STRUCTURAL PARCELLATION OF THE THALAMUS USING SHORTEST-PATHTRACTOGRAPHY

Niklas Kasenburg1, Sune Darkner1, Ute Hahn2, Matthew Liptrot1,3, Aasa Feragen1

1DIKU, University of Copenhagen; 2Department of Mathematics, Aarhus University; 3DTU Compute

ABSTRACTWe demonstrate how structural parcellation can be imple-mented using shortest-path tractography, thereby addressingsome of the shortcomings of the conventional approaches. Inparticular, our algorithm quantifies, via p-values, the confi-dence that a voxel in the parcellated region is connected toeach cortical target region. Calculation of these statisticalmeasures is derived from a rank-based test on the histogramof tract-based scores from all the shortest-paths found be-tween the seed voxel and each voxel within the target region.Using data from the Human Connectome Project, we showthat parcellation of the thalamus results in p-value maps thatare spatially coherent across subjects. Whilst some agreementto a hard segmentation as in previous studies is observed, thesoft segmentation exhibits better stability for voxels con-nected to multiple target regions.

1. INTRODUCTION

Structural parcellation [1, 2, 3, 4, 5, 6] is the data-driven seg-mentation of a source region defined by structural connectiv-ity to a set of pre-defined target regions of interest (ROIs).We propose a parcellation algorithm which statistically quan-tifies the degree of connectivity to each target ROI for eachvoxel in the region to be parcellated. In contrast to classicalmajority voting methods [7], our method outputs a soft seg-mentation based on these confidence measures. We demon-strate the approach on 5 subjects from the Human Connec-tome Project [8].

Our approach is based on shortest-path tractography(SPT) [9] and overcomes all of the following problems,which remain unsolved by traditional approaches to struc-tural parcellation:

a) Traditional fibre tracking methods start at a sourcepoint and randomly walk in the most likely direction asdefined by a fibre orientation function (fODF). Thesemethods may have a hard time retrieving connectionsfrom a voxel v to a specific cortical region C if thevoxel v is also connected to other regions, whose con-nections are easier to track. In particular, fibre tracking

A.F. is funded in part by the Danish Council for Independent Research(DFF), Technology and Production Sciences.

has a bias towards finding connections between nearbyregions and will typically find more connections tolarge target regions than to small ones, even if this isnot supported by the data [10, 11].

b) Traditional structural parcellation assumes that everyvoxel in the region to be segmented (in our case, thethalamus) is connected to one of the target ROIs. How-ever, this is not always the case. Moreover, there maynot be enough signal in the data to support such anassignment even if the connections exist. This is es-pecially true for inside the thalamus, where fibres areusually hard to resolve.

c) Traditional structural parcellation performs voting toobtain a hard segmentation of the region to be seg-mented based on connections to the cortical ROIs.However, this is suboptimal for several reasons [2]:First, if a source voxel is not physically connected toeither of the target regions, the found connections arebased only on noise and may therefore be very unsta-ble. Second, both anatomically and because of partialvolume effects, many voxels are with high probabilityconnected to multiple cortical regions. Since a hardsegmentation is not able to model this, it may again,lead to unstable results.

Shortest-path tractography finds the most likely trajectoryfor a fibre connecting any two locations in the brain. Thisformulation lends itself well to structural parcellation, whichprecisely seeks connections between two pre-specified re-gions of interest: The source region to be segmented (in thispaper, the thalamus) and the cortical target region, one at atime. Since SPT will always find a most likely fibre connect-ing any two voxels, it avoids problem a): Connections will befound both to small regions and between regions that are anydistance apart.

However, this property of SPT also introduces a new prob-lem: A connection will always be found, even if it is not there.Many SPT methods assign a score to the found paths whichcan be used to threshold unlikely paths, but such a thresholdwill reintroduce the biases from problem a). In this paper,we therefore propose an alternative approach which, assum-ing some thalamic voxels are physically connected to eachtarget region, aims to define these as statistically more sig-

nificant than those which are not. This is obtained through astatistical test over source region voxels.

