Detecting Beta-amyloid Plaque via Low Rank Based Orthogonal Projection and Spatial-spectrum Detector Using High-resolution Quantitative Susceptibility Mapping for Preclinical Studies

Abstract

Detecting beta-amyloid (Aβ) plaques at different stages is crucial for accurate assessment and effective intervention in Alzheimer’s disease (AD). In this study, we developed a novel method for reliably identifying Aβ plaques, characterized by sparse negative susceptibility values, in preclinical studies using high-resolution quantitative susceptibility mapping (QSM), named QSM-PLAQUE Aβ Detector. This approach decomposes a high-resolution QSM MRI image into three components: L (representing the background subspace), S (representing the signals subspace), and N (representing the noise). Subsequently, we established an orthogonal subspace based on L to eliminate the background from the sum of L and S. Finally, a plaque detection process was conducted, where Aβ plaques were identified based on the neighbor spectrum (NS) of a voxel being tested rather than just analyzing the voxel itself alone. Experiments demonstrated that the proposed method effectively detects Aβ plaques of varying shapes and intensities across the entire mouse brain. It shows robust performance across histology, high-resolution QSM MRI, and synthesized datasets, without requiring training samples. The QSM-PLAQUE Aβ Detector provides a practical framework for identifying and visualizing Aβ plaques in preclinical studies, offering a new strategy for quantitative assessment of Aβ plaques and may guide the development of advanced techniques for preclinical AD research.

I. Introduction

ALZHEIMER’S disease (AD) represents the most common neurodegenerative disorder and ranks as the primary contributor to the onset of dementia and sixth most common cause of death in the United States in 2019 [1]–[3]. AD is characterized by gradual and progressive nature rather than being an acute illness and is influenced by multiple risk factors such as genetic factors [4], cardiovascular factors [5], cerebrovascular dysfunction [6] etc. However, regardless of the initiating factors, the pathological processes may ultimately converge on the hallmarks of AD. One of the primary hallmarks of AD is the beta-amyloid (Aβ) protein, which serves as the main component of plaques and plays a critical role in the disease’s progression [7]. The onset of the disease initiates in the brain 1–3 decades before individuals experience memory losses, such as forgetting the location where a car has been parked [8]. Therefore, accurate assessment of AD is crucial for effective disease management and intervene in the progression of the condition [9].

During the early stage of research on AD, the detection of Aβ plaques was constrained to histological staining using postmortem analysis which prevented early diagnosis [10]. During the 1980s to 1990s, antibodies were used for the detection and quantitation of Aβ peptide [11], [12], or biochemical methods like ELISA (Enzyme-Linked Immunosorbent Assay) were developed to quantify Aβ levels in cerebrospinal fluid (CSF) [13], [14]. In the 2000s, medical imaging advanced significantly for detection of Aβ, techniques such as positron emission tomography (PET) imaging with amyloid-specific tracers and MRI have shown their potential to non-invasively detect Aβ plaques [15]–[18]. After 2010, the analysis of cerebrospinal fluid (CSF) biomarkers, such as Aβ42, total tau (t-tau) and phosphorylated tau (p-tau) were well established and recognized as standard for diagnosis for AD [19]. Plasma levels of Aβ42 and Aβ40, particularly their ratio (Aβ42/Aβ40), have shown promise as minimally invasive biomarkers for AD [20], [21].

However, although PET imaging and CSF analysis offer safe and accurate diagnosis for AD, PET imaging is unable to detect individual plaques due to its limited spatial resolution [22], and CSF cannot provide any spatial information. More recently, various imaging modalities, particularly in MRI encompassing both structural and functional modalities, have revealed characteristic changes in the brains of patients with AD [23]. Techniques like functional MRI (fMRI) and resting-state fMRI (rs-fMRI), which are advanced forms of MRI, are utilized to detect changes in functional connectivity that result in distinctive patterns associated with amyloid plaque deposition [24], [25]. However, these techniques do not directly visualize plaques themselves.

Quantitative susceptibility mapping (QSM) is an advanced MRI technique that reconstructs the spatial distribution of magnetic susceptibility in tissue by processing phase data from gradient echo sequences [26], [27], and it has emerged as a promising method for imaging Aβ plaques. In particular, QSM provides superior detail about local tissue composition, allowing for the detection of substances such as iron, calcium, and protein aggregates [28], [29]. Beyond Aβ deposition, iron accumulation is another critical pathological hallmark of AD and can significantly influence QSM signal intensity [30]. While Aβ plaques exhibit diamagnetic properties, iron deposits are strongly paramagnetic [31]. The coexistence of these two materials within the same voxel can result in partial signal cancellation, complicating the interpretation of the net magnetic susceptibility measured by QSM. To address this challenge, methods have been developed to separate the paramagnetic and diamagnetic contributions to susceptibility. These approaches often combine QSM with additional relaxation measurements such as $R_2^{\prime }$ or $R_2^{*}$, enabling algorithms like APART-QSM [32], $\chi$-separation [33] and DECOMPOSITION-QSM [34] to find the paramagnetic susceptibility, which primarily reflects iron, and a diamagnetic susceptibility, which is associated with substances like myelin or Aβ plaque [35]. Accurate interpretation of QSM signals is essential when investigating neurodegenerative changes in the brain. Given the characteristics of QSM MRI, it has emerged as a computational technique of significance for revealing the pathology of AD or other diseases [36], [37].

Although QSM provides significant information in assessing iron accumulation and other tissue properties at the millimeter scale [27], detecting Aβ plaques requires spatial resolution at micrometer scale [38]. In this paper, high-resolution QSM at 18 μm was acquired to detect Aβ plaques in 5xFAD mice [39]. However, the QSM image suffers from complex background signals, which pose a challenge to the accurately detect the plauqes. To address this issue, we introduced Go Decomposition (GoDEC) [40], which is one of implementation of low-rank and sparsity matrix decomposition (LRaSMD), which decomposes the slice of high-resolution QSM MRI image into three parts, one is low rank, $\mathbf{\text{L}}$, which representing the background, $\mathbf{\text{S}}$ is sparse, which representing the signals, and $\mathbf{\text{N}}$, which representing the noise. Subsequently, an orthogonal subspace based on $\mathbf{\text{L}}$ was established to eliminate the background from the sum of $\mathbf{\text{L}}$ and $\mathbf{\text{S}}$. The analysis then conducted on the subspace which is orthogonal to $\mathbf{\text{L}}$.

The subsequent question relates to the methodology for detecting biomarkers, such as Aβ, on high-resolution QSM MRI. Unfortunately, there are few literatures of detecting and counting the Aβ plaques accurately using high-resolution QSM. Existing methods for detecting Aβ plaques primary focus on machine learning, deep learning, threshold methods or their combination using microscopic images [2], PET images [41], or optical images [42]–[44] etc. Yoon, et. al [2] proposed segmentation based method to detect Aβ plaques using microscopic images for deep learning training to find the somatosensory cortex detection and then overlaid the original image for calculating the the load of Aβ plaques. Reith et. al [41] suggested that use of deep learning to automate the quantification of standardized uptake value ratio (SUVR) and classify amyloid status in ¹⁸F-florbetapir PET scans, which are used to detect amyloid plaques in AD. Icke et. al [45] investigated the use of confocal fluorescence images combined with a U-Net architecture for the automated detection of Aβ plaques. The performance of the deep learning methodologies varied depending on the choice of training dataset and the structural configuration of the model.

