A General B\'ezier Tree Encoding Counterfactual Framework for Retinal-Vessel-Mediated Disease Analysis

The geometry of the retinal vessel is a key biomarker of vascular diseases, yet clinical evidence remains primarily observational. Existing generative counterfactuals intervene only at the image-level disease label, failing to isolate explicit anatomical structure. To address this limitation, we propose the B\'ezier Tree Encoding Counterfactual Framework (BTECF). By abstracting vascular networks into interconnected cubic-B\'ezier segments, BTECF establishes a disease-agnostic representation in which structural topology is explicitly preserved and atomically perturbable. Coupling this encoding with a diffusion-based generator enables parameter-level do-interventions on explicit geometric axes (e.g., tortuosity, caliber) while preserving background fundus textures. We validate BTECF on diabetic retinopathy, together with independent cohorts for ischemic stroke and Alzheimer's disease. Isolated counterfactual interventions produce dose-responsive shifts in classifier predictions; a matched pixel-drop control attenuates this response by an order of magnitude or more, ruling out out-of-distribution generation artifacts. By enforcing causal isolation between vessel topology and pixel-level confounders, BTECF provides a unified generative paradigm for hypothesis verification across systemic diseases. To support reproducibility, the code will be publicly released upon acceptance.