MRI Distance Measures as a Predictor of Subsequent Clinical Status During the Preclinical Phase of Alzheimer's Disease and Related Disorders

ABSTRACT

Brain atrophy over time, as measured by magnetic resonance imaging (MRI), has been shown to predict subsequent cognitive impairment among individuals who were cognitively normal when first evaluated, indicating that subtle brain atrophy associated with Alzheimer's disease (AD) may begin years before clinical symptoms appear. Traditionally, atrophy has been quantified by differences in brain volume or thickness over a specified timeframe. Research indicates that the rate of atrophy varies across different brain regions, which themselves exhibit complex spatial and hierarchical organizations. These characteristics collectively emphasize the need for diverse summary measures that can effectively capture the multidimensional nature of degeneration. In this study, we explore the use of distance measurements to quantify brain volumetric changes using processed MRI data from the Preclinical Alzheimer's Disease Consortium (PAC). We conducted a series of analyses to predict future diagnostic status by modeling MRI trajectories for participants who were cognitively normal at baseline and either remained cognitively normal or progressed to mild cognitive impairment (MCI) over time, with adjustments for age, sex, education, and APOE genotype. We consider multiple distance measures and brain regions through a two‐step approach. First, we build base models by fitting individual mixed‐effect models for each distance metric and brain region pairing, using follow‐up diagnosis (normal vs. MCI) as the outcome and volumetric changes from the baseline, as summarized by a given distance measure, as predictors. The second step aggregates these individual region‐distance base models to derive an overall estimate of diagnostic status. Our analyses showed that the distance measures approach consistently outperformed the traditional direct volumetric approach in terms of predictive accuracy, both in individual base models and the aggregated models. This work highlights the potential advantage of using distance measures over the traditional direct volumetric approach to capture the multidimensional aspects of atrophy in the development of AD and related disorders.

This study investigates the use of distance measurements to characterize brain atrophy over time based on MRI data. Our analyses showed that the distance measures consistently outperformed the traditional direct volumetric approach for predicting follow‐up diagnosis, highlighting its potential advantage in capturing the multidimensional aspects of ADRD atrophy.

1. Introduction

Substantial evidence (e.g., Bateman et al. 2012; Buchhave et al. 2012; Fagan et al. 2014) suggests that Alzheimer's disease (AD) related brain changes can begin years or even decades prior to the manifestation of clinical symptoms. This pre‐symptomatic phase presents a critical window for early detection and intervention. Prior research has highlighted the value of using volumetric measures derived from magnetic resonance imaging (MRI) biomarkers to detect early neurodegenerative changes in AD before clinical symptoms become apparent (e.g., Fennema‐Notestine et al. 2009; Jack et al. 2010; Miller et al. 2013).

Neurodegeneration in AD occurs non‐uniformly across the brain, leading to differential atrophy rates across regions. Previous studies indicate that AD‐related atrophy often begins in medial‐temporal lobe regions, particularly the entorhinal cortex and hippocampus (e.g., Jack et al. 2010; Ledig et al. 2018; Miller et al. 2013), coinciding with areas of very early tau protein accumulation (Arnold et al. 1991; Montine et al. 2012). Changes in the frontal and occipital lobes tend to manifest later in the course of AD (Scahill et al. 2002).

Brain atrophy is generally assessed by measuring regional volumetric changes directly and comparing volumes over time, quantified as difference scores. However, the hierarchical organization of the brain allows larger areas to be segmented into smaller subregions, further enabling the simultaneous assessment of changes in these smaller subregions. Combining these changes provides an alternative method to quantify volumetric changes in a brain region and may be more sensitive to temporal changes. This study expands prior work by evaluating whether measuring brain atrophy using distance measures provides a better approach for detecting structural brain changes on MRI. A distance measure, which is a numerical description of how close or distant two objects are, summarizes differences in each direction and could potentially offer a more precise and nuanced assessment of these structural changes.

To conceptualize the application of distance measures for characterizing brain atrophy, consider a map analogy, using the temporal lobe as an example. The temporal lobe can be segmented into subregions that include the inferior, lateral, and supratemporal gray matter, along with white matter. The volume of each subregion at a particular time represents a direction in multidimensional space. Collectively, these volumes form a unique configuration in this space, similar to identifying a location on a map. As brain atrophy progresses, these volumes change, resulting in the relocation of this configuration within the map. Measuring the distance between the positions at different points in time provides a summary of the volumetric changes occurring within each subregion, which allows us to track the progression of atrophy in the temporal lobe as a whole. Due to the challenges of higher‐dimensional visualization, Figure 1 illustrates this concept on a 2D map. Specifically, it shows changes from baseline in the temporal lobe, where gray matter and white matter define the two subregions.

FIGURE 1.

Conceptualization of changes in regional brain volumes as movements of points on a map: Each point represents the volumes of grey matter (Subregion 1) and white matter (Subregion 2) in the temporal lobe from a participant in the PAC data set, tracked over six MRI scans. Annotations $V_1^{t_1},V_2^{t_1}$ indicate baseline volumes; $V_1^{t_2},V_2^{t_2}$ and similar subsequent labels denote follow‐up measurements.

Numerous distance measurements are available, each offering a unique perspective to quantify changes over time. To the best of our knowledge, distance measures have not been explored within the context of MRI or AD and related disorders (ADRD). Consider two points, each representing the volume of a brain region at a specific time, with coordinates corresponding to subregional volumes. For example, Euclidean distance calculates the straight‐line distance between two points. The “city block distance,” also known as the Manhattan distance, sums the absolute differences along each coordinate for a piecewise linear calculation. Additionally, the angular distance measures the angle between two points projecting from the origin. Leveraging such diverse metrics may enable the identification of more subtle changes within brain structures.

This study introduces a novel method for characterizing brain atrophy in the context of ADRD through volumes, focusing on the temporal dynamics of brain structural changes during the early phase of ADRD, which may have particular potential for early diagnosis prior to the onset of clinical impairment. Using longitudinal MRI data from the Preclinical AD Consortium, we examine nine distance measurements across six regions of interest to track changes in brain volumes over time among individuals who were cognitively unimpaired at baseline and subsequently developed clinical symptoms of mild cognitive impairment (MCI) or remained cognitively unimpaired. Our study builds on previous research in several ways. First, we simultaneously examine volumetric changes across multiple subregions of a specific region of interest (ROI) and summarize them into a single metric that represents the overall volumetric dynamics of the ROI. This approach not only effectively reduces the data complexity in quantifying brain atrophy of multiple subregions, but also serves as a method for dimension reduction that preserves the specific details of local variations while circumventing the problem of multicollinearity. Second, considering the spatial and hierarchical organization of brain regions and their differential atrophy rates, we integrate individual summaries of volumetric changes over time, where each summary is defined by a combination of a specific distance measure and a particular ROI. This aggregation process is guided by each summary's predictive contribution to the clinical diagnostic status and can further enhance the detection of brain atrophy, facilitating the tracking of ADRD in its early stages. We hypothesized that this method would provide a robust framework for tracking the progression of atrophy among individuals progressing from normal cognition to MCI.

2. Materials and Methods

2.1. Participants

The analyses used data from the Preclinical Alzheimer's Disease Consortium (PAC), which was established in 2014 to develop harmonized data sets for enhancing the understanding of the earliest phases of AD. The PAC data integrates data from five ongoing cohort studies: the Adult Children Study (ACS) (Coats and Morris 2005), the Australian Imaging, Biomarker, and Lifestyle (AIBL) study (Ellis et al. 2009), the Biomarkers of Cognitive Decline Among Normal Individuals (BIOCARD) study (Albert et al. 2014), the Neuroimaging Substudy of the Baltimore Longitudinal Study of Aging (BLSA) (Resnick et al. 2000; Shock et al. 1984), and the Wisconsin Registry for Alzheimer's Prevention (WRAP) (Johnson et al. 2017). These longitudinal studies have been collecting clinical and cognitive data, as well as biomarkers, including blood samples, cerebrospinal fluid (CSF), MRI, and positron emission tomography (PET) scans at regular intervals (e.g., annually or every 24 months, based on each study's design). Details of these studies can be found in the aforementioned references. All participants were middle‐aged and older at enrollment and provided written informed consent with study protocols approved by each site's local institutional review board.

