# Exploring Brain Imaging and Genetic Risk Factors in Different Progression States of Alzheimer's Disease Through OSnetNMF-Based Methods

Min Gao, Wei Kong, Kun Liu, Gen Wen, Yaling Yu, Yuemin Zhu, Zhihan

Jiang, Kai Wei

► To cite this version:

Min Gao, Wei Kong, Kun Liu, Gen Wen, Yaling Yu, et al.. Exploring Brain Imaging and Genetic Risk Factors in Different Progression States of Alzheimer's Disease Through OSnetNMF-Based Methods. Journal of Molecular Neuroscience, 2025, 75 (1), pp.7. 10.1007/s12031-024-02274-8. hal-04926870

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Submitted on 3 Feb 2025

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<figure><img src="image_1.png" /></figure>
Exploring Brain Imaging and Genetic Risk Factors in Different Progression States of
Alzheimer's Disease Through OSnetNMF-Based Methods

Min Gao¹†, Wei Kong¹†*, Kun Liu¹, Gen Wen², Yaling Yu²,³, Yuemin Zhu⁴, Zhihan Jiang¹, Kai Wei⁵

¹College of Information Engineering, Shanghai Maritime University, 1550 Haigang Ave., Shanghai 201306, P. R. China

²Department of Orthopedic Surgery, Shanghai Sixth People's Hospital Affiliated to Shanghai Jiao Tong University School of Medicine, Shanghai, 200233, China

³Institute of Microsurgery on Extremities, Shanghai Sixth People's Hospital Affiliated to Shanghai Jiao Tong University School of Medicine, Shanghai, 200233, China

⁴University Lyon, INSA Lyon, CNRS, Inserm, IRP Metislab CREATIS UMR 5220, U1294, Lyon 69621, France.

⁵Bio-Med Big Data Center, CAS Key Laboratory of Computational Biology, Shanghai Institute of Nutrition and Health, University of Chinese Academy of Sciences, Chinese Academy of Sciences, Shanghai, China

†The first two authors should be regarded as joint first authors.

*Correspondence: author: Wei Kong (e-mail: weikong@shmtu.edu.cn)

**Abstract:** Alzheimer's disease (AD) is a neurodegenerative disease with multiple factors for which there is currently no effective treatment. Mild cognitive impairment (MCI) is an early disease that may progress to AD. Multimodal imaging genetics can integrate imaging data and genetic data to gain a deeper understanding of the progression of complex diseases and the individual variations. Exploring the pathogenic mechanisms from normal progression to MCI and ultimately leading to AD will contribute to understanding the pathogenesis of AD and providing insights for early diagnosis. As an effective joint feature extraction and dimensionality reduction method, non-negative Matrix Factorization (NMF) and its improved methods, especially the netNMF method, have been widely used in the field of multimodal analysis for mining brain imaging and genetics data by considering the interactions between different features. However, many of these methods overlook the importance of the coefficient matrix and do not address issues related to data accuracy and feature redundancy. To cope with this problem, we propose an orthogonal sparse network nonnegative matrix factorization (OSnetNMF) algorithm by adding orthogonal and sparse constraints based on netNMF. By establishing linear relationships between structural magnetic resonance imaging (sMRI) and corresponding gene expression data and with orthogonal and sparsity constraints as conditions, the proposed OSnetNMF is able to reduce the redundancy of features and decrease the correlation between the data, resulting in more accurate and reliable biomarker extraction. Experiments showed that the OSnetNMF algorithm allows us to accurately identify the risk regions of interest (ROIs) and key genes that characterize the progression of AD, as well as some obvious trends of ROI pairs, such as l4thVen-HIF1A, rBst-MPO, rBst-PTK2B, etc. Meanwhile, comparative experiments showed that the proposed algorithm can find more disease-related biomarkers with better reconstruction performance.

# 1. Introduction

Alzheimer's disease (AD) is a severe neurodegenerative disease, which is the most common type of dementia. It is mainly manifested as cognitive decline and non-cognitive neuropsychiatric symptoms[1]. While the precise cause of AD remains elusive, it is preceded by mild cognitive impairment (MCI), a condition featuring symptoms like memory loss and potential cognitive
deficits[2]. Therefore, studying how MCI progresses to AD and the correlation between the two disorders will greatly contribute to the exploration of AD biomarkers and their early diagnosis. Previous research suggests a connection between structural and functional brain abnormalities and molecular irregularities associated with AD[3]. In recent years, the field of brain diseases has witnessed growing interest in imaging genetics analysis. The present study investigates how genetic variations impact brain structure, metabolism, and function by analyzing radiomics data (such as sMRI, PET, fMRI) alongside genetic data (including DNA methylation, SNPs, gene expression), contributes to the study of neurodegenerative diseases such as AD[4-5].

As a traditional imaging genomics method, canonical correlation analysis (CCA) and its improved algorithms, such as sparse canonical correlation analysis (SCCA), sparse multi-view canonical correlation analysis (SMCCA), adaptive sparse multi-view canonical correlation analysis (AdaSMCCA) and multi-constrained uncertainty-aware adaptive sparse multi-view canonical correlation analysis (MC-unAdaSMCCA), can find the maximum correlation between multiple sets of variables to discover the correlation between datasets; they are widely used in the correlation analysis of multimodal data[6-9]. However, the CCA-based algorithm can only find a pair of canonical correlated variables at a time, which means that it can only analyze one module. To obtain more canonical correlated variables, it is necessary to progressively establish new CCA models based on existing research results, making the entire modeling process more cumbersome and time-consuming.

As an effective joint feature extraction and analysis method, the Non-negative Matrix Factorization (NMF) algorithm and its extended models can simultaneously obtain multiple sets of modules and weights, overcoming the limitations of CCA-based methods, which are restricted to a single pair of canonical variables. Original NMF is a low-dimensional reduction method that has been widely used for analyzing imaging genomics datasets[10], which can project data matrices from different modalities onto a common expression space, thereby integrating different knowledge into a standard module consisting of variables with more significant coefficients in the same projection direction. Since NMF does not consider the lack of prior information in the data, the generated results may be restricted. Wang et al. proposed the group sparse joint non-negative matrix factorization (GSJNMF). to address this issue[11]. GSJNMF, can identify multiple sets of canonical correlated variables simultaneously. However, this NMF-based method does not consider the correlation between base vectors, which may extract irrelevant features from the coefficient matrix. To address this, Peng et al. proposed the group sparse of joint non-negative matrix factorization on orthogonal subspace (GJNMFO)[12]. GJNMFO enhances the orthogonal and group sparse constraints in matrix decomposition and projects multimodal data into a low-dimensional orthogonal space, resulting in more accurate feature extraction. To further explore the correlation and regulatory relationship between imaging data and genomic data, the Network-based Nonnegative Matrix Factorization (netNMF) method was proposed[13]. netNMF combines non-negative matrix factorization (NMF) technology with network data to reveal latent patterns and structures in high-dimensional data. Its core idea is to decompose the input matrix into non-negative submatrices containing pattern and structural information[14]. However, it overlooks the interactions between different features. Ding et al. proposed a Non-negative Matrix Factorization Network Analysis (NMFNA)[15], which introduces a graph regularization constraint
into NMF. This method decomposes integrated data into submatrices for network construction,
providing a better integration and decomposition of different types of data.

