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

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

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

arXiv:2608.19932v1 [cs.AI] 20 Aug 2026

Matej Gazda iD , Jakub Gazda iD , Juraj Gazda iD , and Peter Drotár iD

how accurately the missing structures can be reconstructed from the ones that are present, and which observed structures carry the most information for doing so. Analyses that need the heart as a whole, cardiac simulation [1]–[4], morphometric shape analysis [5], and population-scale phenotyping [6], all work on the cardiac structures as 3D meshes in consistent vertex correspondence, which statistical shape models (SSMs) supply together with population-level variability [7], [8]. Throughout, whole-heart means joint coverage of the main cardiac structures rather than of every cardiac tissue: ten of our eleven structures are blood-pool or luminal surfaces and the eleventh is the left-ventricular wall. Prior work has established ventricular and four-chamber correspondence at scale [9]–[11], and has demonstrated fourchamber reconstruction and masked completion; recent additions include sparse-view reconstruction from cine MRI [12], simulation-oriented template growth [13], and an age-specific paediatric atlas [14]. Section II reviews this literature, and Table I compares the closest released CT geometry resources. What none of them supplies is the anatomical representation, the baseline design, and the variable-label conditioning that the completion task above requires. Three gaps follow, and they are what this paper addresses. First, the closest released resources use opening tags or cut rings where this task needs the appendage body, the pulmonary-vein bodies, and the caval stumps as full surface Index Terms—Computed tomography, conditional-Gaussian shape completion, graph neural network, statistical shape model, blocks in one shared topology. Second, completion benchmarks compare deep models against a least-squares projection variational autoencoder, whole-heart cardiac anatomy. onto shape modes, an estimator weaker than the conditional posterior available from the same fitted model, so the reported I. I NTRODUCTION margins say more about the choice of baseline than about the Public cardiac datasets rarely annotate the same parts of the deep architecture. Third, existing reconstruction and compleheart. A study that pools two cohorts can therefore be missing tion models fix their input and output anatomy to a prescribed a chamber or a vessel that the other cohort provides, and re- acquisition or four-chamber topology, which is the assumption annotating the images is seldom an option. This paper asks that fails when cohorts are annotated differently from one another. Matej Gazda is with the Department of Mathematics and TheoretiThe fragmentation is concrete. Short-axis cine magnetic cal Informatics, Faculty of Electrical Engineering and Informatics, Tech- resonance imaging (MRI) benchmarks annotate the left vennical University of Košice, Letná 1/9, 042 00 Košice, Slovakia (e-mail: tricle, myocardium, and right ventricle. Other public CT matej.gazda@tuke.sk). Jakub Gazda is with the 2nd Department of Internal Medicine, Faculty of and cardiac magnetic resonance (CMR) benchmarks cover Medicine, Pavol Jozef Šafárik University and L. Pasteur University Hospital, at most seven structures (four chambers, myocardium, aorta, Trieda SNP 1, 040 11 Košice, Slovakia. Juraj Gazda and Peter Drotár are with the Department of Computers and pulmonary artery) and omit the left atrial appendage, pulInformatics, Faculty of Electrical Engineering and Informatics, Technical monary veins, and venae cavae. TotalSegmentator v2 [15], [16] University of Košice, Letná 1/9, 042 00 Košice, Slovakia. produces eleven-structure CT segmentations, but most existing This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may cohorts have partial labels. A model that completes the missing no longer be accessible. structures from whatever subset is available would unify these

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 elevenstructure cardiac computed-tomography (CT) statistical shape model, built from 383 automatically labelled cases in 11 571vertex 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 maskconditioned 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.

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

datasets under one eleven-structure representation. Cardiac mesh reconstruction methods assume a fixed input and output structure set [4], [17], and the closest completion work masks chambers during training but emits four-chamber anatomy with no corresponding appendage, pulmonary-vein, or caval blocks [18]. The task addressed here is different: one explicit eleven-block topology, conditioned on whichever named blocks a cohort provides. This paper presents three contributions:

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TABLE I C LOSEST MULTISTRUCTURE CT GEOMETRY RESOURCES . “P UBLIC PACKAGE ” MEANS A DIRECT PUBLIC DOWNLOAD OF A REUSABLE MODEL OR MESH COHORT, ASSESSED FROM THE CITED PUBLICATIONS AND THEIR LINKED REPOSITORIES AT THE TIME OF WRITING ; BOUNDARY TAGS OR CUT RINGS ARE NOT COUNTED AS SEPARATE ANATOMICAL SURFACE BODIES . N ONE OF THE PRIOR RESOURCES SUPPORTS MISSING - LABEL CONDITIONING ; THIS WORK CONDITIONS ON ANY OBSERVED STRUCTURE - BLOCK SUBSET.

