A Metric Space of Spatial Graphs: Two-Sample Testing, Data Depth, and Application to Cardiac Fibrosis

2026-08-13

AI Summary

Cardiac fibrotic plaques exhibit geometric and topological structures that significantly influence arrhythmogenesis, yet their inter-individual variability lacks effective quantitative characterization. This work addresses this gap by skeletonizing plaque structures from tissue sections into spatial graphs and introducing the first metric framework tailored for spatial graphs with varying numbers of nodes and edges—based on a rotation-invariant fused Gromov–Wasserstein distance. The approach further incorporates data depth in metric spaces and a DepthPlot visualization tool, enabling distribution-level two-sample testing and interpretable analysis. Applied to patient data, the method reveals patient-specific fibrotic architectures alongside partial geometric similarities across hearts, yielding statistically and clinically meaningful characterizations of both central and peripheral regions.

Abstract

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.

Problem

• spatial graphs
• cardiac fibrosis
• two-sample testing
• data depth
• arrhythmia

Innovation

• Spatial Graph Space
• Fused Gromov-Wasserstein Metric
• Data Depth
• Two-Sample Testing
• DepthPlot