Inspired by Kasenburg et al. [12] we quantify the confi-dence with which a voxel is significantly connected to eachof the target regions, by assigning a p-value to each sourcevoxel. This also solves problem b), as we allow voxels withinthe source region to have a p-value close to 1 for every singletarget region, i.e. not being strongly connected to any targetregion. Using the p-values as confidence maps, we obtain asoft structural parcellation of the thalamus, which allows foroverlap between the segments corresponding to different tar-get ROIs. In this way, we also solve problem c).

2. METHODOLOGY

Shortest-path tractography (SPT). In SPT [9], the diffusionweighted image (DWI) is turned into a graph where all grey-and white matter voxels from the brain are nodes. Edges areformed between pairs of voxels in a 3×3×3 neighbourhood,whenever at least one of the voxels is white matter. Eachedge ~e pointing out of a voxel v is given a weight reflectingthe probability of a fibre bundle tangential to ~e. This prob-ability is defined by integrating the fODF estimated for thevoxel v over the set of directions out of v which are closerto ~e than to any other edge pointing out of v. The integral isestimated by sampling. To obtain an undirected graph, edgeweights are averaged over their start and end nodes, obtainingprobabilistic edge weights p(e).

From this weighted graph, tractography between twopoints v and w is phrased as finding the most probablepath from v to w in the graph, that is, the path maximiz-ing p(π) =

∏ni=1 p([vi−1, vi]), where π = [v0, v1 . . . , vn]

is a path in the brain graph from v0 = v to vn = w, and[vi−1, vi] is the edge connecting vi−1 and vi. Most proba-ble paths are computed using Dijkstra’s shortest path algo-rithm after log-transforming the weights into edge lengthsl(e) = − log(p(e)).

Significance of SPT tracts. The main disadvantage ofSPT is that for any two brain voxels v and w, SPT will finda path connecting them, whether the data really supports thisor not. This can be alleviated by thresholding the path-lengthcorrected path probability score s(π) = [p(π)]

1|π| , where |π|

is the number of nodes on π. However, this requires a choiceof a threshold. A test for the statistical significance of SPTtracts would therefore be desirable. In this paper, we presenta statistical test for the significance of connectivity from asource voxel v ∈ R1 to a target region R2.

Assumptions. We seek a segmentation of a given sourceregion R1 (the thalamus) defined by its connection to a setof cortical target regions. Let us focus on one target regionR2. Consider all paths found by SPT from a voxel v to anyvoxel w in the target region R2. We assume that the major-ity of voxels in R1 are not physically connected to the targetregion R2 and that the scores of SPT paths found from such

vunc ∈ R1 to voxels w ∈ R2 describe noise. This noise maybe specific to the regions R1 and R2, but we assume that itis independent of the source voxel vunc. We also assume thatvoxels vcon ∈ R1 that actually are physically connected toregion R2 show more high scoring SPT paths than expectedfor vunc. Thus, their score distribution should be skewed to-wards the right, as the distribution is a mixture of noise fromthe target voxels w in R2 that are not connected to vcon, andhigh-scores from those target voxels w which are connectedto vcon.

Histograms and cumulative histograms. For eachsource voxel v ∈ R1 we extract a histogram Hv of scorescorresponding to SPT paths from v to any w ∈ R2 as fol-lows: Scores are divided into N bins and the number Hv(i)of scores falling into bin i is counted for i = 1, . . . , N . Ex-amples of these histograms are shown in the top row of Fig. 1together with the average histogram over all source voxels.Some histograms (top right in Fig. 1) indeed represent a dis-tribution of scores that is more skewed to the right comparedto the average histogram, suggesting that the correspondingvoxel is physically connected to the target region.

