Characterizing heterogeneity in Alzheimer’s disease progression: a semiparametric model | Scientific Reports

Scientific Reports volume 15, Article number: 7660 (2025) Cite this article

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The progression of Alzheimer’s disease (AD), a leading cause of dementia worldwide, is known for its variability and complexity, challenging the conventional methods of monitoring and predicting disease trajectories. This study introduces a semiparametric modeling approach to analyze longitudinal cognitive and imaging data. We studied two different outcome variables from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database: the Alzheimer’s Disease Assessment Scale-Cognitive Subscale 13 (ADAS13) scores and ventricular volumes \({(\text{mm}}^{3})\). Unlike traditional linear mixed effects models, semiparametric models do not assume a linear AD progression over time. Semiparametric models offer the advantage of capturing the non-linear features of AD progression, such as cognitive decline and neurodegeneration, represented by changes in ADAS13 scores and ventricular enlargement, respectively. By integrating regression splines and mixed modeling techniques, we provide a nuanced understanding of AD progression that captures the heterogeneity of disease trajectories. Our analysis reveals variations in the timing and degree of cognitive decline and neurodegeneration among AD patients, underlining the need for personalized approaches for monitoring and managing AD. This study’s findings contribute to the modeling of AD progression and offer potential implications for interventions and prognostic assessments in clinical and research settings.

Alzheimer’s disease (AD), a leading cause of dementia, presents a significant global health challenge, affecting millions of individuals and their families1. It is characterized by progression of cognitive decline and neurodegeneration, leading to severe impairment in daily functioning and independence2. Among the tools available to assess the progression of AD, the Alzheimer’s Disease Assessment Scale-Cognitive Subscale 13 (ADAS13) and measurements of Ventricular Volume have emerged as pivotal indicators3,4,5,6,7. The ADAS13, an enhanced measure of cognitive performance, extends beyond traditional cognitive assessments to include a wider array of cognitive domains, thereby providing a more comprehensive evaluation of the cognitive deficits in AD5. It plays a crucial role in the clinical assessment of individuals with AD, serving as a benchmark for the severity and progression of cognitive impairments5. Parallel to cognitive assessments, neuroimaging advancements have highlighted the importance of Ventricular Volume as a biomarker of AD progression. Enlargement of the ventricular volumes indicates brain atrophy, a hallmark of Alzheimer’s pathology, offering a window into the structural changes occurring in the brain8. The quantification of Ventricular Volume over time thus provides invaluable insights into the neurodegenerative process, complementing cognitive measures like the ADAS139. Over time, both outcome variables may change rapidly or exhibit increased variability due to various factors (Fig. 1). Despite the utility of ADAS13 and Ventricular Volume measurements in monitoring AD progression, the complex relationship between cognitive decline and neuroanatomical changes is not fully understood10,11. Variations in the rate and pattern of progression among individuals highlight the need for advanced analytical methods capable of capturing and characterizing the heterogeneous nature of AD progression.

Spaghetti plots show ADAS13 and Ventricular Volume \(({mm}^{3})\) trajectories over time (age in years) and time since baseline observation (in years) in the top and bottom row, respectively. These plots show the heterogeneity and non-linearity in both outcome variables. The black lines represent observed outcome variables for randomly selected 20 patients, while the gray dots represent the observed outcome variables for all subjects in our study cohorts. The blue lines show the average trend of ADAS13 and Ventricular Volume scores as Age and Time since baseline increase. More specifically, (a) presents trajectories of ADAS13 scores against patients’ age, and (b) presents trajectories of the Ventricular Volume versus patients’ age. The multitude of gray points in these upper panels reflects the individual data points across the patient cohort, illustrating the broad spectrum of disease manifestations with advancing age. (c) demonstrates the progression of ADAS13 scores over time since baseline, and (d) similarly shows the progression of Ventricular Volume from the initial baseline measurement.

Conventional statistical analyses have primarily explored the links between cognitive decline, yielding significant insights. However, the application of advanced statistical models that can capture the complex dynamics of AD progression has been limited. Techniques such as ordinal regression and progression models for repeated measures have demonstrated potential in refining our understanding of AD by modeling cerebral atrophy and treatment effects over time with greater nuance12,13. Additionally, tensor-based methods for integrating multi-modality data have emerged, offering improved diagnostic accuracy and biomarker identification through the analysis of complex datasets14. A broader literature review also highlights the evolving use of sophisticated statistical approaches to better characterize AD progression. This indicates a shift towards more dynamic and precise modeling techniques in epidemiological studies15. Although such parametric models effectively identify broad changes in AD progression, they do not adequately capture the individual trajectories of ADAS13 and Ventricular Volume16,17. Recently, joint models have become instrumental in predicting AD progression using one or more longitudinal biomarkers linked to time-to-event data18,19,20. While joint models are valuable for predicting AD progression, they come with several disadvantages, such as complexity in implementation and computational intensity.

