BrainODE: Neural Shape Dynamics for Age- and Disease-aware Brain Trajectories
Authors: Wonjung Park
Suhyun Ahn
Maria C. Valdes Hernandez
Susana Muñoz Maniega
Jinah Park
Venue: NeurIPS 2025 poster ()
Keywords: Shape analysis
Medical application
Prediagnosis
Primary area: applications

Abstract: We present BrainODE, a neural ordinary differential equation (ODE)-based framework for modeling continuous longitudinal deformations of brain shapes. BrainODE learns a deformation space over anatomically meaningful brain regions to facilitate early prediction of neurodegenerative disease progression. Addressing inherent challenges of longitudinal neuroimaging data-such as limited sample sizes, irregular temporal sampling, and substantial inter-subject variability-we propose a conditional neural ODE architecture that models shape dynamics with subject-specific age and cognitive status. To enable autoregressive forecasting of brain morphology from a single observation, we propose a pseudo-cognitive status embedding that allows progressive shape prediction across intermediate time points with predicted cognitive decline. Experiments show that BrainODE outperforms time-aware baselines in predicting future brain shapes, demonstrating strong generalization across longitudinal datasets with both regular and irregular time intervals.

BrainODE: Neural Shape Dynamics for Age- and Disease-aware Brain Trajectories
Wonjung Park* KAIST
Suhyun Ahn" KAIST
Fabiola@kaist.ac.. ac.
www.kaist.ac.... ac.
Maria C. Valdes Hernandez The University of Edinburgh m. valdes headde--rrnnnedd.com ac.
Susana Muñoz Maniega The University of Edinburgh s m. maniega@ed ac.uk uk
Jinah Park KAIST
jinahpark@kaist.co ac.
Abstract
We present BrainODE, neural ordinary differential equation (ODE)-based framework for modeling continuous longitudinal deformations of brain shapes. BrainODE learns deformation space over anatomically meaningful brain regions to facilitate early prediction of neurodegenerative disease progression. Addressing inherent challenges of longitudinal neuroimaging sizes, irregular temporal sampling, and substantial inter-subject variability-we propose a conditional neural ODE architecture that models shape dynamics with subject-specific age and cognitive status. To enable autoregressive forecasting of brain morphology from single observation, we propose a pseudo-cognitive status embedding that allows progressive shape prediction across intermediate time points with predicted cognitive decline. Experiments show that BrainODE outperforms time-aware baselines in predicting future brain shapes, demonstrating strong generalization across longitudinal datasets with both regular and irregular time intervals.
as limited sample
Introduction
Precise modeling disease progression neurodegenerative disorders, particularly Alzheimer's disease (AD), is imperative for enabling early diagnosis and effective intervention [2] 28 29). The progressive and heterogeneous nature these disorders poses significant challenges for longitudinal modeling 4 22]. Besides, many studies have shown that the hippocampus, key brain region implicated in AD, plays critical role in accurate diagnosis and monitoring of disease trajectories. These highlight the potential of longitudinal models support emerging therapeutic advancements toward targeted and personalized strategies [19]. Consequently, developing robust longitudinal modeling approaches has become important in improving clinical outcomes in AD.
To model neurodegenerative progression, directly learning from 3D brain MRI data is an intuitive approach. However, this strategy is often impractical due the severe scarcity of longitudinal medical imaging data (5) and the inherently high dimensionality of MRI scans. Recent studies on the biological trustworthiness of generative models for brain imaging report that these limitations hinder the ability to capture complex brain changes driven by both aging and disease processes [I. Furthermore, the hippocampus-a key brain region for diagnosing cognitive status-has only 1.7% of the full brain MRI volume. Although autoencoder architectures have been used to manage high-dimensional images in latent spaces alleviate unaffordable GPU memory consumption, auto-encoded inputs
equal contribution
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

---

never exactly reconstructed after being decoded [14]. This poses a significant difficulty to deal with full brain MRI as cognitive status is often reflected subtle structural changes within specific brain regions [6] such as the hippocampus. Therefore, efficient yet expressive representations with high precision such as surface meshes are required to address the challenge.
Beyond representations, modeling clinically plausible disease progression requires addressing several components of longitudinal medical data: 1) irregular time intervals, 2) medical priors, and 3) practical usage. First, longitudinal medical data consist of irregular time points, requiring methods that handle variable time intervals (i.e., conventional inductive bias for uniform interval times is not suitable). Moreover, these methods should support updating predictions based on sequential longitudinal data. Second, driving clinically meaningful results can be achieved by incorporating demographic information, cogmitive status, and morphological priors for deformations at scale 4 Especially, since neurodegenerative progression gradual, not reversible, and not discrete, the method should be able to condition intermediate representations for the cognitive status c E [0, 1] spanning normal to AD. Lastly, given the constrained acquisition environment, the method should predict progression from single time point, enhancing practical applicability. In addition, this capability is critical for overcoming the limitation requiring multiple data points for interpolation at intermediate time points and extrapolation to future time points.
In this paper, we present a pioneering approach, BrainODE fV Cr, t,, which progressively models brain dynamics using neural ordinary differential equations (neural ODEs), simultaneously addressing the key components of longitudinal modeling. including irregular time points, medical priors, and practical usage. BrainODE learns the deformation space (i.e., morphological prior) in brain shapes on longitudinal data of irregular time intervals under supervision of an ODE solver, enabling efficient modeling in shape spaces with the constant memory cost of neural ODEs 77. For clinically plausible modeling, demographic information := age and cognitive status e of subjects are integrated to condition our model. Contrary to using discrete or future cognitive status in previous studies, Brain- ODE is capable of embedding the continuous states in modeling hidden states ((t) - (Ve, Cr, t), consistent with the gradual and continuous nature of neurodegenerative progression.
Beyond plausible conditioning, we further propose the pseudo-cognitive status embedding method, which integrates pseudo-cognitiy status shape sampling and cognition estimator. Especially, by leveraging the knowledge of the gradual conversion and condition -injetiinn-jjeciviityy of BrainODE, the sampling generates intermediate shapes VE and the corresponding status é through interpolation between the forward and the backward trajectory of the converted subject whose diagnosis is changed from normal cognition to AD. Afterward, we employ a cognition estimator Tg. trained on pseudo-data and observation (V,, c. Vi, ct), to provide smooth conditions for enhancing the fidelity of cognitive status for BrainODE. Moreover, through this estimator To, our novel method has unique property that enhances the practical feasibility of clinically progressive modeling. Since the estimator predicts the severity of cognition from the geometry of the brain shape, BrainODE can seamlessly incorporate these predictions to model longitudinal progression.
the
In summary, our main contributions are as follows.
1. We propose a neural ODE-based brain shape deformation model specifically designed to address key challenges of longitudinal medical data, including irregular intervals, medical priors, and practical applicability.
2. Through extensive experiments comparing various methods datasets with both regular and irregular time intervals, we validate and demonstrate the effectiveness of BrainODE in modeling brain shape dynamics.
3. We show that the bility in longitudinal medical domains, enabling a robust and stable modeling for BrainODE in patients transitioning to neurodegenerative disorders, especially AD.
status embedding strengthens the BrainODE's applica-
2 Related Work
To model longitudinal progression in neurodegenerative diseases, two primary approaches have been widely adopted: generative models and deformation-basee methods. BrainODE emerges as a novel methodology that bridges the gap between deformation-basee approaches, particularly ODE-b methods and flow models, offering unique framework for learning brain dynamics.
2

---

Generative models. Building upon advancements in generative model, diffusion model-based methods have been proposed, embedding the aging prior in longitudinal progress modeling. For instance, BrL.P [24] learns the Brain MRIs distribution and spatial-temporal consistency in the latent space of a VAE, covering various cognitive statuses of AD. However, its conditioned statuses are discretized and reflect future states, thus making it challenging to model continuous disease progression with the current status, Additionally, LoCI-DiffCom [35], another longitudinal generation method, models healthy infant aging on brain MRI scans. However, these image-based generative have common limitations: substantial computational costs, inconsistencies in subject identity due to Gaussian sampling (i.e., subject-specificit is lost due to stochasticity in generation). and coarse resolution that fails to capture subtle yet clinically important changes. Furthermore, none of these methods is capable of modeling the clinical continuum of the AD trajectory as they rely on discrete condition values for the disease laheling. ConDOR [8, a recent study on longitudinal neurodegeneratiie diseases, generates volume values of brain regions using conditional diffusion model. It employs ordinal regression for pseudo-intermeddite sampling across longitudinal observations. While ConDOR addresses irregular intervals and predicts future values from a single point input, it produces less representative scalar values than volume grids or shapes and also relies on discrete ordinal categories for disease statuses.
approaches [5 24 [K]
Neural ordinary differential equations. Neural ODEs define the dynamics of hidden states by parameterizing their temporal derivatives with neural network, enabling continuous time modeling of complex dynamics [7] To handle irregular time intervals, Latent ODEs [27] employ variational autoencoders (VAEs) to model the likelihood of observations. However, autocncoding methods yield imperfect preservation due to the information loss [4] and are hard to train in modeling prior distribution when the data is scarce. For nigh-dimensional data, NODEO [31 is designed for 21) image registration tasks by learning continuous deformation fields between moving and fixed images.
In the medical image domain, LaTiM (34] performs NeuralODE-based classification tasks for diabetic retinopathy in 2D fundus images, predicting severity grades. For 3D brain MRIs, Lachinov et. al. 17 leverage NeuralODE to model longitudinal progression on optical coherence tomography and brain MRI volumes for a single disease. Since directly modeling in voxel spaces requires huge computation costs, they project 3D feature representations into 2D space to solve ODE solver. NODER 3l aims to predict missing timepoints in brain image sequences, requiring at least two prior timepoints as input to regress deformation fields. However, -basdtd methods such as NODEO and NODER limits their practical applicability for predicting future shapes from a single observation. Moreover, it requires separate training for each individual brain image sequence, making it unsuitable for learning generalizable longitudinal dynamics. Furthermore, none of these approaches model the progressive transition from healthy diseased states across continuous trajectories. and their reliance on image grids rather than volumetric shape spaces distinguishes them from our work.
Flow models. Continuous normalizing flow models have been recently proposed [TT [1]. employing a differential ODE solver to generate bijective mappings through learned fluid dynamics. For instance, PointFlow 32 maps point cloud geometries to a captured prior distribution with VAE, often resulting in loss of geometric details. Another representative work, ShapeFlow [14], directly learns the deformation function between geometries, preserving shape details. In contrast
BrainODE, these flow-based methods assume uniform time intervals for one-to-one mappings between shapes and are thus not readily suited for irregularly sampled time series. ImageFlowNNe (18) models subject- specific spatiotemporal dynamics in 2D medical images using neural ODEs/SDES with multi-scale representations based on U-Net architecture. The use of multi-scale representations in a joint space of neural ODEs/SDES and pixel-level reconstruction results in substantial memory consumption, making it infeasible in 3D voxel space (c.g., over 80 GB for volumes of size 128*). In contrast, BrainODE employs mesh representations and PCA-based 3D shape parameterization, which signifi- cantly reduces computational requirements (~ 375 MB for training) while achieving high-precision shape modeling. In medical images, TimeFlow [3 learns a deformation function for registration in longitudinal brain MRIs with varying time intervals. However, generating transformations requires two time points of MRI scans for interpolation and extrapolation, limiting its applicability compared to BrainODE, which can predict disease progression from a single time point. Multi Marti-largiial Flow Matching (MMFM) 25 shares several core similarities with our BrainODE. Both methods employ single shared model to learn across diverse conditions, support continuous dynamies modeling, handle irregular and multi tett--tme--poin data, and incorporate techniques for imputing mission observation.
3

---

However, MMFM is designed to model distributional translations, making it well-suited for capturing general patterns and group-level dynamics in low-dimensional scalar spaces. Consequently, it does not estimate explicit individual trajectories nor incorporate structural parameters that reflect anatomical geometry. contrast, BrainODE embeds brain morphology directly into parameter space and models continuous shape trajectories at both individual and disease group levels. This enables precise tracking and predicting of morphological changes for individual subject.
Method
3.1 BrainODE
Definition of brain shapes. Our BrainODE learns a continuous deformation trajectory of brain shapes from longitudinal datasets. We denote the longitudinal brain shape sequence for each subject as == (V.. V,,,.., Vi J, where N 2 2. Here, i; are irregular time points of the age at medical imaging acquisition. Each shape is reconstructed
triangular mesh X; FV, where v== {U,, ..., Un denotes the ordered set of mesh vertices, and F = {f1, f2,.... fm] is the shared face connectivity across all shapes. Details mesh reconstruction from MRI scans provided in Appendix A.
To model brain shape deformation efficiently given scarce longitudinal datasets, further reduce the dimensionality of the vertex space using principal component analysis (PCA), a widely used technique in shape modeling 66 [6]. Our approach is motivated by the following factors: (i) the necessity of preserving key shape features blocky and noisy mesh surfaces derived from mm3 resolution MRIs, and (ii) the observation that local shape features do not change abruptly despite overall volume reduction caused by brain cell loss. Each brain shape is projected onto the PCA basis as V1 ~ Σj= Xie; where A, = {Af X2.... and & e2.. are the set of PCA coefficients and eigenvectors, respectively, and & is the number of retained components. To ensure accurate reconstruction from the reduced space, we empirically determined the optimal dimension based on the explained variance ratio and reconstruction loss analysis (see Appendix A.2)) Finally, the shape representations to train the longitudinal trajectory of brainODE is A (Ar. ......,, Ar,
150
Training deformation space of brain shapes. BrainODE fa(A,, Ct () models shape deformation function of the individual current shape A,, cognitive status o [0, 1], and normalized age t t [0, 1]. defined as:
So(A,, c2, = dAs(cr,t)
dt
Under the NeuralODE framework, brainODE estimates the shapes A4tar at the timepoint / + Al from the timepoint & by solving the following formulation:
A--:= = - J. ++D1
So(Ne, Ct, l)dt.
(2)
By using ODE solver, BrainODE model longitudinal data of irregular time intervals. It also incorporates the prior knowledge of brain atrophy and ventricle enlargement by leveraging cognitive status e and age For pair of longitudinal deformation from time Lj. BrainODE is trained to
seek a mapping 1 R + R² that minimizes the distance between predicted deformed shapes 2 4:4 (Ar, and the target shape (A,). The model is supervised using L2 loss:
L = 11 |0 1:5 (A:, A,
Combinatorial training samples. To fully utilize the longitudinal datasets with arbitrary N obser- vations for alleviating the data deficiency, we design the training process for BrainODE to use the entire combinations of time points. Furthermore, BrainODE learns not only a forward trajectory of neurodegenerative disease progression but also the backward trajectory to lead the invertible shape deformation (i.e, $ $55 (T 5/4 (AGGG) BrainODE.
A.). Algorithm elaborates the training process for

---

Algorithm BrainnDE training process
1: Input:
2: A set of observed shapes A [Ao, A1,..., AN-1).
3: corresponding cogmitive status |co, ................................ CN CN-11. age t = [to. t tN-
4: N +- length(A). (A), loss <-
>> Forward trajectory
5: for 0 to N 2 do
6: Aprea [ N] -- ODEintegral (input (A])). myget_time- t/i N))
D Eq. (2)
7: loss loss C2 (Aped i Mi,A[i N])
8:
end for
Backward trajectory
9: for i
1 to N do
Apred [0 i] -- ODEiinegrallinpu = ([).)0))))) target_time = 100 :
Eq.(2)
11: loss < loss C2 (Aped i), A(0 i])
12: end for
3.2 Pseudo-cognitive status embedding
Forward trajectory of normal
Cognition estimator
Aesqua
Shape knowledge guidancs
Rese,
falAs,
BrainODE
Shape space
ODE Solver
Ausan-rrar
Pseudo- sreudd-cogniiveeco status shape sampling
brain dynamics
Backward trajectory Alzhehimer's Disease
(s) pseudoo-ogmitive status shupe sampling
(b) The pseudo cognitive status embedding
Figure Illustration of the proposed methods of BrainODE. (a) The framework of the pseudo- cognitive status shape sampling for continuous progression modeling of neurodegenerative disorders. (b) The pseudo-cognitive status embedding of BrainODE is guided by a cognition estimator Tg.
Pseudo-cognitive status shape sampling. Beyond a longitudinal prediction for the unchanged cognitive status, to achieve clinically useful brain shape progression for early diagnosis, it is essential to reflect the gradual decline in cognitive status in modeling. However, discrete diagnostic labels such normal and disease as well the severe scarcity of converted cases make difficult to model the conversion toward disease. To address this, we introduce pseudo-cognitee sampling, bridging the absence of explicitly intermediate cognition data. Using condition innjiicviiieiiiitty and both trajectories of BrainODE. in subsection 3.1 we sample intermediate brain shapes along their shape trajectory A, At+At by assuming smooth morphological transition over time the prior knowledge progressive brain atrophy by aging and disease. The intermediate shape at time f + fin (where 0 < tin <. At) is interpolated between predicted shapes from the forward trajectory of A, with C and the backward trajectory of Ar+ with Cz+At 1, using the learned deformation &a of BrainODE. The pseudo-cognitive status E j0, 1] is defined as the relative position between the two time points. The final formulation for the pseudo-cognitive status and its shape is:
status shape
Aetto = (0/ 1.1t (A,, C. = 0) sL+ a (A++ C+ = =1)) Ett
tin
(4)
At
5

