Modeling Brain Activation in fMRI Using Group MRF - CiteSeerX

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Modeling Brain Activation in fMRI Using Group MRF Bernard Ng*, Ghassan Hamarneh, and Rafeef Abugharbieh  Abstract—Noise confounds present serious complications to functional magnetic resonance imaging (fMRI) analysis. The amount of discernible signals within a single dataset of a subject is often inadequate to obtain satisfactory intra-subject activation detection. To remedy this limitation, we propose a novel Group Markov Random Field (GMRF) that extends each subject’s neighborhood system to other subjects to enable information coalescing. A distinct advantage of GMRF over standard fMRI group analysis is that no stringent one-to-one voxel correspondence is required. Instead, intra- and inter-subject neighboring voxels are jointly regularized to encourage spatially proximal voxels to be assigned similar labels across subjects. Our proposed group-extended graph structure thus provides an effective means for handling inter-subject variability. Also, adopting a group-wise approach by integrating group information into intra-subject activation, as opposed to estimating a single average group map, permits inter-subject differences to be characterized and studied. GMRF can be elegantly implemented as a single MRF, thus enabling all subjects’ activation maps to be simultaneously and collaboratively segmented with global optimality guaranteed in the case of binary labeling. We validate our technique on synthetic and real fMRI data and demonstrate GMRF's superior performance over standard fMRI analysis. Index Terms—fMRI, functional neuroimaging, group regularization, inter-subject variability, Markov random field

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I. INTRODUCTION

magnetic resonance imaging (fMRI) has become a predominant modality for studying human brain function. By exerting different stimuli and examining which brain regions activated in response, a mapping of brain regions to functions can be generated [1]. The standard approach [2] for inferring intra-subject brain activation from fMRI data involves comparing each voxel’s intensity time course UNCTIONAL

Manuscript received January 23, 2012. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada and in part by the Michael Smith Foundation for Health Research U.S. Asterisk indicates corresponding author. *Bernard Ng is with the Biomedical Signal and Image Computing Lab (BiSICL), The University of British Columbia, Vancouver, BC, Canada, V6T 1Z4 (e-mail: [email protected]). Rafeef Abugharbieh is with BiSICL, The University of British Columbia, Vancouver, BC, Canada, V6T 1Z4 (e-mail: [email protected]). Ghassan Hamarneh is with the Medical Image Analysis Lab, Simon Fraser University, Burnaby, BC, Canada, V5A 1S6 (e-mail: [email protected]). Copyright (c) 2012 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected].

independently against an expected response to estimate what is referred to as “activation effect”. Statistics, such as t-values, pertaining to the probabilities (p-values) of observing the estimated effects given that no activation occurred are then computed and assembled into an activation statistics map. Voxels with t-values above a user-defined threshold are declared as activated. This inference framework is equivalent to performing n independent hypothesis tests, where n is the number of voxels, typically in the range of 105 to 106. To control the number of false positives, one must account for multiple comparisons during threshold selection [3]. A. Intra-subject Activation Detection Traditional Bonferroni correction typically results in overly stringent thresholds due to the assumption that the voxels are independent [4], where in reality, voxels are certainly correlated due to factors, such as physiological noise, interpolation during motion correction, and the functional integration property of the brain in general [1]. To account for correlations between spatially neighboring voxels, Worsley et al. proposed a thresholding technique based on Gaussian random field (GRF) theory [3]. Deriving this GRF-based threshold requires estimating the smoothness of the activation statistics maps, which is non-trivial. The standard way is to spatially smooth the fMRI volumes using a fixed Gaussian kernel and assume the smoothness of the activation statistics maps is equal to the kernel width [1]. Spatial smoothing also helps increase SNR based on the matched filter theorem [1], provided that the Gaussian kernel matches the shapes of the active spatial clusters. However, the shapes of the active clusters are likely to vary across the brain. To remedy this limitation, scale-space approaches have been proposed [5-7], in which kernels of varying widths are applied with the threshold adjusted for the additional scale dimension [5, 7]. The main drawback of GRF thresholding is the need for spatial smoothing, which blurs the details of the activation statistics maps [5]. The use of permutation tests [4] has been proposed to ameliorate this drawback at the expense of longer computational time. Alternatively, one may model the activation statistics maps using mixture models [8, 9], which assumes the activation statistics of the active and non-active voxels are drawn from two distinct probability distributions. Under this setting, activated voxels are delineated by comparing the posterior probabilities of a voxel being active or non-active given data, which mitigates the need for multiple comparisons correction [8]. However, mixture models alone do not account for spatial correlations. To mitigate this

