Energy-based Tissue Manifolds for Longitudinal Multiparametric MRI Analysis

Kartikay Tehlan; Lukas Förner; Sina Wendrich; Nico Schmutzenhofer; Michael Frühwald; Matthias Wagner; Nassir Navab; Thomas Wendler

cs.CVApr 8, 2026

Energy-based Tissue Manifolds for Longitudinal Multiparametric MRI Analysis

Authors: Kartikay Tehlan, Lukas Förner, Sina Wendrich, Nico Schmutzenhofer, Michael Frühwald, Matthias Wagner, Nassir Navab, Thomas Wendler

Organizations: Dept. of diagnostic and interventional Radiology and Neuroradiology, University Hospital Augsburg, Germany · Digital Medicine, University Hospital Augsburg, Germany · Chair for Computer Aided Medical Procedures and Augmented Reality, Technical University of Munich, Germany · Bavarian Center for Cancer Research (BZKF) Augsburg, Germany · Dept. of Pediatrics and Adolescent Medicine, University Hospital Augsburg, Germany · Center for Advanced Analytics and Predictive Sciences, University of Augsburg, Germany

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 $E_θ(\mathbf{u})$ 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.

Figures & tables

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 ( $t_{0}$ ): 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 $E_{\theta}(\bm{u})$ , visualised here in 2D for clarity. The vector $\bm{d}$ connecting the healthy tissue centroid to the tumour centroid defines a one-dimensional axis in sequence space along which longitudinal analysis is performed. Centre: For each time point, brain voxels are projected onto $\bm{d}$ and their energies $E(\bm{u},t_{k})$ are displayed as scatter plots. The shift $\Delta E(t_{k})$ relative to the baseline manifold $E(\bm{u},t_{0})$ quantifies how far follow-up tissue states deviate from the original configuration. Right: The corresponding gradient magnitude $\|\nabla E(\bm{u},t_{k})\|$ along $\bm{d}$ reflects changes in basin boundaries and ridge stability over time. Tracking the distribution of voxel energies and their displacement along the healthy–tumour axis enables detection of tissue at risk.

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.

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 re-emergence of tumour-associated sequence states prior to clear anatomical delineation.

Patient	$\delta E_{t1}$	$\delta E_{t2}$	$\delta E_{t3}$	Drift
Stable disease	$0.827$	$0.684$	$-0.063$	$-0.081$
	$(\pm 0.009)$	$(\pm 0.008)$	$(\pm 0.360)$	$(\pm 0.005)$
Recurrence	$1.119$	$+0.880$	$+0.925$	$+0.258$
	$(\pm 0.009)$	$(\pm 0.008)$	$(\pm 0.006)$	$(\pm 0.004)$

Table 1: Mean energy change (

\delta E

) and sequence-space drift relative to baseline for both patients at three time points: immediately post-resection (

t_{1}

<

48h), two months post-resection (

t_{2}

), and last available scan (

t_{3}

; 2 years and 13 months post-resection for stable and recurrence patients, respectively). Drift is projected onto the healthy–tumour centroid axis in

\mathbb{R}^{5}

; negative values indicate displacement away from the tumour basin, positive values toward it. All differences are statistically significant (

p\ll 0.05

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