Studying APOE-e4 Allele Dose Effects with a Univariate Morphometry Biomarker

Abstract

This paper proposes a subspace decomposition method capable of exploiting a univariate morphometry biomarker to represent the hippocampal morphological changes associated with APOE-e4 genetic risk for Alzheimer’s disease (AD) based on magnetic resonance imaging (MRI) data of the elderly cognitively unimpaired (CU) individuals. Rank minimization and sparsity constraint theory drive our method. Rank minimization mechanism combined with improved partial singular value decomposition (SVD) is used to extract group common structures and reduce the computational cost of SVD of the large-scale matrix. Further, sparse constraint considering the local continuity of the hippocampal atrophy regions is applied to recover individual morphological differences. Based on the group common structures of amyloid-β (Aβ) positive AD patients and Aβ negative CU subjects, we identified the regions-of-interest (ROI), which reflect significant morphometry changes caused by the AD development. Then univariate morphometry index (UMI) is constructed from these ROIs. Capitalizing on the study of a large number of cognitively unimpaired late middle-aged and older adults with two, one, and no APOE-e4 alleles, we set up three experiments to study the differences of hippocampal morphological changes among the longitudinal APOE-e4 homozygotes (HM, N=30), heterozygotes (HT, N=49) and non-carriers (NC, N=61). The results show that the proposed UMI enjoys improved computational efficiency comparing to our prior work and demonstrates a more vital ability to identify APOE-e4 dose effects than the raw surface and volume measurements. Our work may provide a cost-effective UMI that helps evaluate AD burden, progression, and response to interventions, potentially facilitating AD clinical trial enrollments.

1. Introduction

Alzheimer’s disease (AD) is an irreversible neurodegenerative disorder characterized by progressive cognitive impairment that interferes with memory, thinking, and behavior. For AD prevention therapies to have a maximal effect, they might need to be initiated before measurable impairments in cognition, at which time extensive pathology likely already exists [1, 2]. Therefore, there is a need to study biomarkers closely related to AD progression in its earliest non-symptomatic and symptomatic stages to observe AD development and evaluate the effectiveness of early interventions. It is well known that apolipoprotein E (APOE) e4 allele is a major genetic risk factor for late-onset AD [3, 4]. Its discovery has made it possible to study large numbers of genetically at-risk individuals before the onset of symptomatic memory impairment, and has led to the concept of the preclinical stage of AD [5]. Therefore, studying the association between APOE genotype with neurodegenerative pathology in healthy adults may benefit the development of preclinical AD biomarkers.

Numerous studies have revealed the pathological correlations between probable AD and the APOE-e4 allele by using a variety of biomarkers, including magnetic resonance imaging (MRI) studies of whole brain atrophy rates of APOE-e4 carriers [6], fluorodeoxyglucose positron emission tomography (FDG-PET) studies of APOE-e4 carriers that have revealed AD-like patterns of reduced CMR glucose [7, 8], measures of increased amyloid plaque burden determined from increases in the standard uptake value ratio (SUVR) of florbetapir PET [9], and distinct default mode network (DMN) alterations observed in the resting-state functional MRI analysis [10], etc. Among them, structural MRI (sMRI) has been widely used to identify brain structure changes, including cortical atrophy [11, 12], hippocampal atrophy [13–15], or ventricular enlargement [14, 16], which can serve as the indicative diagnostic biomarkers of early AD, owing to their close relationship between neurodegeneration and cognition. Hippocampal atrophy measures from structural MRI are widely used for investigating the genetic influence of APOE-e4 on hippocampal morphology because the changes in hippocampal morphometry become apparent in the early stages of memory decline [13, 17–20]. However, the genetic influence of APOE-e4 on hippocampal morphometry research has reported mixed results. Some studies have elucidated a dose-dependent disease vulnerability on the hippocampal structure level [21–26], including our own work [27], e.g., significantly reduced hippocampal volume or accelerated longitudinal hippocampal atrophy has been found in healthy APOE-e4 homozygotes (HM), as compared to heterozygotes (HT) and non-carriers (NC). Some other studies [28–33] did not find significant group differences between HM, HT, and NC. Therefore, developing sensitive and reliable brain imaging biomarkers to validate the dose-dependent impact of the APOE-e4 allele on hippocampus among cognitively unimpaired individuals would be highly advantageous to preclinical AD diagnosis and prognosis.

While much APOE genetic influence on hippocampal morphometry research adopted multivariable imaging biomarkers (e.g., [26, 27]), a univariate neurodegeneration biomarker (UNB) based on an individual patient’s brain scans with high diagnostic accuracy would be highly desirable for clinical use [34] and patient screening in clinical trials [2]. Such a personalized measure may overcome inflated Type I error due to multiple comparisons [30, 31]. For example, for randomized clinical trials (RCT), regulatory agencies, including the Food and Drug Administration (FDA), requires conventional univariate hypothesis testing and its associated statistical power analysis [2]. Meanwhile, a single MRI-based measure of cerebral atrophy was used as a neurodegeneration biomarker in the recently proposed AD amyloid/tau/neurodegeneration (A/T/N) framework [35]. In our recent work [36], we proposed a subspace decomposition framework to generate a univariate morphometry index to quantify the hippocampal morphological changes induced by AD based on the reliable and informative region-of-interest (ROI), which takes into account the intense image noise and the significant within-group variance on the obtained structural MR images. However, our previous algorithm for solving SPCP has few limitations. First, our previous work’s low-rank subspace decomposition algorithm is the alternating splitting augmented Lagrangian method (ASALM). The disadvantage of ASALM is that its convergence is slow for real-time application [37]. Second, the singular value decomposition (SVD) in our previous algorithm used a low-rank matrix factorization mechanism to factorize a large-sized matrix into two small-sized matrices generated by implementing matrix column orthogonal strategy and low-rank constraint. Although this mechanism can effectively reduce the SVD computation cost for a large-scale matrix, the excessive intermediate processing steps require further optimization.

In this work, we propose an improved subspace decomposition framework to establish univariate and personalized neurodegeneration MRI biomarkers with strong statistical discrimination power [38]. We hypothesize that the proposed univariate morphometry biomarker may distinguish hippocampal surface morphological changes related to the APOE-e4 dose effects in the cognitively unimpaired population. In our experiments, we first used sMRI images of 120 Aβ positive AD patients and 257 Aβ negative CU subjects from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) cohort [39] to establish the AD-specific abnormality patterns over the hippocampal surface. We then compute a univariate similarity index to examine the degree of similarity to this AD-specific surface pattern for a given individual subject from an independent Arizona APOE cohort [40], including 140 cognitively unimpaired individuals (30 HMs, 49 HTs, and 61 NCs). We aim to test whether the proposed univariate morphometry biomarker gains improved computational efficiency and whether it has the superiority to the raw surface and volume measurements in reflecting the morphological atrophy rates caused by different APOE-e4 allele loads in the CU individuals.

