An integrated geometric quantification and shape analysis framework for axillary lymph node metastasis in breast cancer patients

1. Introduction

• Metastatic involvement may produce spatially heterogeneous changes that global measurements of size, elongation, or compactness may not fully reflect.Examples include asymmetric enlargement, focal contour bulging, lobulation, and contour irregularity.Source: §1 · p. 1
• CT-derived meshes can contain geometric and topological defects, so reliable surface analysis requires control of artifacts, topology, mesh quality, and geometric preservation.Genus alone does not ensure connectedness, closedness, manifoldness, or consistent orientation.Source: §1 · pp. 1–2
• SH degree is a methodological question because the degree needed for faithful reconstruction may differ from the degree that best represents morphology associated with metastatic status.Higher degrees recover finer detail but may also retain variation weakly associated with the clinical endpoint.Source: §1 · p. 2
• Geometric feature families measure different properties and neighborhoods, motivating feature-dependent reconstruction scales rather than one universally optimal SH degree.The families include differential-geometric, spectral, geodesic, shape-distribution, and symmetry descriptors.Source: §1 · p. 2
• The study combines topology-aware processing and multi-resolution geometric analysis to evaluate surface validity, SH resolution effects, robustness, and transportability.The planned evaluations include feature-family-specific resolution preferences and independent multicenter lymph-node and cross-organ experiments.Source: §1 · p. 2

2.1. Study cohort and input data

• The primary cohort comprised 1,769 nodes from 214 women, with 1,172 nodes from patients without metastasis and 597 from patients whose dissected nodes were metastatic.An independent four-hospital cohort had no node-level metastasis labels, while 59 labeled LIDC-IDRI lung nodules supported a separate cross-organ experiment.Source: §2.1 · pp. 2–3

2.2. Overview of the geometric processing workflow

• The workflow combined context-aware smoothing, adaptive voxelization and topology correction, geometry-constrained smoothing, and local artifact repair before spherical analysis.Natural voxelization was preferred when valid; topology repair and then convex-hull reconstruction served as fallbacks when needed.Source: §2.2 · p. 3, Figure 1 · p. 4

2.3. Context-aware smoothing for staircase-artifact suppression

• Context-aware smoothing targeted staircase-prone regions by weighting nearby vertices according to artifact likelihood and distance, rather than smoothing the entire mesh uniformly.It was applied for 10 iterations before voxelization and five afterward.Source: §2.3 · pp. 3–4

2.4. Adaptive voxelization and strict genus-0 topology correction

• Voxelization used an adaptive natural-first search, retaining the first candidate that passed strict validity checks and invoking repair or convex-hull fallback only if necessary.Validity required a closed, consistently oriented, connected genus-0 mesh with no listed mesh-integrity defects.Source: §2.4.2 · p. 4, §2.4.1 · p. 4, §2.4 · p. 4

2.5. Geometry-constrained adaptive smoothing

• Adaptive Laplacian and Taubin smoothing reduced residual irregularities while constraining geometric deviation, with iteration acceptance and rollback governed by smoothness improvement and fidelity limits.Stopping used a 0.20% improvement threshold with two-evaluation patience; volume, principal-axis extent, and normalized HD95 limits constrained changes.Source: §2.5 · p. 5, §2.5.3 · p. 5

2.6. Detection of local surface artifacts

• Local-artifact detection addressed defects that topology checks could miss through α-shape candidate localization followed by ray-based fold detection, small-hole detection, and CT-informed screening.Targets included inward folds, steep-walled pits, and CT-related surface artifacts.Source: §2.6 · p. 6

2.6.2. Ray-based inward-fold detection

• The α-shape stage defined a sensitive candidate search domain, while ray-based detection identified inward folds using repeated intersections and joint count-and-proportion thresholds.Candidate faces were not removed solely because they fell within the α-shape search domain.Source: §2.6.1 · p. 6, §2.6.2 · p. 6

