Longitudinal Relationship Between Brain Atrophy Patterns,Cognitive Decline,and Cerebrospinal Fluid Biomarkers in Alzheimer’s Disease Explored by Orthonormal Projective Non-Negative Matrix Factorization

Data sets

We downloaded 3D T1-weighted MRIs from 44 young controls from the Autism Brain Imaging Data Exchange (ABIDE, available at: http://fcon_1000.projects.nitrc.org/indi/abide/). These images were used to characterize brain structures in young healthy individuals and were used to develop reference templates (gender-specific or gender-nonspecific) for our atrophy analysis.

The ADNI Data were downloaded from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database (http://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 MRI, positron emission tomography, other biological markers, and clinical and neuropsychological assessment can be combined to measure the progression of mild cognitive impairment (MCI) and early AD. For up-to-date information, see http://www.adni-info.org. Our ADNI3 data consists of 3T T1-weighted sMRI for 1025 subjects, along with their levels of AD biomarkers. The 1,025 subjects comprise 83 with AD, 405 with MCI, and 537 controls (CN). We applied OPNMF to 83 AD subjects’ baseline sMRI scans and used the resulting atrophy patterns, relative to the normal template, to calculate atrophy coefficients for 1,884 longitudinal image datasets from all 1,025 subjects. Subjects were imaged between 1 and 6 times. The longitudinal analysis was limited to subjects with more than 1 visit. Among AD subjects, 43.4% had more than 1 visit with a mean (SD) follow-up time of 1.73 (0.09) years. Among MCI subjects, 53.6% had more than 1 visit and their mean (SD) follow-up time was 2.36 (0.07) years. Finally, 51.6% of controls had more than 1 visit and the mean (SD) follow-up time was 2.51 (0.06) years.

Finally, we analyzed a validation dataset consisting of 506 subjects’ baseline 3T T1-weighted sMRI from ADNI2, excluding subjects who were also in the ADNI3 dataset.

Summary information of the data set is presented in Table 1. Data for the ADNI3 subjects with two or more visits which were used for longitudinal analysis are summarized in Table 2. Summary information for the ADNI2 dataset is presented in Supplementary Tables 4 and 5.

Measurements of atrophy severity

MRI datasets from both ABIDE and ADNI were first pre-processed with the SPM12 VBM toolbox (Voxel-based morphometry, available at: http://www.fil.ion.ucl.ac.uk/spm/, London, UK) to derive voxel-wise gray matter volumes for each subject of the dataset using default parameters. T1-weighted structural brain images were segmented into gray and white matter. The gray matter images were normalized to MNI152 space by the high-dimensional DARTEL transformation with size 121 * 145 * 121 [27]. To construct gray matter loss images for each subject, we created two gender-specific global “normal” templates obtained by averaging the gray matter images of 22 young females and 22 young males from ABIDE. Then we subtracted each raw image data set from the gender-specific young template to generate “atrophy maps”. A threshold of 0.1 was applied to eliminate negative values for the gray matter loss and reduce the influence of voxels with partial volume effect. In our previous paper [25], instead of using gender-specific templates, we employed the gender-nonspecific young template by averaging the gray matter images of all young controls from ABIDE. Through thorough investigations by comparing the outcomes of these two approaches (for details, refer to Supplementary Material Section 5), we decided that a more effective approach to address potential gender difference is by utilizing gender-specific templates.

OPNMF [28] produces a low-rank approximation factorization of a non-negative matrix into a product of two non-negative matrices, the first of which has orthonormal columns. Our data are represented by a tall matrix X that is organized by arraying each subject’s vectorized image into a single column. Starting from the data matrix X of size M × N representing gray matter loss images, each with M = 2122945 voxels, from N = 1025 subjects, OPNMF approximates X as a product of two non-negative matrices W and H, where W is of size M × K and H is K⪡N where K⪡M is the number of components. The columns of W are referred to as components and H contains the per-component coefficients for each subject. We fixed K = 6 on the basis of our previous study [25], which showed that there is a sharp decrease in the magnitude of the reconstruction error gradient when moving from K = 5 to K = 6 and the components derived from K = 6 are more biologically interpretable on the same training dataset. Values of K > 6 yielded less easily interpretable components. We observed a similar error curve pattern in the present dataset and therefore kept K = 6.

