Development and validation of consensus clustering-based framework for brain segmentation using resting fMRI

Srikanth Ryali; Tianwen Chen; Aarthi Padmanabhan; Weidong Cai; Vinod Menon

doi:10.1016/j.jneumeth.2014.11.014

. Author manuscript; available in PMC: 2016 Jan 30.

Published in final edited form as: J Neurosci Methods. 2014 Nov 29;240:128–140. doi: 10.1016/j.jneumeth.2014.11.014

Development and validation of consensus clustering-based framework for brain segmentation using resting fMRI

Srikanth Ryali ¹, Tianwen Chen ¹, Aarthi Padmanabhan ¹, Weidong Cai ¹, Vinod Menon ^1,^2,³

PMCID: PMC4276438 NIHMSID: NIHMS645759 PMID: 25450335

Abstract

Background

Clustering methods are increasingly employed to segment brain regions into functional subdivisions using resting-state functional magnetic resonance imaging (rs-fMRI). However, these methods are highly sensitive to the (i) precise algorithms employed, (ii) their initializations, and (iii) metrics used for uncovering the optimal number of clusters from the data.

New Method

To address these issues, we develop a novel consensus clustering evidence accumulation (CC-EAC) framework, which effectively combines multiple clustering methods for segmenting brain regions using rs-fMRI data. Using extensive computer simulations, we examine the performance of widely used clustering algorithms including K-means, Hierarchical, and Spectral clustering as well as their combinations. We also examine the accuracy and validity of five objective criteria for determining the optimal number of clusters: Mutual Information, Variation of Information, Modified Silhouette, Rand Index, and Probabilistic Rand Index.

Results

A CC-EAC framework with a combination of base K-means clustering (KC) and hierarchical clustering (HC) with Probabilistic Rand Index as the criterion for choosing the optimal number of clusters, accurately uncovered the correct number of clusters from simulated datasets. In experimental rs-fMRI data, these methods reliably detected functional subdivisions of the supplementary motor area, insula, intraparietal sulcus, angular gyrus, and striatum.

Comparison with Existing Methods

Unlike conventional approaches, CC-EAC can accurately determine the optimal number of stable clusters in rs-fMRI data, and is robust to initialization and choice of free parameters.

Conclusions

A novel CC-EAC framework is proposed for segmenting brain regions, by effectively combining multiple clustering methods and identifying optimal stable functional clusters in rs-fMRI data.

Keywords: Parcellation, Segmentation, Data clustering, Consensus clustering, resting-state fMRI

Introduction

Resting-state functional magnetic resonance imaging (rs-fMRI) is increasingly used to characterize the functional organization of the human brain (Barnes et al., 2010; Barnes et al., 2012; Beckmann and Smith, 2004; Calhoun et al., 2001; Cohen et al., 2008; Deen et al., 2011; Kelly et al., 2012; Kelly et al., 2010; Kim et al., 2010; Nelson et al., 2010; Ryali et al., 2012; Shen et al., 2010; Shen et al., 2013; Wig et al., 2013a; Wig et al., 2013b; Yeo et al., 2011; Zhang et al., 2010). Segmenting brain regions into functionally homogeneous subdivisions is critical for understanding the role of individual brain areas in perception and cognition (Ryali et al., 2012; Wig et al., 2013a; Yeo et al., 2011). Human brain parcellations have typically been constructed using anatomical features, such as cellular organization, gyral folding patterns, or neurotransmitter profiles (Brodmann, 1994; Zilles and Amunts, 2010). However, such segmentations are based on post-mortem brains in the elderly, which may differ from functional subdivisions in normal healthy individuals. Furthermore, they cannot be used to characterize individual differences in functional brain organization. Segmentation schemes using rs-fMRI have been shown to be informative for: (i) determining the functional boundaries of brain regions in normal healthy adults (Cohen et al., 2008; Kim et al., 2010; Ryali et al., 2012; Shen et al., 2010), (ii) characterizing the development of these boundaries from childhood to adulthood (Barnes et al., 2012; Uddin et al., 2010b), (iii) elucidating aberrations in the functional organization in clinical populations (Zarei et al., 2012), and (iv) selecting nodes for further seedbased whole brain functional connectivity analysis (Deen et al., 2011; Shen et al., 2013; Supekar and Menon, 2012; Uddin et al., 2011).