Participants were included in the PAC data sets if they were cognitively normal at baseline and had at least one molecular biomarker (derived from CSF or PET) collected while they were cognitively normal. The present analyses focused on T1‐weighted structural MRI (sMRI) data collected from 542 participants who were cognitively unimpaired at their first available MRI scan, underwent a minimum of three MRI scans, and had relevant demographic information (i.e., age, sex, and education), and Apolipoprotein E (APOE)‐$\mathrm{ϵ}$4 carrier status, the most important genetic risk factor for late‐onset AD (Corder et al. 1993). For the purposes of this manuscript, the term “baseline” refers to the participants' first available MRI scan. Molecular biomarkers were not examined in this study.

2.2. MRI Processing

The processing and harmonization pipeline for the PAC sMRI scans has been published previously (Soldan et al. 2024). Briefly, the PAC sMRI scans were processed using a fully automated pipeline. Using participants' T1‐weighted scans, brain extraction was first performed based on a multi‐atlas registration framework (Doshi et al. 2013). To maintain longitudinal consistency across multiple scans taken over time, each brain mask was merged with a probabilistic mask derived from the baseline image. Skull‐stripped images were then corrected for magnetic field intensity inhomogeneity (Tustison et al. 2010). Following a multi‐atlas, multi‐warp label‐fusion method, MUSE (Doshi et al. 2016), each image was segmented into 145 ROIs spanning a dense parcellation of the cortex, white matter lobes, and deep structures. Information regarding quality control of processed scans and more details regarding sMRI processing are provided in Supporting Information S1: Section E.

The volumes of these 145 individual ROIs were then calculated. To address variations attributable to different studies and scanner types, individual ROI volumes were harmonized following the ISTAGING statistical harmonization framework, as described in Pomponio et al. (2020), designed to remove site effects via regression‐based methods. We further normalize ROI volumes by dividing each regional volume by the intra‐cranial volume (ICV). This is a standard approach to account for individual differences in head size (e.g., Jack Jr. et al. 2012; Westman et al. 2013) (see Supporting Information S1: Section E.3 for ICV estimation).

Our analysis primarily considers six larger anatomical regions: the temporal lobe, frontal lobe, parietal lobe, occipital lobe, limbic system, and ventricles, with each one representing the sum of volumes of individual ROIs. The limbic system regions included the hippocampus, amygdala, cingulate gyrus, parahippocampal gyrus, thalamus, fornix, and basal ganglia. A list of individual ROIs within each of these six regions is given in Supporting Information S1: Table F6.

2.3. Diagnosis

As described in detail previously (Gross et al. 2017), each study implemented a rigorous process for arriving at a consensus diagnosis (e.g., cognitively unimpaired (CU), impaired not MCI, MCI, or dementia) for each participant, at each visit. This process utilized published criteria for MCI (e.g., Albert et al. 2011) or dementia (McKhann et al. 2011), and incorporated information from cognitive assessments, the Clinical Dementia Rating (CDR; Morris 1993), medical, neurological, and psychiatric assessments, and informant interviews from the CDR. Clinical diagnoses were made without consideration of biomarker measures. The mean absolute time difference between the date of the MRI scan and the corresponding clinical diagnosis was 0.2 years (SD = 0.23, range = $-2.18$ to 1.11 years).

While our focus is on comparing individuals who remained CU to those who progressed to MCI, we increased our sample size by including 10 individuals who progressed to dementia. For these 10 individuals, we only included data from the period before they reached a dementia diagnosis (i.e., data from the baseline MRI scan through each datapoint up to, but not, including visits with a dementia diagnosis). Individuals with a diagnosis of impaired, not MCI, were included in the CU group given they do not meet clinical criteria for MCI. As a result, participants were classified into the following categories: CU–CU and CU–MCI, reflecting diagnosis at the baseline MRI scan and diagnosis at the last available MRI scan (or diagnosis prior to the first recorded dementia status, for the subset who progressed to dementia).

2.4. Distance‐Based Modeling

2.4.1. Motivation and Preliminary Illustration

The brain's hierarchical nature allows for larger regions to the parcellated into several smaller ones. For example, the limbic system can be divided into distinct partitions, including the hippocampus, amygdala, parahippocampal gyrus, and anterior, middle, and posterior cingulate gyrus. A “composite ROI” is an anatomical area formed by combining multiple smaller ROIs; for the present study, all ROIs were derived from 145 individual MUSE ROIs. For a composite ROI, the volumes of its subregions can be arranged into a numeric array, represented by $\mathit{V}=\left(V_1,\dots ,V_p\right)\in {\mathrm{ℝ}}^p$, where $p$ denotes the number of subregions; $V_j$ is the volume of subregion $j$ for $j=1,\dots ,p$; $\mathrm{ℝ}$ represents the set of real numbers. Depending on the brain parcellation method used, p can differ for the same brain region. For instance, in this work, the frontal lobe is divided into 46 different subregions (see Supporting Information S1: Table 10 for the subregion names). Having a large $p$ allows for a more detailed representation of a composite ROI, and this can be advantageous for tracking brain structural changes. However, it results in analytical challenges due to the curse of dimensionality (e.g., Bellman 1957). This common issue arises when the number of variables or predictors (in this case, the subregional volumes) is relatively large compared to the number of observations or samples.

Longitudinal volumetric MRI data provides a collection of ${\mathit{V}}_t$ at different times t for a composite ROI. Each ${\mathit{V}}_t$ is a numerical representation of the volumes within the subregions of this ROI. To assess changes in brain volume within this composite ROI, we measure the distance between volume arrays at time 1 vs. time 2 (${\mathit{V}}_{t_1}$ and ${\mathit{V}}_{t_2}$). A distance measure summarizes volumetric changes over time across multiple brain areas (i.e., all regions within a composite ROI) simultaneously. It operates by first examining changes in each subregion separately and then calculate the corresponding distance measures based on these changes. Numerous measurements are available to calculate the distance between ${\mathit{V}}_{t_1}$ and ${\mathit{V}}_{t_2}$. The Euclidean distance calculates the straight‐line distance, providing an overall or global measure of change while disregarding the path of that change. In contrast, the city block distance computes changes in a stepwise, grid‐like manner, emphasizing incremental changes in each dimension, emphasizing local changes. Additionally, angular distance, which measures the angle between ${\mathit{V}}_{t_1}$ and ${\mathit{V}}_{t_2}$, is sensitive to subtle changes. Even a small shift in direction can lead to a substantial change in the location in a multi‐dimensional space, a feature captured by angular distance but potentially overlooked by others like Euclidean and city‐block distances. See a graphical illustration of the aforementioned three distance measures in Figure 2. Furthermore, transforming data can sometimes enhance the detection of changes. One example is the Bhattacharyya (log) distance, which examines the distance between ${\mathit{V}}_{t_1}$ and ${\mathit{V}}_{t_2}$ at a square‐root scale.

FIGURE 2.

Graphical representation of Euclidean, City block, and angular distances used to calculate changes in regional volumes. In the illustration, a region is divided into two subregions, where the two points $\left(V_1^{t_1},V_2^{t_1}\right)$ and $\left(V_1^{t_2},V_2^{t_2}\right)$ represent the volumes of these subregions at times $t_1$ and $t_2$, respectively. Each distance measure quantifies the change in volume for the entire region between $t_1$ and $t_2$ in a distinct way.

In this study, we focus our analysis on changes from baseline to more clearly assess the effectiveness of distance measures in tracking brain atrophy. All participants were CU at baseline, which provides a definitive reference point for each individual's condition. Comparing against the baseline makes it easier to identify changes in brain structure in individuals who progress to cognitive impairment and those who do not, compared to analyzing changes between successive visits. When distance increases over time, it indicates progressive atrophy—meaning these regions are undergoing shrinkage as the disease progresses. Biologically, this atrophy reflects neurodegenerative processes such as neuronal loss and synaptic and axonal degeneration. Thus, increasing distance values signify damage to, or loss of, brain cells and their connections, providing a quantitative marker for underlying disease progression.

We consider nine different distance measurements to summarize changes, ranging from linear metrics like the “city block” distance to nonlinear metrics such as angular distance. These measures and their formulas for calculation are listed in Table 1 and were applied to six composite ROIs: the temporal lobe, frontal lobe, parietal lobe, occipital lobe, limbic system, and ventricles. A detailed segmentation of the six broader regions into finer subregions is provided in Supporting Information S1: Table F6.

