Tau lesions and amyloid plaques (short as Aβ) are two of the most important biomarkers in the AT(N) system, which is widely used to classify individuals with Alzheimer’s disease (AD) (Jack et al., 2016). Toxic tau and Aβ proteins deposition impacts neuronal activity and correlates with AD plasma biomarkers and grey matter atrophy (Sanchez-Rodriguez et al., 2024). There is significant evidence that cognitive unimpaired individuals with positive tau and Aβ are more likely to develop mild cognitive impairment (Ossenkoppele et al., 2022). The accurate prediction of tau and Aβ is therefore important to help further understand the disease progression. Positron Emission Tomography (PET) scans for both tau and Aβ serve as tools for analyzing the longitudinal progression of these proteins (Johnson et al., 2016). It is recognized that tau misfolding precedes neurodegeneration in the brain, and the prevalent hypothesis is that Aβ promotes abnormal tau aggregation (Jack et al., 2010, Jack et al., 2013). Our goal is to develop a mechanism-based model that captures the spatio-temporal dynamics of tau, Aβ, and their interactions in the brain.

Summary of the method: Our methodology comprises two key components: a mathematical model (the “forward model”) and an inversion/parameter estimation algorithm (the “inverse problem”). The forward model simulates the propagation of tau and Aβ in the brain. In the inverse problem, we estimate the model’s unknown parameters by matching longitudinal PET observations to the ODE solutions. We describe these components below in more detail.

❶ Forward model: We propose an ODE-based, biophysics-inspired model characterized by a few scalar parameters and unknown sparse initial conditions (ICs). The model accounts for the spatial spreading, growth, and clearance of tau and Aβ. We include a term to model an Aβ promoted tau growth. We also include a basic model for tau-induced relative-to-the-first-scan atrophy. Following Young et al. (2024), we define a normalized disease age that is subject specific but yet allows for a cohort-level analysis.

❷ Inverse problem: To estimate the unknown parameters of the ODE model, we minimize the ℓ2 mismatch function, where the data are region-based PET-derived biomarker measures and MRI-derived atrophy. This data is compared to the ODE solutions. We use a gradient based optimization algorithm to reconstruct these parameters.

Contributions. This study is an extension of the work in Wen et al. (2024a). There we introduced the scaffold of our methodology: the disease age normalization, the coupled formulation for tau and Aβ, and conducted a preliminary evaluation using the ADNI (Petersen et al., 2010). Our contributions in this paper are summarized below.

We conduct a detailed study of the inverse problem formulation and find that it is significantly ill-posed. Since no prior information about the parameters is available, we propose a methodology to address this ill-posedness.

We extend the model to account for abnormal relative atrophy and examine the feasibility of reconstructing the parameters. We propose a simple model and we discuss challenges related to incomplete information.

We extend our study to the larger dataset including ADNI-4 data and explore tau–Aβ temporal correlations.

Related Work. Most efforts in interpreting and analyzing PET data use machine learning. Karlsson et al. (2025) evaluated various machine learning models and composite features derived from clinical variables, plasma biomarkers, and structural MRI to predict temporal tau load, while Leuzy et al. (2022) investigated the optimal combination of biomarkers. Similar studies can be found in Jack et al., 2023, Cai et al., 2023 and Cody et al. (2024). In addition, Ghazi et al. (2021) utilized logistic functions to forecast the progression of scalar biomarkers—including total Aβ-PET and Tau-PET from cohort data.

Mechanism-based models use diffusion-reaction ODEs to simulate the spreading of tau and Aβ in the brain, as well as brain atrophy associated with neurodegeneration. The diffusion models spatial spreading and the reaction models growth and clearance. The ODE models are typically parameterized by a handful of parameters and are easy to explain and interpret. Schafer et al. (2021a) developed a diffusion-reaction ODE model for tau and incorporated brain atrophy dynamics into the tau model. Raj et al. (2025) investigated a few diffusion-reaction models for tau and Aβ, testing different interaction terms and concluding that a one-way Aβ-to-tau interaction is most appropriate. A similar model was presented in Bertsch et al. (2023). More recently, Alexandersen et al. (2025) proposed using a biophysical model to explore the initial condition of tau. Despite these advancements, several limitations remain: (1) Chronological age is typically used as the time variable, which fails to align personalized data and models to a common time axis (Therneau et al., 2021, Schafer et al., 2022). Recent studies have highlighted the importance of this alignment (Young et al., 2024, Ren et al., 2025), with Young et al. (2024) proposing a conceptual framework for aligning personalized disease (biomarker) trajectories with a global progression timeline. (2) A robust inversion solver is needed for the ill-posed biophysical forward model—one that accounts for the uncertainty in the estimated parameters. (3) Some definitions of atrophy rely on relative volume loss compared to baseline (Jack et al., 2005, Schafer et al., 2021a, Schafer et al., 2022, Xie et al., 2023). As we will see, working with relative atrophy is possible but requires careful interpretation and comparison with imaging data.

Another consideration in defining tau abnormality is whether to include subcortical regions in the analysis or not. On one hand, subcortical regions are known to exhibit abnormal tau deposition (Braak et al., 2011, Soleimani-Meigooni et al., 2020, Leuzy et al., 2019). On the other hand, Flortaucipir – the PET tracer used in ADNI – has exhibited off-target binding in subcortical areas. Consequently, some studies exclude these regions (Vogel et al., 2020), while others include them (Scheufele et al., 2020, Wen et al., 2023, Bertsch et al., 2023). We present results for both scenarios, with a particular focus on identifying which ROIs are most frequently selected as ICs. The paper’s main focus is presenting a forward model describing the dynamics of tau and Aβ, the inversion algorithm to estimate the parameters, and an evaluation of the methodology on ADNI data.