2.1 Individual-level simulations and predictions

Fig 2 demonstrates the model’s performance for three representative individuals. The model achieves prediction accuracies of 96.15%, 93.97%, and 93.35% for A, tau, and N, respectively. Training fits (red rectangles), test evaluations (green), and future predictions (blue) are shown, with curves representing accumulated biomarker levels across 68 brain regions. By visualizing only two representative fitting points per biomarker, we simplify interpretation while preserving the spatial distribution of pathology. These results illustrate the model’s capability to forecast individualized biomarker trajectories, providing actionable information for clinicians to anticipate disease progression and personalize interventions.

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Fig 2. Model simulations of three spatially dependent biomarkers (A, tau, N) for three individuals.

Training fits, test evaluations, and future predictions are color-coded as red, green, and blue, respectively. Regional curves show accumulated biomarker levels across 68 brain regions.

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2.2 Population-level evaluation for one time point prediction

To assess generalizability, we evaluated performance across all subjects (Table 1). The proposed nonhomogeneous (NH) PDE model, which allows region-specific parameters to capture spatial heterogeneity, consistently outperforms the homogeneous (H) benchmark in both training and testing. On the test set, the NH model achieves higher mean accuracy for every outcome, with average gains of 1.87% (A), 1.44% (tau), 4.40% (N), and 4.88% (C). In addition, the NH model shows smaller standard deviations, indicating more stable and robust predictions across subjects.

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Table 1. Summary reported as (median, mean, std) for each biomarker for homogeneous (H) model and nonhomogeneous (NH) model for one-time-point prediction.

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Boxplots (Fig 3) and histograms (Fig 4) further confirm that most subjects achieve over 80% accuracy across biomarkers under the NH model. Together, these results support the utility of incorporating regional heterogeneity for reliable, personalized predictions at the population level.

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Fig 3. Boxplots of model accuracy across all subjects for H and NH models (interquartile range and median shown).

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Fig 4. Distributions of fitting accuracies (top) and prediction accuracies (bottom) of the nonhomogeneous (NH) model across all subjects for the four biomarkers: A, tau, neurodegeneration (N), and cognition (C).

Each histogram shows the empirical distribution of subject-specific accuracies for the corresponding biomarker. The y-axis represents the probability (normalized frequency) of observations within each accuracy bin.

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2.3 Population-level evaluation for long term prediction

Based on the two-step prediction results summarized in Table 2, we further evaluated the model’s multi-step forecasting ability by training on the first time points and predicting the last two visits. This two-step-ahead validation provides a more stringent test of long-term predictive performance compared to the one-step-ahead setting. Consistent with the one-step results, the nonhomogeneous (NH) model consistently outperforms the homogeneous (H) benchmark across all biomarkers in both fitting and testing. On the test set, the NH model achieves higher mean accuracy for every biomarker, with average gains of 1.63% for A, 15.46% for tau, 1.43% for N, and 2.11% for C. Notably, the improvement for tau is substantial, increasing from 72.20% (H) to 87.66% (NH), indicating that region-specific parameters are critical for capturing the heterogeneous progression of tau pathology. For N and C, the NH model also demonstrates more stable predictions, as reflected by smaller standard deviations (2.30% vs. 2.63% for N; 5.99% vs. 8.87% for C on the test set). The two-step prediction results are consistent with the one-step findings, confirming that the NH model’s superiority is not limited to short-term forecasting but generalizes to longer-term trajectory prediction. This multi-step validation directly addresses the concern regarding overestimation of predictive performance, demonstrating that the proposed model maintains robust accuracy even when tasked with predicting multiple future time points. The consistency across both evaluation settings (one-step and two-step) provides strong evidence for the model’s ability to capture underlying disease progression dynamics and its potential utility for clinical trajectory forecasting.

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Table 2. Summary reported as (median, mean, std) for each biomarker for homogeneous (H) model and nonhomogeneous (NH) model for two-time-point prediction.

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2.4 Stability analysis of thresholds for creating functional connectivity network

We further assessed the sensitivity of the model to the choice of FC threshold used for network construction. As summarized in Table 3, the prediction accuracy remained consistently high across all thresholds, with median test accuracies ranging from 88.76% to 90.15%. Notably, the standard deviations of test accuracies decreased substantially at thresholds of 0.75 and 0.80 (from 7.51% to approximately 2%), indicating stable and reliable predictions. These findings demonstrate that the proposed nonhomogeneous model is highly robust to the specific FC threshold selection, maintaining strong predictive performance for A deposition across a reasonable range of network configurations.

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Table 3. Summary statistics of fit and prediction accuracy of nonhomogeneous model for predicting A with different thresholds.