TABLE 1.

Distance measurements. Definitions of various distances $d\left(\mathit{x},\mathit{y}\right)$ between two numeric arrays $\mathit{x},\mathit{y}\in {\mathrm{ℝ}}^p$, where each has $p$ components, are presented here. In particular, $x_k$ represents the $k$th element of $\mathit{x}$ and $y_k$ is defined similarly; $g\left(\mathit{x}\right)$ denotes the geometric mean defined by $g\left(x\right)={\left({\prod }_{k=1}^p{x}_k\right)}^{1/p}$.

Distance	$d\left(\mathit{x},\mathit{y}\right)$
Aitchison	${\left[{\sum }_{k=1}^p{\left\{\log \left(\frac{x_k}{g\left(\mathit{x}\right)}\right)-\log \left(\frac{y_k}{g\left(\mathit{y}\right)}\right)\right\}}^2\right]}^{1/2}$
Angular	$\arccos \left({\sum }_{k=1}^p\sqrt{\frac{x_k^2}{\parallel x{\parallel }^2}}\sqrt{\frac{y_k^2}{\parallel y{\parallel }^2}}\right)$
Bhattacharyya (arccos)	$\arccos \left({\sum }_{k=1}^p\sqrt{x_k}\sqrt{y_k}\right)$
Bhattacharyya (log)	$-\log \left({\sum }_{k=1}^p\sqrt{x_k}\sqrt{y_k}\right)$
City block	${\sum }_{k=1}^p\left|x_k-y_k\right|$
J‐divergence	${\left[{\sum }_{k=1}^p\left\{\log \left(x_k\right)-\log \left(y_k\right)\right\}\left(x_k-y_k\right)\right]}^{1/2}$
Euclidean	${\left\{{\sum }_{k=1}^p{\left(x_k-y_k\right)}^2\right\}}^{1/2}$
Matusita	${\left\{{\sum }_{k=1}^p{\left(\sqrt{x_k}-\sqrt{y_k}\right)}^2\right\}}^{1/2}$
Minkowski	${\left\{{\sum }_{k=1}^p{\left(x_k-y_k\right)}^q\right\}}^{1/q}\left(\text{integer}q\right)$

In addition to providing a multifaceted quantification of brain structural changes over time, the use of distance measurements can also act as a method for dimension reduction in statistical modeling for longitudinal volumetric data. Specifically, applying distance measurement translates volumetric changes in multiple areas into a univariate representation while ensuring that detailed local information is retained. Consequently, this summary of changes within a composite ROI can then be used as a predictor in modeling outcome variables like diagnostic status. In addition to possibly providing more information than using regional volumes alone, this strategy reduces the number of predictors needed compared to the direct use of volumes from all subregions, thereby addressing the common problem of multicollinearity in such analyses.

For illustrative purposes, we first examine distance measures in a randomly selected subgroup of participants who have progressed from normal cognition to MCI due to ADRD ($n$ = 10) and had at least five MRI measurements, compared to a subgroup that has remained cognitively unimpaired over follow‐up to date ($n$ = 10) and also had at least five MRI scans. These two groups were equivalent in terms of age ($\pm 5$ years, in case of multiple, individuals with the closest age were selected), sex, and APOE‐$\mathrm{ϵ}$4 carrier status (60% for each group). To motivate our later, comprehensive analysis using all available CU‐CU and CU‐MCI participants in the PAC data set, this focused exploration serves as a preliminary illustration of the potential benefits of employing distance measurement for assessing volumetric changes in a brain region, as opposed to directly analyzing brain volumes over time. Additionally, this initial analysis compares two different distance measurements within the same brain region, as well as different regions with the same distance measure applied, in terms of how well they distinguish between participants who remained normal vs. those who progressed.

As an example, we first calculated volumetric changes within the limbic system from baseline by applying the “city block” and “Bhattacharyya (log)” distances, respectively. Then, we plot volumetric change vs. age at MRI scan for each distance measure (the center panel of Figure 3 corresponds to “city block” and the right panel corresponds to “Bhattacharyya (log)”). To compare, we also plot volume vs. age at the MRI scan (left panel of Figure 3). As described above, these two distance measurements are made between the volumes of individual ROIs or subregions (see Supporting Information S1: Table F6 for the subregion names) that compose the limbic system at baseline and these volumes at follow‐up scans. As shown in Figure 3, the “city block” distance appears to better differentiate between CU and MCI diagnosis than simply analyzing limbic system volume over time. This conclusion is based on the observation of greater increases in distances over time from the baseline to the last scan in the individuals who progress to MCI vs. those who remain CU. This suggests that volumetric changes in the limbic system, as characterized by the “city block” distance, could serve as an informative indicator for monitoring the early stages of ADRD. On the other hand, the “Bhattacharyya (log)” distance exhibits a monotonic trend in both diagnostic groups, which may reflect changes attributable to normal aging (rather than disease) or non‐ADRD neurodegenerative processes. This example indicates that different distance measurements can reveal varying aspects of volumetric changes in the same brain region.

FIGURE 3.

Within the limbic system: The left plot presents volume vs. age at MRI scan; The middle plot presents volumetric change from the baseline over time, which is quantified using the “city block” distance; The right plot presents volumetric change from the baseline over time which is quantified using the “Bhattacharyya‐log” distance. The diagnosis of CU and MCI are represented by blue and red dots. Observations from the same participant are connected, with the black dashed line indicating those who remained CU and the black solid line denoting participants who have progressed to MCI.

With Euclidean distance applied, Figure 4 suggests that changes in brain volume in different regions, as characterized by a single distance measurement, are distinct. In the limbic system, those who progressed to MCI demonstrate an overall upward trend over time in the volumetric changes observed within the limbic system, compared to those who remained CU, and these changes seem to allow differentiation between the CU and MCI follow‐up diagnoses. In contrast, the parietal lobe, as measured by the Euclidean distance, does not exhibit a clear difference in volumetric changes between the two diagnostic groups. This comparison also highlights the heterogeneous nature of regional atrophy associated with ADRD.

FIGURE 4.

Volumetric changes from the baseline over time calculated via the Euclidean distance within the limbic system (left) and parietal lobe (right), respectively. The diagnosis of CU and MCI are represented by blue and red dots. Observations from the same participant are connected, with the black dashed line indicating those who remained CU and the black solid line denoting participants who have progressed to MCI.

2.4.2. A Two‐Step Model for Longitudinal Diagnostic Status

To determine whether distance measurements better capture changes in brain volumes over time, we model longitudinal clinical diagnostic status using data from participants who have undergone at least three MRI scans in the full PAC data set, all of whom were CU at baseline and who either remained CU or progressed to MCI. The requirement for a minimum of three MRI scans per participant is driven by the calculation of volumetric change from baseline and an additional observation for the purpose of model validation. Specifically, we examine the volumetric change from baseline within a composite ROI, which summarizes changes across all subregional volumes, vs. using the total volume of the composite ROI itself as a predictor of the diagnosis.

We begin our analysis by fitting random‐effects logistic regression models to each pair of composite ROI and distance measures. Each pair is dedicated to the longitudinal modeling of volumetric changes from baseline, which are calculated using the specified distance measurement within the given region, as a predictor of diagnostic status (i.e., remained CU or progressed to MCI). For comparison, we similarly applied these models to each composite ROI, using its volume as the predictor. The results, presented in Table 3, consistently show that distance‐based models outperform direct volume‐based models for all considered ROIs. A more detailed discussion is deferred to Section 3.1.

TABLE 3.

Base model assessment within temporal, occipital, and frontal lobes.