OPNMF alone typically produces components that are approximately orthogonal but multiple components may have positive scores for a single voxel, leading to overlaps between components. OPNMF with refinement [25, 26], which produces at most one nonzero component in any given voxel, was applied to 83 AD baseline sMRI datasets to compute W^*, with each of the K columns representing one AD-associated atrophy component and normalized to sum to 1. To apply these component scores to additional image volumes, the predictor matrix W^*, of size M × K, is fixed and non-negative least squares (NNLS) applied with outcomes Y being the M × 1 vectorized loss image of a subject. The resulting 1 × K coefficient vector β is the estimate of atrophy severity for each component and each image. Due to the normalization of W^*, the entries in the vector β are proportional to the amount of total volume loss attributable to the corresponding component.

In examining the relationship between the estimated coefficients from baseline AD subjects, both general and partial correlations were assessed. For partial correlations [29], Pearson correlation was applied after adjusting for age, sex, and APOE4. The Gaussian Copula Graphical Model (GCGM) was further implemented for more robust results, with adjustments for the same variables [30]. 95% Confidence intervals for GCGM partial correlations were established using 1000 bootstrap samplings and 10,000 GCGM iterations.

The detailed procedure for obtaining gray matter loss images, additional information on the use of OPNMF to extract distinct atrophy regions, how NNLS is used to get the image specific coefficients and why GCGM was chosen are described in our previous paper [25]. There are two modifications of our previous methodology pipeline: 1) the use of gender-specific templates instead of one combined template, 2) the normalization of the atrophy components coefficients. All these procedures were performed using MATLAB (The MathWorks, Inc. (2023), version: 9.11.0 (R2021b) on a computer with 32GB of RAM and 2.8 GHz processor.

Demographic and clinical data

Demographic and clinical information including gender, age, and educational level, APOE4 genotype, cognitive assessment scores, diagnosis information, and CSF biomarkers for all analyzed visits were downloaded from the ADNI website. The cognitive assessments we included in this analysis are Mini-Mental State Examination (MMSE), Montreal Cognitive Assessment (MoCA), and Clinical Dementia Rating Sum of Boxes (CDR-SB). CSF biomarkers consisted of CSF total tau (TAU), phospho-tau (PTAU), and Aβ₄₂ (referred to as ABETA in the following).

Statistical analysis

All statistical analyses were performed with R version 4.2.0. A two-tailed p-value<0.05 was considered statistically significant for single tests. Associations between the baseline atrophy severity and baseline variables were assessed using multiple linear regression models. Longitudinal outcomes were modeled by linear mixed-effects models with random intercepts and slopes for the time variable, and subject-specific predicted slopes of atrophy severity and cognitive scores were used to study correlations with diagnostic groups and biomarkers. Kendall’s tau rank correlation, which is a non-parametric measure of correlation based on ranks, was employed to determine associations between the slopes for atrophy severity and slopes for cognitive scores. In the multiple linear regression models and linear mixed-effects models, adjustments for the potential confounders of age, sex, educational level, and APOE4 genotypes were made. For the linear mixed-effects models, we only included subjects with more than 1 visit. To address potential issues of multiple testing, we applied false discovery rate (FDR) analysis based on Benjamini-Hochberg’s method to obtain adjusted p-values where appropriate [31].

Validation analysis

A validation analysis was performed by computing the β coefficients of preprocessed ADNI2 images using the six atrophy components we obtained from the ADNI3 dataset.

Baseline characteristics

Subject characteristics in our ADNI3 sample are shown in Table 3. The CSF biomarkers (ABETA, TAU, and PTAU) are not available for some of the ADNI3 subjects (see the “Missing” rows in Table 3).

Atrophy components

The six components are shown in Fig. 1, arranged in a descending order based on their average contribution to total gray matter volume loss. Briefly, component 1 represents volume loss primarily in bilateral temporal, medial frontal, lateral parietal lobes and cerebellum, component 2 covers primarily the bilateral frontal and parietal lobes, component 3 represents both bilateral medial parietal and occipital, component 4 is the bilateral anterior frontal lobes, component 5 is primarily bilateral anterior medial temporal lobes (hippocampus) and component 6 represents primarily the bilateral cerebellum. The striking hemispheric symmetry of the components emerges from the data and reflects the hemispheric symmetry of the common form of late onset AD.

