Quantitative comparison and analysis of sulcal patterns using sulcal graph matching: A twin study

NeuroImage 57 (2011) 1077–1086

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NeuroImage j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / y n i m g

Quantitative comparison and analysis of sulcal patterns using sulcal graph matching: A twin study Kiho Im a, b, Rudolph Pienaar b, c, Jong-Min Lee d, Joon-Kyung Seong e, Yu Yong Choi f, Kun Ho Lee g, P. Ellen Grant a, b, c,⁎ a

Division of Newborn Medicine, Children's Hospital Boston, Harvard Medical School, Boston, MA, USA Center for Fetal Neonatal Neuroimaging and Developmental Science, Children's Hospital Boston, Harvard Medical School, Boston, MA, USA Deptartment of Radiology, Children's Hospital Boston, Harvard Medical School, Boston, MA, USA d Department of Biomedical Engineering, Hanyang University, Seoul, Republic of Korea e School of Computer Science and Engineering, Soongsil University, Seoul, Republic of Korea f Department of Computer Engineering, Hanyang University, Seoul, Republic of Korea g Deptartment of Marine Life Science, College of Natural Sciences, Chosun University, Gwangju, Republic of Korea b c

a r t i c l e

i n f o

Article history: Received 10 February 2011 Revised 14 April 2011 Accepted 29 April 2011 Available online 6 May 2011 Keywords: Graph matching Sulcal pattern Sulcal pit Twin

a b s t r a c t The global pattern of cortical sulci provides important information on brain development and functional compartmentalization. Sulcal patterns are routinely used to determine fetal brain health and detect cerebral malformations. We present a quantitative method for automatically comparing and analyzing the sulcal pattern between individuals using a graph matching approach. White matter surfaces were reconstructed from volumetric T1 MRI data and sulcal pits, the deepest points in local sulci, were identified on this surface. The sulcal pattern was then represented as a graph structure with sulcal pits as nodes. The similarity between graphs was computed with a spectral-based matching algorithm by using the geometric features of nodes (3D position, depth and area) and their relationship. In particular, we exploited the feature of graph topology (the number of edges and the paths between nodes) to highlight the interrelated arrangement and patterning of sulcal folds. We applied this methodology to 48 monozygotic twins and showed that the similarity of the sulcal graphs in twin pairs was significantly higher than in unrelated pairs for all hemispheres and lobar regions, consistent with a genetic influence on sulcal patterning. This novel approach has the potential to provide a quantitative and reliable means to compare sulcal patterns. © 2011 Elsevier Inc. All rights reserved.

Introduction The human cerebral cortex exhibits a complex and variable pattern of sulcal and gyral folds. Although the developmental mechanisms underlying the formation of human cortical folding remain largely unknown, a number of observations and hypotheses support the idea that the folding geometry is an important macroscopic feature for deeper architectural organization or developmental events of the brain. It has been postulated that the cortical folding patterns are related to the underlying fiber connectivity (Van Essen, 1997) and cytoarchitectonic map of the cerebral cortex (Fischl et al., 2008). During early brain development, impairments in neuronal proliferation, migration and differentiation can lead to abnormal cortical convolutions (Clark, 2001; Gaitanis and Walsh, 2004). There have been various approaches developed for quantifying the morphology of the cortical folds. The degree of gyrification and cortical folding complexity have been measured with the gyrification index, fractal ⁎ Corresponding author at: Children's Hospital Boston, 300 Longwood Avenue, Boston, MA 02115, USA. Fax: + 1 617 730 4671. E-mail address: [email protected] (P.E. Grant). 1053-8119/$ – see front matter © 2011 Elsevier Inc. All rights reserved. doi:10.1016/j.neuroimage.2011.04.062

dimension and curvature at global or lobar level (Armstrong et al., 1995; Cachia et al., 2008; Im et al., 2008; Im et al., 2006; Pienaar et al., 2008). Other approaches are sulcal-based, using automatic sulcal extraction and labeling techniques. In these approaches, the overall shape of each sulcus has been analyzed using sulcal features such as area, depth and length (Kochunov et al., 2010; Mangin et al., 2004; Ochiai et al., 2004). Local shape studies using parametric sulcal meshes have been proposed for one of the major sulci, the central sulcus (Cykowski et al., 2008a; Li et al., 2010). Local curvature and depth analyses at the vertex level for the entire cortical area have also been proposed based on the technique of sulcal pattern matching of cortical surfaces (Hill et al., 2010; Luders et al., 2006). Although these studies capture various morphometric features, they do not provide an accurate means to quantify the spatial, geometric and topological relationship between sulci, or the sulcal pattern. Here the term, sulcal pattern, is used to describe the global pattern of positioning, arrangement, number and size of sulcal segments and their relationship. The variability in sulcal patterns leads to difficulty in defining the precise anatomical correspondence and analyzing the local sulcal shape across different brains. Ono's atlas contained a description of sulcal pattern variability with sulci