ROI	Base model	ROC AUC	Bootstrap CI	PR AUC	Bootstrap CI
Temporal lobe	Model volume directly	0.71	[0.64, 0.78]	0.24	[0.17, 0.35]
	Model volume change from baseline
	Angular	0.88	[0.82, 0.93]	0.62	[0.49, 0.75]
	Bhattacharyya (acos)	0.86	[0.80, 0.91]	0.61	[0.48, 0.75]
	Bhattacharyya (log)	0.85	[0.80, 0.91]	0.61	[0.48, 0.74]
	J‐divergence	0.88	[0.84, 0.93]	0.62	[0.48, 0.75]
	Matusita	0.88	[0.84, 0.93]	0.62	[0.48, 0.75]
	Aitchison	0.84	[0.78, 0.90]	0.59	[0.46, 0.73]
	City block	0.88	[0.83, 0.92]	0.59	[0.45, 0.73]
	Euclidean	0.86	[0.80, 0.91]	0.61	[0.47, 0.73]
	Minkowski	0.87	[0.81, 0.91]	0.59	[0.46, 0.72]
	Average among distance	0.87	[0.83, 0.92]	0.61	[0.50, 0.76]
Occipital lobe	Model volume directly	0.69	[0.62, 0.76]	0.24	[0.16, 0.34]
	Model volume change from baseline
	Angular	0.83	[0.76, 0.89]	0.50	[0.37, 0.66]
	Bhattacharyya (acos)	0.84	[0.78, 0.90]	0.60	[0.47, 0.75]
	Bhattacharyya (log)	0.84	[0.78, 0.90]	0.60	[0.47, 0.74]
	J‐divergence	0.83	[0.77, 0.90]	0.54	[0.41, 0.69]
	Matusita	0.83	[0.78, 0.89]	0.55	[0.41, 0.69]
	Aitchison	0.83	[0.77, 0.89]	0.58	[0.44, 0.72]
	City block	0.83	[0.77, 0.89]	0.54	[0.41, 0.68]
	Euclidean	0.83	[0.76, 0.89]	0.53	[0.41, 0.68]
	Minkowski	0.83	[0.76, 0.89]	0.54	[0.41, 0.69]
	Average among distance	0.83	[0.77, 0.90]	0.55	[0.44, 0.72]
Frontal lobe	Model volume directly	0.71	[0.65, 0.78]	0.26	[0.18, 0.37]
	Model volume change from baseline
	Angular	0.82	[0.76, 0.88]	0.55	[0.41, 0.70]
	Bhattacharyya (acos)	0.84	[0.78, 0.90]	0.58	[0.45, 0.73]
	Bhattacharyya (log)	0.84	[0.78, 0.90]	0.58	[0.45, 0.73]
	J‐divergence	0.84	[0.78, 0.89]	0.55	[0.43, 0.70]
	Matusita	0.85	[0.79, 0.90]	0.58	[0.45, 0.72]
	Aitchison	0.82	[0.76, 0.87]	0.39	[0.28, 0.51]
	City block	0.85	[0.79, 0.90]	0.58	[0.44, 0.72]
	Euclidean	0.82	[0.76, 0.89]	0.54	[0.41, 0.70]
	Minkowski	0.84	[0.78, 0.89]	0.55	[0.43, 0.70]
	Average among distance	0.84	[0.79, 0.89]	0.54	[0.44, 0.72]

Note: Assessment of base models via area under the ROC curve (ROC AUC) and area under the precision‐recall curve (PR AUC), as well as their corresponding bootstrap 95% confidence intervals across nine distance measurements, and six regions: Temporal, occipital, frontal and parietal lobes, limbic system and ventricle. Within each region, “Model volume directly” refers to the model that directly uses volume as a predictor. Under the heading “Model volume change from baseline,” the results are from models that use volumetric change from the baseline as a predictor, and this change is quantified by various distance measurements, with their names displayed in the second column; “Average among distance” represents the average AUC across all models that use distance measures. The models estimated to positively contribute the prediction of diagnostic status via the wisdom‐of‐crowds are highlighted in bold text.

As illustrated in Section 2.4.1, different distance measurements reveal varying temporal dynamics of brain structural changes in the same region, and the representation of brain atrophy through individual distance measurements can vary across regions. This variation suggests that each specific pairing of a distance measure and a brain region provides unique information regarding age‐ and disease‐related brain changes. Additionally, some pairings may hold the potential to identify subtle changes that might be overlooked by others. To further investigate the effectiveness of using distance measures to quantify volumetric changes, we aim to explore how various distance measures applied to different ROIs can collectively explain brain atrophy, rather than selecting a specific measure.

The most straightforward approach to link diagnosis and various representations of volumetric change, defined by a combination of a distance measurement and a composite ROI, is to incorporate all such pairs as predictors in a single model. However, this method risks encountering the curse of dimensionality (e.g., Bellman 1957). Though penalized approaches can be used to select the most informative combinations, these techniques might not be computationally efficient and could lead to misleading results due to the strong collinearity among these variables and complex correlation structures in longitudinal data. To overcome these issues, we instead used a two‐step procedure. The first step involved fitting individual models for each (region, distance) pair, already described earlier in this section, and referred to as “base models.” Then, the second step aggregates base models together according to each individual's predictive contribution to the diagnostic status. See Figure 5 for an overview.

FIGURE 5.

Flowchart of the two‐step model.

We formalize the two‐step model as follows. Suppose we observe $n$ subjects and each subject $i$ has $J_i+1$ measurements. Let $Y_{\mathit{ij}}$ be a binary variable indicating diagnostic status, with $Y_{\mathit{ij}}=1$ denoting MCI and $Y_{\mathit{ij}}=0$ denoting cognitively unimpaired. We first build up base models. Each model is defined by a combination of a composite region $R_k$ for $k=1,\dots ,K$ and a distance measurement $d_l$ for $l=1,\dots ,L$. We construct a dynamic predictor by calculating volumetric change at follows‐up scans from the baseline, denoted by $D_{\mathit{ij}}^{\left(k,l\right)}=d_l\left({\mathit{V}}_{\mathit{ij}}^{\left(k\right)},{\mathit{V}}_{i0}^{\left(k\right)}\right)$ for $i=1,\dots n$ and $j=1,\dots ,J_i$. Here ${\mathit{V}}_{i0}^{\left(k\right)}$ is an array consisting subregional volumes in region $R_k$ at the baseline and ${\mathit{V}}_{\mathit{ij}}^{\left(k\right)}$ is defined similarly. Alongside $D_{\mathit{ij}}^{\left(k,l\right)}$, we also consider covariates ${\mathit{X}}_{\mathit{ij}}$ including including age (z‐scored), sex, education (z‐scored separately for each cohort), APOE‐$\mathrm{ϵ}$4 carrier status and composite ROI volume at baseline. We adjust for baseline volume to account for variations in change from baseline measurements, as these differ among individuals.

For each pair of region $R_k$ and distance measure $d_l$, a mixed‐effect logistic regression model is then applied for analyzing the regression relationship between longitudinal diagnostic status and the dynamic predictor together with the covariates:

$$\log \left(\frac{P\left(Y_{\mathit{ij}}=1\right)}{P\left(Y_{\mathit{ij}}=0\right)}\right)={\alpha }_{\mathit{kl}}+{\beta }_{\mathit{kl}}{D}_{\mathit{ij}}^{\left(k,l\right)}+{\mathit{X}}_{\mathit{ij}}^{\mathrm{T}}{\gamma }_{\mathit{kl}}+u_i^{\left(k,l\right)}$$ (1)

for $j=1,\dots ,J_i$ and $i=1,\dots ,n$, where ${\alpha }_{\mathit{kl}}$ is the fixed intercept, ${\beta }_{\mathit{kl}}$ denotes the coefficient of the dynamic predictor $D_{\mathit{ij}}^{\left(k,l\right)}$, ${\gamma }_{\mathit{kl}}$ refers to the coefficients of the covariates ${\mathit{X}}_{\mathit{ij}}$. The random intercept $u_i^{\left(k,l\right)}$ represents subject‐specific effects. Further, assume $u_i^{\left(k,l\right)}\stackrel{\mathit{iid}}{\sim }N\left(0,{\sigma }_{\mathit{kl}}^2\right)$.

Each base model indexed by $\left(k,l\right)$ for $k=1,\dots ,K$ and $l=1,\dots ,L$, can estimate probability of the diagnostic status $Y_{\mathit{ij}}$ through

$${\widehat{P}}^{\left(k,l\right)}\left(Y_{\mathit{ij}}=1\right)=\frac{\exp \left\{{\widehat{\alpha }}_{\mathit{kl}}+{\widehat{\beta }}_{\mathit{kl}}{D}_{\mathit{ij}}^{\left(k,l\right)}+{\mathit{X}}_{\mathit{ij}}^{\mathrm{T}}{\widehat{\gamma }}_{\mathit{kl}}+{\widehat{u}}_i^{\left(k,l\right)}\right\}}{1+\exp \left\{{\widehat{\alpha }}_{\mathit{kl}}+{\widehat{\beta }}_{\mathit{kl}}{D}_{\mathit{ij}}^{\left(k,l\right)}+{\mathit{X}}_{\mathit{ij}}^{\mathrm{T}}{\widehat{\gamma }}_{\mathit{kl}}+{\widehat{u}}_i^{\left(k,l\right)}\right\}},$$ (2)

and moreover, the diagnostic status can then be obtained via a thresholding step ${\widehat{Y}}_{\mathit{ij}}^{\left(k,l\right)}=\mathbb{1}\left\{{\widehat{P}}^{\left(k,l\right)}\left(Y_{\mathit{ij}}=1\right)>\delta \right\}$ with a predefined threshold $\delta$.

