white matter tract analysis in 454 adults using maximum...

White Matter Tract Analysis in 454 Adultsusing Maximum Density Paths

Gautam Prasad1, Shantanu H. Joshi1, Neda Jahanshad1, Julio Villalon1,Iman Aganj2, Christophe Lenglet3, Guillermo Sapiro2, Katie L. McMahon4,

Greig I. de Zubicaray5, Nicholas G. Martin4, Margaret J. Wright4,Arthur W. Toga1, and Paul M. Thompson1

1Laboratory of Neuro Imaging, Department of Neurology, UCLA School ofMedicine, Los Angeles, CA, USA

2Department of Electrical and Computer Engineering, University of Minnesota,Minneapolis, MN, USA

3Department of Radiology - CMRR, University of Minnesota Medical School,Minneapolis, MN, USA

4Queensland Institute of Medical Research, Brisbane, Australia5School of Psychology, University of Queensland, Brisbane, Australia

Abstract. We introduce a framework for population analysis of whitematter tracts in diffusion weighted images. Our pipeline computes fibersfrom high angular resolution diffusion imaging (HARDI) images; it clus-ters the fibers incorporating prior knowledge from an atlas; the methodthen represents each fiber bundle compactly using a path following pointsof highest density (maximum density path; MDP); and registers thesepaths together using geodesic curve registration to find local correspon-dences across a population. Our framework is tested with 454 subjectsimages using 4-Tesla HARDI to find localized statistics across 50 whitematter tracts based on fractional anisotropy (FA) and sex differences.In addition, we compared the results with those derived with streamlinetractography.

Keywords: HARDI, tractography, MRI, brain, clustering, atlas, Dijk-stra, shortest path, geodesic distance, Hough, connectivity

1 Introduction

Diffusion weighted imaging (DWI) captures the diffusion of water in the brainin vivo. The brain’s structure may be understood by analyzing the local diffu-sion information these images provide. These images are used by tractographyalgorithms to follow the dominant directions of diffusion to reconstruct fibers ori-ented along axons that correspond to the brain’s major white matter pathways.We can then study these white matter regions in individuals and populations togain localized understanding of the effects of disease [22], changes with age [1],hemispheric differences [10], and sex differences [17].

2 Authors Suppressed Due to Excessive Length

The wealth of information from high angular resolution imaging (HARDI)images affords us a better understanding of the brain’s structure when comparedto the single-tensor model [4]. The single-tensor model does not account forfiber crossing or mixing, while the orientation distribution function (ODF) [23]derived from HARDI images allow us to discriminate multiple fibers passingthrough a voxel with different orientations. In contrast to previous tractographyapproaches our method leverages all the information rendered by our HARDIimages to generate fibers using a novel tractography algorithm.

Tractography algorithms generate a huge number of fibers in the image spaceand need to be clustered for analysis. A wealth of clustering methods have beenapplied to tractography results including fuzzy clustering [18], normalized cuts[5], k -means [15], spectral clustering [16], Dirichlet distributions [14], and hier-archical clustering [24], and a Gaussian process framework [26]. Many of thesemethods will benefit from prior anatomical information provided by an atlas oflikely locations of the tracts in the brain, helping to provide insight into whento split or combine clusters to conform to known anatomy.

In [6] the authors model fiber results with parametrized curves enabling coor-dinate based population analysis. This analysis has also been widely done usingtract based spatial statistics (TBSS) [20], which uses an alignment-invarianttract skeleton representation. Our approach provides additional analysis powerbecause it finds homologies across our representations through registration al-lowing localized comparisons in a population.

We create a framework to generate fibers from HARDI images, cluster themby incorporating prior anatomical knowledge, find correspondences across ourwhite matter regions, and use them to compute tract-based measures of FA andgeometric variation by examining hemispheric and sex differences in fifty majorwhite matter regions. Our method is tested on 454 subjects captured with 4-Tesla HARDI, to our knowledge the largest whole-brain tractography study todate.