Resource

Anatomy outside the four chamber Public package walls

Hoogendoorn Detailed multiregion whole-heart at- Atlas and SSM reported; no 1) A controlled completion benchmark. Prior cardiac et al. [20] las, including great vessels reusable model package locompletion work compares against a least-squares procated Rodero AO and PA walls; PV and caval SSM parameters and 1 000 jection onto PCA modes. We instead compare standard et al. [2] openings tagged; LAA body omit- synthetic finite-element estimator families under one frozen development/test ted meshes Strocchi AO and PA walls; LAA, PV, SVC 24 patient-specific finitesplit, the same named-block panels, and one prespecified et al. [3] and IVC at cut rings element meshes geometric endpoint: a regularised conditional-Gaussian This work AO, PA, LAA, PV, SVC and IVC Template, per-case displaceas separate surface blocks in one ment fields and reconstrucposterior, a mask-conditioned graph β-VAE, a mixshared topology tion code ture of probabilistic PCAs under prespecified validity rules, a structure-local conditional-Gaussian ablation, and nearest-neighbour and population-mean floors. Two further non-linear arms outside the prespecified family, a cludes. published mesh convolution operator substituted into the II. R ELATED W ORK same architecture and a published completion method run on our own fitted decoders, are reported descriptively A. Cardiac Statistical Shape Models as additional non-linear baselines. The individual estiBai et al. [9] built a 1 000-case biventricular SSM from mators are established; the contribution is their matched high-resolution cardiac MRI. Qi et al. [14] recently added an evaluation on eleven-block cardiac correspondence. age-specific biventricular atlas from 50 cardiac MRI studies 2) Mask-agnostic completion for cohort unification. One in children aged 10–18 years. Ugurlu et al. [10] scaled to closed-form operator accepts any coordinate subset for ≈55 000 biventricular meshes with open tooling, and Ma which the regularised conditional solve is defined, withet al. [11] extended dynamic modelling to four chambers on out mask-specific retraining. We evaluate named-block ≈96 000 UK Biobank participants. panels at k ∈ {1, 3, 5, 9} and axial partial-vertex masks. The closest public CT resources contain more anatomy A fixed biventricular CARE panel supplies a worked than the shorthand “four-chamber model” suggests; Table I cohort-unification example. The contribution is reduced documents them resource by resource. Rodero et al. [2] and geometric error under the tested aligned-CT masks, not Strocchi et al. [3] release aortic and pulmonary-artery walls a validated clinical measurement. with their four-chamber models, but the appendage, pulmonary 3) An open eleven-structure correspondence resource. veins, and venae cavae enter as tags or cut rings rather than We release an SSM built from 383 CT cases (11 571 as separate surface bodies, and Hoogendoorn et al. [20] built corresponding vertices) with its template, displacements, a detailed multiregion spatio-temporal CT atlas from 138 and code. The contribution is the open correspondence sequences without a reusable public package. The distinction across these eleven structures together, not a new segrelevant here is a task-enabling choice of representation, not mentation or registration method. Two external CT coan anatomical count or a qualitative first: every subject uses horts, CARE2026 and MM-WHS 2017 [19], provide expert labels for seven overlapping structures, of which the same face graph and vertex indices within each block, with the reference-closeness rule admits five and three respec- eleven named blocks concatenated so their joint covariance can tively for native comparison; the appendage, pulmonary be fitted, whereas these models emit no LAA, PV, SVC, or veins, and caval stumps carry only automatic (silver- IVC surface body. None of them can therefore be scored on the eleven-block panel without refitting its own correspondence standard) labels. pipeline, so our positioning against the shape-modelling literaThe remainder of this paper is organised as follows. Sec- ture is Table I on representation, together with the probabilistiction II reviews related literature. Section III describes the PCA (PPCA) and structure-local conditional-Gaussian family data, mesh extraction, and ANTs SyN registration that produce fitted here on identical data. the eleven-structure SSM. Section IV defines the conditionalGaussian and graph β-variational-autoencoder (β-VAE) estimators and their prespecified comparators. Section V reports B. Whole-Heart Mesh Reconstruction Image-domain restoration and classification are separate the evaluation protocol, frozen-split internal completion, external expert-surface and structure-wise evidence, and the ge- stages from geometry extraction: DenoMamba [21] denoises ometry, support, truncation, and uncertainty audits. Section VI low-dose CT with fused spatial and channel state-space moddiscusses interpretation and limitations, and Section VII con- ules before any geometry is extracted, and shrunken-feature

