Cortical surface parcellation based on intra-subject white matter fiber clustering
Narciso López-López, Andrea Vázquez, Cyril Poupon, Jean-François Mangin, Pamela Guevara
CCORTICAL SURFACE PARCELLATION BASED ON INTRA-SUBJECT WHITE MATTERFIBER CLUSTERING
Narciso L´opez-L´opez , , Andrea V´azquez ,Cyril Poupon , Jean-Franc¸ois Mangin and Pamela Guevara Universidad de Concepci´on, Faculty of Engineering, Concepci´on, Chile Universidade da Coru˜na, LBD group, Dept. of Computer Science,Facultade de Inform´atica, CITIC, Campus de Elvi˜na, 15071, A Coru˜na, Spain Neurospin, I2BM, CEA, Gif-sur-Yvette, France
ABSTRACT
We present a hybrid method that performs the completeparcellation of the cerebral cortex of an individual, based onthe connectivity information of the white matter fibers froma whole-brain tractography dataset. The method consists offive steps, first intra-subject clustering is performed on thebrain tractography. The fibers that make up each cluster arethen intersected with the cortical mesh and then filtered todiscard outliers. In addition, the method resolves the overlap-ping between the different intersection regions (sub-parcels)throughout the cortex efficiently. Finally, a post-processingis done to achieve more uniform sub-parcels. The output isthe complete labeling of cortical mesh vertices, representingthe different cortex sub-parcels, with strong connections toother sub-parcels. We evaluated our method with measures ofbrain connectivity such as functional segregation (clusteringcoefficient), functional integration (characteristic path length)and small-world. Results in five subjects from ARCHI data-base show a good individual cortical parcellation for each one,composed of about 200 sub-parcels per hemisphere and com-plying with these connectivity measures.
Index Terms — Parcellation, clustering, white matter,connectivity, fibers, tractography.
1. INTRODUCTION
Advances in brain imaging have allowed the study of thestructure and connectivity of white matter (WM), a re-search area that is constantly growing. One of the mostused techniques to understand the anatomical connectivityof the brain is the diffusion-weighted Magnetic ResonanceImaging (dMRI). It is a non-invasive and in-vivo technique,
This work has received funding by the CONICYT PFCHA/DOCTORADO NACIONAL/2016-21160342, CONICYT FONDECYT1190701, CONICYT PIA/Anillo de Investigaci´on en Ciencia y Tecnolog´ıaACT172121, Basal Center FB0008, and by the European Union’s Horizon2020 research and innovation programme under the Marie Sklodowska-Curiegrant agreement No 690941. based on measurements of the movement of hydrogen mo-lecules present in water [1]. Tractography algorithms usedMRI information to estimate the main trajectories of theWM tracts [2]. When applied to the whole-brain, resultingdatasets contain a large amount of 3D polylines, called fibers,that represent the main brain WM connectivity.Understanding how the brain works requires a detaileddescription of the network of connections that form it [3].A cortical parcellation represents a way to divide the braincortex into macroscopic regions, according to their structureor functioning, in order to study brain connectivity [4]. Thebest known parcellation is Brodmann’s atlas, based on post-mortem cytoarchitecture study, focused on the size, density,shape and distribution of cell bodies in cortical layers [5]. Incontrast, in-vivo techniques based on MRI, enable the devel-opment of other parcellations, based on anatomical structures[6], functional MRI (fMRI) [7] or a multi-modal approach [8].Performing a cortical parcellation is a difficult task due to thehigh variability that exists between subjects in terms of whitematter and gray matter, as well as the disadvantages of eachimaging modality.The most common approaches to estimate brain con-nectivity are diffusion tractography, structural covariance,resting-state functional connectivity and meta-analytic con-nectivity modeling [9]. Diffusion tractography provides in-formation about structural connectivity, but has the limita-tions of not being able to delimit the beginnings and termina-tions of fiber bundles [10] and produce false positives due tothe large number of fibers that cross between the WM tracts.In addition, short association fiber connections can be lost dueto the limited resolution of the tractographic methods [11].Two strategies can be used to perform diffusion-based corticalparcellations. One approach first determines correspondingconnections across subjects and then creates a parcellationaccording to the main connections in all the subjects. Forexample, a fiber bundle atlas of superficial WM connectionswas used to segment bundles in a group of subjects and getsome consistent parcels for the 10 analyzed subjects [12]. a r X i v : . [ q - b i o . N C ] F e b he difficulty here is to detect a representative set of the com-mon connections for a population of subjects and create thefinal parcels. The second approach detects robust individualparcels from the whole tractography dataset, and then man-ages to find and delineate consistent parcels across subjects[13].In this work, we propose a new hybrid method of indi-vidual cortical parcellation based on WM connectivity andintra-subject fiber clustering, with automatic parcel labeling.Our goal is to perform a good quality individual cortical par-cellation to be used for a group-wise parcellation in the future.We applied the method to a group of subjects and evaluatedseveral measures of brain connectivity. We demonstrate thatthe resulting networks for each subject comply with the in-tegration and functional segregation as well as with the small-world definition.
