Using resting-state functional magnetic resonance imaging (fMRI) technology to assist in identifying brain diseases has great potential. In the identification of brain diseases, graph-based models have been widely used, where graph represents the similarity between patients or brain regions of interest. In these models, constructing high-quality graphs is of paramount importance. Researchers have proposed various methods for constructing graphs from different perspectives, among which the simplest and most popular one is Pearson Correlation (PC). Although existing methods have achieved significant results, these graphs are usually fixed once they are constructed, and are generally operated separately from downstream task. Such a separation may result in neither the constructed graph nor the extracted features being ideal. To solve this problem, we use the graph-optimized locality preserving projection algorithm to extract features and the population graph simultaneously, aiming in higher identification accuracy through a task-dependent automatic optimization of the graph. At the same time, we incorporate supervised information to enable more flexible modelling. Specifically, the proposed method first uses PC to construct graph as the initial feature for each subject. Then, the projection matrix and graph are iteratively optimized through graph-optimization locality preserving projections based on semi-supervised learning, which fully employs the knowledge in various transformation spaces. Finally, the obtained projection matrix is applied to construct the subject-level graph and perform classification using support vector machines. To verify the effectiveness of the proposed method, we conduct experiments to identify subjects with mild cognitive impairment (MCI) and Autism spectrum disorder (ASD) from normal controls (NCs), and the results showed that the classification performance of our method is better than that of the baseline method.
The global incidence of brain diseases, such as autism spectrum disorder and Alzheimer’s disease, has been increasing in recent years, which poses a huge burden to both families and society [
Utilizing resting-state functional magnetic resonance imaging (rs-fMRI) to study brain activity during rest has become an independent field of research, due to its non-invasive, operable, and easily shareable advantages, as well as its potential to serve as the biomarker for psychiatric disorders [
In the exploration of graph estimation methods, various approaches have been proposed and attempted. One popular method is the utilization of similarity measures, such as Pearson’s correlation (PC) [
However, the graphs constructed by these methods are fixed and often not linked to downstream tasks, which can result in both unsatisfactory constructed graph and extracted features, affecting the final performance and effect. In addition, when using certain methods, a considerable amount of manual adjustments and optimization are required to match the constructed graph with downstream tasks. To address this limitation, we propose a novel adaptive graph estimation method for brain diseases that jointly learns the graph and projection matrix [
In the subsequent sections, this paper will further elaborate on the methods, experiments, and conclusions in order. The method section begins by introducing the data source and describing the preprocessing steps. It then proceeds to review the existing baseline works before presenting our proposed method, including the underlying motivation, model, and algorithm. Moving on to the experiment section, we conducted identification experiments on MCI and ASD to validate the feasibility of our method and compare it with the baseline methods. Finally, in the conclusion section, a concise summary of the paper is provided, along with a discussion on potential future research directions.
This study utilized two benchmark databases to validate the effectiveness of the proposed method, including the publicly available ADNI database and ABIDE database with rs-fMRI data. For the ADNI database, we selected 137 subjects (68 MCIs and 69 NCs) and preprocessed the data according to the most recent study [
| Datasets | Class | Gender (M/F) | Age (Years) | MMSE | GCDR | FIQ | VIQ | PIQ |
|---|---|---|---|---|---|---|---|---|
| ADNI | MCI | 39/29 | 76.50 ± 13.50 | 26.77 ± 1.23 | 0.48 ± 0.02 | - | - | - |
| NC | 17/52 | 71.50 ± 14.50 | 28.85 ± 1.15 | 0 | - | - | - | |
| ABIDE | ASD | 68/11 | 18.58 ± 11.45 | - | - | 107.92 ± 3.15 | 105.81 ± 1.23 | 108.81 ± 2.10 |
| NC | 79/26 | 19.13 ± 11.85 | - | - | 113.15 ± 2.45 | 113.13 ± 1.15 | 115.07 ± 2.08 |
For the preprocessing pipeline of each subject in ADNI dataset, the scanning time was 7 min, corresponding to 140 volumes. To address errors caused by magnetic field and signal instability, we removed the first 10 volumes. Then, we performed motion correction and calculated frame-wise displacement (FD) based on head motion parameters, and the subjects with more than 2.5 min of FD larger than 0.5 mm were excluded from the dataset. To reduce the influence of signal from head motion parameters, white matter, and cerebrospinal fluid (CSF), we adopted nuisance regression based on the Friston 24-parameter model. Subsequently, the corrected images were registered to the standard Montreal Neurological Institute (MNI) space and underwent spatial smoothing with band-pass temporal filtering (0.015 - 0.150 Hz). Finally, we divided the brain into 116 regions of interest (ROIs) based on AAL atlas [
For the ABIDE dataset, all fMRI data were acquired on a standard echo-planar imaging sequence using a clinical routine 3.0 Tesla Allegra scanner. 184 subjects (including 79 ASDs and 105 NCs) from the largest site (i.e., NYU) were used in our study. The imaging parameters are as follows: TR/TE is 2000/15 ms with 180 volumes, the number of slices is 33, and the slice thickness is 4.0 mm. The preprocessing pipeline mainly consisted of four steps: 1) volume slice and head motion correction, 2) interference signal regression (including ventricle, white matter signal, and the high-order effects of head motion described by the Friston 24-parameter model), 3) registration to MNI space, and 4) temporal filtering (0.01 - 0.10 Hz). Finally, the brain was segmented into 116 ROIs based on the AAL atlas.
