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Journal of Neuroscience Methods journal homepage: www.elsevier.com/locate/jneumeth

Functional clustering of whole brain white matter ﬁbers Zhipeng Yang

a,b

a

a

a

, Xiaojie Li , Jiliu Zhou , Xi Wu , Zhaohua Ding

c,d,e,

T

*

a

Department of Computer Science, Chengdu University of Information Technology, Chengdu, 610225, PR China College of Electronic Engineering, Chengdu University of Information Technology, Chengdu, 610225, PR China Vanderbilt University Institute of Imaging Science, Nashville, TN, 37232, United States d Department of Electrical Engineering and Computer Science, Vanderbilt University, Nashville, TN, 37232, United States e Department of Biomedical Engineering, Vanderbilt University, Nashville, TN, 37232, United States b c

A R T I C LE I N FO

A B S T R A C T

Keywords: Fiber clustering White matter ﬁbers Functional MRI Multi-modality

Background: Large numbers of ﬁbers produced by ﬁber tractography are often grouped into bundles with anatomical interpretations. Traditional clustering methods usually generate bundles with spatial anatomic coherences only. To associate bundles with function, some studies incorporate functional connectivity of grey matter to guide clustering on the premise that ﬁbers provide the basis of information transmission for cortex. However, functional properties along ﬁber tracts were ignored by these methods. Considering several recent studies showing that BOLD (Blood-Oxygen-Level Dependent) signals of white matter contain functional information of axonal ﬁbers, this work is motivated to demonstrate that whole brain white matter ﬁbers can be clustered with integration of functional and structural information they contain. New methods: We proposed a novel algorithm based on Gaussian mixture model and expectation maximization to achieve optimal bundling with both structural and functional coherences. The functional coherence between two ﬁbers is deﬁned as the average correlation in BOLD signal between corresponding points. Whole brain ﬁbers under resting state and sensory stimulation conditions were used to demonstrate the eﬀectiveness of the proposed technique. Results: Our in vivo experiments show the robustness of proposed algorithm and inﬂuences of weights between structure and function, and repeatability of reconstructed major bundles across individuals. Comparison with existing methods: In contrast to traditional methods, the proposed clustering method can achieve structurally more compact bundles, which are speciﬁcally related to evoking function. Conclusion: The proposed concept and framework can be used to identify functional pathways and their structural features under speciﬁc function loading.

1. Introduction One of the challenges in neuroscience is to answer the question about the relationship between function and structure of the human brain. Many methods aim to describe how the structure of the neural connectional network transfers information between functional regions. Diﬀusion magnetic resonance imaging (dMRI) is one of the major established in-vivo mapping techniques for probing white matter (WM) structural connectivity (Basser et al., 2000). Within brain WM, the anisotropic nature of diﬀusion signals can be well characterized and extracted to estimate the orientation of the underlying coherent neuronal axon populations (Johansen-Berg and Behrens, 2013). Based on the estimated orientation information, axonal ﬁbers can be virtually reconstructed or traced throughout the brain using computational methods widely known as tractography. To date several ﬁber ⁎

tractography algorithms have been proposed and a subset of these methods are summarized in Fillard et al. (2011). Fiber tractography algorithms typically produce a sheer number of ﬁber tracts that populate densely throughout the brain. To facilitate further characterization of physical or topological properties of these ﬁbers, they are often grouped into a disjoint set of bundles with anatomically distinct interpretations (Brun et al., 2004; Gerig et al., 2004; O’Donnell and Westin, 2007; Maddah et al., 2008a). To this end, a plethora of ﬁber bundling methods have been developed, which, in their rudimental forms, could be divided into two categories: parcellation- and clustering-based methods. Firstly, parcellation-based methods attempt to group ﬁbers with references to their connections to speciﬁc regions in parcellated cortices. A key step in these approaches is the deﬁnition of regions of interest in the cortex. Most of them opt for a standard template, such as Talairach atlas (Cammoun et al., 2012),

Corresponding author at: Vanderbilt University Institute of Imaging Science, 1161 21st Avenue South, MCN AA-1105, Nashville, TN 37232-2310, United States. E-mail address: [email protected] (Z. Ding).

https://doi.org/10.1016/j.jneumeth.2020.108626 Received 7 August 2019; Received in revised form 28 December 2019; Accepted 3 February 2020 Available online 04 February 2020 0165-0270/ © 2020 Elsevier B.V. All rights reserved.

