Calibrated Alzheimer's Conversion Risk in Mild Cognitive Impairment: Persistent Homology of Clinical Trajectories with Conformal Guarantees

Background. Predicting conversion from mild cognitive impairment (MCI) to Alzheimer's disease (AD) is central to trial enrichment and care planning, yet existing models provide no individual-level uncertainty estimates and rarely include transparent leakage audits. We introduce the first application of persistent homology to longitudinal clinical trajectory point clouds for this task, and the first split-conformal individual risk guarantee for any AD-conversion model. Methods. We analysed 741 MCI subjects (240 converters, 32.4%) from ADNI with a uniform 4-year follow-up cap. Five leakage sources were corrected; without them a naive pipeline achieved AUC=0.934, inflated by +0.075. Vietoris-Rips persistent homology and sublevel-set proxies were combined with trajectory slopes and engineered features (76 total) in a stacking ensemble evaluated by 5-fold cross-validation. Results. Cox and Random Survival Forest models with TDA features achieved concordance C=0.799 and C=0.826 versus C=0.753 and C=0.812 without (+0.045 and +0.014). The primary nested AUC is 0.840 (same-fold bound 0.866); external AUC was 0.879 on a zero-overlap ADNI-2/GO/3 cohort. H0 persistence entropy was the top SHAP feature and significantly associated with APOE4 dosage (Spearman r=-0.191, p<0.0001, Bonferroni-corrected). Cross-conformal coverage was 90.4%+-2.2% (target 90%); empirical external coverage 96.9%. Maximum fairness gap in false-negative rate across seven subgroups was 0.092. Conclusions. We propose H0 persistence entropy as a topological biomarker of cognitive decline and demonstrate that a leakage-audited, conformally calibrated pipeline reaches competitive accuracy with individual-level uncertainty quantification not previously available for this task.

Discrete Ricci Curvature on Protein Contact Graphs for Lightweight Fold Classification

Protein fold classification can be approached via sequence-based representations or structural descriptors, but direct comparisons between lightweight handcrafted descriptors and pretrained protein language model embeddings remain limited. We investigate discrete Ricci curvature on Calpha contact graphs as a lightweight structural descriptor for fold classification. Each protein domain is represented by a 22-dimensional fixed-length feature derived from summary statistics and quantiles of Ollivier-Ricci and Forman-Ricci edge curvature distributions. We evaluate on CATH top-10 Topology classification and on the ASTRAL 40%-identity SCOPe top-10 Fold benchmark, comparing against geometry, contact-graph statistics, persistent homology, and mean-pooled ESM-2 (150M) baselines. On both datasets, lightweight structural descriptors substantially outperform mean-pooled ESM-2 embeddings, with a larger performance gap on the ASTRAL 40% SCOPe benchmark. Ricci alone uses 22 dimensions, or 3.4% of the ESM-2 baseline dimensionality, and already outperforms mean-pooled ESM-2 on both datasets. Combining Ricci with persistent homology yields the strongest performance, achieving macro-F1 of 0.71 on CATH and 0.68 on SCOPe with a 112-dimensional feature vector. These results identify a regime where lightweight interpretable graph descriptors offer a practical alternative to pretrained protein language model embeddings.

TopoAgent: An Agentic Framework for Automated Topology Learning in Medical Imaging

Topological data analysis (TDA), particularly persistent homology (PH), captures geometric structural properties in medical images (e.g., connected components, loops, shape characteristics), which conventional pixel-level deep learning approaches often neglect. While many topological descriptors are known for converting persistence diagrams (PDs) or raw images into topological feature vectors, existing methods mostly default to a single fixed descriptor (e.g., persistence images), leaving the diversity of topological representations largely unexplored. To the best of our knowledge, there is no known large language model (LLM)-based agentic framework that can automatically determine the most suitable topological descriptors for a given image dataset and produce the corresponding topological feature vectors for downstream tasks. To fill this gap, we propose \textbf{TopoAgent}, an LLM-based agentic framework that automates topology learning for medical image analysis.TopoAgent operates through a Perception--Reasoning--Action--Reflection loop supported by 21 domain-specific tools and dual memory that accumulates experience across runs. Its skill set is distilled from systematic evaluation of 15 topological descriptors across 26 datasets with six classifiers. TopoAgent analyzes input images and their topological characteristics, reasons about which topological descriptors best suit the input, and determines the optimal descriptor and its configuration, all without task-specific training.