Functional Clustering of Whole Brain White Matter Fibers

Abstract

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 et al., 2009). 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 et al., 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), Freesurfer parcellation scheme (Xia et al., 2005; Hagmann et al., 2012) 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 et al., 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., 2009; Maier-Hein et al., 2013).

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. (2005) 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., 2012; Ge et al., 2013; Yoldemir et al., 2014). Venkataraman et al. (2012) 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 (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; Messe et al., 2015) 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 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., 2017; 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.

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:

$$\theta =\underset{\mu ,\sigma ,\pi }{\text{arg}\max }p(\mu ,\sigma ,\pi \mid X)\propto \underset{\mu ,\sigma ,\pi }{\text{arg}\max }p(X\mid \mu ,\sigma ,\pi ).$$ (1)

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

$$\begin{gathered} p(X\mid \mu ,\sigma ,\pi )=\prod _{i=1}^{N}p(x_i\mid \mu ,\sigma ,\pi ) \\ =\prod _{i=1}^N\sum _{k=1}^K{\pi }_{k}p(x_i\mid {\mu }_k,{\sigma }_k), \end{gathered}$$ (2)

where

$$p(x_i\mid {\mu }_k,{\sigma }_k)=\prod _{j=1}^M\frac{1}{(2\pi {)}^{3∕2}{\mid {\sigma }_{k,j}\mid }^{1∕2}}\times \text{exp}(-(x_{i,j}-{\mu }_{k,j}){\sigma }_{k,j}^{-1}(x_{i,j}-{\mu }_{k,j}{)}^T),$$ (3)

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 μ_k of the k_th fiber bundle. So μ_k,j denotes the coordinates of the j_th point on the central fiber of the k_th bundle, σ_k,j is the 3×3 covariance matrix of the distribution of the points corresponding to μ_k,j. Variable x_i,j denotes the points in the i_th fiber corresponding to μ_k,j.

The central fiber of bundle μ_k 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 μ_k,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 μ_k in each iteration, which can be computationally expensive. In this work, we adopt a simple approach to associate x_i and μ_k, which matches the starting and ending points of x_i to those of μ_k, and resamples x_i to a set of equally spaced points with the same number as in μ_k. This allows point-wise correspondence between x_i and μ_k to be established in a highly efficient manner.

2.2. Optimization

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:

$$E(\mu ,\sigma ,\pi )=\sum _{i=1}^N\log (\sum _{k=1}^K{\pi }_k\prod _{j=1}^M\frac{1}{(2\pi {)}^{3∕2}\mid {\sigma }_{k,j}{\mid }^{1∕2}}\times \text{exp}(-(x_{i,j}-{\mu }_{k,j}){\sigma }_{k,j}^{-1}(y_{i,j}-{\mu }_{k,j}{)}^T)).$$ (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.

1. Expectation Step: In the n_th iteration, the cluster membership probability $m_{i,k}^n$ which denotes each fiber x_i belonging to fiber bundle (π_k, μ_k, σ_k) is estimated using last estimation of θ^n-1

   $$m_{i,k}^n=\frac{{\pi }_k^{n-1}p(x_i\mid {\mu }^{n-1},{\sigma }^{n-1})}{\sum _{k=1}^K{\pi }_k^{n-1}p(x_i\mid {\mu }^{n-1},{\sigma }^{n-1}}.$$ (5)

2. Maximization Step: In the n_th iteration, the parameters (μ, σ) are optimized to minimize the objective function below:

$$Q(\mu ,\sigma )=\sum _{i=1}^N\sum _{k=1}^K{m}_{i,k}^n\sum _{j=1}^M(\frac{1}{2}\log \mid {\sigma }_{k,j}^n\mid +(x_{i,j}-{\mu }_{k,j}^n)({\sigma }_{k,j}^n{)}^{-1}(x_{i,j}-{\mu }_{k,j}^n{)}^T)).$$ (6)

By solving the equations dQ / dμⁿ = 0, dQ / dσⁿ = 0 and $\sum _{k=1}^K{\pi }_k=1$, we could obtain the optimal solutions (μⁿ,σⁿ) of function (6) below:

$${\mu }_{k,j}^n=\frac{\sum _{i=1}^N{m}_{i,k}^n{x}_{i,j}}{\sum _{i=1}^N{m}_{i,k}^n},$$ (7)

$${\sigma }_{k,j}^n=\frac{\sum _{i=1}^N{m}_{i,k}^n(x_{i,j}-{\mu }_{k,j}^n)(x_{i,j}-{\mu }_{k,j}^n{)}^T}{\sum _{i=1}^N{m}_{i,k}^n},$$ (8)

$${\pi }_k^n=\frac{N_k}{N}.$$ (9)

Parameters ${\mu }_k^n$ and ${\sigma }_k^n$ denote the central fiber and covariance matrices of k_th bundle in n_th iteration that are obtained by maximizing the probability (2), N_k as the effective number of fibers assigned to bundle k.

