An Algebraic Topological Method for Multimodal Brain Networks Comparisons
Tiago Simas, Mario Chavez, Pablo Rodriguez, Albert Diaz-Guilera
AAn Algebraic Topological Method forMultimodal Brain Networks Comparisons
Tiago Simas ∗ , Mario Chavez , Pablo Rodriguez , andAlbert Diaz-Guilera Department of Psychiatry, University of Cambridge,Cambridge, UK Telefonica I+D, Barcelona, Spain CNRS-UMR-7225, Hˆopital Piti´e Salpˆetri`ere, Paris, France Departament de Fisica Fonamental, Universitat de Barcelona,Barcelona, SpainApril 13, 2015
Abstract
Understanding brain connectivity has become one of the most im-portant issues in neuroscience. But connectivity data can reflect eitherthe functional relationships of the brain activities or the anatomicalproperties between brain areas. Although one should expect a clearrelationship between both representations it is not straightforward.Here we present a formalism that allows for the comparison of struc-tural (DTI) and functional (fMRI) networks by embedding both in acommon metric space. In this metric space one can then find for whichregions the two networks are significantly different. Our methodologycan be used not only to compare multimodal networks but also to ex-tract statistically significant aggregated networks of a set of subjects.Actually, we use this procedure to aggregate a set of functional (fMRI) ∗ [email protected] a r X i v : . [ q - b i o . N C ] A p r etworks from different subjects in an aggregated network that is com-pared with the anatomical (DTI) connectivity. The comparison of theaggregated network reveals some features that are not observed whenthe comparison is done with the classical averaged network. In the last decade, the use of advanced tools deriving from neuroimagingand complex networks theory have significantly improved our understandingof brain functioning [24]. Notably, connectivity-based methods have had aprominent role in characterising normal brain organisation as well as alter-ations due to various brain disorders [1, 3, 5]. Most of the recent works aimto quantify the role of connectivity in the communication abilities of neuralsystems. However, the very same notion of connectivity is controversial sincedata used in brain connectivity studies can reflect functional neural activi-ties (electrical, magnetic or hemodynamic/metabolic) or anatomical proper-ties [1, 4]. Neuroanatomical connectivity is meant as the description of thephysical connections (axonal projections) between two brain sites [4], whereasfunctional connectivity is defined as the estimated temporal correlation be-tween spatially distant neurophysiological activities such as electroencephalo-graphic (EEG), magnetoencephalographic (MEG) functional magnetic reso-nance imaging (fMRI) or positron emission tomography (PET) recordings [1].In recent years, the concept of “brain networks” is becoming fundamentalin neuroscience [2, 4, 3, 5]. Within this framework, nodes stand for differentbrain regions (e.g. parcelated areas or recording sites) and links indicate thepresence of an anatomical path between those regions, or a functional depen-dence between their activities. In the last years, this representation of thebrain has allowed to visualize and describe its non-trivial topological prop-erties in a compact and objective way. Nowadays, the use of network-basedanalysis in neuroscience has become essential to quantify brain dysfunctionsin terms of aberrant reconfiguration of functional brain networks [2, 3, 5].Experimental evidence has revealed, for instance, alterations in functionaland anatomical brain networks in normal cognitive processes, across develop-ment, and in a wide range of neurological diseases (see [4, 5] and referencestherein). Despite its evident interplay, comparison of anatomo-functionalbrain networks is not straightforward [6, 7]. Theoretical studies providesupport for the idea that structural networks determine some aspects of2unctional networks [6], but it is less clear how the anatomical connectiv-ity supports or facilitates the emergence of functional networks. Althoughnodes with similar connection patterns tend to exhibit similar functionality,the functionality of an individual neural node is strongly determined by thepattern of its interconnections with the rest of the network [7].Correspondences between functional and structural networks remains thusan active research area [8, 9, 10]. A better understanding of how anatomicalscaffolds support functional communication of brain activities is necessary tobetter understand normal neural processes, as well as to improve identifica-tion and prediction of alterations in brain diseases.In this paper we address this relationship between anatomical and func-tional connectivity. In previous studies, the correspondence of these net-works has been often assessed by the difference in an Euclidean space ofvectors containing connectivity measures such as the clustering coefficient,shortest path length, degree distribution, etc. Here, we propose a radicallydifferent framework for studying brain connectivity differences. Instead ofextracting a vector of features for each network (anatomical or functional),we jointly embed all of them in a common metric space that allow straight-forward comparisons. Before embed functional and anatomical networks intothe common metric space, we aggregate group of subjects (e.g. functionalnetworks) according to [19] to obtain a group representation network. There-fore the method employed in this work allows to preserve in the aggregationconnected components and to identify among different subjects, a commonunderlying network structure. Our approach may provide a useful insight forthe analysis of multiple networks obtained from multiple brain modalities orgroups (healthy volunteers versus patients, for instance).
