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.

In longitudinal brain MRI of multiple sclerosis patients, treating each segmented lesion (or lesion-voxel centroid) at baseline as a point in 3D space, the H0 (connected-component) persistence barcode of a distance-based Vietoris-Rips/Čech filtration over the full lesion point cloud — whose death times equal the pairwise inter-lesion distances realized in the minimum spanning tree (MST) of that cloud — contains features (e.g., persistence entropy, number of bars below threshold d, bar-length distribution statistics) that, when added to a predictive model, improve AUROC by ≥0.05 over a baseline using only raw nearest-neighbor inter-lesion distances, for the binary classification task "will lesion i merge with or split from its baseline identity by follow-up timepoint t" — evaluated on a held-out cohort of ≥100 patients not used in training.

Minimum viable test (MVT):

• Cohort: n=100 patients minimum (held-out test), n≥300 total recommended (train/val/test 60/20/20) with paired baseline + ≥1 follow-up 3D FLAIR/T1-Gd scans.
• Pipeline: (1) lesion segmentation → centroid extraction; (2) build point cloud per timepoint; (3) compute H0 persistence diagram via Vietoris-Rips filtration (GUDHI/Ripser); (4) derive feature vector (persistence entropy, total persistence, bar-count histogram at k thresholds, Betti curve summary stats); (5) compute NN-distance baseline features (min, mean, std of nearest-neighbor distances per lesion); (6) label each baseline lesion as "merged," "split," "stable," or "resolved" by matching to follow-up lesion mask via deformable registration + overlap/centroid-tracking heuristic (ground truth defined by radiologist-adjudicated correspondence on a 20-patient subsample, kappa≥0.75 required); (7) train logistic regression + gradient-boosted tree (XGBoost) classifiers on (a) baseline-only features, (b) H0+baseline features; (8) compare AUROC with DeLong test and bootstrap CI.
• Pre-registration of exact feature set and classifier hyperparameters before touching test set (lock file in repo, hash-committed).

# Phase A: data prep
for patient in cohort:
    seg_base, seg_fup = segment_lesions(mri_base), segment_lesions(mri_fup)
    seg_fup_reg = register(seg_fup, target=mri_base)  # ANTs SyN
    centroids_base = extract_centroids(seg_base)
    centroids_fup  = extract_centroids(seg_fup_reg)
    labels = match_correspondence(centroids_base, centroids_fup)  # merge/split/stable/new

# Phase B: feature extraction
def h0_features(points):
    rips = gudhi.RipsComplex(points=points, max_edge_length=T_max)
    st = rips.create_simplex_tree(max_dimension=0)
    diag = st.persistence()
    return {
        'entropy': persistence_entropy(diag),
        'total_persistence': sum(death-birth for (b,death) in diag),
        'bar_hist': histogram_bars(diag, bins=[5,10,20,40]),  # mm
        'betti0_auc': betti_curve_auc(diag)
    }

def nn_baseline_features(points):
    D = pairwise_distances(points)
    return {'nn_dist': D.min(axis=1), 'mean_dist': D.mean(), 'std_dist': D.std()}

X_baseline = [nn_baseline_features(p) for p in all_patients]
X_full     = [concat(nn_baseline_features(p), h0_features(p)) for p in all_patients]
y          = labels  # binary: merge_or_split vs stable

# Phase C: model + eval
model_base = XGBClassifier().fit(X_baseline_train, y_train)
model_full = XGBClassifier().fit(X_full_train, y_train)
auc_base = roc_auc_score(y_test, model_base.predict_proba(X_baseline_test))
auc_full = roc_auc_score(y_test, model_full.predict_proba(X_full_test))
delong_test(auc_base, auc_full)  # significance
bootstrap_ci(auc_full - auc_base, n=2000)