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Coclustering Based Parcellation of Human Brain Cortex Using Diffusion Tensor MRI

Coclustering Based Parcellation of Human Brain Cortex Using Diffusion Tensor MRI

Cui Lin1,Shiyong Lu1,Danqing Wu1,Jing Hua1,and Otto Muzik2

1Department of Computer Science,Wayne State University

{cuilin,shiyong,dqwu,jinghua}@http://www.wendangku.net/doc/1ab371df5022aaea998f0f03.html 2PET Center at Children’s Hospital of Michigan,Radiology at Wayne State University

otto@http://www.wendangku.net/doc/1ab371df5022aaea998f0f03.html

Abstract.The fundamental goal of computational neuroscience is to discover

anatomical features that re?ect the functional organization of the brain.Investiga-

tions of the physical connections between neuronal structures and measurements

of brain activity in vivo have given rise to the concepts of anatomical and func-

tional connectivity,which have been useful for our understanding of brain mech-

anisms and their plasticity.However,at present there is no generally accepted

computational framework for the quantitative assessment of cortical connectiv-

ity.In this paper,we present accurate analytical and modeling tools that can re-

veal anatomical connectivity pattern and facilitate the interpretation of high-level

knowledge regarding brain functions are strongly demanded.We also present

a coclustering algorithm,called Business model based Coclustering Algorithm

(BCA),which allows an automated and reproducible assessment of the connec-

tivity pattern between different cortical areas based on Diffusion Tensor Imaging

(DTI)data.The proposed BCA algorithm not only partitions the cortical mantel

into well-de?ned clusters,but at the same time maximizes the connection strength

between these clusters.Moreover,the BCA algorithm is computationally robust

and allows both outlier detection as well as operator-independent determination

of the number of clusters.We applied the BCA algorithm to human DTI datasets

and show good performance in detecting anatomical connectivity patterns in the

human brain.

1Introduction

With ever-improving imaging technologies,the complexity and scale of brain imaging data has continued to grow at an explosive pace.Recent advances in imaging tech-nologies,especially that of Diffusion Tensor Imaging[1–3],have allowed an increased understanding of normal and abnormal brain structure and function[4,3].It is well understood that normal brain function is dependent on the interactions between spe-cialized functional areas of the brain which process information within local and global networks.Perhaps the most promising approach to parcelate the cerebral cortex into such distinct functional areas originates from the notion that functionally discrete ar-eas of the cortical mantel are characterized by cortico-cortical connectivity patterns, which represent functionally integrated neural subsystems and determine the region’s functional properties[5]and also allow their anatomical delineation and mapping.

At present,no generally accepted parcellation scheme exists for the human cortex, although circumstantial evidence points to a distinct arrangement of functional terri-tories within the cortex.As illustrated in Figure1,most cortical voxels in one region are strongly connected to a particular region of the cortex and the connections to any other regions are relatively weaker.For example,most voxels on the top of the cortex congregating in cortical region C2are connected to voxels of cortical region C1,with only few connections to other cortical regions.Therefore,in order to perform an accu-rate in-vivo analysis of the cortico-cortical connectivity,what is needed is a partitioning procedure that not only simultaneously partitions voxels into groups,but also identi?es the corresponding strong connectivities between the two classes of groups.

Traditional clustering algorithms[6–9]are suboptimal in incorporating anatomical constraints and as a result will fail to identify accurately the corresponding connectiv-ity between cortical regions.Moreover,our focus in this paper is to assess the neural connections within a hemisphere(intra-hemispheric connections)sine these connec-tions are relatively weak compared to the connections between the left and right hemi-sphere,but represent crucial neural pathways which are abnormal in neurological dis-ease.Consequently,we consider only clusters which connect cortical areas within one hemisphere.

Coclustering Based Parcellation of Human Brain Cortex Using Diffusion Tensor MRI

Fig.1.The coclustering process

The main contributions of this paper are:

1.We are the?rst to propose a new coclustering model for de?ning cortico-cortical

connectivity analysis as a computational problem.

2.Our BCA coclustering algorithm is able to de?ne functional cortical areas based on

cortico-cortical?ber tract connections taking into account anatomical constraints.