In the second step, we combine estimations from each base model using three aggregation rules. The most straightforward methods are best‐of‐crowds and majority‐of‐crowds. The “best‐of‐crowds” approach finds the optimal base model with the highest accuracy of estimating group membership (CU vs. MCI) at the observation level, which is evaluated through micro average $f_1$‐score (Müller and Guido 2016). Then, the final estimation $\widehat{P}\left(Y_{\mathit{ij}}=1\right)$ is given using the optimal model. The “majority‐of‐crowds” gives a state estimation agreed on by the majority of base models. However, given that MCI diagnosis is a rare event and many base models possibly are biased toward the majority class, which is “CU,” the “best‐of‐crowds” and “majority‐of‐crowds” rules might be flawed, we then consider another alternative “wisdom‐of‐crowds” (Budescu and Chen 2015). Specifically, for a given $Y_{\mathit{ij}}$, the overall performance of base models is evaluated using a score defined as

(3)

Then, the contribution of base model $\left(k,l\right)$ at the $j$th measurement is calculated by

$$C_j^{\left(\mathit{kl}\right)}=\frac{1}{n}\sum _{i=1}^n\left\{S_{\mathit{ij}}-S_{\mathit{ij}}^{\left(-k,-l\right)}\right\},$$ (4)

where $S_{\mathit{ij}}^{\left(-k,-l\right)}$ denotes the score for estimating the probability of $Y_{\mathit{ij}}=g$ for $g=0,1$ without the base model $\left(k,l\right)$, which takes the following form

The final estimation $\widehat{P}\left(Y_{\mathit{ij}}=1\right)$ is a weighted sum of predictions from the base models with positive contribution $C_j^{\left(\mathit{kl}\right)}$, that is,

(5)

To compare the distance‐based method with traditional volumetric analysis, which uses composite ROI volume as a predictor for diagnostic status, we apply the same two‐step procedure. This way of directly leveraging regional volumes evaluates volumetric differences over time within the model's structure. To be more specific, each base model in the first step is now defined by a composite ROI. The measurement index $j$ starts from $j=0$, which denotes the baseline and yields $J_i$ observations for each subject i in the modeling process, while the distance‐based method uses $J_i-1$ observations, which reflect changes from the baseline. The predictor $D_{i,j}^{\left(k,l\right)}$ is replaced by $V_{\mathit{ij}}^{\left(k\right)}$, representing the volume of composite ROI $R_k$, for $j=0,1,\dots ,J_i$ and $i=1,\dots ,n$. As above, the covariates are age, sex, education, and APOE‐$\mathrm{ϵ}$4 carrier status. All other steps remain the same as those in the distance‐based method.

2.4.3. Model Validation

To demonstrate the advantages of leveraging distance measurements for quantifying brain atrophy and tracking ADRD development compared to direct brain volume assessment, we evaluate their respective performance in predicting future diagnostic status. This evaluation uses historical data and follows the two‐step model detailed in Section 2.4.2. To proceed, we use fivefold cross‐validation. Instead of directly splitting the data into five folds at the subject level, we made a group‐wise division, aligning with the two diagnostic groups defined earlier in this section, followed by combining these groups. In each iteration of the cross‐validation, we take one fold as a holdout. To achieve the objective of predicting an individual's future diagnostic status based on their historical data, we construct testing and training data in the following manner: The testing data comprised the last observations from individuals in this holdout set, representing the “future” state of each person that will be predicted. Our training data includes two parts: Training Set 1 and Training Set 2. The remaining four folds form Training Set 1. Training Set 2 consists of all earlier observations from the holdout set, excluding those in the test data. This data‐splitting strategy ensures that training and testing data include subjects with comparable disease progression. In addition, it also makes the prevalence of the latest diagnostic status in Training Set 1 similar to that in the test set. Further, we include Training Set 2 to enhance predictive accuracy for the testing subjects by learning from their individual‐specific disease progression trajectories. Within the training set, the parameters for each base model are estimated. Additionally, the aggregation schemes, specifically how each base model is integrated to produce a final estimation, are established. Then, this refined ensemble is applied to the testing set.

3. Results

Analyses included a total of 542 PAC participants, each with at least three MRI scans. As noted above, this included two groups based on each participant's diagnostic status at the time of their first MRI scan and the last scan (or the diagnosis prior to the first recorded dementia status): CU–CU ($n$ = 480) and CU–MCI ($n$ = 62). Baseline demographic information for each group is provided in Table 2. On average, participants were late middle‐aged at their baseline MRI scan, and 32% were APOE‐$\mathrm{ϵ}$4 carriers. The number of follow‐up scan visits ranged from 2 to 17, with an average of 4.06 and standard deviation of 2.92. The time span between the baseline and the last MRI scans was 10.1 years on average (range = 2.1–24.4 years).

TABLE 2.

Baseline demographics of groups categorized based on the diagnosis closest to the first and the last MRI scans.

	CN–CN	CN–MCI
Sample size, $n$	480	62
Age at baseline MRI scan	65.77 (9.78)	67.92 (11.27)
Female (%)	59%	53%
Z‐scores of education a	−0.06 (1.00)	0.00 (0.89)
APOE‐$\mathrm{ϵ}$4 carriers (%)	31%	44%
MMSE (baseline) b	29.19 (1.11)	29.15 (1.13)
Years between baseline MRI scan and last MRI scan c	9.73 (5.78)	12.50 (6.43)

Note: Values reflect mean (SD) unless otherwise indicated.

^a Z‐scores of education are calculated separately based on each site's “Years of education” variable; range = [–2.35, 2.23].

^b Mini‐Mental State Examination (MMSE) – Total Score.

^c Here, the last scan represents the last MRI scan used in the analysis, since only MRI data prior to the first recorded dementia status is included for those who progressed to dementia.

3.1. Distance vs. Volume

Figure 6 displays the receiver operating characteristic (ROC) curves for predicting follow‐up diagnostic status (remained CU vs. progressed to MCI) in the test set. The predictions are obtained by using two different kinds of predictors in the two‐step model, respectively: (1) volumetric change from the baseline calculated through distance measurements and (2) composite ROI volume. Across three different aggregation methods employed to combine diagnostic status predictions from individual base models, the models utilizing distance measurements consistently show a higher area under the ROC curve (AUC), with significant differences confirmed by the p values from bootstrap test (Carpenter and Bithell 2000): wisdom‐of‐crowds (p value $<0.001$), best‐of‐crowds (p value $=0.02$), and majority‐of‐crowds (p value $<0.001$). This demonstrates the superior performance of distance measurements in tracking brain atrophy in the context of ADRD, in comparison to directly analyzing regional brain volumes.

FIGURE 6.

ROC curves for CU vs. MCI prediction. From left to right: Results from aggregation rules wisdom‐of‐crowds, best‐of‐crowds, majority‐of‐crowds, respectively. The orange ROC curve refers to the model with volumetric change as a predictor, calculated through distance measurement. The blue ROC curve represents the model that directly uses volume as a predictor.

To further demonstrate the benefits of using distance measurements for quantifying brain volumetric changes over time, we report diagnostic predictions (i.e., AUC) from each base model, separately for composite ROIs calculated using the individual distance measures vs. models that use the composite ROI volume as a predictor. As shown in Tables 3 and 4, the models that used distance measures exhibit higher AUCs compared to the models using composite ROI volume directly. Notably, this advantage is consistent across different brain regions.

TABLE 4.

Base model assessment within parietal lobe, limbic system, and ventricle.