In order to efficiently solve the interaction between multimodal data modules and features, based
on the above approaches, we propose in the present study a novel method, called OSnetNMF,
which introduces orthogonal and sparse constraints into the netNMF to better integrate biological
information and reduce redundancy for exploring the relationship between genes and sMRI of AD
and MCI. In OSnetNMF, the orthogonal constraints imposed by forcing certain columns of the
coefficient matrix to be orthogonal aim to reduce information redundancy. And the sparse
constraints aid in selecting the most important features for data representation, focusing attention
on critical biological information. By reducing the density of the coefficient matrix, sparse
constraints allow us to enhance the interpretability of the algorithm, making the features obtained
more biologically meaningful. Through the OSnetNMF algorithm, factors in imaging genetics
reflecting the progression of AD, such as l4thVen-HIF1A, rBst-MPO, rBst-PTK2B, were
discovered. Furthermore, by comparing the enrichment quantity of the most significant modules
and their reconstruction functions with OSnetNMF, netNMF and NMFNA, the effectiveness of the
proposed algorithm was validated.

## 2.Methods

2.1 The architecture of the proposed OSnetNMF

The architecture of the proposed OSnetNMF consists of two parts, as shown in figure 1. The one
is the imaging genetic modal extraction based on OSnetNMF, the other is the model comparison
and bioinformatics analysis. In the first part, after data preprocessing, the OSnetNMF algorithm is
employed. Firstly, using individual and mutual Pearson correlation coefficient matrices, three
correlation networks are constructed: the sMRI network, gene network, and their mutual
correlation network. Then, sparse and orthogonal constraints are introduced to better integrate
biological information, enhance result accuracy and identify key genes and brain regions in the
most significant network modules (a focal point of OSnetNMF). In the second part, one aspect
involves contrast experiments to validate the effectiveness of the algorithm. Another aspect is
bioinformatics analysis, which includes, Gene Ontology (GO) and Kyoto Encyclopedia of Genes
and Genomes (KEGG) enrichment analysis, protein interaction network (PPI), Related Heatmaps,
the study seeks key gene-brain region pairs undergoing continuous changes from CN to MCI and
progressing to AD.
<figure><img src="image_2.png" /><figcaption>Figure 1 | The architecture of the Orthogonal Sparse Network Non-negative Matrix Factorization (OSnetNMF) integrating genes and ROIs data of Alzheimer's disease (AD)/mild cognitive impairment (MCI).</figcaption></figure>

## 2.2. Network non-negative matrix factorization (netNMF)

The original NMF (Non-negative Matrix Factorization) is a non-negative matrix decomposition algorithm used for data dimensionality reduction and feature extraction[10]. It is widely applied in fields such as signal processing, image processing, and text mining. It decomposes the matrix $X \in \mathbf{R}^{m \times n}$ to obtain the basis matrix $W \in \mathbf{R}^{m \times k}$ and the coefficient matrix $H \in \mathbf{R}^{k \times n}$. Its objective function is as follows:

$$ \min_{W,H} \| X - WH \|_F^2 \quad \text{s.t. } W, H \ge 0 \qquad (1) $$

The traditional NMF algorithm can only decompose single-modal data and cannot analyze the network control module composed of multi-modal data. In addition to the two-factor NMF, the three-factor NMF also plays an important role in matrix factorization, which constrains scales of $W$ and $H$ by an extra factor $S$, i.e., $X \approx WSH$. This factored matrix $S$ not only provides an additional degree of freedom to make the approximation tight, but also indicates associations between identified modules[13]. A three-factor NMF variant TriNMF is defined as[16],

$$ \min_{W,H} \| X - WSH \|_F^2 \quad \text{s.t. } W, S, H \ge 0 \qquad (2) $$

which minimizes the objective function by,

$$ w_{ik} \leftarrow w_{ik} \frac{(XHS^T)_{ik}}{(WW^T XHS^T)_{ik}} \qquad (3) $$
$$h_{kj} \leftarrow h_{kj} \frac{(X^TWS)_{kj}}{(HH^T X^TWS)_{kj}} \quad (4)$$

$$s_{kk} \leftarrow s_{kk} \frac{(W^T X H)_{kk}}{(W^T W S H^T H)_{kk}} \quad (5)$$

Particularly, if $X$ is the symmetric similarity matrix, it could be decomposed into $USU^T$. Chen and Zhang proposed the netNMF algorithm to decompose multiple matrices simultaneously[13]. It is worth noting that the algorithm can be used to analyze the network modules of various data. For example, their respective correlation networks and mutual correlation networks can be calculated for analysis for the two expression matrices $X_1 \in R^{n \times p}$, $X_2 \in R^{n \times q}$. The objective function of the netNMF algorithm is formulated as:

$$\min_{G_1, G_2, S_{11}, S_{22}} \|R_{11} - G_1 S_{11} G_1^T\|_F^2 + \|R_{12} - G_1 G_2^T\|_F^2 + \|R_{22} - G_2 S_{22} G_2^T\|_F^2 \\ \text{s.t. } G_1, G_2, S_{11}, S_{22} \ge 0 \quad (6)$$

Among them, $R_{11} \in R^{p \times p}$ and $R_{22} \in R^{q \times q}$ are the respective association networks of $X_1$ and $X_2$. $R_{12} \in R^{p \times q}$ is the cross-correlation network of $X_1$ and $X_2$. $G_1 \in R^{p \times k}$, $G_2 \in R^{q \times k}$, $S_{11} \in R^{k \times k}$ and $S_{22} \in R^{k \times k}$ are non-negative factor matrices. The algorithm can be analyzed from the network module level constructed by the data, and $\|R_{12} - G_1 G_2^T\|_F^2$ is used to identify the one-to-one relationship between the two types of modules.

In fact, $R_{11} \in R^{p \times p}$ and $R_{22} \in R^{q \times q}$ are the symmetric similarity matrices corresponding to the two types of features, respectively, and $R_{12} \in R^{p \times q}$ is the nonnegative similarity matrix between the two types of data. $S_{11}$ and $S_{22}$ represent the matrix describing the similarity of the two networks obtained from the decomposition of $R_{11}$ and $R_{22}$, respectively. Off-diagonal elements in $S_{11}$ and $S_{22}$ indicate the importance of relationships between ROIs and between genes. The netNMF minimizes the objective function by

$$(g_1)_{ik} \leftarrow (g_1)_{ik} \frac{(2R_{11}G_1S_{11} + R_{12}G_2)_{ik}}{(2G_1S_{11}G_1^T G_1S_{11} + G_1G_2^T G_2)_{ik}} \quad (7)$$

$$(g_2)_{kj} \leftarrow (g_2)_{kj} \frac{(2R_{22}G_2S_{22} + R_{12}G_1)_{kj}}{(2G_2S_{22}G_2^T G_2S_{22} + G_2G_1^T G_1)_{kj}} \quad (8)$$

$$(s_{11})_{kk} \leftarrow (s_{11})_{kk} \frac{(G_1^T R_{11} G_1)_{kk}}{(G_1^T G_1 S_{11} G_1^T G_1)_{kk}} \quad (9)$$

$$(s_{22})_{kk} \leftarrow (s_{22})_{kk} \frac{(G_2^T R_{22} G_2)_{kk}}{(G_2^T G_2 S_{22} G_2^T G_2)_{kk}} \quad (10)$$