In contrast to traditional clustering paradigms,the BCA algorithm is not only able to partition image voxels within the cortical mantel into well-de?ned clusters,but also is able to maximize the connectivity strength between such clusters.Moreover, the BCA method is able to identify outliers as well as the number of cortical clusters with high ef?ciency.

3.The application of the BCA algorithm to human DTI dataset allows automated and

reproducible assessment of the connectivity patterns in the human brain.

Organization.The rest of the paper is organized as follows:Section3formalizes the coclustering model for the cortico-cortical connectivity analysis.Section4proposes our coclustering algorithm,BCA,in order to assess?ber tract connectivity between remote cortical areas.Section5presents3-D visualization of the obtained results followed by a discussion of the application in patient groups.Finally,Section6concludes the paper and comments on future work.

2Background and Related Work

The cerebral cortex sends connections(efferents)and receives connections(afferents) from many subcortical structures,but the largest part of the connections arriving at the cerebral cortex comes from the cerebral cortex itself.Assessing connectivity patterns of cortico-cortical?ber tracts is important for our understanding of the mechanisms involved in human brain functions and might provide clues towards the identi?cation and characterization of many neurological diseases.

Recently,Diffusion Tensor Imaging(DTI)tractography has been shown to produce results that are consistent with known pathways formed by major white matter?ber tracts in the human brain[10,2],although limitations in data acquisition and process-ing algorithms[11]related to clinical constraints produce data which cannot resolve crossing or intersecting?bers.DTI is based upon the ability of MRI to evaluate in vivo the direction and magnitude of water diffusion in tissues[2].These attributes of in vivo water diffusion depend upon microscopic tissue architecture[12].Therefore, changes in these parameters serve as markers for changes in tissue micro-architecture. The principal eigenvector obtained from DTI provides information about the prefer-ential direction of water diffusion in each imaging voxel.This direction corresponds to the direction of the nerve?ber bundles,which predominantly constitute the given voxel. Hence,different nerve?ber bundles can be identi?ed and used to assess the integrity of white matter tracts throughout the brain[13,12].Despite some success in delineating functional cortical areas using DTI,a systematic framework allowing functional par-cellation of the neocortex based on quantitative assessment of?ber tract connectivity has not yet been produced,and the relationship among cortical territories,?ber tracts, and neuronal connections remains controversial.Consequently,there is a need to fur-ther develop advanced clustering algorithms that allow better characterization of brain connectivity patterns and as a result improve our understanding of process interactions in a complex biological system.

Traditional partitioning relocation clustering algorithms,such as the K-means[6], K-medoids[7]are simple and ef?cient,however,their?nal results may be overly sen-sitive to the initial cluster set and the presence of outliers.In addition,it is dif?cult to implement when no information exists about the likely cluster number.Hierarchical clustering algorithms[8,14]do not require the number of clusters K as input,but they require a termination condition.In addition,they do not support reclassi?cation of ob-jects to new clusters.Density-based algorithms[9,15,16]have good performance with respect to noise handling and one-scan ef?ciency,but are suboptimal for the cortico-cortical problem,as they do not consider the connectivity strength between clusters,

hence fail to identify accurately the corresponding strongest connectivity between cor-tical regions.

Even though our?rst coclustering algorithm GCA[17]was effective in the analy-sis of thalamo-cortical connectivity,it is not directly applicable to the cortico-cortical connectivity problem as each?ber connects to two different cortical voxels.Direct ap-plication of the GCA algorithm to cortico-cortical connectivity analysis might lead to the following undesirable results:(1)the same voxel can be classi?ed simultaneously into several clusters,(2)two end voxels of a?ber tract might be classi?ed into the same cluster,and(3)two partitions of voxels given by GCA might be inconsistent and the resolution of this inconsistency is not obvious.

3The Coclustering Model

In this section,we present our coclustering model,which models the cortico-cortical connectivity problem.The structure of the cerebral cortex and its cortical connections is initialized as a graph G(V,F),as illustrated in Figure1,where V represents all cortical voxels,and F represents all of the cortico-cortical connections between each other.