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Fig. 1

Six atrophy components derived from OPNMF. Note that the orthogonality and nonnegativity of the components mean that only one component can be nonzero in a given voxel. The images in the figure are heatmaps of each column in W^*; a high value (yellow/white) means that, for that component, volume loss is particularly high in the corresponding voxel.

Atrophy severity correlations

The entries in the coefficient vector β (1 X 6), termed beta1 through beta6, are markers of regional brain atrophy. As seen in Fig. 2a, the general correlation matrix shows strong positive correlations between different pairs of components. Notably, beta1 displayed particularly strong correlations with other components, especially beta2 (0.86, p < 0.001) and beta3 (0.92, p < 0.001). After adjusting for age, sex, and APOE4, the Pearson partial correlations (Fig. 2b) revealed a mix of positive, negligible, and negative associations between component pairs. In examining the atrophy severities estimated by our six components, particularly strong positive Pearson partial correlations were detected between component 1 and component 5 (0.64, p < 0.001). Conversely, there were significant negative Pearson partial correlations observed, especially between beta2 and beta6 (–0.42, p = 0.006). The averaged GCGM partial correlations were computed over multiple iterations to account for initialization influence. Significant positive GCGM partial correlations were observed among beta1 with beta4 (0.47, CI = [0.23,0.65]) and beta1 with beta5 (0.41, CI = [0.16, 0.62]) (Fig. 2c). Additional results, which adjust only for age and gender, can be found in the Supplementary Figure 1 and are congruent in general to those observed when also accounting for APOE4. We conclude that a large portion of the observed correlations between the beta coefficients are due to the influence of age, gender, and APOE4 genotype.

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Fig. 2

Pearson General Correlations (a), Pearson Partial Correlations (b), and GCGM Partial Correlations of Atrophy Components (c). Partial Correlations are calculated after adjusting for Sex, Age, and APOE4. Colored boxes denote significant correlations where adjusted p-value<0.05. All the p-values shown here are adjusted for multiple comparisons using the Benjamini-Hochberg method. UL and LL signify the upper limit and lower limit of the 95% confidence intervals generated from the bootstrapping.

Cross-sectional analysis

Association between baseline atrophy severity and diagnosis group

Figure 3 shows evidence of overall trends between diagnosis group (CN < MCI<AD) and baseline atrophy severity in all six components (all p-values < 0.001).

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Fig. 3

Baseline atrophy severity by diagnosis group. The p-values are the results from two-sample t-tests comparing atrophy in different groups. (‘NS.’ Means not significant, ‘^****’ means 0 < p-values < 0.0001, ‘^***’ means 0.0001 < p-values < 0.001, ‘^**’ means 0.001 < p-values < 0.01, ‘^*’ means 0.01 < p-values < 0.05).

Baseline association between atrophy severity and cognitive scores

Figure 4 illustrates the strength of associations between atrophy and cognitive scores with p-values (for numerical results, see Supplementary Table 1). In the whole cohort, atrophy in all six components is significantly associated with all three cognitive scores. Among AD subjects, baseline atrophy is not significantly associated with cognitive score, except for atrophy in component 5 (medial temporal lobes) and CDR-SB. This may be due to homogeneity within AD group. For MCI subjects, atrophy in each of the six components is significantly associated with all three cognitive scores. In control subjects, notable associations were observed between atrophy in component 5 and three cognitive scores. In summary, among all six components, baseline atrophy in component 5 exhibits the strongest associations with all three cognitive test scores.

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Fig. 4

Baseline associations between atrophy coefficients and cognitive scores. Since a higher value on the CDR-SB is indicative of poorer cognitive function, while a lower value on both MMSE and MoCA indicates poorer cognitive function, when plotting the heat map, we multiplied the correlations between CDR-SB and atrophy by –1. Therefore, the heatmap only indicates the magnitude of the correlations with p-values. All the p-values shown here are adjusted for multiple comparisons using the Benjamini-Hochberg method. (‘^****’ means 0 < p-values < 0.0001, ‘^***’ means 0.0001 < p-values < 0.001, ‘^**’ means 0.001 < p-values < 0.01, ‘^*’ means 0.01 < p-values < 0.05).