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categorized based on the connection and interruption patterns to neighboring sulci (Ono et al., 1990). Using this method, the number of interruptions in the superior temporal sulcus was shown to be asymmetric between hemispheres, a feature that might relate to language lateralization (Ochiai et al., 2004; Ono et al., 1990). Recent studies have attempted to perform automatic clustering of cortical folding patterns (Sun et al., 2009; Sun et al., 2007). However, these techniques cannot quantitatively compare individual brains and have not provided a quantitative method to investigate group differences in folding patterns. Abnormal sulcal arrangement, connection and interruption, or an unusual orientation have been shown in several sulcal areas in various disorders: schizophrenia (Kikinis et al., 1994; Nakamura et al., 2007), temporal lobe epilepsy (Kim et al., 2008), obsessive–compulsive disorder (Shim et al., 2009), bipolar disorder (Fornito et al., 2007), persistent developmental stuttering (Cykowski et al., 2008b), and Turner syndrome (Molko et al., 2003). Differences in global sulcal patterns may reflect variations in early brain development and manifest as individual variability in cognitive function, personality traits or psychiatric disorders (Kim et al., 2008; Nakamura et al., 2007). However, these sulcal pattern studies have been built around qualitative analysis methods, based on visual inspection with observer-dependent criteria, or cannot quantify relationships between sulcal segments. This lack of quantification makes it difficult to analyze sulcal patterns over areas larger than one specific sulcus. Therefore lobar, hemisphere and whole brain comparisons become difficult and overwhelmingly complex. The qualitative visual methods are also very laborious and time consuming, making assessment of large groups difficult. Furthermore, it might be important to examine sulcal patterns within individual brains because the formation of gyri and sulci may have both local and long-range effects on sulcal patterns and functional compartmentalization. The optimal arrangement and positioning of cortical areas and resulting sulcal pattern could be explained by an evolutionary design strategy for the minimization of axonal length (Klyachko and Stevens, 2003). In the experiment of genetic manipulations during embryonic development, to decrease or increase the size of somatosensory and motor areas resulted in significant deficiencies at tactile and motor behaviors. Such findings suggested that areas have an optimal size and position for maximum behavioral performance (Leingartner et al., 2007; O'Leary et al., 2007). Hence, cortical areas might not develop independently, but develop in relation to other functional areas giving rise to specific sulcal patterns. With this point of view, intersulcal distance was measured in the Ono's atlas (Ono et al., 1990). It was performed manually and intersulcal relationships with other measurements such as depth, length or area were not measured. In order to perform a comprehensive and quantitative analysis of sulcal patterns, we need to consider not only the geometric features of sulcal folds but also their patterning and geometric and topological relationships. To achieve this end, we suggest a novel method where the sulcal pattern is represented as a sulcal pit-based graph structure that can be automatically compared using a spectral-based matching algorithm. Sulcal pits are defined as the deepest local regions of sulci, and are thought to be the first cortical folds that occur during radial growth of the cerebral cortex. Although sulcal patterns exhibit various forms in different brains, the sulcal pits show relatively invariant spatial distribution, which may be closely related to functional areas under tight genetic control (Im et al., 2010; Lohmann et al., 2008: Regis et al., 2005). Hence, the sulcal pits may be appropriate and biologically meaningful markers to include in a graph structure. We computed the similarity between graphs by determining the optimal match using the spectral method (Leordeanu and Hebert, 2005) which exploits features of nodes and their relationships. We applied our method to a twin study to investigate the genetic effect on the sulcal patterns from the perspective of our sulcal pit-based graph approach.