We applied various semiparametric mixed effects models as alternatives to traditional random intercept and slope mixed effects models widely used for estimating and predicting AD disease progression by utilizing various outcome variables that enable monitoring disease progression. Semiparametric models utilize penalized regression splines and mixed effects modeling to achieve a more flexible mean response structure to analyze longitudinal data with non-linear trajectories21,22. This approach allows for obtaining smooth estimates of non-linear mean functions and derivatives of the models while avoiding overfitting. This work aims to build a flexible and accurate semiparametric mixed effects model and compare it with other alternative mixed effects models to show. Its improved performance in predicting AD progression. This resulted in constructing a robust and precise model for predicting AD progression and rate of change using widely available measures such as ADAS13 score and Ventricular Volume.

We used ADNIMERGE data from the Alzheimer’s Disease Neuroimaging Initiative (ADNI) database, which is publicly available at https://adni.loni.usc.edu/. The ADNI database comprises a vast collection of clinical/demographic characteristics of subjects along with magnetic resonance imaging (MRI) data from patients with different levels of cognitive impairment. While others have explored other outcome variables to assess AD progression, our study focused on two primary outcome variables: ADAS13 and ventricular volume. We applied our models separately and independently to each of these two outcomes. The team at the University of California, San Francisco (UCSF) utilized FreeSurfer (http://surfer.nmr.mgh.harvard.edu/) to process the ADNI database, extracting features such as volumetric segmentations and cortical reconstructions. AD causes a gradual decline in cognitive function, tracked by increased ADAS13 and other measures23. In addition, Ventricular Volume enlargement is a hallmark of AD. Hence, higher values of ADAS13 and Ventricular Volume indicate more severe AD. The spaghetti plots in Fig. 1 illustrate the variability in ADAS13 and Ventricular Volume trajectories over time for our study cohorts.

In each panel of Fig. 1, twenty patients’ trajectories have been prominently marked with black lines, selected randomly to present the heterogeneity within the cohort. These black lines serve as a narrative, weaving through the gray data points to highlight that while some paths follow a more aggressive trajectory, others suggest a slower pace of progression, reinforcing the understanding that AD affects patients differently. This distinction in the progression patterns emphasizes AD progression’s non-linearity, complexity, and heterogeneity and reflects the underlying biological variability during the disease. The blue lines loess curves across the four figures provide insights into the average trends of ADAS13 scores and Ventricular Volumes over time. They are crucial for understanding aging and cognitive health.

For the semiparametric analysis, we used the predictors from the ADNI database and identified from previous research2. These predictors included age, Mini-Mental State Examination (MMSE) scores, gender, changes in diagnosis, the presence of the APOE4 gene, Functional Activities Questionnaire (FAQ) scores, Rey Auditory Verbal Learning Test immediate recall (RAVLT-immediate), Rey Auditory Verbal Learning Test learning scores (RAVLT-learning), baseline diagnosis status, and the patient’s education level. Table 1 provides our subjects’ demographic and clinical characteristics for each outcome variable of interest. This table encompasses a range of baseline diagnoses, including AD, Cognitive Normal (CN), and various stages of Mild cognitive impairment (MCI), alongside educational backgrounds, gender distribution, and cognitive assessments. The left panel of the table presents the characteristics of the cohort used to analyze ADAS13, while the right panel presents the characteristics of ventricular volume.

Traditional linear mixed effects (LME) models have been commonly used for predicting AD progression by utilizing ADAS13 or other continuous outcome variables24,25,26. These models allow for the inclusion of fixed effects for population-level estimates and random effects to estimate how each patient varies from the population27. In our analysis, we employed cubic truncated power basis penalized regression splines22,28 to accurately capture the non-linearity in longitudinal trajectories of ADAS13 and Ventricular volume. The model was previously extensively discussed elsewhere for predicting rapid lung function decline for a chronic lung disease22. By integrating fixed effects terms with these penalized splines, we formulated a continuous function capable of estimating the trends of ADAS13 and Ventricular volume over time, denoted in age (in years). Allowing subject-specific random effects enables characterizing and accounting for the between and within-subject variability in the outcome of interest.

