Functional connectivity networks (FCNs) are important in the diagnosis of neurological diseases and the understanding of brain tissue patterns. Recently, many methods, such as Pearson’s correlation (PC), Sparse representation (SR), and Sparse low-rank representation have been proposed to estimate FCNs. Despite their popularity, they only capture the low-order connections of the brain regions, failing to encode more complex relationships ( i.e. , high-order relationships). Although researchers have proposed high-order methods, like PC + PC and SR + SR, aiming to build FCNs that can reflect more real state of the brain. However, such methods only consider the relationships between brain regions during the FCN construction process, neglecting the potential shared topological structure information between FCNs of different subjects. In addition, the low-order relationships are always neglected during the construction of high-order FCNs. To address these issues, in this paper we proposed a novel method, namely Ho-FCN Tops, towards estimating high-order FCNs based on brain topological structure. Specifically, inspired by the Group-constrained sparse representation (GSR), we first introduced a prior assumption that all subjects share the same topological structure in the construction of the low-order FCNs. Subsequently, we employed the Correlation-reserved embedding (COPE) to eliminate noise and redundancy from the low-order FCNs. Meanwhile, we retained the original low-order relationships during the embedding process to obtain new node representations. Finally, we utilized the SR method on the obtained new node representations to construct the Ho-FCN Tops required for disease identification. To validate the effectiveness of the proposed method, experiments were conducted on 137 subjects from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database to identify Mild Cognitive Impairment (MCI) patients from the normal controls. The experimental results demonstrate superior performance compared to baseline methods.
Resting state functional magnetic resonance imaging (rs-fMRI) is an imaging technique used to study brain function by measuring the level of neural activity in the resting state. It employs magnetic resonance imaging to observe changes in the blood oxygen level-dependent (BOLD) signal, revealing patterns of synchronization and connectivity between different brain regions. In the field of medicine, rs-fMRI finds extensive applications such as researching brain networks, neuropsychiatric disorders, individual differences, and assessing brain injuries. It holds significant clinical and research value in understanding brain diseases and the working mechanisms of the brain [
Functional connectivity networks (FCNs) refer to networks formed by the interconnections between different regions of interest (ROIs) within the brain. These connections represent interactions and information transfer between different ROIs. FCNs based on rs-fMRI can help enhance our understanding of brain function and have been widely used in the research of various neuropsychiatric disorders, such as major depressive disorder [
Up to now, researchers have proposed various effective methods for estimating FCNs, including Pearson correlation (PC) [
Although the mentioned methods provide a comprehensive estimation of FCNs, from another perspective, these methods sequentially estimate FCNs for individual subjects. This leads to the oversight of the similarity in brain topology among different subjects. Consequently, it fails to fully capture all the information contained in the BOLD signals, potentially impacting subsequent classification performance. To address this limitation, Wee et al. proposed incorporating the L 2,1 -norm regularization term into the FCN estimation model [
To address the aforementioned issues, we propose a novel method for estimating FCNs. Given our intention to incorporate potential shared topological structures among subjects during the estimation of FCNs, we initially employ the GSR method to perform a preliminary estimation of FCNs using BOLD signals, resulting in primary FCNs. This stage ensures that all subjects exhibit similarity and sparsity in the topological structure of brain networks. Although the preliminary estimated FCNs can be directly utilized for disease identification, they still contain low-order relationships from original node features and some potential noise. Therefore, we further process them by re-encoding low-order FCNs using Correlation-Preserving Embedding (COPE) [
1) Proposal of a new FCN estimation method. The FCNs we estimate not only integrates low-order and high-order information to make it more comprehensive but also incorporates potential shared topological information in the brain, allowing the obtained FCNs to more accurately reflect the true state of the brain.
2) To validate the effectiveness of the proposed method, we conducted classification experiments on the ADNI dataset. The experimental results demonstrate outstanding performance of the proposed method.
The remaining sections of this paper are organized as follows. The Section 2 begins with an introduction to preprocessed data and a comprehensive review of the most pertinent studies. Subsequently, we propose a novel approach for estimating FCN, encompassing its mathematical model and algorithm. In Section 3, we describe the experimental setup and report the experimental results. In Section 4, we discuss these findings. Finally, in Section 5, we conclude this paper.