Material and methods Participants The study protocol was approved by the relevant Institutional Review Board (Seoul National University, Catholic University of Korea), and written informed consent was obtained from participants. A total of 48 young healthy twin volunteers were recruited, consisting of 14 female and 10 male monozygotic twin pairs, with ages ranging from 18.3 to 24.9 years (mean ± standard deviation: 20.7 ± 1.8 years). Zygosity Blood or hair samples were taken at the date of scanning or cognitive testing. Zygosity was determined by DNA analysis using the 15 highly polymorphic markers: D3S1358, TH01, D21S11, D18S51, PentaE, D5S818, D13S317, D7S820, D16S539, CSF1PO, PentaD, vWA, D18S1179, TPOX, and FGA. MRI acquisition and image processing Contiguous 0.9 mm axial MPRAGE images were acquired with a 1.5 T MR scanner (Magnetom Avanto, Siemens) with TR = 1160 ms; TE = 4.3 ms; flip = 15°; FOV = 224 mm; matrix = 512 × 512; number of slices = 192. Two images were acquired and averaged for each subject. The images were processed to extract cortical surfaces using the FreeSurfer pipeline (Dale et al., 1999; Fischl et al., 1999). This processing includes removal of non-brain tissue (Segonne et al., 2004), Talairach transformation, tissue segmentation, intensity normalization (Sled et al., 1998), tessellation of the gray matter white matter boundary, automated topology correction (Fischl et al., 2001; Segonne et al., 2007), and surface deformation following intensity gradients to optimally place the gray/white matter and gray matter/cerebrospinal fluid boundaries (Dale et al., 1999). Once the cortical models were reconstructed, they were automatically parcellated into anatomical regions based on lobar and gyral/sulcal structure (Desikan et al., 2006; Fischl et al., 2004). We used the left and right lobar regions for sulcal pattern analysis in this study. Extraction of sulcal pits on the cortical surface The sulcal pit is the locally deepest point in a sulcal catchment basin, and it can be identified by using a sulcal depth map on the cortical surface. We used the white matter surface to extract sulcal pits (Im et al., 2010). FreeSurfer computed sulcal depth measures as described previously (Fischl et al., 1999), by integrating the dot product of the movement vector during inflation with the surface normal vector at each vertex. We used a watershed algorithm based on a depth map to extract the sulcal pits on triangular meshes. To prevent over-extraction of the pits, we first reduced noisy depth variations with surface-based diffusion smoothing with a full-width half-maximum value of 10 mm (Chung et al., 2003). Subsequently we performed segment merging in the watershed algorithm using the area of the catchment basin, the distance between the sulcal pits, and the ridge height. If one of the areas of two or more catchment basins was smaller than a threshold (20 mm 2) when they met at a ridge point, the smaller catchment basin below the threshold was merged into the adjacent catchment basin with the deepest pit and its sulcal pit was removed. If the distance between any two pits was less than a 15 mm threshold, the shallower pit was also merged into a deeper one. Finally, merging was executed if the ridge was lower than a threshold of 2.5 mm. We explained methodological procedures in more detail and evaluated that the smoothing and merging thresholds cooperate well in detecting the appropriate sulcal pits in our previous study (Im et al., 2010).

K. Im et al. / NeuroImage 57 (2011) 1077–1086

Graph construction using sulcal pits for representing sulcal pattern We obtained sulcal pits, their corresponding sulcal basins, and ridge points on the white matter surface and constructed a graph to represent a given sulcal pattern. Each sulcal pit corresponds to a node in the graph representation. When sulcal basins meet, the sulcal pits i and k in those basins are connected with an undirected edge ei,k (Fig. 1). We used a 3D position (x, y, z) (mm), sulcal depth d, area of sulcal basin s (cm 2) for sulcal geometry, and the number of connections with 1st neighborhood nodes c (the number of edges) for graph topology as features (feature vector F) on each node i. F ðiÞ = ðxi ; yi ; zi ; di ; si ; ci Þ The weights of edge e of i and k were the 3D Euclidean distance Ed between the two nodes and the depth of the ridge point (Fig. 1).

 F ei;k = Edi;k ; di;k

Similarity measure between graphs using a spectral matching method In order to compare quantitatively different sulcal graph sets, we adapted a spectral matching technique in computer vision to determine the optimal match and compute the similarity value between them (Leordeanu and Hebert, 2005). Given a pair of graphs, P and Q, containing m and n nodes respectively, we estimated their similarity by exploiting not only the features of the nodes themselves but also their relationship. P and Q can be represented as follows: P = fp1 ; p2 ;…; pm g and Q = fq1 ; q2 ;…; qn g: The affinity of an assignment between two nodes, a =(pi, qj), measures how well features of node pi in P match the features of qj in Q. Letting b=(pk, ql) be another distinct assignment, there is an affinity that measures how compatible the two pairs of nodes, (pi, pk) in P and (qj, ql) in Q, are with each other. We constructed a matrix M to store the affinities of candidate assignments and their pairwise affinities as follows:  Mða; bÞ =

the affinity of a; if a = b the affinity of ða; bÞ; otherwise

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M(a, a) is the affinity at the level of an individual assignment a = (pi, qj). M(a, b) describes how well the relationship between two nodes (pi, pk) within P is preserved after putting them in correspondence with the other two nodes of Q, (qj, ql). We set all the affinity values to be nonnegative. As P and Q have m and n nodes, respectively, there are mn assignments and mn * mn pairs of assignments. However, we filtered out the assignments that are unlikely to be correct as suggested in (Leordeanu and Hebert, 2005; Lyu et al., 2010). Since all images and cortical surfaces were processed in a stereotaxic space, an assignment a = (pi, qj) was rejected if the distance between the two feature points pi and qj is greater than a given threshold value of 50 mm. We constructed the M by measuring affinity values based on geometric features and shapes as described by the graph structure. An individual affinity M(a, a) was computed using the features F on nodes pi and qj. For an assignment a = (pi, qj), we defined the displacement vector D(a) as follows:

 DðaÞ = F pi –F qj 
DðaÞ = xa ; ya ; za ; da ; sa ; ca
 = xpi –xqj ; ypi –yqj ; zpi –zqj ; dpi –dqj ; spi –sqj ; cpi –cqj : Let W be the weight vector that gives the importance of every element in D(a), where every element of W is nonnegative. (wx, wy, wz), wd, ws and wc are weights for 3D position, sulcal depth, area of sulcal basin and topology of graph structure, respectively. The weighted norm of D(a), ||D(a)||W was used as an input for calculating the affinity.   DðaÞ 

‖

= W






‖ w x ;w y ;w z ;w d ;w s ;w c ‖ x a

y a

z a

d a

s a

c a

Then, the affinity value was measured using exponential function which shows a maximum value of 1. The individual affinity M(a, a) is

M ða; aÞ = exp

−

X1 σ

X1 = ‖DðaÞ‖W where σ is a user-provided regularization parameter.