In our study, we employed parametric linear mixed models that can be described by the following formulation:

Here, the summation over q (from 0 to K) reflects the use of polynomial terms in the regression, with our analyses considering K = 1, 2 3, to capture linear, quadratic, and cubic trends, respectively. In this model, we did not incorporate penalized regression splines, so the population-level mean is modeled as a linear function of time. Thus, the overall longitudinal trend for outcomes such as ADAS13 scores and Ventricular Volume is given by:

We compared these conventional linear mixed models with the proposed semiparametric model based on fit statistics. To account for stochastic variability around the mean response, the model includes three components:

Random Intercept (\({\beta }_{0i}\)): Each patient i has a unique intercept \({\beta }_{0i}\), which captures individual differences in the overall trajectory of ADAS13 and Ventricular Volume. This random intercept is assumed to follow a normal distribution with mean \(0\) and variance \({\upsigma }_{{\upbeta }_{0}}^{2}\), denoted as: \({\beta }_{0\text{i}} \sim \text{ N}(0,{\upsigma }_{{\upbeta }_{0}}^{2})\)

Within-Patient Variability \({\upvarepsilon }_{\text{i}}\left({\text{t}}_{\text{ij}}\right)\): The term \({\upvarepsilon }_{\text{ij}}\left({\text{t}}_{\text{ij}}\right)\) represents the within-patient variation, and decomposed in two parts which is comprised of an exponential correlation function and measurement error, expressed as \({\upvarepsilon }_{\text{ij}}\left({\text{t}}_{\text{ij}}\right)= {\upeta }_{\text{i}}\left({t}_{ij}\right)+ {\varphi }_{ij}\); where \({t}_{ij}\) is the time (measured as age, in years) of the \({j}^{th}\) encounter for the \({i}^{th}\) patient \((i=1,\dots I;j=1,\dots ,{n}_{i})\).

Correlated Component (\({\upeta }_{\text{i}}\left({t}_{ij}\right)\)): This component is modeled as a stationary Gaussian process with mean 0 and variance \({\upsigma }_{{{\upvarphi }}_{0}}^{2}\). It exhibits an exponential correlation function \(\lambda ({t}_{ij})\). In the longitudinal model, this correlation function, \(Corr\left({\eta }_{i}\left({t}_{0}\right), {\eta }_{i}(t-{t}_{0})\right)= \text{exp}\left\{-|t|/\varsigma \right\},\) where \(\varsigma = \text{exp}\left\{-|t|/\varsigma \right\}\) describes the rate of decay for the correlation function for time between observations of \(|t|\). This structure assumes that observations farther apart in time are less correlated. The joint distribution of these values is multivariate normal, denoted as, \({\updelta }_{\text{i}}\left({\text{t}}_{\text{ij}}\right)\sim MVN(0,\Sigma )\); Where mean 0 and variance–covariance matrix \(\Sigma\).

Measurement Error: \({\varphi }_{ij}\) is the “white noise” component, assumed to be normally distributes, \({\varphi }_{ij}\sim N(0, {\upsigma }_{\upomega }^{2})\).

Covariate Effects: The second summation, \(\sum_{p=1}^{R}{\theta }_{p}{x}_{ijp}\) accounts for the effects of additional covariates on the outcome.

For a more flexible approach, we then applied semiparametric mixed effects models with cubic truncated power basis penalized regression splines22. This method allows for smooth estimation of the longitudinal trajectories of ADAS13 and Ventricular Volume over time. In this framework, the population-level mean response is modeled by a continuous function. Then, our main semiparametric model has the following form,

where the coefficients.

\({\theta }_{0}\), \({\theta }_{1}\), \({\theta }_{2}\), and \(\theta\) correspond to the standard terms for intercept, linear, quadratic, and cubic effects, respectively.

The sum \({\sum }_{a=1}^{A}{b}_{a}{\left(t-{\delta }_{a}\right)}_{+}^{3}\) comprises the penalized spline basis functions, are the coefficients and \({\delta }_{a}\) (for a = 1,…,A) are the chosen knot locations, selected based on model fit criteria.

The smooth functions, denoted as \(f\left(t\right)\), is incorporated into a linear mixed effects model framework. The integration yields a semiparametric mixed effects model that combines both fixed and random effects to capture the complexity of the data structure.

Here, \({Y}_{ij}\) is the outcome of interest (the ADAS13 or Ventricular Volume) for the \({i}^{th}\) patient at time point \({t}_{ij}\), where \(\left(i=1,\dots I;j=1,\dots ,{n}_{i}\right)\). The term \({\varepsilon }_{i}\left({t}_{ij}\right)\) captures the residual variability within patients, modeled using an exponential correlation function along with a measurement error component. This model rely on penalized regression splines, contained in the term \(f({t}_{ij})\), to characterize ADAS13 and Ventricular Volume over time. Additional information regarding the full model formulation is available in the Supplementary Material.