ROI	Base model	ROC AUC	Bootstrap CI	PR AUC	Bootstrap CI
Parietal lobe	Model volume directly	0.69	[0.61, 0.76]	0.25	[0.17, 0.35]
	Model volume change from baseline
	Angular	0.83	[0.77, 0.89]	0.53	[0.40, 0.68]
	Bhattacharyya (acos)	0.83	[0.77, 0.89]	0.59	[0.46, 0.74]
	Bhattacharyya (log)	0.83	[0.77, 0.90]	0.59	[0.46, 0.74]
	J‐divergence	0.87	[0.81, 0.91]	0.61	[0.46, 0.74]
	Matusita	0.85	[0.80, 0.91]	0.59	[0.45, 0.73]
	Aitchison	0.85	[0.78, 0.90]	0.58	[0.45, 0.73]
	City block	0.84	[0.79, 0.90]	0.57	[0.43, 0.71]
	Euclidean	0.83	[0.77, 0.89]	0.57	[0.43, 0.71]
	Minkowski	0.84	[0.78, 0.90]	0.57	[0.43, 0.71]
	Average among distance	0.84	[0.80, 0.91]	0.58	[0.46, 0.74]
Limbic system	Model volume directly	0.69	[0.62, 0.76]	0.26	[0.17, 0.36]
	Model volume change from baseline
	Angular	0.85	[0.79, 0.91]	0.58	[0.45, 0.73]
	Bhattacharyya (acos)	0.83	[0.77, 0.89]	0.59	[0.46, 0.74]
	Bhattacharyya (log)	0.83	[0.76, 0.89]	0.59	[0.46, 0.73]
	J‐divergence	0.85	[0.80, 0.91]	0.57	[0.44, 0.72]
	Matusita	0.86	[0.80, 0.91]	0.58	[0.45, 0.72]
	Aitchison	0.83	[0.77, 0.89]	0.54	[0.41, 0.70]
	City block	0.86	[0.81, 0.91]	0.58	[0.45, 0.73]
	Euclidean	0.86	[0.81, 0.91]	0.58	[0.45, 0.73]
	Minkowski	0.86	[0.81, 0.92]	0.59	[0.46, 0.73]
	Average among distance	0.85	[0.80, 0.91]	0.58	[0.47, 0.75]
Ventricle	Model volume directly	0.69	[0.62, 0.76]	0.22	[0.15, 0.32]
	Model volume change from baseline
	Angular	0.83	[0.77, 0.88]	0.46	[0.34, 0.59]
	Bhattacharyya (acos)	0.84	[0.78, 0.90]	0.60	[0.47, 0.75]
	Bhattacharyya (log)	0.83	[0.77, 0.90]	0.59	[0.46, 0.74]
	J‐divergence	0.84	[0.78, 0.90]	0.58	[0.44, 0.71]
	Matusita	0.84	[0.78, 0.90]	0.58	[0.44, 0.71]
	Aitchison	0.82	[0.76, 0.88]	0.41	[0.30, 0.54]
	City block	0.83	[0.77, 0.89]	0.54	[0.41, 0.69]
	Euclidean	0.82	[0.76, 0.88]	0.54	[0.40, 0.68]
	Minkowski	0.82	[0.76, 0.88]	0.54	[0.40, 0.68]
	Average among distance	0.83	[0.78, 0.90]	0.54	[0.44, 0.72]

Note: Assessment of base models via area under the ROC curve (ROC AUC) and area under the precision‐recall curve (PR AUC), as well as their corresponding bootstrap 95% confidence intervals across nine distance measurements, and six regions: Temporal, occipital, frontal and parietal lobes, limbic system and ventricle. Within each region, “Model volume directly” refers to the model that directly uses volume as a predictor. Under the heading “Model volume change from baseline,” the results are from models that use volumetric change from the baseline as a predictor, and this change is quantified by various distance measurements, with their names displayed in the second column; “Average among distance” represents the average AUC across all models that use distance measures. The models estimated to positively contribute the prediction of diagnostic status via the wisdom‐of‐crowds are highlighted in bold text.

Considering the imbalance between the CU and MCI groups, we conducted additional precision–recall (PR) curve analyses to validate the robustness of our method. The results, presented in Figure 7 and Table 3, demonstrate that our distance‐based approach achieves the highest AUC under the PR curve, with a PR‐AUC of approximately 0.58 across different aggregation methods. Given the positive rate of 11.4% for the CU‐MCI group in our study, a PR‐AUC of 0.58 indicates that the distance‐based model effectively distinguishes CU‐MCI cases from CU‐CU cases, significantly outperforming a random classifier. Furthermore, it represents a significant improvement over the PR‐AUC obtained from volume‐based methods (all p values < 0.001). These findings consistently support the conclusion that distance measurements offer a more effective approach to quantifying brain volumetric changes.

FIGURE 7.

Precision–recall curves for CU vs. MCI prediction. From left to right: Results from aggregation rules wisdom‐of‐crowds, best‐of‐crowds, majority‐of‐crowds, respectively. The orange precision–recall curve refers to the model with volumetric change as a predictor, calculated through distance measurement. The blue precision–recall curve represents the model that directly uses volume as a predictor.

We also compared our method with a benchmark where the prediction is the last available diagnosis, as described by Marinescu et al. (2020). This “no change” benchmark assumes that a patient's future diagnosis will be the same as their most recent diagnosis recorded in the data. The benchmark achieves an ROC‐AUC of 0.56 (bootstrap CI: [0.52, 0.61]) and a PR‐AUC of 0.23 (bootstrap CI: [0.14, 0.33]). Our distance‐based method demonstrates significantly superior performance compared to this “no change” benchmark across all three aggregation methods. For ROC‐AUC, the differences between the “no change” prediction and the distance‐based method are statistically significant, with p values of 0.02 for WOC and 0.04 for both BOC and MOC. Similarly, for PR‐AUC, the distance‐based method outperforms the “no change” benchmark with p values < 0.001 across all aggregation methods. These results highlight the ability of the proposed distance‐based approach to effectively use patient history to predict future diagnoses.

Moreover, to evaluate our model without using Training Set 2 during model development, we conducted additional experiments in Supporting Information S1: Section A, where we completely held out Training Set 2 for testing and trained the models exclusively on Training Set 1. The results confirm that the distance‐based model, particularly with the “WOC” aggregation, maintains its predictive advantage.

3.2. Evaluating Random Effect and Covariate Effect in the Distance‐Based Two‐Step Model

This section evaluates the random effect and covariate effect within the two‐step model, focusing on distance‐based measurements. A key aspect of modeling the longitudinal diagnostic status is the consideration of individual's prior trajectory, characterized by the subject‐specific random intercept in model (1). To illustrate the importance of including random effects, we compared the prediction performance (i.e., AUCs) described above to those using models with an average random effect, which is zero by assumption. For these latter models, all steps were identical to the examination in Section 3.1, with the exception that $u_i^{\left(k,l\right)}=0$ in Equation (2) is set to zero when predicting the test set. As anticipated, the ROC curves and AUCs (orange curves) in Figure 6, reflecting subject‐specific random effects, outperform the ROC and AUCs derived from average random effects (red curves in Figure 8). The differences are significant based on p values < 0.001 across all three aggregation methods. This comparison suggests evaluating an individual's current state in isolation may not yield as comprehensive information as considering each individual's prior trajectory when predicting diagnoses associated with ADRD.

FIGURE 8.

ROC curves for CU vs. MCI prediction. From left to right: Results from aggregation rules wisdom‐of‐crowds, best‐of‐crowds, majority‐of‐crowds, respectively. The red ROC curve refers to the model that used volumetric change as a predictor and average random effect. The green ROC curve represents the model that used volumetric change as a predictor and excluded the covariates age, sex, education, and APOE‐$\mathrm{ϵ}$4 carrier status.

To additionally assess covariate effects, we re‐ran the models with the primary covariates (i.e., age, sex, education, APOE‐$\mathrm{ϵ}$4 carrier status) omitted, but all other aspects of the two‐step modeling procedure unchanged. The corresponding ROC curves (green curves) and AUCs are presented in Figure 8. While these ROC curves and AUCs outperform the results from the model with confounders but no random effects (red curves, Figure 8), they are similar (no statistically significant differences, p value $>0.05$ across all three aggregation methods) to the ROC curves and AUCs from the models that include both random effects and confounders (orange curves, Figure 6). This again highlights the predictive strength of prior trajectory and further suggests that the predictive information provided by confounding covariates might be captured by the prior trajectory. This conclusion is further supported by the PR analyses presented in Figure 9.

FIGURE 9.