2 limitation, Woolrich et al. proposed a spatial mixture model based on continuous Markov random field (MRF) [10, 11]. Discrete MRF approaches have also been proposed [12-14], which convert activation detection into a class labeling problem, i.e. active or non-active, with discrepancies in labels between neighboring voxels penalized. Spatial continuity is thereby encouraged, which helps suppress false declarations of isolated voxels as being activated. In addition to the above methods that model spatial correlations at the inference stage, approaches that incorporate such correlations during activation effect estimation have also been put forth. For instance, Ruttimann et al. [15] proposed fitting spatial wavelets to fMRI volumes and performing statistical inference in the wavelet domain. By exploiting the decorrelation property of wavelet transform, the independence assumption in Bonferroni correction would deem more appropriate. However, the optimal way for transforming the results back to the spatial domain is not obvious. If inverse wavelet transform is naively applied to the activation statistics of the significant wavelet coefficients, most voxels will be assigned non-zero activation statistics in the spatial domain [16]. Thus, thresholding will again be required. To alleviate this limitation, Wink et al. [17] proposed using wavelet transform as a denoising procedure and performing standard statistical inference on the denoised data. However, this approach neglects the effects introduced by wavelet denoising [16]. To overcome this drawback, Van De Ville et al. [16] proposed a thresholding technique that explicitly accounts for the effects of wavelet processing to enable proper inference. Besides the wavelet approach, substantial research efforts have been placed on using Bayesian approaches [18-23] to model spatial correlations through incorporation of priors. For instance, Penny et al. proposed using a spatial Laplacian prior on the activation effects to encourage spatial smoothness [18]. Extending Penny et al.’s work, Harrison et al. employed a diffusion-based prior that accounts for how the spatial correlation structure is non-stationary across the brain [19]. Vincent et al. [23] proposed a spatial mixture model that jointly estimates activation effects and the hemodynamic response function (HRF). Other approaches include using spatiotemporal autoregressive models to account for correlations in noise [21]. Most of these methods, however, require deploying approximate inference or computationally expensive sampling techniques. The effectiveness of the methods described above in detecting intra-subject activation is contingent on the amount of discernible signals within the data. In typical task-based fMRI studies, only about 20% of the variability in the intensity time courses pertains to signal [24] with typical signal-tonoise ratios (SNRs) lying between 0.2 and 0.5 [25, 26]. Hence, the amount of discernible signals within a single dataset of a subject is very limited. One way of improving SNR is to increase the number of experimental trials. However, this approach is not always practical, especially when considering patient populations, since requiring patients to maintain the same performance level over 15 to 20 minutes can be challenging. Restricting experimental tasks to only those with

“good contrast” can also help improve SNR but limits the neuroscience questions that can be addressed. For typical clinical studies comprising ~5 minutes runs with subjects undergoing tasks that might not provide good contrast, other means for dealing with low SNR are necessary. Since most fMRI studies focus on finding common patterns across a group of subjects, exploiting the group dimension may present a direct, intuitive way of enhancing intra-subject activation detection, which we explore in this work. B. Group Activation Detection In standard fMRI group analysis [2], brain images of all subjects are first spatially normalized to a common brain template to create a voxel correspondence. Activation effects at each voxel location are then statistically compared across subjects to generate a group activation statistics map. The resulting group map is subsequently thresholded to identify the commonly activated voxels. The implicit assumption is that a one-to-one correspondence exists between the active voxels of the subjects, and that all voxels are perfectly aligned in the template space after spatial normalization. However, the large anatomical inter-subject variability renders whole-brain spatial normalization prone to mis-registration errors [27]. In fact, even if perfect anatomical alignment can be achieved, whether a one-to-one functional correspondence exists between voxels across subjects is debatable [28, 29]. The amount of functional inter-subject variability observed in past studies [30, 31] suggests that such a one-to-one voxel correspondence is rather unlikely. Nevertheless, active voxels are typically found within the same anatomical regions across subjects, albeit the exact location being different [30]. The true level of functional inter-subject variability is very difficult to determine. The amount of variability suggested by past studies may not accurately reflect the true variability level since past results are mainly based on standard univariate analysis [31], which has lower detection sensitivity than many recent methods [10-14, 16, 18-23]. Hence, moderately activated voxels with activation statistics just below the user-defined threshold might be mistakenly declared as non-active. Coupled with anatomical misalignment, the level of intersubject variability may appear much larger than actual. The conventional way to account for functional intersubject variability is to spatially smooth the fMRI volumes to increase the functional overlaps across subjects. Smoothing also serves the purpose of deriving a GRF-based threshold and increasing SNR [3]. However, spatial smoothing tends to smear the activations statistics maps. To alleviate this drawback, Keller et al. proposed introducing a “spatial jitter” variable into GLM and marginalizing out this variable to account for the various ways that the activation statistics maps may be misaligned across subjects [32]. Another approach, as proposed by Thirion et al. [33], is to first threshold the activation statistics map of each subject and segment out activated voxel clusters using watershed. Clusters that are consistently present in most of the subjects based on the location of the activation maxima of the clusters are then declared as activated. The advantage of this approach is that it