The experiments in this chapter are based on the ADNI dataset, which includes 137 subjects, comprising 69 normal controls (NCs) and 68 individuals with Mild Cognitive Impairment (MCIs). The scanning parameters are as follows: the in-plane image resolution is 2.29 to 3.31 mm; slice thickness = 3.31 mm; echo time (TE) = 30 ms; repetition time (TR) is 2.2 to 3.1 s; each subject’s scanning duration is 7 minutes (with a total of 140 volumes). Demographic information about the subjects can be found in
Subsequently, the subjects involved in this chapter were processed using the standard pipeline of the DPARSF (Data Processing Assistant for Resting-State fMRI) toolbox for rs-fMRI. The specific steps included: 1) removing the first three volumes from the fMRI time series of each subject to ensure signal stability; 2) performing motion correction on the remaining time series to achieve consistent slice acquisition time and mitigate the impact of head motion; specifically, subjects with frame-wise displacement (FD) greater than 0.5 mm for more than 2.5 minutes were excluded; 3) implementing nuisance regression to reduce the influence of ventricular and white matter signals. Subsequently, 4) the motion-corrected images were registered to the Montreal Neurological Institute (MNI) space, and a band-pass filter (0.015 Hz to 0.150 Hz) was applied to remove low-frequency and high-frequency noise. A spatial smoothing process with a 4 mm full-width at half-maximum (FWHM) Gaussian kernel was also applied. It is important to note that scrubbing was not performed in this chapter’s experiments to avoid introducing additional artifacts. Finally, 5) using the Automated Anatomical Labeling (AAL) template, the fMRI scans’ brain space was parcellated into 116 predefined ROIs through a deformable registration method, and the BOLD signals within the gray matter were extracted to calculate the average time series for each ROI.
| Dataset | Class | Gender (M/F) | Age (years) | MMSE |
|---|---|---|---|---|
| ADNI | MCI | 39/29 | 76.50 ± 13.50 | 26.77 ± 1.23 |
| NC | 17/52 | 71.50 ± 14.50 | 28.82 ± 1.15 |
As mentioned previously, PC, outlined in [
In PC-based FCNs, the edge weight w i j between two ROIs is defined as follows:
w i j = ( x i − x ¯ i ) T ( x i − x ¯ i ) ( x i − x ¯ i ) T ( x i − x ¯ i ) ( x j − x ¯ j ) T ( x j − x ¯ j ) (1)
where x i ∈ R T ( i = 1,2, ⋯ , M ) is the BOLD signal of the ith ROI. T is length of the time series and M is numbers of ROI.
Without loss of generality, we redefine x i = ( x i − x ¯ i ) / ( x i − x ¯ i ) T ( x i − x ¯ i ) . Then, Equation (1) can be simplified as w i j = x i T x j , which corresponds to the optimal solution of the following model:
min w i j ∑ i , j M ‖ x i − w i j x j ‖ 2 (2)
Equation (2) can be further expressed in matrix form, whose form is:
min W ‖ W − X T X ‖ F 2 , (3)
where X = [ x 1 , x 2 , ⋯ , x M ] ∈ R T × M is the BOLD signals matrix. Here, W = ( w i j ) ∈ R M × M is the FCN estimated using PC, and ‖ ⋅ ‖ F represents the Frobenius-norm of a matrix.
Different from PC, which measures the full correlation, partial correlation estimates the functional connectivity between a pair of ROIs while considering the effects from other ROIs [
min w i j ∑ i = 1 M ( ‖ x i − ∑ j ≠ i w i j x j ‖ 2 + λ ∑ j ≠ i | w i j | ) s .t . w i i = 0 , ∀ i = 1 , 2 , ⋯ , M (4)
where λ is the regularization parameter that controls the sparsity of the brain network W.