Segmentation of brain regions using rs-fMRI can be posed as a data clustering problem. Data clustering algorithms are unsupervised machine learning methods that group objects or voxels (in the case of brain imaging data), into clusters having high intra-cluster and low inter-cluster similarity (Kelly et al., 2012; Kelly et al., 2010; Kim et al., 2010; Shen et al., 2010). Several data clustering approaches including K-means (Bellec et al., 2010; Chang et al., 2012; Deen et al., 2011; Kim et al., 2010), spectral clustering (Kelly et al., 2012; Kelly et al., 2010; Shen et al., 2010; Shen et al., 2013), and hierarchical clustering (Cauda et al., 2011; Salvador et al., 2005) have been used to segment brain regions using rs-fMRI. Unlike supervised machine learning problems, data clustering problems are highly ill-posed (Duda, 2001; Hastie et al., 2001; Jain et al., 1999) and the clusters obtained from a given dataset are strongly dependent on the clustering method used (Fred and Jain, 2005). For example, the widely used K-means algorithm, with Euclidean as a distance measure, imposes equal-sized hyper-spherical cluster shapes and performs poorly when the true underlying clusters are arbitrarily shaped. More broadly, the results of a clustering critically depend on (i) initialization, (ii) choice of parameters such as the number of clusters, (iii) choice of the distance metric (Euclidean, Manhattan or correlation), and (iv) choice of features used. Other widely used clustering methods including hierarchical and spectral clustering are also plagued with similar issues. Critically, determining the optimal number of arbitrary shaped stable clusters in rs-fMRI data remains a major challenge.

Recent advances in data clustering and machine learning have led to the suggestion that consensus clustering can overcome many of the aforementioned limitations of current methods (Carpineto and Romano, 2012; Fred and Jain, 2005; Monti et al., 2003; Strehl and Ghosh, 2002; Topchy et al., 2005, 2003). Consensus clustering is similar to ensemble methods in supervised learning where multiple weak classifiers are combined to build a strong classifier (Freund and Schapire, 1997; Hastie et al., 2001; Kittler et al., 1998; Quinlan, 1996). A key goal here is to discover clusters that are stable across multiple partitions obtained by applying different clustering algorithms, parameters, initializations and data features (Fred and Jain, 2005; Topchy et al., 2005).

Previous neuroimaging studies have suggested that consensus clustering is a potentially useful approach for functionally segmenting brain regions (Bellec et al., 2010; Kelly et al., 2012; Kelly et al., 2010). These approaches have used different clustering algorithms and objective measures for selecting the optimal number of clusters from the data. However, these methods have not been validated adequately, even using realistic simulated datasets where the underlying clusters are known. Therefore, it is unclear whether these methods are sensitive to the choice of the underlying clustering method, its initialization, the choice of its free parameters, or the metrics used for choosing the optimal number of clusters. A commonly used approach for combining multiple partitions in the computer vision literatures is based on evidence accumulation (Fred and Jain, 2005). Using such an approach, Fred and Jain showed that clusters obtained by “weak” methods, such as K-means, can be effectively combined to discover stable clusters of arbitrary shape and size.

Here we develop and validate a novel consensus clustering evidence accumulation (CC-EAC) framework for segmenting individual brain regions. We first validate our approach on several simulated datasets and then apply them on experimental rs-fMRI datasets. We apply CC-EAC to determine functional subdivisions in five brain areas in the right hemisphere: (i) supplementary motor area (SMA) and pre-SMA, (ii) insular cortex, (iii) intraparietal sulcus (IPS), (iv) angular gyrus (AG), and (v) striatum. While SMA-preSMA, insular and striatal regions have been previously segmented using different clustering approaches (Chang et al., 2012; Deen et al., 2011; Kelly et al., 2012; Kim et al., 2010; Ryali et al., 2012), to our best knowledge functional subdivisions of the IPS and AG have not been previously investigated.