Longitudinal analysis

Association between longitudinal atrophy severity and diagnosis group

Figure 5 depicts the estimated marginal means (obtained with the R package ‘emmeans’ [32]) and 95% confidence intervals of the atrophy from the linear mixed modeling results, which indicates that over time, the atrophy increases in all diagnostic groups, except for cerebellar atrophy (component 6) in the control group. Additionally, the rates of increase of atrophy over time were significantly different comparing AD subjects with MCI subjects in components 2, 3, and 5, and were also significantly different between AD and controls in all components (for numerical results, see Supplementary Table 2).

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Fig. 5

Estimated marginal means and 95% CIs of atrophy coefficients in different diagnostic groups from linear mixed-effects models adjusted for age, sex, education, and APOE4 genotypes. Since the values in the plot are marginal estimations from modeling results along over, the estimates can are extended beyond the exact follow up time when necessary.

Correlations between baseline atrophy and cognitive declines

Based on the linear mixed model results, across the whole cohort, subjects with higher baseline atrophy in components 1–5 (again with the exception of the cerebellum) tend to perform worse in all three cognitive tests (i.e., lower MMSE and MOCA scores, higher CDR-SB score) and have more rapid cognitive degeneration than those with lower levels of baseline atrophy. This is demonstrated in Fig. 6, where the subjects were divided into three groups with low, median, and high baseline atrophy, in order to visualize the interaction effect between the rate of cognitive changes with time and the baseline atrophy values.

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Fig. 6

Associations of baseline atrophy with longitudinal cognitive scores: estimated marginal means and 95% CIs from linear mixed-effects models adjusted for age, sex, educational level, and APOE ɛ4 genotypes. Brown line corresponds to the group with baseline beta value to be the mean minus standard deviation, orange line corresponds to the group with baseline beta value to be the mean and navy line corresponds to the group with baseline beta value to be the mean plus standard deviation. Since a higher value on the CDR-SB is indicative of poorer cognitive function; a lower value on both MMSE and MoCA indicates poorer cognitive function, to decrease confusion, the y-axis of the CDR-SB z-score is inverted. The p-values are derived from the interaction of time and group in linear mixed models.

Correlations between change rate of atrophy and cognitive declines

The associations of longitudinal changes in atrophy with cognitive decline measured by Kendall’s tau are shown in Fig. 7. From this, we see that, within the entire cohort, a more rapid increase in atrophy is significantly associated with swifter declines in cognition (i.e., decrease in MMSE and MoCA scores and increases in CDR-SB scores) in all six components. For the AD cohort, a significant association between the rates of changes was only found between MoCA scores and component 6. For MCI cohort, notable longitudinal associations were identified between atrophy in components 2, 4, 5, and all three scores. For controls, such associations exhibit between component 2–5 and MMSE scores, and component 5 with CDR-SB and MoCA scores. Across the entire cohort, including MCI subjects and controls, the rate of atrophy increases within component 5 (hippocampus) has the strongest association with the rate of change in cognitive scores. The values of Kendall’s tau rank correlation coefficients can be found in Supplementary Table 3.

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Fig. 7

Kendall’s tau rank correlation coefficient between change rate of atrophy and change rate of cognitive assessments scores in all three groups. Since a higher value on the CDR-SB is indicative of poorer cognitive function; a lower value on both MMSE and MoCA indicates poorer cognitive function, when plotting the heat map, we changed the sign of the Kendall’s tau results between the changing rate of CDR-SB and atrophy. Therefore, the heatmap only indicates the magnitude of the Kendall’s tau rank correlation coefficients. All the p-values shown here are adjusted for multiple comparisons using the Benjamini-Hochberg method. (‘^****’ means 0 < p-values < 0.0001, ‘^***’ means 0.0001 < p-values < 0.001, ‘^**’ means 0.001 < p-values < 0.01, ‘^*’ means 0.01 < p-values < 0.05).

Correlations between change rate of atrophy and baseline CSF biomarkers

Figure 8 shows the association between change rate of atrophy and baseline CSF biomarkers. As in Fig. 6, subjects were categorized into three groups based on their baseline biomarkers. From the graph, we can see that subjects with higher baseline TAU (Fig. 8a) and PTAU (Fig. 8b) tend to have a faster increase in brain atrophy for all six components, and subjects with lower baseline ABETA tend to have a faster increase in brain atrophy for components 1-5 (Fig. 8c).