## 2.3. Orthogonal Sparse Network Non-negative Matrix Factorization (OSnetNMF)

This study employs the method proposed in to impose $l_1$-norm constraints on matrix $H$[17]. Additionally, by effectively integrating biological information, we implement orthogonal
constraints on matrix **H**, reducing the correlation between coefficient matrices. This not only minimizes redundancy between different coefficient matrices but also imparts sparsity to the matrix decomposition obtained from the feature matrix. Consequently, more features relevant to biological learning are discovered. Finally, the objective function obtained is as follows:

$$ \min_{G_1, G_2, S_{11}, S_{22}} \|R_{11} - G_1 S_{11} G_1^T\|_F^2 + \|R_{12} - G_1 G_2^T\|_F^2 + \|R_{22} - G_2 S_{22} G_2^T\|_F^2 + \\ \alpha \sum_{I=1}^2 e_{1 \times k} H_I H_I^T e_{I \times k}^T + \beta \|H_I H_I^T - I\|_F^2 \quad \text{s.t. } G_1, G_2, S_{11}, S_{22} \ge 0 \quad (11) $$

Where **R**<sub>11</sub> and **R**<sub>22</sub> are the ROI and gene co-expression networks, **R**<sub>12</sub> is the ROI-gene co-expression network, α is used to constrain **H**, and β is the orthogonal constraint Hyperparameters. Other notation meanings and parameter settings are the same as those in the netNMF.

The multiplicative iterative update algorithm is adopted here to minimize the objective function of the NMFNA[15]. L<sub>1</sub> and L<sub>2</sub> are the hypergraph Laplacian matrices of **X**<sub>1</sub> and **X**<sub>2</sub>, respectively. Further, let **Ψ**<sub>1</sub>, **Ψ**<sub>2</sub>, **Φ**<sub>1</sub> and **Φ**<sub>2</sub> be Lagrangian multipliers, respectively, constrain **S**<sub>11</sub> ≥ 0, **S**<sub>22</sub> ≥ 0, **G**<sub>1</sub> ≥ 0, and **G**<sub>2</sub> ≥ 0, the Lagrange function f of the NMFNA is,

$$ \begin{aligned} L = & \operatorname{tr}((R_{11} - G_1 S_{11} G_1^T)^T (R_{11} - G_1 S_{11} G_1^T)) \\ & + \operatorname{tr}((R_{12} - G_1 G_2^T)^T (R_{12} - G_1 G_2^T)) \\ & + \operatorname{tr}((R_{22} - G_2 S_{22} G_2^T)^T (R_{22} - G_2 S_{22} G_2^T)) \\ & + \alpha \operatorname{tr}((e_{1 \times k} H_1 H_1^T e_{1 \times k}^T)^T (e_{1 \times k} H_1 H_1^T e_{1 \times k}^T)) \\ & + \alpha \operatorname{tr}((e_{1 \times k} H_2 H_2^T e_{1 \times k}^T)^T (e_{1 \times k} H_2 H_2^T e_{1 \times k}^T)) \\ & + \beta \operatorname{tr}((H_1 H_1^T - I)^T (H_1 H_1^T - I)) \\ & + \operatorname{tr}(\Psi_1^T S_{11}) + \operatorname{tr}(\Psi_2^T S_{22}) + \operatorname{tr}(\Phi_1^T G_1) + \operatorname{tr}(\Phi_2^T G_2) \end{aligned} \quad (12) $$

The following formula can be obtained using (12) to calculate the partial derivatives of **S**<sub>11</sub>, **S**<sub>22</sub>, **G**<sub>1</sub> and **G**<sub>2</sub>, respectively.

$$ \frac{\partial f}{\partial S_{11}} = -2G_1^T R_{11} G_1 + 2G_1^T G_1 S_{11} G_1^T G_1 + \Psi_1 \quad (13) $$

$$ \frac{\partial f}{\partial S_{22}} = -2G_2^T R_{22} G_2 + 2G_2^T G_2 S_{22} G_2^T G_2 + \Psi_2 \quad (14) $$

$$ \frac{\partial f}{\partial G_1} = 4(G_1 S_{11} G_1^T G_1 S_{11} - R_{11} G_1 S_{11} + \beta G_1 G_1^T G_1 - \beta G_1) + 2(G_1 G_2^T G_2 - R_{12} G_2) + \Phi_1 \quad (15) $$

$$ \frac{\partial f}{\partial G_2} = 4(G_2 S_{22} G_2^T G_2 S_{22} - R_{22} G_2 S_{22} + \beta G_2 G_2^T G_2 - \beta G_2) + 2(G_2 G_1^T G_1 - R_{12} G_1) + \Phi_2 \quad (16) $$
Further, through the Karush-Kuhn-Tucher (KKT) condition, the iteration rules of $S_{11}$, $S_{22}$, $G_1$ and $G_2$ can be obtained, as given below. The specific algorithm flow is depicted in Figure 1 OSnetNMF section.

$$ (g_1)_{ik} \leftarrow (g_1)_{ik} \frac{(2R_{11}G_1S_{11}+R_{12}G_2+2\beta G_1)_{ik}}{(2G_1S_{11}G_1^T G_1S_{11}+G_1G_2^T G_2+\alpha G_1+2\beta G_1G_1^T G_1)_{ik}} \quad (17) $$

$$ (g_2)_{kj} \leftarrow (g_2)_{kj} \frac{(2R_{22}G_2S_{22}+R_{12}^T G_1 2\beta G_2)_{kj}}{(2G_2S_{22}G_2^T G_2S_{22}+G_2G_1^T G_1+\alpha G_2+2\beta G_2G_2^T G_2)_{kj}} \quad (18) $$

$$ (s_{11})_{kk} \leftarrow (s_{11})_{kk} \frac{(G_1^T R_{11} G_1)_{kk}}{(G_1^T G_1 S_{11} G_1^T G_1)_{kk}} \quad (19) $$

$$ (s_{22})_{kk} \leftarrow (s_{22})_{kk} \frac{(G_2^T R_{22} G_2)_{kk}}{(G_2^T G_2 S_{22} G_2^T G_2)_{kk}} \quad (20) $$

## 2.4. Network module selection method

Two types of modules, $G_1$ and $G_2$, can be identified from the ROI matrix $R_{11}$ and gene network matrix $R_{22}$ constructed in the present study. Specifically, zscore is introduced here to calculate the weights of genes and ROIs. The genes and ROIs with weights above the threshold are treated as members of the corresponding community.

$$ Z_{ij} = \frac{(G_l)_{ij} - \mu_{(G_l)_j}}{\sigma_{(G_l)_j}} \quad (l = 1, 2) \quad (21) $$

Among them, $\mu_{(G_l)_j} = \frac{1}{N} \sum (G_l)_{ij}$, $\sigma^2_{(G_l)_j} = \frac{1}{N} \sum ((G_l)_{ij} - \mu_{(G_l)_j})^2$. In the present study, the threshold was set to 1.