In this study we considered anatomical and functional brain connectivities(extracted from diffusion-weighted DW-MRI and fMRI data, respectively)defined on the same brain regions. Brain images were partitioned into the 90anatomical regions ( N = 90 nodes of the networks) of the Tzourio-Mazoyerbrain atlas [12] using the automated anatomical labeling method.The anatomical connectivity network is based on the connectivity matrix3btained by Diffusion Magnetic Resonance Imaging (DW-MRI) data from 20healthy participants, as described in [11]. The elements of this matrix repre-sent the probabilities of connection between the 90 brain regions of interest.These probabilities are proportional to the density of axonal fibers betweendifferent areas, so each element of the matrix represents an approximation ofthe connection strength between the corresponding pair of brain regions.The functional brain connectivity was extracted from BOLD fMRI restingstate recordings obtained as described in [13]. All acquired brain volumeswere corrected for motion and differences in slice acquisition times usingthe SPM5 software package. All fMRI data sets (segments of 5 minutesrecorded from healthy subjects) were co-registered to the anatomical dataset and normalized to the standard MNI (Montreal Neurological Institute)template image, to allow comparisons between subjects. As for DW-MRIdata, normalized and corrected functional scans were sub-sampled to theanatomical labeled template of the human brain [12]. Regional time serieswere estimated for each individual by averaging the fMRI time series over allvoxels in each of the 90 regions. To eliminate low frequency noise (e.g. slowscanner drifts) and higher frequency artifacts from cardiac and respiratoryoscillations, time-series were digitally filtered with a finite impulse response(FIR) filter with zero-phase distortion (bandwidth 0 . − . x i ( t ) and x j ( t ) (normalizedto zero mean and unit variance) was defined by means of the linear cross-correlation coefficient computed as r ij = (cid:104) x i ( t ) x j ( t ) (cid:105) , where (cid:104)·(cid:105) denotes thetemporal average. For the sake of simplicity, we only considered here cor-relations at zero lag. To determine the probability that correlation valuesare significantly higher than what is expected from independent time series, r ij (0) values (denoted r ij ) were firstly variance-stabilized by applying theFisher’s Z transform Z ij = 0 . (cid:18) r ij − r ij (cid:19) (1)Under the hypothesis of independence, Z ij has a normal distribution withexpected value 0 and variance 1 / ( df ij − df is the effective numberof degrees of freedom [14, 15, 16]. If the time series consist of independentmeasurements, df ij simply equals the sample size, N . Nevertheless, autocor-related time series do not meet the assumption of independence required bythe standard significance test, yielding a greater Type I error [14, 15, 16].
4n presence of auto-correlated time series df must be corrected by the fol-lowing approximation df ij ≈ N + N (cid:80) τ N − τN r ii ( τ ) r jj ( τ ), where r xx ( τ ) is theautocorrelation of signal x at lag τ . From Eq. (1) our networks weights are in a non-normalised interval Z ij ∈ [ a, b ] ⊂ R . In order to apply the framework described in [18], we normaliseour networks weights into the unit interval I = [0 ,
1] by means of a uniquelinear function: w ij = (1 − (cid:15) ) Z ij + (2 (cid:15) − · M IN ( Z ij ) M AX ( Z ij ) − M IN ( Z ij ) + (cid:15) (2)where (cid:15) in general is set to 0 .
