A Differential-Geometric Index For Three-Dimensional Crown Adaptation: Implications For Bioactive Cement Distribution And Therapeutic Restorative Interface Optimization
European Journal of Prosthodontics and Restorative Dentistry (2026) 34 (2s), 100-110
KeywordsCAD/CAM, marginal adaptation, differential geometry, zirconia, cement thickness, digital dentistry, internal fit, mesh metrology
AuthorsAbstractBackground: Hypothyroidism Conventional assessment of crown adaptation relies on one- or two-dimensional linear measurements that discard the geometric information inherent in three-dimensional intraoral scan data. This study introduces and validates a novel differential-geometric metric—the Conformal Adaptation Index (CAI)—for quantifying the internal fit of CAD/CAM ceramic crowns, and benchmarks it against simpler three dimensional comparators. Materials and methods: The CAI was derived from the Dirichlet energy of the signed-distance discrepancy field between the prepared abutment and the crown intaglio. Theorem 1 proved rigid-body (SE(3)) invariance. The index was discretised for triangular meshes and implemented in open-source software (available from the corresponding author upon reasonable request). An in silico validation study was conducted on 120 mathematically generated maxillary first-molar preparations (surfaces of revolution with controlled perturbations) plus 30 asymmetric buccolingual variants. Baseline comparators included mean 2D marginal gap (MG), mean absolute thickness (MAT), root-mean-square thickness (RMS), total variation (TV), and volumetric discrepancy per unit area (VD/A). Finite-element stress peaks (SPEAK) under occlusal load served as a biomechanical association benchmark. Because all metrics violated normality (Shapiro–Wilk p < 0.001), Spearman rank correlations and non-parametric ROC analyses were used. Results: CAI demonstrated very strong correlation with volumetric discrepancy (ρ = 0.96, p < 0.001) and significantly outperformed all simpler metrics in association with peak ceramic stress (ρ = 0.90 for CAI / RMSG versus 0.84 for RMS, 0.81 for TV, 0.79 for MAT, 0.71 for MG; all p < 0.001). For discriminating high-stress geometric profiles (SPEAK > 400 MPa), area under the ROC curve was 0.91 for CAI versus 0.83 for RMS, 0.80 for TV, 0.78 for MAT, and 0.74 for MG. The asymmetric preparation subset yielded comparable CAI–SPEAK association (ρ = 0.88). Mesh-convergence analysis showed CAI stabilized within 0.5 % at mean edge lengths below 90 μm. An empirical scaling law relating maximum local cement excess to the square root of CAI (R² = 0.93) was established for rotationally symmetric forms. Conclusion: The CAI provides a mathematically rigorous, rotation-invariant scalar measure of three-dimensional crown adaptation that captures spatial heterogeneity inaccessible to simpler thickness statistics. Its strong association •••••••••••••••••••••••••••• ejprd.org - Published by Riset Publishing Services LLC
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