether generating highly compact trajectory bundles [Panel6(g)]. In this case, the crossing is correctly recovered [see Panels6(h and i)]; however, a large G value may produce incomplete re-sults at bifurcations because it reduces the exploration capability.
5.3. Fiber cup diffusion phantom
The next experiment is performed using data acquired from aphysical diffusion phantom, which was designed and built for theFiber Cup Contest 2009 (Poupon et al., 2008). For these data, weuse 2 repetitions of 64 measures with a b-value equal to 2000.
In the Fiber Cup challenge was only reported a single fibertrajectory per given seed point. A total of 16 seed points were used[see Panel Fig. 7(a)]. Panel 7(b) presents the Ground Truth (GT) foreach seed on the phantom. For more information about the Fiber
Cup we refer the reader to http://www.lnao.fr/spip.php?rubrique79.
5.3.1. Evaluation metricsTo evaluate our results, we first use the metrics reported in Fil-
lard et al. (2011). Following, we detail these metrics in order tostimulate a discussion on their pertinence in Section 5.3.3:
� Spatial metric (sm): it is the L2 norm between two corresponding3D trajectory points jjp2 � p1jj2, where p1 and p2 are a pair ofcorresponding points between the trajectories f1 and f2
respectively.� Tangent metric (tm): it is defined as arccos vT
1v2
�� �� 180p
� ��� ��, where
v1 ¼f 01ðp1Þjjf 01ðp1Þjj
and v2 ¼f 02ðp2Þjjf 02ðp2Þjj
are the normalized tangent vectors
at p1 and p2.� Curve metric (cm): the curvature at a position p of f is given by
jðpÞ ¼ jjf 0 ðpÞ�f 00 ðpÞjjjjf 0 ðpÞjj . Then, the curve metric is jjðp1Þ � jðp2Þj.
Previous metrics focus on the point-to-point performance, i.e.,from a local perspective. However, from a global point of view,the connections generated by the estimated trajectories are rele-vant. Recently in Côté et al. (2013), a set of global metrics wereintroduced, which take into account the resulting connectivity.Those metrics are: the Average Bundle Coverage (ABC), the ValidConnections (VC), the Invalid Connections (IC), the No Connections(NC), the Valid Bundles (VB) and the Invalid Bundles (IB). In theexperiments, we evaluate the results with both local and globalmetrics.
5.3.2. Comparisons and resultsIn order to report a single estimated fiber per seed point, given
the tracking result, we use the average trajectory of the largestcluster in G; we call this pathway the main trajectory per seedpoint. Fig. 8 illustrates the selection process of the bundle withthe biggest number of trajectories for the seed number 12. Giventhat our tractography method has a stochastic behavior, the maintrajectory can be different among outcomes. For this reason, wecompute the final single pathway as follows: first, we perform100 Montecarlo computations of the tractography process (usingthe same parameters) for each seed point. Then, we estimate themain trajectory of each outcome. Next, the 100 independent maintrajectories are clustered using the same procedure in Section 4.2.The final result is the mean pathway of the largest cluster; i.e.,the most likely trajectory. Note that the strategy of computing atractography experiments each one with N particles significantlyreduces the computational cost w.r.t. computing one experimentwith a large number of particles aN. This is due to the complexityorder OðN2Þ of the computation of the collective intelligence.
Fig. 10. Fiber tracking results of our proposal from the seed number 5 (white circle) by using 512 particles on both the MIVDO field with and without denoising. (a–c) areobtained by using a G ¼ 1e�4 and the noisy orientation field. (d–f) are obtained without flocking interaction and using the denoised orientation field. (b and e) show thebundle with the largest number of fibers from (a and d), respectively. (c and f) show the main trajectories from (b and e), respectively.
Table 2Percentage of main trajectories contained by the cluster with largest number over 100repetitions per seed point by using tracking with and without flocking interaction.
a This trajectory was incorrectly estimated according with the GT.
R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530 523
In order to show that a simple denoising process is not enoughto produce correct fiber tracking and a sophisticated procedure isrequired, we performed the following experiment. Fig. 9 showsthe DBF MIVDO field by using the raw DW-data and the DW-datafiltered with a homogeneous-neighborhood mean denoising filter.In this figure one can see how the filtered MIVDO field is morehomogeneous in the orientations (even at the crossings). Fig. 10shows the fiber tracking from the seed point number 5 on boththe MIVDO field with and without the denoising process. Eventhough the orientation field in Panel 9(b) is less noisy, the fiber
Table 3References of the 3 methods with the best ranking in the Fiber Cup.