2 Methods

Our method computes population statistics of white matter tracts from HARDIsubject images of the brain by finding fibers, clustering the fibers, representingthe clusters compactly, and finding correspondences across these representationsthrough registration. The framework is designed to use a novel tractographyalgorithm on each subject image to calculate a set of fibers. Those fibers thatintersect an atlas of white matter tract regions of interest (ROI) are selectedas a cluster. The clusters are represented by a path following points of highestdensity in the bundle and these paths are brought into correspondence usinggeodesic curve registration for point-wise comparisons. The pipeline uses this tofind localized differences in white matter tract fractional anisotropy (FA) andgeometry across hemispheres and sex. We summarize the method in Figure 1.

White Matter Fiber Tract Analysis 3

Fig. 1: Schematic of the pipeline for statistical analysis of white matter tracts.

2.1 HARDI Tractography using the Hough Transform

We compute fibers in each HARDI subject image using a novel tractographyalgorithm [3] based on the Hough transform. It utilized the extensive informationprovided by HARDI at each voxel, parametrized by the orientation distributionfunction (ODF).

Our tractography method selects fibers in the diffusion image space by gener-ating scores for all possible curves. The method incorporates fractional anisotropy(FA) from the single-tensor model of diffusion [4] into a prior probability to gen-erate seed points. These seed points are used to generate curves that receivea score estimating the probability of their existence, computed from the ODFsfrom the voxels the curve passes through.

The ODFs at each voxel from our HARDI images were computed using anormalized and dimensionless estimator derived from Q-ball imaging (QBI) [2].In contrast to previous methods, this method views the Jacobian factor r2 incomputing the solid angle (CSA) ODF as

14π

+1

16π2FRT

{52

b ln(− ln

S(u)S0

)}. (1)

In this equation, S(u) is the diffusion signal, S0 is the baseline image, FRTis the Funk-Radon transform, and 52

b is the Laplace-Beltrami operator. Thisestimate is more accurate mathematically and outperforms the original QBIdefinition [23], meaning it provides superior resolution for detecting multiplefiber orientations [2, 8] and more information for our scoring function.


The Hough transform tractography chooses fibers from all curves generatedin the image space. The method scores as many fibers as possible arising froma seed point and uses the voting process provided by the Hough transform toselect the best fitting curve through each point. These filtered curves comprisethe final fibers produced by the method.

The method was run on each subject image and generated 2000 to 5000 fibers(Fig. 2 shows a representative example with our data), which we subsequentlyclustered using a white matter atlas.

2.2 Fiber Clustering with White Matter ROI Atlas

The fibers we calculated using tractography based on the Hough transform areclustered by selecting those that intersect an atlas of white matter tract ROI.We use the atlas to incorporate prior anatomical information into our clustering.

We utilize the Johns Hopkins University (JHU) atlas, which delineates 50white matter regions of interest (ROI). This ROI atlas is registered to our subjectspace using an affine transform provided by FMRIB’s Linear Image RegistrationTool (FLIRT) [11].

For our analysis pipeline, the intersection of fibers with the atlas specifies afiber bundle or cluster. We select an ROI in the atlas and compute the locationsour fibers intersect. The intersection is quantified by a set of voxels each fiberpasses through and we count the number of voxels that coincide with the selectedROI to determine a fiber intersection score. This score is used to select fibersthat belong to a ROI and thus a white matter tract. The ROI will select manyfibers that spuriously intersect it. We address this by a threshold that removesfibers with a relatively low score, the threshold is dependent on how many andwhat type of fibers the tractography algorithm samples. In our case, we tunedthe threshold to fibers computed with our Hough transform based method.

2.3 Cluster Representation using Maximum Density Path

Once we have selected a fiber bundle to represent a white matter tract we com-pute a compact representation. This representation follows points of maximumfiber density in the bundle to create a path. We refer to this as the maximumdensity path (MDP) and it provides a basis for population comparisons andallows localized statistics and analysis.

We constructed a graph (a set of nodes and undirected edges) to representthe voxel-wise density in our fiber bundle. We computed a density volume ofour fiber bundle to characterize our search space. This space specifies the countof fibers that intersect each voxel. These voxels are used as nodes in a graph (aset of nodes and undirected edges connecting them) with those of positive valueconnected to its 26 neighbors by undirected edges. Each edge was weightedinversely by the sum of the voxel densities it connected as

1di + dj

, (2)


with di and dj as node i and j’s corresponding voxel density value.We found a path through this graph following nodes with highest density

using Dijkstra’s algorithm [7]. Dijkstra’s algorithm is a graph search methodthat finds the shortest path from a source node to every other node. However,the number of nodes in the graph may be large and when the algorithm is usedfor a single destination node it may be stopped once that path is found.