2.6.4. CT-informed geometric screening

• Small-hole screening combined triangle size and shape, adjacent-face normal variation, and multiscale smoothing response, while CT-informed screening required multiple artifact-evidence branches.Staircase evidence alone was insufficient for CT-informed removal; selected faces from the ray, hole, and CT stages were combined as a union.Source: §2.6.3 · p. 6, §2.6.4 · p. 7, §2.7 · p. 7

2.7. Repair and reconstruction of local surface artifacts

• The full repair sequence combined artifact excision, boundary-aware triangulation, conforming subdivision, quadratic reference-surface estimation, constrained cotangent bi-Laplacian fairing, and local Taubin smoothing.The sequence is summarized in Fig. 4.Source: §2.7 · p. 7

2.7.1. Local excision and initial patch construction

• Boundary openings were filled with constrained Delaunay triangulation when valid, otherwise conflict-aware ear clipping or a centroid-fan fallback; long patch edges were split conformingly.Long edges exceeding 1.5 times the input mesh’s median edge length were midpoint-split for at most three passes.Source: §2.7.1 · pp. 7–8

2.7.2. Quadratic local reference surface

• A distance-weighted quadratic reference surface was fitted from five rings of measured faces; only interior patch vertices were projected, with displacement clipped to twice the median edge length.The fit was omitted when fewer than 12 measured reference vertices were available or the local frame was numerically degenerate.Source: §2.7.2 · p. 8

2.7.3. Constrained cotangent bi-Laplacian fairing

• Cotangent bi-Laplacian fairing jointly refined the patch and four surrounding face rings while fixing the outer boundary to limit deformation propagation.Reconstructed vertices were allowed to adapt to local geometry, while measured surrounding vertices were strongly constrained to their original positions.Source: §2.7.3 · p. 8

2.7.4. Local smoothing and post-repair validation

• Five local Taubin-smoothing iterations followed fairing; backtracking rejected updates that could invert affected triangles, and strict validation confirmed connected, closed genus-0 meshes without listed topology or integrity defects.The outer boundary of the smoothing region remained fixed; validation checked for boundary edges, non-manifold edges, orientation conflicts, duplicate or degenerate faces, and unreferenced vertices.Source: §2.7.4 · p. 8

• Three candidate mappings are considered: conformal with Möbius area correction, area-preserving density equalization, and spherical Tutte mapping as a robust alternative.Source: §3.1 · pp. 8–9
• Conformal and area-preserving maps must pass orientation, triangle, vertex, and spherical-area checks; Tutte mapping is the fallback when geometry-related validity issues occur.Source: §3.1 · p. 9
• For each node and SH degree, the admissible map with the smallest normalized symmetric bidirectional nearest-neighbor RMS distance is selected using geometry alone, not metastasis labels or predictive outcomes.Symmetric mean nearest-neighbor distance and HD95 are recorded for quality assessment but do not determine selection.Source: §3.1 · p. 9

• Real spherical harmonics provide a hierarchical spectral representation of genus-0 surfaces using the degree-specific spherical parameterization.Source: §3.2 · p. 9
• SH coefficients are estimated after weighted-centroid removal using spherical-triangle areas as quadrature weights, then the centroid is restored after reconstruction.Area weighting reduces sensitivity to nonuniform vertex sampling on the spherical parameterization.Source: §3.2 · p. 9
• Primary lymph-node analyses evaluate degrees 5, 8, 10, 15, 20, and 34; each degree-L expansion contains (L + 1)^2 three-dimensional coefficient vectors.Source: §3.2 · p. 9

3.3. Surface feature extraction

• The same feature-extraction procedures and numerical settings are applied independently to SH reconstructions at different degrees.Source: §3.3 · p. 9

3.3.1. Conventional geometric shape features

• Py14 comprises the 14 three-dimensional PyRadiomics shape features, extracted from original, processed-reference, and SH-reconstructed surfaces as the global-morphology baseline.Source: §3.3.1 · p. 9