After observing the Aβ plques, we found it exhibits characteristics by three points that are seldom mentioned in existing literatures:

• 1. The voxels it occupied are relatively small.

• 2. The occurrence of Aβ plaque is relatively rare.

• 3. Most importantly, they are significant different from its surrounding voxels.

Given these characteristics, we can treat Aβ plaques as anomaly, and detect Aβ plaques as anomaly detection. Anomaly detection can be considered passive target detection and it does not assume any prior knowledge of signals of interesting is given, and has developed well in Hyperspectral imaging [46]–[49]. The Reed-Xiaoli Detector (RXD), developed by Reed and Yu, is one of the most widely used classical anomaly detection methods [49], [50].

RXD design involves the characterization of the background through a multivariate Gaussian distribution model. Despite the distinctive spatial characteristics, like intensive value, of a test sample compared to its surrounding neighbors, is the spatial information provided by a voxel sufficient to differentiate the test sample from its neighbors? This paper introduces the concept of a neighbor spectrum as the spectral characteristics that capture distinctive features between a voxel to be tested and its surrounding neighbors, rather than focusing on a single voxel. Ultimately, the Generalized Likelihood Ratio Test (GLRT) can be derived as the optimal detector for determining the qualification of a data sample as an anomaly.

Finally, we summarized the contribution of this paper as follows:

• We present a novel methodology firstly that quantifies and counts Aβ plaques in whole mouse brain using high-resolution QSM.

• We present the initial application of low-rank and sparsity matrix decomposition (LRaSMD) in high-resolution QSM imaging.

• We propose a novel orthogonal projection to eliminate the background effects.

• We present anomaly detection for identifying Aβ plaques, incorporating a novel concept called neighbor spectrum as a feature of a data sample. This innovation enables the application of classical anomaly detection theory in medical imaging.

This article is organized as follows. Section II provides a brief overview of GoDec and RXD, while Section III details the derivation of the QSM-PLAQUE Aβ Detector. Section IV and VI present and discuss the results of tree experiments and one simulation. Section V discusses the Sensitivity Analysis of GoDec parameters. Section VII highlights the key contributions and discusses the limitations as well as the significance of preclinical studies of AD.

II. Related Work

This section will introduce a well-established technique for matrix decomposition known as GoDec, commonly applied in video and hyperspectral images but not yet explored in high-resolution QSM. Subsequently, there will be a brief discussion on RXD anomaly detectors. Finally, the section will conclude with a discussion on QSM.

A. Go Decomposition (GoDec)

GoDec [40] is a matrix decomposition algorithm originally developed for robust data recovery and data denoising application [51], [52]. It decomposes a data matrix into three parts. This decomposition can be represented as:

$$\mathbf{\text{X}}=\mathbf{\text{L}}+\mathbf{\text{S}}+\mathbf{\text{N}}.$$ (1)

where $\mathbf{\text{L}}$, a low-rank matrix, $\mathbf{\text{S}}$, a sparse matrix, and $\mathbf{\text{N}}$, a noise matrix. To acquire an optimal solution in (1), an optimal problem was introduced by GoDec to find the low-rank, the sparse as following constrained condition:

$$\mathbf{\text{X}}=\mathbf{\text{L}}+\mathbf{\text{S}}+\mathbf{\text{N}}\mathit{\text{s.t.}}\mathit{\text{rank}}\left(\mathbf{\text{L}}\right)\le r,\mathit{\text{card}}\left(\mathbf{\text{S}}\right)\le k$$ (2)

where $\mathit{\text{rank}}\left(\mathbf{\text{L}}\right)$ is defined as the rank of low-rank matrix, $\mathbf{\text{L}}$, and $\mathit{\text{card}}\left(\mathbf{\text{S}}\right)$ is interpreted as sparsity cardinality of $\mathbf{\text{S}}$. Nevertheless, GoDec did not estimate these parameters, suggesting their dependence on practical application. Chang et. al [49] have shown the LRaSMD in (1) is performed better than RPCA [53]. In order to achieve (2) in an optimal manner, first, an input dense matrix, using random matrices, ${\mathbf{\text{A}}}_1$ and ${\mathbf{\text{A}}}_2,\mathbf{\text{X}}$, is projected onto its column space as ${\mathbf{\text{Y}}}_1=\mathbf{\text{X}}{\mathbf{\text{A}}}_1$, followed a projection onto its row space as ${\mathbf{\text{Y}}}_2={\mathbf{\text{X}}}^T{\mathbf{\text{A}}}_2$, and this progress from compressive sensing and is named as bilateral random projection (BRP), which was utilized for estimating low-rank matrix, $\mathbf{\text{L}}$, which is derived as follows:

$$\mathbf{\text{L}}={\mathbf{\text{Y}}}_1{\left({\mathbf{\text{A}}}_2{\mathbf{\text{Y}}}_1\right)}^{-1}{\mathbf{\text{Y}}}_2.$$ (3)

Followed by the low-rank matrix approximation in (3), the sparsity matrix $\mathbf{\text{S}}$ and noise matrix, $\mathbf{\text{N}}$ can be derived based on the cardinality of $\mathbf{\text{S}}$.

To address the challenge of slowing-decay singular values, both the randomized power iteration algorithm [54] and GoDec [40] operated on $\widetilde{\mathbf{\text{X}}}={\left({\mathbf{\text{XX}}}^T\right)}^{\gamma }\mathbf{\text{X}}$ instead of the original input dense data matrix, $\mathbf{\text{X}}$. Using BRP, GoDec solves (3) can be reformulated as:

$${\left.\mathbf{\text{L}}={\mathbf{\text{Q}}}_1{\left[{\mathbf{\text{R}}}_1{\left({\mathbf{\text{A}}}_2\right)}^T{\mathbf{\text{Y}}}_1\right)}^{-1}{\mathbf{\text{R}}}_2^T\right]}^{1/(2\gamma +1)}{\mathbf{\text{Q}}}_2^T.$$ (4)

(4) indicated the bracket and exponential part can be used for solve the singular slow decay issue by BRP several times, where the ${\mathbf{\text{Q}}}_1,{\mathbf{\text{Q}}}_2$ are orthogonal matrix, ${\mathbf{\text{R}}}_1$, and ${\mathbf{\text{R}}}_2$ are upper triangular matrix, are the QR decomposition of ${\mathbf{\text{Y}}}_1$ and ${\mathbf{\text{Y}}}_2$.