2. Materials and Methods

2.1. Subjects

Data used in the preparation of this article were partially obtained from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (adni.loni.usc.edu). The ADNI was launched in 2003 as a public-private partnership, led by Principal Investigator Michael W. Weiner, MD. The primary goal of ADNI has been to test whether serial magnetic resonance imaging (MRI), positron emission tomography (PET), other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early Alzheimer’s disease (AD). For up-to-date information, see www.adni-info.org.

Specifically, the sMRI data used to generate AD-specific abnormality patterns, including 120 amyloid-β (Aβ) positive AD patients and 257 Aβ negative CU subjects, were downloaded from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database ([39], adni.loni.usc.edu). The reason for choosing Aβ positive subjects is that the accumulation of beta-amyloid plaques (Aβ) is one of the hallmarks of AD in human brains and a positive Aβ reading is now accepted as ‘dementia due to AD’ together with the presence of clinical symptoms [41].

To validate the effects of APOE-e4 allele load on hippocampal morphology, we used 140 cognitively unimpaired individuals from the Arizona APOE cohort [40] and studied longitudinal brain imaging changes via statistical group difference analysis. The Arizona APOE cohort was established since January 1, 1994, which contains cognitively normal residents of Maricopa County aged 21 years and older. Demographic, family, and medical history data were obtained on each individual undergoing APOE genotyping, and their identity was coded by a study assistant. All individuals gave their written, informed consent-approved by the Institutional Review Boards of all participating institutions-and agree to have the results of the APOE test withheld from them as a precondition to their participation in this study. Genetic determination of APOE allelic status was performed using a ploymerase chain reaction (PCR) based assay [42]. All subjects were genotyped and classified as APOE e4 carriers (N_HM=30, N_HT=49) or non-carriers (N_NC=61). The study excludes subjects with potentially confounding medical, neurological, or psychiatric problems (such as prior stroke, traumatic brain injury, memory, or other cognitive impairment, parkinsonism, major depression, or substance abuse). No subject included in the study met the published criteria for MCI, AD, any other form of dementia [43].

2.2. Algorithm Pipeline

We briefly summarize our overall sequence of steps used to compute the univariate morphometry biomarker. The following sections are detailed explanations of each step. Figure 1 shows the processing procedures in our system. First, we segment hippocampi from T1-weighted MR images and reconstruct hippocampal surfaces. We register hippocampal surfaces across subjects and compute surface morphometry features, radial distance (RD) [44, 45] (Sec. 2.3). Second, we construct two group observation matrices (i.e., Aβ positive AD observation matrix and Aβ negative CU observation matrix) where all RD features of each registered subject are stacked as a column vector. To effectively extract the robust group common morphological structure contained in all individual RD features, we further decompose each group observation matrix into low-rank component, sparse component and noise using a subspace decomposition algorithm. The low-rank component is the group common structure contributed by all the subjects in the same group, the sparse component is the sparse diversities which belong to specific individual subjects. With the subspace decomposition framework, we may obtain the two intrinsic group common morphological structures of Aβ positive AD group and Aβ negative CU group, which takes into account the intense imaging noise and the significant within-group variance on the obtained structure MR images. Third, we compute AD-specific abnormal surface patterns on the ROIs which are identified as significantly different areas induced by AD between the low-rank components of two groups, Aβ positive AD group and Aβ negative CU group. We further use the labeled ROIs and the registration result of an individual hippocampal surface to compute the univariate morphometry index (UMI) (Sec. 2.5). Finally, as application demonstrations of the computed UMIs, we apply them to study the longitudinal brain imaging changes of the three APOE-e4 genotype groups and dose effects of APOE-e4 alleles on the hippocampal morphology (Sec. 3.2 and 3.3).

Figure 1.

Processing procedures in our univariate morphometry biomarker system

2.3. Image Acquisition and Morphometry Feature Extraction

High-resolution brain structural MRI scans were acquired using 3 Tesla MRI scanners manufactured by General Electric Healthcare, Siemens Medical Solutions, and Philips Medical Systems. For each subject, a high-resolution T1 magnetization-prepared spoiled gradient (SPGR) scan was obtained in the sagittal plane. A T1-weighted pulse sequence (radiofrequency-SPGR recall acquisition in the steady state, repetition time = 33 msec, echo time = 5 msec, alpha = 30°, number of excitations = 1, field-of-view = 24 cm, imaging matrix = 256 × 93, slice thickness = 1.5 mm, scan time = 13:36 min) was used to acquire 124 congiguous horizontal MRI slices with in-plane voxel dimensions of 0.94 × 1.25 mm.

Based on the T1-weighted MRI scans, hippocampal substructures were segmented by using FIRST in the FMRIB Software Library (FSL) [46] and 3D triangular hippocampal surface was reconstructed based on marching cubes algorithm [47]. With the generated conformal grid for each surface [48], we computed the hippocampal surface’s conformal representation, which represents the intrinsic and extrinsic features of the surface, respectively. In addition, we carried out a surface fluid registration algorithm [49] and an inverse-consistent surface registration framework to increase robustness [48]. Then all the surfaces have a common reference frame and radial distance (RD) statistical morphological measurements were computed directly on these registered surfaces. RD is the distance from each surface point to its medial core (analogous to the centerline in a tube), representing hippocampal atrophy or enlargement. The entire hippocampal surface morphometry pipeline is described in our prior work [48] and is publicly available at https://www.nitrc.org/projects/mtsms_2020.

2.4. Subspace Decomposition Algorithm

When all individual RD features of the same group are stacked into column vectors to form an observation matrix $A=(a_1,\dots ,a_n)\in {\mathfrak{R}}^{m\times n}$ ($a_i$ denotes the column vector which represent RD features of i-th subject, $m$ is the dimension of each subject RD features and $n$ is the number of the subjects within the same group), the stable principal component pursuit (SPCP) algorithm enables us to decompose $A$ into low-rank component $L\in {\mathfrak{R}}^{m\times n}$, sparse component $S\in {\mathfrak{R}}^{m\times n}$ and noise $N\in {\mathfrak{R}}^{m\times n}$. In order to improve the computation efficiency, we apply Non-Smooth Augmented Lagrangian Algorithm (NSA) [50] based on partial variable splitting to solve this non-smooth convex optimization problem. It is known that the recovery is possible even when the data matrix, $A$, is corrupted with a dense error matrix, ${‖N‖}_F\le \delta$, by solving the SPCP problem [51].

$${\mathit{min}}_{L,S\in R^{m\times n}}\left\{{‖L‖}_{\mathrm{*}}+\gamma {‖S‖}_1,s.t.:{‖L+S-A‖}_F\le \delta \right\}$$ (1)

where ${‖·‖}_{\mathrm{*}}$ is the nuclear norm and ${‖·‖}_1$ is the $l_1$ norm, which are given by ${‖L‖}_{\mathrm{*}}=\sum _i{\sigma }_i\left(L\right)$ and ${‖S‖}_1={\sum }_{i,j}\left|S_{i,j}\right|$, respectively. And $\gamma$ controls the relative importance of the low-rank term $L$ vs. the sparse term $S.{\sigma }_i\left(L\right)$ is the ith largest singular value of $L$.

