NeuroImage 52 (2010) 1302–1313

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Network-level analysis of cortical thickness of the epileptic brain

A. Raj a,⁎, S.G. Mueller b, K. Young b, K.D. Laxer c, M. Weiner b

a Department of Radiology, Weill Cornell Medical College, New York, NY 10021, USAb Department of Radiology and Biomedical Imaging, University of California at San Francisco and Center for Imaging of Neurodegenerative Diseases, San Francisco, CA 94121, USAc Pacific Epilepsy Program, California Pacific Medical Center, San Francisco, CA 94115, USA

⁎ Corresponding author. Weill Medical College of CAvenue, New York, NY 10044, USA.

E-mail address: ashish@med.cornell.edu (A. Raj).

1053-8119/$ – see front matter © 2010 Elsevier Inc. Adoi:10.1016/j.neuroimage.2010.05.045

a b s t r a c t

a r t i c l e i n f o

Article history:Received 22 April 2010Accepted 16 May 2010Available online 27 May 2010

Temporal lobe epilepsy (TLE) characterized by an epileptogenic focus in the medial temporal lobe is the mostcommon form of focal epilepsy. However, the seizures are not confined to the temporal lobe but can spread toother, anatomically connected brain regionswhere they can cause similar structural abnormalities as observed inthe focus. The aim of this study was to derive whole-brain networks from volumetric data and obtain network-centric measures, which can capture cortical thinning characteristic of TLE and can be used for classifying a givenMRI into TLE or normal, and to obtain additional summary statistics that relate to the extent and spread of thedisease. T1-weighted whole-brain images were acquired on a 4-T magnet in 13 patients with TLE with mesialtemporal lobe sclerosis (TLE–MTS), 14 patients with TLE with normal MRI (TLE-no), and 30 controls. Meancortical thickness and curvature measurements were obtained using the FreeSurfer software. These values wereused to derive a graph, or network, for each subject. The nodes of the graph are brain regions, and edges representdisease progression paths. We show how to obtain summary statistics like mean, median, and variance definedfor these networks and to perform exploratory analyses like correlation and classification. Our results indicatethat the proposed network approach can improve accuracy of classifying subjects into two groups (control andTLE) from 78% for non-network classifiers to 93% using the proposed approach. We also obtain network“peakiness”valuesusing statisticalmeasures likeentropyand complexity—this appears tobea good characterizerof the disease and may have utility in surgical planning.

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Introduction

Temporal lobe epilepsy (TLE) is characterized by an epileptogenicfocus in the mesial temporal lobe structures and is the most commonform of partial epilepsy with a prevalence of 0.1% in the generalpopulation. Based on imaging and histopathology, two types ofnonlesional TLE can be distinguished: (1) TLE with mesial–temporallobe sclerosis (TLE–MTS, about 60%–70%), characterizedbyanatrophiedhippocampus with clear MRI abnormalities and severe neuronal loss.Depth EEG shows a relatively circumscribed epileptogenic zone inmesial temporal structures. (2) TLE with normal-appearing hippocam-pus on MRI (TLE-no, about 30–40%) and mild or no neuronal loss onhistological examination. In TLE-no, depth EEG shows a morewidespread, lesswell-defined epileptogenic area in themesial temporallobe extending into the inferior and lateral temporal lobe (Vossler et al.,2004). In both TLE types, however, seizures are not restricted to thetemporal lobe but can spread to other anatomically connectedextratemporal brain regions and can cause similar if less severeabnormalities. The pathophysiology of these extratemporal changes is

not entirely clear, and possiblemechanisms include secondary neuronalloss due to deafferentiation and neuronal loss/dysfunction due to localexcitotoxic effects of seizure spread with propagation of epileptogenicactivity to secondary regions (Zumsteg et al., 2006; Huppertz et al.,2001).

The anatomically defined projections of the epileptogenic focus toother temporal andextratemporal regionswill determine thedistributionand severity of extrafocal abnormalities. Therefore, a description of thedistribution of extratemporal abnormalities might provide additionalinformation regarding the exact location of the epileptogenic focus.Recently, regionally unbiased whole-brain cortical thickness measure-ments based on T1-weighted images were reported (Fischl and Dale,2000; Thompson et al., 2005), and their use in TLE has shown them to bevery sensitive for the detection of distributed atrophic effects in TLE–MTSand TLE-no (Mueller et al., 2009a).

However, these studies have not considered the network-levelarchitecture (i.e., topology) of the observed pattern of extrafocal corticalthinning. The overall goal of this study, therefore, was to test if it ispossible to derive, for each patient, a TLE-related network within thebrain by looking at extrafocal abnormalities in cortical thinning. Thenodes of this network are the regions significantly affected by thedisease, and the link between any two nodes represents the probabilityof the two regions having a disease progression path between them.Prior studies have already demonstrated that it is possible to derive

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connectivity between cortical regions in terms of statistical associationsbetween brain regions across subjects in terms of cortical thickness(Lerch et al., 2006; He et al., 2007, 2008; Chen et al., 2008) or volume(Bassett et al., 2008).

In this work, we propose a generalized Gibbs probability model fornetwork connectivity, attempt to derive individual brain networksfrom this model (rather than group networks proposed by previousauthors), and propose a graph-theoretic method for performingconventional statistical analysis like correlation and classificationusing network-level information. Finally, we explore traditionalnetwork features such as degree distribution and clustering, andnovel features such as network entropy and complexity. Our chosendisease model is TLE, for which the following questions wereaddressed. (1) Do networks derived by this approach differ betweenTLE–MTS and TLE-no? (2) How do these networks contribute to thefocus localization in TLE–MTS and TLE-no? (3) Can the informationcontained in the structure of these networks be used to correctlyclassify a subject into disease group?