Methods

CC-EAC

We develop a CC-EAC framework (Fred and Jain, 2005) to find the stable and robust clusters in a given region of interest (ROI) using rs-fMRI time series as features. One approach would be to use the functional connectivity profiles of each voxel in an ROI to the rest of the voxels in the brain as features for segmentation (Kelly et al., 2012; Kelly et al., 2010; Wig et al., 2013a; Yeo et al., 2011). In this segmentation approach, the similarities of the whole-brain functional connectivity profiles of each voxel in the ROI are used to segment the ROI. CC-EAC is a general framework wherein similarity between functional connectivity features can also be used to segment a given ROI. Here, we use the time series of each voxel in an ROI as features for segmentation, thus allowing voxels with similar time series profiles to be clustered together.

CC- EAC uses two clustering methods: base and consensus to obtain stable clusters. The base clustering method is used to generate several partitions or groupings P = {p¹,p²,…‥,p^N} for a given set of observations Y^s using different initializations. The consensus clustering method is then used to find stable clusters across all these partitions that are not sensitive to the initializations of the base clustering method. These partitions could have been obtained by (i) applying different clustering methods, (ii) applying the same method with different parameter settings, (iii) applying clustering with different distance metrics, (iv) using different initializations of the same clustering method, (v) using different features of the data, (vi) or a combination of some or all of the above. In this work, we will generate different partitions of data by applying the same base clustering method several times with different initializations for each value of k (the number of clusters). We will then use several objective criteria to select the value of k for which the partitions are most stable, and thus determine the optimal number of clusters. The various steps that are involved in CC-EAC are outlined in Figure 1. Detailed descriptions of CCEAC are as follows:

CC-EAC can segment a given brain region (ROI) using rs-fMRI data. Step 1: extract time series from a given ROI for each subject; Step 2: generate several base clusters for each subject and for each k ranging from 2 to 10 using a base clustering algorithm starting from different initializations using K-means or Spectral Clustering methods; Step 3: compute the co-association matrices C_s,K for each k and s, which indicates how many times a pair of voxels are clustered together across the base clusters; Step 4: apply consensus clustering method with C_s,K as the similarity matrix to find stable clusters across all the initialization for each k and s; Step 6 compute cluster comparison criteria: Normalized Mutual Information (NMI), Variation of Information (VI), Rand Index (RI), Probabilistic Rand Index (PRI) and Modified Silhouette across all the subjects for each k; Step 6: Identify the optimal number of clusters k* based on the cluster comparison criteria; Step 7: Compute co-association matrix at the group level for k = k* by averaging subject level co-association matrices *C_s,k*,; Step 8: Find the consensus stable and k* optimal number of clusters by applying consensus clustering method using the group level coassociation matrix as the similarity matrix.

In step 1, we extract the time series in each ROI voxel for every subject. Let $Y^{S} = {y_{i}^{s}}_{i = 1}^{M}$ be the observed voxel time series, where M is the number of voxels in the ROI, $y_{i}^{s}$ is the time series consisting of T fMRI observations at each voxel in ROI for a given subject s and S is the total number of subjects.

In step 2, we generate 100 different partitions of data from each subject for each k ranging from 2 to 10 by applying a base clustering method with different initializations. In this study, we use two clustering approaches (K-means and spectral clustering) as base clustering methods since these have been the two most widely used approaches in neuroimaging research. In both of these methods, we use the correlation between voxel time series as similarity (or distance) measures.

In step 3, we compute a co-association matrix C_s,k, for each k and s that finds similarities between these 100 different partitions. C_s,k is a M × M matrix with the (i,j)-th entry defined as

C_{s, k} (i, j) = \frac{n_{i j}}{N}

Where, n_ij is the number of times the pair of voxels i and j are clustered together in N = 100 partitions of the data for a given k and S. An entry in C_s,k(i,j) close to 1 means that the specific pair of voxels are clustered together consistently across the N partitions and the opposite is true when this is close to 0.