De?nition1(Outlier).Given a partition C of G(V,F),C={C1,C2,···,C K},an outlier is de?ned as:

o={v|v∈V,?C i∈C,v∈C i}

De?nition2(Connection strengthθ(C i,C j)).Given a cortical cluster C i and C j,the connection strength between C i and C j is de?ned as:

N ij

θ(C i,C j)=

i

where N ij equals to the total number of connections between C i and C j.

De?nition3(Spouse cluster).Given a partition P of G(V,F),P={C1,C2,...,C k}, SP(C i)={C j|?C k∈P,θ(C i,C j)≥θ(C i,C k)}

De?nition4(Cocluster set).is called a cocluster set iff C j is the spouse cluster of C i.

Example1.In Figure1,C2is C1’s spouse cluster,they form a cocluster set coclusterA< C1,C2>,forms another cocluster set coclusterD.worth mentioning,the two elements in a cocluster set are not commutative.thus,does not neces-sarily implies.For instance,is an invalid cocluster set,because only C2can be identi?ed as C1’s spouse cluster.

The goal of our coclustering procedures is to partition objects into groups while minimizing the cross-connectivity costs between those groups.More speci?cally,the coclustering procedures will separate objects into K groups so that(1)similar objects are within the same group,while dissimilar objects are in different groups,(2)there is a one-to-one correspondence/one-to-many correspondence between one cortical cluster

to another /other clusters;and (3)the total cross-connectivity cost between each cluster and its non-spouse cluster is minimized.

To achieve the above goals,we de?ne several notions.First,we de?ne the centroid of a cluster and its Within-Cluster Variation (W CV )to quantify the similarity of objects within one cortical cluster.

De?nition 5(Centroid ).Given a cortical cluster C k ,its centroid ?→μk

is de?ned as:?→μk =

?→X n ∈C k ?→X n

|C k |

where |C k |represents the number of cortical voxels in cluster C k .

De?nition 6(WCV ).We de?ne Within-Cluster Variation of cortical cluster C k as:

W CV (C k )=

?→X n ∈C k d (?→X n ,?→μk )

where d (?→X n ,?→μk )is the Euclidean distance between the cortical voxel ?→X n and the cen-troid ?→μk of cortical cluster C k

.Second,we de?ne the Total Within-Cluster Variation(TWCV )to quantify the quan-tity of a particular partitioning.

De?nition 7(TWCV ).The Total Within-Cluster Variation of a cortical partition (C 1,···,C K )is de?ned as

T W CV (C 1,···,C K )

=K

k =1W CV (C k )

=K k =1 ?→X n ∈C k D

d =1

(X n d ?μk d )2=K

k =1D d =1X n d 2?K k =11k D d =1

(SCF k d )2where SCF k d is the sum of the d th feature of all voxels in C k .

Third,in order to minimize the cross-connectivity cost,for each cortical cluster,we de?ne the set of cortical voxels that are connected to it as its shaded cortical cluster .De?nition 8(Shaded cluster ).Given a cortical partition (C 1,···,C K ),the shaded cluster SC k (k =1,···,K )is de?ned as:

SC k ={sc |sc ∈C,?v ∈C k ,?v ∈C,(v ,v )∈F,sc ∩C k =?}

Example 2.In Figure 2,all the cortical voxels that are connected to voxels in cortical cluster C 2forms the shaded cluster SC 1,while all voxels that are connected to the voxles in cortical cluster C 1forms the shaded cluster SC 2.

Coclustering Based Parcellation of Human Brain Cortex Using Diffusion Tensor MRI

Fig.2.Shaded clusters

In an ideal coclustering,as CoclusterB,a shaded cluster should coincide with the corresponding spouse cluster.However,this is not always the case in general.The cross-connectivity cost can be characterized by the disagreement between shaded clus-ters and spouse clusters and quanti?ed by the Within-Cluster Variance of shaded clus-ters with respect to their corresponding spouse clusters,called Shaded Within-Cluster Variation(SWCV),that is de?ned as follows.