Fig. 1. Extraction of sulcal pits and graph construction. The sulcal pits (black or white spheres) and their corresponding sulcal basins (colored patches) are automatically extracted on the white matter surface (left). The inflated surface (right) is provided for better visualization of the sulcal pits. As an example, graph structure is constructed with the sulcal pits which are marked with white spheres. Each sulcal pit corresponds to a node in the graph. When sulcal basins meet, sulcal pits in those basins are connected with an undirected edge.

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In order to measure the pairwise affinity M(a, b), a ≠ b for two distinct assignments, a = (pi, qj) and b = (pk, ql), we define the pairwise displacement vector D(a, b) as follows:






 D a; b = F pi –F pk – F qj –F ql







 D a; b = xpi –xpk – xqj –xql ; ypi –ypk – yqj –yql ; zpi –zpk – zqj –zql ;







 dpi –dpk – dqj –dql ; spi –spk – sqj –sql ; cpi –cpk – cqj –cql :

We added penalty terms, P, to measure the difference in the topology of the two graphs. These terms caused the affinity value in the matrix M to decrease, resulting in decreased similarity. First, we searched paths, V, from pi to pk in graph P (Vpi → k) and chose the shortest paths using the distance weight of edges min(Vqj → l) and from qj to ql in graph Q min(Vqj → l). Then we compared the number of edges (Ne) in the shortest paths to construct an edge number penalty, Pen.






 Pen a; b = Ne min Vpi→k –Ne min Vqj→l ( ) 
min Vpi→k = e ∑ Ede = MIN e∈Vpi→k ( )
 min Vqj→l = e ∑ Ede = MIN

j

j j

j

penalty Pnc was used to decrease the similarity according to the number of nodes which were not finally matched as follows: Pnc = m = Nc + n = Nc –2 where Nc is the number of the consistent assignments and m is the number of nodes in graph P and n is the number of nodes in graph Q. In particular, the more two sulcal graphs are unlike in the number of nodes (most likely due to differences in the underlying sulcal pattern), the larger the decrement of the similarity becomes. This penalty does not have an influence on the matching process, but it does on measuring similarity. The similarity between P and Q graphs AP,Q was computed by ∑ M ða; bÞ

AP;Q =

a;b∈C

N 2c

0

1 X3 −@1− exp σ A −

X3 = wnc Pnc where wnc denotes the weight of the penalty Pnc. Fig. 3 presents the overall process of similarity measure between P and Q graphs and supplementary data (Fig. S1) shows its example using two simple graphs.

e∈Vqj→l

Experiments for choosing weight values The penalties Pen focus on the difference of sulcal arrangement and patterning. However, the variability in the depth of the ridge point also plays a major role in the classification of sulcal patterns (Kim et al., 2008; Ochiai et al., 2004). Shallow ridges, which are superficial gyral banks, can interrupt a sulcal segment. In other cases, sulci can be continuous along their lengths with no interruption. Therefore, we additionally compared the mean values of the depths of the edges between paths to create a depth penalty, Ped.

Ped




  ∑ de  de   e∈∑  min Vpi→k e∈ min Vqj→l    −  a; b =   Ne minðVpi→k ÞÞ Ne min Vqj→l 

j

Examples of Pen and Ped are shown with a schematic diagram in Fig. 2. The pairwise affinity M(a, b) was finally set as follows: X2 − M ða; bÞ = exp σ X2 = ‖Dða; bÞ‖W + wc Pen ða; bÞ + wd Ped ða; bÞ where wc and wd are the weights for graph topology and depth, used for Pen and Ped respectively. As explained in Leordeanu and Hebert (2005), we set M(a, b) = 0 if the two assignments a and b are incompatible (e.g. i = k and j ≠ l). Next, the affinity matrix M was used as described below to choose a subset of consistent assignments, referred to as C, to measure the similarity between the two graphs, P and Q. Based on the Raleigh ratio theorem, subset C was built guided by the principal eigenvector of M while enforcing the one-to-one correspondence constraint for the chosen assignments (Leordeanu and Hebert, 2005; Lyu et al., 2010). Each element of the eigenvector of M gave the confidence of the corresponding assignment between a pair of nodes. Let L be the set of all candidate assignments from graphs, P and Q. We chose the pair (pi, qj) with the maximum confidence from L as a consistent assignment in C. This pair together with all conflicting pairs with this pair was removed from L. Each conflicting pair contained either pi or qj as its element. For the remaining pairs in L, we repeated this process to construct the consistent assignment set C until L became empty. Given the consistent assignment set C between graphs P and Q, their similarity A was defined as the mean of affinity values from C. The last

