A Strong Linear Baseline for Whole-Heart Cardiac Shape Completion on CT, with an Open Eleven-Structure Statistical Shape Model

Abstract: Public cardiac cohorts annotate different subsets of the heart, so shapes from separate sources cannot be pooled without shared correspondence. Among released cardiac shape resources, none we identified carries the atrial appendage, pulmonary veins, and caval stumps as separate blocks in one mesh. Completion benchmarks also compare deep models against a least-squares projection onto shape modes, not the conditional estimator the same fitted model implies. We release an eleven- structure cardiac computed-tomography (CT) statistical shape model, built from 383 automatically labelled cases in 11 571-vertex correspondence, and compare completion estimators under one frozen internal split and endpoint. On a 76-case internal list held out from fitting, a closed-form conditional-Gaussian estimator reconstructed the missing non-chamber structures at 3.717 mm mean per-vertex error, averaged equally over one, three, five, and nine observed structures. A five-refit mask-conditioned graph variational autoencoder reached 5.248 mm and nearest-neighbour retrieval 8.931 mm. The paired difference was 1.531 mm (95% confidence interval 1.384 to 1.711), and the ordering held in a raw-coordinate sensitivity arm. Expert manual labels exist for 58 external CT cases, but our registered reference is close enough to score only five structures. There the closed-form estimator again had lower average surface distance, 95th-percentile Hausdorff distance, and Chamfer error for both completed atria. On a second public benchmark of 20 cases the reference was close enough for three of four completed structures, and the same ordering held there. Four structures have no expert reference. The released model and its completion operator support cohort-unification research on aligned CT, not clinical use.

• The paper introduces an open eleven-structure cardiac statistical shape model with 11,571 corresponding vertices, created from 383 quality-controlled CT cases spanning chambers, vessels, and the left atrial appendage.
• The paper shows that regularised conditional-Gaussian completion outperforms a matched graph β-VAE, achieving 3.717 mm versus 5.248 mm mean per-vertex error internally and retaining the advantage on eligible external structures.
• The paper provides a mask-agnostic closed-form completion operator and an openly scoped benchmark, while showing that uncertainty estimates remain poorly calibrated and outputs should not yet support clinical or simulation-ready use.

Motivation and problem statement

Public cardiac imaging cohorts annotate different subsets of the heart, so meshes from separate sources cannot be pooled without a shared correspondence across a common anatomy. Existing released cardiac shape resources stop short of what this task requires: none carries the left atrial appendage, pulmonary veins, and caval stumps as separate, fully meshed surface blocks within one shared topology, and prior completion benchmarks compare deep models against a least-squares projection onto PCA modes rather than the conditional posterior estimator that the same fitted shape model implies. The paper addresses this gap with three contributions: an eleven-structure cardiac CT statistical shape model (SSM) built from 383 automatically labelled cases in 11,571-vertex correspondence; a controlled completion benchmark comparing estimator families under one frozen internal split, matched observation panels, and a single prespecified geometric endpoint; and a mask-agnostic closed-form completion operator that accepts any coordinate subset for which the regularised conditional solve is defined.

The eleven-structure correspondence resource

The SSM is built from 631 TotalSegmentator CT volumes with silver (automatic) labels, reduced to 383 cases after segmentation quality control, registration QC, and exclusion of field-of-view truncations. Eleven structures are represented — five chamber blocks (LV, MYO, RV, LA, RA) and six non-chamber blocks (AO, PA, LAA, PV, SVC, IVC) — with vertex counts proportional to mean surface area, totalling 11,571 vertices. Ten of the eleven are blood-pool or luminal surfaces; the myocardium is the only wall object, so the model carries no right-ventricular or atrial wall and does not represent total myocardial mass.

Correspondence is obtained by warping a single medoid-derived template into each patient via ANTs symmetric diffeomorphic registration, with a coordinate-map propagation of template vertices. A centre-of-mass prealignment handles scanner-origin offsets of up to 870 mm, and no isotropic-scale or Procrustes normalisation is applied, so displacements encode size together with shape. A sparse "R-local" repair fixes detached non-chamber components, touching only 868 of 4,431,693 case-vertex opportunities and no chamber vertex by construction. Warp-overlap Dice on the external CARE2026 cohort reaches medians above 0.96 for the chambers and myocardium but falls to 0.53 for the IVC, reflecting frequent external FOV truncation of the great vessels. Two caveats are disclosed plainly: ECG gating status is absent everywhere, so cardiac phase enters as unstructured population variation, and the study mix is predominantly non-cardiac (336 of 383 cases).

Completion estimators and evaluation design

The benchmark compares, under one frozen 307/76 development/evaluation partition and one endpoint (mean per-vertex Euclidean distance, MPVED, over missing non-chamber vertices, equally averaged over $k\in\{1,3,5,9\}$ observed structures): a regularised conditional-Gaussian posterior over PCA coefficients (Cond-G), a mask-conditioned graph $\beta$-VAE extending CoMA with Chebyshev convolutions, a mixture of probabilistic PCAs under prespecified validity rules, a structure-local Cond-G ablation, and nearest-neighbour and population-mean floors. The conditional mean given observed coordinates $\mathbf{x}_o$ is computed in coefficient space with a relative diagonal ridge on the posterior precision; observed coordinates are copied to the output.

Two features of the design deserve emphasis. First, the deep model is not compared against a weak projection baseline but against the full conditional posterior, which is the estimator prior benchmarks omitted. Second, two additional non-linear arms — SpiralNet++ substituted at parameter parity, and latent optimisation (Litany et al.) run on the locked $\beta$-VAE decoders — are reported descriptively to test whether the deep arm is representative of its class rather than a strawman.