3.3.2. Extended surface descriptors and feature-family organization

• The extended set contains 352 variables organized a priori into 20 families spanning global, local differential, intrinsic spectral, distributional, symmetry, and SH spectral descriptors.Together with Py14, this yields 366 candidate variables for multi-resolution analysis.Source: §3.3.2 · p. 9

3.4. Quantitative evaluation and statistical analysis

• Quantitative analyses assess geometric validity, multi-resolution representation, prediction, perturbation robustness, and transportability; external lymph-node and lung-nodule cohorts are analyzed separately for label-free replication and cross-organ validation.Source: §3.4 · pp. 9–10
• Table 1 presents the feature families used in the multi-resolution SH analysis.Source: Table 1 · p. 10
• Patient is the grouping and resampling unit: all nodes from one patient remain in the same cross-validation fold, and training-only preprocessing and tuning help prevent information leakage.Source: §3.4 · p. 10

3.4.1. Topological validity and downstream computational quality

• Strict topology validity requires a single connected, closed, consistently oriented genus-0 surface without listed mesh defects; computational quality is assessed on 500 paired surfaces using triangle quality and adjacent-face normal variation.Higher triangle quality and lower normal jump indicate improved numerical suitability for differential-geometric computation.Source: §3.4.1 · p. 10

3.4.2. Univariate discrimination of conventional shape descriptors

• Each Py14 descriptor is evaluated independently on original and six SH-reconstructed surfaces with single-feature logistic regression and repeated patient-grouped cross-validation.Source: §3.4.2 · p. 10

3.4.3. Degree-wise spectral localization of discriminative information

• Degree-wise SH energy and folded AUC are examined jointly to distinguish represented geometric information at each spectral scale from its relevance to metastasis-status discrimination.The degree-0 term is excluded because it represents the constant component rather than shape variation.Source: §3.4.3 · pp. 10–11

3.4.4. Geometric fidelity and high-resolution representation efficiency

• Geometric fidelity is assessed on 500 surfaces using four errors relative to processed references and pooled rank-based scores; degree 34 is also examined for representation efficiency and preservation of conventional morphology.The composite score is cohort-relative, not an absolute physical error measure; coefficient counts are compared with mesh vertices as representation elements, not runtime or byte-level storage.Source: §3.4.4 · p. 11

3.4.5. Fixed-resolution and family-specific mixed-resolution prediction

• Fixed-resolution models use all 366 variables at one degree, while the mixed-resolution model selects a degree separately for each feature family using training data and evaluates held-out predictions.Comparisons use locked outer-fold assignments; the original-surface Py14 model is the conventional global-morphology baseline.Source: §3.4.5 · p. 11

3.5. Resolution-dependent perturbation robustness

• The analysis applied one perturbation realization to 324 lymph-node surfaces across six SH degrees, with each degree using its clean-surface-selected spherical mapping.Between-degree differences therefore reflect both representation resolution and its clean-selected mapping, rather than spectral truncation alone.Source: §3.5 · p. 11

3.5.1. Controlled geometric perturbation

• The controlled perturbation used a spatially correlated field and fixed amplitude of 0.2, with the same standardized random field applied across degrees for each node.Candidate perturbations were retained only if affected triangles kept positive area and no face orientation reversed.Source: §3.5.1 · p. 12

3.5.2. Geometric perturbation transmission across SH degrees

• Geometric perturbation transmission was measured as the ratio of reconstructed to input RMS displacement, where 1 indicates complete transmission and values below 1 indicate attenuation.Clean-selected spherical parameterizations were held fixed within each clean–perturbed pair so within-degree differences reflected the imposed perturbation rather than remapping.Source: §3.5.2 · p. 12, §3.5.1 · p. 12