The expression provided in Equation (4) can be reformulated in the following matrix form:
min W ‖ X − X W ‖ F 2 + λ ‖ W ‖ 1 s .t . w i i = 0, ∀ i = 1,2, ⋯ , M (5)
where ‖ ⋅ ‖ 1 denotes the L 1 -norm of a matrix for modeling sparsity prior into the constructed FCN. It is important to note that the constraint w i i = 0 ensures that the diagonal elements of W are entirely zeros, preventing degenerate solutions.
As mentioned earlier, researchers have proposed various high-order FCN estimation methods [
As previously mentioned, researchers have developed various methods for estimating FCNs, including some that can only simulate low-order relationships between ROIs, such as PC and SR, as well as some high-order FCNs construction methods like SR + SR and Ho-FCNCOPE. However, these methods have certain limitations. In light of this, our research motivation has two main aspects: 1) to integrate low-order and high-order information between ROIs, fully harnessing the advantages of high-order FCNs, enhancing the quality of FCN estimation, and making it more conducive to subsequent disease recognition tasks; 2) some common FCN estimation methods only utilize spatial relationships between ROIs, overlooking the potential shared topological structure among subjects. To address this issue, we propose a high-order FCN estimation method based on brain topological structure. The specific model is shown in
Compared to the basic low-order FCN construction methods described earlier, high-order FCN construction methods can consider deeper interactions between ROIs, thereby extracting more information useful for recognition tasks. However, such methods overlook the potential topological structure among subjects. Based on this, our proposed high-order FCN estimation method, which integrates brain topological structure, comprises three main steps. Firstly, we replace the commonly used SR method in high-order FCN construction with the GSR method for the preliminary estimation of low-order FCN between different ROIs. In contrast to SR methods, GSR introduces a group-level prior, assuming that all subjects share the same topological structure. This process relies on the introduction of the L 2,1 -norm, a constraint that integrates the shared topological structure into the FCN construction process, ensuring that the estimated FCN has a certain degree of sparsity [
min H i ∑ n = 1 N ( 1 2 ‖ x i n − X − i n h i n ‖ 2 2 ) + λ 1 ‖ H i ‖ 2,1 (6)
where N is the number of subjects, p is the number of ROIs and x i n is the BOLD signal time series of the ith ROI of the nth subject. Besides, h i n is the weight vector that denotes the effect of other ROIs to the ith ROI for the nth subject, and H i is the corresponding weight matrix composed of the ith ROI of all N subjects. The notation ‖ ⋅ ‖ 2,1 is the L 2,1 -norm.
As mentioned earlier, current high-order estimation methods involve directly applying second-order correlation representations to the adjacency matrix of low-order FCN. This approach has drawbacks, as it not only introduces noise from the low-order FCN into the estimation process of the high-order FCN, affecting the quality of the high-order FCN estimation, but also fails to determine whether the low-order relationships among valuable ROIs in the low-order FCN are fully preserved during the high-order FCN estimation process. Following Su et al.’s research [
Ψ ( E ) = ∑ i = 1 p ‖ e i − ∑ j = 1 p h i j e j ‖ 2 (7)
where e i is the new node representation for node i in the low-dimensional embedding space, and h i j represents the edge weights of the previously estimated low-order brain functional network in the preceding step. In Equation (7), the weights h i j in the low-order brain functional network serve as the encoding of the low-order information required for the high-order FCN. Therefore, to ensure the effective preservation of low-order information, it is only necessary to find a suitable embedding space through COPE. We can apply mathematical knowledge to simplify Equation (7) into the matrix form of Equation (8) and impose constraints on t r ( E E T ) = p to avoid the occurrence of trivial solutions.
Ψ ( E ) = ∑ i = 1 p ‖ E I i E H i ‖ 2 = t r ( E ( I − H ) ( I − H ) T E T ) (8)
Thus, we obtain the following COPE model, where L = ( I − H ) ( I − H ) T :
min W t r ( E L E T ) s .t . t r ( E E T ) = p (9)
Ultimately, we can transform Equation (9) using the Lagrange multiplier method and then complete the optimization to find the optimal values. The solution, when expressed in matrix form, is shown in Equation (10):
min W ‖ E − E W ‖ F 2 + λ 2 ‖ W ‖ 1 s .t . w i i = 0, ∀ i = 1,2, ⋯ , p (10)
where λ 2 is a regularization parameter, used to control the balance between the two terms in the objective function, and w i j represents the high-order correlation coefficient between the i-th and j-th ROIs in the finally obtained Ho-FCNTops.
