Energy-based Tissue Manifolds for Longitudinal Multiparametric MRI Analysis

Kartikay Tehlan; Lukas F\"orner; Matthias Wagner; Michael Fr\"uhwald; Nassir Navab; Nico Schmutzenhofer; Sina Wendrich; Thomas Wendler

arxiv: 2604.07180 · v2 · pith:JJQ2ZVBEnew · submitted 2026-04-08 · 💻 cs.CV · cs.AI

Energy-based Tissue Manifolds for Longitudinal Multiparametric MRI Analysis

Kartikay Tehlan , Lukas F\"orner , Sina Wendrich , Nico Schmutzenhofer , Michael Fr\"uhwald , Matthias Wagner , Nassir Navab , Thomas Wendler This is my paper

Pith reviewed 2026-05-22 10:45 UTC · model grok-4.3

classification 💻 cs.CV cs.AI

keywords energy-based modelinglongitudinal mpMRItissue manifoldsimplicit neural representationsneuro-oncologychange detectionmultiparametric MRI

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The pith

Energy landscape from one baseline MRI scan tracks tissue changes over time without retraining

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that an energy function over multi-sequence intensity vectors can be learned from a single baseline scan and then used as a fixed geometric reference for later scans. Voxels from follow-up images are evaluated under this energy landscape to detect shifts in their position and energy value within the space of intensity combinations. In a paediatric recurrence case, these shifts showed progressive movement toward the original tumor-associated low-energy region before the tumor reappeared visibly on radiology, while a stable case showed voxels staying confined to the same basins. This matters because it offers a way to monitor disease evolution by the geometry of sequence space rather than by segmenting or classifying images each time. The approach avoids retraining the model on new scans and does not rely on explicit tissue labels.

Core claim

The central claim is that the baseline energy manifold, learned from a single scan via denoising score matching, encodes the set of observed contrast combinations as a fixed geometric reference. Longitudinal assessment is performed by evaluating how the distribution of MRI sequence vectors evolves under this energy function, with local minima defining tissue basins. In the presented paediatric case with later recurrence, follow-up scans showed progressive deviation in energy and directional displacement in sequence space toward the baseline tumour-associated regime before clear radiological reappearance. In contrast, a case with stable disease showed voxel distributions remaining confined to

What carries the argument

The energy function E_θ(u) over the multi-sequence intensity vector, trained by denoising score matching on the baseline scan alone, which supplies local minima as tissue basins, gradient magnitude as proximity to regime boundaries, and Laplacian curvature as local constraint structure.

Load-bearing premise

The baseline energy manifold learned from one scan remains a valid fixed reference across later scans, and shifts in voxel distributions under this function reflect biological tissue changes rather than imaging variability or artifacts.

What would settle it

A recurring tumor case in which follow-up voxel distributions show no progressive rise in energy or directional movement toward the baseline tumor regime before radiological reappearance would falsify the reported pattern.

Figures

Figures reproduced from arXiv: 2604.07180 by Kartikay Tehlan, Lukas F\"orner, Matthias Wagner, Michael Fr\"uhwald, Nassir Navab, Nico Schmutzenhofer, Sina Wendrich, Thomas Wendler.

**Figure 1.** Figure 1: Overview of the proposed energy-based longitudinal tissue tracking framework. Left: Two regions of interest (ROIs) are manually placed on the baseline scan (t0): one in healthy tissue (cyan) and one covering the tumour (magenta). Their centroids in 5- dimensional sequence space define two basin attractors of the learned energy function Eθ(u), visualised here in 2D for clarity. The vector d connecting the h… view at source ↗

**Figure 2.** Figure 2: Longitudinal projection of voxel energies along the baseline healthy–tumour axis for a patient with stable disease. The baseline manifold (top) defines the reference geometry; follow-up scans (middle, bottom) show preserved basin structure and stable energy distribution without progressive shift toward the baseline tumour regime, consistent with absence of recurrence. Recurrence: In a second patient, prio… view at source ↗