3 provides subject-specific activation maps, in addition to a group map, which enables inter-subject differences to be studied. Towards this end, a number of hybrid “group-wise” methods that integrates group information into intra-subject activation detection have been explored. C. Group-wise Activation Detection Adopting a group-wise approach facilitates modeling of subject differences while exploiting group information to better disambiguate signal from noise. Methods under this category include group ICA, proposed by Calhoun et al. [34], which involves concatenating the subjects’ voxel time courses and decomposing the concatenated matrix into group independent spatial components and associating time courses. A back reconstruction procedure is then performed to extract subject-specific spatial component maps. The optimal way to delineate active voxels from these intra-subject spatial component maps is nontrivial. One approach is to apply a threshold to pseudo z-scores estimated from the intra-subject spatial component maps [35]. However, these pseudo z-scores do not have a clear statistical interpretation, in contrast to activation statistics generated from GLM [35]. Therefore, the spatial component maps are more often used for generating group maps instead [34]. A tensor extension of ICA [36] has also been proposed for extracting intra-subject spatial components from multi-subject data, but suffers from the same limitation as group ICA. Recently, Bathula et al. [37] proposed a leave-one-out strategy, where all subjects except one are analyzed to compute a group map, which is then used as a prior to segment the active voxels of the left-out subject. However, mis-registration arising from spatial normalization, as required for generating group maps, may limit the accuracy of this strategy. To relax the one-to-one voxel correspondence criterion, Kim et al. [38] proposed using a hierarchical Dirichlet process (HDP) model with “random effects” to extract Gaussian-shaped functional clusters, as opposed to detecting active voxels. Adding “random effects” to the HDP model permits slight inter-subject variations in the location of the detected clusters without losing cluster correspondence between subjects. However, considering the possible shapes that a functional cluster may take, the assumption that all clusters are Gaussian-shaped can be limiting. In this paper, we present a new group-wise approach, Group MRF (GMRF) [39], for detecting intra-subject activation1. Modeling each subject’s activation statistics map as an MRF, we extend the neighborhood system to other subjects so that information can be coalesced across subjects. The intuition behind this approach is that, although the existence of a oneto-one functional correspondence between voxels is very unlikely, true brain activation should presumably appear in similar proximal locations across subjects [30], whereas false positives are more likely to be randomly scattered across the 1

A preliminary version of this work appeared in [39]. Extending [39], synthetic data experiments were performed on real anatomical ROIs, as opposed to 2D square grids. Also, we included repeatability analyses with voxel time courses temporally split in half as well as with subjects randomly split into subsets. Further, we applied GMRF to Parkinson’s disease subjects’ data and showed a spatial focusing effect of medication on regional activation.

brain with little cross-subject consistency. Thus, regularizing inter-subject neighbors reinforces brain areas that are consistently recruited across subjects, while suppressing false positives. An advantage of our approach is that if intra-subject neighbors are deemed unreliable due to noise, additional neighbors from other subjects are available to better approximate the state of each voxel. Also, since only voxels that are spatially proximal to each other are hypothesized to be in similar state, our approach alleviates the need for finding an exact one-to-one voxel correspondence across subjects. Instead, a relaxed condition of requiring only the brain structures of the subjects to be approximately aligned is in place. We note that GMRF is very different from the existing approaches for handling functional inter-subject variability [32, 33, 38]. In Thirion et al.’s structural approach [33], a liberal statistical threshold is recommended during the initial cluster extraction stage to reduce the number of false negatives, and the cluster correspondence procedure only ensures that the activation maxima of the clusters are commonly present across subjects. Hence, not all voxels within the detected clusters are necessarily activated. In contrast, GMRF directly determines if a voxel is activated based on non-thresholded activation statistics maps of all subjects. As compared to Kim et al.’s approach [38], GMRF does not pose any assumptions regarding the shape of the activation patterns. Moreover, in contrast to Keller et al.’s method that only generates a single group map [32], GMRF provides a separate activation map for each subject, which permits inter-subject differences to be studied. GMRF is thus distinct from [32, 33, 38]. Furthermore, unlike the existing intra-subject activation detection methods, GMRF enables activation statistics maps of all subjects within a group to be jointly and cooperatively segmented as a single MRF with global optimality guaranteed for binary labeling.

voxel q

voxel j subject 1 Subject i

voxel p

subject Ns Fig. 1. Proposed GMRF. Each subject’s regional activation statistics map is modeled as an MRF with its neighborhood system extended to other subjects to enable information coalescing.