Number Method Descripti
2 Fiber Tracking on the Fiber Cup Phantom using ConstrainedSpherical Deconvolution (Jeurissen et al., 2009; Jeurissen et al.,2011)
Streamlin(FOD) pestep is us
5 Filtered Tractography: Validation on a Physical Phantom(Malcolm et al., 2009; Malcolm et al., 2010)
Streamlindiffusionunscente
7 Tracking a Physical Phantom by Global Fiber Reconstruction(Reisert et al., 2009; Reisert et al., 2011)
Global traa parame
tracking does not obtain a valid trajectory. In contrast, using theoriginal noisy MIVDO field in Panel 9(a), our proposal recovers atrajectory similar to the GT. We attribute such a difference to thefact that our approach performs a more sophisticated regulariza-tion process over the trajectories than a simple data smoothing.
Table 2 shows the percentage of pathways in the cluster withthe largest number of main trajectories over 100 repetitions perseed point; it is larger when the flocking behavior is used, indicat-ing that the uncertainly is reduced. For comparative purposes, allthe most likely trajectory per seed point are compared with the out-comes of the 3 competitor methods with the best ranking in the Fi-ber Cup [Fillard et al. (2011)]: Method 2, Method 5 and Method 7(see Table 3). We generated our results using G ¼ 1e�4. Given thatthe participants of the Fiber Cup contest did not know the GT struc-ture, the comparison with these methods is only included to showthe capabilities of our proposal.
Fig. 11 shows the results of the three best methods reported inthe Fiber Cup and our proposal. Since the phantom has a 2Dlayered structure, a visual inspection is enough to appreciate theperformance of the methods. One can see that the best methodsare: Method 2 (the shapes of all the trajectories were similar to
on
e with propagation direction following the Fiber Orientation Distributionak closest to previous direction. An anisotropic denoising pre–processinged before FOD estimatione tractography with filtered estimation of propagation direction. Themodel estimation is guided by the previous propagation direction using
d Kalman filteringctography. Every point and direction of every fiber-segment estimation ister of the model and contribute as a single isotropic Gaussian model
Fig. 11. Visual result of the fiber tracking of the 3 best methods of the Fiber Cup and our approach.
Table 4Symmetric Root Mean Square Error using the spatial metric (L2 norm). The three bestresults by fiber estimation are in bold font.
Table 7Final scores by method per local metric. The three best scores by metric are in boldfond.
Methods
2 5 7 Proposal
Score sm 23 10 27 36Score tm 24 5 33 34Score cm 20 3 42 31
Total 67 18 102 101
Table 8Evaluation of the fiber tracking algorithms taking into account the global connectivityindexes. The Valid Connections (VC), Invalid Connections (IC), No Connections (NC) aregiven in percentages and the Valid Bundles (VB) and the Invalid Bundles (IB) are givenin number of bundles. The best results by metric are in bold fond.
524 R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530
the GT), Method 7 (only the trajectory 12 was incorrectly esti-mated) and our proposal (only trajectories 7 and 10 were incor-rectly estimated).
From a quantitative point of view, Tables 4–6 show the numer-ical comparison, trajectory by trajectory, between the GT and theresults of the 4 methods according to the local metrics used inthe Fiber Cup contest. These tables show the symmetric Root Mean
Square Error (sRMSE) for each local metric [see Fillard et al. (2011)].We show the three best results per estimated fiber in bold. Table 4shows the comparison of errors using the spatial metric. Here, thescores indicate that our proposal, the Method 2 and the Method 7have the best performance. Table 5 shows the errors according to
Fig. 12. Illustration of the inconsistencies of the curve metric: the yellow pathways (YP) present better results than the blue ones (BP) in that metric, despite the fact that theYP are clearly worse (best values per metric per panel in bold font). The GTs are in red color. (a) YP: sm = 34.82, tm = 24.97, cm = 0.083; BP: sm = 4.95, tm = 24.71, cm = 0.085.(b) YP:sm = 38.65, tm = 67.5, cm = 0.049; BP: sm = 6.45, tm = 13.01, cm = 0.091. (c) YP: sm = 21.57, tm = 40.93, cm = 0.072; BP: sm = 5.21, tm = 23.99, cm = 0.273. These valueswere taking from Fillard et al. (2011).