Based on NSA, the subproblems for solving low-rank component $L$ and sparse component $S$ of Eq. (1) via the following iteration procedure:

$$\left\{\begin{array}{c}{L}_{k+1}\leftarrow {argmin}_L\{{‖L‖}_{\mathrm{*}}+〈Y_k,L-Z_k〉+\frac{{\rho }_k}{2}{‖L-Z_k‖}_F^2\}\\ \left(Z_{k+1},S_{k+1}\right)\leftarrow {argmin}_{\left(Z,S\right)}\{\gamma {‖S‖}_1+〈{-Y}_k,Z-L_{k+1}〉+\frac{{\rho }_k}{2}{‖Z-L_{k+1}‖}_F^2\}\\ \\ Y_{k+1}\leftarrow Y_k+{\rho }_k\left(L_{k+1}-Z_{k+1}\right)\\ {\rho }_{k+1}=\alpha {\rho }_k,\alpha \ge 1\end{array}\right.$$ (2)

Let $Y_k\in R^{m\times n}$ and ${\theta }_k$ be optimal Lagrangian dual variables for $Z_k=L_k$ and $\frac{1}{2}{‖Z+S-A‖}_F^2\le \frac{{\delta }^2}{2}$ constraints, ${\rho }_k>0$ is the penalty parameter for the violation of the linear constraint and $〈.〉$ denotes the standard trace inner product. In procedure (2), the subproblem for computing $L_{k+1}$ is a matrix shrinkage problem which can be solved via thresholding the singular values of the matrix ${(Z}_k-Y_k/{\rho }_k)$[52]. And the subproblem for computing $S_{k+1}$ is an element shrinkage problem which can be solved via $l_1$ norm:

$$S_{k+1}=sign(A-(L_{k+1}+Y_k/{\rho }_k))\odot max\{\left|A-\left(L_{k+1}+\frac{Y_k}{{\rho }_k}\right)\right|-T_{th},0\}$$ (3)

where $T_{th}$ is the threshold which is defined as $T_{th}=({\rho }_k+{\theta }_k)/(\sqrt{\mathrm{m}\mathrm{a}\mathrm{x}(m,n)}·{\rho }_k·{\theta }_k)$, ${\left\{{\theta }_k\right\}}_{k\in {\mathbb{z}}^{+}}$ is a bounded sequence, and ${\theta }_k$ satisfies the following expression $\delta ={‖\mathrm{m}\mathrm{i}\mathrm{n}\{\frac{\mathrm{\gamma }}{{\theta }_k}E,\frac{{\rho }_k}{{\rho }_k+{\theta }_k}\left|A-(L_{k+1}+\frac{1}{{\rho }_k}{Y}_k)\right|\}‖}_F,\odot$ denotes the componentwise multiplication operator. The details about computing $L_{k+1}$ based on improved partial singular value decomposition (SVD) and computing $S_{k+1}$ by considering the local continuity of the hippocampal atrophy regions can be found in Supplementary Materials. Based on the description above, we can use the proposed method to recover the low-rank component and the sparse component with the local continuous constraint from the large-scale data matrix under the complicated environment. In particular, we can use the low-rank component $L$, i.e., the essential group common structure, to detect the group differences without the influence of the individual’s specific diversities S and noise.

2.5. Univariate Morphometry Index (UMI) for AD-specific Abnormal Surface Pattern Similarity Measure

A Univariate Morphometry Index (UMI) reflecting individual AD-specific abnormal surface pattern similarity is generated by summarizing information in ROIs. Our UMI generation procedure is similar to our prior work [36]. First, for each registered vertex on the hippocampal surface, we used permutation t-test [53] to analyze the group difference for the low-rank components of the RD features between Aβ positive AD group and Aβ negative CU group. Given the level of significance $\alpha$, the accepted vertices to form the ROIs can be obtained for their permutation p-maps with the probabilities $<\alpha$. Since the extracted low-rank components represent the common morphological structure of the group, the extracted ROIs based on low-rank components obtained by subspace decomposition algorithm can accurately reflect the hippocampal morphological structure changes from CU to AD.

Secondly, we use a vertex-wise two-sample independent t-test for the mean differences between the low-rank components of the Aβ negative CU and the Aβ positive AD groups on the predefined ROIs. The obtained t-map is then transformed into a z-score map which is called as AD-specific abnormal surface pattern. And a vertex-wise two-sample independent t-test is applied for the differences between raw RD measures of individual subject and the mean low-rank components of the Aβ negative CU group on the predefined ROIs. The obtained t-map is then transformed into a z-score map which is called as individual atrophy degree. At last, the UMIs can be computed through the vertex-wise summation across all the vertices on the predefined ROIs with the production of AD-specific abnormal surface patterns and the atrophy degrees of the individual subject. Thus,

$$UMI=\frac{\sum _{i=1}^m(D_{Ti}·D_{Wi})}{100}$$ (4)

Here $D_{Ti}$ denotes the atrophy degree at vertex $i$ of the individual subject, $D_{wi}$ is the AD-specific abnormal surface pattern at vertex $i$ between the AD group and the CU group. From the Eq. (4), we can see that the UMI measures the degree of similarity to this AD-specific abnormal surface pattern for a given individual subject on the selected ROIs. In other words, the greater the UMI, the closer the atrophy degree of the individual subject is from AD morphological characteristics.

2.6. Evaluation of the Univariate Biomarkers

For longitudinal cognitively unimpaired APOE-e4 carriers (HM and HT) and non-carriers (NC) groups, we analyze the longitudinal group changes of the univariate biomarkers, including the UMIs, Mini-Mental State Examination (MMSE) [54] scores, Auditory Verbal Learning Test Long-Term Memory (AVLT-LTM) [55] scores and hippocampal volume measures, to compare the statistical discrimination abilities of different univariate biomarkers based on analysis of variance (ANOVA) method. All subjects of each longitudinal group underwent two tests, including the baseline test and a 24-months test. Meanwhile, Cohen’s d test is applied to evaluate the effect size of the univariate biomarkers above. Cohen’s d test takes the difference in means between two groups and is divided by the pooled standard deviation of the groups.