Precision–recall curves for CU vs. MCI prediction. From left to right: Results from aggregation rules wisdom‐of‐crowds, best‐of‐crowds, majority‐of‐crowds, respectively. The red precision–recall curve refers to the model that used volumetric change as a predictor and average random effect. The green precision–recall curve represents the model that used volumetric change as a predictor and excluded the covariates age, sex, education, and APOE‐$\mathrm{ϵ}$4 carrier status.

To further demonstrate the robustness of the proposed distance‐based method, we conduct additional analyses on unharmonized data, perform subgroup analyses (female vs. male), and examine the impact of different subregion sizes, as detailed in Supporting Information S1: Sections B, C, and D.

4. Discussion

This study evaluated whether distance measurements, applied to longitudinal MRI data, can better quantify changes in brain volumes over time than traditional approaches to calculating volume change over time in the early phases of ADRD. To our knowledge, distance measures have not been evaluated within the context of MRI atrophy and ADRD. Our findings indicate that using distance measurements to quantify changes in brain volumes is a superior method for assessing brain atrophy as a predictor of progression from CU to MCI. Importantly, these measures of volumetric change, when employed as a predictor, were more effective in predicting ADRD progression than the conventional approach of direct volume analysis. Our results also suggest that an individual's prior trajectory outweighs their current state in predicting diagnostic status, and that the trajectory‐based model might account for individual differences typically controlled for by including model covariates such as age, sex, education, and APOE carrier status. Additionally, analyses (see Supporting Information S1: Section A) conducted without using Training Set 2 in model development emphasize the importance of accounting for individual‐level variation and incorporating historical data when predicting future outcomes. Together, these findings highlight the importance of historical data in understanding the progression of ADRD and the heterogeneous nature of its trajectory. Furthermore, considering the complex nature of brain structural changes over time, distance measures can serve as a dimension reduction tool to simultaneously summarize changes in multiple brain regions beyond those known to be strongly associated with the early AD process.

4.1. Implications From Base Models

In the proposed two‐step procedure, estimations from the base models were combined into one final diagnosis using three aggregation rules: best‐of‐crowds, majority‐of‐crowds, and wisdom‐of‐crowds. The wisdom‐of‐crowds, in particular, assigns a contribution score to each base model, selecting only those with a positive impact for the construction of final estimations, while excluding the rest. The individual contribution score is basically calculated through the difference of prediction performance with and without this base model, which is designed to mitigate the bias toward the prevalent membership as CU in the PAC study cohort. On the other hand, the best‐of‐crowds and majority‐of‐crowds methods are particularly susceptible to encountering this bias issue. However, our focus is not to select the best way of aggregating diagnosis prediction from individual models. The consistency of results across these three aggregation rules, as discussed in Section 3, supports our conclusion that applying distance measures could potentially better characterize brain atrophy.

With respect to the base model results obtained from using volumetric change as a predictor, a few specific distance measures stand out as most accurately predicting CU/MCI diagnoses. These are highlighted in bold in Table 3, specifically because they have positive contribution scores under the “wisdom‐of‐crowds” rule, including distance measures from the limbic system, temporal lobe, occipital lobe, and ventricles. Given that AD is likely to be a primary or contributing factor for clinical progression for most individuals within the PAC data set, these regions align with prior AD research findings, especially concerning the early stages of the disease. For example, medial temporal lobe regions such as the hippocampus, entorhinal cortex, and amygdala are part of the limbic system and are among the earliest regions to be affected in AD (Jack et al. 1997; Miller et al. 2013; Sperling et al. 2014). Furthermore, the temporal lobe's critical role in memory processing and language closely matches early AD symptoms, such as difficulties recalling recent events and impairments in category fluency (e.g., Fabrigoule et al. 1998; Amieva et al. 2005). In parallel, the loss of brain tissue volume as brain degeneration progresses often triggers ventricular enlargement, making it an indirect marker of atrophy. Additionally, although it is less common, there are instances where early AD might affect the occipital lobe, leading to visual disturbances as initial symptoms. Some atypical forms of AD, such as the posterior cortical atrophy (PCA) variant (Crutch et al. 2012), primarily affect the occipital lobe early in the disease course. The involvement of the occipital lobe in predicting diagnostic status may reflect the fact that AD progression can vary significantly among individuals. Furthermore, the exclusion of the frontal lobe and parietal lobe from the contributing base models can be attributed to the less pronounced volumetric changes observed in these regions during the early phases of AD.

The specific distance measures that have shown positive contributions vary among brain regions. For instance, Bhattacharyya‐based measures were selected for the temporal lobe and ventricles but not for the limbic system. This could be because the information is adequately represented by other regions when combined with various distance measurements. Moreover, the angular distance uniquely showed a positive contribution in the temporal lobe, hinting at the presence of subtle volumetric changes within certain subregions of this area. This sensitivity could be due to the fact that even minor alterations in the volume array—a collection of subregional volumes within a particular region—can shift its spatial position in Euclidean space, a change that angular distance is capable of detecting. It is important to note that not every distance measure applied to regions with known early changes is automatically included in the final prediction model. The aggregation process evaluates each distance measure conditionally, determining whether it provides additional predictive information beyond what is already contributed by other distance metrics. The exclusion of certain region‐distance pairs is likely due to their predictive value already being captured by other combinations.

4.2. Implications of ROC‐AUC Findings

The ROC curves in Figure 6 primarily illustrate the advantages of using distance measurements for delineating brain atrophy and tracking ADRD progression, over the direct analysis of brain volumes. Figure 8 additionally underscores the critical role of an individual's clinical trajectory in establishing diagnostic status. However, the large ROC‐AUC values presented there warrant caution in interpretation, given the majority of participants in the PAC data set were cognitively normal at their last available MRI scan. This represents a significant imbalance in diagnosis groups that might have inflated the ROC‐AUCs, because the high ROC‐AUCs might be partly attributed to model correctly identifying the larger group (rather than the smaller number of individuals in the progression groups). We therefore conduct additional PR analyses to further support our study's findings. A more comprehensive evaluation of the model's performance in identifying the minority class is provided in a later section.

4.3. Train/Test Data Construction

To compare the predictive performance between using volumetric changes, characterized by distance measurements, and using the volume itself, we evaluated the two‐step model. This was done by selecting the last observation of each individual in the held‐out set as the “future” state to assess how accurately each method predicts an individual's future diagnostic status based on their historical data. The rest of the data was used as the training set. Although it is possible to take every visit after the second one as a “future” state, this approach might not provide a sufficiently detailed historical context for accurate prediction. Our decision to use the last observation as the “future” state is driven by the need to balance having cumulative information gathered over time and accurately predicting upcoming health conditions.

4.4. Further Discussion of Distance Measurements

In our context, when calculating the distance between two volume arrays, each element of the array represents a proportion. This proportion is derived from dividing the original volume by the ICV in order to adjust for head size. Hence, the input to the distance calculation is a set of proportions, each ranging between 0 and 1. Among the various distance measurements under consideration, the angular distance, defined as the angle between two vectors (two sequences of numbers) in Euclidean space, inherently normalizes these arrays to a unit norm as part of its calculation, a requirement stemming from the domain constraints of the arccos function. In contrast, other distance measurements do not necessitate this normalization to a unit norm.

Unlike the other distances presented in Table 1, the Bhattacharyya measurements are not a strict mathematical distance metric because they do not satisfy the triangle inequality when the volume arrays are not unit‐normed. Despite this, Bhattacharyya measures are still effective for tracking volumetric changes in our context. We focus on the application of Bhattacharyya measurements to the ventricle and frontal lobe for better illustration. In our analysis, the ventricle has $p=6$ subregions, and the volume in each subregion increases over time but not uniformly across subregions. Let ${\mathit{x}}_0\in {\mathrm{ℝ}}^p$ be the baseline volumes of $p$ subregions. Then the dot product $⟨{\mathit{x}}_0,{\mathit{x}}_j⟩$ increases over time for $j\ge 1$, where ${\mathit{x}}_j$ denotes volume measures from later scans. Using the fact that the functions $t\mapsto \arccos \left(t\right)$ and $t\mapsto -\log \left(t\right)$ are decreasing for $t\in \left[0,1\right]$, then by definition, the Bhattacharyya measurements exhibit a decreasing trend over time, as illustrated in the first row of Supporting Information S1: Figure G17. In contrast, the second row in Supporting Information S1: Figure G17 suggest that volumetric changes from the baseline within the frontal lobe increase over time. Different from the ventricle, there are totally $p=46$ subregions in the frontal lobe, and there is not a consistent, universal pattern of volume change in each subregion. Generally, the dot product $⟨{\mathit{x}}_0,{\mathit{x}}_j⟩$ decreases over time, leading to the upward trend observed after applying the decreasing functions $-\log$ or $\arccos$.