01 in order to avoid merging and isolate verticeswith weights at the boundaries of Z ij ∈ [ a, b ]. As proved in [18], since thenormalisation is done by a unique linear function this does not affect networksproperties. Among many ways to aggregate a group of networks here we employed atopological algebraic way to aggregate a group of networks. The networksgroup possess the same nodes but different edges values and can mathemat-ical be represented by a weighted graph G = ( N, E ). N is the set of nodesrepresenting the brain ROI’s ( N = 90 in this study) and E is the set ofedges values (connections) between ROI’s, e.g. ∀ e i,j ∈ E : e i,j ∈ [0 ,
1] in theproximity space or ∀ d i,j ∈ E : d i,j ∈ [0 , + ∞ ] in the distance space.For mathematical notation simplicity, we denote a network with the samenotation we use to the set of nodes N . That is, a set of n networks (e.g. groupof subjects) is represented by N k with k ∈ { , , . . . . , n } .One possible way to aggregate a group of n networks is simply by aver-aging the homologous edges values. Obtaining in this way a group represen-tative network, N ∗ . N ∗ i,j = e ∗ i,j = (cid:80) nk =1 e [ k ] i,j n (3)where e [ k ] i,j is the edge e i,j from network N k .5nother way to aggregate networks as explained in Simas et. al. [19], isby considering all networks as a multilayer network (often called multiplex ),which can be represented as fourth-order tensor [19]. This tensor can berepresented as a extended matrix [21]. The work of Simas and Rocha [18],introduces a framework to aggregate networks in an algebraic way, relating itwith fuzzy logic reasoning, and in [19] this work was extended for multilayernetworks. In order to work algebraically with networks we have to set analgebra (defined as a vector space equipped with a bilinear product). Thisalgebra allows us to perform algebraic operations with networks in the sameway we perform algebraic operations in other contexts with other algebras(such as adding and multiplying real numbers). In short a network can berepresented by an adjacency matrix and a multilayer network by a tensor.Considering a set of tensors working under the algebra L = ( I, ⊕ , ⊗ ), wherethe weights (tensor entries) of the tensors in I ⊆ ¯ R (subset of extendedreal line ) and ⊕ and ⊗ two binary operators we may represent a multilayernetwork with tensor T in this algebra. In Simas et al. [19] we have seen for theparticular case of multiplex networks where layers are connected with weights w i,i,L k ,L j = 1 (in the proximity space), the representative group network (e.g.functional) can be represented by N ∗ in the distance space (see below andEq. 6), as: N ∗ ≡ N ⊕ N ⊕ · · · ⊕ N k (4)and the respective embedding by the following equation: N embedded = N ∗ ⊕ N ∗ ⊕ · · · ⊕ N ∗ r (5)where N ∗ is defined in Eq. (4) and r , is the convergence parameter [18, 19].Figure 1 summarises the metric embedding of a multiplex network describedabove.Embedding a network of networks or, in our specific case, a a multiplexfMRI network, allows us to determine which edges in the several layers con-tribute to the aggregation. We can therefore determine the subjects thatcontribute more/less or none to the aggregated network, and identify in eachsubject the sub-graphs for which they contribute more.For our particular case, we embed our networks using the Metric Closure[18] defined by the algebra L = ([0 , + ∞ ] , min, +), where ⊕ = min and ⊗ = +. The metric closure or metric embedding of a given network into ametric space, is a generalisation of All Pairs Shortest Paths Problem (APSP)6igure 1: Schematic representation of the main steps for the described net-works aggregation and metric embedding (defined here for the algebra L ).as shown in Ref. [18], e.g. Johnson Algorithm can be used to calculate themetric closure [20].Note that to calculate the metric closure (or Johnson algorithm) of anetwork we have to translate our networks from a proximity space into adistance space. There are many possible mappings to map a similarity spaceinto a distance space, see [18]. Applying Eq. (6) to all network weights, w i,j ∈ [0 ,
1] (for more details see [18]), we obtain the isomorphic distancenetwork with weights d i,j ∈ [0 , + ∞ ]. d i,j = 1 w i,j − In general networks have been compared using statistical measures of lo-cal and global properties of networks, such as: clustering coefficient, small-worldness, degree distributions, etc. We can find in the literature some exam-ples of such techniques to compare multiple networks [4, 5]. Our approachin this work is different. After embedding networks into the same metricspace defined by the applied algebra, in our case L = ([0 , + ∞ ] , min, +), weare able to compared them topologically. However, since networks generallycome from different modalities (e.g. fMRI and DTI) it requires a previousstep. We need to normalise the embedded edge weights distributions fromthe different modalities to the same average and variance to remove scalefactors. One possible way to normalise both