The Pearson correlation is the simplest and widely adopted method for constructing graph, and in this study, we used it as a baseline task to develop our work. Assuming the features of each subject is represented by x i ∈ R t × 1 , where t is the number of the features. The graph edge values based on PC can be defined by the following formula:
w i j ( P C ) = ( x i − x ¯ i ) T ( x j − x ¯ j ) ( x i − x ¯ i ) T ( x i − x ¯ i ) ( x j − x ¯ j ) T ( x j − x ¯ j ) (1)
where x ¯ i ∈ R t × 1 is the mean vector of x i .
If x i is centered by x i − x ¯ i and standardized by ( x i − x ¯ i ) T ( x i − x ¯ i ) denoted as x i , then the above formula can be expressed as an inner product form:
W ( P C ) = X T X (2)
here, X = [ x 1 , x 2 , ⋯ , x r ] ∈ R t × r is the preprocessed data matrix.
SR is one of the commonly-used methods for calculating partial correlation. The mathematical model for SR is expressed as follows:
min w i j ∑ i = 1 n ( ‖ x i − ∑ i ≠ j w i j x j ‖ 2 + δ ∑ i ≠ j | w i j | 1 ) , s .t . w i i = 0 , ∀ i = 1 , ⋯ , n (3)
which can be further rewritten by the following matrix form:
min W ‖ X − X W ‖ F 2 + δ ‖ W ‖ 1 , s .t . w i i = 0 , ∀ i = 1 , ⋯ , n (4)
where ‖ X − X W ‖ F 2 is a data fitting term for capturing the partial correction information, ‖ W ‖ 1 is an l 1 -regularized term for obtaining sparse solutions of W, and δ is a regularization parameter for controlling the balance between these two terms. Note that the constraint w i i = 0 is used here to avoid the trivial solution (i.e., W = I , the identity matrix) by implicitly removing x i from X.
As mentioned earlier, the quality of the graph we construct is crucial for successful classification task. However, due to the lack of fundamental facts, we can only explore the real graph through research. Although previous methods have achieved good results in constructing graph, they are usually fixed once constructed and are generally separated from subsequent downstream task. This separation may lead to suboptimal graph, and extracted features are not ideal. Moreover, some methods cannot naturally integrate supervised information, which may limit application of the method in certain areas such as medicine. In
To address the aforementioned issues, this paper utilizes subject-level graphs in the alternating optimization process, allowing the projection matrix to fully utilize the potential relationships between subjects in the graph, and synchronously learn with graph optimization. Such projection matrix constructs the population graph that merges the subsequent feature extraction and is closely linked
to the downstream task. At the same time, in order to balance the uniformity (or smoothness) and locality of the graph, we regularize the objective function by adding an entropy term. Then, we introduce a pairwise constraint (in the form of must-link/cannot-link constraints) to integrate supervised information into the joint optimization objective, for the construction of subject-level graphs and calculation of the projection matrix, thereby achieving more flexible modeling. This will be detailed in the following sections on algorithms and models.