Journal of Neuroscience Methods 335 (2020) 108626

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signals. A potentially more promising solution is to consider functional information along WM tracts instead of from cortical GM regions. Integration of functional and structural connectivity in this manner allows the rich information in the WM to be exploited for improved construction of functional pathways. However, functional signals in WM have been almost ignored by the fMRI research community to date. A primary reason is an absence of signiﬁcant hemodynamic changes within WM in response to changes in electrical activity, so that any corresponding weak BOLD signals are not reliably detectable. Recently, growing evidences have demonstrated that WM activations could be detected on the basis of BOLD signals (Wu et al., 2017; Marussich et al., 2017; Courtemanche et al., 2018; Ding et al., 2018;). And some studies have taken a step further by utilizing WM BOLD signals to improve image registration or to classify patients with mild cognitive impairment (Zhou et al., 2018; Chen et al., 2017). Meanwhile, Peer et al. were able to derive functional structures by clustering BOLD signals, which bore gross similarity with ﬁber bundles obtained by diﬀusion tensor imaging (DTI), but the consistency between the two structures was not highly impressive due to the discrepancy in the clustering schemes employed (Peer et al., 2017). We have earlier demonstrated the possibility of constructing functional pathways in WM by multi-modality fusion (Yang et al., 2018), and proposed a global optimization scheme to enhance reconstructed ﬁbers (Xiao et al., 2019). To further harness the complementary information from the two modalities, in this proof-of-concept study, we propose a novel fusional clustering algorithm by incorporating functional and geometric information of whole brain ﬁbers produced by DTI tractography. The fusion may yield more structurally plausible and functionally interpretable WM bundling, and also possesses the potential of identifying functional network evoked by speciﬁc function loading. In the remainder of this paper, we ﬁrst describe processing procedures implemented in this study and details on in vivo imaging. The technique we proposed on the basis of Gaussian Mixture Model (GMM) and Expectation Maximization (EM) is presented next, followed by in vivo experiments that demonstrate the eﬀectiveness of the proposed technique in identifying brain functional circuits. Our speciﬁc focus will be given to ﬁber connections of the whole brain under resting state and sensory stimulation conditions. Finally, we discuss some technical issues regarding applications of the proposed technique to studies of structure-function relation in the human brain.

Freesurfer parcellation scheme (Xia et al., 2005; Hagmann et al., 2008) and AAL template (Gong et al., 2009). Although parcellation-based methods handily allow interpretations of the identiﬁed bundles in terms of their functionalities, these functionalities are merely derived from knowledge of functional neuroanatomy on the cortex, which may not be consistent with structural connectivity obtained by DWI-based tractography (Damoiseaux and Greicius, 2009; Ge et al., 2013). To add to the complications, the parcellation of brain cortex is still an open problem, such that erroneous parcellations would lead to invalid ﬁber bundles that compromise meaningful interpretations (Smith et al., 2011; Maier-Hein et al., 2017). Secondly, clustering methods based on ﬁber geometry and anatomical similarity describe WM connections as clusters of ﬁber trajectories. These methods begin with a deﬁnition of similarity measures between ﬁbers and then apply clustering procedure to divide them into sub-groups. Ding et al. (2003) deﬁned ﬁber similarity as the mean Euclidean distance between corresponding points in a pair of ﬁbers, and employed a traditional clustering strategy to group ﬁbers into distinct bundles. The concept of ﬁber similarity has later been extended beyond the Euclidean distance and its derivations. O’Donnell et al. (2006) proposed a distance measure by the maximum of point-wise minimum distance and Zhang et al. (2008) generalized it to the mean of the distances between all points. For simplicity, the number of times that two ﬁbers shared the same voxel was also used to represent similarity (Jonasson et al., 2005; Klein et al., 2007). Meanwhile, Hausdorﬀ distance has been applied for ﬁber clustering as well (Maddah et al., 2008b). Quite a few studies resorted to statistical bundle modelling for representation of ﬁber features, which included a 9-D feature vector (Brun et al., 2004), B-Splines (Maddah et al., 2005), polynomials (Klein et al., 2007), among others. Recent work is focused on group clustering of large populations to beneﬁt the neuroscience community (Guevara et al., 2012; Ros et al., 2013; Garyfallidis et al., 2017; Zhang et al., 2018; Siless et al., 2018). Furthermore, Gupta et al. (2017) modeled the clustering problem with a deep learning framework using learned shape features of ﬁber bundles, and achieved a clustering that is arguably at the state-of-the-art level. These methods are eﬃcient, and often yield rather appealing eﬀects, but a common limitation is that the functionality of resulting bundles is overlooked in its entirety, since only geometric and spatial information is considered in the clustering procedure. To address this limitation, a number of recent studies have incorporated functional Magnetic Resonance Imaging (fMRI) data to achieve ﬁber bundling with functional homogeneity within the bundle structure (Venkataraman et al., 2011; Ge et al., 2013; Yoldemir et al., 2014). Venkataraman et al. (2011) computed functional connectivity as Pearson correlations between each pair of voxels in cortical regions connected directly by ﬁbers, and subsequently used a probabilistic framework to combine functional and structural connectivity. Ge et al. (2013) improved the ﬁber similarity metric by adding correlations in fMRI time series between the terminals. Galinsky and Frank (2017) developed a new general computational approach for combining three MRI modalities to estimate structural-functional brain modes. These multi-modal studies achieved the goal of functionally-structurally coherent clustering with which to construct a whole brain network (Deligianni et al., 2013; Messé et al., 2014) and have been applied to pathological studies to demonstrate the eﬀectiveness (O’Donnell et al., 2017). These approaches, however, did not make full use of the multimodal information available. In essence, the functional connectivity of grey matter (GM) regions and structural connectivity of WM bundles are simply combined, with the relation between them derived from statistical hypothesis testing of the two individual connectivities. While conceptually attractive, this scheme may end up with functional structures that have no correspondence to the pathways actually involved in the speciﬁc function of the connecting cortical regions because of the diﬀerence in biophysical origins between dMRI and fMRI

2. Method In this section, we will ﬁrst introduce the GMM method and its implementation for ﬁber clustering. Then we will present a novel approach for measuring the similarity of ﬁber pairs for clustering a target ﬁber under speciﬁc function loading. 2.1. Problem deﬁnition In this work, we treat each ﬁber as a sampled curve that contains a sequence of discrete 3D points in the native space. The corresponding points of ﬁbers in each bundle are assumed to follow a Gaussian distribution, based on which, clustering can be deﬁned as an optimization problem to ﬁnd optimal bundles with similar structures from the ﬁber set. Let X denote all the ﬁbers of the whole brain. This work aims to divide X into K ﬁber bundles that belong to a certain anatomical and functional structures. For each bundle, μ denotes the central ﬁber of the bundles, σ the covariance matrices, π the mixture proportions of each bundle Gaussian model. We assume that whole brain ﬁbers, are independent and identically drawn from the GMM model (π , μ, σ ) . The ﬁnal goal is to ﬁnd the optimal Gaussian distribution for each bundle (π , μ, σ ) that maximizes the posterior probability:

θ = arg max p (μ, σ , π|X ) ∝ arg max p (X |μ, σ , π ) μ, σ , π

2

μ, σ , π

(1)

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The probability of X conditioned on (π , μ, σ ) can be simpliﬁed as the product of the likelihood of each ﬁber as follows:

min, k =

πkn − 1 p (x i |μn − 1 , σ n − 1) K ∑k = 1 πkn − 1 p (x i |μn − 1 , σ n − 1)

(5)

N

∏ p (xi |μ, σ , π )

p (X |μ, σ , π ) =

i Maximization Step: In the nth iteration, the parameters (μ, σ ) are optimized to minimize the objective function below:

i=1 N

K

= ∏ ∑ πk p (x i |μk , σk )

N

Q (μ, σ ) =

(2)

i=1 k=1

K

M

1

∑ ∑ min,k ∑ ( 2 log |σkn,j| + (xi,j − μkn,j )(σkn,j)−1 (xi,j − μkn,j )T )) i=1 k=1

j=1

where

(6) M

p (x i | μk , σk ) =

∏ j=1

K

1 × exp(−(x i, j − μk, j ) σk−, j1 (x i, j − μk, j )T ) (2π )3/2|σk, j |1/2

By solving the equations dQ/ dμn = 0, dQ/ dσ n = 0 and ∑k = 1 πk = 1, we could obtain the optimal solutions (μn , σ n ) of function (6) below: (3)

N

μkn, j =

where k, i and j respectively index the ﬁber bundles, ﬁbers in the ﬁber set and the points along each ﬁber. There are K bundles, N ﬁbers of X and M points on the central ﬁber μk of the kth ﬁber bundle. So μk, j denotes the coordinates of the jth point on the central ﬁber of the kth bundle, σk, j is the 3 × 3 covariance matrix of the distribution of the points corresponding to μk, j . Variable x i, j denotes the points in the ith ﬁber corresponding to μk, j . The central ﬁber of bundle μk is deﬁned to be the ﬁber in bundle k that has a minimum average Euclidean distance to the remaining ﬁbers in k. An ideal approach to deﬁne x i, j for μk, j is to ﬁnd a spatially closest and functionally most correlated point in x i . But this requires a correspondence ﬁnding procedure for all the points in x i and μk in each iteration, which can be computationally expensive. In this work, we adopt a simple approach to associate x i and μk , which matches the starting and ending points of x i to those of μk , and resamples x i to a set of equally spaced points with the same number as in μk . This allows point-wise correspondence between x i and μk to be established in a highly eﬃcient manner.

πkn =

N

K

M

i=1

k=1

j=1

∑i = 1 min, k (x i, j − μkn, j )(x i, j − μkn, j )T N

∑i = 1 min, k Nk N

(8) (9)

and denote the central ﬁber and covariance maParameters trices of kth bundle in nth iteration that are obtained by maximizing the probability (2), Nk as the eﬀective number of ﬁbers assigned to bundle k.

σkn

2.3. Implementation and similarity measure Parameter initialization is the key step for EM to determine the central ﬁber and covariance matrix. The WM template JHU-ICBMTRACTS (Mori et al., 2008) was used in this work to initialize all ﬁbers into 48 bundles. The template in the MNI space was coregistered to subject space ﬁrstly using the reference of b = 0 diﬀusion weighted image (DWI). Secondly, each ﬁber was classiﬁed into kth bundle if it has the largest number of overlapped points with k. In order to eliminate ﬁber spurs, ﬁbers with an overlap of fewer than 10 points with all the 48 bundle templates were discarded. This threshold is an empirical value to balance the number of valid ﬁbers and allowable variation for the subsequent iterations. Then the statistical parameter (μk , σk , πk , k = 1, 2, ..., 48) of each bundle can be calculated. The initial procedure and the result of one subject is shown in Fig. 1. Each representative bundle can be distinguished in this step, which ensures a fast and accurate iteration in the iterative procedure. At each iteration step, the central ﬁber and covariance can be updated by the following steps: (1) The ﬁber which has the minimum distance to all the other ﬁbers is determined as the central ﬁber. (2) Resampling all ﬁbers of each bundle with the same number of consecutive points as the central ﬁber. This allows points correspondence between each ﬁber to be naturally established. (3) Then the mean and covariance of each bundle can be calculated as the central ﬁber and

1 × exp(−(x i, j (2π )3/2 |σk, j |1/2

− μk, j ) σk−, j1 (yi, j − μk, j )T ))

(7)

μkn

The maximization of probability (2) would lead to optimal bundling of the ﬁbers into K clusters. A frequently used approach to maximization of (2) is to incorporate the Expectation-Maximization (EM) algorithm, to ﬁnd the local maximum of the likelihood function:

∑ log( ∑ πk ∏

N

∑i = 1 min, k N

σkn, j =

2.2. Optimization

E (μ, σ , π ) =

∑i = 1 min, k x i, j

(4)