distributions, if we assume nor-mality, is by calculating the z-score of the edge weights distributions (zeroaverage and standard deviation set to the unit).The embedded networks represent a hyper-grid in a multi-dimensionalspace with dimension equal or below to the number of nodes. In order tosimplify and have some visual insight we can downgrade linearly this multidi-mensional grid into a 3D grid. This can be achieved applying to the embed-ded networks any technique for dimensionality reduction such as linear/non-linear Multi-Dimensional Scaling (MDS). MDS procedures refer to a set ofrelated ordination techniques used in information visualisation, in particularto display the information contained in a distance matrix [23]. These tech-niques guarantee, with a given distortion, that the relative distance betweennodes is preserved in both multi-dimensional and low-dimensional reductionspace. Plotting this low-dimensional grid (e.g. in 3D) we can use any statis-tical technique to fit a continuous surface into the data (see below Figs. (3,4)). Its is natural to think that the difference between two surfaces obtainedfrom different networks will emphasise topologically differences between thetwo connectivities. In this work we performed this operation in the multi-8imensional space by subtracting homologous embedded edges weights andtake the absolute value of both embedded hyper-grids. This is, we subtracthomologous embedded edges pairwise according to the formula: M = | M fMRI ∗ − M DT I | ≡{ e diffi,j : ∀ e ∗ i,j ∈ M fMRI ∗ ∧ ∀ e ∗∗ i,j ∈ M DT I : e diffi,j = | e ∗ i,j − e ∗∗ i,j |} (7) M is the difference grid in the multi-dimensional space. Because the M -gridrepresents the difference between the two grids from different modalities (seeabove), the relative distance between nodes in M (given by Eq. (7)) shouldbe concentrated at the origin if they are topological similar, otherwise widelydistributed in the multi-dimensional space. Nodes at a distance from theorigin of k × σ are statistical different of k standard deviations. Moreover,since we z-scored both embedded edge distributions this give us some degreeof statistical significance when we compare both networks. All nodes thatrelay outside of a hyper-sphere with centre at the origin with radius R = k × σ ,are statistical different. Here we had set σ = 1 for both distributions (z-scorevariables are estimated from the distributions of the embedded weights).Figure 2 illustrates this process. After applying to both fMRI ∗ and DTInetworks the same algebra and the metric embedding described above, bothnetworks rely on the same metric space, therefore comparable. Topologicaldifferences can be visually seen in a linearly downgraded to 3D dimensionsusing a multi-dimensional scaling technique, which preserves the relative dis-tance between points in the grid (nodes or brain areas). In figure 3 we illustrate the results of different aggregation procedures onthe ensemble of fMRI networks. Compared with a fMRI connectivity matrixfrom a single subject (Fig. 3(A)), one can notice the difference of a single av-eraging across subjects (Fig. 3(B)) and our proposed algebraic topologicallyaggregated connectivity network (Fig. 3(C)). It is clear that the averagingprocedure tends to blur connectivity values between nodes. In contrast, thetopologically algebraic aggregation can preserve components that are com-mon across subjects. As other multilinear algebra or tensor-based analysis,our approach provide a natural mathematical framework for studying con-nectivity data with multidimensional structure. For illustrative purposes, we9igure 2: Topological algebraic networks comparison. Connectivity fromdifferent modalities (here fMRI and DTI) are firstly embedded (black dotpoints on the manifolds indicate the brain nodes) and then compared in alow-dimensional space. Black points outside the sphere correspond to nodeswith a topological difference (at a given threshold) in the two modalities.also show the DTI connectivity matrix in figure 3(D)). It worths noticingthe similarity of the anatomical connectivity structure with the aggregated(multiplex) connectivity obtained in figure 3(C). Moreover, since each layerencodes the functional network for a given subject, each subject contributesto the tensor aggregation/embedding with some or none connections (edges),as depicted in metric closure, Eq. (5). If a layer do not contribute for theaggregation/embedding, we may consider this layer (subject network) as anoutlier. Moreover, we are also able to identify the specific sub-network con-10igure 3: (A) fMRI single subject network (B) Average aggregated fMRInetwork (C) fMRI Algebraic Topologically aggregated (multiplex) network(D) DTI network.tribution (edges) of a given layer to the aggregation/embedding.Low-dimensional embeddings of different aggregated networks are illus-trated in figure 4. High-dimensional data, such as the information containedin the distance matrix obtained for the different networks, can be difficultto