Internal results: the linear estimator wins

On the frozen 76-case evaluation list in the primary R-local representation, Cond-G attains a four-$k$ MPVED of 3.717 mm against 5.248 mm for the $\beta$-VAE and 8.931 mm for nearest-neighbour retrieval. The paired primary difference is 1.531 mm (95% CI 1.384–1.711; $p<0.001$), and Cond-G has the lowest value at every individual $k$ (5.304/3.732/3.007/2.825 mm for $k=1/3/5/9$). The ordering holds in the raw-coordinate sensitivity arm (3.898 vs 5.319 mm), across all five refit seeds, and in all 24 structure-$k$ combinations for both representations. All seven supportive contrasts survive Holm adjustment. Single-component PPCA trails Cond-G by only 0.047 mm, indicating that the advantage is shared by regularised Gaussian conditioning generally rather than unique to one implementation.

The additional non-linear arms strengthen rather than undermine the finding: latent optimisation beats the feed-forward $\beta$0-VAE on both internal and MM-WHS endpoints (by 0.204 mm and 0.368 mm respectively) yet still loses to Cond-G by 1.327 mm and 1.426 mm, and SpiralNet++ underperforms both. Because the graph architecture predates the partition — its earlier sweep may have seen cases now in the evaluation list — the paired bootstrap is disclosed as descriptive conditional on this design, and the authors correctly decline to call it confirmatory. A development-only sensitivity sweep over ten nearby configurations (5.983–6.433 mm) and an EdgeConv substitution (5.774 mm under operator-specific retuning) show the shared training protocol is not neutral across architectures, a limitation of the matched-protocol design.

External validation and per-structure evidence

On CARE2026 ($\beta$1 expert-labelled CT cases), a prespecified reference-closeness rule admits five of seven overlapping structures; AO fails because expert annotations cover a shorter aortic extent than the whole-aorta reference, and PA misses the strict threshold by 0.026 mm. For the eligible structures, Cond-G has lower pooled ASSD, HD95$\beta$2, and Chamfer RMS than the $\beta$3-VAE for both atria under biventricular input (e.g. LA 3.275/7.846/4.138 mm vs 3.934/8.990/4.860 mm) and in the leave-one-out setting, with one subset–metric reversal (subset-A RV HD95). On MM-WHS ($\beta$4), the rule admits three of four completed targets (LA, RA, PA — the latter only under R-local), and Cond-G is lower on all three metrics for all three; AO again fails the closeness rule. LAA, PV, SVC, and IVC carry no external expert reference at all, so their evidence is silver-target only. The authors are explicit that these absences and qualifications are results, not values to be averaged away.

Downstream measurement, robustness, and uncertainty

Morphometric retention from the biventricular panel shows that five of seven internal structures beat a development-only ridge regression, the mean shape, and the median volume; pulmonary-vein and caval volumes are not separated from the regression internally, and external caval volume is worse than a constant ($\beta$5 mL), with calibration slopes of 0.64 (PV) and 0.73 (SVC) indicating reversion toward the mean for the smallest blocks. These are measurement-agreement results, not clinical validation.

A partial-observation experiment truncating an observed chamber axially finds all six primary cells non-inferior at the 1.531 mm margin; LV removal costs $\beta$6 mm once the myocardium is observed, implying that under the tested masks the left-ventricular blood pool contributes nothing measurable given its wall.

Uncertainty audits reveal opposite failure modes: Cond-G's ellipsoids are overconservative (nominal 95% coverage actually 0.991–0.999), while the $\beta$7-VAE's 160-draw ensemble is overconfident (coverage 0.026–0.203). A post-hoc stratification by a frozen Mahalanobis out-of-support threshold (64 typical, 12 atypical cases) shows Cond-G's conservatism is consumed on atypical anatomy while the $\beta$8-VAE remains most overconfident there; RMSE roughly doubles across strata for both models. Neither construction is calibrated, and no conformal layer is evaluated — the authors note that no development-free calibration set of sufficient size exists in their data.

Validity audits are candid about geometric defects: self-intersections and cross-structure triangle contacts occur in completed meshes (and in targets, where structures share anatomical interfaces), so zero predicted degeneracy does not imply anatomical validity, and fixed axial truncation curves are non-monotone, establishing neither truncation robustness nor registration performance on partial surfaces.

Limitations

The paper concedes five substantive limitations. The internal split is drawn from the same source pool as development and the graph architecture predates the partition, so inference is descriptive rather than confirmatory. The internal endpoint scores against a silver, SyN-registered target, not anatomical truth, and single-template propagation may introduce template bias. The cohort is CT-only, ungated, and of unknown health and ancestry composition, so the released model is not a normative reference and cross-phase conditioning is untested. The model comparison is finite and contains no published external system, since none emits the eleven blocks scored; invalid multi-component mixture cells are missing evidence, not evidence against multimodality. Finally, outputs are point completions with miscalibrated uncertainty in opposite directions, and no clinical, simulation-ready, or collision-free claim is supported.

Conclusion

The paper delivers a narrow, well-controlled finding: on a frozen internal split, matched external cohorts, and a prespecified endpoint, a closed-form regularised conditional-Gaussian estimator outperforms a matched graph $\beta$9-VAE and published non-linear baselines for whole-heart cardiac shape completion, by 1.531 mm internally with the ordering retained on eligible external structures. Its durable contribution is arguably the open eleven-structure correspondence resource and the benchmark design itself, which corrects the systematic use of projection baselines in prior completion evaluations. The released model and completion operator support cohort-unification research on aligned CT; the authors explicitly scope all results away from clinical use, and the open questions they leave — whether a better nonlinear model, registration strategy, or larger expert-labelled cohort would reverse the ordering, and whether calibrated per-patient uncertainty is achievable at all in this setting — remain unresolved.