A GENERALIZED FRAMEWORK OF PATHLENGTH ASSOCIATED COMMUNITY ESTIMATION FOR BRAIN STRUCTURAL NETWORK

Abstract

Diffusion MRI-derived brain structural network has been widely used in brain research and community or modular structure is one of popular network features, which can be extracted from network edge-derived pathlengths. Conceptually, brain structural network edges represent the connecting strength between pair of nodes, thus non-negative. The pathlength. Many studies have demonstrated that each brain network edge can be affected by many confounding factors (e.g. age, sex, etc.) and this influence varies on each edge. However, after applying generalized linear regression to remove those confounding’s effects, some network edges may become negative, which leads to barriers in extracting the community structure. In this study, we propose a novel generalized framework to solve this negative edge issue in extracting the modular structure from brain structural network. We have compared our framework with traditional Q method. The results clearly demonstrated that our framework has significant advantages in both stability and sensitivity.

1. INTRODUCTION

Non-invasive MRI is one of the most popular tools to study the brain. Traditional MRI studies focus on voxelwise or region-of-interest (ROI)-based measures. Considering most brain disease-induced deterioration is a systematic process [1], these MRI measures are thus suboptimal. The brain functional [2] and structural networks [3], derived from functional MRI and diffusion-weighted MRI respectively, have the potential to gain system-level insights into the mechanisms of brain diseases, thus providing a novel platform for developing new diagnostic strategies [4]. Over the last decade, the brain network approach has emerged as a new omics approach that can potentially revolutionize our understanding of the brain by showing how complex brain activities interact in both time and space.

Currently, standard network analyses commonly involve the comparison of summary graph-theoretical metrics such as clustering coefficient, etc. While easy to compute and widely popular among network researchers, these simple scalar statistics defined at either the nodal or the global level are problematic in that they reduce the complex network data to a dimension of 1 and discard important informative graph structure encoding complex network inter-relationship for every intermediate scale between nodes and the entire graph. Just as social networks can be divided into cliques that describe modes of association (e.g. family, school) [5, 6], a brain network can be divided into modules or communities. Modules contain a collection of nodes that are densely interconnected (via edges) with one another but weakly connected with nodes in other modules. Thus, modularity or community structure best describes the intermediate scale of network organization, rather than the global or local scale. In many networks, modules can be divided into smaller sub-modules, thus can be said to exhibit hierarchical modularity or near decomposability.

The concept of community structure has been extensively used in many areas, not just brain network [7, 8]. For example, [9] adopted this concept to understand species geographic distributions with the relative abundance of coexisting species. Bader et al. [10] detected the phylogenetic and morphological global desert rodent communities using community structure patterns. Moreover, email networks provide a precise and nonintrusive depiction of the human organization information which analyzed as the social network of selfsimilar community structure [11]. Therefore, identifying the community structure in large networks is always a hot topic [12, 13]. However, there is one important issue, especially in brain network, which has not been well addressed in the literatures.

In the network, each single edge represents some mathematical relationship between pair of nodes and it’s a common sense that each edge can be affected by different confounding factors to some degree. For example, the edge in brain structural network is extracted based on the fiber bundles and represents the connecting strength between pair of brain ROIs and this connecting strength can be affected by age [14, 15], sex [16, 17], health status [18, 19], and biological knowledge [20] etc. Therefore, directly extracting the community structure from raw network without accounting for these confounding factors could be biased. The most popular approach to remove confounding factors is using generalized linear regression (GLR). However, GLR-adjusted edge values may become negative, which leads to barriers in extracting the modular structure[21].

Currently the Louvain algorithm [22] is one of the most popular methods for the community extraction. Although the implementation of Louvain algorithm in Brain Connectome Toolbox (BCT, https://sites.google.com/site/bctnet/) [23] can take into account the negative edges in the network, however, the result is not stable. (We will demonstrate this in Section 3.3). In this paper, we proposed a new generalized framework to extract the community structure from GLR-adjusted network.