interpret. Here, multidimensional scaling (MDS) was used for visualisingthe level of similarity of individual nodes of each -aggregated- network. TheMDS algorithm aims to place each node in a low dimensional space such thatthe between-nodes distances are preserved as well as possible. This repre-sentation into a low-dimensional space enables an exploratory analysis andmakes data analysis algorithms more efficient. Indeed, from the differentplots of figure 4 one can identify brain areas that are topologically close inthe aggregated network as those points that are close on the 3D grid. Thisis clearly illustrated by the MDS representation of the multiplex functionalnetwork (Figure 4 (C)). Nodes from the occipital regions form a compactgroup of nodes topologically close (with similar connectivity structure), asrevealed by the blue points depicted on Figure 4 (C). We also notice that acompact group of nodes is formed by regions of the temporal lobe, putamenand insula, which are indicated by the red circle. Similarly, the anatomicalnetwork in Figure 4 (D) clearly displays a natural organisation, i.e. nodes ofthe two hemispheres lie on both sides of the dotted black line. Further, nodesfrom occipital regions in the anatomical network, indicated by the blue cir-cles (including calcarina, cuneus, precuneus, . . . ), are distantly located fromthe group of frontal brain areas indicated by the red marks.Finally, figure 5 displays the difference grid M in a low-dimensional space.11igure 4: Multi-Dimentional Scaling (MDS) of the embedded networks (a)fMRI single subject (b) fMRI average embedded network (c) fMRI AlgebraicTopological aggregation (multiplex) embedded network (d) DTI embeddednetwork. Black dots indicate the embedded nodes. In plots (C-D), blue andred points indicate the groups of brain areas discussed in the text.As defined in Eq.( 7), M corresponds to the relative distance between nodesin networks from different modalities. Differences between brain areas arerepresented as points widely distributed in the low-dimensional space. Thosenodes from different modalities (fMRI and DTI) that share an identical topo-logical structure are located at the origin. The larger the difference in theconnectivity structure, the larger the distance from the origin. By settinga threshold k × σ , one can identify brain areas with similar connectivity asthose points that lie inside of the hyper-sphere of radius k × σ with centreat the origin.The number of brain regions (ROIs) with similar anatomical and func-12igure 5: Comparisons between DTI and all other embedded fMRI networks.(A) 3D projections from Eq. (7). Only points outside the sphere are plotted.(B) Number of ROI’s inside the sphere of radius of k × σ . Results from asingle subject, average connectivity and multiplex networks are representedby the red, blue and green points and curves, respectively. We consider theregions statistically different for k > k < k × σ . Curves correspond to the number of regions inside a hyper-sphereof various radius. We notice that the number of regions differ as a functionof the aggregated network’s type. It is worthy to mention that the differ-ences above k -standard deviations are the important ones, since is above thisthreshold that the ROI’s or nodes become statistical different when comparenetworks. In our example, the fluctuations below one standard deviationsmay give us some trend but all nodes in the networks are statistical equalfor all types of aggregation. For our specific case, as an example, the brainareas located outside the hyper-sphere of radius k = 1 . The recent prevalence of applications involving multidimensional and multi-modal brain data has increased the demand for technical developments in the13 veraged aggregated network
Calcarine L Lingual L Occipital Sup LCalcarine R Lingual R Occipital Sup RCuneus L Temporal Sup L Occipital Mid LCuneus R Temporal Sup R Occipital Mid ROccipital Inf Insula R
Multiplex aggregated network
Cingulum Post R Occipital Mid R Thalamus LAmygdala R Occipital Inf L Heschl RPostcentral R Occipital Inf R Temporal Sup R
Table 1: ROIs with connectivity differences from DTI at 1.2 standard devi-ationanalysis of such complex data. Indeed, the discrepancy between structuraland functional brain connectivity is a current challenge for understandinggeneral brain functioning. In this paper, we presented a method for charac-terising the correspondences between functional and anatomical connectivity.To summarise, the main steps of our method are:1. Metric network embedding: This procedure embed a group of con-nectivity graphs in a common space allowing straightforward compar-isons. In contrast with simple averaging of connectivity matrices, thetopologically algebraic aggregation can preserve components that arecommon across different subjects or different neuroimaging modalities.This tensor-based aggregation allows enhancing the common underly-ing structures providing a natural mathematical framework for study-ing connectivity data with multidimensional