In step 4, we apply the consensus clustering method using C_s,k as the similarity matrix for each k and s to generate consensus clusters across the 100 base partitions for each k and s.

In step 5, we quantify the stability of partitions for each k across all the subjects, using five different cluster similarity measures: normalized mutual information (NMI), variation of information (VI), rand index (RI), probabilistic rand index (PRI), and modified silhouette. The details of these measures are given in the next section.

In step 6, we choose the optimal number of clusters (K*) where the similarity between the partitions is optimal. We compare the performance of each method in finding the optimal number of clusters from the data.

In step 7, we compute the group level average co-association matrix by averaging the coassociation matrices, corresponding to k = k*, across all subjects (S).

Finally in step 8, we apply a consensus clustering method with the group co-association matrix as the similarity matrix to find k* optimal number of clusters from the data. We use two different consensus clustering approaches: hierarchical with average linkage and spectral clustering, and also compare their relative performance.

In summary, we use the following combinations of base and consensus clustering methods within the CC-EAC framework: (1) K-means combined with hierarchical clustering (KC-HC) as in (Bellec et al., 2010), (2) K-means combined with spectral clustering (KC-SC), and (3) spectral clustering combined with spectral clustering (SC-SC) as in (Kelly et al., 2012). For each combination, we apply five objective criteria to find optimal number of clusters from the data. We finally compare the performance of these methods in finding stable clusters of several ROIs using both simulated and real rs-fMRI datasets. We run K-means and spectral clustering methods 100 times with different random initializations for each k (number of clusters) to generate base partitions. In the K-means clustering method, we use the correlation between two voxel time series as a “similarity” or “distance” metric. In this setting, correlation is equivalent to the Euclidean distance metric as the voxel time series have been normalized by subtracting their temporal means and dividing by their standard deviations. Further details of the rs-fMRI data preprocessing procedures are outlined below. We also note that the spectral clustering method used in this study is based on the work of Ng et al., (2002) and that the sample correlation matrix of the voxel time series is used as the similarity matrix in the spectral clustering method.

Objective criteria for determining the optimal number of clusters

We compare the following criteria that are used in determining the optimal number of clusters from the data (step 5 in Figure 1). Let L^i,k and L^j,k be the voxel labels for two subjects i and j for a given number of clusters k. The following criteria define the similarity between these two different partitions.

(a) Normalized mutual information (NMI) is an information theoretic criterion that can be used to measure the similarity between two partitions L^i,k and L^j,k (Fred and Jain, 2005; Strehl and Ghosh, 2002)

N M I (L^{i, k}, L^{j, k}) = \frac{2 I (L^{i, k}, L^{j, k})}{H (L^{i, k}) + H (L^{j, k})}

(1)

Where I(L^i,k, L^j,k) is the mutual information between the two partitions, and H(L^i,k) and H(L^j,k) are their entropies. The mutual information between these two partitions is defined as

I (L^{i, k}, L^{j, k}) = \sum_{p = 1}^{k} \sum_{q = 1}^{k} \frac{m_{p, q}^{i, j}}{M} {log}_{2} \frac{\frac{m_{p, q}^{i, j}}{M}}{\frac{m_{p}^{i, k}}{M} \frac{m_{q}^{j, k}}{M}}

(2)

where, $m_{p, q}^{i, j}$ is the number of shared voxels between the clusters $L_{p}^{i, k}$ and $L_{q}^{j, k}, m_{p}^{i}$ is the number of voxels in the cluster $L_{p}^{i, k}$ and $m_{q}^{j}$ is the number of voxels in the cluster $L_{q}^{j, k}$ . The entropy H(L^i,k) is defined as

H (L^{i, k}) = \sum_{p = 1}^{k} \frac{m_{p}^{i}}{M} {log}_{2} \frac{m_{p}^{i}}{M}

(3)

NMI is the normalized measure of mutual information such that 0 ≤ NMI ≤ 1, NMI is 0 if there is no similarity between the partitions and is 1 if they are identical. We compute the average NMI by averaging the NMI between all pairs of subjects for a given k. We then select the optimal number of clusters k* for which the average NMI is maximized.