As evident in our method, the matching and similarity measure between two sulcal graphs depends explicitly on the choice of weights, (wx, wy, wz), wd, ws, wc and wnc. Consequently, the choice of weight values is a strong driver of matching and similarity confidence. We constructed a set of experiments to explore the weight space for reliable matching as follows: 1. Choose a set of known sulcal graphs extracted from a subset of subjects 2. Disrupt the geometric features or graph topology of each known graph - The addition of extra nodes in the graph structure (a) an extra node having a small area and a different position compared to an original node (b) an extra node having a similar position but shallow depth compared to an original node - Spatial transformation of the graph structure (c) random scaling, translation, and rotation of the 3D position in the graph 3. For each disrupted graph, modify weights in such a manner until the original graph and its disrupted version match correctly. (a) As shown in Fig. 4A, we added an extra node r on the edge e between original nodes i and k. The original node i was changed to i′, which maintained the area and position of i ((xi′, yi′, zi′) = (xi, yi, zi), si′ = si). The node r was directly connected to nodes j and k as well as node i′. In order to match the topology of the graph structure, i should be matched with r because matching i with i′ would increase the penalty Pen. However, the area of r was set to be quite small and its position was entirely different from i ((xr, yr, zr) ≠ (xi, yi, zi), sr ≪ si). Hence, matching i and i′ was regarded as a correct match. For a correct match, we increased the weights of the position (wx, wy, wz) and area ws, but decreased the weight of the graph topology wc. (b) In the second experiment, we added an extra node r on the edge e between original nodes i and k. The original node i was changed to i′ and it was spatially translated somewhat, but maintaining the depth and area of the node i ((xi′, yi′, zi′) ≠ (xi, yi, zi), di′ = di, si′ = si). The node r was connected to nodes j

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Fig. 2. Examples with schematic diagrams for the penalties of the number and depth of edges in the shortest path, Pen and Ped.

and k and its position is the same as the position of the node i ((xr, yr, zr) = (xi, yi, zi)). However, the area and depth values of r were set to be quite small (dr ≪ di, sr ≪ si) (Fig. 4B). Practically, a very shallow and small sulcal basin is a noisy fold of the cortical surface. Therefore, matching i and i′ was set for a correct match. For the right match, we increased the weights of the depth wd and area ws, but decreased the weights of the position (wx, wy, wz) and graph topology wc. (c) We deformed the x, y, z position of the nodes in the original graph with random transformation of scaling, translation and rotation (Fig. 4C). The range of the transformation is from 0.9 to 1.1, from − 10 mm to 10 mm, and from −10° to 10° for scaling, translation and rotation respectively. The other geometric features were not changed. Because the 3D position of sulcal pits has a large effect on the graph match, their transformation can cause some mismatches. However the topology of a graph is independent of the transformation. An appropriate weight of graph topology can efficiently prevent probable mismatches. To avoid a mismatch in this case, we increased the weight of graph topology wc, but decreased the weight of position (wx, wy, wz). For these experiments, we randomly picked 10 subjects and constructed original and corrupted sulcal graphs for three cases (a), (b), and (c) in the frontal and temporal lobes. We qualitatively modified the weights for correct matches of them as described above.

Then, the weights were finally tuned by visual observation through many experiments of matching between different brains. We modified the weights in case wrong matching of spatial arrangement, area, and position of sulcal folds were recognized visually. We set wx = 0.025, wy = 0.025, wz = 0.025, wd = 0.8, ws = 0.03, wc = 0.1, and wnc = 0.3. The value of 2 was assigned to the parameter σ in the function for the affinity measure.

Evaluation of sulcal graph matching and similarity measure In order to assess the effectiveness and performance of our sulcal graph matching and similarity measure, we experimented with all 48 subjects comparing the original graph to one where we overextracted sulcal pits and distorted 3D positions of sulcal pits by using altered parameters. As explained above, the area of the catchment basin was used for the segment merging process in the watershed algorithm with a threshold of 20 mm 2 to avoid extraction of noisy sulcal pits. We extracted sulcal pits with the lower thresholds of 15, 10, 5, and 0 mm 2. It resulted in the over-extraction of noisy sulcal pits added to original sulcal pits from the threshold of 20 mm 2. In addition, to distort the 3D position of the sulcal pits, they were randomly transformed with scaling, translation and rotation (Fig. 5). The range of the transformation was from 0.95 to 1.15, from − 5 mm to 5 mm, and from −5° to 5° for scaling, translation and rotation,

Fig. 3. The overall process of similarity measure between P and Q graphs.