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2.5 Impact of training strategies

We next evaluated the impact of the proposed hierarchical structured training strategy with homotopy regularization technique on parameter estimation and predictive performance. As shown in Fig 5, homotopy regularization consistently outperforms conventional training across a range of regularization weights. Compared with vanilla optimization, the proposed strategy improves both fitting and prediction accuracy, particularly in the intermediate regularization regime, while maintaining stable performance throughout the continuation process. These results suggest that homotopy regularization effectively stabilizes optimization of the nonlinear PDE-constrained inverse problem and reduces susceptibility to poor local minima. Besides, as demonstrated in Table 1 and Fig 3, the proposed hierarchical structured training strategy in Fig 9 consistently yields equal or improved fitting and prediction accuracy compared with the preceding stage. This coarse-to-fine strategy enables efficient exploration of the high-dimensional parameter space while preserving information learned at earlier stages.

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Fig 5. Comparison of homotopy regularization (solid lines with circles) and vanilla training (dashed lines with squares) across optimizer iterations.

Homotopy training maintains consistently high fitting and prediction accuracy as the regularization weight is gradually reduced, whereas vanilla optimization exhibits substantially lower performance.

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2.6 Effectiveness of low-rank approximation for patient-specific functional connectivity network

We further evaluated the effectiveness of the proposed low-rank representation for personalized functional connectivity estimation. Fig 6 compares the approximation error of the learned patient-specific FC matrices with that of the optimal rank-2 SVD approximation. Despite being inferred indirectly from biomarker observations rather than FC measurements, the proposed representation achieves an error distribution comparable to the corresponding rank-2 SVD reconstruction. Since rank-2 SVD provides the optimal low-rank approximation in the least-squares sense, these results suggest that the learned representation captures the dominant subject-specific connectivity patterns relevant to disease progression.

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Fig 6. Distribution of relative approximation errors for patient-specific FC matrices.

The proposed rank-2 latent representation achieves approximation accuracy comparable to the optimal rank-2 SVD reconstruction of the corresponding FC matrices, indicating that the dominant subject-specific connectivity structure can be captured using only a small number of latent parameters.

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To rigorously isolate and quantify the contribution of patient-specific functional connectivity (FC) to predictive performance, we conducted a controlled ablation study. Specifically, we held all learned nonhomogeneous model parameters and initial conditions fixed and compared two FC configurations: (i) population-average FC, implemented by setting , and (ii) patient-specific FC, in which and were optimized from individual neuroimaging data. By keeping all other model components identical, this design ensures that any difference in predictive performance can be attributed specifically to the incorporation of personalized connectivity information.

Using the biomarker A as a representative case, Table 4 reports the meanstandard deviation of test accuracy and the maximum individual-level improvement achieved by incorporating patient-specific FC. At the population level, incorporating patient-specific FC results in only a modest improvement in predictive accuracy, with the mean test accuracy increasing from to . This relatively small change indicates that population-average FC provides a strong baseline for A prediction at the cohort level. Nevertheless, the individual-level analysis reveals substantially larger benefits for specific patients, with the maximum improvement in test accuracy reaching 9.73 percentage points. These results suggest that the predictive value of patient-specific FC is heterogeneous across individuals: while population-average connectivity may adequately represent patients whose functional connectivity patterns are close to the population norm, personalized FC can provide substantial additional predictive information for patients whose connectivity patterns deviate more substantially from the population average. Thus, although patient-specific FC does not substantially improve aggregate predictive performance, it can yield meaningful individualized gains for a subset of patients.

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Table 4. Comparison of prediction accuracy for A with and without patient-specific in the non-homogeneous model. Accuracy is reported as meanstd, together with the maximum individual-level improvement in accuracy.

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2.7 Two-level sensitivity analysis

We performed a two-level sensitivity analysis to identify critical model parameters and brain regions. First, assuming homogeneous parameters across all 68 DK regions, total sensitivity indices were calculated over age (Fig 7A), identifying six highly influential parameters: , , , , , and (Fig 7B). Second, we analyzed the top five region-specific parameters to pinpoint sensitive brain regions (Fig 7C), providing a quantitative basis for targeted monitoring and intervention.

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Fig 7. A: Dynamics of total sensitivity indices for 14 model parameters over age.

B: Importance bar plot of sorted model parameters based on total sensitivity indices over age. C: The 68 Desikan–Killiany (DK) regions are grouped into six anatomical lobes. The second-level sensitivity analysis of key region-specific parameters is shown across three time stages (t = 60, 80, and 100). The first-order Sobol index (S₁), represented by the circle radius, quantifies each lobe’s direct contribution to the model variance, while the second-order index (S₂), encoded by edge width, captures the strength of pairwise interactions between lobes. During computations, the model’s initial condition is set to the population mean at t = 50, and the parameter ranges are defined by the minimum and maximum values of each parameter across the 68 regions. Each second-level sensitivity analysis is performed for a single region-specific parameter, with all other parameters fixed at the mean of the optimized values across all subjects.