To further assess the statistical powers of the univariate biomarkers, we calculate the required minimum sample sizes (MSS) by comparing the univariate biomarker means of each longitudinal APOE-e4 genotype groups at two test time points (baseline test and 24-months test). Suppose the means and standard deviations of the specific univariate biomarker of each longitudinal group at two test time points are $\left({\mu }_1,{\sigma }_1\right)$ and $\left({\mu }_2,{\sigma }_2\right)$. To conduct a two-sided test with significance level $\alpha$ and power of $1-\beta$, the appropriate sample-size for the specific univariate biomarker is as follows [56]:

$$MSS=\frac{\left({\sigma }_1^2+{\sigma }_2^2\right){\left(z_{1-\alpha /2}+z_{1-\beta }\right)}^2}{{\mathrm{\Delta }}^2}$$ (5)

where $\mathrm{\Delta }=\left|{\mu }_2-{\mu }_1\right|$. The significance level $\alpha$ is set as 0.05 and the power $1-\beta$ is set as 0.8. For each longitudinal group, if the MSS derived from A biomarker is smaller than B biomarker, we can conclude that A biomarker is more differentiated than B biomarker. This indicates that MSS is an indicator for characterizing the statistical identification ability of the biomarkers.

3. Experimental Results

3.1. ROIs Extraction

In order to extract the significant morphological group differences induced by AD, we apply the subspace decomposition to a cohort consisting of 120 Aβ positive AD patients and 257 Aβ negative CU subjects from the ADNI database. Demographic and clinical data of this cohort is compared using a one-way analysis of variance, and the gender data was analyzed by a chi-square test. Table 1 indicates that the factors of age, gender and education of these two groups are matched, while the MMSE is significantly different between these two groups.

Table 1.

Demographic information of Aβ positive AD and Aβ negative CU groups, from the ADNI cohort.

	AD	CU	Inferential Statistics
Sample Size	120	257
Gender(M/F)	64/56	125/132	χ2 = 0.7214; p = 0.40
Age Education	74.15 ± 7.47 15.61± 2.69	75.24 ± 6.45 16.10± 2.50	F = 2.13; p = 0.15 F = 2.94; p = 0.09
MMSE	22.53 ± 3.25	28.93 ± 2.20	F = 503.67; p <0.0001

Because low-rank components correspond to group common morphological structures, we apply subspace decomposition algorithm to extract the low-rank component of Aβ positive AD and Aβ negative CU observation matrices, respectively. Each observation matrix is formed by stacking individuals of each group. Considering the bias and variance of the ROI generation model, we randomly divide the subjects of each group into 10 folds and choose 9 folds from each group as the training sets respectively. This process is repeated 10 times. The purpose of this process is to validate if the generated ROIs each time is consistent. The total performance indicators for the left and the right hippocampus based on our algorithm are shown in Table 2. Here we set the parameter $\gamma$ as 0.0082. From the performance indicators in Table 2, the parameters setting ${\rho }_0,\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left(L\right)$ and $‖L+S-A‖/‖A‖$ of each group (Aβ positive AD group or Aβ negative CU group) is similar whether it is on the left or the right hippocampus. Here the rank of $L$ is obtained by counting the number of the eigenvalues of $L$ whose values are greater than three times the $\delta$. Here $\delta =0.15$.

Table 2.

The performance indicators for the left and right hippocampus based on subspace decomposition algorithm (with data from the ADNI cohort).

		${\rho }_0$	rank(L)	$\parallel \mathit{S}{\parallel }_{\mathbf{\Omega },1}$	$\parallel \mathit{L}+\mathit{S}-\mathit{A}\parallel /\parallel \mathit{A}\parallel$
Left Hippocampus	CU	0.09	11~12	142~150	0.056
	AD	0.13	9~11	209~225	0.057
Right Hippocampus	CU	0.08	11~12	146~153	0.057
	AD	0.12	9~11	218~226	0.058

After extracting the $L$ and $S$ from the observation matrix $A$, we can define the ROIs by using permutation t-test between the $L$ of Aβ positive AD group and the $L$ of Aβ negative CU group. As the training samples are taken randomly each time (the changing rate is about 10%), each ROI obtained will be inconsistent if the extracted $L$ components are not robust.

In order to verify whether the ROIs generated by the $L$ components are robust, we imitated the process of generating the ROIs based on the $L$ components, and formed the ROIs based on the raw RD features of Aβ positive AD group and Aβ negative CU group, then verify whether the ROIs generated by each type of data ($L$ components or raw RDs) are consistent when the training data is different at each time.

In addition, to improve the computational efficiency, we run Monte Carlo to generate two random groups for each registered sample point on the surface, which takes relatively small random permutation processes of the possible replicates. Here the number of random permutation processes is set as 5,000. Moreover, the generated ROIs are the vertices whose permutation t-test p-values (uncorrected for multiple comparisons) representing the group differences are smaller than 0.00001. The generated ROIs of the left hippocampus (LH) and right hippocampus (RH) based on the low-rank components and the RD data are shown in Figure 2. Figure 2 (a) and (b) are the statistical ROI results for LH. (c) and (d) are the statistical ROI results for RH. Furthermore, (a) and (c) are the statistical ROI results based on the low-rank components. (b) and (d) are the statistical ROI results based on the raw RD data. Non-blue colors show the number of times a vertex is selected. The red color denotes that the vertex is selected 10 times after 10 ROI generation processes. For LH, the total number of ROI vertices based on the low-rank components is 2162, and the vertices having been selected 10 times account for 75.28% of the total number of ROI vertices, while they are 2542 and 52.41% based on the raw RD data. For the right hippocampus, they are 2249 and 74.23% based on the low-rank component, 2518 and 54.23% based on raw RD data. From the results, we can see that the selected ROIs based on the low-rank component are more robust than the ROIs based on the raw RD data. It suggests that the low-rank component can represent the essential common structure from the subjects without the interference of the unique individual structure and the noise.

Figure 2.

The extracted ROIs for the group differences between Aβ positive AD group and Aβ negative CU group based on the low-rank data and the RD data. (a) and (b) are the statistical ROI results for LH. (c) and (d) are the statistical ROI results for RH. And (a) and (c) are the statistical ROIs based on the low-rank components. (b) and (d) are the statistical ROIs based on the RD data. All the statistic ROI results are obtained by the permutation t-test (p-value<0.00001, uncorrected for multiple comparisons). Non-blue colors show the number of times a vertex is selected.