(b) Variation of information (VI) is an information theoretic metric introduced by (Meila, 2007) to quantify the dissimilarity between two partitions. VI for two partitions L^i,k and L^j,k is defined as

V I (L^{i, k}, L^{j, k}) = H (L^{i, k}) + H (L^{j, k}) - 2 I (L^{i, k}, L^{j, k})

(4)

As in the case of NMI, we compute the average VI between all pairs of subjects for a given k and choose the optimal number of clusters k* for which the average VI is minimized.

(c) Rand Index (RI) is a measure for comparing similarity between two sets or between data partitions as in our case. RI for two partitions L^i,k and L^j,k can be defined as

R I (L^{i, k}, L^{j, k}) = \frac{n_{11} + n_{00}}{n_{00} + n_{01} + n_{10} + n_{11}} = \frac{n_{11} + n_{00}}{(_{2}^{N})}

(5)

Where

n₁₁: the number of pairs of voxels that are in the same cluster in both L^i,k and L^j,k
n₀₀: the number of pairs of voxels that are in different clusters in both L^i,k L^j,k
n₁₀: the number of pairs of voxels that are in the same cluster L^i,k in but in different clusters in L^j,k
n₀₁: the number of pairs of voxels that are in different clusters in L^i,k but in the same cluster in L^j,k

We compute the average RI between all pairs of the subjects for every k and choose the optimum k* for which the average RI is maximized.

(d) Probabilistic Rand Index (PRI) proposed in (Carpineto and Romano, 2012) is a modified measure of RI for quantifying the similarity between two data clustering solutions. This measure gives different weights to n₁₁, n₀₀,n₁₀, and n₀₁ depending on the number of clusters present in that particular partition. The rationale behind the different weightings is that if the number of clusters is small then the probability of grouping two voxels in a cluster is greater than when the number of clusters is higher. In other words, the information content of two voxels being clustered together is greater when the underlying number of clusters is larger. Therefore, PRI (Carpineto and Romano, 2012) gives more weight to n₁₁,n₁₀, and n₀₁ for partitions having a higher number of clusters compared to the partitions having lower cluster numbers, and less weight to n₀₀. These weights are defined using the entropy criterion (Carpineto and Romano, 2012) as follows

w_{h} = - {log}_{2} p_{h}

(6)

Where, h ∈ {00,01,10,11} is a binary indicator variable. We compute probabilities (p_h=n₁₁,n₀₀,n₁₀ and n₀₁) as defined in (Carpineto and Romano, 2012). Accordingly, PRI between two partitions L^i,k and L^j,k for subjects i and j for k number of clusters is defined as:

P R I (L^{i k}, L^{j, k}) = \frac{w_{11} n_{11} + w_{00} n_{00}}{w_{00} n_{00} + w n_{01} + w_{10} n_{10} + w_{11} n_{11}}

(7)

We compute average PRI for selecting the optimal number of clusters from the data.

(e) Modified Silhouette (Bellec et al., 2010) is a cluster measure that is used to select the optimal number of clusters from the data and to quantify the stability of the clusters at both subject and group levels. Modified silhouette on a co-association matrix C^k is defined as

Modified Silhouette (k) = \frac{1}{M} \sum_{m = 1}^{M} (w {(C_{k})}_{m} - max_{C \neq C_{k}} w {(C)}_{m})

(8)

Where, w(C_k) is the stability map generated for each cluster C as

w {(C)}_{m} = \sum_{n \in C, n \neq m} C_{m, n}^{k}

(9)

We select the optimal number of clusters k* for which the modified silhouette is the largest.

Resting-state fMRI data

Resting-state fMRI data were acquired from 21 adult participants. The study protocol was approved by the Stanford University Institutional Review Board. The participants ranged in age from 19 to 22 y (mean age 20.40 y) with an IQ range of 97 to 137 (mean IQ: 112). The participants were recruited locally from Stanford University and neighboring community colleges.

