空间图的度量空间：两样本检验、数据深度及其在心脏纤维化中的应用
A Metric Space of Spatial Graphs: Two-Sample Testing, Data Depth, and Application to Cardiac Fibrosis
AI总结：
本研究针对心脏纤维化相关空间图，提出带旋转不变Fused Gromov Wasserstein度量的空间图度量空间，结合两样本检验、DepthPlot可视化等方法，揭示纤维化纹理的患者特异性及潜在病理关联。
AI中文摘要：
心脏纤维化会降低电导率，是心律失常的主要诱因。心律失常波通常围绕无传导性的纤维化斑块旋转，因此这些斑块（心肌内空间孤立的纤维化组织区域）的几何与拓扑结构对心律失常动力学具有重要作用。尽管这些结构具有临床相关性，但人们对其仍知之甚少。我们利用受心脏纤维化影响的人类心脏的组织病理图像来解决这一公开问题。通过骨架化将每个斑块表示为空间图，其中节点作为欧几里得空间中的点嵌入，边编码底层组织的几何属性。本研究的核心方法学贡献是引入空间图空间，这是一种配备旋转不变的融合格罗莫夫-瓦瑟斯坦（Fused Gromov Wasserstein）度量的度量空间，可对节点和边数量不同的空间图进行比较。在此基础上，我们对空间图开展分布层面的统计检验和深度度量。为便于解释空间图样本分布，我们引入DepthPlot，这是一种用于度量空间中深度度量的新型可视化工具。将我们的方法应用于比较患者及斑块在心室中的空间位置时，我们发现纤维化纹理表现出强烈的患者特异性特征，而部分心脏显示出显著的几何相似性，这可能反映了共同的病理突变或其他未知因素。通过定量深度度量，我们通过中心和外围空间图来表征检验结果，证明所提框架能为纤维化纹理表征提供具有统计和临床意义的见解。
英文摘要：
Cardiac fibrosis reduces electrical conductivity and is a leading cause of arrhythmia. Arrhythmic waves typically rotate around non-conducting fibrotic patches, so the geometry and topology of these patches (spatially isolated regions of fibrotic tissue within the heart muscle) play an important role in arrhythmia dynamics. Despite their clinical relevance, these structures remain poorly understood. We address this open problem using histopathological images of human hearts affected by cardiac fibrosis. Each patch is represented as a spatial graph via skeletonization, where nodes are embedded as points in Euclidean space and edges encode geometric properties of the underlying tissue. The core methodological contribution of this work is the introduction of spatial graph space, a metric space equipped with a rotation-invariant Fused Gromov Wasserstein metric that enables comparison of spatial graphs with differing numbers of nodes and edges. Building on this, we perform a distribution-level statistical testing and depth measures for spatial graphs. To enable the interpretation of the spatial graph sample distribution, we introduce DepthPlot, a novel visualization tool for depth measures in metric spaces. Applying our methodology to compare patients and the spatial position of patches within the ventricles, we find that fibrotic textures exhibit strong patient-specific features, while some hearts display notable geometric similarities, potentially reflecting shared pathological mutations or other unknown factors. Through quantitative depth measures, we characterize test outcomes via central and peripheral spatial graphs, demonstrating that the proposed framework yields statistically and clinically meaningful insights into fibrotic texture characterization.