After the low-rank approximation was obtained, the (2) can be solved by formatting it as a minimizing decomposition error problem:

$$\begin{gathered} \underset{{\mathbf{\text{L}}}_{N\times L},{\mathbf{\text{S}}}_{N\times L}}{\mathit{\text{min}}}{‖{\mathbf{\text{X}}}_{N\times L}-{\mathbf{\text{L}}}_{N\times L}-{\mathbf{\text{S}}}_{N\times L}‖}_F^2\cdot \\ \mathit{\text{s.t.}}\mathit{\text{rank}}\left({\mathbf{\text{L}}}_{N\times L}\le r,\mathit{\text{card}}\left({\mathbf{\text{S}}}_{N\times L}\right)\le k\right. \end{gathered}$$ (5)

where $\Vert \cdot {\Vert }_F$ is the Frobenius norm. As (5) lacks an analytical solution, it must rely on a numerical solution through the following iterative processes:

$$\begin{gathered} {\mathbf{\text{L}}}_{N\times L}^{\left(t\right)}=\underset{\mathit{\text{rank}}\left({\mathbf{\text{L}}}_{N\times L}\right)\le r}{\mathit{\text{arg}}\mathit{\text{min}}}{‖{\mathbf{\text{X}}}_{N\times L}-{\mathbf{\text{L}}}_{N\times L}-{\mathbf{\text{S}}}_{N\times L}^{(t-1)}‖}_F^2. \\ {\mathbf{\text{S}}}_{N\times L}^{\left(t\right)}=\underset{\mathit{\text{card}}\left({\mathbf{\text{S}}}_{N\times L}\right)\le k}{\mathit{\text{arg}}\mathit{\text{min}}}{‖{\mathbf{\text{X}}}_{N\times L}-{\mathbf{\text{L}}}_{N\times L}^{\left(t\right)}-{\mathbf{\text{S}}}_{N\times L}‖}_F^2. \end{gathered}$$ (6)

The implementation details of GoDec are described as Algorithm 1.

Algorithm 1.

GoDec

Input: ${\mathbf{\text{X}}}_{N\times L}$ input original dense data matrix

$r$: the rank of the low-rank matrix $\mathbf{\text{L}}$

$k$: the sparsity cardinality for sparsity matrix $\mathbf{\text{S}}$

$ϵ$: the pre-determined error threshold

$\gamma$: power scheme

Output:

${\mathbf{\text{L}}}_{N\times L}$: Low-rank matrix

${\mathbf{\text{S}}}_{N\times L}$: Sparsity matrix

Initialisation : ${\mathbf{\text{L}}}_{N\times L}^{(0)}={\mathbf{\text{X}}}_{N\times L},{\mathbf{\text{S}}}_{N\times L}^{(0)}={\mathbf{\text{O}}}_{N\times L},t=0$

LOOP Process

1: while $\parallel {\mathbf{\text{X}}}_{N\times L}-{\mathbf{\text{L}}}_{N\times L}-{\mathbf{\text{S}}}_{N\times L}{\parallel }_F^2\ge ϵ$ do

2:  $t=t+1;$

3: $\widehat{\mathbf{\text{L}}}={\left[\left({\mathbf{\text{X}}}_{N\times L}-{\mathbf{\text{S}}}_{N\times L}^{(t-1)}\right]{\left[{\mathbf{\text{X}}}_{N\times L}-{\mathbf{\text{S}}}_{N\times L}^{(t-1)}\right)}^T\right]}^{\gamma }\left({\mathbf{\text{X}}}_{N\times L}-{\mathbf{\text{S}}}_{N\times L}^{(t-1)}\right)$

4: ${\mathbf{\text{Y}}}_1=\widehat{\mathbf{\text{L}}}{\mathbf{\text{A}}}_1,{\mathbf{\text{A}}}_2={\mathbf{\text{Y}}}_1$

5: ${\mathbf{\text{Y}}}_2={(\widehat{\mathbf{\text{L}}})}^T{\mathbf{\text{Y}}}_1={\mathbf{\text{Q}}}_2{\mathbf{\text{R}}}_2,{\mathbf{\text{Y}}}_1=\widehat{\mathbf{\text{L}}}{\mathbf{\text{Y}}}_2={\mathbf{\text{Q}}}_1{\mathbf{\text{R}}}_1$

6: ${\left.{\mathbf{\text{L}}}_{N\times L}^{(t)}={\mathbf{\text{Q}}}_1{\left[{\mathbf{\text{R}}}_1{\left({\mathbf{\text{A}}}_2\right)}^T{\mathbf{\text{Y}}}_1\right)}^{-1}{\mathbf{\text{R}}}_2^T\right]}^{1/(2\gamma +1)}{\mathbf{\text{Q}}}_2^T$

${\mathbf{\text{S}}}_{N\times L}^{(t)}=P_{\Omega }\left({\mathbf{\text{X}}}_{N\times L}-{\mathbf{\text{L}}}_{N\times L}^{(t-1)}\right)$

where $\Omega$ is the nonzero subset of the first $k$ largest entries of $\left|{\mathbf{\text{X}}}_{N\times L}-{\mathbf{\text{L}}}_{N\times L}^{(t)}\right|$

7: end while

The input parameters, $r,k,ϵ$, and $\gamma$, in the algorithm 1 should be predetermined in advance as initial setting for running. There is no prior knowledge for setting these parameters in Godec, we addressed this issue by using singular value decomposition (SVD) and energy threshold, typically involving the SVD of a slice of high-resolution QSM image, and determine rank based on predetermined energy (square of singular value) threshold. For sparsity, considering susceptibility of Aβ plaques, and the sparsity can be identified when susceptibility less than −0.02 ppm in QSM. The detail of setting is discussed in section III.

B. RXD

Diverging from target detection, which employs binary hypothesis testing to formulate an anomaly detector through Likelihood Ratio Test (LRT), an anomaly detector utilizes a binary composite hypothesis testing formulation, leading to the derivation of Generalized Likelihood Ratio Test (GLRT).

In the implementation, two assumptions were considered. The first assumes the existence of a secondary dataset for estimating the target signal, which is subsequently employed for detection. The second assumption pertains to the Gaussian distribution assumed for both background and signal. If the two assumptions satisfied, RXD detector can be derived based on GLRT:

$$\delta (\mathbf{\text{r}})=(\mathbf{\text{r}}-\mathit{\mu }){\mathbf{\text{K}}}^{-1}(\mathbf{\text{r}}-\mathit{\mu }{)}^T.$$ (7)

where $\mathbf{\text{r}}$ is the signal to be tested, $\mathit{\mu }$ is mean of singals, and $\mathbf{\text{K}}$ is covariance matrix.

It is crucial to highlight that applying equation (7) to medical images is not feasible due to substantial differences. A key distinction arises from the fact that in medical images, the signal being tested pertains to a single voxel, where anatomy undergoes changes. This is unlike other fields where a vector represents measurements for the same voxel with diverse physical properties. To tackle this challenge, we introduced a novel concept called neighbor spectrum (NS), which examines the neighboring voxels as a vector instead of the original signal voxel. The utilization of NS enables the application of classical detection theory to medical images. The more detail of setting is discussed in section III.

III. PROPOSED METHODS

A. Overview

Figure 1 presents the acquired raw wrapped phase image, which was preprocessed using STI Suite [35], and illustrates the QSM-PLAQUE Aβ Detector, designed to identify Aβ plaques, characterized by sparse negative susceptibility values in high-resolution QSM images via a low-rank orthogonal projection method. The process decomposes a high-resolution QSM image into three components: $\mathbf{\text{L}}$ (background subspace), $\mathbf{\text{S}}$ (signal subspace), and $\mathbf{\text{N}}$ (noise). An orthogonal subspace is then constructed based on $\mathbf{\text{L}}$ to remove the background from the combined $\mathbf{\text{L}}$ and $\mathbf{\text{S}}$. Finally, Aβ plaques are detected by evaluating the neighboring spectrum (NS) of a testing voxel via RXD detector.