**Figure 3.** Figure 3: Longitudinal projection of voxel energies along the baseline healthy–tumour axis for a patient with recurrence. Relative to the baseline manifold, follow-up scans show progressive redistribution of voxel energies and displacement toward the baseline tumour regime, accompanied by deformation of the energy profile, consistent with reemergence of tumour-associated sequence states prior to clear anatomical de… view at source ↗

read the original abstract

We propose a geometric framework for longitudinal multi-parametric MRI analysis based on patient-specific energy modelling in sequence space. Rather than operating on images with spatial networks, each voxel is represented by its multi-sequence intensity vector ($T1$, $T1c$, $T2$, FLAIR, ADC), and a compact implicit neural representation is trained via denoising score matching to learn an energy function $E_{\theta}(\mathbf{u})$ over $\mathbb{R}^d$ from a single baseline scan. The learned energy landscape provides a differential-geometric description of tissue regimes without segmentation labels. Local minima define tissue basins, gradient magnitude reflects proximity to regime boundaries, and Laplacian curvature characterises local constraint structure. Importantly, this baseline energy manifold is treated as a fixed geometric reference: it encodes the set of contrast combinations observed at diagnosis and is not retrained at follow-up. Longitudinal assessment is therefore formulated as evaluation of subsequent scans relative to this baseline geometry. Rather than comparing anatomical segmentations, we analyse how the distribution of MRI sequence vectors evolves under the baseline energy function. In a paediatric case with later recurrence, follow-up scans show progressive deviation in energy and directional displacement in sequence space toward the baseline tumour-associated regime before clear radiological reappearance. In a case with stable disease, voxel distributions remain confined to established low-energy basins without systematic drift. The presented cases serve as proof-of-concept that patient-specific energy manifolds can function as geometric reference systems for longitudinal mpMRI analysis without explicit segmentation or supervised classification, providing a foundation for further investigation of manifold-based tissue-at-risk tracking in neuro-oncology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds a fixed patient-specific energy manifold from one baseline mpMRI scan to track longitudinal voxel shifts without segmentation, but the two descriptive cases give only preliminary support and leave technical comparability unaddressed.

read the letter

The main point is that this work tries to create a stable geometric reference from a single baseline scan by fitting an implicit neural representation to the multi-sequence intensity vectors, then watches how later scans distribute in that energy landscape. In the recurrence example the voxels drift toward the low-energy tumor basin before the tumor reappears on images; in the stable case they stay put. That construction is new enough in the longitudinal mpMRI setting and avoids the usual segmentation or retraining steps at follow-up time points. The geometric reading with minima, gradient magnitude, and curvature is a clean way to describe tissue regimes without labels. Credit for trying to turn the baseline data into a fixed reference that later scans are measured against rather than re-modeled each time. The method itself is coherent on paper and the denoising score matching choice fits the energy-function goal. The soft spots are not small. Only two proof-of-concept cases are shown, with no quantitative metrics, no error analysis, no comparison to existing longitudinal methods, and no statistical test of the observed drifts. The central assumption that the baseline energy function remains a valid fixed reference rests on the intensity vectors staying directly comparable across scans. The abstract and description give no sign of intensity standardization, histogram matching, or scanner harmonization. In pediatric mpMRI even modest gain or B1 changes can translate the whole cloud of vectors and produce exactly the kind of directional shift reported, so the biological interpretation is not yet separable from possible technical artifact. That concern is not minor and needs direct testing. This is for readers already working on manifold or energy-based methods in neuro-oncology imaging who want to explore label-free longitudinal references. Someone looking for a finished clinical tool or strong empirical results will not find it here. The paper deserves peer review because the core construction is internally consistent and the idea has clear potential, but any referee should insist on quantitative validation, artifact controls, and at least one external check on the fixed-manifold assumption before the claims can be taken as more than suggestive.

Referee Report

2 major / 2 minor

Summary. The paper proposes a geometric framework for longitudinal multiparametric MRI analysis based on patient-specific energy modelling in sequence space. Each voxel is represented by its multi-sequence intensity vector u = (T1, T1c, T2, FLAIR, ADC), and a compact implicit neural representation is trained via denoising score matching to learn an energy function E_θ(u) over R^d from a single baseline scan. This baseline energy manifold is treated as a fixed geometric reference for evaluating follow-up scans by analysing how voxel distributions evolve under the energy function, with local minima defining tissue basins. In a paediatric recurrence case, follow-up scans show progressive deviation in energy and directional displacement toward the baseline tumour regime before radiological reappearance; in a stable case, distributions remain confined without drift. The cases serve as proof-of-concept for label-free, segmentation-free tissue-at-risk tracking.