---

These pseudo-labels and their corresponding interpolated shapes are used to train BrainODE to learn deformation dynamics that span the cognitive continuum.
Progressive longitudinal shape prediction with cognition estimator: To predict future brain shapes and their associated cognitive status, we introduce progressive estimation framework for longitudinal shape progression and the corresponding cognitive status. Prior work demonstrated that hippocampal binary masks can serve as effective biomarkers for distinguishing between NC and AD, achieving over 90% accuracy [19]. Inspired by this insight, we developed a cogmition estimator ти trained on an augmented dataset composed of (A,, Ct A,, Gy). Through this training, the cognition estimator learns longitudinal shape changes corresponding to continuous cognitive status.
During inference, using the pscudo-cognitive status shape sampling and cognition estimator, we update the future prediction progressively using the estimated intermediate shapes and cognitive status (i.e., risk probability of AD). We term this progressive modeling as the pseudo-cognitive status embedding, depicted in Algorithm
Algorithm The pseudo-cognitive status embedding
Require: A (t): Current brain shape at time I, N: Number of intermediate steps 1: At: Target time interval, 78 Moel of cognition estimator
2:
3: Initialize Amid_pree. f- Att)
4: for i= == to N - do
5: ē To (Amid_ (redm pred)
6: Amid_ pood + ODEIntegral (inpu (Amid prod 2, tt )), target time ++ (i+ A4
7: end for
Eq. (2)
8: Apred f- Ared_pred
9: return Apred
4
Experiments
In this section, we demonstrate the comparison of BrainODE with baseline methods for time series and then discuss the effect of our proposed components. We conduct experiments using four different longitudinal brain MRI datasets of normal cognition and Alzheimer's 's disease (AD), comprising both regular and irregular time intervals: for regular intervals, the Lothian Birth Cohorts 1936 (LBC1996) [9] and the Australian Imaging, Biomarker and Lifestyle (AIBL) datasets [IO], each collected fixed 3 and 1.5 year intervals, respectively; for irregular intervals, the Alzheimer's Disease Neuroimaging Initiative (ADNI) [12] and the Open Access Series of Imaging Studies (OASIS) datasets 20,, with varying acquisition intervals reflecting real-world clinical settings. We provide detailed implementations and training settings of BrainODE in Apppendix A] Also, the demographies and cognitive statuses of these datasets and preprocessing methods for shape representations are detailed in Appendix B
Baselines. To provide a comprehensive evaluation, we employ baseline models that can jointly model shapes and temporal sequences, including traditional awme- architectures, flow-based model, neural ODE- based model, and generative model. As conventional time-aware baselines, we include recurrent neural networks (RNNs), an RNN variant for irregular intervals (RNN-Decay), and long short-term memory networks (LSTMss). For flow-based modeling, we adopt ShapeFlow [4), which learns a continuous deformation field between pairs of geometric shapes. In addition, we employ Latent ODE [27]. a neural ODE-based model that can handle irregular time intervals by integrating an ODE solver parameterized by neural network. We also employed simple yet plausible baseline. linear extrapolation for subjects with two or more available observations. For image generative models, we adopt BrLP 24 which generates longitudinally predicted MRIs conditioned by discrete cognitive statuses and volume of brain regions. Regarding image registration based approaches, we also report the evaluation results of NODER B3 in Appendix C.6 as it cannot perform prediction from a single time point and is therefore not included in the main comparison.
Evaluation. Given the varying number of longitudinal observations N ranging from 2 to 5 across the datasets, we formulate the brain disease progression modeling into two tasks: (1) predicting the latest shapes for each subject using four observed time points (4-shot prediction), and (2) predicting

---

LBC 9
AIBL 10
Method
s-shot
1-shot
4-shot
1-shot
hippo
LV
hippo
LV
hippo
LV
hippo
LV
Linear extrap.
0.737
2.075
0,666
2.013
RNN
1.060
5.528
12.801
50.829
0.964
5.561
10.914
52.418
LSTM
1.044
5.723
9.978
37.948
0.957
6.745
7.848
39,572
RNN-Decay
1.075
5.549
0.787
5.703
0.967
5,587
0.780
5.601
ShapeFlow 14
0.652
7.112
0.576
4.939
0.776
3.149
0,567
2.796
LatentODE 27
0,880
5.759
0.895
6.663
1.126
8,095
1.079
6.800
BrLPP 24
1.078
2.230
1.054
2.067
1.019
1.893
1.031
2.042
Ours
0.488
1.630
0.365
1.743
0.461
1.635
0.406
1.708
Table 1: Quantitative evaluation of shape prediction performance in Euclidean distance on the LBC1936 and AIBL test sets with regularly sampled time points.
4-shot
1-shot
Shape acc
Method
hippo
TV
hippo
LV
Te acc
NC & AD
CONV
Linear extrap.
0.898
2.240
RNN-Decay
0.984
6.444
0.832
6.306
Baseline
0.636
0.282
0.883
LatentODE 27
1.215
5.300
1.119
4,463
+ To only
0.611
0.310
0.883
BrLP (24
1.017
1.962
1.031
2.085
pseduo only
0.619
0.178
0.875
Ours
0.543
1.959
0.492
1.673
BrainODE
0.606
0.216
0.891
Table Quantitative evaluation of shape predic- tion performance in Euclidean distance on ADNI, OASIS, AIBL test sets with irregular time points.
Table 3: Ablation results of BrainODE. We report
shape prediction performance (mm) and diagnosis accuracy by the cognition estimator. CONV de- notes the converted cases.
the latest shapes from a single observation point (I-shot prediction). These tasks are intended to reflect the practical characteristics of medical longitudinal datasets well as to evaluate whether the baseline methods and BrainODE can address the crucial components for clinically plausible discase progression modeling discussed in |section I The lateral ventricles (LV) and hippocampus (hippo), key brain regions for AD diagnosis, are selected as representative target shapes. To assess the shape modeling quality, we measure the average Euclidean distance between the predicted and ground truth shapes. For image-based prediction with BrLP [24]. Chamfer distances on the boundary points of the segmented brain regions are measured. For pair comparisons, all methods are trained with a single type of time interval between the regular and irregular datasets.
For pair comparisons, all models are trained and evaluated separately for the two settings, and the regular and irregular datasets (i.e., the subjects for each task are different patients).
4.1 Experimental Results
Regular time interval predictions. We first discuss prediction results on datasets with regular sampling intervals, LBC1936 and AIBL. As their intervals differ, we report the results separately.
Across all settings, BrainODE consistently outperforms baseline methods, shown in Table Especially, in the 1-shot tasks, our model demonstrates a clear advantage in capturing both subtle and large-scale morphological changes of the brain progression. Although RNNs and LSTMs benefit from inductive biases under regular intervals, they exhibit limited capability in modeling neurodegenerative trajectories, especially in the 1-shot setting, due to their inability capture continuous temporal dynamics effectively. Among deep learning-basse based methods, RNN-Decay and BrLP achieve competitive results with BrainODE. However, on the LBC1936 dataset, their performance even falls below that of simple linear extrapolation, underscoring the inherent difficulty of modeling shape trajectories in progressive neurodegeneration. ShapeFlow shows notably inferior performance in L.V modeling on LBC1936, which has longer temporal intervals than AIBL, indicating its limitations capturing the brain dynamics driven by aging. This highlights the importance of methods that incorporate both effective representation and domain-specin knowledge.
7

---

33333
Buckdlan Distance
Hippocampus
0 mm
Lateral Ventricles
Extrapolation
LatentODE
BrLP
ShapeFlow
BrainODE
Figure 2: Qualitative results in modeling the LV and hippocampus shapes.
Interestingly, BrI.P exhibits larger discrepancy from BrainODE in the hippocampus than in the L.V. plausible interpretation is that the coopocampus-occuyye a much smaller proportion of the brain compared the LV -ys-ssss challenges for voxel-based autoencoders in capturing its fine-grained morphology. LatentODE also demonstrates inferior performance since it relies on an autoencoder to obtain latent features of mesh representations. In contrast, BrainODE preserves structural fidelity in such subregions by leveraging its deformation space modeling method.
Irregular time interval predictions. We further evaluate BrainODE unified dataset combining ADNI, OASIS, and AIBL, which contain irregular sampling intervals. As described in Table 2 BrainODE again achieves superior performance compared to other methods capable of handling irregular time intervals. However, unlike the previous experiment with regular intervals, BrainODE's 4-shot prediction results exhibit degraded performance relative to Table 1. This decline be attributed to the increased complexity irregular datasets; for instance, ADNI and OASIS include subjects with wide age range (e.g., 55-93 years), introducing greater inter-subject variability in disease progression. In contrast, LBC1936 consists of subjects with identical ages across each 3-year acquisition, and AIBL features shorter intervals of approximately 1.5 years, resulting in more temporally consistent trajectories.
In comparison, BrLP shows performance improvement in the irregular 4-shot setting. However, compared to deformation methods, we found that this image-generative approach cannot preserve the subject consistency due to autoencoding and prior sampling. While iterative sampling of BrLP mitigates this limitation statistically, it cannot guarantee that the generated subject identity matches the input, further discussed in subsection C.5
4.2 Qualitative results and condition fidelity
Qualitative results. For a qualitative comparison, we visualize the Fuclidean distance error maps on the predicted meshes in Figure 2, illustrating the performance of 4-shot predictions using linear extrapolation, LatentODE, BrLP, ShapeFlow, and our BrainODE. Consistent with the quantitative results in Table BrainODE exhibits the smallest Euclidean distance errors for both the LV and hippocampus shapes from the ground truth (GT). with error maps primarily in the blue range. Linear extrapolation yields plausible shapes but requires more than two time points and generates noisier surfaces, as indicated by the yellow and green regions (up to 2 mm errors). Furthermore, when the intral becomes greater, this method produces abnormal shapes and therefore is not appropriate for shape prediction (see Figure 14). BrLP predicts longitudinal individual hippocampal and LV shapes with spatial-temporal consistency each subject, but voxel-based generation introduces significant deviations from the GT, with errors reaching up to 4 mm (yellow areas). LatentODE struggles to

---

Age
Age
Age
Cogritive 00)
Inter 3132
2929
2763
Velitee:
2724
2600
"
Cogribive Status
Atzhelmer's (1
Figure 3: Demonstration of shape prediction by BrainODE with the age and cognitive status condi- tions. The hippocampal shapes the first row from the age of 65 to 77 with 3-year intervals. The second column is shapes from the age of 80 to 92.
capture unique shape features from individuals, often producing nearly identical LV shapes for different subjects, evidenced by uniform error distributions. ShapeFlow inherits individual shape characteristics via its flow-bol fow-based ODE, but its assumption of uniform time intervals during training leads to higher errors (2-4 mm) compared BrainODE, which effectively adapts to varying intervals. Overall, BrainODE demonstrates superior performance with smoother error distributions that closely align with the GT shapes across both regions.
Condition-injective shape modeling. To evaluate the qualitative effects of condition injection in BrainODE, visualize predicted hippocampal meshes under varying injected conditions in Figure Specifically, we compare predictions starting from the same initial hippocampal shape while varying age (65 and 80 years) and cognitive status (c 0 for normal cognition, c for declined cognition). As shown, BrainODE models longitudinal hippocampal shape changes while reflecting medical priors that older age and declined cognition accelerate hippocampal volume degradation [4]. For starting age of 65, a cognitively normal subject (c 0) exhibits a hippocampal volume reduction from 3168 to 2753 mmm3, whereas subject with declined cognitive status (c 1) shows larger reduction from 3168 to 2654 mmn'. Furthermore, for the same cognitive status of AD, older subjects starting at age 80 undergo more significant volume reduction (e.g., from 3168 2460 mmm) compared to their younger counterparts. These results highlight the condition-injectii deformation based on given conditions. Further quantitative analysis of -injetio-injeetivityy is provided in subsection C.4]
of BrainODE to predict
4.3 Evaluation on the Pseudo-cognitive Status Embedding
We perform an ablation study on BrainODE and its pseudo-cognitive status embedding, which includes the pseudo-cognitive status shape sampling and the cogmition estimator тę. Beyond shape modeling, we conduct a diagnosis task for neurodegenerative disorders using т and report the classification accuracy. Tabll 3) summarizes the results, where the vanilla Bascline (described in subsection 3.1 excludes both the pseudo-cognitive status shape sampling and the cognition estimator. The pretrained shows an accuracy of 88 88.3% in diagnosing normal cognition (NC) and AD based on shape representations. Compared to the Baseline, incorporating only the cognitive estimator тa in the BrainODE pipeline improves shape modeling accuracy for NC and AD (from 0.636 to 0.611). but reduces performance for converted cases (CONV, from 0.282 to 0.310). pseudo-cognitive status shape sampling to augment the dataset enhances shape modeling precision in terms of Euclidean distance, particularly for the CONV cases (from 0.282 to 0.178). However, it slightly decreases the diagnosis accuracy (from 0.883 to 0,875) compared to training with only observed data.
contrast, applying
When both approaches are combined, BrainODE achieves the lowest shape modeling errors for NC and AD (0.606) and a competitive performance for CONV (0.216), outperforming the Baseline across all categories. This suggests that the estimated cognitive status è provides geometric cues that enhance BrainODE's understanding of neurodegenerative disorders. enabling the reflection of this knowledge in the shape modeling process within the shape space. Notably, during inference, applying the pseudo-cognitive status embedding method boosts the prediction accuracy 89. the highest among all configurations, highlighting its effectiveness in real-world scenarios.

---

5 Discussion
BrainODE is the first neural ODE-based framework for modeling longitudinal 3D brain shape trajectories. We focus on precise modeling of shape deformation in clinically important subregions, beyond the prior works that mainly targeted whole-brain image prediction.
Our contributions are threefold. First, BrainODE bridges neural ODE-based 3D shape modeling with longitudinal neurodegenerative disease. We propose a pipeline consisting of mesh reconstruction for longitudinal shape representation, PCA-based parameterization, and unified deformation- ODE framework across disease groups. While neural ODEs have shown promise in continuous- time modeling, they have not been applied to predict longitudinal brain subregion shapes. To fill this gap, we convert brain images to fine- teniigraiedd meshes and represent them in a compact PCA space. Then, we introduce BrainODE with continuous conditioning, pseudo-status embedding, and iterative shape sampling, for the model stability and disease-continuun plausibility. Our model achieves precise subject-level trajectory prediction with low computational cost (-375MB for training), while maintaining fidelity at both the individual and group levels.
Second, BrainODE the first large-scale application of longitudinal brain shape modeling. To the best of our knowledge, BrainODE is trained on multi-site datasets with 30 years age span. Through large-scale experiments, we demonstrate that BrainODE effectively captures brain morphological patterns and aging dynamics across both normal and AD trajectories. Furthermore, we identify fundamental limitations in existing generative image-based based models, which fail to precisely capture subregion-level deformation, and we offer straightforward adaptations of ODE methods to address this limitation.
Third, we define and simultaneously address key challenges unique to longitudinal brain shape modeling. These include data scarcity, irregular sampling intervals, the integration of medical priors, and the ability to predict only from a single observation. By defining these considerations upfront, BrainODE provides foundational blueprint for this brain morphology prediction area.
Limitations and Future Work. Despite these advances, BrainODE has several limitations. In this work, we utilize two cognitive status groups: normal cognition (NC) and Alzheimer's disease (AD). Although some datasets include mild cognitive impairment (MCI) as a separate category and intermediate cognition between NC and AD, excluded MCI from BrainODE modeling due its ambiguous clinical definition and variability in progression (e.g., assigning e == 0.5 for MCI seems straightforward yet cannot reflect severity in the values). In future work, aim to extend BrainODE to incorporate NC, MCI, and AD by establishing more robust and discriminative criteria for cognitive status. For subjects with multiple longitudinal observations (n 1). BrainODE can generate n predicted shapes corresponding to each time point. To evaluate n--hoo prediction performance, we simply average these predicted shapes. Developing more advanced aggregation strategies such as attention- or time-aware mechanisms remains an important direction for future research to better capture longitudinal progression. As a pioneering model for brain shape prediction under varying cognitive conditions, BrainODE introduces a pseudo-cognitive status embedding that enables intermediate shape prediction and the corresponding cognition estimation. In this work, we primarily estimate cognitive status using hippocampal shapes, guided by prior research [1]]30] In future work, we plan explore BrainODE and congnition estimator to incorporate additional brain regions.
Conclusion
We introduce BrainODE, a pioneering approach for predicting longitudinal brain shape deformations, taking into account individual demographies and cognitive status. To address the challenges of longi- tudinal prediction, including irregular time intervals, embedding medical priors, and practical usage, we propose BrainODE within neural ODE-based architecture that smoothly deforms input shapes to target shapes at future time points. In addition, we introduce a bbeudo-cognitive status embedding to progressively predict future shapes over intermediate time points with predicted cognitive decline using the proposed cognition estimator. Through both quantitative and qualitative experiments, we demonstrate that BrainODE effectively models brain dynamics over time, outperforming baseline methods. Given the importance of accurately predicting brain shape changes in key regions for disease diagnosis, our work focuses primarily on precise longitudinal brain shape modeling. Future work will extend this approach to other brain regions and to clinical applications of carly diagnosis.
10