Table 9Rankings of the methods per metric. The mode of the rankings is indicated for eachmethod.
Fig. 13. (a) Actual fiber structure of the HARDI Reconstruction Challenge phantom.(b) Connectivity matrix from (a).
R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530 525
the tangent metric. In this metric, our proposal performs similarlycompared to Methods 2 and 7. In the same way, the results of ourmethod by using the curve metric in Table 6 are good, but areslightly exceeded by Method 7.
To provide a objective value to the results per local metric, wescore the errors as in the Fiber Cup contest: three points aregranted for the best result, two points for the second one, one pointfor the third one and zero points otherwise. Thus, Table 7 showsthe summation of the points per metric for each method. Onecan see that for the spatial metric our approach obtains the bestrating. In a similar way, by using the tangent metric, our resultsget the best score. Using the curve metric, the score of our methodobtains the second place (see discussion about the metrics in thefollowing subsection). Summarizing the scores over the three met-rics, we can see that the method with highest score is Method 7,followed very closely by our approach.
Moreover, Table 8 shows the evaluation by using the globalmetrics: VC, IC, NC, VB and IB (the ABC metric is not representativein our case because we only compute one trajectory per seedpoint). To evaluate these metrics, we use the white matter and re-gion of interest masks used in Côté et al. (2013). Those masks areavailable at http://scil.dinf.usherbrooke.ca/tractometer/. The val-ues in Table 8 show that the method with the best performanceis the number 7 followed by our proposal. Note that our proposalpresents competitive results for both quantitative and qualitativemetrics.
5.3.3. Discussion about the metricsHere, we discuss some issues about the definition of the curve
based metric in Section 5.3.1: if the curve value is similar betweenthe corresponding points, the metric reports a good performance,even when the trajectory is wrong in shape. For instance, somecontradictions with the curve metric can be illustrated by theexamples in Fig. 12. A visual inspection clearly favors the blue tra-jectories over the yellow1 ones compared with the GT (trajectories
1 For interpretation of color in Fig. 12, the reader is referred to the web version ofthis article.
in red). In the three cases, the curve metric indicates that the yellowpathways are better estimators than the blue ones. In contrast, thespatial and tangent metrics correctly indicate that the blue pathwaysare better estimators. Note that in some cases [see Panel 12(a)], thetangent metric does not clearly differentiate between the correcttrajectories and the wrong ones. In our opinion, the spatial metric re-flects more adequately the local pathway similarities.
We consider there is not a single metric to indicate which meth-od has the best performance as each metric evaluates differentfeatures of the trajectories. For this reason, we use the mode ofthe rankings as the overall performance. Table 9 shows the rank-ings for each metric and the mode of them per method. As can beseen, our proposal and the Method 7 have as mode the 1st place,however, Method 7 has a slightly better performance. From a com-putational point of view, we notice that our solution presents thefollowing advantage: our proposal recovered the previous resultsin about 30 min, as opposed to Method 7 that requires more thana processing day.
5.4. Phantom of the HARDI Reconstruction Challenge 2013
The following experiments are performed using the data ac-quired from the HARDI Reconstruction Challenge 2013. The phantomconsists of fiber bundles that simulate connections of pairs of cor-tical areas. It includes a wide range of: (a) partial volume configu-rations (branching, crossing, kissing), (b) fiber bundles radii, and (c)fiber geometries.
These data were generated using an approach similar to theNumerical Fiber Generator (Close et al., 2009): for the local modelof diffusion, the signal was simulated considering extra-axonal andintra-axonal diffusion, also, depending on the position in the space,
Fig. 14. (a) Estimated fiber structure for the HARDI Reconstruction Challenge phantom by taking the main trajectory from each seed point. (b) Estimated connectivity matrixfrom (a). (c) Estimated connectivity matrix generated from the average trajectories for each cluster in the set G by using u ¼ 5%.
Fig. 15. Estimated connectivity matrix generated from the average trajectories of the clusters in G repeating the fiber tracking 100 times. (a) With flocking interactionG ¼ 1e�4. (b) Without flocking interaction G ¼ 0.
Fig. 16. Results of the fiber tracking on in vivo human brain data using 512 particles. The seed point is located on the genu of the corpus callosum: blue dot in Panel (a). (b)Without flocking interaction, (c) With flocking interaction, G ¼ 1e�4. (For interpretation of the references to colour in this figure legend, the reader is referred to the webversion of this article.)