Given a set of data points X = [ x 1 , x 2 , ⋯ , x r ] , x i ∈ R D , D is the number of upper triangular elements in the data calculated using Pearson correlation. The paired-constraints linking constraint M = { ( x i , x j ) | x i and x j belongtothesameclass } and the paired-constraints non-linking constraint C = { ( x i , x j ) | x i and x j belongtodifferentclass } cannot be linked. The goal of semi-supervised GoLPP is to optimize both the graph and projection directions within a unified framework while maintaining paired constraints. The objective function is defined as follows:
min W , S i j ∑ i , j = 1 n ‖ W T x i − W T x j ‖ 2 S i j ∑ j = 1 n ‖ W T x i ‖ 2 + φ ∑ i , j = 1 n S i j ln ( S i j / α ) , s .t . ∑ j = 1 n S i j > 0 , ∀ i , j = 1 , ⋯ , n S i j = 1, if ( x i , x j ) ∈ M , ∀ i , j = 1, ⋯ , n S i j = 0, if ( x i , x j ) ∈ C , ∀ i , j = 1, ⋯ , n S i j > 0 , otherwise (5)
Although the objective function is not convex, it can be easily solved through Alternating Optimization (AO). The iterative process includes the following two main steps:
Step 1: Initialize the weights for the graph. For example, a simple initialization method is shown below:
S i j = { 1, if ( x i , x j ) ∈ M 0, if ( x i , x j ) ∈ C exp ( − ‖ x i − x j ‖ 2 / σ ) , otherwise (6)
Then, the optimal projection matrix is obtained using a generalized eigenvalue problem. In fact, the solution obtained in this way is not precise, as it involves ratio tracing and ratio tracking problems, which are beyond the scope of our main focus. For more details, please refer to [
Step 2: Revise W = W ¯ and minimize the objective function, and we can obtain the following problem:
min S i j ∑ i , j = 1 n ‖ W T x i − W T x j ‖ 2 S i j + φ ∑ i , j = 1 n S i j ln ( S i j / α ) , s .t . ∑ j = 1 n S i j > 0 , ∀ i , j = 1 , ⋯ , n S i j = 1, if ( x i , x j ) ∈ M , ∀ i , j = 1 , ⋯ , n S i j = 0, if ( x i , x j ) ∈ C , ∀ i , j = 1 , ⋯ , n S i j > 0 , otherwise (7)
In fact, we only need to consider the following problem first, and then add the part of equality constraints for S i j :
min S i j ∑ i , j = 1 n ‖ W T x i − W T x j ‖ 2 S i j + φ ∑ i , j = 1 n S i j ln ( S i j / α ) , s .t . ∑ j = 1 n S i j > 0 , ∀ i , j = 1 , ⋯ , n S i j > 0 , otherwise (8)
where ( x i , x j ) ∉ M and ( x i , x j ) ∉ C . Then, letting ∂ L ∂ S i j = ‖ W T x i − W T x j ‖ 2 + φ ( ln ( S i j / α ) + 1 ) = 0 , We can obtain:
S i j = α ⋅ exp ( − 1 ) ⋅ exp ( − ‖ W T x i − W T x j ‖ 2 / φ ) (9)
α is a positive parameter used only to obtain a more compact and desirable solution under the new constraint. When the distance between two samples approaches 0, we expect the weight S i j to approach 1. We set α = e , then we get:
S i j = exp ( − ‖ W T x i − W T x j ‖ 2 / φ ) (10)
Finally, when we add the equality constraints, we have:
S ¯ i j = { 1, if ( x i , x j ) ∈ M 0, if ( x i , x j ) ∈ C exp ( − ‖ W T x i − W T x j ‖ 2 / σ ) , otherwise (11)
Finally, we update S i j using S ¯ i j and continue to iterate until convergence.
We use this approach to continuously update the construction of the graph and projection matrix, so as to learn important information between subjects in different transformation spaces, such as subject category information. This information is implicitly contained in the projection matrix, so that when the projection matrix is used to map data in the future, it can better distinguish between subjects. In addition, it implies the possibility that two samples belong to the same class. Specifically, if two samples come from the same class, the edge
weight is 1; if they come from different classes, it is 0; exp ( − ‖ W T x i − W T x j ‖ 2 / σ ) ,
if they are unlabeled, then the edge weight lies between 0 and 1. Specifically, when two samples are very close to each other, this value tends to 1; conversely, when samples are far apart from each other, it tends to 0.
