Persistent homology of the whole lesion set: H0 persistence of the multiple sclerosis lesion cloud across a distance filtration, so the death times are inter-lesion distances, predicts which lesions merge or split between baseline and follow-up MRI. The hypothesis is falsified if this H0 feature set does not beat a nearest-neighbour inter-lesion-distance baseline (AUROC difference of at least 0.05) on a held-out longitudinal cohort of at least 100 patients, because H0 death times of a point cloud are the edge lengths of its minimum spanning tree and may add nothing beyond local distances.

For each patient scan pair (baseline t0, follow-up t1) in a longitudinal MS cohort, represent the set of segmented lesion centroids at t0 as a point cloud in 3D physical space. Compute the H0 persistence barcode of this point cloud under a Vietoris-Rips (equivalently, single-linkage) filtration by Euclidean distance, where death times correspond to edge lengths of the minimum spanning tree (MST) connecting lesions. Construct a feature vector from this barcode (e.g., death-time distribution statistics, persistence entropy, number of bars in distance bins, Betti curves). Train a classifier/regressor using these H0 features to predict, for each lesion or lesion-pair, whether it will merge with another lesion, split into multiple lesions, or remain stable by t1. The hypothesis is CONFIRMED if this H0-feature model achieves AUROC on held-out patients at least 0.05 higher than a baseline model using only raw nearest-neighbor inter-lesion distances (1-NN, 2-NN distances per lesion) as features, evaluated via identical cross-validation folds and lesion-merge/split ground truth derived from registered longitudinal segmentations.

Multi-site validation (MV) design comparing two nested feature sets in a controlled classifier framework, with the baseline model as a strict feature subset test (NN-distances are a special case of pairwise-distance information that H0 also encodes via MST edges) to isolate whether higher-order topological aggregation adds predictive signal.

PIPELINE:

1. load_longitudinal_cohort(cohort_dirs) -> patients[t0_mask, t1_mask, metadata]
2. for patient in patients: t1_reg = deformable_register(patient.t1_mask, patient.t0_mask) centroids_t0 = extract_centroids(patient.t0_mask) correspondence = voxel_overlap_match(patient.t0_mask, t1_reg, thresh=0.5) labels = classify_events(correspondence) # stable/merge/split/new/disappear
3. for patient in patients: D = pairwise_euclidean(centroids_t0) rips_complex = gudhi.RipsComplex(distance_matrix=D) st = rips_complex.create_simplex_tree(max_dimension=1) st.persistence() H0_bars = st.persistence_intervals_in_dimension(0) # death times = MST edges h0_features = extract_features(H0_bars) # entropy, betti_curve(thresholds), total_persistence nn_features = compute_knn_distances(D, k=[1,2])
4. X_h0 = assemble(h0_features, covariates=[lesion_count, lesion_volume]) X_nn = assemble(nn_features, covariates=[lesion_count, lesion_volume]) y = labels.merge_or_split_binary
5. cv = PatientGroupKFold(n_splits=5) model_h0 = XGBClassifier(); model_nn = XGBClassifier() auroc_h0 = cross_val_auroc(model_h0, X_h0, y, cv) auroc_nn = cross_val_auroc(model_nn, X_nn, y, cv)
6. held_out_test: fit on train+val (N>=200), evaluate on test (N>=100) delta_auroc = auroc_h0_test - auroc_nn_test bootstrap_CI(delta_auroc, n=2000); delong_test(h0_scores, nn_scores, y)
7. decision: delta_auroc >= 0.05 AND CI excludes 0 AND p<0.05 -> CONFIRMED else FALSIFIED

KNOWN_FAILURE_MODES:

• H0 death times are mathematically just sorted MST edge lengths; if classifier only uses aggregate statistics (mean/max death time) without positional/order context, it may be information-theoretically near-equivalent to k-NN distances for k=1, producing near-zero delta_auroc by construction (the falsification condition baked into the hypothesis itself).
• Registration error between timepoints can mislabel merge/split events, injecting label noise that suppresses both models' AUROC and obscures true delta.
• Confounding by lesion count/density: patients with more lesions naturally have more merge events AND different persistence statistics (shorter death times), so H0 'predictive power' may be a count proxy, not true topology; must control via stratified/matched sensitivity analysis.
• Centroid-only representation discards lesion shape/volume, which may dominate true merge risk (e.g., large lesions merge more due to size not position), biasing against finding genuine topological signal independent of size.
• Small effective positive class (merge/split events may be <10% of lesions) leading to unstable AUROC estimates even with N=100 patients if lesion-per-patient counts are low.

ABORT_CHECKPOINTS:

• Checkpoint A (Day 10): If lesion correspondence labeling has <70% inter-rater/algorithm agreement with manual expert review on a 20-patient pilot subset, abort/redesign labeling before full-scale feature extraction.
• Checkpoint B (Day 25): If baseline merge/split event rate across pilot cohort (n=30 patients) is <5%, abort or seek higher-risk-enriched cohort (e.g., active/progressive MS subtype) to ensure statistical power.
• Checkpoint C (Day 45): After internal 5-fold CV on training cohort, if delta_auroc (H0 vs NN) point estimate is <0.02 with tight CI, consider early stopping before committing compute to full held-out validation, since replication is unlikely to reverse a near-null internal result.
• Checkpoint D (Day 60): Prior to final external validation, verify covariate balance (lesion count, lesion volume distributions) between train and held-out test sets; if substantially mismatched, results will be uninterpretable and protocol must be revised.