---

Acknowledgement
This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea (OV))) joint research center to develop diagnosis and treatment strategy for neurodegenerative diseases) and Institute for Information & communications Technology Promotion (IITP) grant funded by the Korea government(MSIT) (No.00223446, Development of object-oriented synthetic data generation and evaluation methods).
hata:-00000000S..taaaaaaaaiiaaa of Korea-UK preclinical/cliniaal
References
[1] Suhyun Ahn, Wonjung Park, and Jinah Park. Are medical image generative models biologically trustworthy? Proceedings of the IEEF/CVF International Conference on Computer Vision, pages 633-643, 2025.
[2] Liana G Apostolova, Mona Beyer, Amity E Green, Kristy $ Hwang, Jonathan H Morra, Yi-Yu Chou, Christina Avedissian, Dag Aarsland, Carmen C Janvin, Jan P Larsen, et al. Hippocampal, caudate, and ventricular changes in parkinson's 's disease with and without dementia. Movement Disorders, 25(6):687-695, 2010.
[3] Hao Bai and Yi Hong. Noder: Image sequence regression based on neural ordinary differential equations. In International Conference Medical Image Computing and Computer-Assisted Intervention, pages 673-682. Springer, 2024.
[4] Richard Al Bethlehem, Jakob Seidlitz, Simon White, Jacob W Vogel, Kevin M Anderson, Chris Adamson, Sophie Adler, George S Alexopoulos, Evdokia Anagnostou, Ariosky Areces- Gonzalez, al. Brain charts for the human lifespan. Nature, 604(7906):525-533 2022.
[5] Anna Cascarano, Jordi Mur-Petit, Jeronimo Hernandez-Goonzale,,, Marina Camacho, Nina de Toro Eadie, Polyxeni Gkontra, Mare Chadeau-Hyy Hyam, Jordi Vitria, and Karim Lekadir. Ma- chine and deep learning for longitudinal biomedical data: a review of methods and applications. Artificial Intelligence Review, 56(Suppl 2):1711-1771, 2023.
[6] Joshua Cates, Shireen Elhabian, and Ross Whitaker. Shapeworks: particle-based shape corre- spondence and visualization software. In Statistical Shape and Deformation Analysis, pages 257-298. Elsevier, 2017.
[7] Ricky TQ Chen, Yulia Rubanova, Jesse Bettencourt, and David K Duvenaud. Neural ordinary differential equations. Advances in neural information processing systems, 31, 2018.
[8] Hyuna Cho, Ziquan Wei, Seungjoo Lee, Tingting Dan, Guorong Wu, and Won Hwa Kim. Con- ditional diffusion with ordinal regression: Longitudinal data generation for neurodegenerative disease studies. In The Thirteenth International Conference Learning Representations.
[9] Ian Deary, Alan Gow, Michelle D Taylor, Janie Corley, Caroline Brett, Valerie Wilson, Harry Campbell, Lawrence J Whalley, Peter M Visscher, David J Porteous, et al. The lothian birth cohort 1936: a study to examine influences on cognitive ageing from age to age 70 and beyond. BMC geriatrics, 7:1-12, 2007.
[10] Kathryn A Ellis, Cassandra Szoeke, Ashley I Bush, David Darby, Petra L Graham, Nicola Lautenschlager, S Lance Macaulay, Ralph N Martins, Paul Maruff, Colin L Masters, et al. Rates of diagnostic transition and cognitive change at 18-month follow-up among 1,112 12 participants in the australian imaging, biomarkers and lifestyle flagship study of ageing (aibl). International Psychogeriatrics, 26( 26(4):543-554) 2014.
[11] Will Grathwohl, Ricky T. Q. Chen, Jesse Bettencourt, and David Duvenaud. Scalable re- versible generative models with free-form continuous dynamics. International Conference on Learning Representations, 2019.
[12] Clifford Jack Jr, Matt A Bernstein, Nick Fox, Paul Thompson, Gene Alexander, Danielle Harvey, Bret Borowski, Paula J Britson, Jennifer L. Whitwell, Chadwick Ward, et al. The
11

---

alzheimer's 's disease neuroimaging initiative (adni): Mri methods. Journal of Magnetic Reso- nance Imaging: An Officcaal Journal of the International Society for Magnetic Resonance in Medicine,
200.
[13] Bailiang Jian, Jiazhen Pan, Yitong Li, Fabian Bongratz, Ruochen Li, Daniel Rueckert, Benedikt Wiestler, and Christian Wachinger. Timeflow: Longitudinal brain image registration and aging progression analysis. arXiv preprint arXiv:2501.0866) 2025.
[14] Chiyu Jiang, Jingwei Huang, Andrea Tagliasacchi, and Leonidas Guibas. Shapeflow: Learn- able deformation flows among 3d shapes. Advances in Neural Information Processing Systems, 33:9745-9757, 2020.
[15] Firas Khader, Gustav Müller-Franzes Soroosh Tayebi Arasteh, Tianyu Han, Christoph Haar- burger, Maximilian Schulze-Hagen, Philipp Schad, Sandy Engelhardt, Bettina Bacßler, Sebas- tian Foersch, et al. Denoising diffusion probabilistic models for 3d medical image generation. Scientific Reports, 13(1):3303, 2023.
[16] Jacil Kim, Maria del C Valdes- Hernandez, Natalie A Royle, and Jinah Park. Hippocampal shape modeling based on a progressive template surface deformation and its verification. IEEE Iransactions on medical imaging, 1242-1262,,,, 2014.
[17] Dmitrii Lachinov, Arunava Chakravarty, Christoph Grechenig, Ursula Schmidt-Erfune and Hrvoje Bogunović. Learning caatio-tempora model of disease progression with neuralodes from longitudinal volumetric data. IEEE Transactions Medical Imaging, 43(3): 1665-199551 2023.
[18] Chen Liu, Ke Xu, Liangbo L Shen, Guillaume Huguet, Zilong Wang, Alexander Tong, Danilo Bzdok, Jay Stewart, Jay C Wang. Lucian Del Priore, et al. Imageflownet: Forecasting multiscale image-level trajectories of discase progression with crrgularly-samplee longitudinal medical images. In ICASSP 2025-2025 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pages 1-5. IEEE, 2025.
[19] Manhua Liu, Fan Li, Hao Yan, Kundong Wang, Yixin Ma, Li Shen, Mingqing Xu, Alzheimer's Disease Neuroimaging Initiative, al. A multi-model deep convolutional neural network for automatic hippocampus segmentation and classification in alzheimer's 's disease. Neuroimage, 208: 16459, 2020.
[20] Daniel S Marcus, Anthony F Fotenos, John G Csernansky, John C Morris, and Randy L Buckner. Open access series of imaging studies: longitudinal mri data in nondemented and demented older adults, 2010.
[21] Emile Mathieu and Maximilian Nickel. Riemannian continuous normalizing flows. Advances in Neural Information Processing Systems, 33:2503-2515, 2020.
[22] C DeCarli MD, James V Haxby, JA Gillette, D Teichberg, SI Rapoport, and MB Schapiro. Longitudinal changes lateral ventricular volume in datients with dementia of the alzheimer type. Neurology, 42(10))2029
1992.
[23] Sean M Nestor, Raul Rupsingh, Michael Borrie, Matthew Smith, Vittorio Accomazzi, Jennic Wells, Jennifer Fogarty, Robert Bartha, and Alzheimer's Disease Neuroimaging Initiative. Ventricular enlargement as possible measure of alzheimer's 's disease progression validated using the alzheimer's 's disease neuroimaging initiative database. Brain, 131(9):2443-245 2008.
[24] Lemuel Puglisi, Daniel C Alexander, and Daniele Ravi. Enhancing spatiotemporal disease progression models via latent diffusion and prior knowledge. In International Conference on Medical Image Computing and Computer-Assister Intervention, pages 173-183. Springer, 2024.
[25] Martin Rohbeck, Edward De Brouwer, Charlotte Bunne, Jan-Christian Huetter, Anne Biton, Kelvin Y Chen, Aviv Regev, and Romain Lopez. Modeling complex system dynamics with flow matching across time and conditions. The Thirteenth International Conference on Learning Representations, 2025.
12

---

[26] Javier Romero, Dimitrios Tzionas, and Michael J. Black. Embodied hands: Modeling and capturing hands and bodies together. ACM Transactions on Graphics, (Proc. SIGGRAPH Asia), 36(6), November 2017.
[27] Yulia Rubanova, Ricky TQ Chen, and David K Duvenaud. Latent ordinary differential equations time series. Advances in neural information processing systems, 32,
for
2019.
[28] Xiaoying Tang, Dominic Holland, Anders M Dale, Laurent Younes, Michael I Miller, and Alzheimer's s Disease Neuroimaging Initiative. Shape abnormalities of subcortical and ventricular structures in mild cognitive impairment and alzheimer's is disease: detecting, quantifying, and predicting. Human brain mapping, 35()):370133725, 2014.
[29] Fabio Trimarchi, Placido Bramanti, Silvia Marino, Demetrio Milardi, Debora Di Mauro, Giuseppe lelitro, Barbara Valenti, Gianluigi Vaccarino, Carmelo Milazzo, and Giuseppina Cutroneo. Mri 3d lateral cerebral ventricles in living humans: morphological and morpho- metrical age- gender-reeltee preliminary study. Anatomical science international, 88:61-69, 2013.
[30] Ingrid S van Maurik, Stephanie Vos, Isabelle Bos, Femke H Bouwman, Charlotte E Teunissen, Philip Scheltens, Frederik Barkhof, Lutz Frolich, Johannes Kornhuber, Jens Wiltfang, et al. Biomarker-basee prognosis for people with mild cognitive impairment (abide): modelling study. The Lancet Neurology, 18(1 18(11): 1e11110441000 1044, 2019.
[31] Yifan Wu, Tom Z Jiahao, Jiancong Wang. Paul A Yushkevich, M Ani Hsieh, and James C Gee. Nodeo: A neural ordinary differential equation based optimization framework for deformable image registration. In Proceedings of the IEEE/CVF conference on computer vision and pattern 2022.
recognition, pages
[32] Guandao Yang. Xun Huang, Zekun Hao, Ming- Yu Liu, Serge Belongie, and Bharath Hariharan. Pointflow: 3d point cloud generation with continuous normalizing flows. In Proceedings of the IEEE/CVF international conference on computer vision, pages 4541 14550, 2019.
[33] Jee Seok Yoon, Chenghao Zhang. Heung-Il Suk, Jia Guo, and Xiaoxiao Li. Sadm: Sequence- aware diffusion model for longitudinal medical image generation. In International Conference on Information Processing in Medical Imaging, pages 388 400. Springer, 2023.
[34] Rachid Zeghlache, Pierre-Henri Conze, Mostafa EI Habib Daho, Yihao Li, Hugo Le Boité, Ramin Tadayoni, Pascale Massin, Béatrice Cochener, Alireza Rezaei, Ikram Brahim, al. Latim: Longitudinal representation learning in continuous-time models to predict disease progression. In International Conference on Medical Image Computing and Computer-Asissted Intervention, pages 404-414. Springer, 2024.
[35] Zihao Zhu, Tianli Tao, Yitian Tao, Haowen Deng, Xinyi Cai, Gaofeng Wu, Kaidong Wang, Haifeng Tang, Lixuan Zhu, Zhuoyang Gu, et al. Loci-diffcom: Longitudinal consistency- informed diffusion model for 3d infant brain image completion. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 249-258. Springer, 2024.
13

---

NeurIPS Paper Checklist
1. Claims
Question: Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope?
Answer: [Yes]
Justification: The main claims in the abstract and introduction are consistent with the contributions and scope presented in the paper.
Guidelines:
The answer NA means that the abstract and introduction do not include the claims made the paper.
The abstract and/or introduction should clearly state the claims made, including the contributions made in the paper and important assumptions and limitations. A No or
NA answer to this question will not be perceived well by the reviewers.
The claims made should match theoretical and experimental results, and reflect how much the results can be expected to generalize to other settings.
is fine to include aspirational goals as motivation as long as is clear that these goals not attained by the paper.
2. Limitations
Question:
Does the paper discuss the limitations the work performed by the authors?
Answer: [Yes]
Justification:
We acknowledge the limitation in the discussion section.
Guidelines:
The answer NA means that the paper has no limitation while the answer No means that the paper has limitations, but those are not discussed in the paper.
The authors are encouraged to create a separate "Limitations" section their paper.
The paper should point out any strong assumptions and how robust the results are to violations of these assumptions (e.g., independence assumptions, noiseless settings, model well-specification, asymptotic approximations only holding locally). The authors should reflect how these assumptions might be violated in practice and what the implications would be.
The authors should reflect on the scope of the claims made, e.g., if the approach was only tested on few datasets or with few runs. In general, empirical results often depend on implicit assumptions, which should be articulated.
The authors should reflect on the factors that influence the performance of the approach. For example, facial recognition algorithm may perform poorly when image resolution is low images are taken in low lighting. Or speech-to-text system might not be used reliably to provide closed captions for online lectures because it fails to handle technical jargon.
The authors should discuss the computational efficiency of the proposed algorithms and how they scale with dataset size.
If applicable, the authors should discuss possible limitations of their approach to address problems of privacy and fairness.
While the authors might fear that complete honesty about limitations might used by reviewers as grounds for rejection, worse outcome might be that reviewers discover limitations that aren't 't acknowledged judgment and recognize that individual actions in favor of transparency play impor- tant role in developing norms that preserve the integrity of the community. Reviewers will be specifically instructed to not penalize honesty concerning limitations.
the paper. The authors should use their best
3. Theory assumptions and proofs
Question: For each theoretical result, does the paper provide the full set of assumptions and complete (and correct) proof?
Answer: [NA]
14

---

Justification:
Guidelines:
The answer NA means that the paper does not include theoretical results.
All the theorems, formulas, and proofs in the paper should be numbered and cross- referenced.
All assumptions should be clearly stated or referenced the statement of any theorems.
The proofs can either appear the main paper or the supplemental material, but they appear in the supplemental material, the authors are encouraged to provide short proof sketch to provide intuition.
Inversely, any informal proof provided in the core of the paper should be complemented by formal proofs provided in appendix supplemental material.
Theorems and Lemmas that the proof relies upon should be properly referenced.
4. Experimental result reproducibility
Does the paper fully disclose all the information needed to reproduce the main ex- perimental results of the paper to the extent that it affects the main claims and/or conclusions of the paper (regardless of whether the code and data are provided or not)?
Question:
Answer: [Yes]
Justification: We provide information to reproduce the main experimental results and algorithms.
Guidelines:
The answer NA means that the paper does include experiments. If the paper includes experiments, No answer to this question will not be perceived well by the reviewers: Making the paper reproducible is important, regardless of whether the code and data are provided or not.
If the contribution is dataset and/or model, the authors should describe the steps taken to make their results reproducible or verifiable.
Depending on the contribution, reproducibility can be accomplished in various ways. For example, if the contribution is novel architecture, describing the architecture fully might suffice, or if the contribution is a specific model and empirical evaluation, it may be necessary to either make possible for others to replicate the model with the same dataset, or provide access to the model. In general. releasing code and data is often one good way accomplish this, but reproducibility can also be provided via detailed instructions for how to replicate the results, access to a hosted model (e.g.. in the case of large language model), releasing of a model checkpoint, or other means that are appropriate to the research performed.
While NeurIPS does not require releasing code, the conference does require all submis- sions to provide some reasonable avenue for reproducibility, which may depend on the nature of the contribution. For example
(a) If the contribution is primarily a new algorithm, the paper should make it clear how reproduce that algorithm. (b) If the contribution is primarily
model architecture, the paper should describe the architecture clearly and fully.
(c) If the contribution is a new model (e.g., a large language model), then there should either be way to access this model for reproducing the results or way to reproduce the model (e.g., with an open-source dataset instructions for how to construct the dataset).
(d) We recognize that reproducibility may be tricky in some cases, in which case authors are welcome deseribe the particular way they provide for reproducibility, In the case of closed-source models, it may be that access to the model is limited in some way (e.g., registered users), but it should be possible for other researchers to have some path to reproducing or verifying the results.
5. Open access to data and code
Question: Does the paper provide open access to the data and code, with sufficient instruc- tions to faithfully reproduce the main experimental results, as described in supplemental material?
15

---

Answer: [Yes]
Justification: We provide data information, code, and sufficient instructions in the main and supplemental materials.
Guidelines:
The answer NA means that paper does not include experiments requiring code.
Please see the NeurlPS code and data submission guidelines (https://nips... cc/
for more details.
While we encourage the release of code and data, we understand that this might not be possible, "No" is an acceptable answer. Papers cannot be rejected simply for not including code, unless this is central to the contribution (e.g., for a new open-source benchmark).
The instructions should contain the exact command and environment needed to run reproduce the results. See the NeurlPS code and data submission guidelines (https /pips.ccc/pulic//uuui guwde//ddee/ee/ Cot//ooodeeeiiseeeeeeee hwtpbbmmsssiooooooooiooo
for more details.
The authors should provide instructions on data access and preparation, including how to access the raw data, preprocessed data, intermediate data, and generated data, etc.
The authors should provide scripts to reproduce all experimental results for the new proposed method and baselines. only subset of experiments are reproducible, they should state which ones are omitted from the script and why.
At submission time, to preserve anonymity, the authors should release anonymized versions (if applicable).
Providing as much information possible in supplemental material (appended to the paper) recommended, but including URLs to data and code is permitted.
6. Experimental setting/details
Question: Does the paper specify all the training and test details (e.g., data splits, hyper- parameters, how they were chosen, type of optimizer, etc.) necessary to understand the results?
Answer: [Yes]
Justification: Throughout the main and supplementary materials, we offer training and test details.
Guidelines:
The answer NA means that the paper does not include experiments.
The experimental setting should be presented in the core of the paper to level of detail that is necessary to appreciate the results and make sense of them.
The full details can be provided either with the code, in appendix, or as supplemental material.
7. Experiment statistical significance
Does the paper report error bars suitably and correctly defined or other appropriate information about the statistical significance of the experiments?
Question:
Answer: [No]
Justification: The paper does not report error bars or statistical significance tests for the main experimental results.
Guidelines:
The answer NA means that the paper does not include experiments.
The authors should answer "Yes" if the results accompanied by error bars, confi- dence intervals, or statistical significance tests, at least for the experiments that support the main claims of the paper.
The factors of variability that the error bars are capturing should be clearly stated (for example, train/test split, initialization, random drawing of some parameter, or overall run with given experimental conditions).
The method for calculating the error bars should be explained (closed form formula, call to a library function, bootstrap, etc.)
16

---

The assumptions made should be given (e.g.. Normally distributed errors).
It should be clear whether the error bar is the standard deviation the standard error of the mean.
It is OK to report 1-sigma error bars, but one should state it. The authors should preferably report a 2-sigma error bar than state that they have 96% CI, if the hypothesis of Normality of errors is not verified.
For asymmetric distributions, the authors should be careful not to show in tables or figures symmetric error bars that would yield results that are out of range (e.g. negative emrr rates).
If error bars are reported in tables or plots, The authors should explain in the text how they were calculated and reference the corresponding figures or tables in the text.
8. Experiments compute resources
For each experiment, does the paper provide sufficient information on the com- puter resources (type of compute workers, memory, time of execution) needed reproduce the experiments?
Question:
Answer: [Yes]
Justification:
We offer memory information using the proposed method.
Guidelines:
The answer NA means that the paper does not include experiments.
The paper should indicate the type of compute workers CPU or GPU, internal cluster, or cloud provider, including relevant memory and storage.
The paper should provide the amount of compute required for each of the individual experimental runs well estimate the total compute.
The paper should disclose whether the full research project required more compute than the experiments reported in the paper (e.g., preliminary or failed experiments that didn't make into the paper).
9. Code ethics
Question: Does the research conducted in the paper conform, in every respect, with the NeurIPS Code of Ethics
Answer: [Yes]
Justification: Yes. We preserve anonymity.
Guidelines:
The answer NA means that the authors have not reviewed the NeurIPS Code of Ethics.
If the authors answer No, they should explain the special circumstances that require deviation from the Code of Ethics.
The authors should make sure to preserve anonymity (e.g.. if there is a special consid- eration due to laws or regulations their jurisdiction).
10. Broader impacts
Question: Does the paper discuss both potential positive societal impacts and negative societal impacts of the work performed?
Answer:
[Yes]
Justification: discussed potential societal impacts, and limitations.
Guidelines:
The answer NA means that there no societal impact of the work performed.
If the authors answer NA or No, they should explain why their work has no societal impact why the paper does not address societal impact.
Examples of negative societal impacts include potential malicious or unintended uses (e.g., disinformation, generating fake profiles, surveillance), fairness considerations (e.g., deployment of technologies that could make decisions that unfairly impact specific groups), privacy considerations, and security considerations.
17