In our experiments, we selected two classic methods, PC and SR, as baselines for estimating FCN. Additionally, we adopted three high-order FCN estimation methods based on correlation, including the latest research advancements: SR + SR, GSR + SR, and Ho-FCNCOPE [
As we are aware, the field of medical imaging commonly grapples with the challenge of limited data samples, and our research is similarly affected by this issue. Our dataset comprises 137 subjects, and to validate the effectiveness of the proposed method, we employ a leave-one-out cross-validation (LOOCV) strategy. Specifically, from the pool of 137 subjects, we initially select one sample as the test set, with the remaining 136 serving as the training set. Simultaneously, these 136 subjects undergo internal LOO training and testing iterations. This process repeats iteratively until each experimental sample has completed one testing cycle. Refer to
After completing the estimation of FCNs for all subjects, the next step is to perform disease identification and classification based on the obtained FCNs. For the subject sample used in this study, each subject has 116 ROIs, corresponding to a total of 116 × ( 116 − 1 ) = 13340 features when calculating the correlations between these ROIs. Since the estimated FCNs are symmetric, we only need to consider the upper triangular matrix elements of the FCNs as input features for the classifier, resulting in a final feature count of 116 × ( 116 − 1 ) 2 = 6670 .
It is essential to note that the number of subjects in the dataset we utilized is only 137, significantly smaller than the number of features. This situation often leads to the curse of dimensionality, potentially affecting subsequent classification performance. Although researchers have proposed various feature selection methods to address this issue, our primary objective is to evaluate the effectiveness of the proposed FCNs estimation method in disease identification [
To comprehensively assess the performance of our proposed method in terms of classification, we selected eight metrics for comparative evaluation. This aims to quantify the improvements of our method relative to the control method. The metrics include accuracy (ACC), specificity (SPE), sensitivity (SEN), F1-score, balanced accuracy (BAC), positive predictive value (PPV) and negative predictive value (NPV). The mathematical definitions of these metrics are as follows:
A C C = T P + T N T P + T N + F P + F N (11)
S E N = T P T P + F N (12)
S P E = T N T N + F P (13)
B A C = S E N + S P E 2 (14)
P P V = T P T P + F P (15)
N P V = T N T N + F N (16)
F 1 - s c o r e = 2 T P 2 T P + F N + F P (17)
Please note that in this study, subjects with MCI are considered positive class, while NC samples are considered negative class. In which, TP, TN, FP, FN respectively represent true positive, true negative, false positive, and false negative.
In
| Method | ACC | SEN | SPE | F1 | BAC | PPV | NPV |
|---|---|---|---|---|---|---|---|
| PC | 0.8540 | 0.8676 | 0.8406 | 0.8551 | 0.8541 | 0.8429 | 0.8657 |
| SR | 0.8248 | 0.8088 | 0.8406 | 0.8209 | 0.8247 | 0.8333 | 0.8169 |
| SR + SR | 0.7372 | 0.7059 | 0.7681 | 0.7273 | 0.7370 | 0.7500 | 0.7260 |
| GSR + SR | 0.6642 | 0.5000 | 0.8261 | 0.5965 | 0.6630 | 0.7391 | 0.6264 |
| Ho-FCNCOPE [ | 0.8248 | 0.8088 | 0.8406 | 0.8209 | 0.8247 | 0.8333 | 0.8169 |
| Ho-FCNTops | 0.8905 | 0.8824 | 0.8986 | 0.8889 | 0.8905 | 0.8955 | 0.8857 |
MCI recognition task, the proposed method in this paper demonstrates better classification performance.