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Stage-wise sensitivity analysis reveals a dynamic network propagation mechanism: at early stages (t = 60), first-order sensitivity indices (S₁) are relatively uniform, with modest cross-lobe interactions (S₂). By mid-stage (t = 80), temporal–frontal, temporal–parietal, and temporal–limbic interactions dominate, while at late stage (t = 100), inter-lobar interactions drive model sensitivity, with frontal–temporal and frontal–limbic couplings particularly prominent. These findings quantitatively support a network-based mechanism of disease spread, consistent with connectome-driven hypotheses.

Table 5 summarizes top DK regions for each biomarker and disease stage. The temporal and frontal lobes are most affected, with involvement intensifying over disease progression, whereas the insular lobe remains minimally affected. Aggregated across biomarkers, frontal and temporal lobes show increasing involvement from early to late stages, confirming that network hubs play a central role in AD progression [48].

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Table 5. Top DK regions by biomarkers and three disease stages with lobe distribution. F: Frontal, T: Temporal, P: Parietal, O: Occipital, L: Limbic, I: Insular.

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3.1 Model performance and predictive reliability

Leveraging multimodal neuroimaging data from the ADNI cohort, we parametrized the PDE model to construct individualized, region-specific biomarker trajectories. Unlike conventional AD criteria, which often overlook interindividual heterogeneity and focus on convergent phenotypes [49], our approach emphasizes personalized modeling as a pathway toward precision medicine.

The model demonstrated high accuracy in both fitting and prediction for A, , N, and cognitive decline (Fig 3, Table 1), with consistently low variance between training and testing errors. Homotopy-regularized training improved convergence and optimization stability compared to standard training (Fig 5), mitigating overfitting in this high-dimensional parameter space. Furthermore, low-rank approximations of subject-specific FC matrices achieved comparable accuracy to SVD rank-2 approximations of true FC matrices, demonstrating that computational efficiency can be achieved without compromising predictive performance. These results also support the use of the rank-two parameterization in (7). The low-rank representation reflects a deliberate trade-off between model flexibility and parameter identifiability. While higher-rank parameterizations could potentially capture more complex subject-specific connectivity patterns, they would also introduce substantially more parameters relative to the limited amount of longitudinal rs-fMRI data available for each subject. The observed agreement between the learned FC matrices and the corresponding rank-two SVD approximations suggests that the proposed representation captures the dominant patient-specific connectivity variations while maintaining computational efficiency and robustness. The ablation study of further demonstrates that patient-specific FC provides a tangible, albeit modest, improvement over population-average FC. While the cohort-level gain is limited, the substantial individual-level improvements indicate that patient-specific and can provide meaningful additional predictive information for certain patients, particularly those whose connectivity patterns deviate from the population average. The modest aggregate improvement may partly reflect limitations in both the available neuroimaging data and the current low-rank parameterization, where the rank-two update may be too restrictive to fully capture the complexity of individual-specific FC patterns. With larger, higher-quality, and more longitudinally sampled datasets, together with more expressive personalized FC parameterizations, the benefits of patient-specific connectivity may become more pronounced. Collectively, these results validate both the robustness and the predictive reliability of the model, supporting its use for patient-specific forecasting.

3.2 Regional vulnerabilities and biological interpretation

The sensitivity analysis elucidated key parameters (, , , , , ) that drive biomarker dynamics and highlighted spatiotemporal patterns of vulnerability [46]. Across disease stages, the temporal lobe consistently emerged as the earliest and most affected region, with frontal lobe involvement increasing in mid-to-late stages, whereas the insular lobe remained minimally affected and parietal/occipital lobes were relatively spared (Table 5). These patterns align with established neuropathological observations [50–52], wherein tau pathology initiates in the entorhinal and hippocampal cortices before propagating to temporal and frontal association areas. Imaging studies [53,54] similarly report cortical thinning beginning in medial temporal regions and progressing anteriorly with disease severity. Our findings reinforce the temporal lobe as a neurodegenerative epicenter and the frontal lobe as a late-stage amplifier.

3.3 Amyloid, tau, and neurodegeneration trajectories

The model recapitulates empirical biomarker propagation patterns observed in PET and MRI studies. Early A accumulation was predicted in temporobasal and frontomedial cortices, followed by expansion to frontal and limbic regions [55–57]. Tau propagation persisted in temporal and frontal lobes across stages, consistent with functional-pathway-based stepwise spread [58,59]. Neurodegeneration, reflected in cortical thinning, intensified primarily in frontal and temporal lobes during late stages, in agreement with longitudinal MRI studies [60,61]. These trajectories capture the hallmark spatial hierarchy of AD progression: limbic–temporal initiation, frontal expansion, and relative parietal/occipital sparing.