In our previous work [36], we used the subspace decomposition algorithm (ASALM) to extract low-rank components and sparse components based on stable principal component pursuit which are combined with a low-rank matrix factorization mechanism [57]. This paper used the subspace decomposition algorithm (NSA) to extract low-rank components and sparse components based on a stable principal component pursuit combined with a partial SVD mechanism. Both mechanisms are to speed up the SPCP optimization for large-scale matrices by avoiding the full SVD computation. Moreover, the low-rank components extracted by the two methods have the same rank and high similarity. However, the NSA algorithm has a faster convergence rate, i.e., fewer iteration steps, than the ASALM algorithm. A brief convergence speed comparison of the two algorithms is presented in Figure 3. All the experiments are performed with an Intel Core i7 personal computer, with 3.40 GHz CPU, 16 GB RAM and MATLAB 2016 installed in the Win 7 operating system. With the same termination condition, the evolution of rank ($L_k$) for extracting low-rank components of left hippocampus of AD group shows that the results showed that the extracted low-rank components of AD-LH and CU-LH groups based on the two methods have the same ranks, that is, the rank equals to 10 for AD-LH and 13 for CU-LH. Using Corr2 in Matlab, we obtained that the correlation between the two low-rank components based on the two methods is 0.992 for AD-LH, and 0.994 for CU-LH. To extract the final low-rank components, the iteration number of the NSA method accounts for about 38% of the ASALM method. The NSA with a partial SVD mechanism is more efficient than the ASALM with a low-rank matrix factorization mechanism.

Figure 3.

Evolution of $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}\left({\mathrm{L}}_{\mathrm{k}}\right)$ for ASALM and NSA algorithms

3.2. Longitudinal Study for different APOE-e4 genotype groups

There are 140 cognitively unimpaired individuals with 61 NCs, 49 HTs, and 30 HMs from the Arizona APOE cohort. All subjects underwent two tests, including the baseline test and a 24-months test. Through the comparisons between the demographic and clinical data of the different groups at these two test time points, the statistical results in Table 3 indicate that there are no significant group differences in the factors of gender, education degree, age, and MMSE between the three groups.

Table 3.

Demographic information of the different longitudinal genotype groups, from the AZ APOE cohort. Means are followed by standard deviations for education, age, and MMSE measurements.

		Gender (M/F)	Education	Age	MMSE
HM baseline		8/22	16.21± 2.18	65.01±5.38	29.75±0.58
HM m24				67.37±5.43	29.54±0.81
HT baseline		16/33	15.72± 2.41	68.01±6.69	29.58±0.79
HT m24				70.24±6.64	29.58±1.03
NC baseline		18/43	15.92± 2.24	66.64±7.01	29.71±0.59
NC m24				68.85±6.94	29.68±0.66
Inferential Statistics	baseline	χ2 = 0.33; p = 0.85	F = 0.38; p = 0.68	F = 1.21; p = 0.30	F = 0.59; p = 0.56
	m24			F = 1.14; p = 0.32	F = 1.28; p = 0.28

Based on the predefined ROIs from the low-rank components of Aβ positive AD group and Aβ negative CU group in Section 3.1, we generated the UMIs of the cognitively unimpaired individuals with different APOE-e4 genotypes (HM, HT, and NC) at two test points (baseline test and 24-months test) and studied the statistical differentiation ability of the UMIs for the different longitudinal groups. Specifically, since the predefined ROIs represent the significant group difference areas induced by AD, the generated UMIs indicate the degree of similarity to the AD-specific abnormal surface pattern for a given individual subject with different APOE-e4 genotypes before the onset stage of AD. Combined the AD-specific abnormal surface patterns and the individual atrophy degrees on the selected ROIs, the UMIs of the cognitively unimpaired individuals with different APOE-e4 genotypes can be computed through Eq. (4). In order to verify whether the generated UMIs have strong statistical power for distinguishing these three longitudinal groups, we introduce the UMIs based on raw RD measures (UMI-RD) and the volume data to compare the discrimination power with the UMIs based on the subspace decomposition. The UMI-RDs of the different APOE-e4 subjects are computed according to Eq. (4) based on the obtained ROIs shown in Figure 2 (b) and (d). The volumetric MRI measurements of the hippocampus are generally accepted as the best-established biomarkers of clinical AD progression [58]. Moreover, the hippocampal volume is computed on our smoothed surfaces after linearly registered to the MNI imaging space [59].

We computed the required minimum sample sizes (MSS) of three longitudinal groups from baseline to 24-months follow-up based on UMIs, UMI-RDs, and volume measures, respectively. The comparison results are shown in Table 4. Paired Cohen’s d measure computes the effect sizes of the three longitudinal groups. For the longitudinal NC, HT, and HM groups, the MSSs and effect sizes for UMIs mean differences are 533 and 0.48, 393 and 0.53, 296 and 0.61. For UMI-RDs, the MSSs and effect sizes of the longitudinal NC, HT, and HM groups are 1061 and 0.21, 823 and 0.31, 552 and 0.46. For hippocampal volume measures, the MSSs and effect sizes of the longitudinal NC, HT, and HM groups are 828 and 0.34, 661 and 0.42, 430 and 0.52. This indicates that UMI, as a univariate morphometry biomarker, has more substantial statistical discrimination power to distinguish the longitudinal groups with different APOE-e4 genotypes than both the UMI-RDs and the hippocampal volume measures.

Table 4.

The longitudinal study from baseline to 24-months follow-up for different APOE-e4 genotype groups (from the AZ APOE cohort) based on different univariate biomarkers.

Biomarker	genotype	MSS	Effect Size (ES)
UMI	NC	533	0.48 0.53 0.61
	HT	393
	HM	296
UMI-RD	NC	1061	0.21
	HT	823	0.31
	HM	552	0.46
Volume	NC	828	0.34
	HT	661	0.42
	HM	430	0.52

3.3. Dose Effects of APOE-e4 Alleles on the Longitudinal Hippocampal Morphological Changes

To further validate the APOE-e4 allele dose effects on the longitudinal hippocampal atrophies, the ANOVA method is applied on the UMI, UMI-RD, and volume measurement changes between NC and HT, NC and HM, and HT and HM, respectively. The biomarker changes are described as the change rates ($R_f$) of the biomarker.

$$R_f=\frac{f_{second}-f_{first}}{f_{first}}$$ (6)

where $f_{first}$ and $f_{second}$ represent the values of the biomarkers at the baseline test and a 24-months test. We computed the $R_f$ values of UMIs, UMI-RDs and volume measurements of three longitudinal groups with different APOE-e4 genotypes, respectively. The statistical results (shown in Table 5) show that the mean $R_f$ values of the UMIs and UMI-RDs have an upward trend while the mean $R_f$ values of the volume measures have a downward trend from NC to HT to HM genotype. This means that different APOE-e4 allele load affects the shrinkage rate of the hippocampus, i.e., HM genotype will cause the largest atrophy rate, followed by HT, and the smallest is NC.

Table 5.

Mean R_f values of different biomarkers comparisons between different longitudinal genotype groups (from the AZ APOE cohort).