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Fig. 4. Artificially altered graphs from an original sulcal graph. The original sulcal graph from real human data is shown in the first row. The other graphs are generated from the original by adding an artificial node r or random spatial transformation. Matching the i with the i′ is preferred.

respectively. We generated the graphs for each case and performed the matching and similarity measure between the original and altered graphs. We further tested our method with two noisy graphs. After extraction of the noisy sulcal pit map, we randomly divided the noisy sulcal pits into two parts and constructed two different graphs, which contained the same original sulcal pits, but had different noisy pits. One of them was spatially transformed as explained above. If surplus noisy pits are not excluded but matched with any node of the original graph, it was considered as the wrong correspondence and a mismatch. We counted the number of mismatches and divided that value by the number of surplus noisy pits to calculate the mismatching rate for each subject according to different noise levels. Our method includes some original features such as the penalty term and the number of edges to reflect the difference in the topology of the two sulcal graphs. To evaluate the behavior of the method with and without the graph topology feature, we set the weight of graph topology to 0 and measured another mismatching rate and similarity in the experiments mentioned above. Similarity analysis of sulcal pattern in twins Sulcal pits were extracted and sulcal graphs were constructed for the left and right hemispheres and lobar regions. To analyze the genetic influence on the sulcal pattern, we measured the similarity of the sulcal patterns in monozygotic twins. We tested whether the similarity between the 24 twin pairs was significantly higher

compared to the similarity of the 24 unrelated pairs randomly taken from the same data pool. The value of the statistic observed in this study was the mean of the similarity. The mean similarity of the twin pairs, mean(Atwins) was compared with the mean similarity from unrelated pairs. The test may not always be true if an observed effect, the difference between two means, occurs simply by chance. We carried out a permutation test based on 10,000 times set of random unrelated pairs. We constructed the permutation distribution of the statistic from this large number of resamples and located the mean (Atwins) on the distribution to get the permutation test P value. The P value was calculated as the proportion of the 10,000 resamples that gave a result at least as great as mean(Atwins). The statistical tests were performed for each hemisphere and lobar region. Once given the optimal match between the nodes in different sulcal graphs, various similarities can be measured by changing the weights to focus on different aspects. For example, to remove the effect of the 3D position, the weight of the 3D position (wx, wy, wz) can be set to 0 when measuring the similarity after matching. In the analysis for twins, we set to 0 or doubled the value for various sets of weights in the similarity measure and performed the same statistical tests for left and right hemispheres. In addition, we evaluated the impact of each individual weight on similarity measures by setting all the other weights to 0. The topology of the sulcal graph and the number of sulcal pits are unrelated to global effects such as overall brain size or normalization to Talairach space. However, the similarity measure using the geometric features of 3D position, depth, or area might be influenced by global effects. In our similarity measurement, we can set all weights to 0 for the individual affinity M(a, a) and compute the pairwise affinity M(a, b). This allows us to observe the effect of intersulcal geometric relationships in isolation and minimize the influence of global factors. We performed this analysis in twins, by measuring the similarity of the intersulcal relationships in the position, depth, or area for the left and right hemispheres and statistically tested whether the twin pairs showed significantly higher mean similarity. The entire set of the weights used in our analysis is shown in Table 1. Results

Fig. 5. Over-extracted sulcal pits and their spatial transformation and matching them with original sulcal pits. Noisy sulcal pits, which are marked with black, are extracted with the low threshold of 10 mm2 for the area of the catchment basin in the merging process. The positions of the sulcal pits from 10 mm2 threshold are randomly deformed. Correct matching is the matching i, j and k with i′, j′ and k′, respectively, excluding noisy sulcal pits.

Sulcal pattern matching and similarity measure The example of optimal sulcal pattern matching and similarity measure in the temporal lobe is shown in Fig. 6. The sulcal basins that

K. Im et al. / NeuroImage 57 (2011) 1077–1086 Table 1 Statistical results for the comparison of the sulcal graph similarity between the twin and unrelated pairs with different weights (left and right hemispheres). Weights

Statistical test (P value)

(wx, wy, wz)

wd

ws

wc

wnc

Left

Right

0 0.025 0.025 0.025 0.025 0.05 0.025 0.025 0.025 0.025 0.025 0 0 0 0

0.8 0 0.8 0.8 0.8 0.8 1.6 0.8 0.8 0.8 0 0.8 0 0 0

0.03 0.03 0 0.03 0.03 0.03 0.03 0.06 0.03 0.03 0 0 0.03 0 0

0.1 0.1 0.1 0 0.1 0.1 0.1 0.1 0.2 0.1 0 0 0 0.1 0

0.3 0.3 0.3 0.3 0 0.3 0.3 0.3 0.3 0.6 0 0 0 0 0.3

0.0001 b 0.0001 b 0.0001 b 0.0001 b 0.0001 b 0.0001 b 0.0001 b 0.0001 b 0.0001 b 0.0001 b 0.0001 0.0029 b 0.0001 0.0002 0.0003