Participants were instructed to keep their eyes closed and try not to move for the duration of the 8-min scan. Functional Images were acquired on a 3T GE Signa scanner (General Electric) using a custom-built head coil. Head movement was minimized during scanning by a comfortable custom-built restraint. A total of 29 axial slices (4.0 mm thickness, 0.5 mm skip) parallel to the AC-PC line and covering the whole brain were imaged with a temporal resolution of 2 s using a T2* weighted gradient echo spiral in-out pulse sequence (Glover and Law, 2001) with the following parameters: TR = 2,000 msec, TE = 30 msec, flip angle = 80 degrees. Slices are sequentially acquired from inferior to superior of the brain. The field of view was 20 cm, and the matrix size was 64×64, providing an in-plane spatial resolution of 3.125 mm. To reduce blurring and signal loss arising from field in homogeneities, an automated high-order shimming method based on spiral acquisitions was used before acquiring functional MRI scans.

Preprocessing of rs-fMRI data

Data were preprocessed using SPM8 (Statistical Parametric Mapping software, http://www.fil.ion.ucl.ac.uk/spm). The first eight image acquisitions of the rs-fMRI time series were discarded to allow for stabilization of the MR signal. Each of the remaining 232 volumes underwent the following preprocessing steps: realignment, slice-timing correction, normalization to the MNI template, and spatial smoothing using a 6-mm full-width half maximum Gaussian kernel. Excessive motion, defined as greater than 3.5 mm of translation or 3.5 degrees of rotation in any plane, was not present in any of the scans. The time series at each voxel were filtered using a band pass filter (0.0083 Hz < f < 0.15 Hz). For each ROI to be segmented, we regressed out the temporal mean and linear trend of each voxel time series within the ROI. Finally we divided each voxel time series by its standard deviation.

Synthetic fMRI datasets

Because the actual functional boundaries are unknown in experimental fMRI data, the performance of clustering methods and the suitability of cluster similarity measures were first validated on realistic simulated datasets in which the precise boundaries between the clusters are known. For this purpose, we constructed synthetic fMRI datasets using real fMRI signals extracted from 6 regions (3 in each hemisphere) in primary auditory cortex using the maximum probabilistic cytoarchitectonic maps available in the SPM Anatomy Toolbox (Eickhoff et al., 2005) (Figure 2). Voxel-wise time series were extracted from the experimental rs-fMRI dataset with 21 adult subjects that were preprocessed only up to the smoothing step to ensure that voxels were highly correlated both within and between regions. To create distinct brain regions or clusters, we destroyed between-region correlations by randomizing the phase of time courses in different regions (Prichard and Theiler, 1994). This manipulation has the advantage of maintaining the spectral information of the original time series (Prichard and Theiler, 1994). Specifically, to destroy the correlation between regions, we added a randomly generated common phase to all time courses within a region, and similarly another random common phase to time series in other regions (Chen et al., 2013). As a result, correlations between voxels within each region still remained high and were spatially varied. In contrast, correlations between voxels in different regions were expected to be zero, thus creating distinct clusters.

ROIs with (A) 2, (B) 3, (C), 4, (D) 5, and (E) 6 clusters with arbitrary sizes and shapes constructed from resting state fMRI. The clusters are formed by adding a constant phase (randomly generated) to each voxel time series within a given cluster. This procedure introduces a greater correlation between voxel time series within a cluster while destroying the correlations between voxel time series belonging to different clusters. Additionally, this procedure preserves the spectral profiles of each voxel.

Brain regions used for segmentation of experimental fMRI data

To investigate the performance of different consensus clustering methods and different similarity measures for obtaining the optimal number of clusters, we examined their ability to functionally segment different brain regions into spatially contiguous subdivisions.

