The Case for Interpretable Geometry: Statistical Shape Models vs. Curvature-Based Descriptors in Aortic Disease Classification

Purpose: Quantifying aortic morphology is central to surgical planning for thoracic endovascular aortic repair (TEVAR), yet no consensus exists on how best to represent three-dimensional aortic shape for outcome prediction. Two broad strategies have emerged: statistical shape analysis (SSA), which relies on statistical methods and dimensionality reduction to capture the most significant shape modes, and geometrically-informed approaches that extract descriptors grounded in differential geometry. Here, we directly compare these paradigms on a cohort of 290 CTA scans classified by surgical outcome (non-pathological, successful TEVAR, failed TEVAR). Methods: For the geometrically-informed approach, we use a two-dimensional feature space using normalized fluctuation in integrated Gaussian curvature $\widetilde{\delta K}$ and mean aortic radius $R$. For SSA, we construct a point-cloud shape model with dimensionality reduction using Principal Component Analysis (PCA) and evaluate classification performance as a function of the number of retained principal components. Results: SSA's leading principal components encode variations in global aortic size and are statistically redundant with ($R$, $\widetilde{\delta K}$), yet they lack a one-to-one correspondence with interpretable anatomical quantities. Testing on an unseen, independent dataset reveals that the geometrically-informed approach provided better generalizability than SSA. Using Gaussian process classification with 10-fold cross-validation, we find that the geometrically-informed approach achieves a higher weighted $F_1$ score than SSA achieves with up to 20 principal components. While SSA's full-dataset accuracy rises above 90\% with increasing dimensionality, this gain is driven by overfitting rather than genuine discriminative power. Conclusion: These results demonstrate that geometrically-informed descriptors offer a more interpretable, robust, and clinically translatable framework for aortic disease classification than data-driven statistical shape representations.