Given an initial parameter θ, this framework iteratively ﬁnds the optimal parameter θ by expectation and maximization step of (4) for all the ﬁbers until convergence. i Expectation Step: In the nth iteration, the cluster membership probability min, k which denotes each ﬁber x i belonging to ﬁber bundle (πk , μk , σk ) is estimated using last estimation of θn − 1

Fig. 1. White matter template and initial 48 bundles. (a) 48 bundle templates; (b) Whole brain original ﬁbers; (c) Initial clustering of 48 bundles. Diﬀerent colors in (a, c) represent diﬀerent bundles. In (b), the whole brain ﬁber is randomly colored for rendering. 3

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Fig. 2. Central ﬁbers and covariance shown by ellipsoids. The ﬁber paths are portions of the splenium of corpus callosum and the retrolenticular part of right internal capsule.

corresponding covariance. Two central ﬁbers and corresponding covariance of two bundles are displayed in Fig. 2 by ellipsoids. The ellipsoid of each point describes the distribution of the points that corresponds to the central point within the same bundle. Due to the divergence of WM ﬁbers approaching the GM, the covariance of the ends of the bundle is greater than the compact middle portion. M Disti, k = ∑ j = 1 (x i, j − μkn, j )(σkn, j )−1 (x i, j − μkn, j )T ) of (6) is the similarity measure between ﬁber x i and central ﬁber μk of kth bundle, and Disti, k, j = x i, j − μkn, j for two corresponding points. In the earlier work, this measure denotes the Euclidean distance in 3D space between two ﬁbers. In our work, we deﬁne a new similarity measure Disti, k, j = (1 − α ) Distispatial + αDisti,functional (10) to combine anatomical , k, j k, j and functional similarity, α is their relative weight. Functional distance Disti,functional is based on the Pearson correlation coeﬃcient of BOLD k, j

established by the local research ethics committee at Vanderbilt University. T2*-weighted images were acquired from seven adults with sensory stimulations, using a T2*-weighted (T2* w) gradient echo (GE), echo planar imaging (EPI) sequence with TR = 3 s, TE = 45 ms, matrix size = 80 × 80, FOV = 240 × 240 mm2, 34 axial slices of 3 mm thick with zero gap, and 145 volumes. Sensory stimuli were prescribed in a block design format, which started with 30 s of right palm stimulations by continuous brushing followed by 30 s of no stimulation, and so on. Resting state data were also acquired prior to the stimulation experiment, with the same imaging parameters as above. DWIs were acquired as well with TR = 8.5 s, TE =65 ms, b = 1000s/mm2, SENSE factor = 3, matrix size = 128 × 128, FOV = 256 × 256, 68 axial slices of 2 mm thick with zero gap, and 32 diﬀusion-sensitizing directions. To provide anatomical references, 3D high resolution T1–weighted (T1w) images were acquired from all the subjects using a multi-shot gradient echo sequence at voxel size of 1 × 1×1 mm3. Once acquired, the dataset from each subject underwent the following procedures using SPM12. All fMRI time series were corrected for slice timing and head motion and smoothed with FWHM = 4 mm. Subjects with head motion more than 2 mm of translation or 2°of rotation in any direction were excluded. The smoothed data were then coregistered with the b = 0 DWI images. Voxels in each time series were band-pass ﬁltered to retain frequencies only of 0.01–0.08 Hz, which contained the principal frequency (0.016 Hz) of the sensory stimuli. Bias correction and segmentation were applied to T1w images to yield GM and WM and cerebrospinal ﬂuid images, all of which along with the corrected T1w images were coregistered with the b = 0 DWI data.

signal of two corresponding points and spatial distance. Distispatial is , k, j deﬁned as the Euclidean distance in DWI space calculating by the coordinates of two points. Disti,functional and Distispatial were normalized ink, j , k, j dividually by the maximum spatial and functional distance of ﬁber x i . The iterative procedure is summarized as follows:

Input: X and (π , μ, σ ) . Output: Optimal (μ, σ ) and cluster result of each ﬁber xi → k 1. Initialize xi → k and (π , μ, σ ) ; 2. Compute membership probability min, k using (3), (5) and (10); 3. Computer updated bundle parameters (μn + 1, σ n + 1) using (7), (8) and (9); 4. Update x in→+k1 5. Repeat steps 2 through 3 until the diﬀerence in clusters between successive iterations is smaller than a predeﬁned parameter T; Otherwise output x in→+k1

2.5. Fiber reconstruction 2.4. Data acquisition Euler method with a tensor model was used to reconstruct whole brain streamlines (Basser et al., 2000), which are termed as ﬁbers throughout this paper. The voxels with fractional anisotropy greater than 0.3 were deﬁned as seed points. Following the direction of the eigenvector associated with the largest eigenvalue, the ﬁber was obtained from the seed points sequentially with step size of 2 mm. The procedure was terminated when voxels with fractional anisotropy below 0.1 or the angle between two consecutive steps exceeded 45°,

Full brain MRI data were acquired from seven healthy (four males and three females), and right-handed adult volunteers (mean age = 27.5 yrs, stdev = 4.3 yrs). All imaging was performed on a 3 T Philips Achieva scanner (Philips Healthcare, Inc., Best, Netherlands) using a 32-channel head coil. Subjects lay in a supine position with eyes closed except when performing functional task. Prior to imaging, informed consent was obtained from each subject according to guidelines 4