# 3. Results

## 3.1. Data sources and preprocessing

The real data we used are all from the ADNI database (http://ADNI.loni.usc.edu/). We collected sMRI imaging and gene expression data from 386 non-Hispanic Caucasian participants in this study; these participants included 25 AD participants, 248 mild cognitive impairment (MCI), and 113 healthy controls (HC). Table 1 gives characterization statistics of the participant sample.

For sMRI data, the original sMRI images were downloaded from ADNI1 in the experiment, which used DiffusionKit software to achieve head movement correction of the original sMRI image[18]. They registered it to the Montreal Neurological Institute (MNI) standard space. Next, the segmentation of MRI images was implemented by using MATLAB software CAT toolkit in the SPM software package[19]. Specifically, the CAT toolkit provided a voxel-based morphometric measurement (VBM) function by scaling the volume change due to spatial registration to adjust the segmentation. The volume of gray matter tissue was calculated in the ROI as a feature[20].
After screening, the gray matter volumes of 140 ROIs were finally extracted as imaging features[21-22].

Significant differential expression analysis was performed using the limma package for gene expression data[23]. Those genes were removed when the p-value was higher than 0. 01; 4026 genes were obtained[21]. Where the samples were grouped as diseased and not diseased (AD and MCI defined as diseased, HC defined as not diseased).

In order to explore the changes in the progression of CN to MCI and eventually to AD, we transported the data of AD and MCI into OSnetNMF separately for the study.

<table><caption>TABLE1. Characterization statistics of the participant sample.</caption><thead><tr><th>Groups</th><th>AD</th><th>MCI</th><th>HC</th></tr></thead><tbody><tr><td>Number</td><td>25</td><td>248</td><td>113</td></tr><tr><td>Gender(M/F)</td><td>10/15</td><td>133/115</td><td>58/55</td></tr><tr><td>Age(mean±std)</td><td>75.99±10.22</td><td>71.900±7.34</td><td>75.06±5.68</td></tr><tr><td>MMSE (mean)</td><td>20.48</td><td>27.90</td><td>29.00</td></tr></tbody></table>

## 3.2. Parameter Selection

Two parameters need to be selected, namely $\alpha$ and $\beta$. The total module similarity as objective function value is used as the selection standard. By iterating from 0. 0001 to 0.1, the parameters are selected based on the magnitude of similarity of the compared modules. Two modules similarity[24] is defined as,

$$mSim = \sum_{x,y} \frac{|c_x \cap c_y|}{\min\{|c_x|,|c_y|\}} \qquad (22)$$

and the modules whose sizes are smaller than five were removed. In the above, $C_x$ is the members set of module x. The schematic diagram of parameter selection is shown in Figure 2(A, B). After several iterations, we selected AD group $\alpha=0.0001$, $\beta=0.0004$; MCI group $\alpha=0.0001$, $\beta=0.0009$.

In addition, according to the selection experience of previous literature, the value of k generally does not exceed one-tenth of the minimum number of samples (or features) for the number of genes-ROIs network modules k. The number of Modules Selection Parameter k is determined by the singular value decomposition method, and its first inflection point is selected to k[25]. In order to reduce the decomposition error, a large iteration number is used. In the study, we set it to 200 since the decomposition error here has already reached a relatively stable state (Figure 2(C, D)). Therefore, in the present study, the value of k was set to 9 for the AD and MCI groups.
<figure><img src="image_3.png" /><figcaption>Figure 2 | Parameter settings and module number selection for the OSnetNMF. A and B are cases with setup parameters for the AD and MCI groups, respectively. C and D are module selections for AD and MCI groups, respectively.</figcaption></figure>

### 3.3. Network module selection

The gene and brain region modules of the AD and MCI groups undergo OSnetNMF will result in 9 gene network modules and 9 sMRI network modules. The two valid network constructions of Gene and sMRI are based on the mean of the values of each element of the reconstruction matrices $R'_{11}$ and $R'_{22}$ based on the respective original matrices $R_{11}$ and $R_{22}$. The two reconstruction matrices $R'_{11}$ and $R'_{22}$ are formulated as

$$ 
\begin{aligned}
R'_{11} &= G_1 S_{11} G_1^T \\
R'_{22} &= G_2 S_{22} G_2^T
\end{aligned} 
\qquad (23) 
$$

When the element value is greater than the mean, it can be determined that there is an interaction between the corresponding gene/brain region. The AD group is discarded because sMRI network module 1 does not contain any elements. The MCI group is discarded because gene network modules 1, 3, 4, 6, and 7 do not contain any elements. In addition, to determine the most significant module, we calculated the relative error between the original values and the reconstructed values of the elements contained in the gene modules, as provided in Table 2. We selected the module with the smallest relative error in the network module as the most significant module. In both the AD and MCI groups, module 2 has the smallest relative error, and the corresponding number of brain regions is also moderate. Therefore, subsequent analyses of AD
and MCI will focus on module 2.

### 3.4. Algorithm comparison

To demonstrate the superiority of the algorithm, we compared it with netNMF and NMFNA on real data, conducting Gene Ontology (GO) and Kyoto Encyclopedia of Genes and Genomes (KEGG) pathway enrichment analyses on the most significant modules identified by each algorithm. The GO and KEGG pathway enrichment analyses were performed using the website https://david.ncifcrf.gov/summary.jsp, with GO analysis utilizing GOTERM_BP_DIRECT, GOTERM_CC_DIRECT, and GOTERM_MF_DIRECT.

<table><caption>TABLE2. Module Related Information.</caption><thead><tr><th>Module Number</th><th>Relative error of AD network modules</th><th>Module Number</th><th>Relative error of MCI network modules</th></tr></thead><tbody><tr><td>2</td><td>2.4856</td><td>2</td><td>2.0971</td></tr><tr><td>3</td><td>16.7726</td><td>5</td><td>3.7638</td></tr><tr><td>4</td><td>3.0879</td><td>8</td><td>4.5449</td></tr><tr><td>5</td><td>5.4154</td><td>9</td><td>2.7389</td></tr><tr><td>6</td><td>7.4458</td><td></td><td></td></tr><tr><td>7</td><td>3.1574</td><td></td><td></td></tr><tr><td>8</td><td>9.6078</td><td></td><td></td></tr><tr><td>9</td><td>3.2247</td><td></td><td></td></tr></tbody></table>

Figure 3 shows the number of significantly enriched GO and KEGG pathways with P-values less than 0.05 for AD and MCI's most significant modules identified by the three comparison algorithms. From the figure, it is evident that OSnetNMF identifies more enriched GO and pathway categories for the most significant modules in AD or MCI compared to the other two algorithms. This suggests that the modules identified by OSnetNMF may contain more biological information relevant to understanding the pathogenic mechanisms of AD and MCI.