---

The conference expects that many papers will be foundational research and not tied particular applications, let alone deployments. However, if there is direct path to any negative applications, the authors should point it out. For example, it is legitimate to point out that an improvement in the quality of generative models could be used to generate deepfakes for disinformation. On the other hand, it is not needed to point out that a generic algorithm for optimizing neural networks could enable people to train models that generate Deepfakes faster.
The authors should consider possible harms that could arise when the technology is being used as intended and functioning correctly, harms that could arise when the technology being used intended but gives incorrect results, and harms following from (intentional or unintentional) misuse of the technology.
If there negative societal impacts, the authors could also discuss possible mitigation strategies (e.g., gated release of models, providing defenses in addition to attacks, mechanisms for monitoring misuse, mechanisms monitor how system learns from feedback over time, improving the efficiency and accessibility of MI..
11. Safeguards
Does the paper describe safeguards that have been put in place for responsible release of data models that have high risk for misuse (e.g., pretrained language models, image generators, or scraped datasets)?
Question:
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper poses such risks.
Released models that have high risk for misuse or dual-use should be released with necessary safeguards to allow for controlled use of the model, for example by requiring that users adhere usage guidelines or restrictions to access the model or implementing safety filters.
Datasets that have been scraped from the Internet could pose safety risks. The authors should describe how they avoided releasing unsafe images.
We recognize that providing effective safeguards is challenging, and many papers do not require this, but we encourage authors to take this into account and make best faith effort.
12. Lieenses for existing assets
Are the creators or original owners of assets (e.g., code, data, models), used in the paper. properly credited and are the license and terms of use explicitly mentioned and properly respected?
Question:
Answer: [Yes]
Justification: Yes. See dataset details.
Guidelines:
The answer NA means that the paper does existing assets.
The authors should cite the original paper that produced the code package or dataset.
The authors should state which version of the asset is used and, if possible, include URL.
The name of the license (c.g., CC-BY 4.0) should be included for each asset.
For scraped data from a particular source (c.g., website), the copyright and terms of service of that source should be provided.
If assets are released, the license, copyright information, and terms of use in the package should be provided. For popular datasets, paperswi thcode. com com/datasets has curated licenses for some datasets. Their licensing guide can help determine the license of dataset.
For existing datasets that are re-packaged, both the original license and the license of the derived asset (if it has changed) should be provided.
18

---

If this information is not available online, the authors are encouraged to reach out to the asset's creators.
13. New assets
Question: Are new assets introduced in the paper well documented and is the documentation provided alongside the assets?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not release new assets.
Researchers should communicate the details of the dataset/code/mmoel as part of their submissions via structured templates. This includes details about training, license, limitations, etc.
The paper should discuss whether and how consent obtained from people whose asset is used.
At submission time, remember to anonymize your assets (if applicable). You can either create an anonymized URL or include an anonymized zip file.
14. Crowdsourcing and research with human subjects
Question: For crowdsourcing experiments and research with human subjects, does the paper include the full text of instructions given to participants and screenshots, if applicable, as well details about compensation (if any)?
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the paper does not involve crowdsoureing nor research with human subjects.
Including this information in the supplemental material is fine, but if the main contribu- tion of the paper involves human subjects, then as much detail as possible should be included the main paper.
According the NeurIPS Code of Ethics, workers involved in data collection, curation, or other labor should be paid at least the minimum wage in the country of the data collector.
15. Institutional review board (IRB) approvals or equivalent for research with human subjects
Question:
Does the paper describe potential risks incurred by study participants, whether such risks were disclosed to the subjects, and whether Institutional Review Board (IRB) approvals (or an equivalent approval/review based on the requirements of your country or institution) were obtained?
Answer: [Yes]
Justification: We properly use open dataset with access.
Guidelines:
The answer NA means that the paper does not involve crowdsourcing nor research with human subjects.
Depending the country in which research is conducted, IRB approval (or equivalent) may be required for any human subjects research. If you obtained IRB approval, you should clearly state this in the paper.
We recognize that the procedures for this may vary significantly between institutions and locations, and expect authors to adhere to the NeurlPS Code of Ethics and the guidelines for their institution.
For initial submissions, do not include any information that would break anonymity (if applicable), such the institution conducting the review.
16. Declaration of LLM usage
19

---

Does the paper describe the usage of LLMs if it is important, original, or non-standard component of the core methods this research? Note that if the LLM used only for writing, editing, or formatting purposes and does not impact the core methodology, scientific rigorousness, or originality of the research, declaration is not required.
Question:
Answer: [NA]
Justification:
Guidelines:
The answer NA means that the core method development in this research does not involve LLMs any important, original, non-standard components.
Please refer to our LLM policy for what should or should not be described.
20

---

A Implementation details
This section provides an overview of the design choices and implementation details for BrainODE. We begin by detailing the longitudinal brain shape reconstruction process from 3D brain MRI scans, which involves segmentation and reconstruction of the hippocampal and lateral ventricle (LV) shapes. Subsequently, we evaluate the effectiveness of principal component analysis for representing variable brain shapes, reducing dimensionality while preserving key shape features. Afterward, we present details of the neural architecture of BrainODE and training strategies. Our source code, including data preprocessing scripts and model configurations, is publicly available at httpp://gii https://github... tighhb......
wwapjjgggg///DDDDD BrainODE
A.1 Longitudinal brain shape reconstruction
Longitudinal brain reconstruction
Vertex optimization model method for timms
IMSIGT ecracioon
Source mosh Miss
Target stape P
up30c0
Beg-wtraion
= (spr
Ag Use P
Pgr = (ayp)
You (h toc
Duforee much
M,
Vatien Opeiniiing
Vatie opennring
Vatiee Optimiieeg
AP
Templeie Mach M, = (P,.))
Ni (9.
Nr - 1l
aetomatrr
(8)
(b)
Figure 4: Illustration of brain shape reconstructions for single subject by iteratively deforming template mesh to match longitudinal MRI scans.
We reconstruct the longitudinal brain shapes == (Vi, V2, V. by modifying the brain shapes modeling method proposed by Park er al. [1I. Their approach reconstructs cross-sectional individual brain shapes by iteratively deforming template mesh to align with target shapes, which repre- sented as point clouds extracted from the boundaries of segmented 3D brain MRI masks. To adapt this cross-sectional method for longitudinal analysis, over time, using the deformed mesh from the previous time point, M;-1, as the initial mesh for the next step. M,. The process begins with the first observed MRI scan and proceeds iteratively through subsequent time points, optimizing vertex positions to minimize both distance and regularization loss, as illustrated in Figure 4
sequentially update template mesh Mo
As part of this pipeline, we use SynthSeg [ to segment brain regions from MRI scans. From the boundaries of the segmented masks, point clouds P, representing the target shapes constructed. The initial shape V1 obtained by deforming the template mesh vertices Vi to fit the target point cloud P,. Then, each subsequent shape V, is reconstructed by deforming the previous mesh V to align with the corresponding target P,, using a vertex optimization model.
The vertex optimization model (Figure 4 4b)) follows the optimization strategy of Point2Mesh 4. which iteratively updates the vertices to align with the target point cloud. The objective function consists of two components: 1) distance loss Laists which measures the discrepancy between the deformed mesh Vú and the target point cloud P., and 2) a regularization loss Lieg, which encourages smooth and plausible deformations.
Specifically, the distance loss Lain combines the Chamfer distance Ly between V, and P,, and the point-to fo-tt distance Lpm between P: and the mesh surface M; (i.e., Ldist = XgLa + Apm Cpm) The regularization loss Lryg includes vertex displacement ||Averi]|2, normal displacement ||Anorm]] edge length variance Ledge prevent mesh distortion, normal consistency Lrome (norm),))) and Laplacian loss Liop for surface smoothness. The final form of the loss function used the vertex optimization model is defined as follows:
21

---

L = Lidist + Love
(5)
where Leg Averllr Anomm |nnormllz Aedge AconsCoom (norm) Atap Ctup-
We empirically set hyperparameters for each loss as (det, Apm: Avert, Anom Aedges Acoos, Map) {0.5, 3, 1, 1500, 1,
A.2 Effeet of PCA on shape reconstruction
Row vertes poattio
gervell
Compactrass medeling
Pradoon
Thm
True
(a) Longitudinal data visualization
(b) Compactness PCA
:
;
}
Original shapes
PCA processed shapes
Original shapes
cccrr
processed shapos
(c) Longitudinal shapes after PCA processing
Figure 5: Impacts of shape dimensionality reduction using PCA. (a) Visualization of vertex positions and the first PCA eigenvalue across longitudinal observations from the LBC1936 dataset. (b) Com- pactness measures of hippocampus and L.V across varying numbers of PCA modes. (c) Qualitative comparison between original shapes and their PCA-reconstructed shapes with 150 modes.
Motivation of PCA. We adopt principal component analysis (PCA) to parameterize brain region shapes for longitudinal modeling, as described in subsection 3.1 Rather than using raw vertex- level inputs or latent codes from autocncoders, we choose PCA-based representations to facilitate reproducible and scalable research. This choice is inspired by prior work such as MANO (13). which models shape variation hands using a low-dimensional parametric space. Building on this foundation, subsequent research has demonstrated the effectiveness of directly predicting such parameters for shape estimation tasks [5] [4 [6]. In similar spirit, we leverage parametric modeling of brain shapes construct compact and anatomically meaningful shape spaces to support the leamning of the longitudinal shape dynamics.
Another key motivation for using PCA is its efficiency in shape representation. We visualize the vertex positions of brain shapes and the first eigenvalues across longitudinal time points in Figure 5)a). The raw vertex positions of hippocampal and LV shapes exhibit irregular deformations over time,
22

---

making it difficult to identify clear trends. This irregularity arises from the complex geometry of the original shape representation, which includes high-frequency artifacts (e.g., bumpy surfaces due to image-level noise) and substantial inter-subject variability,, illustrated in Figure 5[c) In contrast, PCA-derived eigenvalues show more consistent trends across longitudinal time points, enabling stable progressive modeling of brain shape dynamics. Moreover, shapes reconstructed using PCA exhibit smoother surfaces by filtering out high-frequency noise while preserving essential geometric features (Figure S(c)). This dimensionality reduction and smoothing effect of PCA not only simplifies the shape representation but also supports plausible modeling by providing stable inputs for capturing temporal progression in neurodegenerative disease.
Quantitative results of PCA. To quantitatively eval- uate the effectiveness of PCA-based shape representa- tion, we measured the compactness of the statistical shape models for both the hippocampus and LV,, where
Shapes
PCA Modes
Dist. (mm)
0.2642
50
0.3028
Compactness(#) (t) = مد for the first k modes. As depicted in Figure 5jb), the compactness reaches 0.980 for the hippocampus and 0.989 for the LV with k = 150 modes, indicating that over 98% of the total shape vari- ability is preserved in both cases.
100
0.3005
2i=1 A.
0.2640
200
0.2829
300
0.2701
Table 4: Prediction loss in Euclidean dis- tances (mm) when using brain shapes V as-is and using PCA coefficients A on hippocampal shapes of LBC1936.
In addition, validate the use of PCA coefficients as shape representations, we compare BrainODE performance us- ing original vertex-based shapes V and PCA-projected shapes A in modeling the hippocampus. As shown in Table the use of PCA coefficients not only results in smoother reconstructed surfaces but also improves the prediction accuracy. The ablation study shows that using 150 components provides favorable trade-off between reconstruction fidelity and dimensionality reduction.
A.3 Neural architecture of BrainODE
BrainODE. We design the BrainODE function fe as neural network with self-attention mechanism to model condition-injectiv shape dynamics. The input to fe consists of a shape embedding V e 1550 a scalar time t E RI, and scalar cognitive status E 1 These are concatenated into single vector and passed through three linear projections to form the query, key, and value. The attention output then processed by fully connected layers with GeLU ac- tivation. The output l(-)) a 150-dimensional vector representing the velocity in the shape space at time I.
Shape Feature (PCA)
Self Attention
A
Cognitive Status
Metmal
Scale
Age
Mormil
FCN
Cognition estimator. We adopt the cognition estima- tor тy as a simple 3D CNN architecture, followed by a prior study diagnose Alzheimer's disease with the hippocampus masks [9]. The input to To is shape in voxel representation, and the output the estimated cognitive status E [0, 1].
Linear 1x22
Prediction ODE
GeLLi
Figure 6: BrainODE architecture
A.4 Training details of BrainODE
The input to BrainODE fo (A, e) consists of PCA-reduced shape coefficients A, normalized age and cognitive status c. Each brain shape V; {fi, V2,,.., Un is represented as vertices of triangular mesh, where the coordinates are in real-world brain sizes with units in mm. The typical spatial extents of the hippocampus and LV are approximately (50 mm, 65 mm, 35 mm) and (60 mm, 100 mm, 80 mm), respectively. These vertex coordinates are projected onto a PCA basis, and the resulting coefficients A = A2, An) are used as shape inputs for training. To improve generalization with limited longitudinal data, we apply data augmentation via random scaling of training shapes by factor uniformly sampled from the range [0.98, 1.02].
23

---

The temporal imput t is derived by normalizing the age of subjects. Specifically, we select subjects aged between 65 and 95 years and linearly normalize their age (i.e., (age 65)/30 such that /. E (). The cognitive status is continuous value in the range [0, 1]. as described in subsection 3.2
BrainODE is trained for 100 epochs using the AdamW optimizer with learning rate of 0.0005. For solving the neural ordinary differential equations, we employ the fourth-order Runge- Runge-Kutta (RK4) integration method. The training process minimizes the L2 loss function, as derived in Equation 3)
B
Dataset details
Dataset
Subjects #
NC
AD
CONV
1st observed age
# of observ.
Intervals (yrs)
LBCT936
516
516
73.12 = 0.79
3.59 1.19
3.15 0.78
AIBL
211
134
21
73.87 = 5.25
3.04 = 1.15
1.66 + 0.57
ADNI
829
558
271
0
75.35 = 5.81
5.05 - 3.12
1.07 + 0.89
OASIS2
80
65
11
4
77.42 = 7.11
2.64 + 0.78
2.02 + 0.70
OASIS3
282
253
14
15
71.62 4.87
2.91 + 1.27
2.74 1.50
Table Composition longitudinal datasets. NC, AD, CONV denote whose cogmitive statuses are normal cognition, Alzheimer's disease, and converted from NC to AD, respectively.
Validation data
hippo
LV
Time interval
Dataset
4shot
Ishot
4shot
Ishot
LBC
20
20
Regular
AIBL
18
19
27
34
Irregular
AIBL + ADNI + OASIS2 + OASIS3
65
79
63
170
Table 6: Composition of validation dataset used in subsection 4.1 Table and Table 2
B.1 Dataset Composition
To learn longitudinal brain shape dynamics, we the LBC1936 2 and AIBL datasets regular time interval datasets, whereas AIBL [3]. ADNI (6), OASIS2 [1O), and OASIS3 g are used as irregular time interval datasets. As shown in Table S] each dataset varies in the number of subjects, age at the first brain image acquisition, number of observations, and intervals between scans.
The LBC1936 data are provided by the Lothian Birth Cohort 1936 Study database Most participants underwent brain magnetic resonance imaging between 2008 and 2010 at an average age of 72.6 years, with follow-up scans conducted every three years over five waves. The AIBL dataset collects scans every 18 months across five waves. Since AIBL includes subjects with varied ages and provides exact MRI acquisition dates, it is utilized for both regular and irregular time interval analyses. Furthermore, because AIBL,, OASIS2, and OASIS3 contain longitudinal data where cognitive status transitions from normal Alzheimer's disease, their shape data are used for pscudo-cognitive status shape sampling in subsection 3.2]
Validation data. The validation dataset composition is Table 6 used for the quantitative results in Table I and Table 2] Specifically, we selected subjects whose longitudinal observations are more than four times for the 4-shot prediction.
B.2 Medical priors in brain shapes.
In this section, we elaborate on the medical priors for longitudinal progression modeling of the brain, as discussed in sectccioon I Specifically, we analyze the volumes of the hippocampus and LV using all longitudinal datasets (ADNI, OASIS, AIBL, and LBC1936). Additionally, we visualize the rate of volume changes over time intervals to highlight the progression dynamics.
hhtps://oohhinn birtebiitthhhhheie birth-cohorts.ed ed.ac ul/daat-aces--collaartttoo
24