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SPM12 registration tool and with the FA images of Subject 1 as reference. Then the central ﬁber of each bundle was warped using the deformed matrix derived anatomically above. Based on the registration, a Hausdorﬀ distance was computed to quantify correspondence between the bundles of diﬀerent subjects (Li et al., 2010). Table 1 lists the Hausdorﬀ distances (mm) between pairs of corresponding central ﬁbers from Subject 1 and other six subjects, with small distances denoting high correspondences. It can be observed from Table 1 that 18 ﬁber bundles had a Hausdorﬀ distance below 20 for all the subjects, which are denoted by boldface and asterisk (*). It is also noticed that, using the strict inclusion criterion of 20, some major bundles were not retained. This was the price it paid to ensure the reliability of subsequent statistical analysis between two functional states. Nonetheless, the remaining 18 ﬁber bundles still included some major ones, such as the corpus callosum (inter-hemispheric ﬁbers), and the inner capsule that unilaterally radiates through multiple complex regions (internal capsule). Fig. 5 visualizes seven representative bundles of one randomly selected subject. All the ﬁbers of each bundle exhibit a similar shape with good concentricity, which is reasonably consistent with the neuroanatomy. Close inspections reveal, however, that there are certain discrepancies between these clusters and the actual neuroanatomy. These discrepancies actually arise from the imperfect white matter template used rather than from the algorithm itself. For example, the proposed method produced three portions of corpus callosum (CC) each with compact ﬁbers, but a small number of ﬁbers were missing from the body of the CC where the genu transits to the splenium. This is due to the limitation of 48 prototypical bundles used in this study, with which it is rather diﬃcult for ﬁbers in the transition zone to ﬁnd a host bundle. There is also another issue about JHU template that needs to be addressed as well. In the JHU template, the posterior limb of internal capsule (PLIC) and corona radiata (CR) are connected, which then combine cortcospinal tract (CST) to constitute a complete pathway. Reconstructed ﬁbers must pass through these three regions at the same time. Therefore, as shown in Fig. 5, the PLIC and superior CR were similar in shape but connect diﬀerent cortex. PLIC is located at the posterior of superior CR. The visualizations above demonstrate that the proposed clustering method can achieve consistent bundles with incorporation of functional information from white matter, suggesting close relationships and intrinsic coherences between structure and function of white matter bundles. This in fact underlies the neuroscience principle of the proposed multi-modal ﬁber clustering methodology.

Fig. 3. Convergence curves under four weighting values of α . Horizontal axis shows the number of iterations and vertical axis represents the smaller number of diﬀerent ﬁber assignments from two preceding iterations. Note that comparisons with two preceding iterations remove the eﬀect of oscillating in ﬁber assignments between consecutive iterations.

based on which approximately 15000 ﬁbers were reconstructed for each subject. 3. Result 3.1. Parameter evaluation To evaluate the eﬀects of parameter α that regulates the proportion of functional distance, we tested the performance of the proposed algorithm under four weighting conditions: α= 0, 0.2, 0.5 and 0.7 respectively. Fig. 3 shows the smaller number of diﬀerent ﬁber assignments compared with two preceding iterations. As can be seen, the number of new ﬁber assignments was large at the beginning, but quickly became smaller when the number of iterations increased and approached zero after 100 iterations for all four α values, signifying a convergence state was reached. Comparisons among the four convergence curves show that α= 0 has the fastest convergence due to the absence of functional information, 0.2 is similar to 0.5 whereas the trend of 0.7 was unstable presumably due to the presence of heavily weighted functional information. Therefore, α= 0.5 was adopted in all the in-vivo experiments in this work.

3.3. Functional information performance Quantitative compactness (CP) of the 18 clusters obtained with spatial information only and multi-modal information under sensory and resting state are provided in Table 2 (averages of the seven subjects studied) (Liu et al., 2010). To measure the compactness of these 18 bundles, the mean and standard deviations (std) of in-bundle distance is calculated, expressed as the mean distance of all the ﬁbers to their bundle central ﬁber and the standard deviations. The CP was compared between three pairings using paired t-tests (p < .05) and false discovery rate (FDR) correction. It can be seen from Table 2 that the CP of 13 out of all the 18 bundles is greater for the initial clustering than that under the other experiment conditions (denoted by * in Table 2, PFDRcorrected < .05). Four bundles clustered under sensory stimulation have smaller CP than with spatial information and under resting state (denoted by # in Table 2, PFDR-corrected < .05). Anatomically, these four bundles are substantiated paths for transmitting sensory information. For example, the medial lemniscus is an important skin ﬁne sensory transmission pathway (Dong et al., 2009), and the PLIC connects the sensory cortex and thalamus for passing information down to the spine (Jang, 2009). The sensory experiments indicate that the clustering algorithm beneﬁts from incorporating functional information by yielding

3.2. Cluster visualization Fig. 4(a) shows the clusters obtained under sensory stimulations in the seven subjects studied. Approximately 40 bundles were clustered for each of the subjects, and in spite of inter-subject variations in neuroanatomy, the overall shape and location of these bundles were largely consistent across all the subjects. The other 8 bundles that were discarded due to the absence of ﬁbers for some subjects. To facilitate close visual inspections, each bundle was represented by the central ﬁber derived from the ﬁnal iteration, which is shown in Fig. 4(b). Because of spatial variabilities across subjects and path uncertainties in ﬁber tractography, only 30 bundles were identiﬁed to have good correspondence among the seven subjects, the central ﬁbers of which are shown in Fig. 4(b) by colored curves. The bundles that do not have correspondence among all the seven subjects are excluded from subsequent analysis, which are shown is Fig. 4(b) by white curves. To further determine consistent bundles across the subjects, we ﬁrstly estimated the deformation matrix of Subjects 2–6 using the 5