(i) SMA, pSMA

We chose the right medial frontal cortex because it consists of two spatially contiguous cytoarchitectonically distinct subdivisions – the Supplementary Motor Area (SMA) (Brodmann area 4), and the preSMA (pSMA: Brodmann area 4, 6) (Ryali et al., 2012; Zilles et al., 1996). Importantly, the right SMA and pSMA are well demarcated anatomically by a vertical line, passing through the anterior commissure, perpendicular to the anterior-posterior commissure axis (Zilles et al., 1996). The right SMA and pSMA, shown in Figure 3A, were defined using a procedure similar to that used in previous studies (Aron et al., 2007; Johansen-Berg et al., 2004). The automated anatomical labeling (AAL) template (Tzourio-Mazoyer et al., 2002) was used to demarcate the SMA (y < 0) and pSMA (y > 0).

(A) Right SMA and pSMA (SMApSMA), (B) right Insular cortex, (C) right Intraparietal sulcus (IPS), (D) right Angular Gyrus (AG) and (E) right Striatum.

(ii) Insular cortex

The insula is a heterogeneous multimodal brain area situated deep within the lateral sulcus between the frontal and temporal lobes, encompassing Brodmann areas 13, 14, and 16. We used the right insular cortex as defined by the automated anatomical labeling (AAL) atlas (Tzourio-Mazoyer et al., 2002) (Figure 3B).

(iii) Intraparietal Sulcus (IPS)

The right IPS is key area involved in visuospatial attention (Corbetta and Shulman, 2002). Cytoarchitectonic mapping studies in post-mortem brains have identified three subdivisions in the IPS (Choi et al., 2006; Scheperjans et al., 2008). In the current study, we merged the maximum probabilistic maps of three right IPS subdivisions using the Anatomy Toolbox in SPM (http://www.fil.ion.ucl.ac.uk/spm/ext/#Anatomy) to create a single IPS ROI (Figure 3C).

(iv) Angular Gyrus (AG)

The angular gyrus is a ventral posterior parietal cortex region involved in semantic memory, speech, and reasoning (Binder and Desai, 2011; Seghier, 2013). The subdivisions within this right ROI (PGa and PGp) were identified using cytoarchitectonic mapping (Caspers et al., 2006) (Figure 3D).

(v) Striatum

The striatum is a subcortical region that includes the caudate, putamen, and nucleus accumbens, which have distinct as well as overlapping functional properties and connectivity with the rest of the brain (Alexander et al., 1986; Di Martino et al., 2008; Polderman et al., 2006). We used the Harvard-Oxford subcortical atlases in FSL and merged the maximum probabilistic maps of the right caudate, putamen, and nucleus accumbens to form a striatal ROI (Figure 3E).

Results

Synthetic fMRI data: Identification of number of clusters

Our first goal was to determine whether CC-EAC could accurately identify the number of clusters in simulated rs-fMRI data, where the cluster boundaries are known. Within the CC-EAC framework, we investigated the performance of three clustering approaches: KC-HC, KC-SC, and SC-SC for segmenting brain regions into functionally distinct subdivisions. We used five different metrics for determining the optimal number of clusters at the group level: Modified silhouette, PRI, RI, VI, and NMI.

Figure 4 shows the performance of the cluster comparison metrics for each combination of clustering methods on ROIs containing 2, 3, 4, 5, and 6 clusters. The first row of Figure 4 demonstrates that for ROIs with two clusters, all the five metrics for the three cluster combinations correctly identify the optimal number of clusters as 2. For ROIs with three clusters, except VI, all the other metrics correctly identified the optimal number of clusters to be 3, as shown in the second row of Figure 4. VI increased monotonically from 2 to 10 suggesting that 2 is the optimal number of clusters in this ROI. For ROIs with 4 clusters, similar performance was observed except for the cluster combination SC-SC as shown in the third row of Figure 4. None of the cluster comparison metrics was able to identify the correct number of clusters using SC-SC. In the case of cluster combinations KC-HC and KC-SC, with the exception of VI and RI, all other metrics were able to estimate the correct number of distinct clusters as 4 (third row of Figure 4). For ROIs with 5 clusters, the modified silhouette identified the correct number of clusters only for the cluster combination KC-HC while PRI was able to find the correct number of underlying clusters with both KC-HC and KC-SC (fourth row of Figure 4). In the case of ROIs with 6 clusters, all the metrics other than RI gave 6 clusters for KC-HC and KC-SC as shown in the fifth row of Figure 4. Although RI and NMI identified the correct number of clusters in some of the cases, they increased monotonically with increase in the number of clusters. These simulations also suggest that none of the cluster comparison metrics were able to find the optimal number of underlying clusters in ROIs with more than three subdivisions using spectral clustering (SC-SC).