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Fig. 8

Association between change rate of atrophy and baseline TAU (a), PTAU (b), and ABETA (c): estimated marginal means and 95% CIs from linear mixed-effects models adjusted for age, sex, educational level and APOE ɛ4 genotypes. Purple line corresponds to the group with baseline biomarker value to be the mean minus standard deviation, yellow line corresponds to the group with baseline biomarker value to be the mean and green line corresponds to the group with baseline biomarker value to be the mean plus standard deviation. The p-values are from the interaction of time and biomarker values in linear mixed models.

Validation analysis

Data presented in the supplementary materials (Supplementary Figures 2–5, Tables 4–8, and associated text) shows that atrophy coefficients derived from ADNI2 follow a similar pattern among different groups as ADNI3. This suggests that our methodology [26] can characterize the anatomic changes seen in a different dementia cohort with different baseline characteristics and further verified our findings upon the cross-sectional and longitudinal association results with cognitive decline and CSF biomarkers.

The use of gender-specific young templates

In our previously published study [25], we created “atrophy maps” by taking the preprocessed MR images from the baseline AD subjects and subtracting them from a global “normal” young template derived by averaging the images in ABIDE data (available at: http://fcon_1000.projects.nitrc.org/indi/abide/). Instead of using older controls from ADNI, we selected healthy young controls to serve as the reference for calculating AD-associated atrophy. This choice is based on the fact that young controls tend to be more homogeneous than old ones, which leads to more clustered and interpretable atrophy patterns through OPNMF. To differentiate our atrophy patterns from age-related changes, comparisons were conducted between atrophy trajectories in old controls and those in AD subjects.

In the current analysis, we created two gender-specific “normal” templates obtained by separately averaging the gray matter images of 22 young females and 22 age-matched young males from ABIDE and generated gender-specific atrophy maps by subtracting each registered and processed MR images for each subject from the corresponding gender-specific template. We did this because male brains have been found to be, on average, 10% to 15% larger than female brains [50] and, consequently, use of a gender-neutral template tends to exaggerate volume loss in female subjects. Some previous studies used total gray matter volume to normalize the atrophy [20, 51], but it was also found that several regional sex differences remain even after adjusting for overall brain size [50, 52]. Details on our comparison of gender-specific to gender-neutral templates are in Supplementary Material Section 5 and Supplementary Figures 6 and 7; in conjunction with the considerations listed above, we felt that a better approach to account for such sex difference was to use gender-specific young templates.

Longitudinal methods and settings

Longitudinal MRI studies in AD are among the most dependable methods for monitoring changes in the brain throughout the progression of the disease and are employed widely in atrophy related studies [21, 54]. In the ADNI datasets, biomarkers and MR images are generally obtained at approximately, but not exactly, the same time. Therefore, we fit linear mixed models with random slopes separately upon two sets of dependent longitudinal variables to derive individual change rates of atrophy and change rate of cognitive scores. This approach simplifies longitudinal modeling and deals with the problem that biomarkers and MR images might not be measured at the same time.

Certain prior studies concentrating on longitudinal brain atrophy have implemented restricted inclusion criteria, requiring participants to have at least available two scans with a specified time interval between the scans [9, 55]. For present study, we incorporated all available scans from ADNI2 and ADNI3 for atrophy pattern generation and cross-sectional analysis, with the goal of attaining a more extensive dataset and mitigating any selection bias. We limited the analysis to subjects with more than one visit solely for the longitudinal portions of the study. We also attempted to include all subjects, whether they had only one visit or multiple visits, into the model. The findings revealed that subjects with only one visit primarily contribute to the cross-sectional effects of the linear-mixed model. Therefore, we can suggest that excluding individuals with one visit would likely lead us to similar results as driven in the present study. Furthermore, the capability to effectively handle subjects with unbalanced visits provides reasonable validation to our analysis, particularly when dealing with three diagnosis group that exhibit significantly varying follow-up times. Similar approach can also be found in the previous study [4].

In conclusion, the present study has generated six spatially distinct gray matter atrophy components in AD based on a relatively new feature extraction technology called orthonormal projective non-negative matrix factorization (OPNMF) and depicted both cross-sectional and longitudinal patterns of these atrophy components with respect to cognitive decline and CSF biomarkers. One of these independently derived components, consisting of the anterior medial temporal lobes, was found to correlate most strongly with the progression of AD and decline in cognitive function. The results may enrich current findings upon atrophy patterns regarding AD and may help us better characterize its underlying pathology. Further analysis could focus on transferring this methodology framework in other types of dementias on MRI.