Such raw histograms are, however, very jagged and noisy,therefore thez are rarely used for statistical testing. Instead,one usually transforms them into cumulative normalized his-tograms (also known as empirical cumulative distributionfunctions) prior to testing without losing information. Thecumulative normalized histogram of voxel v is given by avector (Cv(1), . . . , Cv(N)), where

Cv(i) =

i∑j=1

Hv(j)

norm(H), norm(H) =

N∑i=1

Hv(i). (1)

Assigning a p-value to histograms. To quantify if andhow much the histogram of voxel v deviates from the null dis-tribution, we compare it to a simulated sample of s histogramsrepresenting noise. When a histogram is skewed to the right,the corresponding cumulative histogram lies below a typicalcumulative histograms representing the null distribution (seethe right of Fig. 1).

Inspired by the envelope rank tests of Myllymaki etal. [13], we present a one-sided rank based test for the cu-mulative histograms. Our test computes a p-value for eachsource voxel v given by

p =1

s+ 1

s+1∑k=1

1(rankk ≤ rank1) , (2)

where 1(x) returns 1 if x is true and 0 otherwise, and therankk is the average rank from below over all bins of the kth

sample from the null distribution. That is,

rankk =1

N

N∑j=1

]{k′ = 1, . . . , s+1|Ck′(j) < Ck(j)} , (3)

p = 0.93769 p = 0.99933 p = 0.87104 p = 0.98201 p = 1.00000 p = 0.02799p = 0.00033 p = 0.00033 p = 0.03432 p = 0.00033

Score

Score Score Score ScoreScore Score ScoreScoreScore Score

Fig. 1. Top: Normalized histograms of SPT tract scores for single source region voxels. The dotted green line is the averagehistogram over all source voxels. Bottom: The corresponding cumulative histograms. The dotted green lines are the minimumand maximum envelopes of the cumulative histograms resulting from the normalized, bootstrapped samples. The first fivevoxels were found to have a p-value very close to 1, whereas the latter five have lower p-values.

where k = 1 is the index of the observed histogram. Therank thus measures how skewed the corresponding originalhistogram is to the right, compared to the other histograms inthe sample of s + 1 histograms. The original test by Mylly-maki et al. [13] was based on minimum ranks rather than theaverage rank, which led to an envelope interpretation of thetest. For our application, the mean rank provides additionalstability, but lacks the envelope interpretation.

Drawing from the null distribution of cumulative his-tograms. In order to define a statistical test for significantlyconnected source voxels, we need to be able to draw samplesfrom the null distribution. Under the assumption that the ma-jority of source voxels are not physically connected to the tar-get region, this could be solved by drawing entire histogramsfrom the population. However this could lead to an unnec-essarily conservative test, since we risk to draw histogramsfrom voxels that actually are physically connected.

Instead, we propose bootstrapping the histograms as fol-lows: To draw a cumulative histogram C from the null distri-bution, we first draw a histogram H from the null distributionof score histograms and then obtain a cumulative sample Cfrom H as in (1). The sample H is drawn by randomly draw-ing a bin value H(i) from the ith bin values in the entire pop-ulation for every bin i in H . This gives a histogram H whosebin valuesH(i) are ith bin values drawn from different sourcevoxels v.

3. EXPERIMENTS

We used the preprocessed DWIs [14, 15] from 5 subjects(Q3 release) of the Human Connectome Project (HCP)data [8, 16]. Fibre orientation distribution functions (fODFs)were computed for every voxel using constrained sphericaldeconvolution [17] with order 8 implemented in the DiPy [18]package. The graph required for SPT was constructed fromthe given fODFs. The voxelwise diffusion parameters nec-essary for probabilistic tractography were generated usingFSL’s BedpostX [19] with the zeppelin model and 3 fibresper voxel.

The thalamic source region was extracted from the MNIatlas [20] provided in FSL [21]. Target regions were extractedboth from the MNI atlas and the Juelich atlas [22] similar

to Behrens et al. [7]: prefrontal/temporal zone (frontal andtemporal lobe from the MNI atlas), motor zone (primary mo-tor cortex and premotor cortex from the Juelich atlas), so-matosensory zone (primary somatosensory cortex from theJuelich atlas) and parieto-occipital zone (occipital and pari-etal lobe from the MNI atlas). The atlas in MNI space waswarped into the respective subject-specific spaces using thewarps provided in the HCP data.