---

500
4010
Figure 7: Illustration of brain atrophy progression. (a) Regional volumes across longitudinal obser- vations. (b) Rate of volume changes over time intervals. The first and second rows represent the volumes of the hippocampus and LV, respectively, across all datasets (ADNI, OASIS, AIBL, and LBC1936).
Brain volume changes. The human brain undergoes brain atrophy (i.e., the loss brain cells) due to normal aging and neurodegenerative disorders. In particular, Alzheimer's 's Disease (AD) causes more rapid brain atrophy. As brain atrophy progresses, brain volumes shrink, while the LVs enlarge as the empty space surrounded by brain subcortical structures. Fiiuure 7[a) illustrates brain atrophy and associated volume changes across longitudinal observations for each subject in the brain MRI datasets. Hippocampal volumes show gradual decline from observation 0 to 4, with the trend less clear at the 5th visit due to limited data availability. Conversely, LV volumes consistently increase over time, reflecting ventricular enlargement caused by the brain atrophy.
To further explore the dynamics of brain atrophy, we calculate the rate of volume changes by dividing the volume difference by the corresponding time intervals (e.g., 1.5 years for AIBL, 3 years for LBC1936, and respective years for irregular time intervals). This rate, expressed in mmm/yeer, quantifies the speed of atrophy or enlargement. Figure 7]b) visualizes these rates across age groups, defined based on the dataset distributions. The hippocampus shows a higher rate of volume reduction in older age groups compared to the 47-57 group, whereas LV enlargement accelerates with age. Notably, since older brains are smaller in volume, similar absolute rate of volume loss implies relatively more severe atrophy in later life stages (e.g., 76-86 vs. 86-96 age groups). These trends align with the medical priors, confirming that older age exacerbates the progression of atrophy.
To assess the impact of cognitive status, Figure 8 compares the rate of volume changes between normal cognition (NC) and AD in the same age groups We visualize the change rate of the left and right brain regions separately. AD (red) exhibits a significantly faster and more irregular rate of volume change than NC (blue). For example, the 67-76 age group, the LV volume reduction rate approximately -700 mm2/yyer for NC, compared to -1600 mm with greater variability in AD. This heterogeneous pathological progression of AD presents challenges for modeling as it requires capturing complex and non-linear patterns that diverge from the more stable trends observed in NC.
C Additional analysis and visualization
In this section, we comprehensively evaluate longitudinal brain shape prediction on multiple perspec- tives. We first present additional experimental results from soossutttttiio 4.1 including the qualitative results of the I-shot prediction in/sussettion C.T. qualitative performance across benchmarks in subsection C.2 the effectiveness of the pseudo-cognitive status embedding in subsection C.3 and
25

---

Liipp Change Rute by Age Group and MCAD
1PMpps Charge tats by Age Gruup and NCAO
-00
O Change tite Ag: Dreas and HCAD
PLYY Change hans to Age Group and MCAD
200
400
ChiPpi
Figure 8: Rate of volume changes over time intervals for normal cognition (NC) and Alzheimer's Disease (AD). The age groups are split based on the data distribution from ADNI, OASIS, AIBL, and LBCI936. NC and AD groups are visualized in blue and red, respectively
BrainODE's condition fidelity in modeling Alzheimer's Disease (AD) in subsection C.4 Furthermore, we examine the subject inconsistency inherent in image generative approaches for brain region pro- methods in in/subsection C.5 compare the prediction method in subssectiion C.6] Finally, we evaluate the long-range
gression modeling compared to deformation-base performance with registration-Be shape prediction capabilities of baseline methods and BrainODE in subsection C.7
C.1 Additional analysis of experiments
We present qualitative results for brain shape modeling using 1-shot prediction in Figure 9 Unlike the 4-shot setting in Figure 2] which leverages multiple time points to guide the deformation trajectory, the 1-shot setting lacks temporal deformation information. As result, conventional extrapolation methods are excluded.
Qualitative evaluation of 1-shot prediction. To implement prediction on arbitrary time points using ShapeFlow 77. which does support irregular time intervals, we train the model on the AIBL [3] dataset using fixed 1.5-year intervals. The starting time point is set to f = 0, and 1.5 years is mapped 1. Consequently, future time targets are normalized relative this interval (e.g., 2-year prediction corresponds to = 2/1.5).
Other baseline methods that handle irregular time intervals also included for comparison. This setup allows us to evaluate the ability of each method to predict future shapes under varying temporal gaps. Notably, addressing irregular time intervals is critical in longitudinal medical imaging, as patient visit schedules often vary. Robust modeling across inconsistent time gaps is essential for accurately capturing disease progression.
Figure visualizes qualitative results of 1-shot prediction for hippocampus and LV across varying ages and cognitive statuses, including normal cognition (NC) and Alzheimer's disease (AD). Overall, BrainODE outperforms baseline methods modeling both brain regions. Specifically, for the hippocampus, both BrainODE and ShapeFlow achieve small Euclidean distances from the ground truth. However, BrainODE uniquely excels in LV prediction, demonstrating the lowest errors.
Regarding the baseline results, ShapeFlow accurately predicts hippocampus shapes but struggles to model the L.Vs, showing significant errors (> mm) in several regions. LatentODE [5] produces incorrect LV shapes by altering their topology and fails to capture subject-specific features, generating
26

---

Conditona
3 ; 3
Hippocampus
71to 74.5 e:1(AD)
cc0.NN))
Lateral Ventricle
CcRel Laten1DDE
IMC)
GT
BrainODE
ShapeFlow
BrLP
Figure 9: Qualitative results of 1-shot prediction for hippocampus and LV shapes. The color map indicates Euclidean distance errors (up to mm for hippocampus, 4 mm for LV) between ground truth (GT) and predicted shapes.
nearly identical shapes regardless of input conditions. This suggests that LatentODE struggles to model the prior distribution of brain progression dynamics using autoencoders, leading to statistically plausible yet trivial solutions (e.g., converging to average shapes) that disregard the given inputs. Similarly, BrLP distorts the anatomical features of the input shapes. For instance, over-enlargements are observed on the occipital horn, frontal horn, and body of the LV in NC subjects, and topological changes appear in occipital horn of the AD subject. In contrast, despite the challenges of model- ing brain progression with irregular time intervals, BrainODE consistently demonstrates superior performance, validating robustness and accuracy.
C.2 Qualitative evaluation across benchmarks
Dataset
4-shot
1-shot
AIBL
0.52 0.07
0.46 0.05
ADNI
0.59 0.59 0.14
0.54 0.15
OASIS
0.52 0.07
0.48 + 0.08
Table 7: Quatitative evaluation of hippocampal shape prediction performance in Euclidean distance (mm) on datasets with irregular time intervals.
The detailed qualitative results in Table by datasets are illustrated in Table 7 From these results, although the discrepancies are not substantial, it underscores the need for further validation across benchmarks. To this end, we conducted the following cross-benchmark experiments.
For Expl, BrainODE is trained on AIBL (with regular time intervals) and validated on ADNI and OASIS (with irregular time intervals). For Exp2. BrainODE is trained on AIBL, ADNI and OASIS (with irregular time intervals) and validated on LBC1936 (with regular time intervals).
These experiments allow
evaluate the robustness of BrainODE under both temporal and benchmark distribution shifts. As shown in Table 8, while performance under these shifts is slightly degraded compared intra-dataset benchmarks, BrainODE consistently outperforms other baseline models, demonstrating robustness to unseen individual shape morphologies and progression patterns. In terms of generalizability, each cohort (dataset) follows different temporal sampling intervals, ranging from 0.5 years (e.g., ADNI) to over 3 years (e.g., LBC1936). Due to this temporal discrepancy, BrainODE exhibits performance decreases in both shift scenarios. These findings underscore the importance of training on large-scale longitudinal datasets with varied time intervals to enhance
27

---

generalization across cohorts. Leveraging broader and more diverse training data is essential for improving robustness and capturing realistic progression dynamics.
Table Cross benchmark validation results for hippocampus.
Train
Validation
Euclidean dist. (mm)
Expl Exp2
AIBL
ADNI OASIS
0.518
AIBL ADNI + OASIS
LBC1936
0.522
C.3 Sampling longitudinal shapes with intermediate timepoints
7ays
70yes
(Dums)
1-op pred
1-hpp 73
c5
1ebp poed 75
1-bre 73
Hop pred 78
1-bp pred 78
(+) hippocampus
(0) Latecal Ventiscles
Figure 10: Qualitative results of sampling longitudinal shapes with intermediate time points for (a) hippocampus and (b) LVs.
In this section, we explore the effects incorporating pseudo-cognítive status embeddings and gener- ating shapes on intermediate time points to enable progressive longitudinal modeling. Specifically, we investigate how this approach impacts the accuracy and stability of shape predictions. To this end, we compare the qualitative results of shapes generated every 3 years over 12-year period with those produced in a single step (i.e., I-hop prediction).
Figure 10 illlustrats the results, showing the deformation of hippocampus and LV shapes from age 73 to 85, along with the Euclidean distance maps from the initial shape at age 73, Predicting brain shapes over long intervals in single step is challenging for BrainODE, as such large temporal gaps are not seen during training, where the average interval between the first and last acquisitions is 4.08 + 2.84 years. To address this, BrainODE adopts an iterative prediction strategy, forecasting future shapes through intermediate time steps, as shown the first row of the figure.
Direct prediction over a 1 12-year interval results in unrealistic deformations and surface artifacts; especially, the LV surface is almost entirely destroyed. In contrast, when intermediate predictions are made every 3 years, the final 12-year prediction maintains anatomical consistency and surface smoothness. This experiment highlights the effectiveness of progressive pseudo-shape sampling in modeling longitudinal brain progression.
C.4 Tendency of shape changes by age and cognitive state
Cop
To further demonstrate the condition-injective of BrainODE, we visualize shape predictions under varying cognitive statuses c= {0,00 0.5, 1.0) in Figure 11 BrainODE predicts larger deforma- tions both the hippocampus and LVs as cognitive status declines, consistent with clinical progression patterns observed in longi- tudinal data (subsection B.2). For instance, the shape predicted with intermediate status 0.5 shows deformations that lie be- tween those of normal cognition and Alzheimer's ' disease, as also reflected in their estimated volumes.
350
Figure 12: Brain Atrophy ten- dency by cognitive status e mod- eled by BrainODE.
Beyond qualitative visualization, we analyze volumetric trends across varying cognitive statuses for 100 baseline hippocampal
28

---

Ag
Age
Ag
- Normal
Normal
Vitee?)
prodromal
prodromal
3982
3474
3415
9393
3298
24063
25771
38103
41122
8982
3448
8378
3815
8262
24063
26730
33176
41261
(e) Hippocampus
(b) Lateral Verztidee
Figure 11: Longitudinal shape predictions with varying cognitive status = (0.0.5, 0.5, 1) every 3 years for hippocampus and LVV The number below each shape denotes the volume of the predicted shape.
shapes in Figure at the age of 65, Even though explicit medical priors is not injected during training. BrainODE captures clinically plausible volume trajectories by leveraging the spatiotemporal supervision from longitudinal datasets. This supports the ability of BrainODE to encode implicit medical priors through learning dynamics across both age and cognitive status c.
C.5 Subject inconsistency of generative approaches
Axial
mm
plane
Hippocampus
Euclidean distance
plane
LV
-2 mm
Segittal plane
Input
(a) Image comparison
(b) Shape comparison
Figure 13: Subject inconsistency of the BrLP 112 generative model. (a) Image comparison shows the input brain MRI scan (axial, coronal, sagittal planes) and the residual difference after BrLP generation with At O. (b) Shape comparison highlights Euclidean distance errors (up =2 mm) between the input and generated hippocampus and LV shapes.
In this study, we examine two primary approaches for modeling longitudinal progression in neu- rodegenerative diseases: generative models and deformation-ball based methods, which exhibit distinct characteristics. Generative approaches, such as BrLP [12), model medical priors using the distribution of given data and its associated conditions (e.g.. g., age and cognitive status). During inference. these models generate statistically plausible samples from the learned distribution. However, as discussed in subsection 4.1 we hypothesize that this approach is unsuitable for longitudinal progression modeling of medical data, as it is challenging to ensure spatial consistency of each subject (i.e., maintaining sampled data's identity). generative model by inputting a brain MRI scan with no time difference (Ai 0) and measuring the differences between the input and generated outputs both image and shape spaces, as depicted in
test this, we conducted a preservation experiment using the BrLP
Figure 13
29

---

In the image-level comparison (Figure 13 a)), a3),),, although the output is expected to be identical to the imput, noticeable differences are observed in high-freeuency details and anatomical structures such as the I.V, as highlighted by their residuals in the color map. We further visualize shapes of the hippocampus and LVs, which are of two key brain regions to determine the cognitive status, in the shape-level comparison (Figure Figuree 13)))). The Euclidean distance is calculated with the input subject's shapes, indicating significant deviations in the hippocampus and LV shapes. In addition, the LV has seamless structure in the brain, however, BrLP generates a disconnected LV due to limited voxel grid resolution. These inconsistencies could compromise longitudinal diagnosis, potentially perturbing the AD decision boundary of deterministic classification models sensitive to subtle changes, as reported in [9]. This highlights a fundamental limitation of generative models in maintaining subject-specific fidelity. In contrast, deformation based approaches such BrainODE offer improved consistency by preserving anatomical identity when A/. = 0, making them more suitable for reliable longitudinal modeling.
C.6 Comparison with registration-based method
We additionally experimented with NODER B. which aims to predict missing timepoints in brain image sequences. NODER requires at least two prior timepoints as input for sequence regression, which limits its practical applicability for predicting future shapes from a single observation. More- over, it requires separate training for each individual brain image sequence, making it unsuitable for learning generalizable longitudinal dynamics.
We conducted this experiment using the LBC1936 dataset under a 4-shot validation setting. Although NODER utilizes deformation information from earlier timepoints, its voxel-based regression approach and lack of explicit conditioning on age and disease lead to inferior predictive accuracy. This highlights the effectiveness of BrainODE's design, which captures shape dynamics among populations in mesh representation and explicitly conditions on age and cognitive status, enabling more accurate modeling of longitudinal shape changes.
Table 9: Comparison the hippocampal shape prediction performance in Euclidean distance (mm)
between registration-bass NODER B and ours the LBC1936 dataset.
Method
hippo
LV
NODER 3
0.898
4.152
Ours
0.488
1.630
C.7 Longitudinal shape prediction capacity
From a practical perspective, an important question arises in longitudinal modeling: how far into the future can models reliably predict brain shape progression while maintaining anatomical consistency? To investigate this, we evaluated the longitudinal prediction capabilities of five models-ShappFlow Extrapolation, BrLP, LatentODE, and BrainODE -uiin the LBCC193 dataset, as shown in Figure 14 The figure presents the predicted hippocampus shapes of single subject spanning ages 65 to 95 (within the training distribution) and extending 107 years (beyond the training distribution), demonstrating each model's 's ability to predict long-term brain shape progression.
Specifically, ShapeFloow fails predict beyond 89 years, the brain shapes "explode", making visualization infeasible. Extrapolation, while able to generate shape predictions, shows progressive collapse when forecasting further beyond the two input time points (e.g., 65 and 71 years), with no- ticeable distortions at 95 years. BrLP appears plausible at a glance, but as discussed in infuubsection C.5) exhibits subject inconsistency; the predicted shape at 71 years already shows changes in overall geometry and new appearances, not in the shape at 65 years. LatentODE, consistent with the main experimental results, struggles with prior modeling within its VAE framework, repeatedly predicting similar shapes (e.g., minimal variation in LV shape across all ages) due to training difficulties. In contrast, BrainODE demonstrates robust performance within the in-distribution age, progressively transforming shapes while preserving anatomical geometry features, such as the curvature of the hippocampus and the expansion patterns of the LV, even up to 95 years. This highlights superior capacity of BrainODE for long-tern longitudinal prediction compared to other models.
30

---

In-distribution
Out-of-distribution
Age (yrs)
95
C
ShapeFlow
Extrapolation
BrLP
LatentODE
BrainODE (Ours)
Figure 14: Comparison of longitudinal shape prediction over time for the hippocampus and L.V. Each column represents different model, showing shapes predicted from ages 65 to 107 at uniform intervals. Colors indicate the Euclidean distance from the first shape at age 65.
References
[1] Benjamin Billot, Douglas N Greve, Oula Puonti, Axel Thielscher, Koen Van Leemput, Bruce Fischl, Adrian V Dalca, Juan Eugenio Iglesias, et al. Synthseg: Segmentation of brain mri scans of any contrast and resolution without retraining. Medical imuge analysis, 16:102789, 2023.
[2] Ian J Deary, Alan Gow, Michelle D Taylor, Janie Corley, Caroline Brett, Valerie Wilson, Harry Campbell, Lawrence J Whalley, Peter M Visscher, David J Porteous, et al. The lothian birth cohort 1936: study examine influences on cognitive ageing from age 11 age 70 and beyond. BMC geriatrics, 7:1-12, 2007.
[3] Kathryn A Ellis, Cassandra Szocke, Ashley I Bush, David Darby, Petra L Graham, Nicola Lautenschlager, $ Lance Macaulay, Ralph N Martins, Paul Maruff, Colin L Masters, al. Rates of diagnostic transition and cognitive change at 18-month follow-up among 1,112 participants in the australian imaging, biomarkers and lifestyle flagship study of ageing (aibl). International Psychogeriatrics, 26(4):543-554, 2014.
[4] Rana Hanocka, Gal Metzer, Raja Giryes, and Daniel Cohen-Or. Point2mesh: self-prior for deformable meshes. ACM Trans. Graph., 39(4), 2020.
[5] Lin Huang, Jianchao Tan, Jingjing Meng, Ji Liu, and Junsong Yuan. Hot-net: Non-autoregressive transformer for 3d hand-object pose estimation. In Proceedings of the 28th ACM international conference on multimedia, pages 3136 3 3145, 2020.
[6] Clifford R Jack Jr, Matt A Bernstein, Nick C Fox, Paul Thompson, Gene Alexander, Danielle Harvey, Bret Borowski, Paula J Britson, Jennifer L. Whitwell, Chadwick Ward, et al. The alzheimer's 's disease neuroimaging initiative (adni): Mri methods. Journal of Magnetic Reso- nance Imaging: An Official Journal of the International Society for Magnetic Resonance in Medicine, 2744)6685-611, 2008
[7] Chiyu Jiang, Jingwei Huang, Andrea Tagliasacchi, and Leonidas J Guibas. Shapeflow: Learn- able deformation flows among 3d shapes. Advances in Neural Information Processing Systems, 01199755-7777 2020.
[8] Pamela LaMontagne, Tammie LS Benzinger, John C Morris, Sarah Keefe, Russ Hornbeck, Chengjie Xiong, Elizabeth Grant, Jason Hassenstab, Krista Moulder, Andrei G Vlassenko, al. Oasis-3: longitudinal neuroimaging. clinical, and cognitive dataset for normal aging and alzheimer discase. medrxiv, pages 2019-12, 2019.
31