3.5.3. Harmonic-domain perturbation analysis

• Harmonic-domain analysis used degree 34 as a common high-resolution reference to quantify retained perturbation energy and perturbation energy relative to the clean signal at each degree.The cumulative retained-energy measure describes perturbation preserved by truncation, while the degree-specific ratio is interpreted jointly with it.Source: §3.5.3 · pp. 12–13

3.5.4. Predictive robustness of curvature-derived features

• Predictive robustness was assessed on 92 curvature-derived descriptors using clean-training models applied without refitting to held-out clean and perturbed nodes.Stability measures included AUC change, mean absolute probability shift, and correlation between paired clean and perturbed probabilities.Source: §3.5.4 · p. 13

3.6.1. Multicenter label-free lymph node replication

• The multicenter lymph-node replication used all 2,644 external surfaces for degree-effect analysis, but no node-level pathological metastasis labels were available.Replication was therefore geometric rather than a test of clinical prediction performance.Source: §3.6.1 · p. 13
• The label-free comparison prespecified convexity, mean curvature, and central asymmetry at common SH degrees 5, 10, and 34, including internally specified family preferences of 5, 10, and 34, respectively.The external cohort tested transfer of prespecified degree effects rather than selecting new optimal degrees.Source: §3.6.1 · p. 13

3.6.2. Cross-organ validation in LIDC-IDRI lung nodules

• The cross-organ LIDC-IDRI experiment retained 59 labeled lung nodules from 56 patients and evaluated three feature families at six prespecified SH degrees.Nodules with unknown diagnosis or metastatic lesions were excluded; patient-grouped repeated cross-validation and AUC were used to assess performance.Source: §3.6.2 · pp. 14–15

4.1. Geometric processing establishes a valid computational surface domain

• Strict topology validity increased from 78.18% (1383 /1769) before processing to 100.00% (1769/1769) afterward, with all reported component criteria reaching 100%.Processing also improved triangle quality and reduced adjacent-face normal jumps in the computational-quality subset.Source: §4.1 · p. 15

4.2.2. Degree-wise spectral discrimination is strongest at intermediate harmonic degrees

• The strongest degree-wise spectral discrimination occurred around degrees 8–12, with degree 10 reaching folded AUC 0.645; discrimination declined to 0.591 at degree 15 and 0.536 at degree 20.Conventional Py14 descriptor discrimination remained largely preserved across SH representations, with median ΔAUC near zero from degrees 5 to 20 and a modest reduction at degree 34.Source: §4.2.2 · p. 15, §4.2.1 · p. 15, Figure 5 · p. 16

4.2.3. Geometric fidelity increases monotonically, whereas predictive utility does not

• Composite geometric fidelity increased monotonically with SH degree, while fixed-resolution model AUC rose to intermediate degrees and declined at degree 34.Across feature–degree combinations, recovery of conventional shape measurements had only a weak association with discrimination (Spearman ρ≈0.25).Source: §4.2.3 · p. 15, Figure 7 · p. 16

4.2.5. High-resolution SH achieves compact geometric representation with high fidelity

• Preferred SH degree varied across the 20 geometric feature families, and high-resolution degree 34 provided a compact representation with high fidelity despite lower predictive utility than intermediate resolutions.At degree 34, 1,225 coefficient vectors represented a median of 38,492 mesh vertices, a 31.42-fold reduction in representation elements.Source: §4.2.4 · p. 15, §4.2.5 · pp. 16–17, Figure 8 · p. 17

4.2.4. Feature-dependent SH resolution preferences

• The family-specific mixed-resolution model achieved the highest overall predictive performance among evaluated representations; after multiple-comparison correction, it was comparable to fixed-resolution models at degrees 8, 10, 15, 20, and 34, but significantly exceeded the low-resolution degree-5 model.Source: §4.2.4 · pp. 15–16

4.3. Resolution-dependent perturbation robustness

• Higher SH degrees transmitted more perturbation, with median transmission ratio increasing from T5 = 0.794 to T34 = 0.992, and produced greater AUC degradation and less stable curvature-based predictions.Perturbation-induced AUC degradation was significantly greater than at degree 5 for degrees 10, 15, 20, and 34 after FDR correction.Source: §4.3 · p. 17, Figure 9 · p. 18