To further validate the performance of the algorithm, we compared the correlation coefficient results of the OSnetNMF algorithm with those of the other two algorithms in the AD and MCI groups using real data. Specifically, we calculated the Pearson correlation coefficients between the reconstructed matrices $R'_{11}$, $R'_{22}$ and $R'_{12}$ and their respective original matrices $R_{11}$, $R_{22}$, $R_{12}$ as shown in Table 3. With respect to $R'_{11}$ and $R'_{22}$ given above, $R'_{12}$ is formulated as

$$R'_{12} = G_1 G_2^T \qquad (24)$$

The corr in Table 3 indicates the Pearson correlation coefficient between the two. From Table 3, it can be seen that the proposed algorithm outperforms the other algorithms in the reconstruction of AD groups $R_{11}$ and $R_{22}$. In MCI group, the algorithm outperforms other algorithms in reconstructing $R_{11}$ and $R_{12}$ matrices. This confirms that the orthogonal sparse constraints help to improve the reconstruction performance of the algorithm. In addition, we plot in Fig. 4 the scatter plots of the three algorithms between the three original matrices and the three reconstructed matrices in the AD and MCI groups. From Fig. 4, it can be seen that the two algorithms have comparable reconstruction performance in reconstruction $R_{11}$. However, the performance is
better in reconstruction $R_{22}$ in AD group, the algorithm performs better in reconstruction $R_{12}$ in MCI group.

<figure><img src="image_4.png" /><figcaption>Figure 3 | Number of GO and KEGG pathway enrichment analyses for selected significant modules 2. A and B are number of modules analyzed for the AD and MCI groups, respectively.</figcaption></figure>

<table><caption>TABLE3. Pearson correlation coefficients between the three original values obtained by the three algorithms and their respective reconstructed values.</caption><thead><tr><th></th><th>AD</th><th>netNMF</th><th>NMFNA</th><th>OSnetNMF</th></tr></thead><tbody><tr><td rowspan="3">AD</td><td>corr(R<sub>11</sub>, R<sub>11</sub>)</td><td>0.9348</td><td>0.9285</td><td>0.9349</td></tr><tr><td>corr(R<sub>22</sub>, R<sub>22</sub>)</td><td>0.8662</td><td>0.8455</td><td>0.9194</td></tr><tr><td>corr(R<sub>12</sub>, R<sub>12</sub>)</td><td>0.9211</td><td>0.9238</td><td>0.8548</td></tr><tr><td rowspan="3">MCI</td><td>corr(R<sub>11</sub>, R<sub>11</sub>)</td><td>0.9334</td><td>0.9302</td><td>0.9353</td></tr><tr><td>corr(R<sub>22</sub>, R<sub>22</sub>)</td><td>0.8968</td><td>0.8912</td><td>0.8546</td></tr><tr><td>corr(R<sub>12</sub>, R<sub>12</sub>)</td><td>0.8531</td><td>0.8520</td><td>0.8879</td></tr></tbody></table>

# 4. Discussion

## 4.1. GO and KEGG enrichment analysis

To explore the meaningful biomarkers in the modules derived from the OSnetNMF model, we performed KEGG and GO enrichment analyses on the genes in the modules with the most significant AD and MCI, screened out the gene pathways with P values less than 0.05 and with significant effects, and plotted the enrichment analyses as shown in Figure 5.
<figure><img src="image_5.png" /><figcaption>Figure 4 | The associated scatter plots of the original matrices R<sub>11</sub>, R<sub>12</sub>, R<sub>22</sub> and their reconstructed matrices R'<sub>11</sub>, R'<sub>12</sub>, R'<sub>22</sub> for the AD and MCI groups. A, B and C are the three scatter plots obtained by netNMF in the AD group. D E and F are the scatter plots of NMFNA in the AD group. G H and I are the OSnetNMF scatter plots in the AD group. J, K and L are the three scatter plots obtained for netNMF in the MCI group. M, N and O are the scatterplot of NMFNA in MCI group. P, Q and R are the OSnetNMF scatterplot in the MCI group.</figcaption></figure>

Figure 5(A, B) lists the top 15 pathways ranked by P-value of KEGG enrichment in the most significant modules for AD and MCI identified according to OSnetNMF, where the size and color of the nodes represent the number of genes enriched in the pathway and the importance of the pathway, respectively. The enrichment results showed that the AD/MCI modules derived from the OSnetNMF model were indeed signaling pathways closely related to AD/MCI. In the AD group, AD risk genes were overrepresented in the complement and coagulation cascade was relatively enriched in microglia, endothelia, and pericytes of the human brain[26]. Air pollution (TRAP) alters gut microbial abundance and bile acid metabolism in a transgenic model of Alzheimer's disease[27]. Immune-related pathways in the cerebral cortex through phagosomes as well as chemotactic signaling pathways have been shown to be associated with AD through transcriptomic meta-analysis[28]. Platelet Activation Can Lead to Elevated Plasma Aβ Levels and Promote Alzheimer's Disease[29]. Overexpression of kinesin superfamily motor proteins is associated with Alzheimer's disease[30].

In the MCI group, it is possible that complementary and coagulation cascades involving dopaminergic neurodegeneration can lead to motor dysfunction and ultimately cognitive impairment[31]. The “complementary and coagulation cascades” pathway is present in both AD and MCI, indicating that there are genes that may play a role in both MCI and AD, potentially leading to the progression from MCI to AD. Formation of neutrophil extracellular traps (NETs) or adhesion of neutrophils, which may disrupt the blood-brain barrier and damage brain parenchyma, which in turn affects mild cognitive impairment[32]. Cognitive impairment can regulate the
inflammatory cascade through the STAT3 signaling pathway and affect adaptive immunity by regulating Th17/Treg cell differentiation[33]. Cognitive dysfunction occurs in 40% of patients with acute myelogenous leukemia[34]. Platelet activity was found to be a significant predictor of the degree of cognitive impairment disorder[35]. The "Platelet activation" pathway is present in both AD and MCI, suggesting that genes influencing this pathway may contribute to the progression from MCI to AD.

To further analyze the biological information related to the pathological mechanism, we conducted GO pathway analysis in the most significant module of AD, including three pathways: GOTERM_BP_DIRECT, GOTERM_CC_DIRECT, and GOTERM_MF_DIRECT, as shown in Figure 5(C, D). Among the molecular functions of the most prominent modules of AD and MCI, Innate Immune Respons: Innate immune response that plays an important role in the pathogenesis of AD/MCI can be regulated by IL-33[36]. Extracellular Exosome: Extracellular vesicles (EVs), including exosomes and microvesicles, have been shown to play a role in the central nervous system, leading to the removal of unwanted biomolecules as well as the dissemination of pathogenic proteins associated with neurodegenerative diseases, among others[37]. Actin Filament Binding, which is closely related to neuronal development and cognitive function, is the plastic activity of synaptic structure and function. It is mainly controlled by the interaction of actin filaments (F-actin) with various actin-binding proteins (ABPs) and postsynaptic signaling messengers[38]. The three mentioned GO enrichment pathways appear in both the AD and MCI modules. It is possible that the genes associated with these pathways play a crucial role in the transition from MCI to AD.

**Oxygen Transport:** Reactive oxygen species signaling and intracellular calcium dysregulation are linked to a number of neurodegenerative illnesses, such as Alzheimer's diseases[39].

**Plasma Membrane:** Alterations in membrane lipid biochemistry may predispose to Alzheimer's disease[40].

**Actin-binding:** Gelatin (GSN) can be used for the treatment of Alzheimer's disease due to its involvement in the immune process and interaction with different cells of the immune system[41].

**Erythrocyte differentiation:** Erythropoiesis correction of anemia through the use of erythropoiesis-stimulating agents can restore cognitive impairment[42].