2. MATERIALS AND METHODS2.1. Database and tractography datasets
We used the ARCHI database, composed of 79 healthy sub-jects [14], and acquired with a 3T MRI scanner (Siemens,Erlangen). The MRI protocol included the acquisition ofa T1-weighted dataset using an MPRAGE sequence (160slices; matrix=256x240; voxel size=1x1x1.1 mm) and a SS-EPI single-shell HARDI dataset along 60 optimized DWdirections, b=1500 s/mm (70 slices, TH=1.7 mm, TE=93ms, TR=14,000 ms, FA=90, matrix=128x128, RBW=1502Hz/pixel, echospacing ES=0.75 ms, partial Fourier factorPF=6/8; GRAPPA = 2).By using BrainVISA/Connectomist-2.0 software [15],data were pre-processed. The outliers were removed and thesources of artefacts were corrected. Next, to obtain ODFfields in each one of the voxels, the analytical Q-ball modelwas computed, and streamline deterministic tractography wasperformed on the entire T1-based brain mask, with a forwardstep of 0.2 mm and a maximum curvature angle of 30 ◦ .Moreover, to convert the data between the spaces Talairach,T1 and T2, the transformation matrices are available in thedatabase. We perform the parcellation of the cortex with a hybridmethod based on WM connectivity given by fiber clusters,leading to an automatic labeling of cortical regions. In thefollowing, we explain the whole method (see Figure 1), whichis composed of five steps: (1) fiber clustering, (2) intersectionwith the mesh, (3)
WM fiber filtering, (4) parcellation of thecortex and (5) sub-parcel post-processing.
STEP 1: Fiber clustering : We apply an intra-subject clus-tering to all the fibers of the whole-brain tractography dataset. The objective is to create clusters with similar fibers, accord-ing to their position and shape, which we call fiber bundles.First, the tractography obtained from the database is pre-processed (see Figure 1(a) . ). Fibers are resampled with 21equidistant points, a number of points big enough to repres-ent all the brain fibers. Next, the tractography datasets aretransformed from T2 to T1 space.Then, an intra-subject clustering algorithm is applied,which is composed of three main steps [16]. First, a Mini-batch k-means clustering [17] is performed in parallel overa subset of fiber points from all the fibers in the dataset. Wechoose minibatch k-means since it has low spatial and tem-poral complexities obtaining good quality of the clusters. TheElbow method [18] is used to determine the optimal numberof clusters for k-means. Next, resulting point clusters areused to create fiber clusters by mapping, i. e. fibers withpoints sharing the same point clusters are grouped. This isconstructed using a dictionary where each fiber has the clusterlabels as a key, and the value in the dictionary is the groupof fibers that share the same key. Finally, a merge is made ofthe final clusters that share the central point and are nearby.Next, for each group we calculate the maximum direct ( d E )and flipped ( d Ef ) distance for all the fibers, and thus be ableto compute the maximum Euclidean distance ( d ME see equa-tion 3) between the relevant points. Equations 1 and 2 showthe computation of the Euclidean distance d E and d Ef , whileequation 3 describes the maximum Euclidean distance: d E ( a, b ) = || a − b || = ( k (cid:88) i =1 ( a i − b i ) ) / (1) d Ef ( a, b ) = d E ( a, b f ) = d E ( a f , b ) (2) d ME = min ( max ( d E ( a, b )) , max ( d Ef ( a, b ))) (3)where a and b are the coordinates of the fiber points inthe 3D space, and k =
21 because it is a sufficient numberof points to consider for obtaining a good result. We use theequation 3 since the orientation of the fibers obtained fromthe tractography is unknown. Finally, all the cluster centersthat comply with d ME < d max are merged. Figure 1(a) . shows an example of the results after applying the clusteringto a tractography dataset. STEP 2: Intersection with the mesh : This step determinesthe intersection of the fiber clusters with the cortical mesh,and was modified from [19]. Figure 1(b) shows an exampleof intersection points of the fibers with the mesh.First, a subdivision of 3D space into small cubic cells(around 1.5 mm) is implemented, to optimize spatial searchesin the mesh. Then, some projection points are calculated fromeach fiber endpoint, detecting the cells and the neighborhoodthat touch the endpoints. Finally, the M¨oller-Trumbore al-gorithm [20] is used to calculate the intersection of the fibers ig. 1 . Parcellation method.