3.4 Network interactions and clinical implications

Second-order sensitivity analysis revealed that inter-lobe interactions dominate over intra-lobe effects, suggesting that AD spreads as a coordinated network disruption rather than isolated regional atrophy. This observation supports connectome-driven propagation frameworks [45,48,62–64], highlighting the role of functional connectivity in mediating cross-lobar pathology. By leveraging functional rather than purely structural connectivity, our model captures dynamic interactions that may underlie symptom evolution: frontal–temporal interplay could correspond to transitions from memory to executive dysfunction, while late insular involvement may relate to emotional and interoceptive deficits. These insights emphasize the potential for network-targeted therapeutic strategies that aim to preserve functional resilience rather than focusing solely on single regions.

3.5 Limitations and future directions

Several limitations of the current framework should be acknowledged.

First, the proposed biomarker cascade model represents a simplified description of AD progression. The current formulation assumes a predominantly sequential pathway from amyloid-beta accumulation to tau pathology, neurodegeneration, and cognitive decline, consistent with the amyloid cascade hypothesis and ATN framework which has been adopted in numerous mathematical and computational models of disease progression [33,35,38]. However, accumulating evidence suggests bidirectional interactions and parallel pathological processes involving tau, neuroinflammation, vascular dysfunction, and metabolic alterations [65,66]. Direct amyloid-mediated neurotoxicity, biomarker feedback loops, and neuroinflammatory mechanisms are not explicitly represented in the current model. Future extensions will incorporate additional state variables and interaction pathways as richer longitudinal multimodal datasets become available.

Second, the model assumes a fixed patient-specific graph Laplacian throughout the observation period. Although functional connectivity evolves during aging and disease progression, reliable estimation of dynamic subject-specific connectivity requires denser longitudinal rs-fMRI data than are currently available. Consequently, the learned graph Laplacian should be interpreted as an individualized network substrate capturing dominant disease-relevant coupling patterns. Future work will investigate time-dependent graph Laplacians and dynamic network reorganization.

Third, another limitation concerns our incorporation of FC into the diffusion term. Our formulation is motivated by the graph Laplacian reaction–diffusion framework, in which FC naturally represents inter-regional coupling and governs the propagation of pathology through the diffusion process across brain regions. Nevertheless, alternative formulations are also possible. For example, FC may additionally influence local biological processes and could be incorporated into the reaction term to modulate regional disease dynamics. Although the current formulation is consistent with a large body of mathematical modeling studies on graph-based reaction–diffusion systems and is further supported by our robustness analyses, we acknowledge that it is not the only biologically plausible modeling strategy. Future work should systematically compare diffusion-driven and reaction-driven FC formulations to better understand their biological implications and improve the mechanistic interpretability of the resulting model parameters.

Fourth, uncertainty in model predictions was not quantified. The current framework provides point estimates of subject-specific parameters and future biomarker trajectories without corresponding confidence intervals or prediction intervals, limiting our ability to assess the reliability of individualized forecasts. Future work will incorporate uncertainty quantification through Bayesian inference, probabilistic parameter estimation, and ensemble forecasting approaches, enabling more rigorous assessment of prediction confidence and improving the reliability of patient-specific forecasting.

Beyond these limitations, future work will integrate additional biological pathways, dynamic network evolution, uncertainty quantification, and richer multimodal datasets, including molecular, vascular, and neuroinflammatory biomarkers. Such extensions will further improve model personalization and provide a more comprehensive mechanistic framework for studying Alzheimer’s disease progression and therapeutic interventions.

5.1 Data description

We accessed the multimodal data from the ADNI website following approval of our data use application (http://adni.loni.usc.edu/). The files titled “UC Berkeley - AV45 analysis [ADNI1,GO,2,3] (version:2020-05-12)” and “UC Berkeley - AV1451 analysis [ADNI1,GO,2,3] (version:2022-04-26)” compiled by ADNI were utilized to obtain A-PET and tau-PET regional standardized uptake value ratios (SUVRs), Regional SUVRs were determined by dividing the standardized uptake values (SUVs) of the target regions by the SUV of the whole cerebellum, which served as the reference region because of its low specific binding and consistent uptake across the study population.

In this study, we quantify neurodegeneration using cortical thickness, a well-established MRI-based biomarker that reflects neuronal loss and structural atrophy in AD. This approach follows previous studies demonstrating that cortical thinning is strongly associated with AD pathology and predicts future cognitive decline [67–72]. Specifically, reduced cortical thickness in AD-signature regions such as the medial temporal, inferior parietal, and posterior cingulate cortices has been shown to correlate with amyloid and tau burden as well as with disease progression. Cortical thickness data was obtained from ADNI as part of the “UCSF-Cross-Sectional FreeSurfer (6.0) [ADNI3]” and “UCSF-Cross-Sectional FreeSurfer (5.1) [ADNI1, GO, 2]” datasets.