Fig. 1.

The raw wrapped phase images were preprocessed using the STI Suite to produce high-resolution QSM images (a). These high-resolution QSM images were then decomposed into three components: low rank (L), representing the background; sparse (S), representing signals of interest; and noise (N) (b). The combined L+S was projected onto the orthogonal subspace of L to remove the background (c). Finally, an RXD detector was applied to generate the detection map (d).

Therefore, the QSM-PLAQUE Aβ Detector operates in three sequential stages:

• 1. Decomposition: In the first stage, the high-resolution QSM image is decomposed into three components: $\mathbf{\text{L}}$ (background subspace), $\mathbf{\text{S}}$ (signal subspace), and $\mathbf{\text{N}}$ (noise).

• 2. Orthogonal Projection: The second stage involves applying an orthogonal projection to the $\mathbf{\text{L}}$ component to isolate and remove the background from the combined $\mathbf{\text{L}}$ and $\mathbf{\text{S}}$ components.

• 3. Plaque Detection: Finally, the third stage performs plaque detection using the refined data, identifying Aβ plaques by analyzing the neighbor spectrum of the target voxels.

B. Neighbor Spectrum (NS)

Our proposed QSM-PLAQUE Aβ Detector utilizes a neighbor spectrum (NS) approach, analyzing the values surrounding a voxel to be tested. In the context of high-resolution QSM, this method leverages the voxel values of neighboring voxel rather than relying on a single-voxel value. This incorporation of spatial information enhances the accuracy of detecting Aβ plaques by providing a more comprehensive analysis of the target area.

In this paper, we explore varying the size of the neighbor matrix, including 3×3, 5×5, and 7×7, corresponding to small, medium, and large matrices, which capture fine details, broader context, and complex structures, respectively. Furthermore, we analyze several methods for converting the neighbor matrix into the neighbor spectrum, including row-wise, column-wise, and diagonal-order flattening, for each matrix size. The varing size of neighbor matrix, and methods for converting the neighbor matrix into the neighbor spectrum are shown in the spatial features of Figure 1 (bottom right). After using a neighbor matrix and methods for converting a voxel into Neighbor Spectrum (NS), the voxel to be tested becomes a vector. The size of this vector depends on the size of the neighbor matrix used. This transformation allows for more detailed analysis by considering the voxel’s surrounding context.

C. Decomposition

GoDec uses LRaSMD [40] models to decompose a data matrix into three spaces, low-rank matrix, $\mathbf{\text{L}}$, a sparse matrix, $\mathbf{\text{S}}$, and noise matrix, $\mathbf{\text{N}}$ with equation (1). There are two parameters that need to be predetermined prior to decomposition. The first one is the rank of the low-rank matrix, which is significantly lower than the original matrix dimensions. One of determining the rank of a matrix is the singular value decomposition (SVD), combing energy threshold, the rank of determining a high-resolution QSM slice image can be obtained as following:

$$\mathit{\text{Energy}}\mathit{\text{Ratio}}(ER)=\frac{\sum _{i=1}^k{\sigma }_i^2}{\sum _{i=1}^n{\sigma }_i^2}$$ (8)

where $\sum _{i=1}^k{\sigma }_i^2$ is a cumulative energy at $k$ and ${\sigma }_i$ is $\mathbf{\text{i}}$th singular value of the diagonal matrix of SVD of high-resolution QSM matrix, and $\sum _{i=1}^n{\sigma }_i^2$ is a total energy, and find the smallest $k$ such that the energy ratio exceeds predetermined threshold (fixed-energy thresholding 25%), which based on experimental results and inspection. The second factor to be determined is the sparsity, which represents the number of non-zero elements in the matrix and is utilized for representing the signals of interest, positioned between background and noise, analogous to a sandwich. Since the susceptibility of Aβ plaques phantom solution has a mean value of −0.024 ppm with a mean standard deviation of 0.002 ppm [31], our method uses the sparsity can be expressed as the largest number of voxels according to QSM value less than −0.02 parts per million (ppm).

D. Orthogonal Projection

The noise matrix from the decomposition will be discarded, and the combination of the low-rank and sparse matrices will be further processed. Despite the fact that the plaques in the sparse matrix are highly contrasted with their surroundings, its sparse nature, with most of its components being zero, poses numerical computation issues and limits its practical application.

While GoDec is often used as a denoising algorithm [40], [51], [52], in our high-resolution QSM-based pipeline, it is repurposed to separate structure background susceptibility variations from localized pathological signals. More specifically, the low rank $\left(\mathbf{\text{L}}\right)$ represents large-scale susceptibility structures, the sparse matrix $\left(\mathbf{\text{S}}\right)$ captures the interest of objects, like Aβ plaques in our case, and the noise matrix $\left(\mathbf{\text{N}}\right)$ represents the noise. Can the background, like noise, be directly removed to isolate the background and pathological signals? The answer is NO since the background is not noise, and the sparse matrix can result in instability during subsequent processing.

We addressed the aforementioned challenge by employing orthogonal projection. This method involves projecting the combination of the low-rank matrix $\mathbf{\text{L}}$, and the sparse matrix $\mathbf{\text{S}}$ onto the orthogonal subspace of $\mathbf{\text{L}}$, denoted as $P_L^{\perp }$. As shown in Fig. 1 (c), the signals, $l+s$, which come from the combination matrix, $\mathbf{\text{L}}+\mathbf{\text{S}}$, and undesired signals, $\mathbf{\text{L}}$, which are all signals from the low-rank matrix, $\mathbf{\text{L}}$. Therefore, the orthogonal subspace of $\mathbf{\text{L}}$ can be expressed as:

$$P_L^{\perp }=\mathbf{\text{I}}-\mathbf{\text{L}}{\left({\mathbf{\text{L}}}^T\mathbf{\text{L}}\right)}^{-1}{\mathbf{\text{L}}}^T.$$ (9)

Where $\mathbf{\text{I}}$ is the identity matrix. If we let $\mathbf{\text{L}}$ be the unwanted signature matrix, we can project all signals $l+s$ onto the space $P_L^{\perp }$ that is orthogonal to the space L by:

$$P_L^{\perp }\times (\mathbf{\text{L}}+\mathbf{\text{S}}).$$ (10)

Using this equation, the resulting image is illustrated in Figure 1 (c). It is evident that the plaques in $P_L^{\perp }(\mathbf{\text{L}}+\mathbf{\text{S}})$ exhibit much higher contrast. This improvement is due to the background suppression and signal enhancement achieved.

E. Plaque Detection

Let ${\mathbf{\text{X}}}^{H\times W\times C}$ presents the $P_L^{\perp }(\mathbf{\text{L}}+\mathbf{\text{S}})$, and defined as,

$${\mathbf{\text{X}}}^{H\times W\times C}=P_L^{\perp }(\mathbf{\text{L}}+\mathbf{\text{S}})$$ (11)

where $H,W$, and $C$ present the hight, width, and number of NS, respectively.

