MRExtrap: Brain MRI Aging Model

• The paper introduces MRExtrap, a method that predicts future 3D brain MRI scans by linearly extrapolating in a learned latent space, capturing subject-specific aging trajectories.
• MRExtrap employs a 3D convolutional autoencoder with various priors—including population, UNet Gaussian, and diffusion—to estimate latent progression rates and forecast age-related structural changes.
• The method demonstrates competitive performance on ADNI data, offers Bayesian posterior updating with multiple scans, and provides latent biomarkers relevant to neurodegenerative disease progression.

Searching arXiv for the named topic and closely related work. MRExtrap is a method for longitudinal aging of brain MRIs that models age-related structural change in the latent space of a 3D convolutional autoencoder via linear extrapolation (Kapoor et al., 26 Aug 2025). It is designed to predict future 3D brain MRI scans at an arbitrary target age from one or more observed scans, with the central premise that autoencoders trained on brain MRIs produce latent spaces in which subject-level aging trajectories appear approximately linear (Kapoor et al., 26 Aug 2025). In this framework, the latent progression rate $\boldsymbol{\beta}$ serves as a subject-specific parameterization of structural change with age, enabling single-scan prediction through population-averaged or subject-specific priors and multi-scan prediction through Bayesian posterior updating (Kapoor et al., 26 Aug 2025).

1. Definition and Scope

MRExtrap was introduced in "MRExtrap: Longitudinal Aging of Brain MRIs using Linear Modeling in Latent Space" (Kapoor et al., 26 Aug 2025). The method addresses the problem of simulating aging in 3D brain MRI scans in order to reveal disease progression patterns in neurological disorders such as Alzheimer's disease (Kapoor et al., 26 Aug 2025). Its stated goal is to predict age-related structural changes in 3D brain MRI and generate plausible future MRIs at an arbitrary target age using an interpretable and efficient model (Kapoor et al., 26 Aug 2025).

The method is framed against prior deep generative approaches that predict a future scan from a single observed scan using nonlinear models, including GANs, VAEs, and diffusion models, often at 2D slice resolution and with limited ability to incorporate multiple longitudinal observations (Kapoor et al., 26 Aug 2025). MRExtrap instead uses a 3D convolutional autoencoder to obtain a low-dimensional latent representation of MRI volumes and then performs linear extrapolation in that latent space (Kapoor et al., 26 Aug 2025).

A central feature of the method is interpretability. The latent progression rate $\boldsymbol{\beta}$ is defined to have the same shape as the latent tensor and is interpreted as a per-latent-dimension rate of structural change with age; its direction and magnitude encode a subject-specific aging signature associated with phenomena such as atrophy and ventricle expansion (Kapoor et al., 26 Aug 2025). This suggests that MRExtrap is not only a forecasting model but also a representation-learning framework for age-related morphometric dynamics.

2. Autoencoder Construction and Latent Representation

MRExtrap begins with a preprocessed 3D T1-weighted MRI volume $$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$ and defines encoder and decoder mappings

$$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$

(Kapoor et al., 26 Aug 2025). The encoder outputs a Gaussian posterior $q_{\phi}(\mathbf{z}\mid \mathbf{x})$ with mean and diagonal variance, while inference uses only the posterior mean as $\mathbf{z}$ (Kapoor et al., 26 Aug 2025).

The latent representation is spatially structured, with size $4 \times 20 \times 24 \times 20$, corresponding to a 4-channel feature volume at $8\times$ spatial downsampling per axis (Kapoor et al., 26 Aug 2025). For vector-form computations, the latent dimensionality is $m = 4 \cdot 20 \cdot 24 \cdot 20$, although the method operates directly on the tensor form (Kapoor et al., 26 Aug 2025).

The autoencoder is trained with a reconstruction-plus-regularization loss:

$$L_{\mathrm{AE}}(\mathbf{x}) = \|\mathbf{x} - \hat{\mathbf{x}}\|_{1} + \mathrm{SSIM}(\mathbf{x}, \hat{\mathbf{x}}) + \mathrm{LPIPS}(\mathbf{x}, \hat{\mathbf{x}}) + \gamma_{1}\,\mathrm{AdvLoss}(\mathbf{x}, \hat{\mathbf{x}}) + \gamma_{2}\,D_{\mathrm{KL}}\bigl(q_{\phi}(\mathbf{z}\mid \mathbf{x}) \,\|\, \mathcal{N}(0, \mathbf{I})\bigr)$$

with $\boldsymbol{\beta}$0 and $\boldsymbol{\beta}$1 (Kapoor et al., 26 Aug 2025). SSIM and LPIPS are used as additive perceptual terms, while the small KL weight is used to avoid collapse while preserving a rich latent (Kapoor et al., 26 Aug 2025).