Resting-state fMRI data were acquired on 3 Tesla MR scanners across multiple ADNI sites using a standardized ADNI imaging protocol (https://adni.loni.usc.edu/data-samples/adni-data/neuroimaging/mri/mri-scanner-protocols/). Each session included a high-resolution 3D T1-weighted MPRAGE scan for anatomical segmentation and spatial registration. Acquisition parameters for the T1-weighted scan were: field of view , voxel size , , , TE = minimum full echo, and scan duration of approximately 6 min 20 s. Resting-state functional images were obtained using the ADNI-3 Basic gradient-echo EPI-BOLD sequence with the following parameters: field of view , , , and flip angle . Each run lasted 10 min and produced approximately 200 volumes.

Preprocessing was performed using a standard resting-state pipeline consistent with prior ADNI studies(han2021reduced, han2024global, mao2026global). The preprocessing steps included motion correction, skull stripping, spatial smoothing with a 4 mm FWHM Gaussian kernel, temporal filtering in the 0.01–0.1 Hz range, and alignment of functional images to the individual T1-weighted scan followed by transformation into MNI-152 standard space. To minimize transient magnetization effects and filtering boundary artifacts, the first five and last five volumes of each run were removed prior to analysis. Parcel-wise resting-state time series were then derived using the Desikan–Killiany–Tourville 68-region atlas [73] by averaging the preprocessed BOLD signal within each cortical parcel. Functional connectivity (FC) data were derived by calculating Pearson’s correlation coefficients between time series extracted from regions defined by the DKT 68 atlas [73], based on rsfMRI data.

The Mini-Mental State Examination (MMSE) scores were obtained from “Mini-Mental State Examination (MMSE) [ADNI1,GO,2,3,4]”. In this study, we did not impose a strict requirement for each subject to have data available for all modalities mentioned. Instead, our inclusion criterion focused on ensuring that each subject had data from at least three separate visits for one of the key measurements: tau-PET, amyloid-beta (A-PET), or cortical thickness.

The study included 1,891 subjects from ADNI who were classified as cognitively normal (CN), mild cognitive impairment (MCI), AD, or of unknown status. The details are shown in Table 6 and the demographic details are shown in Table 7. Because the biomarker-cascade PDE system is sequentially driven (A), we estimate the model in a decoupled, stage-wise manner. Accordingly, the dataset is summarized in terms of task-specific cohorts: for each sub-equation, we include all participants who have at least three longitudinal time points for the modality/modalities required to fit that sub-equation. Importantly, the resulting cohorts are not mutually exclusive. For example, the “A cohort” is the union of all participants with longitudinal A PET data, regardless of whether , N, and/or cognition (C, derived from MMSE) are also available. Table 6 reports the effective sample sizes by diagnosis group for fitting each stage.

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Table 6. Task-specific cohorts used in the sequential (decoupled) estimation of the biomarker-cascade PDE system. For each sub-equation, we report the number of ADNI participants with at least three longitudinal time points for the required modality/modalities. Rows correspond to effective sample sizes for fitting each sub-equation and are not mutually exclusive (e.g., the A cohort is the union of all participants with A available, regardless of whether , N, and/or C are also observed). CN: cognitively normal; MCI: mild cognitive impairment; AD: Alzheimer’s disease.

https://doi.org/10.1371/journal.pcbi.1014751.t006

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Table 7. Subject characteristics after keeping participants with at least 3 longitudinal records for each modality. Values are counts unless stated otherwise.

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5.2 Data preparation

We normalize data measurements across all subjects and brain subregions using:

(1)

where i = 1, 2, ..., N indexes the i-th subject, j = 1, 2, ..., 68 denotes the j-th brain subregion. And is the reference value for in the j-th region. include amyloid-beta (), tau (), neurodegeneration (N) and cognitive impairment C.

To ensure uniform biomarker trajectories and cognitive decline trajectory aligned with disease progression (all increasing with severity), we define linear transformations on the normalized biomarkers and cognitive decline as follows:

• Amyloid and tau proteins: both and increase with disease progression, as measured in PET imaging.
• Neurodegeneration: quantified by cortical thickness which decreases as AD advances. We invert this trend by defining .
• Cognitive decline: Measured via the Mini-Mental State Examination (MMSE) score S, which declines with worsening pathology. We reverse its directionality using .

This normalization and transformation framework ensures consistent biomarker and cognitive decline behavior between subjects and subregions, simplifyingthe interpretation of the model.

5.3 Patient-specific spatiotemporal partial differential equations model

We propose a spatiotemporal generalized Alzheimer’s Disease Biomarker Cascade (ADBC) model that incorporates neuroimaging data and regional dynamics using graph theory and network science. This model can be used to capture the spatiotemporal progression of pathology in amyloid and taupathy, neurodegeneration, as well as cognitive decline. As illustrated in Fig 8A, we represent the brain’s geometry as an undirected weighted graph . Here, 68 nodes (vertices) correspond to DKT 68 atlas, while edges encode functional connectivity between regions. Edge weights, proportional to connectivity strength, are visualized as varying widths. This graph-based topology provides the foundation for our extended mathematical model, enabling spatially resolved modeling of disease progression while retaining the core dynamics of the ADBC framework.