Significance. If the central claims hold after addressing validation gaps, the work could provide a novel differential-geometric tool for longitudinal mpMRI without explicit labels or supervised classification, potentially aiding early recurrence detection in neuro-oncology. Strengths include the use of implicit representations trained via denoising score matching and the formulation of longitudinal assessment as evaluation against a fixed patient-specific manifold rather than anatomical segmentations.

major comments (2)

[Abstract] Abstract, longitudinal assessment paragraph: The claim that observed shifts in voxel distributions under the fixed E_θ(u) reflect biological tissue changes (e.g., progressive deviation toward the tumour-associated regime) depends on the multi-sequence intensity vectors remaining directly comparable across scans. The manuscript provides no description of intensity standardization, histogram matching, or scanner harmonization, which is load-bearing because technical drifts in gain or acquisition parameters could translate the entire cloud of u vectors and induce artifactual movement toward the low-energy basin identified at baseline.
[Abstract] Abstract, case study descriptions: The recurrence and stable-disease cases are presented as descriptive proof-of-concept without quantitative metrics (e.g., mean energy deviation, displacement vector norms, or statistical tests), error analysis, or comparisons to baselines such as direct multi-sequence histogram distances or conventional segmentation-based tracking. This limits the ability to assess whether the directional displacement and energy changes are robust or merely illustrative.

minor comments (2)

[Abstract] The notation for the energy function E_θ(u) and the implicit representation could be expanded with explicit details on network architecture, training hyperparameters, and how gradient magnitude and Laplacian curvature are computed from the learned manifold.
[Abstract] The abstract refers to 'directional displacement in sequence space' without defining the precise computation (e.g., via gradients of E_θ or another operator), which affects reproducibility of the longitudinal analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive and detailed review. We address each major comment below and have revised the manuscript to incorporate clarifications and additional analyses where feasible.

read point-by-point responses

Referee: [Abstract] Abstract, longitudinal assessment paragraph: The claim that observed shifts in voxel distributions under the fixed E_θ(u) reflect biological tissue changes (e.g., progressive deviation toward the tumour-associated regime) depends on the multi-sequence intensity vectors remaining directly comparable across scans. The manuscript provides no description of intensity standardization, histogram matching, or scanner harmonization, which is load-bearing because technical drifts in gain or acquisition parameters could translate the entire cloud of u vectors and induce artifactual movement toward the low-energy basin identified at baseline.

Authors: We agree that comparability of the multi-sequence intensity vectors is essential for interpreting shifts as biological. The original manuscript described basic preprocessing but did not detail it sufficiently. In the revised version we have added an explicit subsection on intensity standardization, specifying per-sequence histogram matching to a common reference distribution derived from the baseline scan together with affine scaling for scanner harmonization. These steps were performed prior to training and evaluation to mitigate technical drift. revision: yes
Referee: [Abstract] Abstract, case study descriptions: The recurrence and stable-disease cases are presented as descriptive proof-of-concept without quantitative metrics (e.g., mean energy deviation, displacement vector norms, or statistical tests), error analysis, or comparisons to baselines such as direct multi-sequence histogram distances or conventional segmentation-based tracking. This limits the ability to assess whether the directional displacement and energy changes are robust or merely illustrative.

Authors: We accept that the presentation was primarily descriptive. In the revision we have added quantitative metrics: mean energy deviation and mean displacement-vector norm (in sequence space) between baseline and each follow-up scan, together with bootstrapped 95% confidence intervals obtained by voxel resampling. We also report the Wasserstein distance between the baseline and follow-up distributions as a simple non-manifold baseline. Direct numerical comparison to segmentation-based tracking is discussed in the limitations section, noting the absence of labels in our framework. These additions are included while retaining the proof-of-concept framing given the small number of cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper trains a patient-specific energy function E_θ(u) via denoising score matching on the multi-sequence intensity vectors from a single baseline scan, then evaluates follow-up scans by computing the same function on their intensity vectors without retraining. This establishes a fixed geometric reference by explicit design choice rather than by any reduction where a claimed result equals its inputs by construction. No steps match the enumerated circularity patterns: there is no self-definitional loop, no fitted parameter renamed as an independent prediction, and no load-bearing self-citation or uniqueness theorem invoked. The longitudinal observations in the case studies are empirical descriptions under the chosen model, not tautological outputs. The framework remains self-contained as a proof-of-concept geometric analysis method.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 1 invented entities

Only the abstract is available, so the ledger is based on stated elements; the energy model is data-fitted and several domain assumptions support the geometric interpretation and fixed-reference usage.

free parameters (1)

implicit neural representation parameters theta
The compact implicit neural representation is trained via denoising score matching on the single baseline scan to learn E_theta(u).

axioms (2)

domain assumption The learned energy landscape provides a differential-geometric description of tissue regimes without segmentation labels.
Invoked to define local minima as tissue basins, gradient magnitude as proximity to boundaries, and Laplacian as local structure.
domain assumption The baseline energy manifold is treated as a fixed geometric reference and is not retrained at follow-up.
Central premise for formulating longitudinal assessment as evaluation of subsequent scans relative to this baseline geometry.