structure.2. Multimodal Networks comparison: the differences between the embed-ded networks are calculated and represented in low-dimensional space.Multi-Dimensional Scaling simply enables to display the informationcontained in the resulting distance matrix allowing thus an exploratoryanalysis of the data.3. Detection of nodes (ROI’s) with different connectivities: from pointswidely distributed in the low-dimensional space one can detect brainnodes that share a similar topological structure as those points are14ocated close to the the origin. One can identify brain areas with thelargest difference between anatomical and functional connectivity asthose points located outside an imaginary hyper-sphere of a radiusgiven by a threshold (Table 1)Our findings suggest that embedding a brain network on a metric spacemay reveal regions that are members of large areas or subsystems rather thanregions with a specific role in information processing. This is clearly illus-trated for the anatomical network in figure 5 D, where frontal and occipitalbrain areas of both hemispheres are situated at distantly and located pointsof the space. Contrary to a classical averaging of connectivity matrices, theembedding of the multiplex functional network reveals brain areas that playa role in large brain system such as the occipital regions, known to be activewhen the subject is at wakeful rest.Although experimental evidence suggests that functionally linked brainregions have an underlying structural core, this relationship does not exhibita simple one-to-one mapping [25]. These correspondences have also beeninvestigated in specific subsystems, must of them focused on the defaultmode network (DMN), which is a group of brain regions that preferentiallyactivate when individuals engage in internal tasks, i.e. when the subject isnot focused on the outside world but the brain is at wakeful rest. Severalstudies report that the DMN exhibits a high overlap in its structural andfunctional connectivity [25, 9]. Nevertheless, strong discrepancies have beenreported and strong functional links can be found between regions withoutdirect structural linkages [9].At a group level, one of the reasons for this discrepancy between struc-tural and functional connectivity has been suggested to be the functionalvariability across subjects [26, 9, 25]. Indeed, clinical studies have providedevidence for a large heterogeneity of the functional connectivity, particularlyin groups of patients with brain disorders such as neuropsychiatric disorders,which strongly alters the structural-functional relationships [25]. Analyticaltools are therefore required to account for this variability in order to enhancethe common underlying network structure.Results suggest that averaged aggregation captures the general differ-ences in regions that play a role in visual, auditory and body self-awarenessprocesses, but fails to identify in detail other specific areas across the sub-jects/groups. In table 1 we observed that the average aggregation essentiallycaptures part of visual (calcarine, cuneus, lingual, occipital), auditory (supe-15ior temporal gyrus), and insula regions that are associated to visual processand body self-awareness. We expected in average that these regions to behighly activated during the scans. This is alined with the fact that rs-fMRIwas acquired with closed eyes and the subjects have some auditory and bodyawareness.From the multiplex aggregation (or algebraic aggregation) shown in table1, we observed that besides capturing the well known visual (occipital areas),primary sensory cortex (postcentral) and auditory regions (Heschl gyrus, su-perior temporal, thalamus), this approach also captures some other networksub-structures involved in touch activation (postcentral gyrus, thalamus) andemotional state activations (amygdala, thalamus, posterior cingulate). Thisalines with our claim that algebraic aggregation preserves better the multi-layer sub-structures across a group of subjects (multilayers) accounting foras much of the variability in the data as possible.Although we cannot definitively a one-to-one mapping of the structuraland functional connectivity, we think that our method could provide newinsights on the organisation of brain networks during diverse cognitive orpathological states. We therefore hope that our approach will foster moreprincipled and successful analysis of multimodal brain connectivity datasets.For all the methods described in this article we provide the correspondingMATLAB software code. Data and code are freely available at the website https://sites.google.com/site/fr2eborn/download
Acknowledgements
The authors thank Y. Iturria-Medina and L. Melie-Garc´ıa for sharing theDTI connectivity data used in the study. This work was supported by theLASAGNE (Contract No.318132) and MULTIPLEX (Contract No.317532)EU projects. A.D.-G. acknowledges support from Generalitat de Catalunya(2014SGR608) and Spanish MINECO (FIS2012-38266).
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