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Fig. 4. Fibers of all the resulting clusters under sensory stimulation with α= 0.5. (a) Fibers of all the bundles; (b) Central ﬁber of each bundle. Note that diﬀerent colors represent diﬀerent bundles. Each row is for one subject, and for each subject, sagittal view, coronal view and axial view are shown from left to right. White curves denote discarded bundles, and colored curves denote 30 bundles retained for subsequent analysis.

more compact ﬁber bundles that are related to the stimulation. In contrast, clustering with resting state signals produces a larger number of compact bundles but without speciﬁc focuses on any pathways, which may be attributable to spontaneous ﬂuctuations of BOLD signals across the entire brain at rest. Fig. 6 visually compares the four compact ﬁber bundles related to the sensory stimulations, with the left column clustered using the spatial information only and the right column from the multi-modal information. The multi-modal clusters have fewer false ﬁbers, while the spatial information only clusters have redundant or missing ﬁbers (marked by the blue arrow in the ﬁgure), thus resulting in the large CPs observed in Table 2. The 18 bundles using functional information get more compact because irrelevant ﬁbers are rejected and assigned to other bundles. These comparisons demonstrate the proposed method is eﬀective in highlighting ﬁber bundles related to the evoking function while maintaining spatial shape at the same time. Eﬀectiveness of the proposed method was evaluated by quantifying the overall coherence of the BOLD signals in the 18 major cluster bundles. Speciﬁcally, we assume that the BOLD signals of all voxels in a bundle, each having 145 time points, follow a multivariate Gaussian random process. Based on this assumption, a 145-dimensional covariance matrix was computed, on which eigen decomposition was performed to yield 145 eigenvalues. The determinant of the covariance matrix, which is a multiplication of the 145 eigenvalues, was deﬁned as dispersion of the Gaussian distribution. A smaller dispersion reﬂects a

tighter distribution of and greater coherence among the BOLD signals in a bundle. The average index of the 18 major bundles for the seven subjects is given in Table 3 for the initial clustering, clustering with spatial information only, and functionally informed clustering under tactile stimulation and resting states. As can be seen, six out of the seven subjects had a smaller dispersion under stimulation and resting states than under the initial state and with spatial information only, which indicates that our multi-modal method in general could enhance functional coherence in the clustered bundles.

4. Discussion To parcellate whole brain ﬁbers into bundles with functional coherence, this work introduces a novel technical concept that incorporates fMRI signals in WM into the process of ﬁber clustering. A hybrid knowledge guided and data-driven framework on the basis of Gaussian mixture modeling is employed to encapsulate both structural and functional information in the ﬁbers, in which a classical white matter template serves as a knowledge guide to ensure the consistency between the derived ﬁber clusters and anatomical structures. Our in vivo experiments demonstrate that, using multi-modal information from ﬁber structure and local fMRI signals, more functionally coherent bundles could be obtained, and more compact bundles speciﬁcally related to evoking function could be identiﬁed. Although BOLD signals in WM are the only functional information used in this exploratory study, 6

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to be slow because noise and other confounds in WM may contribute to the variability of fMRI signals therein, which impacts negatively the converging process. Conversely, a small weight typically gives rise to faster convergence because the shape and spatial location are more consistent than functional correlations. This is also conﬁrmed by Peer’s work, which found the functional clustering with fMRI is not as compact as that of DTI clustering (Peer et al., 2017). Our experiments show that the convergence can reach a reasonable speed when the weight is 0.5, and inter-iteration diﬀerences when converged are the same as without functional information. Thus, to balance the speed of convergence and incorporation of function information, this work chose 0.5 as the weight for all the experiments. Inter-subject consistency of ﬁber clusters was investigated as follows. Through visual and quantitative analysis, 18 major bundles that coincide with neuroanatomy were consistently identiﬁed among the seven subjects studied. The 30 bundles that were excluded from subsequent processing can be divided into two types. First, bundles that do not contain any ﬁbers in initialization. For example, stria terminalis and cingulum are very slender in shape and deep in the brain (Bruni and Montemurro, 2009). Since our initialization is based on spatial coincidence, it is diﬃcult for slender and irregular bundles to have enough coinciding points to “claim” any ﬁbers. In this proof-of concept work, the number of bundles is predetermined by the template used, which reduces the ﬂexibility and possibility of discovering new bundles that are absent from the template. While this tends to be a common drawback of all knowledge based clustering techniques, using an anatomical template however can facilitate functional interpretations of the resulting clusters. Second, bundles for which ﬁbers were not able to be reconstructed in some subjects. For example, the fornix failed to be reconstructed in Subject 2, and thus this bundle was excluded from all other subjects in this study. Another factor that needs to be addressed is the termination criterion, which was set as the diﬀerence in bundle assignments of each iterations compared to the preceding two iteration. In this work, the total number of ﬁbers reconstructed was ∼15,000, among which ∼10,000 remained after initialization, and we have used a very tight threshold of 1 % bundling diﬀerence as a criterion for termination. Because the reconstructed ﬁbers have uncertainty of > = 1 % (MaierHein et al., 2017), the beneﬁt of using smaller changes in ﬁber assignment as the termination criterion is quite marginal. Lastly, it should be emphasized that the comparison between using spatial cues only and multi-modal information demonstrated that functional information enhances the clustering eﬀect for relevant pathways. Notably, the sensory stimulation augments the speciﬁcity of the signal in the sensory information transmission pathway, such that the bundles are structurally more compact and functionally more coherent. Although some small sensory-related connection routes were not captured in this work, complete transmission routes from the sensory cortex to the pons and then to the spinal cord were indeed visualized (Liang et al., 2012). In contrast, spatial improvements were not seen on bundles unrelated to sensory processing, nor on any bundles at rest, which again attest to the fact that functional loading enhances clustering of the bundles related to the speciﬁc function. Of note, the observation that clustering with functional information under a resting state has little diﬀerences from clustering with spatial information alone tends to suggest that the bundles may be structurally shaped by resting state function. It is recognized that, at this stage of technical development, the multi-modal clustering method we proposed still has plenty of room for improvements. As the focus of this work is on the clustering procedure, the ﬁber reconstruction method we used is a very basic and stable ﬁber algorithm on tensor data. Certainly, our algorithm could be applicable to the ﬁbers generated by any tracking algorithm. As HARDI models have become readily available, they will be incorporated for more robust reconstruction of ﬁbers. More accurate ﬁbers along with more elaborate initialization could considerably facilitate clustering. In