Theory

A generalized Gibbs probability distribution model for cortical thickness

Our input consists of the 68 cortical ROIs provided by FreeSurfer(Fischl et al., 2004) and 4 subcortical ROIs from hippocampalvolumetry—details are in the Methods section. Each of these ROIs ismapped to a node in our hypothesized network. Suppose we are givenN subjects each of whom belongs to one of the three groups {control,TLE–MTS, TLE-no}. Let a subject k∈{1,…, N}, have a (n×n) table Tk ofcortical thickness values, where Tk(i) denotes the cortical thickness ofthe i-th ROI of the k-th subject, and i∈{1, …, n}. Let the mean andstandard deviations of each ROI thickness over all known healthysubjects be μT(i) and σT(i). Define the normalized z-scores of thesesubjects as follows

Zthicknessk ðiÞ = TkðiÞ−μT ðiÞð Þ= σT ðiÞ ð1aÞ

Observations of TLE subjects indicated significant cortical thinningand also higher surface irregularity reflected by higher averagecurvature. In Fig. 1, a scatter plot of cortical thickness versus curvature(normalized z-scores) indicates a moderate negative correlationbetween the two for TLE patients and almost no correlation for

Fig. 1. Normalized cortical thickness versus curvature of various ROIs from TLE (a) and healtplotting. Correlation coefficient R and significance value (p) are shown inside the plots.

healthy subjects. Therefore, we define the z-scores corresponding tocortical curvature, given curvature tables Ck:

Zcurvaturek ðiÞ = CkðiÞ−μCðiÞð Þ= σCðiÞ ð1bÞ

For subcortical structures, thickness is difficult and/or inappropriatetomeasure. Instead, we obtain their volume tables Volk using FreeSurferand from hippocampal subfield volumetry (see Methods) and normal-ize them in the same way as thickness and curvature:

Zvolumek ðiÞ = ðVolkðiÞÞ−μvolðiÞ= σvolðiÞ ð1cÞ

Since the data have beennormalized by noise variance of each ROI, itis now possible to use these values irrespective of whether they camefrom thickness, curvature or volume.

Given the preponderance of evidence that statistical association ofcortical thickness or volume denotes cortical connectivity (Lerch et al.,2006; He et al., 2007, 2008; Chen et al., 2008) or volume (Bassett et al.,2008), we now attempt to fit a probabilistic model to these data andask whether individual networks can be created from this model thatcaptures the network properties of the disease. We hypothesize thatthere exists a functionχ:RxR→R such that the probability of a diseaseprogression path between any two nodes i and j of the k-th subject isgiven by

πkði→jÞ∝ exp χ ZkðiÞ; ZkðjÞð Þð Þ: ð2Þ

The distribution over the entire brain network is given by

πk = ∏i;j πk i→jð Þ = 1zexp ∑i;j χ ZkðiÞ; Zk jð Þð Þ

� �ð3Þ

where Z is the normalizing constant and is called Partition Function.This is a generalized version of Gibbs distributions defined over aMarkov Random Field (Geman and Geman, 1984; Li, 1995). Gibbsdistributions were originally used in statistical mechanics and havesince appeared in diverse fields to capture the statistics of images(Geman and Geman, 1984; Besag, 1986), chaotic systems (Beck andSchlogl, 1993), and as natural spatial priors in MRI image reconstruc-tion (Raj et al., 2007).

Clearly, from Eqs. (2) and (3), the set of pairwise potentials χ(Zi, Zj)form a sufficient statistic for the entire distribution, and have a specialconnotation as Potential Functions in statistical mechanics. This paper

hy (b) subjects. Data were condensed by averaging over their respective groups before

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treats them simply as a matrix of connection strength between nodes,but their meaning should always be understood in terms of theprobability distribution shown in Eqs. (2) and (3).

A New Proposal To Create Networks Characteristic of TLE

The correct form ofχ, which depends on the disease process,may bechosen from a set of candidate formulas by optimizing over somereasonable objective function, for instance using a Gibbs learningalgorithm (Li, 1995). In Appendix A, we show that a particular choice ofthemodel,withχ(Zi,Zj)=ZiZj amounts to thePearson correlationwhichhas appeared in previous reports. Here we do not rigorously go into theproblem of model selection. We created a short list (Appendix A) of so-called robust L1-norm models (e.g., Raj et al., 2007) and selected thefollowing formula for χ that gave the highest classifying power, asshown in Fig. 5. Define a (n×n) connectivity matrix ℂk of subject k, foreach pair of nodes i and j in the network as

χkði; jÞ = max jZthicknessk ðiÞj; jZthickness

k ðjÞj� �+ max jZcurvature

k ðiÞj; jZcurvaturek ðjÞj� �

ð4aÞ

For subcortical regions, we have volume but not thickness orcurvature, and a comparable connectivity score is defined as

χkði; jÞ = max jZvolumek ðiÞj; jZvolume

k ðjÞj� �: ð4bÞ

Network pruningNoise reduction is accomplished by imposing the condition that

affected regions exhibit both thinning and curvature increase:

ℂkði;jÞ = χkði; jÞ Zthicknessk ðiÞb 0;Zthickness

k ðjÞb 0;Zcurvaturek ðiÞ N 0;Zcurvature

k ðjÞ N 00 otherwise

8<: ð5aÞ

For subcortical regions:

ℂkði;jÞ = max jZvolumek ðiÞj; jZthickness=volume

k ðjÞj� �Zvolumek ðiÞb 0;Zthickness=volume

k ðjÞb 0;0 otherwise

(

ð5bÞ

Finally, nodes that have zero or very small connections (using auser-defined threshold) to other nodes are eliminated.