526 R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530
there is also an isotropic compartment simulating the cerebro-spinal fluid contamination close to the brain ventricles. The dataand more technical details are available at http://hardi.epfl.ch/static/events/2013_ISBI/index.html. For our experiments, we useone repetition of 64 measures with b ¼ 3000 s=mm2 and a SNRequal to 30.
The simulated cortex of the diffusion phantom is divided in 40regions of interest. A total of 20 fiber trajectories define the
phantom structure [see Panel Fig. 13(a)]. Each trajectory connectstwo different areas on the cortex generating thereby the connectiv-ity matrix shown in Panel Fig. 13(b). To evaluate the performanceof our method, we place one seed in the center of each cortical re-gion. Panel Fig. 14(a) shows the estimated main trajectory fromeach seed point, a total of 40 are shown. A direct visual inspectionreveals the good quality of the results computed with the proposalin this paper. The connectivity matrix generated with the
Fig. 17. Area of 160 seed voxels in the corpus callosum.
R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530 527
estimated main trajectories is presented in Panel Fig. 14(b). One cansee that 34 out of 40 tracts reach their actual connectivity area. Inorder to show that the relevant connectivity information is mainlyassociated to the main trajectories, Panel Fig. 14(c) shows the con-nectivity matrix generated from the average trajectory of each
Fig. 18. Fiber tracking on in vivo human brain data using 160 seed points along the corpuviews of the estimated fibers without outliers and (d–f) show different views of the ave
Fig. 19. Outlier rejection on in vivo real human brain data. Five seed points was seeded on(a) shows the estimated pathways. Panel (b) shows the cleaning effect of the outlier rej
gk 2 G per seed point by using u ¼ 5%. We note that this matrixpresents more false connections than the reconstruction given bythe main trajectories. However, in actual brain analysis, the relevantinformation could not be contained only on the main trajectories;for instance in the case of branchings, the estimation of one singlepathway bundle is not enough.
In order to carefully validate the effect of the flocking behavior,we repeat 100 times the fiber tracking procedure for each seedpoint. Fig. 15 shows both, the estimated connectivity matrix gener-ated taking into account the average trajectories for each cluster inG with G ¼ 1e�4 [Panel Fig. 15(a)] and without the flocking effect[Panel Fig. 15(b)]. We corroborate that the matrix generated usingthe collective interaction is more similar to the actual connectivitymatrix than using G ¼ 0. One can see that the number of false con-nections has been reduced in Panel Fig. 15(a) w.r.t. Panel Fig. 15(b),indicating that the flocking effect reduce the uncertainly.
5.5. In vivo human brain data
The last experiments were achieved using in vivo human braindata with the following features:
s callosum, with G ¼ 1e�4 and 512 particles in each seed point. (a–c) show differentrage trajectory of each cluster in G.
the front-occipital tract in both hemispheres in the circular marks in Panel (a). Panelection procedure.
Fig. 20. Seed points used for the estimation of the CST. 4 voxels are localed in eachhemisphere.
528 R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530
� In vivo human brain data. A single healthy volunteer wasscanned on a Siemens Trio 3T scanner with 12 channel coil.Acquisition parameters were: single-shot echo-planar imaging,five images with b ¼ 0 s=mm, 64 DW images with unique,
Fig. 21. Cortical–Spinal tract estimated by using the seed points in Fig. 20. 512 particles wthe flocking interaction and by using G ¼ 1e�4, respectively. (b and e) show the estimated(c and f) show the average trajectories of each cluster in G.
isotropically distributed orientations ðb ¼ 1000 s=mm2Þ,TR = 6700 ms, TE = 85 ms, 90� flip angle, voxel dimensions equalto 2� 2� 2 mm3. The SNR is, approximately, equal to 26.
Fig. 16 shows the estimated fibers of the frontal forceps. The re-sults are computed using one seed point located in the genu of thecorpus callosum indicated by the blue dot in Panel 16(a). Panel16(b) shows the results when the flocking interaction is deacti-vated ðG ¼ 0Þ and Panel 16(c) shows the tracts using G ¼ 1e�4. Inboth cases we used 512 particles. These images clearly show thatthe flocking effect reduces the incorrected axonal pathway estima-tions (outlier rejection is not performed).
Fig. 18 shows different views of the results computed with ourmethod using 160 seed voxels along the corpus callosum (seeFig. 17): 512 particles were used for each seed point withG ¼ 1e�4. These images show the estimated pathways after remov-ing outliers. The average trajectories of the clusters in the set G areshown in Panels 18(d–f).