Modified silhouette, Probabilistic Rand Index (PRI), Rand Index (RI), Variation of Information (VI), and Normalized Mutual Information (NMI) are distinct criteria used for identifying the optimal number of clusters in ROIs. K-means-Hierarchical (KC-HC) and K-means-Spectral Clustering (KC-SC) and Spectral-Spectral (SC-SC) are three different base and consensus clustering pairs examined in the CC-EAC framework. The performance was examined on 2, 3, 4, 5, and 6 distinct ROIs shown in Figure 3. KC-HC with PRI and modified silhouette consistently identifies the correct number of clusters from each ROI. VI suggests 2 as the optimal number of clusters in majority of the cases while RI and NMI monotonically increases with the number if clusters.

In summary, the modified silhouette and PRI measures were able to correctly identify the number of clusters in all five ROIs with KC-HC as the cluster combination. The other three measures – RI, VI and NMI – were inconsistent in identifying the optimal number of clusters.

Synthetic fMRI data: Functional subdivisions

Our next goal was to examine how well the segmented functional subdivisions matched the simulated data. We segmented the simulated datasets using the KC-HC as the cluster combination and either the modified silhouette or PRI as the criterion for choosing the optimal number of clusters. We made this choice based on the relative performance of these methods (Figure 4). Figure 5 shows the clusters obtained using KC-HC at the group level. Note that the clusters were obtained by applying hierarchical clustering with the co-association obtained for the optimal k (k*) for each ROI as the similarity metric. The obtained stable clusters were similar to the actual clusters within each ROI (Figure 2). Note that the color coding used for different clusters in Figure 2 and Figure 5 is arbitrary. Similar stable clusters are also obtained using KC-SC (data not shown).

Consensus clustering with KC and HC as base and consensus clustering methods respectively (KC-HC) is able to reliably recover the actual functional boundaries on simulated datasets consisting of 2, 3, 4, 5 and 6 clusters in a group of 21 subjects. The clusters obtained with KC-HC is very similar to the actual clusters shown in Figure 2 (color coding in these two figures is arbitrary).

Experimental rs-fMRI data: Identification of number of clusters

Next, we applied CC-EAC to determine the optimal number of clusters in experimental rs-fMRI data. We applied CC-EAC to segment five different brain areas using rs-fMRI data acquired from 21 adult participants.

(i) SMA-pSMA

The first row of Figure 6 shows the performance of each metric in identifying the optimal number of clusters for the SMA-pSMA ROI. The optimal number of clusters according to the modified silhouette, PRI and VI is 2 for each cluster combination method (Figure 6), consistent with the cytoarchitectonic segmentation of this ROI (Zilles et al., 1996). Note that RI and NMI increased monotonically with the number of clusters combinations KC-HC and KC-SC.

Performance of modified silhouette, Probabilistic Rand Index (PRI), Rand Index (RI), Variation of Information (VI), and Normalized Mutual Information (NMI) in identifying the optimal number of clusters in right SMA-pSMA, Insular cortex, intraparietal sulcus (IPS), angular gyrus (AG), and striatum. Modified silhouette, PRI and VI identify the optimal number of correct number of clusters as two in SMA-pSMA using all the three cluster combination methods. The modified silhouette suggests two clusters for Insula and AG while three clusters for IPS and Striatum with KC-KC as the cluster combination. VI is biased towards two clusters while RI and NMI increase monotonically for all the five ROIs and cluster combinations. PRI resulted in three clusters for Insula, IPS, AG and Striatum with KC-HC as the cluster combination.