Graph theoretical features of an individual network

Several network-level summary measures are known (Watts andStrogatz, 1998). The degree of a node is the average number ofconnections to that node. The clustering coefficient of a node is theaverage number of connections between the node's neighbors dividedby all their possible connections. Chen et al. (2008), Bassett et al. (2008)and He et al. (2007, 2008) found these measures to indicate so-calledsmall world properties in the brain. In this study, degree is computed asthe sumof connectivityweights. In contrast topriorwork, our individualsubjects' networks obtained from cortical thickness data simply do notshow enough group-wise differences in terms of the above popularnetwork metrics to make them useful for classification (see Fig. 7). Wetherefore propose three additional summary features that do capturerelevant groupdifferences andmeasure the “peakiness” of the network:network entropy, statistical complexity (Crutchfield and Young, 1989),and monoexponential decay of degree curve.

First, we transform a given network into a Markovian transitionmatrix, where the sum of the connectivity exiting any node adds up to1, and therefore, each edge weight can be thought of as the probability

of moving from that node to its neighbor. Define a new connectivitymatrix ℂ̃k fromℂk whose rows have been normalized to sum to 1. Thetransition probabilities between any two nodes are given directly bythe edge weights of this normalized matrix: pij= ℂ̃k(i,j). It is wellknown (Zhang and Hancock, 2008) that the leftmost eigenvector v1 ofℂ̃k represents the stationary distribution of the network and givesnode probabilities according to pi=v1(i). Then, the new summaryfeatures are as follows:

(a) Network entropy: ∑ipilog2(pi). Low entropy denotes a

“peaky” connectivity curve.(b) Network statistical complexity: ∑

i;jpi∑

i;jpijlog2(pij). Complex-

ity is an information measure representing uniformity inconnection weights, thus low complexity denotes higher“peakiness.”

(c) Mono-exponential decay constant τ of sorted degree curve:s(i)∝ smaxe

−i/τ, where s(i) is the sorted connectivity sum fornode i and smax is its largest value. Small τ denotes peakiness.

A proposal for classifying networks

Let the TLE network of subject k be given by the graph Gk={Vk,Ek},where Vk is the set of nodes or vertices of the graph and Ek is the setof edges. Denote the set of all such graphs by Ω. Since theepileptogenic network of different subjects may vary in terms ofaffected nodes, severity, and spatial distribution, we allow graphsto have varying number of nodes, edges and topologies, in particular,|Gk|≠ |Gk′|, Vk≠Vk′. This is to be contrasted with traditional similar-ity measures defined on scalars or vectors, which require equal sizeof all constituents.

We propose a new method for specifying the “distance” betweenindividual networks, as follows. Suppose we start with the network Gk,and insert or delete nodes as well as edges one by one, until we matchthe network Gk′—such that there exists a graph isomorphism betweenGk andGk′. Each of thesemodifications comes at a predefined cost, in ourcase, a unit cost for inserting or deleting a node, and a cost of |eij| forinserting or deleting an edge of weight |eij|, and a cost of |eij−e′ij| formodifying the cost of edge eij to e′ij. Let us define the edit cost of this editpath fromGk toGk′ as the sumof all thesepredefined costs addedover allthe insert/delete/modify steps. Then the graph edit distance (GED)between Gk and Gk′ is the edit cost of the edit path from Gk to Gk′ whichgives the minimum cost:

dGED Gk;Gk0ð Þ = min GED Gk;Gk0ð Þð Þ: ð6Þ

Details are in Appendix C. The GED allows us to treat individualnetworks like random variables, by a mapping from graphs to non-negative scalar variables, as described in Appendix C. Pairwise GEDdistances are then embedded into a Cartesian feature space using apopular algorithm called multidimensional scaling (MDS) (Seber,1984). MDS has been successful in several biomedical data analysisproblems, particularly genomic data analysis (Tzeng et al., 2008).

The entire process beginning with FreeSurfer analysis on MRimages, to network formation, to GED distance to the MDSmapping isdepicted in a flowchart in Fig. 2. Finally, the Cartesian feature vectorsare fed to a classification routine to obtain class labels for each subject.We chose a classical clustering method for this purpose, called lineardiscriminant analysis (LDA) (Krzanowski, 1988; Appendix C).

NewTLE-specificmeasures: dispersion, severity, and temporal lobe specificity

DispersionWe define a new network measure of “peakiness” called dispersion.

For maximum descriptive power, we propose the first principalcomponent (PC1) of the three previously defined peakiness measures

Fig. 2. Summary flowchart of GED-based network classification. Various MR brain volumes from different subjects are processed by FreeSurfer to get cortical thickness and curvatureof parcellated cortical structures. A network is built for each subject. These networks are fed into the GED algorithm to compute pairwise distances between them. The MDS(multidimensional scaling) algorithm is applied to convert these pairwise distances into Cartesian coordinates in the embedded Euclidean space. Finally, LDA (linear discriminantanalysis) is performed to get disease class for each subject.

Table 1Study subject characteristics. Number of subjects (age range, mean±SD).

Gender Control TLE–MTS TLE-no

Female 20 (38.8±9.7) 8 (39.6±9.9) 9 (35.0±7.5)Male 10 (37.9±9.7) 5 (44.4±6.4) 5 (37.0±5.8)

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(entropy, complexity, exponential decay), each normalized to have unitvariance. PC1 is determined simply as follows (Jolliffe, 2002).

A = ½en; cn; τn�UΣVH = APC1 = first column of U

ð7Þ

To convert PC1 in the range (0,1), we perform the logit transform(Ashton, 1972)

dk =e−PC1k

1 + e−PC1kð8Þ

SeverityOverall brain atrophy or severity can be defined as the sum of

connectivity over all nodes

sk = ∑i;j

jCkði; jÞ j ð9Þ

Temporal lobe specificityThe clinician is also interested in assessing how specific the disease

is to the temporal lobe, and how much it has dispersed outside thetemporal lobe. A simple measure of TL specificity, as a ratio ofconnectivity within ipsilateral temporal lobe, over the entire brain isas follows:

TLspeck =∑i;j∈TL

Ckði; jÞ

∑i;j

Ckði; jÞð10Þ

TL specificity could be an important measure for patients slated fortemporal lobe resection surgery.