Step 1: Fiber clustering . First, the whole tractography is resampled with 21 points and trans-formed to T1 space. Next, a fiber clustering is applied to obtain compact clusters.
Step 2: Intersection with the mesh . Theintersection of the fiber clusters with the cortical mesh is calculated.
Step 3: WM fiber filtering . The fibers of each cluster arelabeled according to an anatomical parcellation. Fibers that do not correspond to the most common connections are filtered out.Also, inverted fibers are realigned.
Step 4: Parcellation of the cortex . Preliminary sub-parcels, from each cluster extremityare created. Next, small preliminary sub-parcels are removed. Finally, the overlap among sub-parcels is solved, by assigningeach triangle to the most connected sub-parcel.
Step 5: Sub-parcel post-processing . The main connected component of eachsub-parcel is kept, in order to remove small isolated areas. Next, an erosion followed by a dilation (opening operation) areapplied to eliminate some imperfections in the perimeter of the sub-parcels.and the mesh triangles. Equation 4 shows the intersection al-gorithm: O + tD = (1 − u − v ) V + uV + vV (4)where O is the ray of origin, t is the distance, D is thenormalized ray direction, ( u, v ) are the coordinates of inter-section in the triangle and V , V and V are the vertices ofthe mentioned triangle. The conditions for the coordinatesare u ≥ , v ≥ and u + v ≤ . Finally, the triangle in-tersects if there are permitted values within the ranges for thevariables t , u and v . STEP 3: WM fiber filtering : This step aims to label theclusters that intersect with the mesh and filter them to delimitanatomical cortical regions (brain circonvolutions). To perform the filtering, we carry out three sub-steps (seeFigure 1(c)):1.
Obtaining the most common connections : First,the fibers of each cluster are labeled according to theDesikan-Killiany cortical atlas labels [21]. For that,we use the mesh vertex labeling information given byFreesurfer (see Figure 1(c) . ). Next, the most com-mon connection, at the beginning and the end of eachcluster, are determined. Figure 2 . shows an examplefor different clusters connecting the Superior-Parietal(SP) and PreCentral (PrC) parcels.2. Removing of uncommon fibers : The next sub-stepaims to remove the fibers that do not correspond to themost common connections, in order to keep only the ig. 2 . Example of WM Fiber filtering for a cluster.
Sub-step3.1 : Obtaining the most common connections. In this case,the most common connections are SP and PrC.
Sub-step 3.2 :Removing of uncommon fibers. The fiber labeled with theIP parcel is removed since it does not belong to SP.
Sub-step3.3 : Alignment of fibers. Fibers that are inverted according tothe most common connections, are swapped.fibers corresponding to the anatomical parcels given bythe Desikan-Killiany cortical atlas. See Figure 2 . foran example.3. Alignment of fibers : Finally, a fiber alignment is ap-plied, where fibers that are inverted by respect to themost common connections are swapped, as shown inFigure 2 . .Once the aforementioned sub-steps have been carried out,white matter fiber clusters are filtered, getting a better delim-itation between bundles. STEP 4: Parcellation of the cortex : This step creates sub-parcels from the preliminary sub-parcels defined by the in-tersection of each cluster. It solves the conflicts between theoverlaps of the preliminary sub-parcels. The creation of thesub-parcels is done in three sub-steps:1.