2. METHODS

The proposed method is a generalized version of Path Length Associated Community Estimation (PLACE) algorithm [24]. The main idea of PLACE is to maximize the difference between inter-community versus intra-community path lengths (Eq. 1):

$$\{\begin{array}{l}{\psi }^{\mathit{\text{PL}}}={\mathit{\text{inter}}}_{\mathit{\text{PL}}}{}^{C_i\leftrightarrow C_j}-\frac{1}{2}\left({\mathit{\text{intra}}}_{\mathit{\text{PL}}}{}^{C_i}+{\mathit{\text{intra}}}_{\mathit{\text{PL}}}{}^{C_j}\right)\hfill \\ {\mathit{\text{inter}}}_{\mathit{\text{PL}}}{}^{C_1\leftrightarrow C_2}=\frac{\sum n\in C_i,m\in C_j{d}_{nm}}{N_i{N}_j}\hfill \\ {\mathit{\text{intra}}}_{\mathit{\text{PL}}}{}^{C_i}=\frac{\sum n,m\in C_i{d}_{nm}}{\left(N_i{}^2-N_i\right)/2}\hfill \end{array}$$ (1)

Here the d_nm denotes shortest distance or path length between node n and node m. Thus, maximizing ψ^PL is searching for a partition with stronger intra-community integration and stronger between-community separation. Since the original PLACE was designed for diffusion MRI-derived brain structural network, the d_nm is derived using network edges, which are all non-negative values. d_nm = 0 if node n cannot be reached from node m. If node n can be reached directly from node m, d_nm = 1/w_nm and w_nm is the connecting strength (e.g. number of fibers) between two nodes. If node n can be reached indirectly from node m, d_nm can be defined by the shortest pathlength between nodes.

Since each network edge can be affected by confounding factors and this influence varies on each edge, we adopt generalized linear regression model to remove confounding’s effects from each edge. (Eq. 2)

$$Y={\beta }_0+{\beta }_1{X}_1+{\beta }_2{X}_2+\dots +{\beta }_n{X}_N+R$$ (2)

Here, $Y={\left[y_{ij}^1,y_{ij}^2,\dots ,y_{ij}^M\right]}^T$ represents the edge value between node i and node j for M subjects. X_k = [x_k1, x_k2, …, x_kM]^T (k = 1,2, …, N) is the k − th confounding factor for M subjects. $R={\left[R_{ij}^1,R_{ij}^2,\dots ,R_{ij}^M\right]}^T$ are the regression residuals for edge (i, j) for M subjects. T represents transpose. All the confounding factors, except categorized variables (e.g. sex, race, etc.), are all normalized using Z score. After applying GLR, the residual R will replace Y to become the new network edge value and in theory R is free of confounding’s influences.

However, the pathlength cannot be computed from the new GLR-adjusted network because of the negative values caused by GLR. Here we propose to use Euclidean Distance (ED) to define the nodal distance (Eq. 3).

$$E{D}_{nm}=\sqrt{{\sum }_{i=1,i\ne n,i\ne m}^P{\left(R_{ni}-R_{mi}\right)}^2}$$ (3)

Here ED_nm is the Euclidean distance between node n and node m, thus non-negative. P is the total number of nodes and R is GLR-adjusted value. The lager ED_nm value means the longer distance between nodes, and ED_nm → 0 means two nodes are closer to each other. Therefore, ED_nm can replace d_nm in Eq. 1.

In summary, the proposed framework adopts GLR to remove confounding factors’ influence from each edge and uses nodal Euclidean distance to replace the nodal distance. The resulted ED matrix will be fed into PLACE algorithm to extract the modular structure. Fig. 1 illustrates the pipeline of the proposed framework.

Fig. 1.

The pipeline of the proposed framework which includes three main steps. The first step is to remove confounding factors’ influence on each network edge separately using the generalized linear regression model. The second step is to quantify the pairwise nodal distance using Euclidean distance; then the last step is to adopt PLACE [24] to derive the community structure.

3. RESULTS AND DISCUSSIONS

3.1. Data description and network reconstruction

In this study, at the final inclusion criteria consisted of an absence of psychiatric symptoms based on the SCID and a score ⩽ 8 on the HAM-D, as well as an MMSE ⩾ 24, we analyzed totally 106 subjects (mean age=67.85±6.72, 55 men) from [25]. For each subject, diffusion-weighed MRI (dMRI) was acquired using 2-D spin-echo EPI sequence (FOV=20mm; voxel size=0.78×0.78×3.0mm³; TR/TE=5,525/93.5ms; flip angle=90°). 40 contiguous axial slices aligned to the AC-PC line were collected in 32 gradient directions with b=1400s/mm² and 6 b0 images. T1-weighted MRIs were acquired using a spoiled gradient recalled-echo (SPGR) sequence (FOV=22cm; 120 interleaved axial slices 1.5mm thick; TR/TE=12.0/5.3ms; flip angle=13°; and voxel size=0.42×0.42×1.5mm³). We used FreeSurfer (V6.0) (https://surfer.nmr.mgh.harvard.edu) to parcellate T1-weighted MRI into 82 ROIs and adopted Fiber Assignment by Continuous Tracking algorithm [26] to extract wholebrain tractography from dMRI. The brain structural network was then reconstructed using the standard pipeline described in [27].