II. METHODS A. Group MRF Given data for a group of subjects, let Gi = (Vi,Ei) be a graph, where Ei = εi U ε~i, Vi and εi are the sets of vertices (voxels) and edges of subject i, respectively, and ε~i is the set of edges between subject i and all other subjects. Also, let jp  εi be the intra-subject edge between voxels j and p and

4 jq  ε~i be the inter-subject edge between voxels j and q. Our proposed group-extended graph structure for a single subject I is shown in Fig. 1. If we connect the voxels of every subject in an analogous manner, yielding a “GMRF”, all subjects’ activation statistics maps can be jointly and collaboratively delineated by minimizing a single MRF energy: Ns  E ( x)     j ( x j )      jp ( x j , x p )  jp i 1 jVi i  , (1)       jq ( x j , x q )  jq ~ i  where xr  L = {0, …, K–1} is the label assigned to voxel r, Ns is the number of subjects, θj(xj) denotes a unary potential, and  jp ( x j , x p ) and  jq ( x j , x q ) are intra- and inter-subject pairwise potentials. λ is used to adjust the contribution of the pairwise potentials while γ balances the contribution of intra- and intersubject pairwise potentials. For binary labeling (i.e. active vs. non-active), we define the unary potential as follows:

 j ( x j  k )  1  p( x j  k | t j ) ,

(2)

where tj is the activation statistic of voxel j and p(xj = k|tj) is the probability of voxel j belonging to class k given tj. The main rationale for (2) is that if voxel j has a low probability of being activated based on tj, we penalize labeling xj as active, and vice versa. Further discussion on our motivation for (2) is provided in Section II-B. p(xj = k|tj) is estimated using a constrained Gaussian mixture model (CGMM) [40], as also detailed in Section II-B. As for the intra-subject and intersubject pairwise potentials, we employ the Pott’s model:



jm

( x j , x m )  1   ( x j  x m ),

(3)

where δ(y) = 1 if y = 0 and 0 otherwise, and m = p or q. Edges are added such that they link every voxel of subject i to its 6connected intra-subject spatial neighbors as well as its c closest inter-subject spatial neighbors for every subject pairs (i,h), i≠h. Each voxel j of a subject i is thus linked to cNs intersubject neighbors, i.e. c edges from each subject h, i≠h. To define inter-subject neighbors, brain structures of all subjects are first aligned, as discussed in Section II-C. The resulting MRF energy (1) can be minimized, e.g. using graph cuts [41]. We note that if one removes the distinction between intra- and inter-subject edges and treats all subjects’ voxels as a single set, (1) can be rewritten in the same form as the standard discrete MRF energy function, which permits a probabilistic interpretation of GMRF since minimizing the discrete MRF energy is equivalent to maximizing the a posterior probability of the corresponding Gibbs distribution [42]. B. Constrained Gaussian Mixture Model To compute p(xj = k|tj), we first estimate the activation statistic, tj, using the standard GLM [2]: y j  X j   j , t j  ˆ j / se( ˆ j ), (4) where yj is the intensity time course of voxel j, ωj is assumed to be white Gaussian noise after preprocessing (Section III), ˆ is the estimate of the true effect, βj , and se( ˆ ) is the j

j

standard error of ˆ j [43]. X is a design matrix with boxcar functions (time-locked to stimuli) convolved with a HRF as regressors [2]. We model tj using a CGMM [40] (Fig. 2). η

a

τ

σk2

tj

µk

b

K

K

α

π

zj

Nv

Fig. 2. Graphical depiction of CGMM for unary potential θj(xj) estimation.

Specifically, tj is assumed to be generated from a mixture of K Gaussians with mixing coefficients πk, means µk, and variance σk2. Conjugate priors are used to constrain these parameters: tj ~

K

  k N (  k ,  k2 ) ,

k 1

 k ~ N ( , 2 ) ,  k2 ~ IG (a, b) ,  ~ Dir ( ) ,

(5) (6)

where IG(a,b) and Dir(α) denote inverse Gamma and Dirichlet distributions, respectively. Adding priors mitigates the singularity problem in maximum likelihood solutions, i.e. µk collapsing onto tj with σk2 approaching 0, thus assigning infinite probability at tj. Also, having priors allows us to encode our knowledge into the model. A discussion of parameter choice is provided in Section II-D. Gibbs sampling is used to estimate p(xj = k|tj), as detailed in the Appendix. Regarding our choice of unary potential, employing GMRF (or discrete MRF in general) requires two penalties for the unary potential: one to penalize labeling a voxel as active and another for labeling the same voxel as non-active. One might envisage other unary potentials apart from (2), such as the negative log data likelihood from GLM, i.e. -ln(p(yj|Xβj,σj2)) for active class, and -ln(p(yj|0,σj2)) for the non-active class. However, the magnitude of the negative log data likelihood can be arbitrarily large for small data likelihood. To ensure the unary potential does not completely dominate the pairwise potentials, it is important to have the values of all potentials to be in the same magnitude range (i.e. [0,1] in the current work). One can normalize the negative log data likelihoods to sum to one, but the resulting formulation is analogous to GMM but with the mixing coefficients assumed equal for the active and non-active labels. In contrast, CGMM learns the model parameters from data and intrinsically provides two penalties with magnitudes lying between [0,1], hence why (2) is a natural choice as unary potential. C. Proposed Neighborhood System To extend each subject’s neighborhood system to other subjects, we need to first align the brain structures of all subjects so that inter-subject spatial neighbors can be properly defined and linked. The conventional strategy is to perform whole-brain warping, but the large anatomical inter-subject variability renders accurate whole-brain registration difficult