**Membrane:** Reduced erythrocyte membrane fluidity has been observed in patients with MCI[43].

**Actin Binding:** Actin-binding proteins are associated with cognitive decline through neuronal activity and PTEN regulation[44]. Through KEGG and GO enrichment analysis, it is indicated that the biological processes of the modules generated by the algorithm are mostly associated with AD/MCI.
<figure><img src="image_6.png" /><figcaption>Figure 5 | KEGG and GO enrichment analyses of selected significant modules 2. A and B are KEGG analyses for AD/MCI, respectively. C and D are GO analysis for AD/MCI, respectively.</figcaption></figure>

## 4.2. ROI and gene selection

To investigate the progression from MCI to AD, we initially identified the most significant modules for MCI and AD separately. Subsequently, we explored the progression between them by finding the intersections and unions of their respective genetic and ROI modules. This approach enhances the study of the pathophysiology of MCI progressing to AD. The intersection of genes yielded 521 genes, and the union generated 967 genes. Figure 6 illustrates the brain regions of the intersection and union. Table 4 lists the ROIs of intersections and unions. There are 26 ROIs in the union and 2 in the intersection. Through literature review, we found that Left Third Ventricle/Right Third Ventricle (l3thVen/r3thVen), Left Fourth Ventricle/Right Fourth Ventricle (l4thVen/r4thVen), Left Brainstem/Right Brainstem (IBst/rBst), Left Cerebellum White Matter/Right Cerebellum White Matter (ICbeWM/rCbeWM), Left Cerebral White Matter/Right Cerebral White Matter (ICbrWM/rCbrWM), Right Lateral Ventricle (rLatVen), Left Pallidum/Right Pallidum (IPal/rPal), Left Putamen (IPut), Right Optic Chiasm (rOC), Left Cerebellar Lobule Cerebellar Vermal Lobules I-V/Right Cerebellar Lobule Cerebellar Vermal Lobules I-V (ICbeLoCbe1-5/rCbeLoCbe1-5), Left Cerebellar Lobule Cerebellar Vermal Lobules VI-VII/Right Cerebellar Lobule Cerebellar Vermal Lobules VI-VII (ICbeLoCbe6-7/rCbeLoCbe6-7), Left Cerebellar Lobule Cerebellar Vermal Lobules VIII-X (ICbeLoCbe8-10), Left Inferior Temporal Gyrus (IInfTemGy), Right Middle Occipital Gyrus (rMidOccGy), Left Inferior Frontal Gyrus (IInfFroGy), Right Superior Occipital Gyrus (rSupOccGy) are related to AD/MCI, while Right
Cuneus (rCun) and Left Lateral Orbital Gyrus (ILatOrbGy) are unrelated to AD/MCI.

<figure><img src="image_7.png" /><figcaption>Figure 6 | Visualization of brain regions. Blue and red parts identified nodes are the union of the most salient modules generated by the OSnetNMF algorithm for the AD and MCI brain regions, where the red parts are intersections. The blue nodes represent the union of the most significant modules 2 in the AD and MCI brain regions, generated separately by the OSnetNMF algorithm, with the intersecting parts marked in red.</figcaption></figure>

<table><caption>TABLE4. Brain regions of union and intersection.</caption><thead><tr><th>Union/Intersection</th><th>ROI</th><th></th></tr></thead><tbody><tr><td rowspan="7">Union</td><td>l3thVen/r3thVen</td><td>lCbeLoCbe1-5/rCbeLoCbe1-5</td></tr><tr><td>l4thVen/r4thVen</td><td>lCbeLoCbe6-7/rCbeLoCbe6-7</td></tr><tr><td>LBst/rBst</td><td>lCbeLoCbe8-10</td></tr><tr><td>LCbeWM/rCbeWM</td><td>lInfTemGy</td></tr><tr><td>LCbrWM/rCbrWM</td><td>rMidOccGy</td></tr><tr><td>rLatVen</td><td>lInfFroGy</td></tr><tr><td>LPal/rPal</td><td>rSupOccGy</td></tr><tr><td></td><td>IPut</td><td>rCun</td></tr><tr><td></td><td>rOC</td><td>lLatOrbGy</td></tr><tr><td>Intersection</td><td>l4thVen/r4thVen</td><td></td></tr></tbody></table>

### 4.3. Biological significance

Through the algorithm, it was found that genes and brain regions in the AD group are mostly associated with AD, while those in the MCI group are mostly related to MCI. However, some are associated with both AD and MCI. Through previous enrichment analysis, it was discovered that
some genes are involved in pathways that are not only present in the MCI group but also in the AD group. Moreover, AD is a progressive disease, starting from normal individuals to MCI and ultimately leading to AD. Exploring the progression is essential. Therefore, we took the genes and brain region data obtained from the AD and MCI modules, performed intersections and unions, aiming to explore the genes and brain regions that may lead from CN to MCI and finally to AD.

Using the Networkanalyze plugin in Cytoscape, protein-protein interaction networks for 967 and 521 genes were constructed for the intersection and union, respectively, as shown in Figure 7. The brightness of gene colors indicates the degree of association with other genes in the network. The MCC and Degree methods of the cytoHubba plugin were employed for network analysis, as depicted in Figure 7 (A, B, C, D). The top genes from each were selected for the intersection, resulting in a stronger protein-protein interaction network. Finally, 27 genes in the union and 26 genes in the intersection were obtained, as shown in Figure 7 (E, F), with specific genes listed in Table 5. Literature search revealed that EPB42, SLC4A1, ALAS2, GATA1, FECH, NME8, CLU, CD33, ANK1, SNCA, MPO, CTSG, DMTN, PTK2B, CAMP, LTF, RETN, LCN2, MMP9, IL1B are related to AD/MCI, while GYPB, AHSP, KLF1, ELANE, OSBP2, EPB41, DCAF12 are unrelated. In the intersection, EPB42, ALAS2, GATA1, DMTN, LTF, MMP9, CAMP, MPO, LCN2, HP, SERPINA1, CA1, PGLYRP1, BCL2L1, HIF1A, CLU, ANK1, IGF1R, MAPK14, FOXO3, VWF, HSPA5 are related, while AHSP, KLF1, HBQ1, STRADB are unrelated. Therefore, we obtained 20 genes in the union and 22 genes in the intersection.