Biomarker	Group Comparisons	Average Rf (%) ± Std (%)	Inferential Statistical
UMI	NC vs. HM	0.30±0.87 vs. 0.98±1.04	F=11.99; p=0.0008
	NC vs. HT	0.30±0.87 vs. 0.64±0.64	F=5.3; p=0.023
	HT vs. HM	0.64±0.64 vs. 0.98±1.04	F=4.05; p = 0.048
UMI-RD	NC vs. HM	0.03±0.08 vs. 0.09±0.20	F=4.14; p=0.045
	NC vs. HT	0.03±0.08 vs. 0.03±0.03	F=2.18; p=0.14
	HT vs. HM	0.03±0.03 vs. 0.09±0.20	F=1.84; p=0.18
volume	NC vs. HM	−0.01±0.04 vs. −0.05±0.08	F=9.89; p=0.0023
	NC vs. HT	−0.01±0.04 vs. −0.03±0.06	F=4.61; p=0.041
	HT vs. HM	−0.03±0.06 vs. −0.05±0.08	F=2.21; p = 0.14

The statistical results for mean $R_f$ values comparisons between different longitudinal genotype groups are computed by the ANOVA method. For the mean $R_f$ values comparisons of NC vs. HM, NC vs. HT, and HT vs. HM, the p-values for UMIs are 0.0008, 0.023, and 0.048, respectively. For UMI-RDs, the p-values are 0.045, 0.14 and 0.18, respectively. For volume measurements, the p-values are 0.0023, 0.041 and 0.14, respectively. The results indicate that the UMIs may have a stronger discrimination ability to distinguish the mean $R_f$ value differences between the three longitudinal groups than both the UMI-RDs and the hippocampal volume measures. With the help of UMI biomarker, i.e., the sensitivity for detecting hippocampal morphological change, we detected the significant group differences for NC vs. HM, NC vs. HM, and HT vs. HM based on the mean $R_f$ values. This indicates that APOE-e4, as a major genetic risk factor for AD, affects the hippocampal morphological changes of elderly cognitively unimpaired individuals. In addition, the mean $R_f$ values of HM are larger than those of both HT and NC, i.e., hippocampal atrophy rate of HM > hippocampal atrophy rate of HT > hippocampal atrophy rate of NC, which indicates a clear dose effect of APOE-e4 alleles on longitudinal hippocampal atrophy rates. This finding is consistent with prior studies [13, 14, 19, 60–65]. However, there are no significant group differences about mean $R_f$ values of HM vs. HT based on volume measurements. There are no significant group differences about mean $R_f$ values of HM vs. HT and HT vs. NC based on UMI-RDs. This demonstrates that the UMIs based on subspace decomposition may outperform the volume measures and UMI-RDs in distinguishing cognitively unimpaired APOE-e4 carriers from NC.

3.4. Multiple Regression Analysis

The effect of APOE-e4 allele load on hippocampal morphology in the healthy population is complicated. For example, the APOE-e4 allele interacts with specific age-associated pathogenic factors [9]. Some significant interactions with age have been reported in the literature [62, 66] that the effects of APOE-e4 allele load on hippocampal morphology of individuals in different aged groups are different. Therefore, the question is whether the calculated UMIs can still reflect the different morphological atrophies associated with different APOE-e4 allele loads, considering the individual age and gender factors. Specifically, it is beneficial to see if $R_f$ values of UMIs vary systematically as the copies of the risk allele change. We generate the multiple regression model to test the relationship between $R_f$ values of UMIs and the influence factors including different APOE-e4 genetic variances (HM, HT and NC), age, and gender, i.e., the $R_f$ values of UMIs are taken as dependent variables and the influence factors are taken as independent variables. In this multiple regression model, the genetic variances (i.e., HM, HT and NC) are transformed into three independent dummy regressors. The estimated regression coefficients and p-values of all the regression factors are shown in Table 6.

Table 6.

The estimated multiple regression coefficients and p-values of the multiple regression model (with data from the AZ APOE cohort).

Regression Factor	Regression Coefficient	p-value
(Intercept)	−0.084	0.18
Genetic-HT	0.040	0.045
Genetic-HM	0.054	0.0046
Age	0.0023	0.020
Gender-male	0.012	0.45

Multiple R-squared: 0.23

As we expected, changes in genetic factors are closely related to $R_f$ values of UMIs. As shown in Table 6, taking the APOE-e4 non-carriers as a reference genetic group, the average $R_f$ values of UMIs of the HT genetic group increase by 0.040, and the $R_f$ values of UMIs of the HM group increase by 0.054. And the regression coefficients of these two regressors are statistically significant. Moreover, the regression coefficient of the HM group is larger than the one of HT group based on the multiple regression model, which indicates that hippocampal atrophy rate of HM > hippocampal atrophy rate of HT > hippocampal atrophy rate of NC under the influence of age and gender. The results also show that the factor of age has a significant influence on the $R_f$ values of UMIs, i.e., the regression coefficient of the age factor is 0.0023, and the p-value of the age factor is 0.020. This indicates that the factor of age also has a significant impact on UMIs, i.e., UMIs tend to increase with aging. In addition, the regression coefficient of male subjects equals 0.012 when female subjects are used as the reference variables, i.e., the average $R_f$ values of UMIs of male subjects with the same age and genetic group is 0.0035 greater than female subjects. The results also indicate that there is no significant correlation between UMIs and gender.

To further verify whether the effects of APOE-e4 allele load on hippocampal morphology of individuals in different aged groups are different, we add a binary indicator variable (Age-group) to each individual, which is transformed from the age variable. When the individual’s age is over 65, the binary indicator is set to high (Age-group-H). Otherwise, it is set to low (Age-group-L). Moreover, we added an interaction term between genetic and Age-group variables to the regression model to test whether the interaction will significantly impact UMIs. The estimated regression coefficients and p-values of all the regression factors are shown in Table 7.

Table 7.

The estimated multiple regression coefficients and p-values of the modified multiple regression model (with data from the AZ APOE cohort).

Regression Factor	Regression Coefficient	p-value
Genetic-HT	0.0012	0.96
Genetic-HM	0.0082	0.68
Age	0.0017	0.31
Gender-male	0.013	0.37
Genetic-HT:Age-group-H	0.062	5.6e-03
Genetic-HM:Age-group-H	0.17	1.11e-05

Multiple R-squared: 0.28

The results show that the multiple R-square values of the new multiple regression model increased from 0.23 to 0.28 compared with the multiple regression model in Table 6, indicating that the new model can better explain the $R_f$ values of UMIs. The regression coefficients of the new multiple regression model show that the interaction between APOE-e4 genetic factors and age factor has a significant impact on the $R_f$ values of UMIs. For example, in the HT group, the average $R_f$ values of UMIs of all the individuals increase 0.0012 compared with the NC group, and APOE e4 HT genotype will produce an additional $R_f$ value of UMIs with 0.062 for individuals of high age (greater than 65 years old). Similarly, in the HM group, the average $R_f$ value of UMIs increase 0.0082 compared with the NC group, and the APOE e4 HM genotype will produce an additional $R_f$ value of UMIs with 0.17 for individuals of high age (greater than 65 years old). These results may indicate that APOE e4 will accelerate the change of UMIs in the high age population, especially those among HM group, thereby causing the deterioration of the individual’s cognitive status, consistent with observations in prior research [67, 68].