On ADNI, the reported reconstruction quality is mean SSIM $\boldsymbol{\beta}$2 and mean Generalized Dice $\boldsymbol{\beta}$3 across four regions via SynthSeg+, which is presented as evidence that the learned latent space is suitable for downstream latent modeling (Kapoor et al., 26 Aug 2025). The autoencoder itself has approximately $\boldsymbol{\beta}$4M parameters and is trained with AdamW at learning rate $\boldsymbol{\beta}$5 for 125 epochs, using a cosine schedule with warmup; adversarial loss is enabled after 25 epochs and ramped with KL weight over 1 epoch (Kapoor et al., 26 Aug 2025).

3. Linear Latent Aging Model

The conceptual core of MRExtrap is the claim that aging trajectories appear approximately linear in the learned latent space (Kapoor et al., 26 Aug 2025). This claim is supported by two empirical observations. First, within a subject, linear interpolation between two time-adjacent latent codes,

$\boldsymbol{\beta}$6

decodes to MRIs whose segmented regional volumes approximately follow the linear interpolation of endpoint volumes for hippocampus, ventricles, gray matter, and white matter (Kapoor et al., 26 Aug 2025). Second, PCA of latents across subjects shows within-subject trajectories lying approximately in linear subspaces, together with a consistent drift direction with age across subjects (Kapoor et al., 26 Aug 2025).

The paper further motivates linearity by combining two empirical relations. The first is an approximately linear relation between regional volumes $\boldsymbol{\beta}$7 and latent codes $\boldsymbol{\beta}$8:

$\boldsymbol{\beta}$9

The second is approximately linear volumetric aging with age $$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$0:

$$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$1

Combining them yields an approximate linear latent trajectory:

$$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$2

(Kapoor et al., 26 Aug 2025).

The operational model is then specified as

$$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$3

where $$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$4 is an anchor latent at age $$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$5 and $$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$6 is the latent progression rate (Kapoor et al., 26 Aug 2025). For two observations $$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$7 and $$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$8 from the same subject, the model implies

$$\mathbf{x} \in \mathbb{R}^{D_d \times D_h \times D_w}$$9

(Kapoor et al., 26 Aug 2025).

This linear structure distinguishes MRExtrap from nonlinear sequence generators. A plausible implication is that the model trades expressivity for tractability and interpretability, particularly in settings where longitudinal observations are sparse but age serves as a reliable temporal coordinate.

4. Estimation of the Latent Progression Rate

MRExtrap estimates $$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$0 from longitudinal latent codes and then uses priors to support prediction in the single-scan regime (Kapoor et al., 26 Aug 2025). For a subject with longitudinal scans $$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$1, the method encodes $$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$2, forms differences in age and latent space, and estimates $$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$3 through a zero-intercept linear fit $$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$4 using L1 loss for robustness to outliers (Kapoor et al., 26 Aug 2025). These estimates define training triplets $$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$5 (Kapoor et al., 26 Aug 2025).

Population-averaged prior

The simplest prior is a Gaussian estimated from all training subjects’ latent progression rates:

$$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$6

with diagonal empirical covariance $$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$7, giving

$$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$8

(Kapoor et al., 26 Aug 2025). In single-scan prediction, the prior mean $$\mathbf{z} = f_{\phi}(\mathbf{x}), \qquad \hat{\mathbf{x}} = g_{\theta}(\mathbf{z})$$9 is used as a point estimate (Kapoor et al., 26 Aug 2025).

Subject-specific amortized priors

MRExtrap also defines subject-specific priors conditioned on a single scan. A UNet-based Gaussian prior parameterizes

$q_{\phi}(\mathbf{z}\mid \mathbf{x})$0

with a 3D UNet of approximately $q_{\phi}(\mathbf{z}\mid \mathbf{x})$1M parameters (Kapoor et al., 26 Aug 2025). The training loss is

$q_{\phi}(\mathbf{z}\mid \mathbf{x})$2

(Kapoor et al., 26 Aug 2025).