<figure><img src="image_8.png" /></figure>

A

<figure><img src="image_9.png" /></figure>

B

<figure><img src="image_10.png" /></figure>

C

<figure><img src="image_11.png" /></figure>

D
<figure><img src="image_12.png" /><figcaption>Figure 7 | PPI network. A, B, C and D are the union and intersection of the most significant module 2 for PPI network analysis. A and C are the top 60 and 40 genes screened by MCC method in PPI for the union and intersection, respectively. B and D are the top 60 and 40 genes screened by Degree method for the union and intersection, respectively. E and F are the top 60 and 40 genes obtained from the previous union and intersection by MCC and Degree method, respectively, and continue to take union and intersection.</figcaption></figure>

<table><caption>TABLE5. Ensemble and intersection genes after PPI network analysis.</caption><thead><tr><th>Union</th><th></th><th>Intersection</th><th></th></tr></thead><tbody><tr><td>GYPB</td><td>MPO</td><td>AHSP</td><td>CA1</td></tr><tr><td>EPB42</td><td>CTSG</td><td>EPB42</td><td>PGLYRP1</td></tr><tr><td>SLC4A1</td><td>DMTN</td><td>KLF1</td><td>STRADB</td></tr><tr><td>ALAS2</td><td>PTK2B</td><td>ALAS2</td><td>BCL2L1</td></tr><tr><td>AHSP</td><td>OSBP2</td><td>GATA1</td><td>HIF1A</td></tr><tr><td>KLF1</td><td>EPB41</td><td>DMTN</td><td>CLU</td></tr><tr><td>GATA1</td><td>CAMP</td><td>LTF</td><td>ANK1</td></tr><tr><td>FECH</td><td>LTF</td><td>MMP9</td><td>IGF1R</td></tr><tr><td>NME8</td><td>RETN</td><td>CAMP</td><td>MAPK14</td></tr><tr><td>CLU</td><td>LCN2</td><td>MPO</td><td>FOXO3</td></tr><tr><td>CD33</td><td>MMP9</td><td>LCN2</td><td>VWF</td></tr><tr><td>ANK1</td><td>DCAF12</td><td>HP</td><td>HSPA5</td></tr><tr><td>SNCA</td><td>IL1B</td><td>SERPINA1</td><td></td></tr><tr><td>ELANE</td><td></td><td>HBQ1</td><td></td></tr></tbody></table>

For the final 20 genes in the union, 22 genes in the intersection, and corresponding ROIs, we created paired correlation heatmaps of ROI-gene pairs during AD, MCI, and normal (CN) periods using MATLAB, as shown in Figure 8 (A, B, C, D, E, F). Figure 8 reveals a strong correlation between ROI-gene pairs. Specifically, from Table 6, the average correlation coefficients for the union in AD, MCI, and CN groups were 0.1551, 0.0642, and 0.0688, respectively. For the intersection, the values were 0.1882, 0.0661, and 0.0559. The sizes in both the union and intersection groups followed the order AD>MCI> CN, demonstrating a strong correlation between ROI-genes in the AD group. To explore the progression mechanism from CN to MCI leading to AD, we used the Pearson correlation coefficient between ROI-genes as an indicator, identifying
<figure><img src="image_13.png" /><figcaption>A</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>B</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>C</figcaption></figure>

<figure><img src="image_16.png" /><figcaption>D</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>E</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>F</figcaption></figure>
<figure><img src="image_19.png" /><figcaption>Figure 8 | Important genes obtained from the PPI network and heat maps of brain regions generated by the algorithm, along with continuously significant changing correlation curves. A, C and E are the heat maps of important genes and brain regions in the union during the AD, MCI, and CN periods, used for correlation analysis. B, D and F are the heat maps of important genes and brain regions related to the intersection at the three time periods. G and H are the trend charts of gene-ROI pairs with continuously significant increases in correlation coefficients from CN to MCI and then to AD for the union and intersection, respectively. I displays gene-ROI pairs with continuously significant decreases in correlation coefficients from CN to MCI and then to AD in the union.</figcaption></figure>

<table><caption>TABLE6. Mean correlation coefficients of AD, MCI and CN for ensembles and concatenated sets.</caption><thead><tr><th></th><th>AD</th><th>MCI</th><th>CN</th></tr></thead><tbody><tr><td>Average correlation coefficient for the union</td><td>0.1551</td><td>0.0642</td><td>0.0688</td></tr><tr><td>Average correlation coefficient for the intersection</td><td>0.1882</td><td>0.0661</td><td>0.0559</td></tr></tbody></table>

<table><caption>TABLE7. Significantly altered genes and brain regions have a significant impact on</caption><thead><tr><th>Intersection (rise)</th><th>14thVen-SERPINA1</th><th>14thVen-HIF1A</th><th>14thVen-HSPA5</th><th>r4thVen-HIF1A</th><th>r4thVen-HSPA5</th></tr></thead><tbody><tr><td>CN</td><td>-0.1015</td><td>-0.1112</td><td>-0.03501</td><td>-0.11241</td><td>-0.03539</td></tr><tr><td>MCI</td><td>-0.0449</td><td>-0.0498</td><td>0.01382</td><td>-0.06655</td><td>0.0090</td></tr></tbody></table>
<table><tbody><tr><td>AD</td><td>0.1213</td><td>0.1326</td><td>0.13736</td><td>-0.02928</td><td>0.08936</td></tr><tr><td>Union (drop)</td><td>l4thVen-GATA1</td><td>r4thVen-ALAS2</td><td>r4thVen-CLU</td><td>rBst-MPO</td><td>rBst-LTF</td></tr><tr><td>CN</td><td>0.0571</td><td>0.087</td><td>0.1678</td><td>0.0799</td><td>0.1474</td></tr><tr><td>MCI</td><td>-0.0381</td><td>-0.0054</td><td>0.0188</td><td>-0.0436</td><td>-0.0525</td></tr><tr><td>AD</td><td>-0.2924</td><td>-0.2220</td><td>-0.1722</td><td>-0.2445</td><td>-0.2958</td></tr><tr><td>Union (drop)</td><td>lCbeWM-ALAS2</td><td>rCbeWM-GATA1</td><td>lCbrWM-MPO</td><td>rCbrWM-MPO</td><td>rCbrWM-CAMP</td></tr><tr><td>CN</td><td>0.0603</td><td>-0.0004</td><td>0.0103</td><td>0.0544</td><td>-0.0071</td></tr><tr><td>MCI</td><td>-0.0505</td><td>-0.0928</td><td>-0.081</td><td>-0.0689</td><td>-0.0971</td></tr><tr><td>AD</td><td>-0.2455</td><td>-0.2802</td><td>-0.4482</td><td>-0.3727</td><td>-0.5602</td></tr><tr><td>Union (drop)</td><td>rLatVen-CD33</td><td>rOC-ALAS2</td><td>rOC-GATA1</td><td>rOC-CLU</td><td>rOC-CTSG</td></tr><tr><td>CN</td><td>-0.0018</td><td>0.1380</td><td>0.0595</td><td>0.1178</td><td>0.0087</td></tr><tr><td>MCI</td><td>-0.1223</td><td>0.0454</td><td>-0.0513</td><td>0.0285</td><td>-0.1838</td></tr><tr><td>AD</td><td>-0.3126</td><td>-0.2686</td><td>-0.2885</td><td>-0.2533</td><td>-0.4592</td></tr><tr><td>Union (rise)</td><td>lCbeLoCbe1-5-CD33</td><td>rBst-PTK2B</td><td></td><td></td><td></td></tr><tr><td>CN</td><td>-0.1242</td><td>-0.1050</td><td></td><td></td><td></td></tr><tr><td>MCI</td><td>-0.0297</td><td>0.0127</td><td></td><td></td><td></td></tr><tr><td>AD</td><td>0.2537</td><td>0.2441</td><td></td><td></td><td></td></tr></tbody></table>