3.5. Influence of Age on the Effect of APOE-e4 Allele Load

As described in Section 3.4, the factor of age has a significant influence on the effects of APOE-e4 allele load on hippocampal morphology of individuals. It is natural to verify what happens to the mean $R_f$ values of different biomarkers comparisons between different longitudinal genotype groups while considering the variable age as a continuous covariate. We further conducted analysis of covariance (ANOCOVA) analysis of experimental data in Section 3.3 and the results were shown in Table 8. Compared with Table 5 of Section 3.3, most of the results are similar except for the detection of more significant group differences between HT and HM groups based on UMIs. For the mean $R_f$ values comparisons of NC vs. HM, NC vs. HT, and HT vs. HM, the p-values for UMIs are 0.0006, 0.029, and 0.025, respectively. For UMI-RDs, the p-values are 0.038, 0.10 and 0.15, respectively. For volume measurements, the p-values are 0.0013, 0.035 and 0.099, respectively. In short, after excluding the influence of individual age differences, we can still find the existence of a clear dose effects of APOE-e4 alleles on longitudinal hippocampal atrophy rates.

Table 8.

Mean R_f values of different biomarkers comparisons between different longitudinal genotype groups based on ANOCOVA method (from the AZ APOE cohort).

Biomarker	Group Comparisons	Inferential Statistical
UMI	NC vs. HM	F=12.82; p=0.0006
	NC vs. HT	F=4.87; p=0.029
	HT vs. HM	F=5.23; p = 0.025
UMI-RD	NC vs. HM	F=4.43; p=0.038
	NC vs. HT	F=2.75; p=0.10
	HT vs. HM	F=2.19; p=0.15
volume	NC vs. HM	F=11.01; p=0.0013
	NC vs. HT	F=4.55; p=0.035
	HT vs. HM	F=2.79; p = 0.099

To answer whether there is a certain age group that whose $R_f$ value is most informative, we divided the individual ages at baseline test into three age groups, i.e., 50∼60, 60∼70 and 70∼80 age groups, and then compared the mean $R_f$ values of UMI biomarker between different longitudinal genotype groups. The comparison results are shown in Table 9. It illustrated that there are no significant group differences about mean $R_f$ values of UMI biomarker for HM vs. NC, HT vs. NC and HM vs. HT within 50∼60 and 60∼70 age intervals. The mean $R_f$ values of UMI comparisons of NC vs. HM, NC vs. HT, and HT vs. HM, the p-values are 0.17, 0.72, and 0.35 within 50∼60 age interval, respectively. Within 60∼70 age interval, the p-values are 0.11, 0.18 and 0.24, respectively. Within 70∼80 age interval, we find more significant group differences between the three longitudinal genotype groups, i.e., the p-values are 0.0004, 0.018 and 0.013, respectively. This may indicate that the dose effect of APOE-e4 alleles on longitudinal hippocampal atrophy rates become more pronounced in the elder age population.

Table 9.

Mean R_f values of UMI comparisons between different longitudinal genotype groups with three age intervals (from the AZ APOE cohort).

Age Group	Group Comparisons	Inferential Statistical
50~60	NC vs. HM	F=1.98; p=0.17
	NC vs. HT	F=0.13; p=0.72
	HT vs. HM	F=0.87; p=0.35
60~70	NC vs. HM	F=2.78; p=0.11
	NC vs. HT	F=1.85; p=0.18
	HT vs. HM	F=1.46; p=0.24
70~80	NC vs. HM	F=20.83; p=0.0004
	NC vs. HT	F=6.61; p=0.018
	HT vs. HM	F=7.57; p=0.013

3.6. Significance of Gene × Age on the $R_f$ Values of UMIs

As described in Section 3.4 and Section 3.5, the factor of age has a significant influence on the $R_f$ values of UMIs, i.e., UMIs tend to increase with aging. We want to study whether the relationship between the $R_f$ values of UMIs and the APOE-e4 allele loads depends on the level of individual age. We examined the interactive effect of APOE-e4 allele load by age on the $R_f$ values of UMIs in our Arizona APOE cohort with regard to APOE-e4 HM, HT and NC by using general linear regression model including the term of age, the term of gene (APOE-e4 HM, HT and NC) and the interaction term of gene by age. The estimated regression coefficients (the corresponding p-values) of age, gene and age×gene are 0.0054 (0.0098), 0.21 (0.028) and −0.0035 (0.026). The results showed that the interaction term of APOE-e4 allele load by age is statistically significant. It indicated that the relationship between the $R_f$ values of UMIs and gene differs depending on the ages of individuals.

3.7. Group Difference Comparisons at Baseline Tests

To verify whether there is any difference among NC, HT and HM on the baseline of UMI, UMI-RD and volume measurements, we used the UMIs, UMI-RDs, and volume measures at the baseline test from the AZ APOE cohort, i.e., 140 cognitively unimpaired individuals including 61 NCs, 49 HTs, and 30 HMs, and applied significant test of group mean differences between NC and HT, NC and HM, and HT and HM based on the ANOVA method. The results shown in Table 10 indicated that there are no significant group mean differences between NC and HT, NC and HM, and HT and HM by using these three biomarkers at baseline test. These cross-sectional findings are consistent with observations in prior research [28–30]. The results completed our study and demonstrated the superior sensitivity gained from our novel framework.

Table 10.

Group difference comparisons of different biomarkers at baseline tests (from the AZ APOE cohort).

Biomarker	Group Comparisons	Average Baseline ± Std	Inferential Statistical
UMI	NC vs. HM	11.11±3.03 vs. 12.45±3.21	F=3.85; p=0.053
	NC vs. HT	11.11±3.03 vs. 11.28±2.98	F=0.09; p=0.77
	HT vs. HM	11.28±2.98 vs. 12.45±3.21	F=2.76; p=0.10
UMI-RD	NC vs. HM	10.83±3.22 vs. 11.42±5.12	F=0.45; p=0.50
	NC vs. HT	10.83±3.22 vs. 11.04±4.27	F=0.08; p=0.80
	HT vs. HM	11.04±4.27 vs. 11.42±5.12	F=0.13; p=0.72
Volume ×1000	NC vs. HM	9.924±0.89 vs. 9.933±0.91	F=0; p=0.96
	NC vs. HT	9.924±0.89 vs. 9.931±1.04	F=0; p=0.97
	HT vs. HM	9.931±1.04 vs. 9.933±0.91	F=0; p=0.99

3.8. Correlation Analysis with Volume and Cognitive Outcomes

Although our work is focused on APOE genetic effect analyses, it is natural to verify whether the UMIs are correlated with other imaging and clinical rating scores, such as hippocampal volume, MMSE scores [54], Auditory Verbal Learning Test Long-Term Memory (AVLT-LTM) scores [55]. Using the Pearson parametric test [69] we explore the correlation analysis between the $R_f$ values of UMIs between the $R_f$ values of hippocampal volume, MMSE, and Auditory Verbal Learning Test Long-Term Memory (AVLT-LTM) measures, to understand which variables are correlated with UMIs in the AZ APOE cohort. The correlation results, i.e., correlation coefficients (CC), 95% confidential intervals of CC, and correlation significance of t-test for CC between the $R_f$ values of UMIs and the $R_f$ values of MMSE, volume, AVLT-LTM measures, are shown in Table 11.