De?nition9(SWCV).The Shaded Within-Cluster Variation(SWCV)of cortical clus-ter SC k is de?ned as:

SW CV(SC k)=

?→

X n∈SC k d(

?→

X n,?→

μk)

Note that,instead of using the centroid of SC k,the centroid of the C k is used to calculate SW CV(SC k).The intuition is that,in an ideal partitioning,the shaded partition SC1,···,SC K should mostly coincide with C1···C K.

De?nition10(STWCV).The Shaded Total Within-Cluster Variation(ST W CV)of cortical partition(SC1,···,SC K)is de?ned as:

ST W CV(SC1,···,SC K)=

K

k=1

SW CV(SC k)

=

K

k=1

?→

X n∈SC k

D

d=1

(X n

d

?μk

d

)2

The variance in distances between voxels is partitioned into variance attributable to differences among distance within clusters and to differences among clusters.W CV(C k) measures the variability within the cluster C i,while we introduce BCV(C k)as a mea-sure of the variability between cortical clusters.

Statement of the problem.Finally,the coclustering problem can be formally stated as follows:given a Graph G=(V,F)and a distance metric d for nodes between v i and v j(i=j),coclustering is required to partition V into K clusters and cocluster sets, as well as a set of outliers,formulated as{,···,

,O},such that the connection strength of each cluster is maximized and the following objective function OT W CV is minimized:

OT W CV(C1,C2,···,C K)=

K

k=1

T W CV(C k)+ST W CV(SC k)

4The BCA Algorithm

In this section,we propose our coclustering algorithm,Business model based Coclus-tering Algorithm(BCA),to solve the coclustering problem.

BCA starts with the density-based initialization,and produces a better solution from the current solution by applying the following three phases,viz.Split,Transfer,and Merge sequentially.Three procedures can run iteratively to produce one solution after another until a termination condition is reached.During each iteration,the current solu-tion S i is associated with the?gure of merits that include a function of OT W CV and the connection strength.

The goal of our algorithm’s initialization is to not only partition cortical voxels into cocluster sets,but also to minimize the distance variance within one cluster while maximizing each cluster’s connection strength.

4.1Density-based initialization

The goal of our density-based initialization is to have an initial clustering of the corti-cal voxels based on the following working hypothesis provided by our domain experts: voxels within one functional cortical region should be close to each other and each functional cortical region should contain at least one dense subregion.The initializa-tion procedure is described by Algorithm Initialize in Figure3.The algorithm takes a cortico-cortical connectivity graph G and two parametersεandδas input and pro-duces an initial coclustering as output.In addition,in the output,a set of voxels O will be identi?ed as outliers that will not be classi?ed into any functional cortical region. Whileεis the maximum radius of a voxel’s neighborhood,δis the minimum number of voxels within theε-neighborhood of a voxel for the voxel to be a core voxel.We?rst introduce the following notions.

De?nition11(ε-neighborhood and core voxel).Given a cortico-cortical connectivity graph G(V,F),theε-neighborhood of a voxel v∈V,denoted Nε(v),is de?ned by Nε(v)={u∈V|dist(v,u)≤ε}.We call v a core voxel iff|Nε(v)|≥δ.

De?nition12(Distance-reachable).A voxel u is directly distance-reachable from a voxel v w.r.t.εandδif u∈Nε(v)and u is distance-reachable from v if there is a chain of voxels v1,···,v n,such that v1=v,v n=u,and v k is directly distance-reachable from v k?1for k=2,···,n.

As shown in Figure3,Algorithm Initialize?rstly calculates N?N distance-connection matrix M to store the Euclidean distances between each pair of cortical voxels(line5).

(1)Algorithm:Initialize

(2)Input:Cortico-cortical connectivity graph G(V,F),maximum radiusε,and minimum number of voxelsδ

(3)Output:initial coclustering{,···,,O}

(4)Begin

(5)Calculate from G the distance-connection matrix M to store dist(u,v)for all u,v∈V;

(6)k=0;

(7)For each voxel v∈V do

(8)If v is classi?ed then

(9)Process the next voxel;

(10)Else/*v is not classi?ed*/

(11)If v is a core voxel then

(12)k:=k+1;

(13)Collect all voxels distance-reachable from v and assign them to C k

(14)Else

(15)Process the next voxel;

(16)End If

(17)End If

(18)End For

(19)Collect all unclassi?ed voxels and assign them to O;

(20)Identify SP(C k)(k=1...K)according to De?nition3.