In our work, we employed parametric and semi-parametric LME models from22 to model longitudinal ADAS13 and Ventricular Volume trajectories and estimate rates of change. Age (in years) was used as the time variable in each model, with rates of change reported per year alongside 95% confidence intervals (CI). Coefficients and their CIs were estimated using the nlme package in R, which employs maximum likelihood (ML) estimation. For fixed effects, CIs were derived from standard errors based on the inverse of the Fisher information matrix, assuming the asymptotic normality of the ML estimates. In contrast, random effects were estimated by maximizing the marginal likelihood of the model parameters.

All analyses were conducted using R (version 4.1.3)29 and the following packages: nlme (version 3.1-163)30, splines (version 4.3.2)31, and ggplot2 (version 3.5.1)32. Further details on the additional models, software implementation, and the subset datasets from the ADNI database used in this work are provided in the online supplement.

We estimated overall evolution in ADAS13, ventricular volume, and the rate of change (slope/ or first derivative with respect to time (age) variable) for each model’s mean response function, holding all other predictors fixed. We compared all the employed models by using model fit statistics such as Akaike and Bayesian Information Criteria (AIC and BIC, respectively), and prediction accuracy metrics such as root mean squared error (RMSE) and mean absolute error (MAE). To evaluate model performance and comparison, we conducted a 20-fold cross-validation analysis. Additionally, a conventional linear mixed effects (LME) model was fitted for comparison (as shown in Tables 2 and 3).

Our analysis cohorts, derived from the ADNI database, encompassed 1633 participants for the ADAS13 scores and 1,499 participants for Ventricular Volume measurements. While there is some overlap between the two groups, they are not identical; the difference in sample sizes reflects variability in data availability, with certain measurements missing for some participants. The dataset accumulated 8489 and 6421 observations for ADAS13 scores and ventricular volumes, respectively. The breakdown of baseline diagnosis (DX_bl) reveals a diverse cohort, including individuals with AD, CN, Early Mild Cognitive Impairment (EMCI), Late Mild Cognitive Impairment (LMCI), and Subjective Memory Complaints (SMC). To clarify, SMC denotes participants classified as CN who report subjective memory complaints but do not meet the clinical criteria for MCI or AD. The distribution across changes in a participant’s diagnosis over time (DXCHANGE) categories are shown in Table 1. Also detailed, indicating a predominance of stable diagnoses in categories 1 (Stable: Cognitively Normal (CN) to CN), 2 (Stable: MCI to MCI), and 3 (Stable: Dementia to Dementia). Additionally, conversion categories are represented, including category 4 (Conversion: NL to MCI), category 5 (Conversion: MCI to Dementia), and category 6 (Conversion: NL to Dementia). Reversions are also noted, with category 7 (Reversion: MCI to NL), and category 8 (Reversion: Dementia to MCI). DXCHANGE was initially listed as a categorical variable in Table 1 for descriptive purposes. However, in the model fittings presented in Tables 2, 3, S5, and S6, DXCHANGE was treated as a continuous variable, as an ordinal variable with five or more categories can be efficiently included as a continuous covariate in regression models33. Educational attainment is predominantly high, with over 94% of participants having more than 15 years of education across both outcomes. The gender distribution is relatively balanced, with a slight male predominance (approximately 56%).

In our study, the semiparametric model with serial correlation incorporates both penalized cubic spline function and exponential serial correlation to account for non-linearity and the temporal correlation structure of the outcome variable, has the best model fit (has the lowest AIC value, 48,184.1) among all models (Table S1) for the outcome variable, ADAS13 score. Additionally, the LME model with serial correlation combines linear mixed effects with exponential serial correlation—allows for the inclusion of both fixed and random effects along with the temporal correlation structure—has the best model fit based on BIC (has the lowest BIC value, 48,329.6) among all models (Table S1) when the outcome variable is ADAS13 score. Furthermore, the LME with Independence model (which assumes linear relationships without considering temporal correlations), the LME with Quadratic Independence model (which incorporates quadratic terms to capture non-linear effects while ignoring temporal dependencies), and the LME with Cubic Independence model (which allows for cubic terms to account for more complex non-linearities but still assumes independence of observations) demonstrated the best prediction accuracy with the lowest RMSE of 4.75 and MAE of 3.68 among all models (see Table S2 for the outcome variable, ADAS13 score).

When the outcome of interest is Ventricular Volume, the semiparametric serial correlation model has the best model fit, with the lowest AIC and BIC values, 122,850.4 and 123,005.8, respectively (Table S3). For the outcome variable, Ventricular Volume, the LME Linear Independence and LME Cubic Independence models have the smallest RMSE (20,016.5) and MAE (15,221.2), respectively. Hence, we can conclude that these two models have better prediction accuracy/ability than other models (Table S4).