---

[9] Manhua Liu, Fan Li, Hao Yan, Kundong Wang, Yixin Ma, Li Shen, Mingqing Xu, Alzheimer's 's Disease Neuroimaging Initiative, al. A multi-model deep convolutional neural network for automatic hippocampus segmentation and classification in alzheimer's disease. Neuroimage, 208:1 16459, 2020.
[10] Daniel Marcus, Anthony Fotenos, John Csernansky, John C Morris, and Randy L Buckner. Open access series of imaging studies: longitudinal mri data in nondemented and demented older adults. Journal of cognitive neuroscience, 22(12):667-2268, 2010.
[11] Wonjung Park, Maria del C Valdés Hernández, Jaeil Kim, Susana Muñoz Maniega, Fraser N Sneden, Karen J Ferguson, Mark E Bastin, Joanna M Wardlaw, Simon R Cox, and Jinah Park. Ai-based deformable hippocampal mesh reflects hippocampal morphological characteristics in relation cognition in healthy older adults. Neurolmage, 310:121 145, 2025.
[12] Lemuel Puglisi, Daniel C Alexander, and Daniele Ravi. Enhancing spatiotemporal disease progression models via latent diffusion and prior knowledge. In International Conference on Medical Image Computing and Computer-Assisted Intervention, pages 173-183. Springer, 2024.
[13] Javier Romero, Dimitrios Tzionas, and Michael J. Black. Embodied hands: Modeling and capturing hands and bodies together. ACM Transactions on Graphics, (Proc. SIGGRAPII Asia), 36(6), November 2017.
[14] Yu Rong, Takaaki Shiratori, and Hanbyul Joo. Frankmocap: A monocular 3d whole-body pose estimation system via regression and integration. In IEEE International Conference Computer Vision Workshops, 2021.
[15] Yulia Rubanova, Ricky TQ Chen, and David K Duvenaud. Latent ordinary differential equations for iregularly-sanpled time series. Advances in neural information processing systems, 32, 2019.
[16] Taeyun Woo, Wonjung Park, Woohyun Jeong, and Jinah Park. A survey of deep learning methods and datasets for hand pose estimation from hand-object interaction images. Computers & Graphics, 1 116:474-490, 2023.
32