Although Bhattacharyya measurements may sometimes contradict the intuition that distances should increase when two objects become further apart, they can still offer meaningful insights. For example, if $\mathit{y}=c^2\mathit{x}$ with $c>0$, the angle between $\sqrt{\mathit{y}}$ and $\sqrt{\mathit{x}}$ is 0, yet Bhattacharyya measurements yield a positive value. Regarding the feasibility of arccos transformation, since volumes are adjusted for ICV and thus always proportions, it holds that $⟨\sqrt{\mathit{x}},\sqrt{\mathit{y}}⟩\le \sqrt{\parallel \mathit{x}{\parallel }_1}\sqrt{\parallel \mathit{y}{\parallel }_1}\in \left(0,1\right)$, making it within the domain of the $\arccos$ function.

4.5. Limitations and Future Directions

This study has limitations. First, it is important to acknowledge that this framework might not consistently detect transitions between diagnostic status due to the inherent challenges of this classification task. Take the results from the distance‐based model along with the wisdom‐of‐crowds aggregation rule as an example. We used disease prevalence estimated from the training data to set a threshold for converting the probability of diagnosis into a definitive diagnosis of CU or MCI. Among the misdiagnosed cases, 27 out of 44 participants experienced a change in diagnoses between the second to last and the last visit, either from CU to MCI ($n$ = 25) or from MCI to CU ($n$ = 2). Regarding the misdiagnosis of individuals reverting from MCI to CU, an explanation could be that these individuals had cognitive impairment due to factors not related to a neurodegenerative process, like AD, but may have experienced an illness or depression that was the primary cause of their cognitive impairment and this condition resolved over time. A possible reason for the misclassification of individuals who progressed to MCI but were CU at their second last MRI scan could be the insufficient number of follow‐up scans, which is important for tracking disease progression. This issue is prevalent in the PAC data set, as illustrated in Supporting Information S1: Figure H18, which shows the irregular timing of MRI scans. Second, it is important to note that information about diagnostic etiology was not available in the current data. We believe that the majority of individuals who progressed to MCI or dementia have AD as a primary or contributing factor. This assumption is based on of the fact that AD is the most common cause of dementia among older persons and the fact that several PAC sites were enriched for a family of history of AD. However, without biomarker confirmation of AD‐specific amyloid and/or tau pathology, we cannot rule out the possibility that some of the observed brain changes and clinical progression are attributable to causes other than AD. Due to limited availability of amyloid PET data, our study's primary focus remains on neurodegeneration more broadly; therefore, any inferences specific to AD should be interpreted with caution. In future studies, it would be valuable to explore how these distance‐based volumetric changes correlate with established AD biomarkers, such as amyloid or tau. Such investigations may further help clarify whether the observed volumetric changes are specific to AD or reflect other neurodegenerative processes across ADRD. This would allow for a more precise understanding of progression patterns specific to each type of dementia. Third, our method requires participants to have at least three MRI scans. This is because we measure changes from baseline, and at least three scans are needed to generate two observations for model fitting. This requirement may be a consideration for other studies with fewer available time points.

Additionally, the proposed distance‐based method provides an alternative indicator of brain atrophy that may capture changes more effectively than conventional volumetric measures. Although this method holds potential for improving our understanding of the progression of atrophy during preclinical disease stages, its current application is best suited for research settings. Future efforts could focus on validating this approach for individual‐level predictions in the clinical context by integrating it with larger data sets and complementary clinical information.

Several factors within this study may affect the generalizability of our findings. The participant pool predominantly consisted of well‐educated, non‐Hispanic Whites. As such, the extent to which these findings can be extrapolated to other socioeconomic statuses, racial, and ethnic groups remains uncertain. Future studies may need to include a more diverse demographic to validate the applicability of these results across different populations. Moreover, the method used to normalize brain volumes in our study was the “ratio” method, wherein each regional brain volume was divided by the ICV. While this is a common practice, another popular approach is the “residual” method, which calculates residuals from a linear regression of the volume of interest on the ICV (O'Brien et al. 2011; Hansen et al. 2015). Furthermore, relatively straightforward linear mixed‐effect models were used in our analyses. Both the choice of models and the normalization method were consistent, whether applying distance measures or direct volumetric approaches, arguably reducing systematic discrepancies in comparisons. Nevertheless, to enhance the robustness of our results, future investigations might consider exploring alternative modeling techniques and adjustment methods. In addition, we chose the PAC data set because it combines data from five different cohort studies, thereby increasing the sample size and allowing us to examine brain volumetric change during the preclinical phase of AD with improved statistical power. Analyzing additional data sets represents an important next step to further validate our approach and assess its generalizability.

Investigating the use of distance measures to assess brain volumetric changes between successive visits could offer further insights into the short‐term dynamics of atrophy progression. Although not included in this study, we acknowledge the potential of this approach and intend to explore it in our upcoming work. While the current study provides valuable insights into the use of distance measurements for assessing brain atrophy, there remain great opportunities for future research to develop more sophisticated models. One promising avenue involves combining MRI data with other biomarkers, such as Fluorodeoxyglucose Positron Emission Tomography (FDG‐PET) and CSF data, in the context of distance measurements. For example, future studies could merge amyloid‐beta (A $\beta$) and tau data with the regional volume arrays, followed by the application of distance measurements to quantify changes. Secondly, in modeling diagnostic status, one can use both volumetric changes quantified by distances and these biomarkers as predictive factors. This multi‐modal approach might provide a better understanding of ADRD progression. Moreover, applying the distance method to examine the conversion from CU to MCI in individuals with confirmed amyloid pathology would provide a more specific evaluation of distance‐based MRI predictors in the context of AD, rather than across the broader range of ADRD. In line with this goal, we plan to incorporate more comprehensive molecular biomarker data in future studies in order to further evaluate the utility of distance‐based measures during the early phases of AD.

Our method focuses on identifying the (distance, composite ROI) pairs that are most predictive for diagnosis. A potential refinement would be to extend this framework to find specific subregions within each composite ROI that exhibit more pronounced changes. This would provide a more detailed understanding of region‐specific trajectories. As another future direction, incorporating the boundary shift interval (BSI) as a complementary approach for volumetric measurement, when data set conditions allow, could provide additional insights into brain atrophy measurement and strengthen the overall analytic framework.

In our current research, we adopted a two‐step procedure to manage the large number of predictors resulting from various combinations of distance measures and brain regions, it would be interesting to leverage state‐of‐the‐art AI models for a more detailed examination. Moreover, utilizing distance measurements on MRI‐derived $\left(X,Y,Z\right)$ coordinates, which define the location of voxels in the brain, enables detailed tracking of spatial changes in brain structures. Further, the use of distance measurements is not restricted to ADRD research, and it holds potential for broader application across various contexts.

5. Conclusion

This work presents a novel approach to characterizing brain atrophy by applying distance measurements to longitudinal volumetric MRI data within the context of the preclinical stages of ADRD. Distance measurements effectively captured the multidimensional nature of brain atrophy. Moreover, volumetric changes quantified by these measures are more sensitive indicators of atrophy in early disease stages compared to direct volume analysis. Plus, using distance measurements facilitates a form of dimension reduction, particularly advantageous when dealing with volumetric data from numerous regions. Furthermore, examining differences in predictions with and without accounting for prior trajectory reveals significant heterogeneity in the progression of ADRD among individuals, which underscores the complexity of the disease and the need for personalized treatment and monitoring. On top of that, our findings also suggest that the prior trajectory may be a more significant determinant of diagnostic status associated with ADRD than the current state alone or confounding factors such as age, sex, education, and APOE genotype. Together, these data suggest that different distance measurements shed light on the multifaceted nature of volumetric changes in the same brain region and highlight the variation in brain atrophy among different regions.

Conflicts of Interest

The authors declare no conflicts of interest.

Supporting information

Data S1. Supporting Information.

Data Availability Statement

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available due to privacy or ethical restrictions.