Methods

Study population

This study was approved by our institution's IRB and writteninformed consent was obtained from each subject according to theDeclaration of Helsinki. Twenty-seven patients suffering from drug

resistant TLE were recruited between mid 2005 and end of 2007 fromthe Pacific Epilepsy Program, California Pacific Medical Center and theNorthern California Comprehensive Epilepsy Center, UCSF, wherethey underwent evaluation for epilepsy surgery. Thirteen patients hadevidence for MTS on 1.5 TMRI (TLE–MTS) and 14 patients had normalappearing hippocampi and normal reads on 1.5 T MRI (TLE-no). Thepresence of MTS in TLE–MTS was confirmed by hippocampal subfieldvolumetry using high-resolution T2-weighted images aimed at thehippocampus. The absence of MTS was confirmed in all but one TLE-no who had a significant volume loss in the ipsilateral subiculum. TLEgroups were matched for lateralization: 9 left/8 right (female), and 5left/5 right (male). Subjects were grouped into ipsilateral andcontralateral, by side-flipping one group, so that all patients had thefocus on the left side. Identification of epileptogenic focus was basedon seizure semiology and prolonged ictal and interictal video/EEG/telemetry (VET) in all patients. The control population consisted of 30healthy volunteers. Table 1 displays subject characteristics.

MRI acquisition

All imagingwasperformedonaBrukerMedSpec4 Tsystemcontrolledby a Siemens TrioTM console and equipped with an eight-channel arraycoil (USA Instruments). The following sequences were acquired: (1) Forcortical thickness and thalamusmeasurements a volumetric T1-weightedgradient echo MRI (MPRAGE): TR/TE/TI=2300/3/950 ms,1.0×1.01.0 mm3 resolution, acquisition time 5.17 min. (2) For themeasurement of hippocampal subfields, a high-resolution T2-weightedfast spin echo sequence: TR/TE=3500/19 ms, 0.4×0.4 mm in planeresolution, 2 mm slice thickness, 24 slices acquisition time of 5.30 min.(3) For the determination of intracranial volume (ICV), a T2-weightedturbo spin echo sequence: TR/TE=8390/70 ms, 0.9×0.9×3mm nom-inal resolution, 54 slices, acquisition time of 3.06 min.

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Cortical thickness measurement

FreeSurfer (version 3.05, https://surfer.nmr.mgh.harvard.edu)was used for cortical surface reconstruction and cortical thicknessestimation. The procedure has been extensively described elsewhere(Fischl et al., 1999; Fischl and Dale, 2000; Fischl et al., 2004). Visualinspection by raters unaware of the clinical diagnosis was usedto manually correct errors due to segmentation miss-classification,if necessary. An automatic parcellation technique was used tosubdivide each hemisphere into 34 gyral labels (Desikan et al.,2006).

Hippocampal subfield volumetry

The method used for subfield marking has been describedpreviously (Mueller et al., 2009b). The marking scheme depends onanatomical landmarks, particularly on a hypointense line represent-ing myelinated fibers in the stratummoleculare/lacunosum (Erikssonet al., 2008) which can be reliably visualized on high-resolutionimages. Therefore, external and internal hippocampal landmarks areused to further subdivide the hippocampus into subiculum, CA1, CA1-2 transition zone, and CA3 dentate gyrus.

Network-level processing

Means of healthymale and female controls were subtracted from allgender-grouped datasets in order to remove systematic gender-relatedbias. Since gender-specific healthymean values are available, it was notnecessary to perform the usual correction involving intracranial volumeas a covariate. Our subjects are approximately age-matched, andtherefore, no age effect adjustment was needed. A connectivity matrixwas then obtained each subject's tables using Eqs. (4a) and (4b) and(5a) and (5b).

Group networksA summary network for the entire group (cf., He et al. (2007)) was

obtained as follows. Across the entire group, a count was maintainedof all non-zero edge weights, and the strongest 15-percentile of edgeswere retained. The summary network has the same nodes as anindividual network, and its edge weight is given by the number oftimes that edge appeared in individual networks.

ClassificationEuclidean embedding of proposed graph edit distance between

subjects was performed using the MDS algorithm. Classification usinglinear discriminant analysis yielded TLE–MTS, TLE-no, and controlgroups. To avoid using the same data for both training andclassification, we adopted the jack-knife or leave-one-out approach.Each subject, whether control or TLE, was classified by using theremaining subjects as training set. Results of this leave-one-outvalidation were summarized by the receiver operator characteristics(ROC) curve (Kraemer, 1992).

LateralizationThe sum of outgoing connectivities (“severity”) of each node was

computed and sorted. Then a lateralization label (“left” or “right”) wasassigned to each subject, depending on which hemisphere had thelarger number of such highly connected nodes. A concordance score(from 0% to 100%) was computed by comparing with the lateralizationobtained by VET.

Network-level metricsTraditional network measures (degree, clustering), new ones

(network entropy, statistical complexity and monoexponential decayparameter fit), and TLE-specific ones (dispersion, severity, TL-specific-ity) were computed. Histograms of these features were compiled, a 2-

sample one-sided Student's t-test was performed between each grouppair, and their p values were obtained.

Results

Cortical thickness and curvature

Normalized thickness maps are consistent with earlier reports(Mueller et al., 2009a) and are not shown here. Briefly, TLE–MTS wasassociated with prominent cortical thinning in ipsilateral medialposterior temporal and lateral prestriatal structures, additionalregions with less severe cortical thinning were found in superiorfrontal, pre/postcentral, and superior temporal regions bilaterally.TLE-nowas associatedwith prominent cortical thinning in ipsilateralanterior inferior, lateral and superior temporal, opercular, andinsular regions. Less prominent cortical thinning was also found insuperior frontal regions and pre/postcentral regions bilaterally. Asreported before, there was more widespread contralateral involve-ment in TLE-no than in TLE–MTS. We did not find significant(pb0.05) curvature changes as a stand-alone measure; however,Fig. 1 shows that moderate correlation exists between thickness andcurvature in TLE but none in healthy subjects.