Reviews:
[{"review_id": "rBpl2fe28P", "reviewer": "Reviewer_ViBW", "summary": "The authors provide motivation from their application (modelling brain deformations due to ageing / dementia using MRI data) for a 3D shape deformation model that can incorporate known covariates (here they chose age and cognitive variables). Neural ODEs are a reasonable approach for this. They compare their method to off-the-shelf methods (RNN-types) and an existing method in the field (BrLP) and demonstrate superior performance under certain metrics.", "questions": "L17 - knowledge of the hippocampus's role in AD is not recent, it has been reported many years ago (at least 1991 in the Braak & Braak paper)\n\nL27 - where do the numbers 160x224x160 with 1mm^3 resolution come from?\n\nL34 - what do you mean by \"medical priors\"?\n\nL40 - what do you mean by \"not invertible\" in this context?\n\nL58 - term \"condition-injectivity\" is italicised as if it has a specific meaning, but it's never defined, just assumed. What does it mean?\n\nL144 - how does age and cognitive status incorporate prior knowledge on brain atrophy and ventricle enlargement? Do you just mean that you expect them to change monotonically wrt. age and cognitive variables?\n\nL148 - need ablation study for effect of doing both forward and backward solvers (i.e., only forward or only backward). It is not clear why you need to include both\n\nL152 - what is \"approximated bijectivity\"?\n\nL152 - I don't quite follow the inline equation; are you saying the transformation matrix \\Phi is orthogonal (inverse equals transpose)?\n\nEquation 1 - what does f_\\theta(\\Lamba_t, c_t, t) equal? The neural ODE? And how is the conditioning on c_t actually implemented? It's not clear from the text.\n\nEquation 1 - does this equation have boundary conditions?\n\nAlgorithm 1 - if I read this correctly, you sequentially solve the ODE over all timepoints, then all timepoints - 1, then all time-points - 2, and so on, with the initial conditions incrementing correspondingly. What's the justification for this? Why not just solve once for the initial timepoint and write the loss with respect to the predictions at each followup data point? The way it's written, you're essentially solving the parts of the same ODE multiple times, instead of a single continuous mapping across all time points; is there not redundancy in the solution here?\n\nAlgorithm 1 - why N-2 going forward (i.e., requiring you need at least 2 timepoints) and N-1 going backward?\n\nAlgorithm 1 - what package and arguments did you use for \"ODEintegral\"? If it's in the Supplementary Material it needs to be moved to the main text, as it has strong implications for solver stability, runtime, etc. \n\nL154 - it may be my limitation, but I can't understand why this is called \"pseudo-cognitive status embedding\". From what I understand, you're just adding training points by interpolating between timepoints using your ODE solver. What does this have to do with \"pseudo-cognitive status\"?", "limitations": "Please see my comments above concerning limitations.\n\nNo comment was made by the authors on societal impact.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strengths\n- Neural ODEs are a reasonable approach for the task\n- Their method can handle irregularly sampled data, which are common in medical analysis\n\nWeaknesses\n- The novelty is unclear; ODE-based methods for this task are common\n- Most of the baseline models are insufficient for the task (RNN-types are not designed for this problem)\n- There are technical flaws in the methodology that need addressing (please see my comments below)", "was_revised": true, "final_justification": "I am satisfied with the authors' responses and have increased my score to \"Borderline Accept\", but not full \"Accept\", due to concerns about the overall novelty of the work.", "rebuttal": "### (W1)\nBrainODE is the first neural ODE-based framework for modeling longitudinal 3D shape trajectories of the brain in individual level and disease group at scale. Here, **we identify a novel problem: the precise modeling of longitudinal shape deformation in clinically important brain subregions**, extending beyond prior works focused on whole-brain image prediction. \n\nWe highlight the novelty through three main contributions:\n\n1. **Bridging neural ODE-based 3D shape modeling with longitudinal neurodegenerative disease**:\n\nWe propose a dedicated pipeline consisting of (i) mesh reconstruction for longitudinal shape representation, (ii) PCA-based parameterization, and (iii) a unified deformation-based ODE framework across disease groups. While neural ODEs have shown promise in continuous-time modeling, they have not been applied to predict longitudinal brain subregion shapes. To fill this gap, we convert brain images to fine-grained meshes and represent them in a compact PCA space. Then, we introduce BrainODE with continuous conditioning, pseudo-status embedding, and iterative shape sampling, to ensure both modeling stability and disease-continuum plausibility. Our model achieves precise subject-level trajectory prediction with low computational cost (~375MB for training), while maintaining fidelity at both the individual and group levels.\n\n2. **First large-scale application of longitudinal brain shapes across age and multi-site datasets**:\n\nTo the best of our knowledge, BrainODE is the first framework to apply Neural ODEs to model longitudinal brain shapes across a wide 30 years age span and multi-site datasets. Through large-scale experiments, we demonstrate that BrainODE effectively captures brain morphological patterns and aging dynamics across both normal and AD trajectories. Furthermore, we identify fundamental limitations in existing generative image-based models, which fail to precisely capture subregion-level deformation, and we offer straightforward adaptations of ODE methods to address this limitation.\n\n3. **Simultaneous identification and resolution of core challenges in longitudinal shape modeling**:\n\nWe define and simultaneously address key challenges unique to longitudinal brain shape modeling. These include data scarcity, irregular sampling intervals, the integration of medical priors, and the ability to predict only from a single observation. Even if beating SOTA performance is not a main goal, by defining these considerations upfront (L33-45), BrainODE provides a foundational blueprint for this research area.\n***\n### (W2)\n- Given the limited research on future brain shape prediction, we selected time-aware architectures as baselines. These include RNN- and ODE-based approaches, motivated by prior works such as LatentODE [22], which specifically addresses irregularly sampled time series—similar to the longitudinal brain data used in our study.\n- In addition, we included BrLP [20], an image-based conditional generative model that also predicts future brain structures conditioned on age and cognitive status. Given its similar objective, BrLP serves as a particularly relevant baseline, and we provide a detailed comparison in Appendix C.4.\n***\n### (W3)\nWe thank the reviewer for pointing out areas of ambiguity. Below, we provide clear and detailed responses to each of your comments. \n***\n>### L17\nWe acknowledge that the hippocampus has long been recognized as critical in AD. The intention to cite a recent study [15] is not to define clinical meaning of hippocampus, but rather to highlight that **hippocampal shape alone can be a strong indicator for AD diagnosis**. It supports the importance of precise modeling of the clinically important subregions.\n\n>### L27\nThe specified image size is a typical brain volume that can **accommodate most adult brains. The 1mm³ voxel spacing is a common setting** used in many representative brain imaging studies (BrainLDM(2022), SynthSeg (2023)).\nHere, we would like to emphasize that the hippocampus occupies only ~0.2% of the whole brain with subtle deformation over time. Since 1mm voxel representations are often insufficient to capture such localized changes, we adopt a mesh-based strategy.\n\n>### L34\nThe term **“medical priors” refers to prior knowledge from clinical studies about brain shape changes** according to age and cognitive status (L38, L268). To support this, we offer this volume analysis on real datasets in Appendix B.2.\nWe observed that hippocampal volume tends to decrease, while that of LV increases, as age progresses or as cognitive status declines (e.g., CN → AD). These monotonic trends highlight that brain shape dynamics are significantly influenced by these variables. Accordingly, **BrainODE is conditioned on age and cognitive status** as well as on the current brain shape to effectively predict within these deformation patterns.\n\n>### L40\nThe term “not invertible” was used to convey that brain structural changes associated with neurodegenerative diseases are typically **not reversible** (i.e., once significant atrophy occurs, the brain does not return to a healthy state). To avoid confusion, we will revise the term “not invertible” to “not reversible”.\\\nThe intention behind L40 was to emphasize our use of a **continuous cognitive status variable**, not relying solely on discrete diagnostic labels (e.g., normal = 0, AD = 1) as done in prior works [20]. This continuous label allows BrainODE to flexibly reflect gradual cognitive decline.\n\n>### L58\n**“Condition-injectivity” refers to BrainODE’s ability to produce different future shapes depending on the given conditions** (age and cognitive status), even from the same current shape. A subject with an AD trajectory produces more severe atrophy than one with a normal (Fig 3 & Appendix C.3). We will clarify this in the final version.\n\n>### L144\nAs addressed in our response to L34, hippocampal volume decreases while that of LV increases over aging with an accelerated pattern (Appendix B.2&Fig7). In addition, AD subjects demonstrate more pronounced deformations compared to normal (Fig8).\nImportantly, **we expect that BrainODE captures these progression patterns, without the need for explicit constraints** enforcing greater atrophy at higher ages or cognitive impairment levels. Fig 12 confirms that the model successfully learns and reflects these trends.\n\n>### L148\n**Since only the forward solver is used for prediction**, the backward training does not improve future shape prediction. Thus, we did not include an ablation experiment for the backward process on predictive performance.\n**However, this backward solver plays a key role in our pseudo-cognitive shape sampling: it is used to predict past shapes from brain shapes with AD trajectory**, enabling denser longitudinal supervision during BrainODE training (Section 3.2, L155).\n\n>### L152\n- Ideally, the forward and backward processes are bijective, i.e., φ_ji(φ_ij(Λ)) = Λ.\nHowever, we use the term **“approximated” bijectivity as shape predictions rely on numerical integration**, which cannot strictly guarantee perfect invertibility (the lemma for the deformation bijectivity is discussed in ShapeFlow).\nThe inline equation is intended to express this bijectivity: if a past shape at time t is reconstructed from a predicted future shape at t + δt, it should recover the original input X using the backward solver.\n- Here, **φ refers to the neural ODE-based shape mapping defined in Equ 2 & L146**.\nWe clarify that φ is not a transformation matrix and nor assumed to be orthogonal. Rather, it represents a learned **mapping from time t to another time t′.** We will clarify the bijectivity and φ in the final version.\n\n>### Equation 1\n- Yes, f_θ is BrainODE (Fig6), and details are in Appendix A.3. \n- As demonstrated in multiple visual examples (Fig1, 3, 5(c)), BrainODE learns subtle shape changes in longitudinal datasets. Therefore, we do **not impose hard constraints**(e.g., enforcing that hippocampal volume remains strictly non-zero in future predictions). One boundary condition is that mapping from **timepoint t to itself should return the identical shape, i.e., φ(X_(t→t)) = X**, which is inherently satisfied by the integral ODE process.\n\n>### Algorithm 1\n- We use all available pairwise timepoints (source&prediction) to maximize the available training data.\n- **There is no redundancy in the for loop.** In the forward process, the initial source shape’s timepoint i ranges from 0 to N−2, and for each i, all subsequent timepoints from i+1 to N−1 are predicted. The final timepoint (N−1) is excluded as an initial point since there are no future timepoints. Conversely, the initial timepoint ranges from N−1 down to 1 in the backward process.\n(cf) We acknowledge an error in Alg1- line 2, where the range should be from 0 to N−1 (not 1 to N). We will correct this, and we appreciate your careful attention.\n- The function odeIntegral is based on the `odeint` function from the torchdiffeq library, used to implement Equ2. While the details are currently in L514 (Appendix A.4), we agree that moving them to the main text would improve clarity.\n\n>### L154\n**Pseudo-cognitive status embedding is a data augmentation strategy to train BrainODE with continuous cognitive status** Our dataset includes only subjects diagnosed as normal (CN) or AD. Importantly, there are several subjects (labeled as CONV in Tab5) that exhibit disease progression from CN to AD. Considering the disease progression continuum (CN→congnition impairment→AD), we leverage the CONV subjects to synthesize intermediate brain shapes using BrainODE, and assign corresponding intermediate cognitive status. Since these values **are not directly observed but inferred from the progression, we refer to them as pseudo cognitive status.** It enables BrainODE to learn a smoother mapping along the cognitive decline continuum.\n***\nWe sincerely appreciate your thoughtful comments, which will help enhance the clarity of our work."}, {"review_id": "kYAvTAfYjA", "reviewer": "Reviewer_VKFD", "summary": "This paper proposes BrainODE, which resorts to a neural ODE-based framework to model continuous brain shape changes, and thereby predicting the neurodegenerative disease progression. It addresses challenges like limited data, irregular sampling, and individual variability by using a conditional architecture influenced by age and cognitive status. A pseudo-cognitive embedding further enables prediction from a single observation. In the experiments, the authors show that BrainODE outperforms existing methods in predicting future brain shapes across several different longitudinal datasets.", "questions": "- How would the model deal with large motion artifacts/shifts if they appear in the flow images. Would it be a potential issue if the adjacent longitudinal images do not include exactly the same viewpoint?\n- How would the model be able to ensure the geometry feasibility/soundness during the transformation from the source to the target domain?\n- Neural-ODE-related frameworks are typically vulnerable regarding training stability, and requires longer time and more computing resources for training. How is the model's efficiency comparing to other non-Neural-ODE models?", "limitations": "The authors have not sufficiently discussed their method's limitations in the submission.", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 2, "strengths_and_weaknesses": "Strength:\n- Neural-ODEs are well-known for continuous modeling of longitudinal changes, the method is well-motivated and the design of incorporating with age and cognitive status is reasonable and sound.\n- Due to the realistic challenges coming from the lack of data in clinical setups, an efficient way to address the longitudinal modeling of shape deformations could be useful for future work.\n- The experiments are sufficient and clearly indicate the effectiveness of the proposed approach, evaluations on both regular and irregular time interval predictions are interesting and insightful.\n- The organization and presentation is clear and easy to follow.\n\nWeakness: \n- Using Neural-ODEs to address topics regarding shape changes and longitudinal data is not novel, and there exist quite many studies on Neural-ODE/fluid-based image registration, non-rigid deformation modeling, and image sequence regression, etc ([1], [2]). However, the authors have not discuss or compare their approach with these related work.\n- How would the model deal with large motion artifacts/shifts if they appear in the flow images. Would it be a potential issue if the adjacent longitudinal images do not include exactly the same viewpoint?\n- How would the model be able to ensure the geometry feasibility/soundness during the transformation from the source to the target domain.\n- Neural-ODE-related frameworks are typically vulnerable regarding training stability, and requires longer time and more computing resources for training. The authors have not mentioned their model's efficiency comparing to non-Neural-ODE models.\n\n[1] Wu et al.: A Neural Ordinary Differential Equation Based Optimization Framework for Deformable Image Registration. CVPR 2022.\n[2] Bai et al.: Image Sequence Regression Based on Neural Ordinary Differential Equations. MICCAI 2024.", "was_revised": true, "final_justification": "With the new results and detailed rebuttal, my concerns have been mostly resolved. I have increased my score as leaning to accept.", "rebuttal": ">### (W1) Neural-ODE Novelity and Comparison with Related Method ([1], [2])\n\nWhile ODE-based models have been explored in medical time-series applications, to the best of our knowledge, BrainODE is the first to model clinically important brain subregions with 3D mesh representations for longitudinal brain prediction. We position our work under the following settings, and selected baselines that can be adapted accordingly: (1) handling time-series data, (2) integrating medical priors, and (3) enabling prediction from a single observation.\n\nHowever, registration-based methods [1,2] are not suitable for **(3) single-shot prediction** as well as (2) unclear conditioning methods on age or cognitive status. Therefore, we exclude direct comparisons with them in the main experiment, but we agree that including a discussion of these methods in the Related Work section would enhance the clarity of our paper.\n- Regarding NODEO[1], it is primarily designed for image registration tasks by learning continuous deformation fields between moving and fixed images. While effective at interpolation between images, **NODEO lacks an explicit age-conditioning method and is not designed for extrapolation/prediction of future brain shapes**, which is central to our problem.\n- **In response to your comment, we additionally experimented with NODER [2]**, which aims to predict missing timepoints in brain image sequences. NODER requires at least two prior timepoints as input for sequence regression. As highlighted in our paper (L35 & L42), this requirement limits its practical applicability for predicting future shapes from a single observation. Moreover, it requires separate training for each individual brain image sequence, making it unsuitable for learning generalizable longitudinal dynamics.  \nWe conducted this experiment using the LBC1936 dataset under a 4-shot validation setting. Although NODER [2] utilizes the deformation information from earlier timepoints, its voxel-based regression approach and lack of explicit conditioning on age and disease lead to inferior predictive accuracy. This highlights the effectiveness of BrainODE’s design, which captures shape dynamics in mesh representation and explicitly conditions on age and cognitive status, enabling more accurate modeling of longitudinal shape changes. We would add this ODE-based regression method in the Appendix of the final version for completeness. \n| Method      | Hippo (mm) | LV (mm)      |\n|:-----------:|:----------------:|:------------:|\n| **NODER [2]**   |      0.898       |    4.152     |\n| **Ours**        |      **0.414**      |    **1.630**     |\n\n***\n>### (Q1&W2) Robustness to viewpoint variation and motion artifacts\n\nThe robustness across varied viewpoints is one of the key advantages of our shape-based approach. As described in Appendix A.1, we construct longitudinal subcortical brain meshes from brain MRIs by aligning them to a template mesh space. This mesh reconstruction step eliminates the influence of viewpoint variation and motion artifacts inherent in raw MRI data. To generate these aligned shapes, we:\n\n1. Segment the target subcortical structure \n2. Rigidly align each shape to a common template mesh M_0​ (Fig. 4) using the Iterative Closest Point algorithm.\n\nAs a result, **all shapes—both intra-subject and inter-subject—are spatially aligned** before being used to train BrainODE. This alignment process ensures that BrainODE focuses **purely on anatomical deformation patterns.**\n\n***\n>### (Q2&W3) Geometry soundness of the deformation\n\nWhen generating brain shapes in mesh representation, we ensure geometric plausibility using multiple mesh-related regularization loss functions (Appendix A.1). These losses promote surface smoothness and prevent skewed meshes. Additionally, we include a regularization term that encourages minimal vertex displacement from the template mesh to reconstruct individual meshes. This approach helps ensure that the shape data, used for BrainODE training, maintain geometric and anatomical soundness.\n\nOn the other hand, during BrainODE training, we do not explicitly enforce geometric constraints on the predicted deformations. This design choice stems from the observation that subcortical areas are subtly changed from the original shapes as demonstrated in Fig 1, 3, and 5(c). In addition, we define and predict shape deformation in PCA spaces which inherently eliminates noisy high frequency surfaces (Appendix A.2). Thus, we do not impose hard constraints (e.g., enforcing non-zero hippocampal volume, surface smoothness), yet the model produces realistic and plausible deformations under PCA-based shape spaces. We only enforce one natural boundary condition: the mapping from time t to itself must yield the identical shape, i.e., φ(X_(t→t)) = X. This identity condition is inherently satisfied by the integral ODE of Equation 2.\n\nWe acknowledge that when the model is queried with extreme out-of-distribution conditions (e.g., 120 years of age), it may produce slightly implausible shapes as discussed in Appendix C.5 and visualized in Fig 14. However, we consider such ages to be beyond practical application scenarios.\n***\n>### (Q3&W4) Training stability and efficiency\n\n- BrainODE adopts a flow-based neural ODE framework that operates in the **deformation space, which inherently supports stable training dynamics**, as discussed in ShapeFlow. Furthermore, our **PCA-based shape representation highlights clear longitudinal deformation patterns**, which facilitate stable learning (Fig 5(a)). By leveraging this representation, BrainODE is able to preserve both global shape topology and fine-grained local variations, while predicting subtle deformations.\n- For comparison, LatentODE shows poor performance, as evidenced by producing similar shapes for an arbitrary subject (Fig 2). One possible explanation is that, despite using a mesh representation, LatentODE fails to capture longitudinal progression correctly due to training instability and the complex shape dynamics. From this perspective, we believe that our design and representation choices ensure training stability in this challenging domain.\n- Furthermore, to empirically validate the stability, we generated future hippocampal shapes from 100 randomly sampled inputs with varied cognitive statuses of 0, 0.5, and 1 (Appendix C.3). The predicted shapes were both plausible and well-reflecting the cognitive status in terms of brain atrophy, even though BrainODE was not explicitly regularized with higher cognitive scores.\n- Regarding computational efficiency, neural ODEs offer efficient gradient computation via the adjoint method as noted in L50 and NeuralODE [5]. In our implementation (Algorithm 1), we train BrainODE over all pairwise combinations of longitudinal shapes (~375MB GPU usage for training). While mini-batch training could be made more efficient with customized implementation (e.g., code of LatentODE [22]), we did not adopt this yet, but we plan to enhance its efficiency.\n- **BrainODE achieves efficient training times with 9 hours for LBC1936 with 40 epochs**, substantially more efficient than the voxel-based generative model, BrLP[20]. BrLP requires multiple training stages, including a KL-regularized VAE-GAN, diffusion model, and conditional generator (e.g., ControlNet and a generalized linear model for covariate prediction), making it significantly computationally demanding (at least one week, due to the difficulty of training a 3D VAE-GAN). We did not provide direct runtime comparisons with RNN- or LatentODE-based models since these methods failed to predict individualized shapes, making fair comparison difficult. **For transparency, we will include BrainODE’s training time in the final version.**\n\n***\n>### (L1) Limitation on methodology\n\nIn addition to the limitations described in Appendix D, one is the lack of explicit geometric constraints as discussed in this rebuttal (Q2&W3), which has a possibility to generate implausible shapes under extreme or out-of-distribution conditions.\n\nAnother limitation is that BrainODE is specifically designed for brain subregions. Although BrainODE can be applied to other brain subregions, it is not directly applicable to modeling deformations of the full brain image at scale. We believe that after precisely modeling brain subregions with BrainODE, whole brains can be generated by conditional generation modules (e.g., ControlNet, VCM(2025)), leveraging the BrainODE outputs. However, we consider it beyond the scope of the present study. We will add a discussion of the above limitations to the final version of the paper.\n\nWe appreciate the reviewer for raising this important point. We will include a discussion of these limitations in the main text of the final version of the paper."}, {"review_id": "mnytkw3adA", "reviewer": "Reviewer_4KRF", "summary": "This paper presents BrainODE, a neural ODE framework for modeling brain shape trajectories conditioned on age and cognitive status. Operating in a PCA space of 3D meshes (hippocampus and ventricles), the model predicts anatomically plausible deformations from sparse longitudinal data. A key contribution is the pseudo-cognitive status embedding, enabling interpolation along the cognitive decline spectrum. Evaluated on four neuroimaging datasets, BrainODE outperforms RNN, latent ODE, and flow-based baselines in both one-shot and multi-shot settings.", "questions": "1. The authors claim strong generalization, but the method is demonstrated only on the hippocampus and lateral ventricles. Could BrainODE be readily extended to other brain regions or full cortical surfaces? Are there computational or modeling limitations that prevent this? Additionally, how can BrainODE be applied to external studies? For example, if the model is trained on studies A and B, how would it perform on study C? Such an analysis is critical to substantiate the claim of \"strong generalization\".\n\n2. Can the authors elaborate on how reliable the pseudo-cognitive status embedding is when applied to unseen subjects? How sensitive is BrainODE’s prediction to errors in the cognition estimator? A sensitivity analysis would help clarify the robustness of this mechanism and strengthen the reader’s confidence in the method’s applicability.\n\n3. Why are MCI subjects excluded from the modeling? Given that MCI is a clinically critical intermediate stage, how do the authors justify its exclusion beyond citing ambiguous labeling? The authors emphasize the practicality of their method, but omitting an entire diagnostic class—particularly one central to early-stage Alzheimer's detection—substantially undermines this claim.\n\n4. Is there a reason why the authors do not compare with ConDOR although they mention it in the prior work? \n\n5. Why do the authors pick the 4-shot prediction and do not experiment with 2-shot, 3-shot etc ? Gradually increasing subject information rather than moving directly to the 4-shot would strengthen the paper.", "limitations": "The authors mention key limitations, but only in the appendix rather than in the main paper. They acknowledge the exclusion of MCI subjects due to ambiguous labeling and discuss future directions for integrating this critical diagnostic group. However, given MCI’s clinical importance and its role in early intervention studies, this limitation should be more prominently discussed. Bringing these limitations into the main text and expanding on their practical implications in case of misdiagnosis would strengthen the paper’s clinical impact. Additionaly the authors. In the limitations section, the authors mention that they average predicted shapes across timepoints for subjects with multiple observations and highlight the need for more advanced aggregation strategies. I encourage the authors to acknowledge recent work on modeling longitudinal trajectories with repeated measures, such as the Adaptive Shrinkage Estimation for Personalized Deep Kernel Regression (ICLR 2025), which addresses the problem of longitudinal prediction of neurodegeneration with repeated measures.", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "**Strengths** \nThe paper addresses an important clinical problem—modeling longitudinal brain shape changes related to neurodegeneration—which has clear relevance for Alzheimer’s research. The authors conduct quantitative and qualitative evaluations on multiple datasets, showing that their model performs favorably compared to selected baselines.\n\n**Weakness** \n1. The paper would benefit from clearer organization and a more unified presentation of the experimental setting. In particular, the distinction between regular and irregular time intervals seems unnecessary; irregular sampling is a general case that naturally subsumes the regular setting. Consolidating the analysis under this more realistic and flexible framework could simplify the narrative and strengthen the practical relevance of the work. Additionally, the manuscript lacks important details regarding the construction of training and test splits across datasets, which limits reproducibility and interpretability of the results. The absence of an appendix in the main paper reduces accessibility to key implementation specifics. Finally, the quantitative evaluation could be significantly improved by reporting results over multiple runs, along with confidence intervals or statistical significance testing, to support the robustness of the performance claims.\n2.  The authors claim strong generalization, but the method is evaluated only on the hippocampus and lateral ventricles. It remains unclear whether BrainODE can scale to more complex anatomical regions or cortical surfaces, and whether there are architectural or computational bottlenecks that limit such extensions. Moreover, there is no assessment of the model’s ability to generalize across studies. Evaluating performance on an unseen dataset (e.g., training on ADNI/AIBL and testing on OASIS) would be critical to support the generalization claims.\n3.  The authors exclude MCI subjects from the modeling due to ambiguous labeling, but MCI represents a clinically important transitional stage with significant relevance to early detection and intervention. Omitting this class substantially reduces the practicality and applicability of the method in real-world clinical settings.\n4.  The evaluation considers 1-shot and 4-shot prediction settings, but does not explore intermediate scenarios (e.g., 2-shot or 3-shot). Gradually increasing the available subject information would provide a more complete picture of model performance and help assess how BrainODE scales with longitudinal depth.", "was_revised": true, "final_justification": "The authors addressed most of my concerns. However, My main concern regarding the handling of the MCI class as well as the sensitivity of the cognition estimator remains. The authors mentioned that will include the sensitivity analysis in the manuscript but I am not able to evaluate it for now as I do not have access to the updated manuscript.  However, my position for this paper tends to be positive, and thus I decide to increase my score to 4.", "rebuttal": ">### (W1a) Regular vs. irregular time intervals\n\nWe distinguished regular and irregular time intervals **to evaluate BrainODE under varying settings**. It allows broad comparisons with baselines: RNN, LSTM, and ShapeFlow are suitable for regular time series and RNN-Decay, LatentODE, and BrLP can handle irregular intervals. BrainODE is designed to handle both, and we report performance respectively.\n\n>### (W1b)Lack of detail in dataset splits and reproducibility & statistical robustness in evaluation**\n\nFor reproducibility, we have **shared the implementation repository**(L415). While we are unable to share the dataset itself due to data usage policies, we provide all relevant code and a toy example to facilitate reproducibility. In addition, we described dataset split details in Tab5&6.  \nFor statistical robustness, we appreciate that sharing your concerns for statistical rigor in our evaluation protocol. Our protocal for Tab1&2 is:\n  1) For 1-shot prediction, we performed three validation rounds with different groups. As a result, their ranking was not changed across validations.\n  2) For 4-shot prediction, since subjects with ≥5 time points are rare, we evaluated on all such subjects of all benchmarks.\n  3) For the pseudo-cognitive status shape sampling experiments (Sec 3.2 & Tab. 3), we conducted leave-four-out cross validation three times using 40 CONV-labeled subjects.\n\nTo clarify, we originally omitted statistical values(e.g., std, significance) since heterogeneous time intervals across validation folds could confound accuracy interpretations. Instead, we focused on **reporting consistent trends in model ranking, which were preserved across multiple resampling scenarios.** But for clarity, we will offer standard deviation for better interpretation following your suggestion. In addition, we will include above validation protocols in the final version.\n***\n>### (W2&Q1) \n### [Generalization to brain structures]\nThe reason why we focus on two subcortical structures is not only because they are highly relevant to AD but also due to their distinct morphological characteristics. The hippocampus is a critical region for AD diagnosis, occupying about 1.7%  of the total brain volume and reflecting conspicuous localized atrophic changes. In contrast, LV serves as a global indicator of brain atrophy by its enlargement with the large-scale morphological variations.\n\nAs the reviewer points out, **BrainODE can be readily extended to other subcortical region** by applying the same pipeline in Appendix A.1. However, extending to cortex presents additional challenges due to their complex folded geometry and establishing reliable point correspondences across inter- and intra- subjects. To model longitudinal cortex shape prediction, tailored methods such as CortexODE(2022) could be integrated with BrainODE. We will include this discussion in the final version.\n\n### [Generalization across datasets]\nWe agree that validating generalization performance across datasets is an important direction, so we conducted the following additional experiments for 1-shot prediction:\n -  Exp1: Training on AIBL (with regular time intervals) and validating on ADNI and OASIS (with irregular time intervals).\n - Exp2: Training on AIBL + ADNI + OASIS (with irregular time intervals) and validating on LBC1936 (with regular time intervals).\n\nThese settings allow us to evaluate the robustness of BrainODE under both temporal and benchmark shifts. While performance under these shifts is slightly degraded compared to intra-dataset benchmarks, BrainODE consistently outperforms other baselines.\n\nIn terms of generalizability, each dataset includes different temporal sampling intervals, ranging from 0.5 years (e.g., ADNI) to over 3 years (e.g., LBC). Due to this temporal discrepancy, BrainODE exhibits marginal performance decreases in both shift scenarios.\n\nThese findings underscore the importance of training on large-scale longitudinal datasets with varied time intervals to enhance generalization across cohorts. Leveraging broader and more diverse training data is essential for improving robustness and capturing progression dynamics. We will include this experiment in the final version.\n\n| Experiment | Training              | Validation     | Hippo    | LV       |\n|:------------:|:-------------------------:|:-------------------:|:-------------------:|:-------------------:|\n| Exp1       | AIBL                    | ADNI + OASIS   | 0.518 (↑0.016) | 2.265 (↑0.554) |\n| Exp2       | AIBL + ADNI + OASIS     | LBC1936        | 0.522 (↑0.172) | 1.988 (↑0.203) |\n\n(Values in parentheses are the surface distance metric (mm) difference from the intra-benchmark result)\n***\n>### (W3 & Q3) Exclusion of MCI\n\nIn the clinical continuum of the AD trajectory (normal(NC) → MCI → AD), determining the severity of MCI and encoding its cognitive status values between NC and AD is highly challenging. In addition, as MCI includes multiple subtypes (e.g., EMCI, LMCI, and unspecified MCI), simply assigning an intermediate value such as 0.5 is insufficient to accurately reflect disease severity, making it difficult to clearly model and validate the BrainODE.\n\nWe carefully designed BrainODE to be conditioned on continuous cognitive status values, enabling a gradual transition from NC to AD, which inherently encompasses the MCI states. Although BrainODE was trained on subjects with clearly defined cognitive status(NC or AD), we leveraged a pseudo cognitive status embedding approach on subjects whose cognition progressed from NC to AD to generate synthetic longitudinal brain shapes with intermediate cognitive status. This allowed us to train and validate BrainODE on these intermediate cognitive status. As a result, BrainODE effectively bridges the gap between NC and AD, demonstrating its capability to predict shapes on intermediate cognitive status such as MCI (Fig12).\n\nThis approach not only highlights BrainODE's advantage over prior methods that rely on discrete conditioning values but also presents promising evidence that the model can flexibly predict under intermediate status such as MCI. In practical terms, this means BrainODE has the potential to be applied in real clinical settings where cognitive status is diagnosed with varying degrees of granularity by medical professionals.\n\nAs the main focus of this paper is **to explore the potential of neural ODEs for modeling brain shape trajectories across a broad age range and disease conditions, we prioritized architectural design and evaluation under clearly distinguishable conditions.** In future work, by quantitatively defining the severity of MCI, our BrainODE would have stronger generalization.\n***\n>### (W4&Q5) Granular longitudinal evaluation between 1- and 4-Shot\n\nTo clarify our evaluation design, the purpose of separating 1- and 4-shot is:\n\n(1) 1-shot prediction reflects a realistic clinical use case where only a single timepoint is available\n(2) 4-shot prediction explores the model’s capability when more longitudinal data are accessible.\n\nWe chose (2) 4-shot as the upper bound since the LBC dataset provides up to five longitudinal scans per subject. To clarify, the subjects used for 1-shot and 4-shot validation are not the same.\n\nTo response the reviewer’s suggestion, we conducted additional experiments for 1-, 2-, 3-, and 4-shot settings in LBC1936. We observed that the prediction accuracy increased with the less standard deviations in more observations. While not the main focus of this work, we acknowledge that optimizing the aggregation of prediction from multiple timepoint is an important future direction(L705). We plan to include this result to provide a more complete picture how BrainODE scales with longitudinal depth.\n| | hippo (mm) | LV(mm)  |\n|:------:|:-----------------:|:-----------------:|\n| 1-shot | 0.46 ± 0.077   | 1.93 ± 0.270 |\n| 2-shot | 0.43 ± 0.075   | 1.84 ± 0.238 |\n| 3-shot | 0.42 ± 0.075   | 1.80 ± 0.229 |\n| 4-shot | 0.41 ± 0.075   | 1.79 ± 0.226 |\n***\n>### (Q2)  BrainODE’s sensitivity to Cognition estimator errors\n\nTo evaluate the reliability of cognition estimator, we provide the accuracy(~88%) in normal/AD classification on unseen subjects, as reported in Tab 3. Before employing the model in our BrainODE, we conducted a preliminary study on 600 subjects (1:1 normal and AD) using only the LV and hippcampus segmentation masks with the same model of Cognition estimator; it achieved over 95% classification accuracy. We interpret the probabiltiy of AD from the cognition estimator as severity of AD.\n\nAbout the sensitivity of BrainODE to the cognitive value, BrainODE predicts the greater atrophy for the same initial shapes under higher values, demonstrating the model’s high sensitivity to the estimator outputs (Fig3&12). This faithful reflection of cognitive status is important in our study, as clinicians can manually specify cognitive status in practical scenarios, rather than relying solely on the estimator.\n\nNonetheless, to ensure realistic modeling without manual intervention, high precision in the cognition estimator remains important. Following your suggestion, to provide better interpretation of our BrainODE with cognition estimator, we will provide a sensitivity experiment under varying cognitive estimator accuracy.\n\n>### (Q4) ConDOR comparison\n\nConDOR focuses on predicting future 1D brain measurements (e.g., cortical thickness as scalar) using a diffusion-based generative model, while our task requires spatially structured shape modeling in 3D mesh format, which ConDOR cannot directly accommodate. Thus, we excluded ConDOR from our baseline comparisons. Instead, we compared with diffusion-based image generative model BrLP.\n\n>### (L)\n\nWe appreciate your suggestion regarding(ICLR2025) the aggregation of predictions from multiple observations. We agree with the reviewer’s comment to move the limitations discussed in appendix to the main paper to more clearly define the scope and boundaries of our study."}, {"review_id": "UugGLQgkUm", "reviewer": "Reviewer_6kif", "summary": "The authors attempt to model the age-related and disease-related deformation of brain anatomy over time. The proposed method, BrainODE, is based on neural ordinary differential equations (neural ODEs). BrainODE modulates shape dynamics with subject-specific age and cognitive status, and is able to forecast from a single observation autoregressively over an embedding space that jointly encodes age and estimated cognitive status. The authors performed quantitative experiments on several brain imaging datasets with subjects following normal aging and cognitive diseases, covering use cases with regular and irregular time intervals. BrainODE shows competitive results compared to simple extrapolation and autoregressive baselines, as well as more relevant methods such as LatentODE and BrLP.", "questions": "I have 3 actionable suggestions that likely does not require additional experiments.\n\n1. See Weakness 1a\n2. See Weakness 1b\n3. See Weakness 1c", "limitations": "I recommend the authors to include a section to discuss the Limitations. I do not foresee any potential negative societal impact.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths:\n1. The authors are tackling an interesting and potentially high-impact problem. The solution they proposed – modeling disease progression using a variant of neural ODE – is very intuitive.\n\n2. The use of shapes and meshes instead of modeling in the 3D voxel space is fairly well explained.\n\n3. I particularly like the visualizations (Figure 2 and 3) where the error in Euclidean distance is highlighted using a visually distinguishing color map.\n\n4. The number and diversity of the datasets chosen for experiments are decent. The suite of baseline methods for comparison is fairly reasonable. The quantitative performances look okay.\n\nWeaknesses:\n1. The presentation of the results can be made stronger, likely without additional experiments.\n    \n    1a. Table 1 and 2 could include additional metrics beyond the error in Euclidean distance: for example, (1) image/mesh similarity metrics such as PSNR, (2) area-of-interest overlap metrics such as dice coefficient, (3) surface smoothness metrics – but note surface smoothness should be compared in reference of the ground truth since it wouldn’t be the smoother the better. I am fairly sure BrainODE can beat the competing methods judging from the other results, but these would provide more comprehensive views of the superiority.\n\n    1b. Table 2 could (and should) be subset into the datasets (ADNI, OASIS and AIBL).\n\n    1c. While Figure 3 looks cool, the authors can potentially enhance it. In my opinion, the authors can additionally show how BrainODE generates smoother trajectories compared to other methods. I would suggest keeping these 4 subfigures as-is as panel A, and additionally include a panel B below, showing line plots quantifying these trajectories, and also include other methods’ trajectories for comparison. Line plots can be grouped by method or grouped by the 4 conditions.\n\n2. There are several prior works in this area that have not been discussed. For example, the most closely related might be ImageFlowNet [1], which uses an adapted neural ODE/SDE to model disease progression in a joint embedding space. Since they do not seem to have a native implementation in 3D volume due to computational cost, I wouldn’t urge the authors to include it for comparison given the amount of additional work. Similarly, a few other techniques have been used to model the progression of diseases in images or volumes, including flow matching [2] and diffusion models [3] [4]. Again, I am not asking for additional experiments, but it would be beneficial to discuss these works and highlight how BrainODE offers a distinct (and more favorable) solution in this context.\n\n[1] ImageFlowNet: Forecasting Multiscale Image-Level Trajectories of Disease Progression with Irregularly-Sampled Longitudinal Medical Images, ICASSP 2025 Oral.\n\n[2] Modeling Complex System Dynamics with Flow Matching Across Time and Conditions, ICLR 2025 Spotlight.\n\n[3] Denoising diffusion probabilistic models for 3D medical image generation, Scientific Reports 2023.\n\n[4] SADM: Sequence-Aware Diffusion Model for Longitudinal Medical Image Generation, IPMI 2023.", "was_revised": true, "final_justification": "In the initial review, I found this work quite interesting but undersold in the current form of presentation, and gave a handful of suggestions regarding presentation and results to report. The authors did a decent amount of work to nicely address all of them. I have also read the reviews of other reviewers, and I personally found them to be sufficiently addressed by the authors.", "rebuttal": "We sincerely appreciate your suggestions (W1) and insightful comments on related works (W2), which have guided us in enhancing the clarity of our paper.\n***\n>### (W1a)\n\nWe agree with your suggestion regarding the inclusion of additional voxel-based evaluation metrics or mesh quality assessments.\n\nIn this work, we initially omitted voxel-based metrics because our method operates directly in the 3D mesh domain. In addition, while voxelization of the hippocampus is straightforward, the lateral ventricle (LV) presents unique challenges: despite its anatomically connected structure, ventricular masks in MRI data are often fragmented or disconnected due to limited resolution. Although we address this by leveraging mesh reconstruction, it makes voxel-based comparisons with mesh occupancy less reliable for LV structures.\n\nNevertheless, as you pointed out, voxel-based metrics can enhance interpretability and comparability with voxel-based approaches. To provide a complementary perspective, we computed **Dice scores, widely used in medical imaging, for the hippocampal shapes.** These results, presented below and to be included in the appendix. The relative ranking of alignment quality remains consistent with Tab 1&2 as you anticipated.\n| Method    | LBC1936 4-shot | LBC1936 1-shot | AIBL 4-shot | AIBL 1-shot |\n|-----------|----------------|----------------|-------------|-------------|\n| ShapeFlow | 0.90 ± 0.02    | 0.92 ± 0.04    | 0.89 ± 0.04 | 0.92 ± 0.02 |\n| BrLP      | 0.84 ± 0.03    | 0.84 ± 0.03    | 0.87 ± 0.03 | 0.86 ± 0.05 |\n| Ours      | **0.93 ± 0.02**    | **0.94 ± 0.01**    | **0.93 ± 0.01** | **0.94 ± 0.01** |\n***\n>### (W1b)\n\nWe agree with your suggestion to provide subset accuracy for Table 2 and will incorporate the results into the final version of the paper.\nFor hippocampal shape analysis, the number of subjects in the 4-shot is 18, 20, and 27, and in the 1-shot is 19, 20, and 40, for AIBL, ADNI, and OASIS respectively. The subset results on our method are presented below, and we will incorporate this to the final version.\n| Dataset | 4-shot       | 1-shot       |\n|---------|--------------|--------------|\n| AIBL    | 0.52 ± 0.07  | 0.46 ± 0.05  |\n| ADNI    | 0.59 ± 0.14  | 0.54 ± 0.15  |\n| OASIS   | 0.52 ± 0.07  | 0.48 ± 0.08  |\n\nFrom these results, although the discrepancies are not substantial, it underscores the need for further validation across benchmarks. To this end, we conducted the following additional experiments:\n -  Exp1: Training on AIBL (with regular time intervals) and validating on ADNI and OASIS (with irregular time intervals).\n - Exp2: Training on AIBL + ADNI + OASIS (with irregular time intervals) and validating on LBC1936 (with regular time intervals).\n\nThese settings allow us to evaluate the robustness of BrainODE under both temporal and benchmark distribution shifts. While performance under these shifts is slightly degraded compared to intra-dataset benchmarks, BrainODE consistently outperforms other baseline models, demonstrating robustness to unseen individual shape morphologies and progression patterns. \n\nIn terms of generalizability, each cohort (dataset) follows different temporal sampling intervals, ranging from 0.5 years (e.g., ADNI) to over 3 years (e.g., LBC). Due to this temporal discrepancy, BrainODE exhibits marginal performance decreases in both shift scenarios.\n\nThese findings underscore the importance of training on large-scale longitudinal datasets with varied time intervals to enhance generalization across cohorts. Leveraging broader and more diverse training data is essential for improving robustness and capturing realistic progression dynamics. We plan to include this validation in the final version of our work.\n\n| Experiment | Training                | Validation     | Hippo    | LV       |\n|:------------:|:-------------------------:|:-------------------:|:-------------------:|:-------------------:|\n| Exp1       | AIBL                    | ADNI + OASIS   | 0.518 (↑0.016) | 2.265 (↑0.554) |\n| Exp2       | AIBL + ADNI + OASIS     | LBC1936        | 0.522 (↑0.172) | 1.988 (↑0.203) |\n\n(Values in parentheses are the surface distance metric (mm) difference from the intra-benchmark result)\n***\n>### (W1c)\n\nWe are truly grateful for your suggestion regarding improved visualization of our results. In response, we added smoother trajectory visualizations compared to the baselines in Fig 14 of Appendix C.5. Additionally, your suggestion to use line plots is an insightful idea for highlighting the smooth longitudinal patterns.\n\nFollowing your suggestion, we will include a smoothed trajectory visualization in Figure 3 to better illustrate the advantages of our method.\n***\n>### (W2)\n\n\nWe appreciate your suggestion to include related works and agree that discussing these models would help better define the scope of our paper. Below, we provide a brief overview of how BrainODE distinguishes itself, particularly in terms of representational effectiveness, computational efficiency, and subject-level modeling.\n\n- ImageFlowNet [1] models subject-specific spatiotemporal dynamics in 2D medical images using neural ODEs/SDEs with multi-scale representations based on a U-Net architecture. Since it performs progression modeling in the pixel space, it presents several limitations in computational cost and precision. The use of multi-scale representations in a joint space of neural ODEs/SDEs and pixel-level reconstruction results in substantial memory consumption, making it infeasible in 3D voxel space (e.g., over 80 GB for volumes of size 128^3). Furthermore, its ability to capture subtle changes, crucial for modeling disease progression, is inherently limited by the resolution of the input image due to the discrete nature of the pixel grid. \nIn contrast, BrainODE employs mesh representations and PCA-based 3D shape modeling, which significantly reduces computational requirements (~375 MB for training) while achieving high-precision shape modeling with a ~0.4 mm average error for hippocampus.\n\n- Multi-Marginal Flow Matching (MMFM) [2] shares several core similarities with our BrainODE. Both methods employ a single shared model to learn across diverse conditions, support continuous dynamics modeling, handle irregular and multi-time-point data, and incorporate techniques for imputing mission observation.  \nHowever, MMFM is designed to model distributional translations, making it well-suited for capturing general patterns and group-level dynamics. Therefore, it does not predict explicit individual-level trajectories and does not incorporate explicit structural parameters. In contrast, BrainODE embeds brain morphology directly into a parameter space and models continuous shape trajectories at both individual and disease group levels. This enables precise tracking and predicting of morphological changes for each subject. \n\n- As we discussed in the main paper, image generative model-based approaches ([3], [4] and BrLP) have several limitations: substantial computational costs, inconsistencies in subject identity due to Gaussian sampling (i.e., subject-specificity is lost due to stochasticity in generation), coarse resolution that fails to capture subtle yet clinically important changes. Furthermore, none of these methods is capable of modeling the clinical continuum of the AD trajectory as they rely on discrete condition values for the disease labeling.\n***\n>### (L1 & Paper formatting)\n\n- We agree with the reviewers' suggestions to move the discussion of limitations from the appendix to the main paper, thus, we plan to include it in the discussion section. This will help appropriately acknowledge the boundaries of our work and clarify where this pioneering effort in longitudinal brain shape prediction is positioned within the medical imaging field.\n- Thank you for pointing out the issue with the table caption. We will adjust its placement accordingly in the final version."}]