4.4. Metastasis-associated geometric phenotypes

• Representative curvedness, curvature, radial morphology, and convexity descriptors separated metastatic from non-metastatic nodes, with univariate AUCs ranging from 0.824 to 0.898.These measurements capture both overall contour departures and regional surface bending, complementing global shape features.Source: §4.4 · p. 17

4.5.1. Multicenter label-free lymph node replication

• In the independent multicenter cohort, feature-level degree effects had pooled Spearman correlation ρ= 0.964 and directional agreement for 121 of 126 matched effects; all six prespecified family–contrast composites were positive and significant after Holm correction.The 2,644 external lymph nodes lacked node-level pathology labels, so this replication addressed geometric effects rather than prediction performance.Source: §4.5.1 · pp. 17–19

5.1. Reliable surface representation enables quantitative geometric analysis

• Preprocessing raised strict topology validity to 100% and improved mesh regularity and adjacent-face normal consistency, establishing a domain suited to spherical and differential-geometric analysis.The improvements were reproduced in an independent multicenter cohort; genus-0 classification alone does not guarantee the connectedness, manifoldness, orientation, or local regularity needed by geometric operators.Source: §5.1 · p. 19

5.2. SH resolution defines a task-dependent geometric scale

• Different geometric feature families favor different SH degrees, and intermediate resolutions can better balance disease-associated morphology against fine-scale uncertainty than uniformly maximizing resolution.The mixed-resolution model was comparable to the best intermediate fixed-degree models, while LIDC-IDRI results also showed that predictive performance depended on SH degree.Source: §5.2 · p. 19

5.3. Resolution-dependent perturbation sensitivity and the fidelity–robustness trade-off

• Higher SH degrees preserve more imposed perturbation and improve geometric fidelity, but curvature-based prediction stability decreases as resolution increases.Lower degrees attenuate more of the perturbation; the appropriate degree therefore depends on both desired geometric detail and downstream stability requirements.Source: §5.3 · pp. 19–20

5.4. Limitations and future directions

• CT-derived surface uncertainty cannot be fully separated from anatomical variation, and preprocessing cannot recover anatomical information absent from the original imaging data.The issue is especially relevant to curvature descriptors; lower-degree representations may remove informative geometry, while multi-resolution representations increase computational cost.Source: §5.4 · p. 20

6. Conclusion

• The processing pipeline establishes a connected, closed, consistently oriented genus-0 domain while improving local mesh quality and limiting unnecessary changes to the underlying geometry.This provides a reproducible geometric basis for differential-geometric and spectral descriptors on clinically derived three-dimensional surfaces.Source: §6 · p. 20
• SH resolution is a task-dependent geometric scale: fidelity improves with degree, but metastasis discrimination is strongest at intermediate scales and feature families favor different resolutions.The family-specific mixed-resolution representation captured discriminative information beyond conventional global shape descriptors.Source: §6 · p. 20
• Higher-degree SH representations preserve more imposed perturbation, whereas spectral truncation attenuates variation and improves the stability of curvature-based predictions.Geometric fidelity, discriminative utility, and perturbation robustness are complementary but non-equivalent objectives.Source: §6 · p. 21

CRediT authorship contribution statement

• Zixi Yi is credited with conceptualization, data curation, formal analysis, investigation, methodology, software, validation, visualization, and the original draft.Yi is also credited with funding acquisition, project administration, and supervision.Source: CRediT authorship contribution statement · p. 21
• Limeng Qu is credited with data curation, methodology, and review and editing of the manuscript.Source: CRediT authorship contribution statement · p. 21
• Gary P. T. Choi is credited with conceptualization, funding acquisition, investigation, methodology, project administration, supervision, and review and editing.Source: CRediT authorship contribution statement · p. 21