Summary group network

Fig. 3 shows summary networks for TLE–MTS and TLE-no groups,displayed using a novel illustration method whereby cortical nodes aremapped to an ellipse except for the cingulate which are mapped as arcswithin the ellipse for ease of visualization. High-threshold (85-percentile) and low-threshold (70-percentile) cases are shown sepa-rately to give a sense of how frequently the edges appear in their group.Localized nature of TLE–MTS is easily distinguished from more diffuseTLE-no. While TLE-no has a larger number of edges at low thresholdthan TLE–MTS, the opposite is true at high threshold. This implies thatTLE–MTS has a few very strong edges, while TLE-no has more, well-distributed, but weaker edges. Table 2 shows that node connectivitystrength is distributed in the brain as expected from prior studies, withdominant ipsilateral temporal involvement in TLE–MTS and bilateralinvolvement in TLE-no.

Classification using individual networks

Fig. 4 shows 2-way (control vs. TLE) and 3-way (control vs. TLE–MTS vs. TLE-no) ROC classification curves from the leave-one-outanalysis. Best accuracy is obtained by graph edit distance measure(93% area under ROC curve, compared to 78% for vector distanceclassifier). The classification performance is higher for the 2-groupcase compared to the 3-group case, as expected. Fig. 5 shows that themax() function (Eqs. (4a) and (4b)) produces the best classifier AUCof all alternatives listed in Appendix A. Supplementary Fig. S1 showsthat classification accuracy increases substantially when one usesboth thickness and curvature rather than thickness alone.

Lateralization

Concordance between lateralization from VET with the networkapproach is plotted in Fig. 6 against the number of highest-connected nodes used in the calculation of laterality. Only 40%concordance is achieved by looking at the first three most well-connected nodes in the networks, rising to 100% with increasingnumber of nodes.

Summary network measures

Degree distribution, clustering coefficient, and sum of outgoingconnection weights are presented in Fig. 7 after sorting from highest

Fig. 3. Epilepsy-derived brain network connectivity of TLE subjects. For visual convenience cortical networks are depicted as 2D circular rings, with cortical ROIs residing on thecircumference, except for cingulate structures which are shown as brown arcs within the circle. Following a dorsal view, frontal lobe is represented in light blue arc, temporal lobe(dark blue), parietal lobe (green), and occipital lobe (orange). A histogram of the five most popular connections across all subjects was compiled and this histogram was thenthresholded to obtain the connectivities above. The straight lines between ROIs denote the presence of the most frequently occurring connections among all subjects in the specifiedgroup. Plots on the left ((A) and (B)) show the first few most frequent connections by setting a high threshold, and on the right (C) and (D) show a few more of these frequentconnections by relaxing the threshold a little.

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to lowest. Only the last measure showed significant group differences.The three new summary measures (network entropy, networkstatistical complexity, and monoexponential decay of sorted degree

Table 2Total node strength in each lobe, as percentage of total. Connections weights lower thanthe low threshold in Fig. 2 were not counted.

TLE–MTS TLE-no

Lobe Ipsi Contra Ipsi Contra

Occipital 0 3 0 5Parietal 3 3 3 5Temporal 47 3 25 20Cingulate 8 8 10 6Frontal 14 11 14 2Total 72 28 52 48

curve are shown in Fig. 8. The p-values corresponding to a 2-tailed t-testbetween the three groups are significant, as shown between each grouppair. Note how these measures impose a specific order on the threegroups: the TLE–MTS patients are distinguished by “peaky” summarycurves, which indicate a well-localized epileptic focus. The TLE-nopatients show greatly reduced peakiness in their curves, as expected.Healthy subjects show the least amount of peakiness, indicating nospecific focus, as expected.

TLE-specific measures

A scatter plot of severity versus dispersion is shown in Fig. 9, whichdemonstrates that while all TLE–MTS fall comfortably within theexpected quadrant of high-severity–low-dispersion, the TLE-no is moredispersed as well as less severe. Fig. 10 shows a scatter plot of dispersionversus TL specificity and again indicates that TLE–MTS subjects fall in theexpected quadrant of low-dispersion–high TL specificity.

Fig. 4. Receiver operator characteristic (ROC) curves for the GED-distance based classification, as well as the standard difference norm-based classification. (A) Two-groupclassification (TLE vs Healthy) and (B) three-group classification (TLE–MTS vs TLE-no vs healthy). Note that there is some loss of performance from the two-group case to the three-group case, but the GED-based method consistently outperforms the standard non-graph approach.

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Discussion

Summary of findings

Network-centric results of Fig. 3 and Table 2 are consistentwithpriorfindings of Mueller et al. (2007, 2009a) and that TLE–MTS showssignificant focal cortical thinning in the ipsilateral temporal lobe, andTLE-no shows weaker but more widespread bilateral involvement.However, the network approach brings added value by improvingclassification and providing new metrics like degree distribution,clustering coefficient, entropy, and complexity. New metrics likeseverity and dispersion may also improve characterization of epilepsy.Network analysis seems to provide especially better characterization ofTLE-no, whichwas traditionally problematic due to lack of measureablesclerosis on MRI exams. Voxel-based morphometry on the 1.5 T(Mueller et al., 2006) found no significant hippocampal or corticalatrophy in TLE-no compared to healthy subjects. Mueller et al., 2009bconfirmed the absence of hippocampal atrophy in TLE-no using high-resolution T2 subfield mapping at 4 T but reported about 17% of TLE-no

Fig. 5. Area under ROC curves, for five different edge weight formulas. The formulaproducing best classification power is indicated by red arrows and boxes. Note the, max()function appears to be most informative edge weight for epilepsy.

to have nonspecific abnormalities which were inadequate for focuslocalization. Network-based classification results in Fig. 4 are signifi-cantly improved compared to the presented volumetric assessment-based conventional classifier using the same volumetric data. However,we have not investigated other contrast mechanisms like T2 relaxation(Mueller et al., 2007) and MRSI (Mueller et al., 2004) which mightprovide additional information not possible by volumetrics alone. Thelaterality result in Fig. 6 showing 100% concordance with VET, ifreplicated in future studies, might be significant for clinical practice.