For every source voxel, SPT was performed to obtain themost likely paths to all voxels in the given target region. Pathscores were binned into a histogram with 1000 bins in therange from 0 to 1. Histograms for all source voxels wereanalysed for each target region separately as described in Sec-tion 2. The corresponding p-value for each voxel is shown onthe left of Fig. 2. In addition, each voxel was classified as be-ing connected to one of the target regions following the hardsegmentation of Behrens et al. [7]. In brief, after probabilis-tic tractography using FSL’s probtrackx [23] (5000 samples,0.5 mm step length, maximum inter-step curvature 80◦) toeach of the target regions, the largest of the four resultant con-nectivity measures determined each voxel’s label (see rightside of Fig. 2).

4. DISCUSSION AND CONCLUSION

We present a new method for structural parcellation based onstructural connectivity as an alternative to hard segmentation.Our method addresses the problems mentioned in Sec. 1:

a) We choose shortest-path tractography (SPT) to avoidproblems like path length dependency common in fibre track-ing methods. We overcome the problem that SPT always findsa path by assigning a confidence p-value to every voxel in theparcellated region.

b) Most voxels in the thalamus are assigned with a lowconfidence (high p-value) for all target regions by our method(see left side of Fig. 2). This occurs mostly inside the tha-lamus and implies that the diffusion signal for those vox-els is not strong or clear enough to assign them to a regionwith high confidence. This is supported by the low fractionalanisotropy (FA) inside the thalamus, suggesting that the res-olution is simply not high enough to resolve the location offibres inside the thalamus.

Fig. 2. Soft (left) and hard [7] (right) segmentations of the thalamus for all target regions for all 5 subjects. Left: Low confidencep-values (blue) reflect regions that are more likely connected to the respective target region then high p-values (red). Right:As in [7] the thalamus is segmented based on which voxel have most streamlines obtained from probabilistic tractography tothe respective target region (white), while the rest of the thalamus is coloured black.

c) Standard hard segmentation of the thalamus [7], asshown in Fig. 2 (right) assumes that each voxel is connectedto only one target region. However, our results indicatethat high confidence regions overlap strongly between themotor and somatosensory zone, as well as between the pre-frontal/temporal and parieto-occipital zone. This makes itespecially difficult for the hard segmentation to pick up anyvoxels connected to the somatosensory zone (see Fig. 2).In contrast, our soft segmentation allows for a voxel to beconnected to multiple target regions.

While the soft segmentations overlap strongly with thehard segmentations (see Fig. 2) for the motor, somatosensoryand parieto-occipital zone, there is only a small overlap for theprefrontal/temporal zone. One possible reason is that a largepart of the hard segmentation lies in the region of high un-certainty, where the resolution of the data is not good enoughto resolve the fibre directions. Additionally, the distributionof histograms can be a mixture of multiple underlying distri-butions corresponding to several major tract that connect the

thalamus to the cortex. Here, this may affect the p-values ofthe projection of the prefrontal and temporal cortex onto thethalamus, indicating that the target regions should be suffi-ciently spatial coherent and reasonably small. A division ofthe prefrontal/temporal zone into two separate regions as inBehrens et al. [7] will potentially resolve this problem.

Here we are not testing which source voxels are signifi-cantly connected to the target region based on the confidencep-values, as this would require to address multiple hypothe-sis testing. Although this could be done via Bonferroni cor-rection, structural connectivity of neighbouring source voxelsis strongly correlated. Bonferroni correction would thereforebe far too conservative and would affect the segmentation inan unknown way. Future work therefore includes multiplehypothesis testing that respects correlation between sourcevoxels to find those significantly connected. Furthermore, itwould be interesting to apply the soft segmentation to otherbrain structures like the corpus callosum and striatum or toextend the method to be able to parcellate the whole cortex

based on whole brain tractography.

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