Being male and changes in diagnostic status are associated with decreased ADAS13 scores, decreasing between -0.62 and -0.57 across all semiparametric and conventional models. Conversely, the presence of the APOE4 gene is associated with an increase in ADAS13 scores, ranging between 0.93 and 1.11. Elevated scores on the Functional Activities Questionnaire (FAQ) are also associated with an increase between 0.32 and 0.33. Higher scores on the Rey Auditory Verbal Learning Test (RAVLT) for both immediate and learning trials indicate a worsening condition, with RAVLT immediate scores associated with a decrease in ADAS13 between − 0.23 and − 0.22 and RAVLT learning scores between − 0.15 and − 0.12. Additionally, being diagnosed with Cognitively Normal (CN), Early Mild Cognitive Impairment (EMCI), Late Mild Cognitive Impairment (LMCI), or Subjective Memory Complaints (SMC) is associated with lower ADAS13 scores, with effects ranging from − 5.82 to − 5.36 for CN, − 5.16 to − 4.76 for EMCI, − 2.46 to − 2.27 for LMCI, and − 6.18 to − 5.66 for SMC. Educational attainment, particularly having between 11 and 15 years of education or higher, is also associated with decreasing scores, with decreases between − 0.88 to − 0.22 for 11–15 years of education, and − 0.63 to − 0.12 for higher education levels. Although the magnitude of the estimated association of covariates with ADAS13 scores differs with model choice, the directions of the estimated associations are similar.

Factors such as being male, with values between 11,261 and 12,506, and changes in diagnosis were associated with increased Ventricular Volume across the models. The presence of the APOE4 gene was associated with a Ventricular Volume change ranging from − 870 to 164, while elevated scores on the FAQ were associated with an increase, ranging from 128 to 296. Higher scores on the RAVLT for both immediate and learning trials indicated a decrease in Ventricular Volume, with RAVLT immediate scores associated with values between − 70 and − 9.68 and RAVLT learning scores between − 14.79 and − 2. Additionally, baseline diagnoses of CN, EMCI, LMCI, and SMC were associated with decreased Ventricular Volume, with values ranging from − 13,431.1 to − 8262.87 for CN, − 9634.89 to − 5438.09 for EMCI, − 4345.93 to − 1595.4 for LMCI, and − 12,723.6 to − 7686.64 for SMC. Educational attainment was also linked with increased Ventricular Volume, with values between 4345.69 and 5511 for those with 11–15 years of education and between 3781.66 and 4451 for higher education levels. These values are provided in Tables 1 and 2 for further reference.

Various statistical models are employed to estimate the progression of ADAS13 scores and ventricular volumes over time, and the estimated evolution of these two outcome variables over age (years) is visualized in Fig. 2 Visual inspections, which demonstrate the impact of specific model assumptions on ADAS13 score and Ventricular Volume increase, indicate that the models are quite similar especially when patients are between 60 and 85 years old. However, the semiparametric models estimated higher ADAS13 scores when patients are younger than 60 than those that ignore the non-linearity in ADAS13 scores. Using various models facilitates a deeper understanding of the patterns and differences in the progression of these biological markers across aging populations with various choices of model assumptions.

The graphs show estimated evolutions of ADAS13 scores (left panel) and Ventricular Volume \(({\text{mm}}^{3})\) (right panel) over age ranging from 54.46 to 97.83 years. Each plot features curves corresponding to different statistical models, including various linear mixed effects (LME) models and Spline Independence (SI) models. The curves are color-coded and mapped to the model types listed in the legend, providing clarity on the population-level estimated trajectories of both outcome variables for each model. For ADAS13 and Ventricular Volume, the LME cubic models are represented by a purple line indicating the model is independent and a light blue line reflecting serial correlation assumption. The LME linear models are delineated with a red line for the independence scenario and a blue line under serial correlation. The quadratic models are also shown in green (independence) and yellow (serial correlation), providing insights into non-linear trends. Lastly, the spline model, adept at capturing local variations and trends, is included in the analysis, although not distinctly colored in the graph.

Figure 3 presents the estimated rates of change over two parameters—ADAS13 scores and ventricular volume—across different ages, employing various statistical models. The estimates of the rate of change varied according to model type for both outcome variables Fig. 3.