Comparison with previous network studies

Network methods have been utilized in different biomedical fields,e.g., to study biochemical and metabolic interactions, epidemiologicalrelationships or protein interaction, and gene expression interactions(Guimerà et al., 2007; Price and Shmulevich, 2007; Steuer, 2007;Martínez-Beneito et al., 2008; Clauset et al., 2008; Charbonnier et al.,2008; Liang andWang, 2008). Similar networkmethods have also been

Fig. 6. Concordance between VET-based lateralization of epileptic focus and MRI-basedlateralization. The x-axis depicts the number of highly connected nodes used to estimatelateralization.

Fig. 7. Plots of three network metrics for three disease classes: TLE–MTS (left column), TLE-no (middle column) and Healthy (right column). Sum of outgoing connectivities for eachnode (top row), degree of each node (middle row), and clustering coefficient of each node (bottom row). No summary statistic was significantly different between groups except thescale difference of connectivity sum.

1309A. Raj et al. / NeuroImage 52 (2010) 1302–1313

applied toMR diffusion tractography (Hagmann et al., 2008), functionalMRI (fMRI) (Achard et al., 2006), andmagnetoencephalography (MEG)(Kujala et al., 2008). In prior thickness- or volume-based studies (Lerchet al., 2006;Heet al., 2007; Chenet al., 2008;Heet al., 2008;Bassett et al.,2008), anatomical connection was defined as statistical associationbetween brain regions, as measured by the Pearson correlationcoefficient across subjects.

Properties of an unweighted version of their network werecharacterized in terms of degree, clustering coefficient, and path length,and confirmed so-called “small-world” attributes in the brain as well asmodular architecture in healthy development (Chen et al., 2008). Thereare several important differences between our approach and thesepapers, which provide insight into disease groups but little practicaldiagnostic information or exploratory analysis of individual subjects.

Fig. 8. Histograms of network metrics. TLE–MTS (left column), TLE-no (middle column),complexity (second row), monoexponential decay parameter fit (third row), and area undcorresponding to a two-way t-test between the three groups, for each of these four summarcharacteristics somewhat between the TLE–MTS and healthy groups. We conjecture that theby the two colors in the middle column: blue for TLE–MTS-like subgroup, and pink for healthterms of their network properties more akin to the Healthy group.

GED methods have recently appeared in the graph processing andmachine learning (Robles-Kelly and Hancock, 2005) where they weremainly used for handwriting detection, face recognition, line drawingmatching, and feature matching. GED does not previously appear tohave been reported in biomedical imaging. Its use in performingstatistical analysis in any context also appears to be novel.

The individual TLE-specific network approach

Unlike previous group networks where ensemble correlationsignifies connectivity, links in our networks come from potentialfunctions (Eqs. (4a) and (4b)) between a pair of ROIs and must beinterpreted as the probability of disease path (Eqs. (2) and (3)) ratherthan anatomic connectivity. Since they come from a single pair of

and healthy (right column). Network entropy for each subject (top row), statisticaler the connectivity sum curves shown in top row of Fig. 5 (bottom row). The p valuesy metrics, are shown between corresponding group pairs. TLE-no group shows networkTLE-no group can be separated into TLE–MTS-like and healthy-like subgroups, as showny-like subgroup. The “Healthy-like” subgroup show high dispersion of epileptic focus in

Fig. 9. Network-derived measures of epilepsy severity and dispersion. These measurescan prove useful to a clinician interested in surgical planning.

1310 A. Raj et al. / NeuroImage 52 (2010) 1302–1313

thickness values, not ensemble average, these probabilities can bevery noisy due to substantial noise embedded at various points in thedata pipeline—in the MRI signal, in the segmentation step, inhomo-geneity correction step, or the registration step, and finally in thecortical measurement step. Further uncertainty is introduced bynatural anatomic variations present in human brains. Yet we are ableto infer individual networks reliably enough to perform accurateclassification and to obtain lateralization information. Why? First,only significantly atrophied ROIs show up as network nodes due tonormalization to z-scores and are further pruned by joint thicknessand curvature constraints. Second, even if individual links in thenetwork are noisy, the network as a whole should still be informative.Since all subsequent computations, whether edit distance classifica-tion or aggregate network measures like degree, clustering, entropy,and complexity are performed on whole networks and depend onoverall network topology, they are quite reliable.

Fig. 10. Scatter plot of dispersion versus temporal lobe specificity of cortical atrophy.The inverse relationship between dispersion and TL specificity of most TLE subjectsindicates that the dispersion is a result of out-migration of epileptogenic regions fromTL to other regions of the brain. Among TLE–MTS patients, this out-migration is limited.However, some TLE-no subjects, as indicated in the figure, exhibit high dispersion aswell as low TL specificity.

What explains the suitability of max() function over otherpotential functions for epilepsy classification? Progression of atrophyin epilepsy is likely caused by excitotoxicity accompanied by greymatter loss and damage of the white matter tracts, which channel thedisease from one cortical or subcortical region to another (Shin andMcNamara, 1994; Zumsteg et al., 2006; Huppertz et al., 2001). Wespeculate that since the more affected node in the node pair willdominate atrophic progression in this model, the likelihood of apathway to become a disease propagation path depends only on themore affected region irrespective of whether the other terminal ishealthy or atrophic. However, we stress that the proposed network-centric approach does not rely on a particular connectivity formula—we simply chose the one that was demonstrably themost informative.