A diffusion prior is also proposed, implemented as a conditional DDPM denoiser $q_{\phi}(\mathbf{z}\mid \mathbf{x})$3 to model $q_{\phi}(\mathbf{z}\mid \mathbf{x})$4 implicitly (Kapoor et al., 26 Aug 2025). It is trained with MSE on noise and uses 500 sampling steps, averaging $q_{\phi}(\mathbf{z}\mid \mathbf{x})$5 samples at test time for stability (Kapoor et al., 26 Aug 2025). The schematic ancestral step is

$q_{\phi}(\mathbf{z}\mid \mathbf{x})$6

with Gaussian noise $q_{\phi}(\mathbf{z}\mid \mathbf{x})$7 and a DDPM schedule (Kapoor et al., 26 Aug 2025).

These priors formalize three levels of specificity: global population behavior, amortized single-scan subject conditioning, and posterior conditioning with multiple scans. This suggests that MRExtrap is organized around progressively stronger subject adaptation rather than around a single monolithic generator.

5. Prediction, Posterior Updating, and Uncertainty

For single-scan forecasting, MRExtrap proceeds by encoding a baseline scan $q_{\phi}(\mathbf{z}\mid \mathbf{x})$8 to obtain $q_{\phi}(\mathbf{z}\mid \mathbf{x})$9, selecting an estimate $\mathbf{z}$0, extrapolating to target age $\mathbf{z}$1 via

$\mathbf{z}$2

and decoding

$\mathbf{z}$3

(Kapoor et al., 26 Aug 2025). The estimate $\mathbf{z}$4 may be the population prior mean, the UNet posterior mean, or an average of diffusion samples (Kapoor et al., 26 Aug 2025).

When multiple scans are available, MRExtrap refines $\mathbf{z}$5 through a linear-Gaussian observation model. For scans at ages $\mathbf{z}$6 with latents $\mathbf{z}$7 relative to anchor $\mathbf{z}$8, the model assumes

$\mathbf{z}$9

with diagonal $4 \times 20 \times 24 \times 20$0 estimated from residuals on training data (Kapoor et al., 26 Aug 2025).

After stacking observations in vector form, the posterior is available in closed form:

$4 \times 20 \times 24 \times 20$1

$4 \times 20 \times 24 \times 20$2

(Kapoor et al., 26 Aug 2025). Posterior samples of $4 \times 20 \times 24 \times 20$3 can then be propagated through latent extrapolation and decoding to obtain a distribution over future MRIs, from which mean and variance of regional volumes or voxel-wise variance maps can be computed (Kapoor et al., 26 Aug 2025).

This Bayesian component is one of the method’s defining properties. Unlike single-shot predictors that bind forecasting to one input image, MRExtrap explicitly treats future generation as posterior inference over an individualized aging rate (Kapoor et al., 26 Aug 2025).

6. Data, Evaluation, and Empirical Findings

MRExtrap is evaluated on ADNI using 9,200 T1-weighted MRIs from 1,700 subjects, split 1411/32/178 for train/val/test, with average age approximately 75 (Kapoor et al., 26 Aug 2025). The preprocessing pipeline consists of ROI clipping, bias field correction, affine registration to MNI152 with 3+3 DoF, skull stripping with ROBEX, and resampling to a final resolution of $4 \times 20 \times 24 \times 20$4 voxels at $4 \times 20 \times 24 \times 20$5 isotropic (Kapoor et al., 26 Aug 2025).

The primary predictive metric is mean absolute error of predicted versus real regional volumes for hippocampus, ventricular CSF, gray matter, and white matter, reported as percent of the subject’s total brain volume at first scan (Kapoor et al., 26 Aug 2025). Segmentation is performed with SynthSeg+ (Kapoor et al., 26 Aug 2025).

