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.

翻译：我们提出了一种基于序列空间患者特异性能量建模的纵向多参数MRI分析几何框架。该方法并非利用空间网络对图像进行操作，而是将每个体素表示为多序列强度向量（T1、T1c、T2、FLAIR、ADC），并通过去噪分数匹配训练紧凑的隐式神经表示，从单次基线扫描中学习定义在 \(\mathbb{R}^d\) 空间上的能量函数 \(E_\theta(\mathbf{u})\)。所学得的能量景观无需分割标签即可提供组织区域的微分几何描述：局部最小值定义组织盆地，梯度幅度反映与区域边界的接近程度，拉普拉斯曲率表征局部约束结构。关键在于，该基线能量流形被固定为几何参考基准：它编码了诊断时观察到的对比组合集合，且在随访中不重新训练。因此，纵向评估被表述为相对于该基线几何结构对后续扫描进行评价。我们并非比较解剖分割结果，而是分析MRI序列向量分布如何在基线能量函数下演化。在一例后续复发的儿科病例中，随访扫描在明确影像学复发前即显示能量演化和序列空间中向基线肿瘤相关区域的定向位移。在一例病情稳定病例中，体素分布始终局限于已建立的低能量盆地，未出现系统性偏移。所呈现病例作为概念验证表明：无需显式分割或监督分类，患者特异性能量流形可充当纵向mpMRI分析的几何参考系统，为神经肿瘤学中基于流形的组织风险追踪进一步研究奠定了基础。