5 [27], especially for diseased subjects. We thus opt to employ a region-based approach, where we extract anatomical regions of interest (ROIs) and perform alignment at the regional level for which highly accurate techniques are available [44]. The region-based alignment approach has several major advantages over whole-brain warping. First, no brain structures will be mistakenly taken as part of another structure. Second, the improvement in alignment by adopting regionbased registration as opposed to whole-brain spatial normalization has shown to enhance activation localization [45]. Third, computational cost will be substantially reduced. We note that GMRF does not require more than usual alignment accuracy that is provided by standard whole-brain warping procedures. In fact, since GMRF relaxes the one-toone voxel correspondence assumption, a higher tolerance for mis-registration errors would be permitted by GMRF than the standard univariate group analysis. We adopted the regionbased alignment approach mainly for reducing inter-subject variability arising from anatomical differences so that any residual inter-subject variability will more likely pertain to true functional differences. However, region-based alignment is by no means a necessity for employing GMRF. Regarding the practicality of the proposed group graph structure, the corresponding adjacency matrix is very sparse, since each voxel connects to only 6+cNs edges (e.g. c = 3, Ns = 10). Hence, typical group sizes of 10 to 20 subjects in clinical studies can be easily handled. D. Parameter Selection From an implementation perspective, our proposed method essentially creates a “mega-subject” by combining voxels of all subjects within a group into a single set with edges added between intra- and inter-subject spatial neighbors. We thus set λ in (1) to 1/Ns to scale the cumulative pairwise potential over subjects to a level similar to that of a single subject, which ensures that the pairwise potentials will not dominate over the unary potential. The choice of γ in (1) depends on the degree of inter-subject variability, which is unknown a priori. We have thus set γ to 0.5 to weight subject i’s own information more than that of other subjects. Nevertheless, varying γ by ±0.2 (i.e. ±40%) had little effects on our results. Also, for the inter-subject pairwise potential in (1), we have set c to 3 in agreement with how a one-to-one voxel correspondence unlikely exists between subjects. Choosing c is analogous to selecting the number of intra-subject spatial neighbors, which is a tradeoff between the amount of spatial regularization for handling noise and localization power. Based on our synthetic data experiments, for a reasonable range of c between 3 to 5, the results are very robust. Setting c to higher values (e.g. 10) will pool information from spatially distant voxels, similar to spatially smoothing with a wide Gaussian kernel, which smears the detected activation patterns, thus not advised. As for the parameters in the CGMM (6), we know that tj of nonactive voxels should theoretically be 0 and the t-threshold for active voxels is typically set between 3 and 4, based on GRF theory [3]. We thus encode this prior knowledge on t-values of active and non-active voxels through η with τ2 set to 1 to model uncertainty in η. K is set to 2 to classify voxels as either active or non-active. We focus on this two-class scenario as in [40], since globally optimal labeling can be obtained using

binary graph cut [41]. As for σk2, we use an uninformative prior by setting both a and b to 0.5 [40], since little is known about this parameter. Finally, we set α to 1/K assuming equal prior class probabilities. E. Empirical Evaluation We generated 500 synthetic datasets to validate our proposed method. Each dataset consisted of 10 subjects with artificial activation injected into real, manually-segmented anatomical ROIs (Section III), as shown in Fig 3(a). The activation pattern comprises a single cluster having a radius of 8 mm as indicated by the red circles in Fig. 3(b)-(g). The cluster centroid was set to be anywhere within 3 mm from the ROI anatomical centroid with the location randomly varied across subjects to simulate functional inter-subject variability. Also, the size of a cluster varied with the location of the cluster centroid (i.e. clusters with activation centroid at the ROI border will be much smaller than clusters with activation centroid near the ROI anatomical centroid). The clusters could thus potentially have less than fifty percent overlap. Synthetic time courses of active voxels were generated by convolving a box-car function, having the same stimulus timing as in our experiment (Section III), with a canonical HRF [2] and adding Gaussian noise. Signal intensity of the active voxels was set to decrease exponentially as a function of distance from the cluster centroid. Our synthetic tests in essence emulate the situation where truly active regions moderately overlap across subjects, but the apparent overlap appears much less due to reduced signal near cluster borders. Four maximum SNR levels were tested: 0.25, 0.5, 0.75, and 1, where SNR is defined as mean signal amplitude over noise variance. For voxels near the cluster centers, their SNRs would be close to the maximum SNR values, whereas for voxels near the cluster borders, their SNRs would be well below the maximum. To compare the proposed GMRF against current state-ofthe-art approaches, we contrast our results to those of: (i) GLM with spatial smoothing and a threshold based on GRF theory at a p-value of 0.05 [2], (ii) CGMM [40], and (iii) MRF [41] separately applied to each subject’s tj map. Furthermore, we tested (iv) group ICA [34] by interpolating yj of all subjects onto an ROI template, concatenating the resulting yj along the temporal dimension, and applying ICA on the concatenated data matrix. We delineated the active voxels of each subject by first finding the associating time course of the IC that maximally correlates with the reference signal (i.e. the time course of active voxels without noise) and performing back-reconstruction to obtain subject-specific spatial component maps [34]. We then transformed these spatial component maps into pseudo z-scores [35] and applied a threshold at 1.96 (i.e. |z| > 1.96). This active voxel delineation approach is more suitable for comparing with our proposed method than finding a group map [34], since a separate activation map can be estimated for each subject, while having group support integrated. We refer to methods (i), (ii), (iii), and (iv) as individual GLM (iGLM), individual CGMM (iCGMM), individual MRF (iMRF), and group ICA (gICA). Fig. 3(b)-(f) show the qualitative results at a maximum SNR of 0.25. Only results from half of the subjects in one of the synthetic datasets are plotted due to space limitations. iGLM missed most of the active voxels, whereas iCGMM identified