We evaluated our proposed method on the MCI and ASD datasets using a leave-one-out (LOO) cross-validation strategy. The performance was measured using four metrics: accuracy (ACC), sensitivity (SEN), specificity (SPE), and the area under the receiver operating characteristic curve (AUC). They intuitively display the overall recognition accuracy, patient recognition accuracy, normal recognition accuracy and confidence level of the model. Here are the definitions for these metrics:
A C C = T P + T N T P + T N + F P + F N (12)
S E N = T P T P + F N (13)
S P E = T N T N + F P (14)
here, TP represents the number of true positive subjects predicted, FN represents the number of false negative subjects predicted, and similarly, TN and FP represent the number of their corresponding experimental subjects.
We compared the proposed the performance of our proposed method with with the baseline methods: PC, SR, the unsupervised form of our method (un-ours) and GCN. Next, we will explain some parameter settings. PC has no parameters, the sparse parameter of SR is selected as the optimal parameter in the range of [2−4, 2−2, 20, 22, 24], and feature extraction uses t-test with a p-value set to 0.01. GCN performs node classification, and the initial graph is constructed using PC. The design of GCN is the same as the original paper [
It is worth noting that our method does not perform feature extraction using t-test because the projection matrix learned during the process of learning the graph can map the data to the optimal position without the need for feature extraction. At the same time, we use thresholding to ensure a certain level of sparsity in the brain constructed by our method, and we discuss the impact of different levels of sparsity on our method. In our method, To simulate the lack of labeled data, we assume that 10% of the training samples are labeled and used to create must-link/cannot-link constraints depending on whether these samples belong to the same class. In this paper, we use leave-one-out cross-validation to split the data. Finally, we use SVM to predict the labels and calculate four evaluation metrics, namely accuracy (ACC), sensitivity (SEN), specificity (SPE), and area under the curve (AUC).
As shown in
| Dataset | Method | ACC | SEN | SPE | ROC |
|---|---|---|---|---|---|
| ASD | PC | 61.41% | 58.23% | 63.81% | 69.83% |
| SR | 66.85% | 46.84% | 81.90% | 68.61% | |
| GCN | 67.39% | 65.82% | 68.57% | 67.20% | |
| Un-ours | 69.02% | 62.03% | 74.29% | 73.14% | |
| Ours | 70.11% | 63.29% | 75.24% | 73.16% | |
| MCI | PC | 78.10% | 77.94% | 78.26% | 85.68% |
| SR | 84.67% | 85.29% | 84.06% | 91.86% | |
| GCN | 85.40% | 86.76% | 84.06% | 85.41% | |
| Un-ours | 85.40% | 83.82% | 86.96% | 89.37% | |
| Ours | 86.13% | 83.82% | 88.41% | 89.49% |
outperforms the comparison method, which illustrates the benefit of considering the co-optimization of the projection matrix of the graph and feature extraction during the construction of the graph. In addition, the inclusion of appropriate supervision information also contributes to the performance improvement. In
In this paper, we construct task-dependent adaptive graph in a semi-supervised scene using a relaxed constrained graph-optimized holdout projection, which fits well into the context of the lack of labelling in medicine. We make use of the projection matrix to map the data to the learned best space, and use the node features in the graph to classify by SVM. The experimental results show that the method achieves 86.13% and 70.11% correct rates for the MCI and ASD recognition tasks based on the leave-one-out method, respectively, and outperforms the baseline method in terms of ACC and SPE. This further illustrates the importance of task-dependent adaptive graph and reliable supervised information to improve the generalization of subsequent classifier. Our model has very few hyperparameters, which avoids the difficulty of parameter selection, but on the other hand, it also reduces the flexibility of the model. In addition, heterogeneity between data from different sites is also a limitation. In the future, we will try to design more powerful and adaptive graph learning methods to overcome the problems of hyperparameter selection and site heterogeneity.
The authors declare no conflicts of interest regarding the publication of this paper.
Zhang, K.P., Zhang, Y.N. and Liu, X.Y. (2023) Semi-Supervised Graph Learning for Brain Disease Identification. Journal of Applied Mathematics and Physics, 11, 1846-1859. https://doi.org/10.4236/jamp.2023.117119