Author rebuttal:

Meta-review: (a) Scientific claims and findings: The paper presents BrainODE, a neural‑ODE framework for longitudinal 3D brain shape modeling at the subregion level (notably hippocampus and lateral ventricles). Shapes are reconstructed as meshes and embedded in a PCA shape space; their temporal dynamics are governed by an ODE conditioned on age and cognitive status. A pseudo‑cognitive embedding augments training to interpolate along a CN→AD continuum and supports one‑shot (single‑visit) forecasting under irregular sampling.

Across four longitudinal MRI cohorts, BrainODE predicts anatomically plausible future shapes and outperforms RNN/LSTM, Latent ODE, flow‑based/registration and image‑generative baselines (including BrLP) under both one‑shot and multi‑shot settings. The rebuttal adds cross‑dataset generalization experiments, 2‑/3‑shot analyses, and statistical reporting (CIs/significance to be included in the final version).

(b) Strengths: 

1. Problem importance & clinical relevance: Modeling subregion shape trajectories is central to aging and AD progression; focusing on anatomically meaningful meshes improves interpretability.
2. Methodological fit: Neural ODEs naturally capture continuous‑time trajectories and irregular intervals; conditioning on age/cognition encodes well‑established medical priors (atrophy/enlargement trends).
3. One‑shot forecasting: The pseudo‑cognitive augmentation enables prediction from a single observation, a realistic clinical scenario.
4. Engineering choices that matter: Working in mesh‑PCA space reduces noise and cost while preserving fine geometric detail; training remains lightweight relative to voxel‑generative pipelines.
5. Empirical breadth: Multiple cohorts, one‑shot and multi‑shot regimes, added cross‑dataset validation, and qualitative analyses (trajectory smoothness, plausibility). Rebuttal commits to 95% CIs/significance tests, subset reporting by dataset, and additional metrics (e.g., Dice).
6. Clarity improvements: Rebuttal resolves many terminology/formulation questions (conditioning, bijectivity, solver choice, algorithmic loops) and moves limitations into the main paper.

(c) Weaknesses / Missing elements
1. Scope of anatomy: Current evaluation is on hippocampus and ventricles; extension to cortical surfaces and more complex regions remains future work.
2. MCI coverage: MCI (a clinically pivotal intermediate stage) is not modeled explicitly; the pseudo‑cognitive continuum is promising but not a substitute for dedicated MCI inference.
3. Geometric guarantees: Predicted deformations are not constrained by hard geometric regularizers; plausibility is aided by mesh‑space priors and data preprocessing, but formal guarantees are absent.
4. Baseline coverage / statistics: Initial draft lacked CIs/significance and some domain‑specific metrics; rebuttal addresses this (to be added) but final inclusion is important.
5. Reliance on cognition estimator: Performance may depend on estimator accuracy; robustness and sensitivity analyses are only partially explored (promised in revision).

(d) Reasons for acceptance 

Timely, high‑impact problem with clear translational value: precise subregional shape forecasting for aging and AD progression.
Strong integration of ideas: clinically informed conditioning + continuous‑time dynamics + mesh‑PCA representation + pseudo‑cognitive augmentation—delivering one‑shot forecasting under irregular sampling.
Compelling empirical evidence: consistent improvements over diverse baselines, plus new cross‑dataset and multi‑shot analyses; qualitative results demonstrate trajectory smoothness and anatomical plausibility.
Practicality and efficiency: orders‑of‑magnitude lighter than voxel‑generative pipelines while targeting a finer (subregional) prediction task.

Thorough, constructive rebuttal resolving conceptual and methodological questions and committing to statistical rigor and clearer presentation.

(e) Rebuttal‑period discussion and how it informed the decision

Key issues raised (anon ids R1…R4) and author responses:

Motivation, terminology, and formulation (R1): Asked to clarify “medical priors,” (ir)reversibility, “condition‑injectivity,” “approximate bijectivity,” and the exact ODE/conditioning mechanics; queried Algorithm‑1 loops and solver choice.
Response: Definitions clarified; “not invertible” → “not reversible”; φ explained as a learned ODE mapping (not an orthogonal matrix); boundary condition specified; torchdiffeq/odeint disclosed; loop‑range bug fixed; justification for pairwise training provided.

Baselines, novelty, and related work (R2/R4): Requested deeper comparisons to neural‑ODE registration, flow‑matching, diffusion, and ConDOR; questioned 4‑shot rationale; suggested stronger metrics/visualizations.
Response: Added NODER comparison (favoring BrainODE); argued registration/ConDOR are mismatched to mesh‑shape forecasting; committed to discussing ImageFlowNet/flow‑matching/diffusion; clarified 4‑shot protocol and added 2‑/3‑shot results; will include Dice and enhanced trajectory plots.

Statistics and reporting (R2): Called for confidence intervals and significance testing; noted cases where extrapolation looked comparable.
Response: Will add 95% CIs + significance; argued qualitative/long‑horizon and one‑shot scenarios favor BrainODE over extrapolation.

Generalization, robustness, and geometry (R3): Asked for cross‑dataset testing, sensitivity to cognition estimator, handling motion/viewpoints, geometric soundness, and extension beyond subcortical regions; raised MCI omission.
Response: New cross‑dataset experiments; mesh alignment/regularization for viewpoint/motion; PCA‑space acts as a soft geometric prior; cognition estimator accuracy reported with planned sensitivity analysis; acknowledged cortical extension as future work; justified MCI handling via pseudo‑continuum but recognized it as a limitation to be moved to the main text.

Weighting in the final decision:
The rebuttal substantively addressed the highest‑impact concerns (formulation clarity, evaluation breadth, statistical rigor commitments, and cross‑dataset evidence). Remaining limitations (explicit MCI modeling, cortical extension, formal geometric guarantees) are reasonable for future work and do not undermine the contribution’s core value. With three score increases and full‑panel acceptance, the paper clears the bar for acceptance.
Decision: Accept (poster)