(21)End Algorithm

Fig.3.Algorithm Initialize

We de?ne dist(u,v)=+∞iff u and v belong to different hemispheres of the brain to implement the constraint that each resulting cluster will not span across different hemispheres.The algorithm will iteratively consider each voxel v(lines7-18).If v is an unclassi?ed voxel,then a new cluster is formed by all the voxels that are distance-reachable from v;otherwise,either v is already classi?ed(lines8-9),or it is a non-core voxel(lines14-15),the processing of v will be skipped.After the iteration completes, all the unclassi?ed voxels will be assigned to a set O as outliers.Finally,for each identi-?ed cluster C k(k=1,···,K),its spouse cluster SP(C k)will be identi?ed according to De?nition3(line20)to produce the initial coclustering result.

Since the analysis performed in the initialization procedure focuses on region den-sity and distances between cortical voxels rather than their connectivity,the BCA al-gorithm further applies operators Split,Transfer,and Merge iteratively to improve the coclustering result by minimizing its OTWCV value.

4.2Split

The split operator attempts to split a cluster into two clusters when such a split will improve the result of coclustering that is characterized by the following split condition. De?nition13(Split condition).Given a coclustering CO={,···,< C i,SP(C i)>,···,}and a cluster C i∈CO,let C i1be the set of voxels in C i that are connected to SP(C i),C i2be C i?C i1,and CO ={< C1,SP(C1)>,···,,,···, },then we say that C i satis?es the split condition iff

1)|C i1|>=δand|C i2|>=δ;

2)OT W CV(CO )≤OT W CV(CO);

3)θ(C1,SP(C1))≤θ(C i2,SP(C i2)).

(1)Algorithm:Split

(2)Input:CO={,···,}

(3)Output:a new version of CO in which no more cluster satis?es the split condition

(4)Begin

(5)While there exists a cluster C i∈CO satisfying the split condition do

(6)Split C i into C i1and C i2;

(7)Recalculate the spouse cluster for each cluster in CO according to De?nition3;

(8)End while

(9)End Algorithm

Fig.4.Algorithm Split

Intuitively,the split condition ensures that after a split,1)the number of voxels in each new cluster is still greater than or equal toδ,2)the OTWCV value for the new coclustering will not increase,and3)the connection strengths of the two new clusters C i1and C i2will be no less than the connection strength of the original cluster C i.This is always true for C i1,and thus we only need to requireθ(C1,SP(C1))≤θ(C i2,SP(C i2))in the above de?nition of the split condition.

Algorithm Split is sketched in Figure4.Basically,it iteratively splits the colustering result until no more cluster satis?es the above de?ned split condition.

4.3Transfer

The transfer operator attempts to reassign each voxel to a new cluster in order to im-prove the result of coclustering that is characterized by the following transfer condition. De?nition14(Transfer Condition).Given a coclustering CO={ ,···,,···,,···,},let v∈C i for some C i in CO,C j be the cluster to whose centroid v is the closest,after transferring v from C i to C j,C i becomes C i,C j becomes C j,and CO becomes CO ={,···,,···,,···,<

C K,SP(C K)>},we say that v satis?es the transfer condition iff

1)|C i|>=δ;

2)OT W CV(CO )≤OT W CV(CO);

3)θ(C i,SP(C i))≤θ(C i,SP(C i))andθ(C j,SP(C j))≤θ(C j,SP(C j)).

(1)Algorithm:Transfer

(2)Input:CO={,···,}

(3)Output:new version of CO in which no more voxel satisfying the transfer condition

(4)Begin

(5)While there exists a voxel v∈C i satisfying the transfer condition do

(6)Transfer v from C i to C j where C j is the cluster to whose centroid v is the closest;

(7)Recalculate the spouse cluster for each cluster in CO according to De?nition3;

(8)End while

(9)End Algorithm

Fig.5.Algorithm Transfer

Intuitively,the transfer condition ensures that after a transfer,1)C i still contains at leastδvoxels,2)the OTWCV value for the new coclustering will not increase,and 2)the connection strengths of the two affected clusters will not decrease.Algorithm Transfer is sketched in Figure5.Basically,it attempts to assign each voxel to a new cluster if it satis?es the transfer condition.The procedure terminates when no more voxel satis?es the above de?ned transfer condition.