0.0001 b0.0001 b0.0001 0.0002 0.0004 0.0003 0.0004 b0.0001 0.0002 b0.0001 b0.0001 0.0062 0.0029 0.0051 b0.0001

affinity 0 0 0

0 0 0

0.0316 0.1348 0.0425

0.0086 0.0496 0.0083

Similarity measure using only pairwise 0.025 0 0 0 0.8 0 0 0 0.03

(wx, wy, wz), wd, ws, wc, and wnc are weights for 3D position, sulcal depth, area of sulcal basin, topology of graph structure, and the number of sulcal pits respectively.

were paired by matching are marked with the same color. In the pair with a high similarity value (Fig. 6A), the geometric features of the nodes and their relationship and sulcal arrangement showed greater

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similarity, and therefore the sulcal patterns were better matched than the pair with low similarity (Fig. 6B). Surplus sulcal pits of one brain were excluded from matching and these decreased the similarity using the penalty Pnc. More examples are provided in supplementary data (Fig. S2). From the evaluation of all 48 subjects described above, we plotted the mean values of the number of surplus noisy pits and mismatches, the mismatching rate and similarity between the original and altered noisy graphs (Fig. 7A), and between two noisy graphs (Fig. 7B) at different noise levels. The mean number of sulcal pits in the original graph set of 48 subjects was 74.1. As the number of noisy sulcal pits increased (74.1 + 6.7, 13.4, 27.0, and 101.7), the number of mismatches also increased as expected, but relatively low mismatching rates were shown in both experiments. Although mean number of noisy sulcal pits was larger than 100, which were even more than the original sulcal pits, the mismatching rates were about 0.15 at the most. The mean similarity value decreased clearly as noise level increased, showing high sensitivity of detecting the changes of the sulcal pattern. When excluding the weight of graph topology (wc = 0), the mismatching rates were quite similar, but the decrement of the similarity was smaller compared to the original results. Twins vs. unrelated pairs Fig. 8 shows the permutation distribution of the statistic, the mean similarity, from 10,000 resamples for the left and right hemispheres and lobar regions. The solid vertical line in the figure marks the location of the statistic for the twins. Twins had significantly larger similarity measures than unrelated pairs for all regions (permutation

Fig. 6. Optimal sulcal pattern matching and similarity measure in the temporal lobe for 2 pairs. The pair having high similarity 0.7590 (A1 and A2) shows more similar geometric features of nodes and their interrelationship and sulcal arrangement than the pair having low similarity 0.5567 (B1 and B2). Surplus sulcal pits excluded from the matching are marked with black. The sulcal basins paired by matching are marked with the same color.

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Fig. 7. Original sulcal graphs were disrupted to have noisy sulcal pits and spatially deformed. Original and disrupted noisy graphs (A) or two noisy graphs (B) were matched and their similarity was measured. The plots show the mean values of the number of surplus noisy pits and mismatches, the mismatching rate, and similarity for 48 subjects with different noise level (the area of the catchment basin for segment merging: 15, 10, 5, and 0 mm2) and different weights (wc = 0.1, wc = 0).

test P values, both hemispheres: b0.0001, left frontal and temporal lobes: b0.0001, parietal lobe: 0.0036, occipital lobe: 0.0005, right frontal: 0.0002, temporal, parietal and occipital: b0.0001). None of the

set of unrelated pairs produced a mean similarity as large as the mean similarity of the twins in both hemispheres, the left frontal and temporal lobes, and the right temporal, parietal, and occipital lobes

Fig. 8. The permutation distribution of the statistic, the mean similarity, from 10,000 resamples for the left and right hemispheres and lobar regions. The solid vertical line marks the mean similarity for the twins. The similarity of the twins is clearly larger than the similarities of unrelated random pairs for all regions.