In the intersection, there were mainly 5 pairs: l4thVen-SERPINA1, l4thVen-HIF1A, l4thVen-HSPA5, r4thVen-HIF1A, r4thVen-HSPA5. All these pairs exhibited a positive correlation trend, indicating that both the brain region and gene need to act together to progress from CN to MCI and finally to AD. The brain regions mainly involved the fourth ventricle. Chronic neuroinflammation is induced by lipopolysaccharide (LPS) administration to the fourth ventricle. AMPA receptor (AMPAR)-mediated synaptic responses were reduced. Mediated synaptic responses were reduced in the LPS infusion group. These results suggest that a NMDAR-dependent reduction of LTD may occur due to alterations in postsynaptic AMPARs and NMDARs, which may be present in mild and early forms of Alzheimer-type dementia[45]. The pathways in which the genes are mainly involved are Complement and coagulation cascades, Central carbon metabolism in cancer and Thyroid hormone synthesis. SerpinA1: Altered isoenzyme expression of SerpinA1 in the cerebrospinal fluid (CSF) of AD patients suggests that
serpinA1 is common to most neurodegenerative disorders, although disease-specific[46]. HIF1A: Modulation of β-catenin activation by HIF-1α may be effective in neurodegenerative diseases[47]. HSPA5: Promoter polymorphisms regulating HSPA5 expression may increase susceptibility to neurodegenerative diseases[48]. After taking the intersection of AD and MCI and conducting biological analysis, the brain regions and gene pairs obtained are both related to MCI and AD. This suggests that under the influence of brain regions and genes, MCI is likely to develop into AD.

In the union, positive correlations mainly consist of 15 pairs: l4thVen-GATA1, r4thVen-ALAS2, r4thVen-CLU, rBst-MPO, rBst-LTF, lCbeWM-ALAS2, rCbeWM-GATA1, lCbrWM-MPO, rCbrWM-MPO, rCbrWM-CAMP, rLatVen-CD33, rOC-ALAS2, rOC-GATA1, rOC-CLU, rOC-CTSG. All these pairs exhibit a positive correlation trend. Both brain regions and genes need to either enhance or diminish to cause the progression from CN to MCI and ultimately to AD. Negative correlations mainly consist of 2 pairs: lCbeLoCbe1-5-CD33, rBst-PTK2B. The progression from CN to MCI and ultimately to AD occurs when one of the brain regions or genes strengthens while the other weakens.

The brain regions involved in the union mainly include the fourth ventricle, Right Brainstem, Cerebellum White Matter, Cerebral White Matter, Right Lateral Ventricle, Right Optic Chiasm, Left Cerebellar Lobule Cerebellar Vermal Lobules I-V. Cerebellum White Matter: Patients with TTM have altered areas of cortico-limbic and cerebellar white matter, which may underlie cognitive and emotional processing deficits, and may contribute to impaired cortico-cerebellar communication, leading to a decreased ability to control hair pulling behavior[49]. Right Brainstem: Assessing epigenetic traits in the brainstem is a cornerstone for interrogating causation in AD[50]. Cerebral White Matter: Brain gray and white matter mononucleotide diglyceride levels rise with the progression of Alzheimer's disease[51]. Right Lateral Ventricle: Adult neurogenesis occurs predominantly in the subgranular region of the hippocampal dentate gyrus and the subventricular region of the lateral ventricles[52]. Right Optic Chiasm: Hesperetin (Hst), an orange flavin analog, greatly reduces demyelination levels in the optic thalamus and possesses beneficial effects in experimental models of Alzheimer's disease[53]. Left Cerebellar Lobule Cerebellar Vermal Lobules I-V: MCI patients have decreased rGMV in the parietal I-V of the cerebellum, making progression to AD more likely[54]. From this, it can be seen that the analyzed brain regions are mostly related to MCI and AD.

The genes are predominantly found in GATA1: GATA1-mediated transcriptional regulation of γ-secretase-activating protein develops high levels of Aβ peptide and Alzheimer's disease-like cerebral amyloidosis[55]. ALAS2: Expression of ALAS2, a transporter of divalent metal species, is significantly increased and has been implicated in the etiology and pathogenesis of several neurodegenerative diseases[56]. CLU: Genome-Wide Association Study Identifies Alzheimer's Disease-Related Variants in the CLU[57]. MPO: MPO gene promoter polymorphisms linked to Alzheimer's disease risk[58]. LTF: LTF is upregulated in AD patients[59]. CAMP: Abnormal cAMP signaling linked to cognitive problems in neurodegenerative diseases such as AD[60]. CD33: The microglial receptor CD33 has been shown to be associated with the risk of AD[61]. CTSG: The discovery of CTSG in LBD brains shows the neurodegenerative lesion[62]. PTK2B:
There is evidence that PTK2B is involved in the pathogenesis of AD[62]. The identified genes are also associated with AD and MCI. This indicates that the gene-brain region pairs we have identified could serve as crucial genes and brain regions in the transition from MCI to AD. Here are the brain regions and genes obtained after taking the union of AD and MCI, so not all of them are present in both AD and MCI. Therefore, some gene and brain region pairs may be potential biomarkers that lead to MCI and subsequently to AD.

# 5. Conclusion

Alzheimer's disease (AD) is an irreversible degenerative brain disorder caused by the loss of nerve cells. In order to understand the pathogenic mechanism from normal cognitive function (CN) to the continuous progression of AD, we explored the correlation between two types of image genetic data (sMRI and gene expression data) of different progression states of AD. In response to the fact that the netNMF algorithm does not consider the issues of redundancy and interpretability of feature data, as well as the accuracy of gene and imaging data extracted by the model, we proposed OSnetNMF to effectively integrate biological information. This was achieved by adding orthogonal and sparse constraints to the coefficient matrix obtained by decomposing imaging and genetic data. The orthogonal constraint reduces redundant information, ensuring that each coefficient vector captures different variances in the data, thereby enhancing the diversity of the results. The sparse constraint helps select the most important features for data representation. By forcing certain weights to be zero, sparse constraints encourage the algorithm to choose the most significant features for the target task, thereby improving the accuracy of the generated modules.

In the modules obtained through the algorithm, we conducted bioinformatics analysis for the most significant modules in the AD and MCI groups: the second module. We found significant genes and brain region pairs. Some exhibit positive correlations, requiring simultaneous action to lead from CN to mild cognitive impairment (MCI) and further to AD progression, such as l4thVen-GATA1. Others show negative correlations, with one enhancing action and the other weakening action, leading to the development from CN to AD, for example, rBst-PTK2B. Understanding their interactions can provide better insights into the pathogenic mechanisms of the progression from CN to AD. Through experimental comparative analysis, the modules we obtained show more enrichment pathways related to the disease, revealing greater possibilities associated with the disease. Our reconstruction outperforms netNMF and NMFNA, demonstrating superior reconstruction performance. Currently, only bimodal data analysis is feasible. In our future work, we will integrate more data types, such as adding SNP data for trimodal analysis, to more comprehensively analyze disease-related regulatory mechanisms.

# Data availability statement

Availability of Data and Materials The authors appreciate the Alzheimer's Disease Neuroimaging Initiative (ADNI, https://adni.loni.usc.edu) for contributing data (gene expression data and sMRI).
## Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

## Author contributions

MG, WK and KL carried out the study conceptualization and design. Data acquisition by KW and ZHJ. MG, KW and GW carried out the experiments, performed analysis and interpreted the results. GM and WK prepared and wrote the manuscript. WK, YY and YZ provided significant assistance and analysis through constructive discussions. All authors have read and agreed to the manuscript.

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