Assume that ${\left\{{\mathbf{\text{r}}}_i\right\}}_{i=1}^{H\times W}$ where $H\times W$ is the total number of data sample vectors and ${\mathbf{\text{r}}}_i={\left(r_{i1},r_{i2},\dots r_{iC}\right)}^T$ is the $i$th data sample vector and the $C$ is the length of NS. According to equation (7), a plaque detector can be defined as:

$$\delta (\mathbf{\text{r}})=(\mathbf{\text{r}}-\mathit{\mu }){\mathbf{\text{K}}}^{-1}(\mathbf{\text{r}}-\mathit{\mu }{)}^T.$$

From a detection point of view, the ${\mathbf{\text{K}}}^{-1}$ is whitening process to suppress target background, which is second order statistics, and centering the whitened data in the origin by the mean $\mathbf{\text{u}}$, which removing the first order statistics. Therefore, the detection process can be interpreted as data sphering for background removing at first stage, and calculating the Mahalanobis distance for each data sample, ${\mathbf{\text{r}}}_i$. Since originated from orthogonal projection data and its signal has been enhanced, the detectability is therefore increased.

IV. EXPERIMENTS AND RESULTS

Animal experiments were conducted in compliance with the local IACUC Committee guidelines. For brain imaging, 5xFAD [39] mice were selected. The animals were sacrificed and perfusion-fixed using a 1:10 mixture of ProHance-buffered formalin to shorten T1 and reduce scan time [55]. The specimens were scanned using a 9.4 T Bruker scanner (Bruker, Billerica, MA) with a modified 3D multi gradient echo (MGRE) pulse sequence at 18 μm isotropic resolution [39], [56]. The repetition time (TR) was 100 ms with two echo times (TEs) of 10 ms and 24.53 ms.

A. Dataset Description

The proposed method’s effectiveness on the following dataset for multiple tasks. Since iron demonstrates paramagnetic properties and the iron-only deposition exhibits paramagnetic susceptibility values on high-resolution QSM, our method focuses on Aβ plaques which exhibit diamagnetic properties (sparse negative magnetic susceptibility values). Accordingly, voxels with paramagnetic susceptibility values on high-resolution QSM were excluded from the analysis.

• 1. High-resolution QSM Slice: In a study focused on an AD mouse model, a single coronal slice was extracted from brain scans acquired over a period of 7.5 months. The scans had dimensions of 640×420. Expert annotations were used to label the plaques within the slice with 5771 voxels, and the annotated slice is illustrated in Figure 2 (b). It is important to note that only plaques located in the cortical region were labeled and detected in this study. The plaques in Figure 2 show its varieties in shape and intensity, and some plaques connected and show cluster properties. To further understand the Aβ plaques annotations, the size distribution of Aβ plaques was established, and shown in Figure 2 (m), illustrates the frequency of occurrence for each plaque size, measured in voxels. This analysis reveals that the majority of plaques are small, with a steep drop-off in frequency as size increases, reflecting the heterogeneity of plaque morphology [57]. To be noted, given that the QSM images were acquired with a resolution with 18 μm × 18 μm × 18 μm, which may pose a significant challenge for detection in clinical QSM due to the spatial resolution is typically much lower.

• 2. Histological Dataset: Histological examinations of the mice’s brains were conducted as previously described [39], [58]. A total of three mice underwent histological analysis, while seven mice were scanned using MRI. Brain section, 30 microns thick, were stained to visualize Aβ plaques and then imaged using a Leica DVM6 digital microscope. As shown in Figure 3 (a), is high-resolution QSM image. Figure 3 (b) illustrates the cortex region in the high-resolution QSM image, which is to be evaluated. In contrast, Figure 3 (c,d) are the image and the cortex region to be evaluated for histology image. It is important to note that the high-resolution QSM and histology images are not from the same animal or spatially co-registered. Rather, the purpose of the histological examination is to assess overall Aβ plaque burden and to evaluate whether the spatial patterns and detection trends observed in QSM are consistent with those seen in histological data.

• 3. Simulation DataSet: A synthesized set of high-resolution QSM images and Aβ plaques was generated by cropping a smoothed background from a high-resolution QSM image with added Gaussian noise with zero mean and a variance of 0.5 was applied to introduce random fluctuations. This was followed by salt-and-pepper noise with a density of 0.1 to mimic impulsive disturbances. To further increase variability, Rayleigh noise was generated using a scale parameter of 0.2, normalized from 0 to 1, and added with a weight of 0.2 to simulate multiplicative effects seen in some imaging modalities. Additionally, Laplacian noise, derived from a zero-mean Laplace distribution with scale 0.05, was incorporated to model heavy-tailed noise distributions [59]. The Aβ plaques were simulated with susceptibility values ranging from −0.0538 to −0.0838, decreasing in increments of −0.01. These values were selected based on the intensity range observed in the cropped QSM images and are intended to represent varying stages of AD progression. These plaques, arranged in a 3×3, 2×2, and 1×1 configuration, were then superimposed on the synthesized high-resolution QSM background with a variety of noises randomly. For each plaque size, four QSM intensity levels (−0.0538, −0.0638, −0.0738, and −0.0838) were assigned, and each combination was repeated five times, resulting in a total of 60 simulated Aβ plaques, and effectively visualizing the progression of plaque development over time, Figure 4 (a). Figure 4 (b) represents the ground truth corresponding to Figure 4 (a).

• 4. High-resolution QSM Dataset: The dataset includes brain scans from female AD mouse 5xFAD models aged 2, 4, 7.5, and 12 months, with 5, 6, 7, and 7 mice in each age group, respectively. Each scan has dimensions of 640 × 420 × 900. From visual inspection, the Aβ plaques exhibit distinct patterns for each age group. Specifically, few Aβ plaques are observed at 2 months, with an increase in number at 4 and 7.5 months, and a dense accumulation at 12 months. For resolution, refer to the first experimental dataset, high-resolution QSM Slice.

Fig. 2.

Detection map comparison of the proposed method under different settings. (c-e) present methods with 3 by 3 neighbors and row-wise (NR), column-wise (NC), or diagonal order (ND), respectively, (f-h) present methods with 5 by 5 neighbors and row-wise (NR), column-wise (NC), or diagonal order (ND), respectively. (i-k) present methods with 7 by 7 neighbors and row-wise (NR), column-wise (NC), or diagonal order (ND), respectively, and (l) nnU-net deep learning method. (m) The size of Aβ plaques distribution.

Fig. 3.

The detection results of Histology vs high-resolution QSM with $p=0.8744$, and The highlighted yellow regions indicate the detection results

Fig. 4.

