Energy-based Tissue Manifolds for Longitudinal Multiparametric MRI Analysis

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.