6

6

‐6 (a) non-smoothed t-maps at SNR = 0.25

(b) iGLM with Gaussian spatial smoothing and GRF threshold

positives than gICA. This adaptability arises from how GMRF draws a balance between subject-specific information on one side, as modeled by the unary potential and intra-subject pairwise potential, and group support on the other side, as captured by the inter-subject pairwise potential. Also, adding inter-subject neighbors provides additional information to better determine the state of voxels near ROI borders, which in our tests completely eliminated false positives at the ROI boundaries as compared to iMRF. We note that since all potentials are normalized between [0,1] and all subjects are weighted equally in GMRF, no single subject will dominate the activation detection process. To quantify and compare the performances of the various methods (Fig. 4), we computed the average Dice similarity coefficient (DSC) over the 500 synthetic datasets (7): DSC 

2TP , 2TP  FP  FN

(7)

where TP, FP, and FN denote the number of true positives, false positives, and false negatives, respectively. 1

(c) iCGMM separately applied to each subject

DSC

0.8 0.6 0.4 0.2

(d) iMRF separately applied to each subject

0 0.25

iGLM

0.5

iCGMM

0.75

iMRF

gICA

1

SNR

group MRF

Fig. 4. DSC at various maximum SNR levels. GMRF consistently outperformed all other methods. Note: DSC of iGLM = 0 at max SNR = 0.25.

(e) gICA projected back onto subjects’ native ROI space

(f) proposed GMRF Fig. 3. Synthetic data results. (a) t-maps. (b) iGLM, (c) iCGMM, (d) iMRF, (e) gICA, (f) GMRF results at SNR = 0.25. Blue dots = detected active voxels. Red circles = ground truth. GMRF correctly detected most of the active voxels with stricter control over false positives than other examined methods.

more active voxels but included many isolated false positives. Enforcing intra-subject spatial regularization using iMRF reduced the amount of false positives, but an ample amount remained especially near the ROI border since fewer intrasubject neighbors are available at those locations. Using gICA decreased the number of false positives near the activation cluster, but also reduced the number of true positives. Moreover, many false positives remained due to lack of explicit spatial regularization. In contrast, the proposed GMRF was able to adapt to subject differences with most of the active voxels correctly detected, while providing much stricter control over false

As evident from the low DSC for iGLM, iCGMM, and iMRF in Fig. 4, information within a single dataset of a subject may be insufficient to obtain satisfactory intra-subject action detection at low SNR, even when contextual information is considered. These results support our argument for incorporating additional information. However, integrating group support with gICA only improved results over iGLM at maximum SNRs of 0.25 and 0.5. We believe the need for a one-to-one voxel correspondence and the assumption that a common spatial component exists across subjects (in one of gICA’s intermediate steps prior to back reconstruction) limit gICA’s effectiveness when moderate functional inter-subject variability is present. This argument is supported by how GMRF achieved higher DSC for all maximum SNR levels compared to gICA, which demonstrates the importance of relaxing the strict one-to-one voxel correspondence constraint in handling inter-subject variability in activation location. We note that even at higher SNR levels, where more reliable information was available within each subject’s data, the addition of group information using GMRF still improved results over using iGLM, iCGMM, and iMRF. We also note that increasing the number of subjects improved DSC (Fig. 5).

7 1

DSC

0.9 0.8 0.7 0.6 0.5 1

2

SNR = 0.25

3

4 5 6 7 Number of Subjects

SNR = 0.50

SNR = 0.75

8

9

10

SNR = 1.00

Fig. 5. GMRF’s DSC vs. number of subjects at varying maximum SNR levels. Increasing the number of subjects consistently increased DSC.