Fig. 19 shows the performance of the outlier rejection proce-dure. Here, five seeds were placed by an expert where the fronto-occipital tract crosses in both hemispheres: black dots in Panel19(a). Also, Panel 19(a) shows the fiber tracking by using 512 par-ticles and G ¼ 1e�4. Then, Panel 19(b) shows the estimated axonalpathways after the outlier rejection stage, one can see that the finalestimations correspond to fronto-occipital tracts.
In the last experiment, in order to estimate cortical–spinal tract(CST), we seed particles at the points marked in Fig. 20. The resultsare shown in Fig. 21: Panel 21(a) shows the fiber tracking estima-tions without the flocking interaction. Panel 21(b) shows the esti-mated fibers after applying the outlier rejection from Panel 21(a).
as used per seed point. Panel (a), and (d) show the estimated fiber pathways withoutfibers after applying the outlier rejection to the pathways in (a and d), respectively.
R. Aranda et al. / Medical Image Analysis 18 (2014) 515–530 529
The average trajectories of the estimated clusters in Panel 21(b) areshown in Panel 21(b). On the other hand, Panels Fig. 21(c–e) showthe results with the flocking interaction ðG ¼ 1e�4Þ. One can seethat when the flocking behavior is applied the CST is better esti-mated, i.e. the pathways do not cross through the corpus callosumreducing false estimations.
6. Conclusions
White matter tractography estimates brain axonal pathways byusing information from the analysis of the DW-MRI. The tractogra-phy problem is complicated by itself: on the one hand, we haveincomplete information because of the partial volume effect prob-lems; on the other hand, we have inconsistencies because of thenoise. In consequence, many fiber paths are erroneously estimated.In order to overcome these difficulties several research groups uti-lize post processing procedures, such as manual elimination of theestimated fiber trajectories. In this paper, we provide two auto-matic tools to aid in this extensive task and, consequently, to re-duce the error rate: a robust flocking based tractography and ashape based clustering method.
To deal with the partial volume effect our tractography can beapplied on any MIVDO field where more than a single diffusionpeak provides valuable information about the structure and mustbe taken into account for a correct tractography. However, in manycases some of those diffusion peaks can be erroneous due to noisein the signal or an improper model fitting. Our tractography meth-od incorporates collective intelligence in the form of flocking inter-action to reduce the recovery of false information; we implement itas a gravitational model which promotes the particles to move ingroups. The method consists of a streamline model that benefitsfrom the advantages from the interaction between neighboring fi-ber-segment estimations, similarly to the strategies of global trac-tography approaches, but without the high computational burden.This strategy allows the particles to estimate dense axonal tractpathways without reducing their exploration capability. In prac-tice, our proposal significantly improves w.r.t. estimations com-puted without the use of collective information. Despite that ourflocking based method reduces false pathway estimations, it isnot exempt from outlier pathways. Our technique uses a low-dimensional feature representation of the estimated fiber path-ways and a shape-based clustering that automatically eliminatesthe spurious axon trajectories with low support. In our experi-ments, this procedure removes erroneous trajectories therebyreducing potential manual elimination.
The building of brain structural atlas is important to improvethe investigation of the general brain structure. The quality ofthose atlas relies on proper tract estimation and the capability ofperforming exhaustive experimentation on large brain-databases.Therefore, this work benefits brain research because it allows toobtain better precision tractography results maintaining low com-putational time.
Acknowledgments
R. Aranda was supported by a Ph.D. scholarship from CONACYT,Mexico. M. Rivera and A. Ramirez-Manzanares were partially sup-ported by SNI-CONACYT, Mexico, grants: 131369, 169178-F and169338. We also acknowledge to Journal NeuroImage to providethe images from Panels 11(a-c).
Appendix A. Sampling of discrete distributions
Let ½qi�i¼1;2;...;I be the event set (in our case the PDDs set in theparticle’s neighborhood) with corresponding probabilities
½pi�i¼1;2;...;I [in our case, computed from (5)], such thatP
ipi ¼ 1and pi > 0. Then Pk ¼
Pki¼1pi (with P0 ¼ 0) is the cumulative dis-
crete probability. Then, a sample q P can be obtained by gener-ating a random number with uniform distribution: r U½0;1�. Thusthe outcome is q ¼ qk such that Pk�1 < r 6 Pk for k ¼ 1;2; . . . ; I.
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