Interpreting TLE-specific measures

Histograms of network peakiness measures (entropy, statisticalcomplexity, and exponential decay) shown in Fig. 8 strongly suggestthat the TLE-no group shows network characteristics somewhatbetween the TLE–MTS and healthy groups. Owing to this mixtureeffect, neither conventional summary network statistics (Fig. 7) norconventional vector distance metrics appear to give good three-wayclassification. Fig. 8 further suggests that the TLE-no group might infact be a mixture of MTS-like and Healthy-like subgroups, as indicatedin the figure by blue and pink. The “Healthy-like” subgroup showshigh dispersion of epileptic focus in terms of their network propertiesmore akin to the Healthy group. This conclusion is also supported byFig. 9, where TLE–MTS subjects uniformly appear to have highseverity and low dispersion, but TLE-no subjects are more dispersedand less severe. Fig. 10 indicates that TLE–MTS subjects fall in theexpected quadrant of low-dispersion–high TL specificity. Within thetemporal lobe, these patients have high severity, but the disease hasnot dispersed outside of the TL. However, a look at the TLE-no showsthat while some of these subjects display TLE–MTS-like properties,others possess high dispersion.

Significance, limitations, and future work

Classification results in Fig. 4 show added clinical value of TLEnetworks compared to volumetric data but cannot serve as replacementfor ictal EEG recordingswhich is the gold standard for focus localization.Instead, our intent was to show how to quantify and analyze theinformation contained in networks derived from volumetry, whetherfor epilepsy or (in the future) for diseaseswithmore challenging clinicaldiagnoses, like Alzheimer's disease, Parkinson's disease, schizophrenia,and multiple sclerosis. Networks analysis could be used to obtain aninitial indication of TLE–MTS, or TLE-no, or of a different formof epilepsyand a strong indication of lateralization. This information may preventunnecessary invasive EEG recordings in inappropriate subjects. Itwill beinteresting to see if there is a difference in success rate of resectionsurgery between the TLE-no in the high-dispersion versus low-dispersion regions of Fig. 9. If network measures are indeed shown tobe predictive of surgical outcome, then they could provide another toolfor surgical planning. These ideas are speculative and need to becarefully validated through a larger study with sufficient statisticalpower and long-term postsurgical follow-up. Future cortical parcella-tion techniques with more ROIs might produce more localized nodeinformation. Functional (fMRI and MEG) data as well as DTI dataelucidating fiber architecture perhaps in combination with corticalthickness might yield larger, more reliable brain networks. Proposedgraph edit distance-based network analysis technique will continue tobe useful in these applications. We also note that applying proposedtechnique to TLE is just a first step. The ultimate goal is to use suchtechniques for focus localization in types of partial epilepsy where thestandard methods fail, e.g., nonlesional neocortical epilepsy.

1311A. Raj et al. / NeuroImage 52 (2010) 1302–1313

Limitations

Many attendant limitations of preprocessing steps that mayadversely affect our results were described by Mueller et al. (2009a).Other limitations include the following:

1. Cortical atrophy is the macrostructural irreversible endpoint of apathophysiological cascade induced by epilepsy. Therefore, proposedapproach is not promising for early detection or characterization ofmore subtle alterations in the brain. Other MRI contrasts (T2, DTI, ASL,and MRSI) might be more sensitive in this respect. T2 relaxation(Mueller et al., 2007) and MRSI (Mueller et al., 2004) were shown toprovide sensitive characterization of epilepsy but it is unclear how toapply proposed network approach on these datasets. However, DTImay be ideal for this purpose andwill be undertaken in future studies.

2. Cortical networks inherit the errors and inconsistencies ofFreeSurfer. Preliminary analysis of back-to-back scans (unpublished)indicate up to 10% variation in thickness of smaller cortical structureseven in healthy subjects. Accuracy of cortical parcellation depends onMR image quality—FreeSurfer did not succeed on a 1.5-T data set fromCornell due to lower SNR and diffuse grey–white boundary.

3. Initial classification into three groups (healthy, TLE–MTS, andTLE-no) was based on MRI exams at 1.5 T, which has limited SNR andresolution. This classification was subsequently supported by (1)histopathological exam on a subset of 10 patients that has undergonesurgery and (2) all subjects also had subfield volumetry as describedin Mueller et al. (2009a,b). The presence of MTS was confirmed in allTLE–MTS. Except for one TLE-no who had a significant volume loss inthe ipsilateral subiculum, absence of MTSwas confirmed in all TLE-no.However, the gold standard to determine if pathology and VETfocus lateralization were correct is the histopathological examinationof the resected tissue and seizure-freedom after surgery. Since not allpatients have undergone surgery yet, we cannot eliminate thepossibility that the three groups are less distinct than initiallyassessed based on 1.5 T scans.

Acknowledgment

This work was partially supported by NIH grants 1R21EB008138and P41 RR023953.

Appendix A. Connection between Gibbs potential and correlation

Let uswrite the thickness correlationbetween two cortical regions as

C i; jð Þ = EðZi⋅ZjÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE Z2

ið ÞE Z2j

� �r , where E xð Þ = 1K ∑

K

k=1x is the ensemble expec-

2 2

tation over all subjects, and E(Zi )=E(Zj )=1 by definition. Nowconsider a candidate Gibbs potential formula χ(Zi,Zj)=ZiZj and assumethat data from different subjects are independently distributed. Thenthe joint probability over all subjects is given by

Pr Z1;…; ZK

� �= ∏k = 1

Kexp ZiZj

� �= exp ∑i = 1

KZiZj

� �= exp KC i; jð Þð Þ

In other words, the Pearson correlation computed in previousanatomical connectivity work reduces to a special case of our proposedGibbs potential, when χ(Zi, Zj)=ZiZj.

Appendix B. Identifying the best connectivity formula

The results described in this paper were obtained using a specificpotential formula: Eqs. (4a) and (4b) because it was found to give thebest classification results. In fact, a number of other formulas wereevaluated, as summarized below.