III. MATERIALS After obtaining informed consent, fMRI data were collected from 10 Parkinson’s disease (PD) patients off and on medication (4 men, 6 women, mean age 66 ± 8 years) and 10 healthy controls (3 men, 7 women, mean age 57.4  14 years). Each subject used their right hand to squeeze a bulb with sufficient pressure to maintain a bar shown on a screen within an undulating pathway. The pathway remained straight during baseline periods and became sinusoidal at a frequency of 0.25 Hz (slow), 0.5 Hz (medium) or 0.75 Hz (fast) during the time of stimulus. Each session lasted 260 s, alternating between baseline and stimulus of 20 s duration. Functional MRI was performed on a Philips Gyroscan Intera 3.0 T scanner (Philips, Best, Netherlands) equipped with a head-coil. T2*-weighted images with BOLD contrast were acquired using an echo-planar (EPI) sequence with an echo time of 3.7 ms, a repetition time of 1985 ms, a flip angle of 90°, an in plane resolution of 128×128 pixels, and a pixel size of 1.9×1.9 mm. Each volume consisted of 36 axial slices of 3 mm thickness with a 1 mm gap. A 3D T1-weighted image consisting of 170 axial slices was further acquired to facilitate anatomical localization of activation. Slice timing and motion correction were performed using Brain Voyager (Brain Innovation B.V.). Further motion correction was performed using motion corrected independent component analysis (MCICA) [46]. The voxel time courses were high-pass filtered to account for temporal drifts and temporally whitened using an autoregressive AR(1) model. No whole-brain registration or spatial smoothing was performed. For testing, we selected the left primary motor cortex (M1) as the region of interest, since the left M1 is known to activate during right-hand motor movements (Fig. 6). Anatomical delineation of the left M1 was performed by an expert based on anatomical landmarks and guided by a neurological atlas. The segmented ROIs were resliced at the fMRI resolution for extracting the preprocessed voxel time courses within each ROI. The ROIs were non-rigidly aligned (Fig. 6) using the point set registration algorithm, “Coherent Point Drift (CPD)” [44], where each voxel is taken as a point with its (x,y,z) coordinates treated as features for alignment. Specifically, CPD treats the voxel locations of the target point set as centroids of a GMM, and fits this GMM to the reference point set by learning the optimal centroids’ positions. This alignment method is more robust to noise and outliers than conventional techniques, such as iterative closest point [44], and hence adopted in this work.

Fig. 6. Homunculus. Hand area of M1 is circled in Homunculus diagram. Approximate corresponding area in the aligned ROIs is indicated. Homunculus courtesy of thebrain.mcgill.ca.

IV. RESULTS AND DISCUSSIONS Results obtained with iGLM, iCGMM, iMRF, gICA, and GMRF on real data of the healthy controls are shown in Fig. 7. As a reference to the proximal expected areas of activation, we also displayed results obtained from standard fMRI group analysis [2], which we refer to as gGLM. Only results for five arbitrary controls during the fast condition are displayed due to space limitations, but we confirm that consistent results were observed across all subjects. The two rows of each subfigure correspond to results obtained using all time points (top) and half of the time points (bottom) for testing repeatability [37]. Only results generated from the latter half of the time points are shown. We note that activation in M1 may be very subtle, since both the task and baseline conditions required motor squeezing. Low contrast is thus expected. iGLM missed most voxels in the hand area (Fig. 6). In contrast, iCGMM detected more voxels in the hand area, but also included many isolated false positives. iMRF reduced the amount of isolated false positives, but mistakenly declared adjacent non-hand areas as active. gICA detected some voxels in the hand area, but the disjoint appearance deviates from the commonly-accepted hypothesis that activation comprises spatially-contiguous clusters. GMRF correctly detected the hand area in all subjects. Furthermore, GMRF achieved higher repeatability compared to all examined methods (Fig. 7).We note that a number of voxels outside the hand area appears to be activated in Fig. 7(a). However, we believe these voxels should not be activated, especially the isolated ones, since squeezing a bulb with the right hand should presumably activate only the hand area in the left M1. Results for PD patients are shown in Fig. 8. The top and bottom rows correspond to PD patients off and on medication, respectively. Again, five arbitrary subjects are shown in favor of space. iGLM detected islands of activation in the hand area, and no obvious differences between PD on and off medication could be identified. iCGMM declared most voxels along the left M1 as active, which likely included many false positives. iMRF detected spatially-contiguous clusters of voxels in the hand area, but also declared the hip and leg areas, which should not be active unless the subjects’ hip and leg moved in sync with the stimulus. Using gICA again resulted in spurious disjoint patterns of activation. In contrast to these methods, GMRF consistently detected the hand area in all subjects