Single-volume prediction

The principal baseline is DANINet, described as a GAN-based 2D slice model (Kapoor et al., 26 Aug 2025). For single-scan prediction, MRExtrap with the Global prior reports the following MAE values:

Region	MRExtrap Global	DANINet
Hippocampus	$4 \times 20 \times 24 \times 20$6	$4 \times 20 \times 24 \times 20$7
Ventricles	$4 \times 20 \times 24 \times 20$8	$4 \times 20 \times 24 \times 20$9
Gray matter	$8\times$0	$8\times$1
White matter	$8\times$2	$8\times$3

MRExtrap Global is reported to outperform the GAN baseline in 3 of 4 regions, with $8\times$4 for hippocampus, gray matter, and white matter (Kapoor et al., 26 Aug 2025). The diffusion prior improves ventricles to $8\times$5 versus $8\times$6 for the Global prior with $8\times$7, while the UNet Gaussian prior is described as close to the Global prior and not consistently better (Kapoor et al., 26 Aug 2025).

Multi-scan conditioning

For posterior updating, the evaluation uses 24 test subjects with at least 6 consecutive years, conditioning on scans from years 1–3 and predicting years 4+ (Kapoor et al., 26 Aug 2025). Adding more conditioning scans reduces standard error but does not consistently reduce average MAE relative to the Global prior baseline (Kapoor et al., 26 Aug 2025). The reported explanation is non-stationarity of progression rates: larger absolute changes in regional progression rates between conditioning and prediction windows correlate with worse posterior-conditioned MAE (Kapoor et al., 26 Aug 2025). Case studies reportedly show both helpful and detrimental posterior updates depending on local stability of progression (Kapoor et al., 26 Aug 2025).

Latent progression rates and disease signal

The norm $8\times$8 stratifies diagnostic groups as healthy $8\times$9 MCI $m = 4 \cdot 20 \cdot 24 \cdot 20$0 dementia, consistent with accelerated atrophy rates reported in volumetric literature (Kapoor et al., 26 Aug 2025). PCA of $m = 4 \cdot 20 \cdot 24 \cdot 20$1 vectors highlights outliers predominantly among MCI and dementia subjects, and age-binned analyses show separation across disease groups, with dementia subjects in the 60–70 age bin exhibiting higher $m = 4 \cdot 20 \cdot 24 \cdot 20$2 (Kapoor et al., 26 Aug 2025).

These findings position $m = 4 \cdot 20 \cdot 24 \cdot 20$3 as more than a forecasting nuisance parameter. A plausible implication is that MRExtrap yields a latent biomarker of disease-associated structural progression.

7. Interpretation, Limitations, and Relation to Broader Work

The principal strength of MRExtrap is its reduction of longitudinal MRI prediction to a simple linear model in a learned latent space (Kapoor et al., 26 Aug 2025). The paper emphasizes that inference is highly efficient: one encoding pass, simple arithmetic in latent space, and one decoding pass, while posterior updates require only closed-form linear algebra (Kapoor et al., 26 Aug 2025). The method also operates on full 3D volumes at $m = 4 \cdot 20 \cdot 24 \cdot 20$4 resolution, thereby avoiding slice-based issues associated with some baselines (Kapoor et al., 26 Aug 2025).

The method’s limitations are explicitly tied to its assumptions. The central assumption is local linearity with a constant $m = 4 \cdot 20 \cdot 24 \cdot 20$5 within a subject, and the paper notes that over long time horizons of approximately 10 years, progression rates can change, especially in MCI and dementia, harming posterior-updated predictions (Kapoor et al., 26 Aug 2025). Some regional aging is mildly nonlinear, so linear models may underfit such dynamics (Kapoor et al., 26 Aug 2025). Domain shift due to site or scanner effects may degrade performance, disease heterogeneity limits the utility of a single population prior, and amortized priors help only modestly from a single scan (Kapoor et al., 26 Aug 2025). The decoder may also smooth subtle high-frequency details due to compression (Kapoor et al., 26 Aug 2025).

In relation to adjacent methodological classes, MRExtrap is explicitly contrasted with GAN-, VAE-, and diffusion-based future MRI prediction methods (Kapoor et al., 26 Aug 2025). Its diffusion component is not a direct image generator but a prior over latent progression rates, which keeps the core longitudinal mechanism linear (Kapoor et al., 26 Aug 2025). This separation between latent evolution and image synthesis is a notable design choice.

The paper also identifies use cases including disease progression modeling, individualized forecasting, data augmentation with uncertainty quantification through sampling of $m = 4 \cdot 20 \cdot 24 \cdot 20$6, and biomarker discovery based on latent atrophy signatures (Kapoor et al., 26 Aug 2025). Code is available at http://github.com/mackelab/mrextrap (Kapoor et al., 26 Aug 2025).