Table 1 Hausdorﬀ distance across subjects for 30 bundles with high correspondence (mm) (* denote Hausdorﬀ distance below 20 between subjects.). Bundle

(1–2)

(1–3)

(1–4)

(1–5)

(1–6)

(1–7)

Middle cerebellar peduncle Pontine crossing tract* Genu of corpus callosum* Body of corpus callosum* Splenium of corpus callosum* Medial lemniscus L* Superior cerebellar peduncle R Superior cerebellar peduncle L* Cerebral peduncle R Cerebral peduncle L Anterior limb of internal capsule R Anterior limb of internal capsule L* Posterior limb of internal capsule R* Posterior limb of internal capsule L* Retrolenticular part of internal capsule R* Retrolenticular part of internal capsule L* Anterior corona radiata R Anterior corona radiata L* Superior corona radiata R* Superior corona radiata L* Posterior corona radiata R Posterior corona radiata L* Posterior thalamic radiation R* Posterior thalamic radiation L External capsule R External capsule L Cingulum (cingulate gyrus) R* Cingulum (cingulate gyrus) L Superior longitudinal fasciculus R* Superior longitudinal fasciculus L

10.2 10.3 9.6 13.0 8.1 7.1 25.6 8.9 43.9 24.0 13.9 6.8

7.1 10.9 9.3 19.1 15.0 11.8 10.8 10.7 21.1 11.2 5.9 9.1

4.6 9.3 11.4 9.5 11.0 10.2 5.1 16.4 42.5 25.1 7.3 13.2

9.4 10.2 12.8 14.2 14.0 4.2 3.2 15.1 48.4 21.2 6.2 15.6

22.4 5.4 5.8 14.0 6.7 11.7 10.5 5.1 43.0 20.6 4.0 18.5

23.4 10.3 7.2 4.9 14.1 15.3 18.1 15.0 19.1 32.9 26.3 12.2

8.3

20.7

15.0

15.7

16.1

16.3

8.6

14.6

8.2

12.4

6.4

15.4

5.9

15.7

5.5

13.4

6.6

11.4

11.8

9.5

11.2

12.1

10.0

14.2

8.8 13.0 9.8 19.6 9.8 12.4 10.5 12.7 25.4 12.0 9.6 12.2 17.6

17.4 9.6 13.4 17.9 13.7 16.4 18.1 28.2 14.1 24.7 15.0 38.5 9.9

14.2 10.5 8.7 9.8 17.9 10.4 17.3 17.0 9.4 26.1 11.0 42.1 11.2

27.5 8.8 13.1 19.1 19.2 5.4 7.2 15.4 24.1 18.5 7.0 23.9 4.1

26.7 12.1 12.5 14.5 20.3 12.7 9.9 8.6 9.2 25.5 6.7 20.0 5.1

17.7 11.9 14.4 19.5 14.9 16.6 15.2 11.0 11.4 30.6 14.1 15.3 13.3

14.0

9.9

23.5

17.1

12.1

6.6

it envisages the possibility of constructing whole brain networks by harnessing BOLD signals in both WM and GM, which may provide more complete knowledge on how structurally segregated brain regions are functionally integrated. Bundling ﬁbers with structural or geometric similarity is the guiding principle of the earliest ﬁber clustering technique (Ding et al., 2003), and is the essence of a variety of sophisticated clustering techniques that were developed subsequently (See Zhang et al., 2018 for literature review), although they come with diﬀerent deﬁnitions of ﬁber similarity and ﬂavors of clustering algorithms. Compared with the traditional methods, our new multi-modal clustering can not only bundle ﬁbers with structural similarity but also identify bundles that have functional coherence. Of particular note, there have been some multimodal approaches that incorporate fMRI signals in GM into ﬁber clustering, which in principle could yield ﬁbers with clear functional interpretations (Ge et al., 2013). However, approaches of this type make no use of functional signals along ﬁber tracts, and thus do not have intrinsic mechanisms to guarantee that the resulting ﬁber clusters are functionally self-coherent. Additionally, by regulating the weighting parameter, our proposed method possesses the ﬂexibility of bundling spatially distant but functionally coherent ﬁbers that belong to the same neural network, without deﬁning cortical regions as anatomical constraints a priori. In this study, our primary focus is on the robustness and intersubject consistency of the clusters obtained. To assess the robustness, an iterative procedure was tested using diﬀerent weights. When the weight factor is > = 0.7, the role of BOLD signals dominates the iteration, and thus a convergence tends to indicate functional similarities of BOLD signals between ﬁbers. Note that convergence with a large weight tends 7

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Fig. 5. Seven representative bundles of subject 4.