8 without falsely declaring the hip and leg areas. In addition, the GMRF results suggest a very interesting trend between PD on and off medication. Specifically, PD off medication seemed to recruit a wider area of the left M1, which normalized back to an extent similar to the controls upon medication. Such spatial focusing effect of levo-dopa medication has also been observed in past studies [47, 48], thus further confirming the validity of our results. iMRF’s results also suggest such trend, but the wider recruitment observed for PD off medication could have arisen from iMRF’s lack of control over false positives, which hindered result interpretations. We note that the inter-subject differences in the detected activation patterns were very subtle (Figs. 7 and 8), but nonetheless, matched our expectation since the hand area should lie within similar proximal location within the left M1 across subjects (Fig. 6). To quantitatively verify our expectation against possible overgroup regularization, we examined the impact of having different random subsets of subjects on the GMRF results [49]. Specifically, we created 1,000 random subsets of 6 to 9 subjects and applied GMRF separately to each subset. We then computed the percentage of subsets for which each voxel is declared as activated (Fig. 9(a) to (c)). Overall, over 80% of the voxels that were declared as activated in Figs. 7 and 8 were correctly detected in majority of the random subsets (Fig. 9(d)-(f)). Over 80% of the non-active voxels in Figs. 7 and 8 were also correctly labeled as non-active for most subsets (Fig. 9(g)-(i)). Our results thus quantitatively demonstrate GMRF’s robustness to different subject subsets. V. CONCLUSIONS We proposed a novel method that incorporates group information for accurately detecting intra-subject fMRI activation. By modeling activation statistics maps of a homogenous group of subjects as a single GMRF, all subjects’ information is jointly exploited. Our formulation enables all subjects’ activation maps to be simultaneously segmented using a binary graph cut, which guarantees global optimality. Moreover, our approach permits information to be coalesced across subjects without having to establish a one-to-one voxel correspondence, as required in standard fMRI group analysis. Applying GMRF to synthetic data for a range of SNR showed superior performance over standard techniques. On real data, using GMRF consistently detected active voxels in areas implicated with the experimental task employed, and also provided additional evidence supporting the spatial focusing effects of medication on PD patients’ regional activation. In contrast, techniques relying on a single dataset of a subject failed to obtain such delineation. Our results thus indicate great promise for the proposed group-wise approach in handling noisy data. GMRF also achieved higher repeatability over time courses of reduced length. Thus, one could envision potential reduction of acquisition time without sacrificing quality of the results using GMRF. Repeatability over different subject subsets was also demonstrated. For future work, there are several directions that need special attention to increase the generality of GMRF. First,

only ten subjects were recruited for each group (PD and controls) in this work. Although, this group size is quite typical for most clinical studies (usually no more than 20 subjects), there are some studies that recruit hundreds of subjects, where direct application of GMRF would be computationally infeasible. Devising methods to sub-group subjects based on factors, such as age, handedness, or even the fMRI data itself, to ease the computational load as well as to ensure group homogeneity [37] would thus be important for applying GMRF to larger scale studies. Second, although we have only demonstrated the power of GMRF by testing on a single ROI, GMRF can be extended to whole-brain analysis without the problem size becoming intractable. Specifically, one can divide the brain into different brain regions using automated brain parcellation tools, such as Freesurfer, and apply the presented single ROI analysis procedure to each ROI separately. This approach bypasses the scalability issue arising from the large number of voxels in the brain, while enabling the whole brain to be studied. However, interactions between boundary voxels of adjacent ROIs would be neglected with this straight forward extension. Third, we have focused so far on binary labeling (active vs. non-active), where global optimality can be achieved e.g. using a binary graph cut. Nevertheless, a number of approximation techniques are available for the multi-label case [50] to include a ‘de-activated’ class if one so desired. Fourth, given only binary labels estimated from GMRF without a one-to-one voxel correspondence across subjects, statistical group comparisons between controls and PDs at the voxel level would not be feasible using the standard univariate analysis. To facilitate group comparisons, one may instead adopt a regional approach by extracting invariant spatial features from the detected ROI activation patterns and statistically comparing these features across populations [51]. APPENDIX To estimate the parameters in CGMM, Gibbs samples were drawn from the following posterior conditional distributions: Nv Nv   p ( | z ,  )  Dir   1   z 1j ,  ,  K   z Kj  , (8) j 1 j 1   p( k | t, z)   Nv k   z j t j  2   k2  2 k2  j 1  , Nv Nv   2  z kj   k2  2  z kj   k2  j 1 j 1 

 ,     

(9)

p ( k2 | t , z , a , b )  Nv Nv  , IG  a  0.5  z kj , b  0 .5  z kj (t j   k ) 2  j 1 j 1  

p ( z j  k | t j ,  k ,  k2 ,  k )   k N (  k ,  k2 ) ,

(10) (11)

where Nv is the number of voxels in subject i’s ROI and zjk is an indicator variable set to 1 if voxel j is estimated to be in state k, and 0 otherwise.

9

6

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(b) iGLM with Gaussian smoothing and GRF threshold

(d) iMRF applied to each subject separately

(c) iCGMM applied to each subject separately

(e) gGLM (f) gICA after back-reconstruction (g) proposed GMRF Fig. 7. Real data results of healthy controls. 5 arbitrary exemplar subjects shown. Top row of each sub-figure corresponds to results obtained using all time points. Bottom row corresponds to using half of the time points. (a) Non-smoothed t-maps. (g) Proposed GMRF correctly detected the hand area in all subjects with greater consistency in the location of active voxels and higher repeatability than (b) iGLM, (c) iCGMM, (d) iMRF, (e) gGLM, and (f) gICA.

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