invented entities (1)

energy function E_theta(u) over R^d no independent evidence
purpose: To encode the set of contrast combinations observed at diagnosis as a patient-specific manifold for tissue regime analysis.
Introduced as the core learned model from baseline data via implicit neural representation.

pith-pipeline@v0.9.0 · 5841 in / 1533 out tokens · 69586 ms · 2026-05-22T10:45:53.082660+00:00 · methodology

Review history (2 revisions) →

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

a compact implicit neural representation is trained via denoising score matching to learn an energy function E_θ(u) over R^d from a single baseline scan. The learned energy landscape provides a differential-geometric description of tissue regimes... baseline energy manifold is treated as a fixed geometric reference
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Local minima define tissue basins, gradient magnitude reflects proximity to regime boundaries, and Laplacian curvature characterises local constraint structure

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

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work page arXiv 2024
[2]

Advances in neural information processing systems32(2019)

Du, Y., Mordatch, I.: Implicit generation and modeling with energy based models. Advances in neural information processing systems32(2019)

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Ho, J., Jain, A., Abbeel, P.: Denoising diffusion probabilistic models. Advances in neural information processing systems33, 6840–6851 (2020)

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Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in neural information processing systems33, 7462–7473 (2020)

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Advances in neural information processing systems32(2019)

Song, Y., Ermon, S.: Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems32(2019)

work page 2019
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How to train your energy-based models

Song, Y., Kingma, D.P.: How to train your energy-based models. arXiv preprint arXiv:2101.03288 (2021)

work page arXiv 2021
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Advances in neural informa- tion processing systems33, 7537–7547 (2020)

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Anahita Fathi Kazerooni, e.a.: The brain tumor segmentation in pediatrics (brats- peds) challenge: Focus on pediatrics (cbtn-connect-dipgr-asnr-miccai brats-peds) (2024), https://arxiv.org/abs/2404.15009

work page arXiv 2024

[2] [2]

Advances in neural information processing systems32(2019)

Du, Y., Mordatch, I.: Implicit generation and modeling with energy based models. Advances in neural information processing systems32(2019)

work page 2019

[3] [3]

Advances in neural information processing systems33, 6840–6851 (2020)

Ho, J., Jain, A., Abbeel, P.: Denoising diffusion probabilistic models. Advances in neural information processing systems33, 6840–6851 (2020)

work page 2020

[4] [4]

Princeton University Press, Princeton (1963)

Milnor, J.: Morse Theory. Princeton University Press, Princeton (1963)

work page 1963

[5] [5]

Advances in neural information processing systems33, 7462–7473 (2020)

Sitzmann, V., Martel, J., Bergman, A., Lindell, D., Wetzstein, G.: Implicit neural representations with periodic activation functions. Advances in neural information processing systems33, 7462–7473 (2020)

work page 2020

[6] [6]

Advances in neural information processing systems32(2019)

Song, Y., Ermon, S.: Generative modeling by estimating gradients of the data distribution. Advances in neural information processing systems32(2019)

work page 2019

[7] [7]

How to train your energy-based models

Song, Y., Kingma, D.P.: How to train your energy-based models. arXiv preprint arXiv:2101.03288 (2021)

work page arXiv 2021

[8] [8]

Advances in neural informa- tion processing systems33, 7537–7547 (2020)

Tancik, M., Srinivasan, P., Mildenhall, B., Fridovich-Keil, S., Raghavan, N., Sing- hal, U., Ramamoorthi, R., Barron, J., Ng, R.: Fourier features let networks learn high frequency functions in low dimensional domains. Advances in neural informa- tion processing systems33, 7537–7547 (2020)

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Maria Correia de Verdier, e.a.: The 2024 brain tumor segmentation (brats) challenge: Glioma segmentation on post-treatment mri (2024), https://arxiv.org/abs/2405.18368

work page arXiv 2024

[10] [10]

Neural computation23(7), 1661–1674 (2011)

Vincent, P.: A connection between score matching and denoising autoencoders. Neural computation23(7), 1661–1674 (2011)

work page 2011

[11] [11]

ACM computing surveys56(4), 1–39 (2023)

Yang, L., Zhang, Z., Song, Y., Hong, S., Xu, R., Zhao, Y., Zhang, W., Cui, B., Yang, M.H.: Diffusion models: A comprehensive survey of methods and applications. ACM computing surveys56(4), 1–39 (2023)

work page 2023