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-ICBM-TRACTS (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 (μ_k, σ_k, π_k, 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.

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.

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 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.

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.

$Dis{t}_{i,k}=\sum _{j=1}^M(x_{i,j}-{\mu }_{k,j}^n)({\sigma }_{k,j}^n{)}^{-1}(x_{i,j}-{\mu }_{k,j}^n{)}^T)$ of (6) is the similarity measure between fiber x_i and central fiber μ_k of k_th bundle, and $Dis{t}_{i,k,j}=x_{i,j}-{\mu }_{k,j}^n$ 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 $Dis{t}_{i,k,j}=(1-\alpha )Dis{t}_{i,k,j}^{spatial}+\alpha Dis{t}_{i,k,j}^{functional}$ (10) to combine anatomical and functional similarity, α is their relative weight. Functional distance $Dis{t}_{i,k,j}^{functional}$ is based on the Pearson correlation coefficient of BOLD signal of two corresponding points and spatial distance. $Dis{t}_{i,k,j}^{spatial}$ is defined as the Euclidean distance in DWI space calculating by the coordinates of two points. $Dis{t}_{i,k,j}^{functional}$ and $Dis{t}_{i,k,j}^{spatial}$ were normalized individually 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 $m_{i,k}^n$ using (3), (5) and (10);

3. Computer updated bundle parameters (μn+1, σn+1) using (7), (8) and (9);

4. Update $x_{i\to k}^{n+1}$

5. Repeat steps 2 through 3 until the difference in clusters between successive iterations is smaller than a predefined parameter T; Otherwise output $x_{i\to k}^{n+1}$

2.4. Data Acquisition

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 3T 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 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 seconds of right palm stimulations by continuous brushing followed by 30 seconds 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 = 1000 s/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 bandpass 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.

2.5. Fiber Reconstruction

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°, based on which approximately 15000 fibers were reconstructed for each subject.

3. RESULT

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.

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.

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.

3.1. 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.

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.

To further determine consistent bundles across the subjects, we firstly estimated the deformation matrix of Subjects 2-6 using the 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).

Table 1.

Hausdorff distance across subjects for 30 bundles with high correspondence(mm)

Bundle	(1-2)	(1-3)	(1-4)	(1-5)	(1-6)	(1-7)
Middle cerebellar peduncle	10.2	7.1	4.6	9.4	22.4	23.4
Pontine crossing tract*	10.3	10.9	9.3	10.2	5.4	10.3
Genu of corpus callosum*	9.6	9.3	11.4	12.8	5.8	7.2
Body of corpus callosum*	13.0	19.1	9.5	14.2	14.0	4.9
Splenium of corpus callosum*	8.1	15.0	11.0	14.0	6.7	14.1
Medial lemniscus L*	7.1	11.8	10.2	4.2	11.7	15.3
Superior cerebellar peduncle R	25.6	10.8	5.1	3.2	10.5	18.1
Superior cerebellar peduncle L*	8.9	10.7	16.4	15.1	5.1	15.0
Cerebral peduncle R	43.9	21.1	42.5	48.4	43.0	19.1
Cerebral peduncle L	24.0	11.2	25.1	21.2	20.6	32.9
Anterior limb of internal capsule R	13.9	5.9	7.3	6.2	4.0	26.3
Anterior limb of internal capsule L*	6.8	9.1	13.2	15.6	18.5	12.2
Posterior limb of internal capsule R*	8.3	20.7	15.0	15.7	16.1	16.3
Posterior limb of internal capsule L*	8.6	14.6	8.2	12.4	6.4	15.4
Retrolenticular part of internal capsule R*	5.9	15.7	5.5	13.4	6.6	11.4
Retrolenticular part of internal capsule L*	11.8	9.5	11.2	12.1	10.0	14.2
Anterior corona radiata R	8.8	17.4	14.2	27.5	26.7	17.7
Anterior corona radiata L*	13.0	9.6	10.5	8.8	12.1	11.9
Superior corona radiata R*	9.8	13.4	8.7	13.1	12.5	14.4
Superior corona radiata L*	19.6	17.9	9.8	19.1	14.5	19.5
Posterior corona radiata R	9.8	13.7	17.9	19.2	20.3	14.9
Posterior corona radiata L*	12.4	16.4	10.4	5.4	12.7	16.6
Posterior thalamic radiation R*	10.5	18.1	17.3	7.2	9.9	15.2
Posterior thalamic radiation L	12.7	28.2	17.0	15.4	8.6	11.0
External capsule R	25.4	14.1	9.4	24.1	9.2	11.4
External capsule L	12.0	24.7	26.1	18.5	25.5	30.6
Cingulum (cingulate gyrus) R*	9.6	15.0	11.0	7.0	6.7	14.1
Cingulum (cingulate gyrus) L	12.2	38.5	42.1	23.9	20.0	15.3
Superior longitudinal fasciculus R*	17.6	9.9	11.2	4.1	5.1	13.3
Superior longitudinal fasciculus L	14.0	9.9	23.5	17.1	12.1	6.6