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(Fig. 8). In the analysis of the similarity with different weights for both hemispheres, the twins showed significantly larger similarity in all cases. It is interesting to note that the results were also statistically significant when using only one weight effect. The plots showing mean similarities from random pairs and the twins which were computed using one weight are provided in supplementary data (Fig. S3). In addition, the similarity of intersulcal relationships alone was significantly higher in the twins at the 0.05 level of P for the 3D position and area in both hemispheres, and the depth in the right hemisphere. All statistical results are reported in Table 1. Discussion We developed a spectral matching-based method that provides comprehensive and quantitative sulcal pattern analysis and comparisons. This method calculates a similarity measure using not only the geometric features of sulcal pits but also their geometric and topological relationships. In particular, we exploit the number of neighbor sulcal pits and the property of the paths between sulcal pits to highlight the interrelated arrangement and patterning of sulcal folds. This method also allows variable weighting of geometric and topological features, to determine their relative importance on sulcal pattern similarity. We then applied this method to twin and randomized pairs to quantify the relative similarity of sulcal patterns. Sulcal patterns had significantly higher similarity in monozygotic twin pairs compared to unrelated pairs for all hemispheres, all lobar regions and all combinations of feature weighting with spatial position having the greatest importance. Methods other than spectral matching can be used to assess the similarity between two sets of feature points. These include an iterative closest point (ICP) method (Besl and McKay, 1992) or a bipartite graph-matching scheme (Cormen et al., 2001). The ICP method finds correspondences between two sets of feature points assuming that there is a one-to-one correspondence between them. However, since the ICP method iteratively searches for a local optimum based on the individual feature similarity, making characterization of the global shape of sulcal patterns difficult and making this method sensitive to noise and outliers (Chui and Rangarajan, 2003). Similarly, the bipartite graph matching scheme exploits only the similarity between the individual feature points, and thus it is sensitive to noise and outliers, too. Cortical areas do not develop independently, but develop in relation to other functional areas (Leingartner et al., 2007; O'Leary et al., 2007), which might give rise to specific sulcal patterns showing geometric and topological relationships of sulcal folds. Unlike other methods, the strength of our spectral-based method is to use those relationships between feature points for sulcal pattern matching and similarity measure. It can reflect global pattern of cortical sulci and is more robust to noise and spatial deformation. Our method also allows quantification of partial matches between two sets. In our experiments, although there were large number of noisy sulcal pits and spatial deformations, our algorithm showed a high rate of correct matches and reliable similarity measures which can detect changes in the sulcal pattern. Our method can effectively extract similarities between sulcal graphs of different individual brains having different number of sulcal pits by measuring the degree of difference in the sulcal pattern (Fig. 6 and Fig. 2 in supplementary data). Furthermore, as the difference of the sulcal graph structure became larger, our novel feature of the graph topology influenced the similarity value and caused more sensitive measurement of the difference (Fig. 7). In our recent study, we developed a method for automatically defining anatomical labels of sulcal pits using previously labeled data as a training set (Im et al., 2011). We labeled main sulcal pits that were commonly present across subjects and measured their presence for a group statistical analysis. However, in that approach, we were interested in group statistics and therefore sulcal pits that were rare across the group were excluded from the labeling for group analysis.

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Here, in this paper, we are interested in quantifying and differentiating individual sulcal patterns, therefore we focus on individual sulcal patterns and use multivariate information, including all sulcal pits. Future work will be aimed at developing another novel analytical method by combining these two approaches. The similarity of sulcal folding in twins has been studied in a few previous studies. The sulcal morphology of the twin pairs was similar enough to enable the observers to distinguish twin pairs from unrelated subjects (Biondi et al., 1998). However, this study was a qualitative analysis of visual observation with a small number of twin subjects. In another study, the sulcal lines of the voxels of the monozygotic twins were significantly more alike compared to unrelated twin subjects (Lohmann et al., 1999). In this study the degree of similarity was measured as the percentage of matching determined using Euclidean distance. Finally, a third study evaluated one major sulcus, the central sulcus, and showed that twins had a significantly smaller sulcal shape distance than unrelated pairs (Le Goualher et al., 2000). However, unlike these studies, in this paper we present a whole brain quantitative assessment of the sulcal pattern similarity in twins that included geometric and topological features, particularly focusing on the relationship and arrangement of sulcal segments. Moreover, our method makes it possible to investigate the effects of multiple factors on the sulcal pattern with various weight sets. When we emphasized or eliminated the weight of each feature, all results remained statistically significant. As we observed the distributions of the mean similarities from unrelated pairs and twins generated with only one weight (Fig. 3 in supplementary data), the effect of the 3D position of the sulcal pits appeared to be the strongest determinant of the similarity in twins. This result supports the hypothesis that the spatial distribution of sulcal pits might be under tight genetic control (Im et al., 2010; Lohmann et al., 2008). For the 3D position, depth and area, the similarity of their intersulcal relationships alone was significantly higher in the twins. This suggests that there may be a genetic influence on the intrinsic sulcal patterns that is less affected by global factors such as overall brain size and shape. Although we excluded all geometric features such as the position, depth, and area in the similarity measure, the topology of the sulcal graph and the number of sulcal pits were also more similar in the twins. It is worth noting that there is a significant twin-related influence on the arrangement of sulcal folds regardless of their geometric information. Although we cannot calculate heritability estimates, our results support a genetic impact on the global sulcal pattern. Future studies will include dizygotic twins to examine the relative contribution of genetic and environmental influences on the sulcal pattern. Future studies will also explore the role of this graph-based sulcal pattern analysis in detecting the severity and extent of cerebral malformations. Such regional quantitative measures of similarity could also support more detailed analysis of the associations between genetic disorders and sulcal structure and may provide a quantitative and reliable means to screen MR images for regional abnormalities. Acknowledgements This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2009-352-D00344). Appendix A. Supplementary Data Supplementary data to this article can be found online at doi:10.1016/ j.neuroimage.2011.04.062. References Armstrong, E., Schleicher, A., Omran, H., Curtis, M., Zilles, K., 1995. The ontogeny of human gyrification. Cereb. Cortex 5, 56–63.

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