To find the shortest path Dijkstra’s algorithm requires the graph to havestart and end nodes, these are specified beforehand in the ROI atlas. The ROIpoints for the start and end locations may not have corresponding positive den-sity values derived from our bundle, we find the closest voxels in the densityvolume as the corresponding start and end nodes in subsequent computationwith Euclidean distance used as the metric.

Dijkstra’s algorithm will find the path once the start and end nodes are fi-nalized. It will first assign a distance value to every node in the graph, zero forthe start node and infinity for the rest. The algorithm marks all nodes as unvis-ited and the start node as current. The current node’s distance to its unvisitedneighbors is calculated and if that distance is smaller then the neighbor’s existingdistance value it is replaced. The method will mark the current node visited andthe current node’s distance is final, meaning the distance from the start node toitself will remain zero. The new current node will be selected as any unvisitednode and the algorithm will continue exploring nodes until the destination nodeis visited, resulting in a path to this node minimized in distance.

If Dijkstra’s algorithm is unable to find the path between the start andend nodes our method automatically identifies this situation and takes stepsto remedy the graph and find the shortest path. The algorithm will be incapableof finding a connection between the two nodes if the graph structure is suchthat there are no edges from the subgraph containing the start node with thesubgraph containing the end node, caused by scanner issues or a deficiency infibers calculated from tractography. In this situation we add edges and nodes tothe graph so all voxels within our ROI are fully connected with their neighbors.The edges are weighted with the smallest positive value our machine is capableof representing and allows gaps between the start and end nodes to be filled inwithout biasing the resulting shortest path.

Dijkstra’s algorithm computes a set of voxel locations in our image spaceconnecting the start and end nodes to create a path. We smooth the path so itis better conditioned for subsequent processing. We convolve the 3D coordinatesof our path with a Gaussian kernel to achieve this, though fitting these pointsto a spline would also have sufficed.

Our resulting maximum density paths (MDPs) are then registered togetherto find correspondences across the set.

2.4 MDP Correspondences through Geodesic Curve Registration

The MDPs represent fiber bundles that characterize a tract in a low-dimensionspace. These compact descriptions of a tract’s scale, location, and high-levelgeometric information are computed for all subjects in a population. We find


correspondences across the paths by bringing them into the same space usinggeodesic curve registration. The registration’s target is the population’s meanMDP.

We use geometric features of the MDPs, based the on our method for match-ing 3D curves [12, 13]. The MDP path is represented by a parametrized curve inR3. Assuming a standard unit interval, the coordinate function of this path isdenoted by β(s) : [0, 1] → R3. The coordinate function is not a good choice toanalyze the shape of the path, since it is confounded by global location, scale,and orientation. While the objective of coordinate registration approaches is theestablishment of spatial correspondences across tracts, our approach here is toestablish shape correspondence between different tracts. As our goal is to ana-lyze the tract geometry independent of the pose, we represent the shape of thecoordinate function of the MDP using a vector valued function as

q(s) =β(s)√||β(s)||

∈ R3 . (3)

Here, s ∈ [0, 1], || · || ≡√

(·, ·)R3 , and (·, ·)R3 is the standard Euclidean innerproduct in R3. We now denote Q ≡ {q|q(s) : [0, 1] → R3|

∫ 1

0(q(s), q(s))R3ds = 1}

as the space of all unit-length curves. On account of scale invariance, the spaceQ becomes a unit-sphere of functions, and represents all open elastic curvesinvariant to translation and uniform scaling. The tangent space of Q is givenas Tq(Q) = {w = (w1, w2, . . . , wn)|w(s) : [0, 1] → R3, ∀s ∈ [0, 1] such that∫ 1

0(w(s), q(s))R3 ds = 0}. Here each wi represents a tangent vector in the tangent

space of Q. We define a metric on the tangent space as follows. Given a curveq ∈ Q, and the first order perturbations of q given by u, v ∈ Tq(Q), respectively,the inner product between the tangent vectors u, v to Q at q is defined as,

〈u, v〉 =∫ 1

0

(u(s), v(s))R3ds. (4)

Now given two shapes q1 and q2, the translation and scale-invariant shapedistance between them is simply found by measuring the length of the geodesicconnecting them on the sphere. To ensure rotational invariance, we search foran optimal geodesic path in a quotient space of the sphere under the specialorthogonal group SO(3).