4.4Merge

Finally,the merge operator attempts to merge two clusters if such a merge will improve the result of coclustering that is characterized by the following merge condition.

De?nition15(Merge Condition).Given a coclustering CO={ ,···,,···,,···,},and two clusters C i,C j∈CO,we merge C i and C j into C m and derive a new coclustering CO ={,···,,···,}.We say C i and C j satisfy the merge condition iff

1)OT W CV(CO )≤OT W CV(CO);

2)θ(C i,SP(C i))≤θ(C m,SP(C m))andθ(C j,SP(C j))≤θ(C m,SP(C m)).

(1)Algorithm:Merge

(2)Input:CO={,···,}

(3)Output:new version of CO in which no more cluster satisfying the merge condition

(4)Begin

(5)While there exists C i,C j∈CO satisfying the merge condition do

(6)Merge C i and C j into C m;

(7)Recalculate the spouse cluster for each cluster in CO according to De?nition3;

(8)End while

(9)End Algorithm

Fig.6.Algorithm Merge

Intuitively,the merge condition ensures that after a merge,1)the OTWCV value for the new coclustering will not increase,and2)the connection strength of the new merged cluster is no less than the connection strengths of the two original clusters. Algorithm Merge is sketched in Figure6.Basically,it merges two clusters into one if the two clusters satisfy the above de?ned merge condition.The algorithm terminates when no more pair of clusters satisfy the above de?ned merge condition.

53-D Visualization of the BCA results

All?ber tracts calculated from DTI data were rendered in relation to the cortical mesh obtained from conformal brain surface mapping[18],as shown in Figure7-(a)-Top.It can be seen that there is a large number of?ber tracts connecting cortical areas.BCA was performed based on the spatial relationship of voxels on the cortical surface and

Figure7-(a)-Bottom exhibits clustered cortical?bers in frontal and lateral view.Figure 7-(b)shows the results of our BCA in a representative subject.Well-know anatomical ?ber tracts in the brain are reproduced such as the colossal?bers(pink)which connect the two hemispheres and the forceps minor of the corpus callosum(yellow)connecting the left and right side of the frontal cortex.Moreover,the intra-hemispheric connec-tions of the arcuate fasciculus connecting Broca’s and Wernicke’s cortical areas can be appreciated.

Coclustering Based Parcellation of Human Brain Cortex Using Diffusion Tensor MRI

Coclustering Based Parcellation of Human Brain Cortex Using Diffusion Tensor MRI

(a)(b)

Fig.7.(a)-Top:frontal and lateral view of cortico-cortical?bers before coclustering;(a)-Bottom: frontal and lateral view of clustered cortico-cortical connectivity;(b)Zoom in view of some speci?c coritco-cortical clusters.

These results indicate that the developed algorithm is consistent with brain anatomy and that it allows automated segmentation of the cortex based on DTI-derived cortical connections within the brain.We therefore believe that our algorithm is well suited to provide an ef?cient framework for further analysis including the quantitative assess-ment of cortico-cortical connectivity.

6Conclusions and Future Work

In this paper,we de?ned the coclustering problem and we applied this approach to the analysis of cortico-cortical connections in the brain.Our approach represents an ef?cient mathematical framework that is computationally robust and is able to be used for quantitative analysis of cortico-cortical?ber tracts.This in turn might be relevant for the identi?cation of secondary epileptic foci in patients with intractable epilepsy and might impact their clinical management.

Although the coclustering problem was initially motivated by the need of cortico-cortical connectivity analysis,we expect that it will have a wide range of applications. In the future,we plan to apply our BCA also to the analysis of thalamo-cortical connec-tivity and the segmentation of thalamic nuclei.

Acknowledgment

This research was partially supported by the Michigan Technology Tri-Corridor basic research grant MTTC05-135/GR686.

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