Simulation experiment: (a) synthesize QSM with noise and different QSM level of Aβ, (b) Ground Truth, (c, e, g) show the detection maps of YOLOv10, Faster R-CNN, and RT-DETR, respectively. (d, f, h) show the detection maps of the same models when the input image is upscaled by a factor of 8. The suffix _UP denotes upscaling, and the number 8 indicates an 8× enlargement of the input image, (f) QSM-PLAQUE Aβ N3 NR

B. Performance Evaluation Methods

We demonstrate the proposed QSM-PLAQUE Aβ Detector on a variety of task, more specifically, we treat plaque detection in high-resolution QSM Slice as objection detection or classification problem using Intersection over Union (IOU), or precision, recall, and F1 score which are defined as below:

$$\mathit{\text{IoU}}=\frac{\mathit{\text{Area}}\mathit{\text{of}}\mathit{\text{Intersection}}}{\mathit{\text{Area}}\mathit{\text{of}}\mathit{\text{Union}}}$$ (12)

$$\mathit{\text{Precision}}=\frac{\mathit{\text{True}}\mathit{\text{Positives}}\left(TP\right)}{\mathit{\text{True}}\mathit{\text{Positives}}\left(TP\right)+\mathit{\text{False}}\mathit{\text{Positives}}\left(FP\right)}$$ (13)

$$\mathit{\text{Recall}}=\frac{\mathit{\text{True}}\mathit{\text{Positives}}\left(TP\right)}{\mathit{\text{True}}\mathit{\text{Positives}}\left(TP\right)+\mathit{\text{False}}\mathit{\text{Negatives}}\left(FN\right)}$$ (14)

$$F1\mathit{\text{Score}}=2\times \frac{\mathit{\text{Precision}}\times \mathit{\text{Recall}}}{\mathit{\text{Precision}}+\mathit{\text{Recall}}}$$ (15)

Precision concerns false positive errors, a higher precision meaning the model diagnosed disease truly correct with low false positives. While recall measures the level of avoiding missing true positives, a higher recall meaning the model identifies patients correctly with low false negatives. Meanwhile, F1 score offers a better balance between precision and recall. A higher IoU suggests better performance in tasks like object detection or segmentation, as it measures the overlap between predicted and ground-truth regions. A threshold, such as 0.7 [60], is often used to determine whether a prediction is considered correct.

Additionally, for the histological dataset, we evaluated the detection performance in both histological data and high-resolution QSM using plaque loading, defined as follows:

$$\mathit{\text{PL}}=\frac{\mathit{\text{Area}}\mathit{\text{of}}\mathit{\text{Plaques}}}{\mathit{\text{Area}}\mathit{\text{of}}\mathit{\text{Region}}}$$ (16)

A closer $\mathit{\text{PL}}$ between histological data and high-resolution QSM indicates a better performance.

For the simulation dataset, we treated the task as a target detection problem, whereas the high-resolution QSM dataset was used to demonstrate a practical application of the proposed method. All plaques across the entire brain were detected using the proposed approach and visualized in a 3D model.

C. Comparative Study Methods

There are a total of ten distinct detectors used, labeled as QSM-PLAQUE Aβ N3 NR, QSM-PLAQUE Aβ N3 NC, QSM-PLAQUE Aβ N3 ND, QSM-PLAQUE Aβ N5 NR, QSM-PLAQUE Aβ N5 NC, QSM-PLAQUE Aβ N5 ND, QSM-PLAQUE Aβ N7 NR, QSM-PLAQUE Aβ N7 NC, QSM-PLAQUE Aβ N7 ND and nnU-net [61]. In this naming convention, the numbers N3, N5, and N7 correspond to the different neighbor sizes (representing 3, 5, and 7 neighbors, respectively). Meanwhile, the labels NR, NC, and ND refer to the order in which the neighbors are considered: NR indicates row-wise, NC indicates column-wise, and ND represents diagonal-wise neighbor order. To compare the proposed approaches, a state-of-the-art deep learning framework, nnU-net, was conducted for a performance comparison due to its effectiveness and adaptability in medical image segmentation. nnU-Net uses a dynamically generated U-Net, with adaptive patch size, batch size, and network depth. Training is done with SGD (momentum 0.99) and a polynomial learning rate schedule starting at 0.01. The loss combines Dice and cross-entropy. Data augmentation and z-score normalization are applied automatically. Training accurately achieves 70% with 1000 epochs and consumes 22 hours on a 5-training dataset with 15750 high-resolution QSM slices. Inference uses sliding windows with 50% overlap and test-time augmentation for robustness.

For object detection task in Simulation DataSet, the YOLOv10 [62], Faster R-CNN [60], and RT-DETR [63], [64] were selected against the QSM-PLAQUE Aβ N3 NR, which was the best performance in high-resolution QSM Slice experiment. Faster R-CNN presents a two-stage object detection framework with conventional backbone network, such as ResNet, while YOLOv10 employs a one-stage with fast speed, and RT-DETR is based on transformer represents the latest advancement in object detection. Training data included 10,000 images were generated from simulation high-resolution QSM data with rotated +15, −15, shear between −0.1 and 0.1 radians, contrast adjustment with sigmoid gain between 0.7 and 1.3, and added Gaussian noise with standard deviation up to 0.05. YOLOv10 was trained using Ultralytics framework with batch size 16, image size 100 ×100, learning rate 0.01, 8 worker threads, and 500 epochs with early stopping True. For Faster R-CNN, the model was trained from scratch, with a batch size of 16, learning rate of 0.00025, and with 10,000 iterations without learning rate decay. The RT-DETR uses a batch size of 8, and 500 epochs, the learning rate was set the default optimizer learning rate in HuggingFace.

D. High-resolution QSM Slice Results

Figure 2 displays the detection maps generated by the QSM-PLAQUE Aβ detector, its variants, and nn-Unet. As illustrated in Figure 2 (c–d), the detectors performed optimally in identifying Aβ plaques, closely matching the ground truth shown in Figure 2 (b) from visual inspection. This success can be attributed to the effective suppression of the complex background. Additionally, the use of 3×3 neighbors accurately captures the small regions surrounding larger QSM values, which aligns with the spatial distribution characteristics of Aβ plaques. However, the QSM-PLAQUE Aβ N5 and QSM-PLAQUE Aβ N7, which utilized larger neighbor sizes to represent the spectrum of Aβ plaques, performed poorly. This is due to the weaker association between distant, smaller QSM values and the actual Aβ plaques. Figure 2 only show qualitative results by visual inspection. To perform a quantitative analysis, Intersection over Union (IoU), precision, recall, and F1 score were calculated based on the results shown in Figure 2. Table I summarizes these metrics, with the best-performing values highlighted in bold. As shown in Table I, the QSM-PLAQUE Aβ N3 NR, QSM-PLAQUE Aβ N3 NC, and QSM-PLAQUE Aβ N3 ND detectors achieved the best performance, while the others produced significantly poorer results. It is worth noting that the differences among the three detectors are not significant, highlighting that the neighborhood direction is not a critical factor. Notably, the performance of the detectors declined as the neighbor size increased. Moreover, nnU-Net exhibited the poor performance across all evaluation metrics. Despite substantial effort invested in annotating Aβ labels, the deep learning model failed to effectively learn their features. This limitation is likely due to the high variability of Aβ, along with its irregular shape and size. The downsampling strategy used by convolutional neural networks to learn the broader context can cause tiny structures such as 1×1, 2×2 voxels, to be lost after several layers. In order to achieve a high performance using deep learning methods, a redesign of the model and a more thoughtful consideration of Aβ feature is needed.

TABLE I.