Table 11.

The correlation analysis between the R_f values of UMIs and the R_f values of hippocampal volume, MMSE, and AVLT-LTM measures (with data from the AZ APOE cohort).

	CC	95% CI of CC	p-value of CC
UMI vs. Hippocampal Volume	−0.80	−0.88~−0.68	3.43e-12
UMI vs. MMSE	−0.13	−0.39~0.16	0.38
UMI vs. AVLT-LTM	−0.11	−0.38~0.18	0.46

Results indicate that the $R_f$ values of UMIs have strong negative correlations with the $R_f$ values of hippocampal volume measures, i.e., CC is −0.80 (p-value=3.43e-12). This is likely due to the UMIs showing an increase with a corresponding hippocampal volume decrease, according to the distributions of UMIs vs. volume measurements. In other words, under the influence of the APOE-e4 risk allele, the hippocampal volume tends to shrink, and the UMIs obtained by our model has an increasing trend. We have known that the volumetric MRI measurements of the hippocampus are generally accepted as the best-established biomarkers of clinical AD progression. The relationship between UMIs and volume measurements indicates that UMIs may depict the morphological changes induced by AD.

In addition, there is a weak negative correlation between the $R_f$ values of UMIs and the $R_f$ values of MMSE scores, i.e., CC is −0.13 (p-value=0.38). With increasing UMIs, the MMSE scores tend to decrease slightly. Similarly, we also did not find a close association between UMIs and AVLT-LTM measures, i.e., CC is −0.11 (p-value=0.46). A reasonable reason may be that the morphological changes caused by AD occur before the cognitive decline in the non-destructive cognitive stage, as previously reported in the literature [70, 71].

4. Discussion

Here we propose a novel univariate morphometry index which is sensitive to AD-induced hippocampal morphometry changes and enjoys improved computational efficiency than our prior work [36]. The newly proposed UMI demonstrated exceptional statistical ability to characterize APOE-e4 gene dose effect on longitudinal hippocampal atrophies of elderly cognitively unimpaired individuals with three different experimental designs. UMIs illustrate AD-related morphological changes using a summary index free from the Type I error associated with multiple comparisons. The more prominent the individual has the atrophic morphological characteristics caused by AD in the defined ROI regions, the larger the corresponding UMI value is. Since APOE is a significant genetic risk factor for developing AD, its discovery has made it possible to study large numbers of genetically at-risk individuals before the onset of symptomatic memory impairment and has led to the concept of the preclinical stage of AD [72]. A UMI which is sensitive to APOE-e4 dose effects will help evaluate AD burden, progression, and response to interventions. Therefore, our work provides a cost-effective and accurate means for prognosis and potentially facilitates enrollment in clinical trials with preclinical AD patients.

In the current work, our main goal is to generate reliable and robust structural MRI ROIs by the analysis of the statistical morphometry difference between the pathology-confirmed AD patients and the pathology-confirmed CU subjects. To address the inhomogeneity of AD pathology, we used the specific AD subjects who had the clinical symptom-based AD diagnosis together with the accumulation of Aβ in brains. As we know, one of the hallmarks of AD is the accumulation of Aβ in human brains and a positive Aβ reading is now accepted as ‘dementia due to AD’ together with the presence of clinical symptoms. In our previous work [36], the generated ROIs based on the group common structures of Aβ positive AD and Aβ negative CU groups could reflect intrinsic morphological changes induced by AD. And we found that Aβ-induced biomarkers are strong indicator for brain structure atrophy. Therefore, we continued this approach that extracts ROIs from the morphometric difference analysis between Aβ positive patients and Aβ negative subjects. On the other hand, with the same framework, whether features extracted from the morphometric analysis between mild AD patients and CU subjects are more representative of the key initial changes deserves more investigation. However, due to AD’s high heterogeneousness, we hypothesize that it is suboptimal to develop a univariate biomarker from the morphometric analysis of mild AD patients.

Even though the proposed UMI based on subspace decomposition could be used to describe the hippocampal morphological changes induced by the APOE-e4 risk allele for AD, there are several limitations. First, the relatively small number of subjects are included as the research objects, e.g., the ROIs are extracted from 120 Aβ positive AD patients and 257 Aβ negative CU subjects), which is not enough to characterize the induced general morphological changes induced by AD fully. For example, there is a certain degree of difference for extracting the common group structure under the condition of the limited number of training samples at each time. Therefore, the generated ROIs may be slightly different each time because of different extracted common group structures. It is conceivable that as the sample size gradually increases, the difference in the ROIs caused by the different training samples will be further reduced. Therefore, considering the limitation of the amount of data and the generalization performance of ROIs in applications, we assign the same weight to each point in the ROIs when characterizing the morphological changing regions induced by AD. Secondly, we just chose the RD as the hippocampal morphological feature because RD has been applied in several subcortical studies and served as an ideal description of the hippocampal structural changes induced by AD. However, multivariate tensor-based morphometry (mTBM) [16] is sensitive to deformations such as rotation, dilation, and shear along the surface tangent direction. Therefore the mTBM can effectively capture hippocampal structural alterations (e.g., atrophy and enlargement) in tensor fields. The ability to describe the morphological features of hippocampi will be enhanced if we combine the mTBM measure and the RD measure. In addition to morphological characteristics, some core AD (i.e., Aβ and tau) biomarkers are needed to construct the biological mechanisms underlying our findings in hippocampal morphology measurements.

5. Conclusion and Future Work

A univariate morphometry biomarker generation method is proposed based on subspace decomposition to effectively depict the hippocampal morphological changes induced by three levels of genetic risk for AD. The empirical results demonstrated the potential that the UMIs may capture the APOE-e4 risk allele-induced brain morphometry abnormalities and reveal the dose effects of APOE-e4 on the hippocampal morphology in cognitively normal individuals. In our future work, we will devise a tensor field data-based univariate morphometry biomarker which may be applied to analyze diffusion tensor imaging (DTI) data for the differential diagnosis, early detection and tracking, and the prediction of clinical decline before the onset stage of AD.