χkði; jÞ = jZthicknessk ðiÞ−Zthickness

k ðjÞj + jZcurvaturek ðiÞ−Zcurvature

k ðjÞj ðA1Þ

χkði; jÞ = jZthicknessk ðiÞ + Zthickness

k ðjÞj + jZcurvaturek ðiÞ + Zcurvature

k ðjÞj ðA2Þ

χkði; jÞ = min jZthicknessk ðiÞj; jZthickness

k ðjÞj� �+ min jZcurvature

k ðiÞj; jZcurvaturek ðjÞj� �

ðA3Þ

χkði; jÞ = max jZthicknessk ðiÞj; jZthickness

k ðjÞj� �+ max jZcurvature

k ðiÞj; jZcurvaturek ðjÞj� �

ðA4Þ

χkði; jÞ = jZthicknessk ðiÞj + jZcurvature

k ðiÞj ðA5Þ

Each of these is a heuristic, and the “correct” formula is at thispoint only justifiable by its classification performance in terms ofarea under the ROC curve (Fig. 5) although some arguments for itssuitability were given in the Discussion. Area under the ROC is agood measure of fitness because it measures how informative anynetwork is regarding the disease mechanism it is supposed tocapture.

Appendix C. Classifying networks using graph edit distance

Specifying a distance between two networks

Starting with network Gk, and inserting or deleting nodes as wellas edges one by one, until we arrive at an isomorphism of Gk′, wedefine the minimum cost of this edit path as the graph edit distance(GED) between Gk and Gk′:

dGED Gk;Gk0ð Þ = min GED Gk;Gk0ð Þð Þ ðC1Þ

where each insert/delete modification comes at a pre-defined cost, inour case, a unit cost for inserting or deleting a node, and a cost of |eij|for inserting or deleting an edge of weight |eij|, and a cost of |eij−e′ij|for modifying the cost of edge eij to e′ij. Note that as defined, thisdistance is not commutative: dGED(Gk, Gk′)≠dGED(Gk′, Gk).

The GED allows us to treat individual networks like randomvariables, by the following mapping from graphs to non-negativescalar variables:

ZðGkÞ = dGEDðGk;GÞ= σGED ðC2Þ

where G ̅ is a centroid graph defined below, and σGED is the variance ofdGED(Gk,G ̅) for all k. The mapping Z:Ω→ℜ converts a given networkinto a distance from some centroid graph. The centroid graph is agraph such that the sum of its GED distances from all subjects' graphsis smallest:

G = argminG

∑N

k=1dGEDðGk;GÞ ðC3Þ

Converting pairwise distances to points in Euclidean space

To completely convert individual networks into random vectorsfor the purpose of statistical analysis, a geometric embedding of thepairwise GED distances into Euclidean space is performed via apopular algorithm called multidimensional scaling (MDS) (Seber,1984). MDS has been successful in several biomedical data analysisproblems, particularly genomic data analysis (Tzeng et al., 2008). MDSconverts a matrix of pairwise GED distances, DGED={dGED(Gk,Gk′)|}into a matrix Δ of size ndims×N,ndimsbN, such that its columns rep-resent Cartesian coordinates of the geometric embedding of i-thsubject's network. The coordinate vectors given by the columns of

1312 A. Raj et al. / NeuroImage 52 (2010) 1302–1313

Δ, i.e. δi=Δ(:, i), faithfully satisfy the GED distance criteria betweenevery subject, such that

‖δk−δl‖2 = dGEDðGk;GlÞ; ∀ k; l∈½1;N� ðC4Þ

Further statistical analysis can be performed on Cartesian vectorsδi. This procedure is useful for classification algorithms that can onlytake feature vectors as input, rather than pairwise distances betweenpoints in feature space. Given a database of graphs, the procedure forGaussian statistical analysis will proceed as follows.

1. Obtain pairwise distance matrix DGED for all graphs in database.2. Perform MDS to obtain Δ=MDS(DGED).3. Determine covariance matrix C=E(ΔΔH).

A number of routine statistical tasks can now be performed, forexample any supervised or unsupervised (clustering) algorithm thatutilizes distance measurements. For instance, suppose it is determinedthat the graphs belong to two distinct groups, such that in Cartesianspace

Δ = ðΔA j ΔBÞ; whereΔA∼NðμA;CAÞ; ΔB∼NðμB;CBÞ

A given graph Gk can be assigned to either group A or B according towhich of

ðδk−μAÞHC−1A ðδk−μAÞ OR ðδk−μBÞHC−1

B ðδk−μBÞ

is smaller. Hypothesis testing to determine whether two graphs Gk

and Gk′ belong to the same Gaussian distribution can similarly beperformed.

Classification using linear discriminant analysis

Multiple-group classification on δi is performed via lineardiscriminant analysis (LDA), which is a classical method for separatingdata points into groups. It works by fitting normal distributions toeach group and finding the classification that maximizes the lineardiscriminant between groups (Krzanowski, 1988). Suppose there areng groups, and the mean and covariance of the embedded coordinates(e.g., δk) for each group are given by μ1, …, μ

ng, and C1, …, C

ng. Let the

overall mean of the data set be given by μ ̅=mean(μ1,…, μng). Then the

LDA method finds the classification Ψ:ℝndims→ℤng that maximizes theso-called linear discriminant:

JðΨÞ = ΨtSBΨΨtSWΨ

ðC5Þ

where the between-class and within-class scatter matrices SB, SW aredefined by

SB = ∑ng

g=1ngðμg−μÞðμg−μ ÞT ;

SW = ∑ng

g=1∑

k:ΨðkÞ=gðδk−μgÞðδk−μgÞT :

ðC6Þ

In this work, LDA was solved using standard eigen-decompositionmethod (Krzanowski, 1988).

Appendix D. Supplementary data

Supplementary data associated with this article can be found, inthe online version, at doi:10.1016/j.neuroimage.2010.05.045.

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