Estimated rates of change in ADAS13 scores (top panel) and Ventricular Volume \({(\text{mm}}^{3})\). (bottom panel) plotted against patient age that ranges from 54.46 to 97.83 years. The graphs indicate the variability in the rate of change in both outcome variables with the choice of statistical model. Each colored line represents a different statistical model. This graph employs the same color scheme as previously described to represent different LME model assumptions and types: cubic (purple for independence, light blue for serial correlation), linear (red for independence, blue for serial correlation), and quadratic (green for independence, yellow for serial correlation). The spline model, effective for local variations, is also depicted, continuing the color conventions from the first graph.

The conventional model (Table 2) enables estimating the rate of change in ADAS13 score as a constant regardless of the subject’s age since this model does not account for non-linearity. The estimated annual change is 0.17 points (95% CI 0.13–0.22) in ADAS13 scores at the population level, indicating a yearly rise. Conversely, when we use the spline serial correlation model, the rate of change (annual change) in ADAS13 scores is now estimated as a function of age (see the brown solid line in the top panel of Fig. 3) since this model enables capturing non-linearity in ADAS13 score trajectories. The penalized cubic splines in this model allow capturing local variations in ADAS13 score, while the conventional model simply ignores these local fluctuations. Figure 3 visually compares the estimated rate of change of ADAS13 for various models. Specifically, the estimated rate of change from the spline serial correlation model increases from age 54.46 to 76.36, and then it starts decreasing, it only gets higher than the estimate from the conventional model between 68.04 and 84.69 years old. The degree of the rise in the maximum ADAS13 score was 0.194 points per year (95% CI 0.100–0.288). The random spline model resulted in the highest rate of change estimate between ages of ~ 72 and –76, but it only slightly differs from other model’s estimates for ADAS13. The LME Cubic Independence, LME Cubic Serial Correlation, Random Spline, Spline Independence, and Spline Serial Correlation models show the survivor effects, and, for patients who remained alive after the age of mid 70 s, the complex models show a lessening of the rate of increase. Additionally, models included in Supplementary Table S5 provide further insights into the variations in ADAS13 score changes under different modeling approaches.

Population-level Ventricular Volume increase in the conventional model (Table 3) was estimated as 1742 \({\text{mm}}^{3}\) per year (95% CI 1689.4–1794.8), indicating a uniform rise of Ventricular Volume over time. As aforementioned for the ADAS13 score in the previous paragraph, the spline serial correlation model (the bottom graph of Fig. 3) provides a rate of change estimate for the Ventricular Volume as a function of the patient’s age. The rate of change in ventricular volume continues at a slower pace from the age of mid-70 s. For those patients surviving until the late 70 s and so on, the increase in Ventricular Volume continues at an even weaker rate. The conventional model underestimates the annual increase in Ventricular Volume compared to the other models (the top graph of Fig. 3). The spline serial correlation model did not have the survivor effect in the estimated rate of increase in Ventricular Volume, while some other models still have that effect (the right graph of Fig. 3). The LME Cubic Independence model resulted in the highest rate of increase estimate for patients aged 70–80. Supplementary Table S6 expands on these findings, providing detailed results for different models applied to Ventricular Volume changes, thus supporting the conclusions drawn from the primary analysis.

The non-linear patterns of ADAS13 trajectories, particularly the acceleration observed between ages 68 and 85, indicate critical periods where targeted cognitive therapies or pharmacological interventions could be prioritized. Similarly, early increases in ventricular volume provide an opportunity to identify individuals at higher risk of rapid neurodegeneration, enabling earlier prognostic assessments and timely clinical interventions.

The results of our study highlight the potential of semiparametric regression in providing a more flexible framework for understanding Alzheimer’s Disease progression, particularly in capturing individual variability that conventional linear mixed-effects models may not fully account for. Our analysis using the ADNI database suggests that AD progression, as measured by ADAS13 scores and ventricular volume, exhibits substantial heterogeneity across individuals. This variability underscores the importance of considering more personalized AD monitoring and management approaches. By employing semiparametric models, our study explored potential non-linear patterns in cognitive decline and neurodegeneration in AD patients. While our findings suggest that these models offer a more flexible representation of disease progression than traditional parametric methods, further validation is needed to assess their broader applicability. Our approach provides insights into temporal disease dynamics, which could contribute to the development of more tailored therapeutic strategies, though additional studies incorporating diverse populations and biomarkers are necessary to strengthen these conclusions.