In addition to computing geodesic paths on the space of shapes, one can alsominimize the geodesic variance that consists of sum of squared pairwise geodesicdistances for a given collection of paths. This optimization procedure enablesone to compute statistical quantities, such as means and covariances of MDPs.

The MDP parametrization afforded by this registration method producescorrespondences that facilitate comparison across a population.


2.5 MDPs Compared across a Population

The registrations step provides features along each MDP that can be comparedacross the population in the form of points. We compute each point’s variationof fractional anisotropy (FA).

3 Experimental Results

3.1 Subject Data

Our subjects comprised of 454 healthy young adult twins and their siblings. Wecaptured these subjects with a 4-Tesla Bruker Medspec MRI scanner, collecting3D 105-gradient high angular resolution diffusion images (HARDI) and standardstructural T1-weighted magnetic resolution images (MRI). The images consistedof 55 slices 2-mm thick with a 1.79 × 1.79 mm2 in-plane resolution. For eachimage, we collected 94 diffusion-weighted images (b = 1159 s/mm2) using auniform distribution of gradient directions on the hemisphere. Additionally, wecollected 11 b0 (non-diffusion encoding) images and corrected all images for eddycurrent distortions and motion with FSL (www.fmrib.ox.ax.uk/fsl). Our cohortconsisted of 289 women and 165 men, ranging from 20 to 30 years of age.

We prepared our image data for our analysis pipeline. In each T1-weighted,we manually removed non-brain tissue and registered them to the Colin27 highresolution brain template using a 9-parameter transformation [9]. These skull-stripped and registered T1-weighted images each have a corresponding averageb0 diffusion weighted image (DWI), combining all eleven we collected. Theseaverage images were masked using BET [19] and this space was used to generateFA maps. Additionally, we used the FA images to compute a geometrically-centered study-specific mean template (mean deformation template; MDT).

We computed MDPs for 50 regions specified in the white matter ROI atlasfor each of the 454 subjects. By registering the MDPs to their sample mean,we computed corresponding points across the MDPs. The FA at these locationswas used for two-sample t-tests comparing MDPs in groups differing by sex.These tests were completed on the fibers derived using the Hough transformmethod and the widely used streamline method [25] that reduces the HARDIinformation to tensors and follows the direction of highest diffusion in the tensori.e. the eigenvector with the largest eigenvalue.

Fig. 3 & 4 show the false discovery rate (FDR threshold < 0.05) [21] adjustedp-values from two-sample t-tests on FA along each subject’s MDP for the Houghtransform and streamline based tractography methods respectively. p-values areoverlaid in color (red being more significant) on the mean MDP for each whitematter region.

4 Discussion

The cingulum, anterior limb of the internal capsule, superior longitudinal fasci-culus, and anterior corona radiata, were some of the white matter regions with


Fig. 2: Fibers extracted using the Hough transform based method.

significant differences between the two sexes. The analysis of the Hough-derivedfibers analysis showed more significant regions than that of the streamline-derived fibers i.e. the Hough results showed in both hemispheres the cingulumand the superior longitudinal fasciculus, hinting that the greater resolution af-forded by HARDI images can help find more significant regions.

We implemented and tested a new pipeline for tract-based analysis of DTI,which uses an atlas to help cluster tracts for point-wise curve-based analysis.Sex differences in the FA of 50 regions of interest delineated in an ROI atlas,suggesting promise of the method for detecting other factors that affect tracts,such as disease and risk genes.

Our method enables population analysis of diffusion-weighted images withoutrelying exclusively on global registration of the images into the same space.The method we introduce uses the registration of the ROI atlas to the subjectspace to roughly initialize seed points which are used to generate maximumdensity paths derived from fibers from tractography. The MDPs are then put intocorrespondence through curve registration, allowing us to focus the registrationspecifically on the white matter region we are interested in without involvingthe rest of the image.

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