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Fig 8. A: Topological representation of the human brain as a graph .

Node set comprises 7 functionally defined brain regions (Left). Edge set is derived from a functional connectivity matrix, where connections between regions are thresholded for denoising (Middle). The undirected weighted graph visualizes connectivity strengths, with edge widths proportional to the magnitude of entries in the functional connectivity matrix (Right). This graph structure enables spatially resolved modeling of Alzheimer’s disease progression. B: A schematic representation of a simplified brain network as a graph, illustrating possible ways to derive a patient-specific adjacency matrix from the population-level adjacency matrix. The left arrow indicates that the topology remains unchanged, but the weights (FC values) on the edges differ, representing variations in the strength of functional connectivity between nodes. The right arrow illustrates a topology change, where a new edge is added to the graph, representing a functional connection strong enough to be included in the patient’s network.

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Now we are ready to extend the ADBC model to be a spatial discretized partial differential equation (PDE) of and the ordinary differential equation (ODE) of C defined on the brain as follows:

(2)

and

(3)

where the integration in last equation is defined as:

(4)

where is the degree of vertices . This term is a global burden summary of neurodegeneration. represents how strongly degeneration burden translates into cognitive decline rate.

In the model, represents amyloid pathology, represents amyloid-related tau pathology (measured by tau-PET), N represents neuronal dysfunction/loss, and C represents cognitive impairment. , , , and are 6 scalar parameters and , , , , , and are 8 region-specific parameters which are graph functions defined on 68 vertices. , and characterize the diffusion properties for , and N, respectively. , , and reflect the logistic growth rates of the various biomarker cascades. , and reflect linear growth rates of the biomarkers and determine the influence of various factors on the time-of-onset of the subsequent biomarker cascades. and , represent the biomarker carrying capacities respectively. We would like to estimate these model parameters using the longitudinal neuroimaging data. is the patient-specific graph Laplacian corresponding to the patient’s specific brain functional connectivity network. Note that the graph Laplacian matrix is symmetric positive semi-definite, which can be treated as the discretized version of using a finite difference method with Neumann or periodic boundary conditions. We will discuss how to learn patient-specific graph Laplacian from limited functional connectivity data below.

5.4 Learning patient-specific functional connectivity matrix

The graph Laplacian of a undirected weighted graph is defined as the difference between the corresponding degree matrix D and adjacency matrix A

(5)

where , the adjacency matrix, which is symmetric, is a collection of weights assigned for connected node pairs .

The corresponding degree matrix D, that is, the number of edges attached to each node defined as follow:

(6)

We use the functional connectivity (FC) matrix of 68 regions to compute the adjacency matrix A of brain graph after statistical truncation for denoising purpose [74]. More precisely, we treat the two subregions are connected where the corresponding Pearson correlation coefficient value in FC matrix exceed 0.75 and the corresponding p value is less than . Consequently, an edge is established between such pairs of nodes in the graph representation of . The corresponding adjacency matrix is weighted by the functional connectivity values after removing self-connections.

FC data were derived by calculating Pearson’s correlation coefficients between time series extracted from regions defined by the DKT 68 atlas [73], based on resting-state fMRI data. fMRI scans require expensive equipment and technical expertise. Conducting repeated scans over time (longitudinal studies) adds significantly to the cost. fMRI generates large datasets, especially when acquired longitudinally. Processing these datasets to extract meaningful information requires substantial computational resources. The size of fMRI datasets necessitates significant storage capacity. Patient data must be handled in compliance with strict privacy regulations, adding to the cost and complexity. Hence, a novel mathematical model/method for learning a patient-specific FC matrix (therefore graph Laplacian) from limited FC data collected from a public dataset is necessary.

We propose a qualitative method to learn a patient-specific adjacency matrix (FC matrix), defined as:

(7)

where represents the elementwise product of and , resulting in a vector. The term denotes the population-level adjacency matrix derived from a limited population group. The rank-two matrix approximates the difference between a specific patient’s connectivity and the population mean, while the diagonal term ensures self-connections are removed. The parameters and , which are unit column vectors in , quantify the patient-specific functional connectivity variation between different node pairs. The latent vectors and are introduced as a parsimonious parameterization of patient-specific deviations from the population-level connectivity network. They should not be interpreted as directly measurable biological quantities or clinical biomarkers. Instead, their role is analogous to latent factors in low-rank matrix factorization, providing a compact representation of individualized functional connectivity variations while maintaining model identifiability and computational tractability. In this framework, and generate subject-specific perturbations of the population adjacency matrix and consequently define personalized graph Laplacians for disease progression modeling.

If , then (7) degenerates to the case with rank one approximation as follows:

(8)

We propose using a rank-two matrix instead of a rank-one matrix to improve expressive capability and to address limitations associated with rank-one representations. A rank-one matrix, such as with the constraint , introduces unnecessary coupling between weights corresponding to different node pairs and . For example:

(9)

with the constraint .