Creation of preliminary sub-parcels from clusters :Each fiber cluster will define two preliminary sub-parcels, one from each extremity of the cluster. The setof preliminary sub-parcels is created by constructing alist of the triangles intersecting each cluster extremity(see Figure 1(d) . ). Note that a triangle can be inter-sected by several cluster extremities.2. Removing of small preliminary sub-parcels : Prelim-inary sub-parcels, that are too small, with a size 10%smaller than the average size of the preliminary sub-parcels within an anatomical parcel, are removed (seeFigure 1(d) . ). With this step a big amount of isol-ated triangles and noisy preliminary sub-parcels are re-moved. 3. Conflict resolution due to preliminary sub-parceloverlap : Some preliminary sub-parcels present anoverlapping within an anatomical region. To solve thisproblem, the conflicting triangles (belonging to severalpreliminary sub-parcels) are analyzed, and assigned tothe sub-parcel with the higher number of intersectingfibers (see Figure 1(d) . ).At the end of this step, the existing conflicts betweenpreliminary sub-parcels disappear, thus obtaining a set ofsub-parcels corresponding to each anatomical parcel (circon-volution). STEP 5: Sub-parcel post-processing : This is the laststep of the cortical parcellation method, that aims to performa refinement of the sub-parcels, in order to get more uniformareas. It consists of three sub-steps:1.
Elimination of connected components : Within ananatomical parcel, a graph is created for each sub-parcel and then, the connected components of eachgraph are calculated. The largest connected componentof each sub-parcel is kept, leading to the removal ofsmall isolated areas (see Figure 1(e) . ).2. Erosion : The edges of each sub-parcel are eroded overthe mesh, to finish the removal of some peaks or protru-sions that deform the sub-parcel perimeter (see Figure1(e) . ).3. Dilation : Finally, the sub-parcels are expanded, fillingthe gaps left by erosion, resulting in smooth, uniformand well-defined sub-parcels (see Figure 1(e) . ). Themorphological operation of erosion + dilation corres-ponds to an opening , which is an operation for noiseelimination.After carrying out the post-processing, we obtain the com-plete parcellation of the cortical mesh. It is defined by thesubdivision of the cortex into sub-parcels, given by a label foreach mesh vertex.
3. RESULTS
Almost all the parcellation steps were implemented in Pythonprogramming language (Python 3.6), with the exception ofthe intersection algorithm, made in C++11, which is paral-lelized with OpenMP. All the experiments ran on a computerwith an 8-core Intel Core i7-6700K CPU running at 4GHz,8MB of shared L3 cache and 8GB of RAM, using Ubuntu18.04.2 LTS with kernel 4.15.0-55 (64 bits).We calculated the cortical parcellation of five subjectsfrom the ARCHI database, using their complete clusteredtractographies. To evaluate the quality of the connectionsamong parcels, we generated a connectivity map for eachsubject, from the resulting parcellation, and evaluated it usingetwork graph metrics. A connectivity map is built up by thetractography of each subject and the mesh parcellated intosub-parcels, performing the following steps:1. The intersection of the complete tractography with theparcellated mesh is calculated.2. A square matrix n ∗ n is created with n equal to the totalnumber of sub-parcels, initialized to 0.3. For each fiber in the tractography, connecting two sub-parcels, a 1 is added to the cells corresponding to bothsub-parcels in the connectivity matrix.Hence, a connectivity map was calculated for each sub-ject, from its individual parcellation and tractography. There are many metrics for the evaluation of the characterist-ics of brain networks [22]. The properties we choose to ana-lyze them are functional segregation (Clustering Coefficient),functional integration (Characteristic Path Length) and Small-World [23]. These properties have been shown to be presentin the brains of the higher vertebrates [24]:1. Functional segregation: It is the presence of strongly in-terconnected groups or clusters in the brain. The metricused to measure this property is the Clustering Coeffi-cient [25]. A value closer to 0 denotes a random net-work, however, a complex-network shows higher clus-tering coefficient values. Equation 5 defines the clus-tering coefficient for undirected graphs: C i = 2 N i k i ( k i − (5)where N i is the amount of links in the neighborhood of i , k i is the degree of a particular node i and C i is theclustering coefficient for node i .The equation 6 measures the average clustering coeffi-cient for the entire network: C = 1 n (cid:88) i ∈ G C i (6)where G is the graph of the undirected network, n isthe total number of nodes in the network and C is theaverage clustering coefficient.2. Functional integration: It is the ability to easily distrib-ute information across the different specialized regionsof the brain. The better the information is distributed,the higher is the functional integration. The measure used to measure this property is the Path Length, spe-cifically the Characteristic Path Length [25] that aver-ages the shortest path length between each pair of nodes(sub-parcels) in the network. Equation 7 describes thecharacteristic path length for an undirected graph: L = 1 n ( n − (cid:88) i,j ∈ G,i (cid:54) = j d ij (7)where G is the graph of the whole undirected network, n is the total number of nodes in the network, d ij between nodes i and j is the smallest distance betweenthem and L is the characteristic path length.3. Small-World: This metric is very relevant, since itcombines the two previous ones. A brain network musthave good functional segregation, keeping functionalintegration a little lower, that is, strongly interconnec-ted internal regions, and in turn, a good amount oflinks to other regions [23]. The σ coefficient is usedto measure the property of small-world, which is theratio between the clustering coefficient and its equival-ent random network divided by the path length and itscorresponding random network. Equation 8 details thesigma coefficient: σ = CC r LL r (8)where C is the clustering coefficient, C r is the cluster-ing coefficient for the equivalent random network, L isthe path length, L r is the path length of its equivalentrandom network, and σ is the coefficient that measuresthe small-world. A network is considered small-worldif C (cid:29) C r and L ≈ L r , then σ > bctpy toolbox for Python, that provides functions to calculate the Clus-tering Coefficient and the Characteristic Path Length met-rics. Moreover, to calculate the Small-World metric, wetransformed our matrices into graphs and used the networkx library for Python [26]. Figure 3 shows the metrics for thefive subjects. A high Clustering Coefficient, while main-taining a lower Characteristic Path Length, and therefore aSmall-World > https://github.com/aestrivex/bctpy ig. 3 . Measures of brain connectivity obtained for the fivesubjects: Clustering Coefficient, Characteristic Path Lengthand Small-World. In this subsection we show the different views of the cerebralcortex of a subject, as well as the different individual parcel-lations for five subjects in the database.Figure 4 shows the coronal, axial, right and left sagittalviews, resulting from the individual parcellation of subject001, consisting of 430 sub-parcels for the whole brain, with209 in the left hemisphere and 221 in the right hemisphere.Finally, Figure 5 displays the parcellation results for the fivesubjects. An average of 400 sub-parcels was obtained for thewhole cortex, with approximately 200 sub-parcels in averageper hemisphere.
4. CONCLUSIONS
We developed a hybrid method for the individual cortex par-cellation, based on the connectivity of WM fiber clusters. Thefiber clustering helps to define compact connections and filterout outliers. The method provides good quality results in theconnectivity maps of the five analyzed subjects, evaluated bynetwork graph metrics. Resulting networks show a high Clus-tering Coefficient, low Characteristic Path Length and Small-World property. These properties indicate good integrationand functional segregation of the brain [23]. As future work,we will explore the implementation of a multi-subject ver-sion of this parcellation method and test it in different data-bases, such as the Human Connectome Project. Hence, wecould obtain an atlas (or model) of cortical parcels with sim-ilar connectivity profiles across a population of healthy sub-jects. Also, other information could be integrated, like fiberssegmented with a bundle atlas [27], or data from other mod-alities, like fMRI.
Fig. 4 . Individual parcellation for subject 001. Coronal, axial,right and left sagittal views are displayed. The parcellationsubdivides the cortex into 430 sub-parcels, 209 in the lefthemisphere and 221 in the right hemisphere.
Fig. 5 . Individual cortex parcellation for five subjects (rightsagittal views). The average of sub-parcels obtained for ahemisphere is 200, with 400 sub-parcels on average of theentire cerebral cortex.
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