Table 2 Compactness of 18 bundles averaged across subjects. Bundle

Initial State

Spatial Information (α = 0)

Tactile Stimulation (α = 0.5)

Resting State(α = 0.5)

Pontine crossing tract *,# Genu of corpus callosum* Body of corpus callosum *,# Splenium of corpus callosum* Medial lemniscus L *,# Superior cerebellar peduncle L Anterior limb of internal capsule L Posterior limb of internal capsule R * Posterior limb of internal capsule L*,# Retrolenticular part of internal capsule R Retrolenticular part of internal capsule L Anterior corona radiata L* Superior corona radiata R* Superior corona radiata L* Posterior corona radiata L Posterior thalamic radiation R Cingulum (cingulate gyrus) R* Superior longitudinal fasciculus R*

Mean 5.17 5.91 8.30 6.24 2.75 3.86 4.92 5.71 6.03 5.74 6.16 7.60 6.75 6.58 4.81 5.03 7.60 7.29

Mean 3.36 3.90 4.11 4.75 3.51 3.49 4.88 4.31 4.43 4.27 4.49 4.60 3.71 3.65 3.55 4.29 4.00 4.29

Mean 2.30 5.06 3.84 5.45 2.20 4.10 5.68 4.01 4.25 5.78 5.98 5.31 4.28 3.32 4.12 4.70 4.81 4.40

Mean 2.78 4.96 4.32 5.25 2.65 3.59 5.85 4.37 4.48 5.32 5.65 4.83 4.06 4.36 4.12 4.72 4.56 4.20

Std 0.75 0.87 0.68 0.55 0.90 1.51 1.43 1.08 0.95 1.13 1.00 0.54 0.92 1.19 1.32 1.22 2.62 1.85

Std 1.78 0.26 0.59 0.79 0.66 1.51 1.30 0.31 0.35 0.72 1.16 0.98 0.55 0.44 0.44 1.03 0.64 0.67

* Denotes PFDR-corrected < 0.05 (Initial state to the other three states). # Denotes PFDR-corrected < 0.05 (Tactile stimulation to the other three states, highlighted). 8

Std 1.17 0.51 0.52 0.84 1.00 1.34 0.98 1.14 0.67 1.28 1.32 1.13 1.24 1.05 0.63 0.46 1.01 0.77

Std 1.12 0.84 0.65 0.90 1.29 1.03 1.09 0.37 0.56 1.05 1.03 0.69 0.34 0.83 1.00 0.80 0.53 0.99

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Z. Yang, et al.

Fig. 6. Comparisons of ﬁber clusters from using spatial information only (left two columns) and incorporation with tactile stimulation (right two columns). Bundles from the top to bottom rows are body of corpus callosum, posterior limb of internal capsule, pontine crossing tract and left medial lemniscus. Table 3 Average dispersion of Gaussian distribution in 18 major bundles under four states (mean ± standard deviations ×105 ).

Initial state Spatial information Tactile stimulation Resting state

Subject 1

Subject 2

Subject 3

Subject 4

Subject 5

Subject 6

Subject 7

2.94 ± 2.43 ± 2.07 ± 2.36 ±

1.17 ± 1.41 ± 1.35 ± 1.53 ±

3.34 ± 3.03 ± 2.44 ± 2.62 ±

1.48 ± 1.20 ± 0.97 ± 1.18 ±

2.03 ± 1.95 ± 1.89 ± 1.91 ±

2.91 ± 2.79 ± 2.43 ± 2.49 ±

0.28 ± 0.28 ± 0.23 ± 0.26 ±

3.07 1.36 1.19 1.52

1.01 1.06 1.05 0.61

4.28 4.26 2.45 3.30

1.49 0.78 0.75 0.75

1.21 1.04 1.35 0.99

2.69 2.12 1.24 1.72

0.30 0.32 0.24 0.24

5. Conclusion

addition, advanced time signal modelling could be employed to improve signal sensitivity, such as extraction of principal components of the fMRI signals to enhance measuring signal coherences. Finally, it should be pointed out that a limited number of subjects were used in this study. Given the small sample size, statistic diﬀerences among genders and age were not signiﬁcant. However, this does not imply that the performance of the proposed algorithm will be the same for both genders of any age ranges. Another factor that could potentially aﬀect the outcome is that all the subjects in this study were right-handed. In our previous work (Wu et al., 2017), we have found lateralized responses in white matter, which may incur diﬀerential clustering performance across the hemispheres. To evaluate the functional structures derived from the proposed algorithm more comprehensively, a much larger sample size with both handedness and wider age ranges is warranted.

This paper proposed a new ﬁber clustering concept and a novel algorithm for clustering whole brain ﬁbers by combining spatial features and functional signals in white matter, which allows for investigations of brain functional structure. This work demonstrates the possibility and beneﬁt of ﬁber clustering using multi-modal information. The proposed concept can be used to identify functional pathways and their structural features under speciﬁc function loading, which may oﬀer an innovative approach of visualizing information transmission pathways in white matter. CRediT authorship contribution statement Zhipeng Yang: Conceptualization, Methodology, Software. Xiaojie 9

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Li: Formal analysis. Jiliu Zhou: Supervision. Xi Wu: Investigation, Validation. Zhaohua Ding: Conceptualization, Writing - review & editing.

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