In contrast, the rank-two representation provides greater flexibility:

(10)

where the unit constraints are imposed to enhance numerical stability and prevent stiffness in solving ODE problems.

This representation offers significantly enhanced expressive capability compared to . For instance, in the rank-one matrix , the weight for the connection between nodes n₁ and n₂ () is tightly constrained by the weight for the connection between n₅ and n₆ () due to the unit norm constraint . In reality, the strength of the connection between n₁ and n₂ should not inherently depend on the strength of the connection between n₅ and n₆. The rank-two representation overcomes this limitation, enabling a more accurate and flexible modeling of patient-specific FC.

As illustrated in Fig 8B, the brain network is represented as a graph with four nodes. The middle panel represents the population-level brain network, while individual deviations from this representation can be categorized into two fundamental cases:

1. (a) edge weight changes: The topology remains unchanged, but the weights on the existing edges differ from the population adjacency matrix. There exits an edge between nodes and , and the weight is modified as with constraint . Note that the entries of can be negative values as long as they satisfy the constraints so that the weight values could be increased or decreased.
2. (b) topology changes: New edges are added, representing additional functional connectivity between regions. A new edge is lighted up between nodes and with the associated weight with constraint .

Luckily, the expression (7) can cover both cases. As shown in Fig 8B: (a) The left arrow indicates that the topology is identical to the population network, but the strength of functional connectivity (edge weights) varies, indicated by a purple color of the edge. (b) The right arrow illustrates a change in topology, where a new edge appears, representing functional connectivity between two regions strong enough to warrant inclusion. With this model, we address the challenges of data collection and computational expense, enabling efficient analysis of patient-specific brain networks. With the graph Laplacian definition (5) and the patient-specific adjacency matrix formulation (7), the corresponding patient-specific graph Laplacian is defined as

(11)

where denotes the population-level graph Laplacian given by

(12)

To ensure symmetry of the connectivity network, we replace each measured adjacency matrix by its symmetrized form which mitigates asymmetries introduced by the measurement noise. For computational tractability and parameter identifiability, we assume that the patient-specific graph Laplacian remains fixed throughout the observation period.

5.5 Parameters inference by novel hierarchical structured training strategy

Let represent the solution vector obtained by solving model (2), where denotes the collection of all model parameters, and is the initial values of this model. is the clinical data of a specific subject for given age , and is the solution of the generalized ADBC model. The model parameters for each patient can be inferred by solving the constrained optimization problem:

(13)

where the objective function is defined as

(14)

where are two unit vectors to learn patient-specific graph Laplacian, is the (k, h) element of adjacency matrix . is imposed as an optimized variable to avoid the overfitting of the measurement noise in .

This optimization problem is to minimize the L² loss on given clinical patient data points and constraint on as the penalty term which is used to ensure the biomarkers and cognitive decline increasing with severity. This optimization problem was solved for each subject in the cohort using all available biomarker and cognitive decline time points .

To address this challenging high-dimensional problem, we introduce a novel hierarchical structured training strategy incorporating a homotopy regularization technique to enhance the stability and efficiency of parameter learning, as shown in Fig 9. This hierarchical strategy consists of four sequential stages, where the solution obtained at each stage serves as the initial condition for the next, enabling a progressively refined learning of the extended ADBC model from neuroimaging data. By initializing each stage with the optimized solution from the previous stage, the framework progressively enlarges the parameter space while preserving previously learned information, thereby providing a monotonic refinement strategy in which additional model flexibility is introduced progressively, allowing the optimization to maintain or improve upon the fitting performance achieved at earlier stages. To illustrate this procedure concretely, we take the A equation as an example and describe the hierarchical approach in detail below. The equations for other biomarkers can be solved in an equation-by-equation manner [75].

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Fig 9. Diagram of the strategies for solving the complex original constrained optimization problem.

The original optimization problem is split into 4 optimization steps with a hierarchical structure as shown vertically. The homotopy regularization technique is applied to every regularization coefficient w, , w₀, , and for every optimization step accordingly, as shown horizontally.