3.2. Extracting the modular structure

Each subject underwent the Judgement of Line Orientation Test (JLO) [28], which measures a person’s ability to match the angle and orientation of lines in space. The Oblique Line Errors is defined as the total number of incorrect oblique lines normalized by the total number of possible oblique line responses. We used the z score of Oblique Line Errors (zOblique) to split 106 subjects into two groups: low vs. high zOblique error group. Low zOblique error group has 52 subjects (mean age=67.84±6.83, 22 females) and high zOblique error group has 54 subjects (mean age=67.87±6.59, 33 females). Each subject’s network dimension is 82×82. We selected Age, Race and Years of Education as the confounding factors. For Race, we use 0 to represent black, 1 as Latinx and 3 as white. Age and Years of Education are Z scored. We applied GLR to remove the confounding factors’ influence on each edge, then applied the Louvain algorithm to extract the modular structure as the baseline result. Then we applied our new framework (Section 2) to extract the modular structure. Fig. 2 illustrates the community structures. The modular structure not only varies between groups but also between methods.

Fig. 2.

The derived community structures. A and B for low and high zOblique error group respectively using our new method; C and D for low and high zOblique error group respectively using Louvain method. Each color indicates a unique community branch and there are totally 8 community branches in each row.

3.3. Stability analysis

We adopted normalized mutual information (NMI) [29] to evaluate the stability of the results. The NMI equals to 1 means two results are exact same, and NMI less than 1 means there are some differences among results. We repeated our method and Louvain method 100 times on each subject, and our method achieved mean NMI = 1.00 ± 0.00 while Louvain method achieved mean NMI = 0.91 ± 0.04 which presents unstable results.This clearly suggests that our method has an advantage in the stability and Louvain result is not reliable, which is consistent with previous literature [30].

3.4. Identifying group differences in modular structure

For each branch that was clustered by our new methods in the community structure, Hotelling T test [31] was used to quantify the group difference and false discovery rate (FDR) [32] was adopted to correct for multiple comparison. By using the proposed method, isthmus cingulate in left hemisphere (highlighted using red color in Fig. 3) shows a significant difference (P value=1.04E-4) in the community assignment or membership. The lh-cuneus, lh-paracentral, lh-postcentral, lh-posteriorcingulate, lh-precentral, and lh-precuneus cortex are shared by both high and low zOblique error groups and highlighted using blue color in Fig. 3. The green regions in Fig. 3 include lh-caudalmiddlefrontal, lh-parsopercularis and lh-superiorfrontal cortex and they are associated with red region only in low zOblique error group. The orange regions in Fig. 3 include lh-frontalpole, rh-isthmuscingulate, rh-postcentral, and rh-precuneus cortex and they are associated with red region only in high zOblique error group.

Fig. 3.

The connectome modular differences between groups. The isthmus cingulate in left hemisphere (highlighted using red color) shows the significant difference (P value=1.04E-4) in the community assignment; the orange region is associated with high zOblique error group; the green region is associated with low zOblique error group; the blue region is shared by both groups.

In the meantime, no significant group difference can be identified in the community structure derived using Louvain method that is it cannot find any different community among those different subject groups.

The proposed method has two advantages. Firstly, this method can be easily modified to test any confounding factors hypothesis, in other words, is very flexible in adjusting the GLR model. In the current study, we only selected three confounding factors (Age, Year of Education and Race) for the GLR model for the purpose of the demonstration and we can examine some other confounding factors (e.g. sex) as well. Secondly, this method can be easily applied to other type of networks for exploring any factor impacts, not just brain structural network. For example, we can apply GLR on functional network and use similar strategy to define the ED on the residue matrix, and then apply PLACE to extract the modular structure.

4. CONCLUSION

In this paper, we proposed a generalized framework to extract the modular structure after accounting for confounding factors’ heterogeneous effects on each brain network edge. Compared with traditional Louvain method, our results demonstrated that the proposed method has significant advantages in stability, sensitivity and application scope.