IoU, Precision, Recall, and F1 Score Metrics for QSM Slice

Detector	IoU	Precision	Recall	F1 Score
QSM-PLAQUE Aβ N3 NR	0.8173	0.8595	0.9433	0.8995
QSM-PLAQUE Aβ N3 NC	0.8113	0.8536	0.9425	0.8958
QSM-PLAQUE Aβ N3 ND	0.8166	0.8599	0.9420	0.8990
QSM-PLAQUE Aβ N5 NR	0.5231	0.5366	0.9541	0.6869
QSM-PLAQUE Aβ N5 NC	0.5246	0.5391	0.9511	0.6882
QSM-PLAQUE Aβ N5 ND	0.5257	0.5389	0.9555	0.6891
QSM-PLAQUE Aβ N7 NR	0.4100	0.4178	0.9567	0.5816
QSM-PLAQUE Aβ N7 NC	0.4110	0.4189	0.9560	0.5825
QSM-PLAQUE Aβ N7 ND	0.4112	0.4190	0.9567	0.5828
nnU-net	0.2196	0.4918	0.2841	0.3602

E. Histological Dataset Results

Figure 3(b,d) shows the detection maps produced by QSM-PLAQUE Aβ N3 NR for high-resolution QSM and histology, respectively. As you can see, the Aβ plaques have been detected in both high-resolution QSM and histology. Quantitative assessment revealed an average plaque load of 13.85% in the histology group, compared to 13.59% in the high-resolution QSM group. As shown in Figure 3(e), a one-way ANOVA [65] test indicated that the difference between the two groups was not statistically significant $(p=0.8744)$. These results demonstrate that amyloid plaques detected using both histology and high-resolution QSM with our proposed methods are closely aligned, highlighting the accuracy of our techniques for assessing disease progression.

F. Simulation DataSet Results

Figure 4 panels (c–f) show the detection maps for YOLOv10, Faster R-CNN, RT-DETR, and our method (QSM-PLAQUE Aβ N3 NR). Among them, YOLOv10 exhibited the poorest detection performance, while our method demonstrated the best results based on visual inspection. More specifically, YOLOv10 failed to detect smaller Aβ plaques of size 1×1. While Faster R-CNN and RT-DETR showed better performance on 1×1 and 2×2 plaques, they were less effective at detecting larger 3×3 plaques. In contrast, our method successfully detected most Aβ plaques across all sizes—1×1, 2×2, and 3×3—demonstrating robust performance across a variety of plaque sizes.

To validate our visual results, a quantitative analysis was conducted using a standard metrics as high-resolution QSM Slice experiment, as summarized in Table II indicating our proposed method, QSM-PLAQUE Aβ N3 NR, has the highest performance across all metrics, an IoU of 0.9214, perfect precision (1.0000), and an F1 score of 0.9591, showing both accurate and consistent detection. However, YOLOv10 achieved perfect precision (1.0000) but poor recall (0.1333), which means it missed almost of 86.67% targets, and thus got a low F1 score of 0.2523. Overall, Faster R-CNN has the lowest overall performance, while RT-DETR performed moderately, with an F1 score of 0.4928. It should be noted that the lower metrics obtained from the three deep learning methods do not imply that these approaches are inherently incapable of detecting Aβ plaques. In our simulation, we tested for tiny objects with extreme dimensions of 1×1, 2×2, and 3×3 voxels, which are representative of tiny Aβ plaques. The results showed that the performance of these deep learning techniques was substantially lower than that of our proposed method. The primary reason for this discrepancy is that all of these deep learning models utilize downsampling strides. Specifically, YOLOv10 and Faster R-CNN use a stride of 32, while RT-DETR uses a stride of 16 or 32, depending on the backbone architecture. This downsampling causes the extremely small objects to be lost during the feature extraction process, which is exactly why YOLOv10, Faster R-CNN, and RT-DETR suffered detecting such small targets and achieved lower performance [66]. Thus, even the best object detectors often miss very small objects, especially without targeted architectural or data-level adaptations. That’s why the performance of these deep learning techniques was substantially lower than that of our proposed method, not tailored for tiny Aβ plaques. Some strategies can be used for dealing with small object detection shortcomings [67]. We also demonstrate the effect of upscaling the input image by 8 times on the three deep learning models, allowing small objects to better survive the CNN downsampling strides. As shown in Table II, by increasing the input size in YOLOv10_UP8, Faster R-CNN_UP8, and RT-DETR_UP8, the apparent size of Aβ plaques is enlarged, and all models’ performance scores rise to approximately 0.90–0.95, approaching those of our proposed QSM-PLAQUE Aβ N3 NR. This demonstrates that appropriately scaling inputs in deep learning models can effectively mitigate small-object detection challenges for tiny Aβ plaques. Thus, employing deep learning for tiny Aβ detection must effectively mitigate small-object issues. Since our proposed method demonstrates robust performance on tiny and small objects, integrating it with deep learning approaches to further address small-target detection could represent a valuable direction for future research.

TABLE II.

IoU, Precision, Recall, and F1 Score Metrics for QSM Slice

Detector	IoU	Precision	Recall	F1 Score
QSM-PLAQUE Aβ N3 NR	0.9214	1.0000	0.9214	0.9591
YOLOv10	0.1538	1.0000	0.1333	0.2523
YOLOv10_UP8	0.9036	1.0000	0.9036	0.9493
Fast R-CNN	0.1009	0.3431	0.1250	0.3431
Fast R-CNN_UP8	0.8750	1.0000	0.8750	0.9333
RT-DETR	0.3269	0.5862	0.4250	0.4928
RT-DETR_UP8	0.9107	1.0000	0.9107	0.9533

G. High-resolution QSM Dataset Results

The quantification of amyloid plaques showed plaque loading of 2.0%, 5.9%, 12.2%, and 16.3% for 2-, 4-, 7.5-, and 12-month-old female mice, respectively. Figure 5 shows 2D slices illustrating the progression of Aβ accumulation at 2, 4, 7.5, and 12 months. Specifically, high-resolution QSM maps were loaded in ITK-SNAP [68] with the detection maps (red) as masks and overlaid them to show the detection of Aβ plaques. Panels (a, b, c, d) in Figure 5 present coronal views and demonstrates a consistent increase in plaque loading, aligning with the progression observed at each time point stage. It is worth noting that the mice in Figure 5 show an obviously sparse distribution of Aβ plaques at 2 months, while 12 months exhibited a significantly dense distribution. The middle months (4 and 7.5) show the transitional stages, with gradually increasing plaque density observable. This finding suggests that the density of Aβ plaques may correlated with the progression of AD and could aid in precisely treatment.

Fig. 5.

2D Plaques Loading Image at 2, 4, 7.5, and 12 months for female. (a-d) are from coronal view

V. Sensitivity Analysis of GoDec

Given that the proposed algorithm fundamentally relies on the matrix decomposition results of GoDec, specifically leveraging Energy parameters for low-rank component and QSM intensive values for sparse component identification—the selection of these hyper-parameters represents a critical determinant of algorithmic performance. While initial parameter values were obstained through empirical observations and from existing literature suggested. To rigorously assess the algorithmic robustness and establish confidence in parameter selection, we conducted a comprehensive sensitivity analysis examining the interaction effects between these two pivotal hyperparameters. The sensitivity analysis encompassed 20 Energy parameter values ranging from 0.05 to 1.00 in increments of 0.05, and 18 Sparse parameter values from −0.005 to −0.09 in decrements of −0.005. This systematic evaluation across 360 parameter combinations. For each configuration, the model performance was evaluated with Intersection over Union (IoU), F1 score, Recall, and Precision.