Moreover, the differences in predictions across models highlight the variability that can arise when applying different statistical approaches to the same dataset. This discrepancy underscores the importance of selecting a model that aligns with the specific characteristics of the data and study objectives. The variation observed among models reflects the complexity of ADNI data and demonstrates how different assumptions about data structure and correlations can influence the interpretation of aging effects on cognitive and anatomical measures. Sigmoid models34, for instance, are often used to characterize the non-linear progression of AD biomarkers, capturing the initial slow progression, an accelerated phase, and an eventual plateau commonly observed in disease trajectories. However, unlike sigmoid models, our semiparametric approach does not impose a predetermined functional form, offering greater flexibility to accommodate individual variations in disease progression within the ADNI dataset. This adaptability may be beneficial for capturing patient-specific trajectories that do not necessarily conform to generalized disease stages. While parametric models provide interpretability and computational efficiency, they may be less suited for AD biomarkers that exhibit inter-individual variability and non-linearity. Similarly, non-linear methods such as neural networks and deep learning models can improve predictive accuracy but often lack transparency, limiting their clinical interpretability. Our semiparametric model seeks to balance flexibility and interpretability, providing an alternative framework for modeling AD biomarker progression. While our approach offers advantages in capturing complex disease trajectories, further validation across diverse datasets and clinical applications is necessary to establish its broader utility.

In AD studies, time-to-event data—such as time to cognitive decline or mortality—are crucial for understanding disease progression. However, semiparametric LME models are not inherently designed to handle survival outcomes alongside longitudinal data, often necessitating additional approaches such as joint modeling. Moreover, exploring the combination of semiparametric models with machine learning techniques, particularly deep learning, could enhance predictive power and facilitate the discovery of novel patterns in large, multidimensional datasets. These advancements could drive forward individualized monitoring and enable more timely, targeted interventions in AD care.

Our findings contribute to the growing body of literature exploring the potential of personalized medicine in treating and managing AD. While our semiparametric model with serial correlation demonstrates improved model fit compared to alternative approaches, we acknowledge that model fit alone does not necessarily translate to superior inference or predictive accuracy. However, better capturing within-subject correlation and temporal dependencies may offer valuable insights for clinical monitoring and the study of dynamic treatment regimes. Although our study highlights semiparametric regression models’ strengths in modeling AD progression, several limitations should be noted. First, our reliance on the ADNI dataset, while extensive, may limit the generalizability of our findings to broader populations, as AD progression may vary across different demographic and clinical cohorts. Additionally, our study primarily focused on two key outcomes—ADAS13 scores and ventricular volume—which, while informative, do not capture the full spectrum of AD symptoms and biomarkers. Future research could extend this work by incorporating additional biomarkers, such as tau and amyloid levels, to provide a more comprehensive understanding of AD progression.

In conclusion, our study highlights the potential of semiparametric regression in modeling AD progression and its applicability in both research and clinical contexts. By offering a more flexible framework to accommodate the heterogeneity inherent in AD progression patterns, this approach may provide insights that contribute to more individualized patient care. Our findings have implications for tailoring interventions and refining prognostic assessments in clinical and research settings. The ability to model non-linear progression patterns with semiparametric approaches highlights critical periods where interventions could be most effective. For instance, the accelerated cognitive decline observed in ADAS13 trajectories between ages 68 and 85 suggests that cognitive therapies or pharmacological treatments should be prioritized during these high-risk stages. Similarly, identifying early and rapid increases in ventricular volume provides a basis for prognostic assessments to identify patients at heightened risk of rapid neurodegeneration, facilitating timely interventions such as clinical trials or lifestyle modifications. The patient-specific trajectories captured by our models enable the design of personalized monitoring schedules and therapeutic strategies, aligning interventions with the individual risk profiles of AD patients. Moreover, the associations between APOE4 status and higher ADAS13 scores emphasize the potential for incorporating genetic information into personalized treatment planning, further advancing precision medicine in Alzheimer’s care.

The datasets used in this study are available on the ADNI website for download (https://adni.loni.usc.edu) and cannot be provided by the authors directly. A code for implementing the approaches has been made available as supplementary information alongside this article (see Online Resource).