https://doi.org/10.1371/journal.pcbi.1014751.g009

1. Homogenized Model (benchmark): To get a good initial guess for model parameters , we homogenize the model by homogenizing the regional variability. Namely, let and be scalars. Then, the model parameters of this homogeneized model is . Because different patients have different onset times, in order to give the initial conditions, we assume that the initial time for all people is 50 years old, and the corresponding initial value is a parameter that needs to be inferred. The optimization problem is formulated as: (15)
   where the coefficient w for the penalty term enforce a sigmoid-like solution. The parameters and are optimized using MATLAB’s fmincon iteratively with randomly generated initial guess with bounds , , , and . And the initial condition of this equation is with . The spatial discreized PDE model is solved by ode45.
2. Non-homogenized model: Extend homogenized parameters to non-homogenized parameters, denoted as to account for regional variability. Let the sparse deviations from the scalar parameters denoted as , the optimization problem is formulated as: (16)
   where and are the solution of homogenized model of step 1. To avoid overfitting issue, an L¹ regularization term is imposed for the model parameters. Regularization weight is tuned to balance fit and sparsity so that to avoid potential overfitting. This optimization problem is solved using MATLAB’s fmincon starting with as the initial of the model parameters.
3. Learnable Initial Conditions: Extend the homogenized initial condition to a non-homogenized vector to account for regional variability, denoted as . To avoid overfitting issues, an L¹ regularization term is imposed for initial condition parameters. Let the sparse deviation from the scalar initial condition be denoted as . The optimization problem is formulated as: (17)
   where is the solution obtained from the previous step and keep fixed during the optimization in this step.
4. Patient-Specific FC matrix: For previous 3 steps, the population FC matrix is keep fixed. Now, we learn a patient-specific FC matrix from PET scan data by incorporating a low-rank variation of , modeled as (11). The constrained optimization problem is formulated as: (18)
   where is the kh-th element of FC matrix , are imposed to ensure stability and interpret ability of the model and optimized by MATLAB fmincon function.

5.6 Homotopy regularization technique for optimization stability

At each optimization step, a regularization term with a corresponding coefficient is introduced to prevent overfitting – a critical requirement for ensuring the learned model’s generalizability. However, directly setting the regularization coefficient to a small target value (often necessary for high model fidelity) leads to training instability due to the complex, non-convex loss landscape. To address this, we propose a homotopy regularization technique inspired by numerical homotopy continuation – a method in computational mathematics that solves hard problems by gradually transforming them from simpler, related problems [76].

The core idea involves solving a sequence of progressively harder optimization problems through a continuous deformation of the regularization coefficient. Specifically:

1. Initialization: Begin with a large regularization coefficient (e.g., ), which simplifies the loss landscape by dominating the objective function, ensuring stable convergence.
2. Progressive Refinement: For each stage, is fixed, and after certain iteration steps of optimization, one can reduce the regularization coefficient by a decay factor (e.g., ). Namely, gradually reduce the regularization coefficient by a decay factor at each stage.
3. Warm-Start Propagation: Use the solution from the previous stage (larger ) as the initial guess for the next stage (smaller ), leveraging the continuity of the homotopy path.

This approach effectively navigates the loss landscape by iteratively transitioning from a heavily regularized, convex-like regime to the target low-regularization regime, mitigating the risk of poor local minima or divergence.

5.7 Training, testing, and prediction evaluation protocol

To evaluate our model’s predictive capability on a per-patient basis, we employ a time-series split strategy using each patient’s individual longitudinal data.

For a given patient with biomarker and cognitive decline measurements at N time points , we implement the following protocol:

1. Training (Parameter Inference): We use the first time points as the training set. The model parameters are inferred by fitting the PDE dynamics to these historical data points.
2. Testing (One-Step-Ahead Prediction): Using the parameters learned in the training step, we solve the PDE to simulate the biomarker dynamics and cognitive decline dynamics forward in time. We then evaluate the model’s prediction at time (the last observed time point, which was held out during training). The testing accuracy is calculated by comparing this predicted value against the actual observed measurement at . The accuracy (%) is defined as a normalized error-based similarity measure

(19)

1. where is the vector of quantities of interest and is the corresponding model prediction.
2. Prediction (Future Forecasting): Once the model’s short-term accuracy is validated on the test point, we use the same inferred parameters to solve the PDE for future times. Specifically, we predict the biomarker value at (e.g., years into the future).

This protocol ensures that the model’s predictive performance is evaluated on data not used during training, providing a robust assessment of its generalization capability for individual patients.

5.8 Sensitivity analysis of model parameters

In this part, we introduce the sensitivity analysis of the model parameters, which provides a better understanding of how the changes in the output of a model can be apportioned to different changes in the model parameters [77]. Sobol method [78], a variance-based sensitivity estimate for nonlinear mathematical models, is employed. It obtains the contribution of each parameter to the variance of the quantities of interest (i.e., the model output C(100) in our case). The sensitivity index quantifies a parameter’s influence on the model output—the larger the index, the greater the parameter’s impact, and thus the higher its importance in the model.

Consider a model in the form with Y a scalar output of the model, is the th parameter and denotes all parameters but . The effect on the model output by varying alone, called the first-order index of , is defined as the normalized variance given below

(20)

The effect on the model output by varying and simultaneously and removing the effect of their individual first-order indexes is called the second-order index and is defined as

(21)

The total index of parameter measures the contribution to the output variance of including all variance caused by its interactions, of any order, with any other model parameters, and it is defined as

(22)

The total index measures the total effects, i.e., first- and higher-order effects (interactions) of parameter .