^*denote Hausdorff distance below 20 between subjects.

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.

Fig. 5.

Seven representative bundles of subject 4

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.

3.2. 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, P_{FDR-corrected} < .05). Four bundles clustered under sensory stimulation have smaller CP than with spatial information and under resting state (denoted by ^# in Table 2, P_{FDR-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 et al., 2009). The sensory experiments indicate that the clustering algorithm benefits from incorporating functional information by yielding 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.

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)
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
Pontine crossing tract *#	5.17	0.75	3.36	1.78	2.30	1.17	2.78	1.12
Genu of corpus callosum*	5.91	0.87	3.90	0.26	5.06	0.51	4.96	0.84
Body of corpus callosum *#	8.30	0.68	4.11	0.59	3.84	0.52	4.32	0.65
Splenium of corpus callosum*	6.24	0.55	4.75	0.79	5.45	0.84	5.25	0.90
Medial lemniscus L *#	2.75	0.90	3.51	0.66	2.20	1.00	2.65	1.29
Superior cerebellar peduncle L	3.86	1.51	3.49	1.51	4.10	1.34	3.59	1.03
Anterior limb of internal capsule L	4.92	1.43	4.88	1.30	5.68	0.98	5.85	1.09
Posterior limb of internal capsule R *	5.71	1.08	4.31	0.31	4.01	1.14	4.37	0.37
Posterior limb of internal capsule L*#	6.03	0.95	4.43	0.35	4.25	0.67	4.48	0.56
Retrolenticular part of internal capsule R	5.74	1.13	4.27	0.72	5.78	1.28	5.32	1.05
Retrolenticular part of internal capsule L	6.16	1.00	4.49	1.16	5.98	1.32	5.65	1.03
Anterior corona radiata L*	7.60	0.54	4.60	0.98	5.31	1.13	4.83	0.69
Superior corona radiata R*	6.75	0.92	3.71	0.55	4.28	1.24	4.06	0.34
Superior corona radiata L*	6.58	1.19	3.65	0.44	3.32	1.05	4.36	0.83
Posterior corona radiata L	4.81	1.32	3.55	0.44	4.12	0.63	4.12	1.00
Posterior thalamic radiation R	5.03	1.22	4.29	1.03	4.70	0.46	4.72	0.80
Cingulum (cingulate gyrus) R*	7.60	2.62	4.00	0.64	4.81	1.01	4.56	0.53
Superior longitudinal fasciculus R*	7.29	1.85	4.29	0.67	4.40	0.77	4.20	0.99

^*denotes P_{FDR-corrected} <0.05 (Initial state to the other three states).

^#denotes P_{FDR-corrected} <0.05 (Tactile stimulation to the other three states, highlighted).

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.

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.

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.

Table 3.

Average dispersion of Gaussian distribution in 18 major bundles under four states (mean±standard deviations × 10⁵)

	Subject 1	Subject 2	Subject 3	Subject 4	Subject 5	Subject 6	Subject 7
Initial state	2.94±3.07	1.17±1.01	3.34±4.28	1.48±1.49	2.03±1.21	2.91±2.69	0.28±0.30
Spatial information	2.43±1.36	1.41±1.06	3.03±4.26	1.20±0.78	1.95±1.04	2.79±2.12	0.28±0.32
Tactile stimulation	2.07±1.19	1.35±1.05	2.44±2.45	0.97±0.75	1.89±1.35	2.43±1.24	0.23±0.24
Resting state	2.36±1.52	1.53±0.61	2.62±3.30	1.18±0.75	1.91±0.99	2.49±1.72	0.26±0.24

4. DISSCUSION

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, 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 multi-modal 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 inter-subject 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 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 et al., 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% (Maier-Hein 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 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.

5. CONCLUSION

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.

Highlights:

• Whole brain streamlines are clustered by combining spatial and functional features.

• Clustered bundles have smaller spatial distance and greater functional coherence.

• Pathways of clustered bundles are consistent with known neuroanatomy.

• More compact pathways are obtained in sensory circuitry under stimulation.