The conclusions drawn from
The embedding dimension plays a crucial role in achieving outstanding classification performance. Therefore, in this section, to assess the impact of the embedding dimension on classification performance, we will explore the effects of different embedding dimensions on the estimation of FCNs and classification performance. Here, the embedding dimension m takes values from [40, 50, ⋯ , 110], encompassing a total of 8 dimensions. As evident from
Furthermore, we observed a trend in the classification performance during the transition from higher to lower embedding dimensions, showing an initial increase followed by a subsequent decrease. When m = 80, the classification performance reached its peak, leading in ACC, SEN, and SPE metrics compared to other embedding dimensions. However, for m < 80, the classification metrics exhibited varying degrees of decline. This might be attributed to excessive reduction in dimensionality, potentially resulting in a significant loss of useful information for classification, thereby affecting subsequent FCNs estimation and classification. Consequently, in the subsequent discussions, we will focus on the case where m = 80.
In practical classification tasks, the ultimate classification accuracy is influenced by various factors, with model parameters being a key factor, a viewpoint validated by many researchers [
Moreover, for hyper-parameter λ 2 , smaller values tend to enhance classification performance. Through the analysis of the estimated FCNs, we discover that as the value of λ 2 increases, the resulting network becomes sparser, potentially leading to the discard of many useful pieces of information for classification, thereby reducing classification performance. Overall, when the hyper-parameter values are concentrated in the region above
Achieving high-quality FCNs is crucial for attaining superior classification performance. Therefore, estimating high-quality FCNs becomes paramount. Visualizing the estimated FCNs helps to intuitively understand the strengths and weaknesses of different methods. It is important to note that, for a unified comparison of different methods, the adjacency matrices shown in
1) The method based on PC presents significant differences compared to other methods. This is because the PC-based method adopts a strategy entirely different
from other methods in capturing relationships between different ROIs, leading to distinct visual differences in FCN representations.
2) For the other five correlation-based methods, they all exhibit varying degrees of sparsity in the visualization of adjacency matrices due to the utilization of SR operations.
3) Our proposed Ho-FCNTops method incorporates the potential shared brain topological structure compared to the Ho-FCNCOPE method. In
In
The proposed method makes a significant contribution by integrating potential shared brain topological structures during the estimation of high-order FCNs. Therefore, a discussion regarding the impact of brain topological structures on FCNs estimation becomes imperative. By examining the classification performance presented in
For the estimated FCNs, they can not only be used for disease identification but also for the detection of potential biomarkers related to diseases. Through feature selection on the estimated FCNs, we can obtain ROIs with higher relevance to disease identification, thereby identifying potential biomarkers associated with MCI. Therefore, we employed a t-test with a significance level of p = 0.0005
for feature selection on the estimated FCNs. After the t-test selection, a total of 36 discriminative features were chosen. Among them, we selected the top 10 most discriminative features and visually presented them in
In this paper, we have focused on addressing several challenges in current FCN research. These challenges include overlooking the potential shared topological structure among different subjects and the difficulties in noise reduction and preservation of low-order relationships during high-order FCN construction. By introducing a novel high-order FCN estimation method based on brain topological structure, we have leveraged the information embedded in the brain’s topological structure. This method integrates low-order information into the construction process of high-order FCN, presenting a novel tool for the diagnosis of MCI. In experiments conducted on the public ADNI database, we have demonstrated significant performance advantages. Compared to baseline methods, our proposed approach achieved superior results in MCI recognition tasks, confirming the effectiveness of our method.
While we have made some progress in FCN estimation, we acknowledge that there are many issues that require further exploration. Therefore, in future research, we will focus on the following aspects: 1) Extend our approach to the diagnosis of other neurological disorders, such as MDD, ASD, etc. 2) Enhance our understanding of brain topological structure and optimize FCN estimation methods accordingly. 3) Further apply machine learning methods to improve the accuracy and generalization of FCN estimation. By conducting in-depth research in these directions, we aim to provide more comprehensive and effective solutions for the early diagnosis of neurological disorders, the development of treatment methods, and a deeper understanding of brain connectivity patterns.
The authors declare no conflicts of interest regarding the publication of this paper.
Zhang, G.Y., Zhang, K.P. and Pang, M.X. (2024) Estimating High-Order Functional Connectivity Networks for Mild Cognitive Impairment Identification Based on Topological Structure. Journal of Computer and Communications, 12, 14-31. https://doi.org/10.4236/jcc.2024.123002