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Data collection and sharing for this project were funded by the Alzheimer’s Disease Neuroimaging Initiative (ADNI), the National Institutes of Health (Grant U01 AG024904), and the DOD ADNI Department of Defense (award number W81XWH-12-2-0012). ADNI is funded by the National Institute on Aging, the National Institute of Biomedical Imaging and Bioengineering, and through generous contributions from the following: AbbVie, Alzheimer’s Association; Alzheimer’s Drug Discovery Foundation; Araclon Biotech; BioClinica, Inc.; Biogen; Bristol-Myers Squibb Co.; CereSpir, Inc.; Cogstate; Eisai Inc.; Elan Pharmaceuticals, Inc.; Eli Lilly and Co.; EuroImmun; F. Hoffmann-La Roche Ltd and its affiliated company Genentech, Inc.; Fujirebio; GE Healthcare; IXICO Ltd; Janssen Alzheimer Immunotherapy Research & Development, LLC; Johnson & Johnson Pharmaceutical Research & Development, LLC; Lumosity; Lundbeck; Merck & Co., Inc.; Meso Scale Diagnostics, LLC; NeuroRx Research; Neurotrack Technologies; Novartis Pharmaceuticals Corporation; Pfizer Inc.; Piramal Imaging; Servier; Takeda Pharmaceutical Company; and Transition Therapeutics. The Canadian Institutes of Health Research provides funds to support ADNI clinical sites in Canada. Private sector contributions are facilitated by the Foundation for the National Institutes of Health (http://www.fnih.org). The grantee organization is the Northern California Institute for Research and Education, and the study is coordinated by the Alzheimer’s Therapeutic Research Institute at the University of Southern California. ADNI data are disseminated by the Laboratory for Neuro Imaging at the University of Southern California.

The author(s) declare financial support was received for this article’s research, authorship, and/or publication. This work was supported by the Institutional Development Award (IDeA) from the National Institutes of General Medical Sciences of the NIH under grant number P20GM121307 to MANB. The National Institutes of Health grants R01HL145753, R01HL145753-01S1, and R01HL145753-03S1 to MSB; Institutional Development Award (IDeA) from the National Institutes of General Medical Sciences of the NIH under grant number P20GM121307 and R01HL149264 to CGK. The Ike Muslow, MD Endowed Chair in Healthcare Informatics of LSU Health Sciences Center Shreveport, partially supported the project.

Division of Clinical Informatics, Department of Medicine, Louisiana State University Health Sciences Center at Shreveport, PO Box 33932, Shreveport, LA, 71130-3932, USA

Fatih Gelir, Steven A. Conrad & Mohammad Alfrad Nobel Bhuiyan

Department of Mathematical Sciences, University of Texas at El Paso, El Paso, TX, 79968, USA

Suneel Babu Chatla

Department of Pathology and Translational Pathobiology, Louisiana State University Health Sciences Center at Shreveport, Shreveport, LA, 71103, USA

Md. Shenuarin Bhuiyan, Christopher G. Kevil & Mohammad Alfrad Nobel Bhuiyan

Department of Pharmacology, Toxicology and Neuroscience, Louisiana State University Health Sciences Center at Shreveport, Shreveport, LA, 71103, USA

Elizabeth A. Disbrow

Center for Brain Health, Louisiana State University Health Sciences Center at Shreveport, Shreveport, LA, 71103, USA

Elizabeth A. Disbrow & Mohammad Alfrad Nobel Bhuiyan

Department of Neurology, Louisiana State University Health Sciences Center at Shreveport, Shreveport, LA, 71103, USA

Elizabeth A. Disbrow

Department of Psychiatry, Louisiana State University Health Sciences Center at Shreveport, Shreveport, LA, 71103, USA

Elizabeth A. Disbrow

Department of Pediatrics, Louisiana State University Health Sciences Center at Shreveport, Shreveport, LA, 71103, USA

Steven A. Conrad & John A. Vanchiere

Department of Molecular and Cellular Physiology, Louisiana State University Health Sciences Center at Shreveport, Shreveport, LA, 71103, USA

Md. Shenuarin Bhuiyan & Christopher G. Kevil

Division of Biostatistics and Epidemiology, Cincinnati Children’s Hospital Medical Center, Cincinnati, OH, USA

Emrah Gecili

Department of Pediatrics, University of Cincinnati, Cincinnati, OH, USA

Emrah Gecili

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MANB conceptualized the study. FG, EG, and MANB were responsible for the primary design of the study. FG and EG carried out implementations; FG extracted and analyzed the data with input from EG and MANB. The first draft of the manuscript was developed by FG, EG, and MANB, and all authors contributed to the finalization of the manuscript. FG had full access to all the data in the study and is the guarantor, taking responsibility for the integrity of the data and the accuracy of the data analysis.

Correspondence to Emrah Gecili or Mohammad Alfrad Nobel Bhuiyan.

The authors declare no competing interests.

The ADNI study was conducted following the ethical standards of the institution.

In the ADNI study, informed consent was obtained from all individual participants.

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Gelir, F., Chatla, S.B., Bhuiyan, M.S. et al. Characterizing heterogeneity in Alzheimer’s disease progression: a semiparametric model. Sci Rep 15, 7660 (2025). https://doi.org/10.1038/s41598-025-92540-5

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Received: 05 July 2024

Accepted: 28 February 2025

Published: 05 March 2025

DOI: https://doi.org/10.1038/s41598-025-92540-5

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