Figure 6 demonstrates that optimal performance for IoU, F1, and Recall metrics (displayed in the top left, top right, and bottom right panels, respectively) is achieved at sparsity values ranging from approximately −0.015 to −0.020 combined with energy values between 0.15 and 0.25. This consistent pattern across multiple evaluation metrics indicates that moderate negative sparsity parameters paired with low-to-medium energy configurations produce superior segmentation results.

Fig. 6.

Sensitivity analysis of GoDec parameters for Aβ plaque detection. Detection performance was evaluated in IoU, F1 score, precision, and recall

While the Precision metric (bottom left panel of Figure 5) reaches its maximum at more extreme parameter settings of sparsity = −0.085 and energy = 0.05, the performance surface remains relatively stable across a broader parameter range (sparsity ≈ −0.015 to −0.085, energy ≈ 0.15–0.5). Given this smooth performance characteristic, adopting the parameter configuration of sparsity ≈ −0.015 to −0.020 with energy ≈ 0.15–0.25 represents a balanced approach that maintains reasonable performance across all metrics while optimizing the majority of evaluation criteria. Notably, this findings are consistent with our setting in the Experiments and Result section.

However, performance degrades significantly when Sparse becomes too negative or Energy is too high, indicating sensitivity to over-sparsification and excessive low-rank relaxation. These findings highlight the importance of proper tuning of both parameters to maintain reliable detection results.

VI. Discussion

Figs. 2–4, and Tables I and II demonstrate the experimental results aligning closely with observations and previous finding.

• First of all, Figure 2 validate a critical observation: Aβ plaques frequently exhibit a spare structure and occupied small area, which is validated by the size of distribution of Aβ plaques in Figure 2 (m), with varying shapes and sizes, making them difficult to detect—even with deep learning approaches. Considering the characteristics of Aβ plaques, as Figure 2, and the Table I shown the smaller neighbor matrix 3×3 is more sensitive to fine local features, which is well-suited for capturing the sparse and irregular structure of Aβ plaques achieves a best performance.

• Also, according to Figure 3, panels (b) and (d) showcase the results generated by QSM-PLAQUE Aβ N3 NR. Panel (b) represents a high-resolution QSM MRI image, while panel (d) displays an optical image. These results demonstrate the versatility of QSM-PLAQUE Aβ N3 NR in generating high-quality outputs across multiple imaging modalities, highlighting its practical applicability in diverse scenarios. It is important to note that the results of the histology image analysis were obtained using only NS and RXD methods. This is because the histology images inherently possess a clean background, which eliminates the need for additional preprocessing for background.

In this study, we conducted three experiments and a simulation based on four dataset to investigate the effects of the proposed method. High-resolution QSM slice focused on object detection and classification tasks, histological dataset examined accuracy of detection according histology image, simulation dataset explored the power of detection of our method in a tiny/small target detection task. Finally, the high-resolution QSM dataset provides a whole-mouse Aβ plaque loading map, offering a more accurate and comprehensive assessment of AD progression. Collectively, this approach enables a visualized and detailed understanding of disease dynamics, enhancing both research and diagnostic capabilities.

Although denoising was the primary application when introducing GoDec, our use in this paper is repurposed beyond conventional denoising. In our case of high-resolution QSM, the background susceptibility variations are not random like noise but structured, and this structured background can be presented as a low-rank matrix. At the same time, our interest of objects, Aβ plaque, is localized and aligned with the sparse matrix. We do not discard the low-rank matrix as in classical denoising; instead, we isolate Aβ plaque by projecting the joint low-rank and sparse matrices onto the orthogonal subspace of the low-rank matrix, and the remainder are disease-relevant features without complex background interference. Thus, GoDec in our paper serves not as a generic denoising algorithm, but as a methodology of structured background separation for high-resolution QSM-based Aβ plaques detection.

Considering the robustness and generality of the GoDec method, both the fixed-energy threshold can be automated through data-driven approaches. Instead of relying on a predetermined 25% energy cutoff, the cumulative energy curve derived from the singular values can be analyzed using elbow detection techniques such as curvature-based or second-derivative analysis [69], [70]. This approach identifies the rank $k$ at which the marginal gain in energy sharply decays, thus selecting the most representative low-rank background without knowing the prior energy threshold.

The ability of our method to identify individual Aβ plaques offers a significant advantage for investigating plaque-level features in AD. This is especially valuable given high-resolution QSM’s inherent limitations in specificity—due to factors like myelin and calcium—and reduced sensitivity in cortical regions. By isolating plaque-level features, our approach enhances the precision of high-resolution QSM-based assessments in AD. Although our high-resolution QSM acquisition was limited to a 2-echo GRE sequence—insufficient for advanced decomposition—emerging techniques such as DECOMPOSE-QSM [34], $\chi$-separation [33], and QSM-ARCS [71] have demonstrated the feasibility of separating paramagnetic and diamagnetic components, and applications in [72]. Integrating these approaches with our study in future work will allow more precise analysis of iron and Aβ contributions, ultimately enhancing the reliability of high-resolution QSM-based detection of AD-related pathology.

Notably, our proposed method was applied to 2D high-resolution QSM slices rather than the full 3D high-resolution QSM dataset, which may not fully capture the spatial correlations between adjacent slices. Therefore, extending the method to 3D high-resolution QSM data would be more physical meaning for detecting Aβ plaques in a volumetric context. While extending from 2D to 3D is theoretically feasible, it introduces significant technical challenges. The greatest challenge of 3D RXD lies in estimating the covariance matrix in a stable and efficient manner within a neighborhood size that is high-dimensional (compared to 2D), non-stationary (unlike 2D). Furthermore, the computational cost associated with covariance matrix estimation is also a critical consideration when the extension from 2D to 3D.

Finally, despite the promising results demonstrated in the high-resolution ex vivo mouse QSM dataset, direct application to in vivo human data presents several challenges. Specifically, human whole-brain QSM data typically suffers from lower spatial resolution, constrained by limitations in SNR, motion sensitivity and acquisition time. These factors result in obscured Aβ plaque and cause the detection to fail. While direct clinical translation is not currently feasible, our method provides a framework that could guide the development of techniques suitable for preclinical study.

VII. Conclusion

This paper proposes a method to quantify Aβ plaques in high-resolution QSM for preclinical studies based on low rank decomposition with spatial information, under the assumption that the plaques are small, sparse, and distinct from their surroundings. Low-rank decomposition addresses the complex background problem in high-resolution QSM MRI, while the covariance matrix helps resolve issues related to variability in size, value, and shape of the features. Our results demonstrate that the proposed method effectively detects Aβ plaques with varying sizes and shapes across histology, QSM, and synthesized images, all without the need for training samples. While this study provides promising results, it is limited by the high-resolution requirement, which by far may not be achievable in clinical settings. The 3D RXD approach shows promise in providing physically meaningful results and could represent a valuable direction for future studies. In conclusion, the proposed approach offers new possibilities for quantifying Aβ plaques and could serve as a foundation for advancing quantitative preclinical neuroimaging studies of AD.

Data & Code Availability Statement

One of mouse brains used in this study, 12 months, together with 166 label Atlas of the mouse brain, code for generating Aβ plaques, are available at Online Drive Folder.