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

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COVID-19 classification from X-ray and CT [22] labels whole a correction to PCA coefficients with a multilayer perceptron images without vertex correspondence or geometry prediction. (AutoSkull). Neither targets cardiac anatomy or conditions on Template-deformation approaches require complete volu- an arbitrary observed-structure subset. metric input [23]. Kong and Shadden [4] predict biharmonic control handles from CT segmentations to deform a temIII. P IPELINE : F ROM CT TO SSM plate into seven structures; their earlier MeshDeformNet [24] predicts per-vertex template deformations with graph con- A. Data, Mesh Extraction, and Registration This study draws on three public CT cohorts. The first, volutions. Pak et al. [25] predict whole-heart meshes from internal in role only (every fit and selection uses it), is 631 CT. MeshGrow [13] combines a seven-region cardiac temvolumes from the TotalSegmentator dataset, labelled for the plate deformation with stepwise aortic tracking to produce eleven structures of Table III by TotalSegmentator v2 [15], simulation-oriented cardiac and vascular meshes, including [16]; it carries silver (automatically generated) labels only an explicit aortic-valve interface. Deng et al. [26] refine and supplies the SSM development and frozen-split internal biventricular meshes from CMR via graph subdivision. On evaluation cases. The first external cohort is the CARE2026 MRI, Beetz et al. [27] reconstruct biventricular point clouds challenge set [38], sixty volumes across three acquisition from sparse cine contours without mesh topology, while Liu subsets (A, B, G), held out entirely from model fitting; seven et al. [12] map sparse multi-view cine MRI to temporally structures (the four chambers, MYO, AO, and PA) additionally coherent four-chamber meshes through differentiable contour rendering. Muffoletto et al. [28] use neural implicit coordinates carry expert manual labels, while LAA, PV, SVC, and IVC for biventricular reconstruction with iterative optimisation at are silver-only. Two volumes fail an affine check (Table II), inference. These methods solve image-to-geometry reconstruc- leaving n=58 split 20/18/20 across A, B and G. The second external cohort is all 20 CT training cases of tion for a prescribed acquisition, rather than conditioning a correspondence model on whichever anatomical structures a MM-WHS 2017 [19], with no exclusions. It provides expert labels for the same seven overlapping structures but none for cohort happens to contain. LAA, PV, SVC, or IVC, and it is held out from every fit and tuning decision, analysed separately as secondary descripC. Shape Completion and Generation tive validation (Section V-A). All three cohorts are analysed PCN [29] and GRNet [30] complete spatial geometry at 1 mm isotropic spacing. Before registration, every cohort from partial point clouds without anatomical part labels. Car- passes an anatomical orientation check on structure ordering diacFlow [17] uses flow matching for four-chamber 3D+t and frame handedness, so that a label map inconsistent with completion and generation. Cardiac Mesh Flow [31] extends its affine is rejected rather than registered. flow matching to explicit four-chamber meshes in vertex correTable II summarises the three cohorts: exclusion cascades, spondence and volume-conditioned generation. Qiao et al. [32] acquisition composition, labels, and split. The internal study generate 3D+t biventricular meshes from demographics. These mix is overwhelmingly non-cardiac (336 of 383 cases are generative models establish strong non-linear shape priors. thorax–abdomen–pelvis, trauma, or staging examinations), and Their conditioning variables and output topology are fixed electrocardiographic (ECG) gating status and cardiac phase are to the structures in their four-chamber task, whereas our absent everywhere; because non-gated acquisitions sample the evaluation varies which named anatomical blocks are supplied cardiac cycle arbitrarily, phase enters as unstructured variation and scores the blocks withheld from that panel. and is treated as a limitation below. The five fixed development Anatomical completion is addressed directly by Vec- folds determine every hyperparameter varied in the matched Heart [18], which masks chambers during training and re- selection grids and the deep-model refit duration. The graph constructs four-chamber anatomy from a hybrid vector-set architecture was fixed before the partition, in an earlier sweep latent decoded to an implicit field. It is the closest task- whose training pool included cases now in the evaluation list level precedent, but its four-chamber implicit output cannot (development-only sensitivity below), and the frozen 76-case represent the appendage, pulmonary veins, or venae cavae list, though withheld from every fit reported here, is not an as corresponding mesh blocks. No compatible eleven-block independent cohort or strict confirmatory holdout (Section VI). implementation or pretrained topology is publicly available The end-to-end pipeline is shown in Fig. 1. The TotalSegfor a matched retraining baseline; our non-linear comparators mentator cardiac labels are predominantly luminal: each chaminstead use the same eleven-block mesh, split, and observation ber (LV, RV, LA, RA) is the blood-pool cavity, and the great panels as the linear estimator. vessels (AO, PA), pulmonary veins (PV), cavae (SVC, IVC), Graph convolutional autoencoders [33] model face and and left atrial appendage (LAA) are the vessel or appendage body meshes, and β-VAEs [34] learn disentangled latent lumen. The single exception is the myocardium (MYO), the representations. Biffi et al. [35] apply cardiac VAEs to disease left-ventricular wall, so the meshes below are blood-pool or classification on 3D cardiac segmentations without completion luminal surfaces for ten structures and one wall object. From conditioning or generation. the 631 training volumes, cases were rejected when any core In cranial modelling, two studies combine a linear shape- structure (LV, MYO, RV, LA, RA, AO) had non-physiological model prior with a learned non-linear correction: Pimentel volume, when the heart bounding box spanned less than et al. [36] refine an SSM fit to a defective skull with a gen- 60 mm craniocaudally, or when structure-centroid separations erative adversarial network, and Milojevic et al. [37] predict exceeded 100 mm, leaving 425 cases.

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

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1 Data preparation ANTs SyN registration

Segmentation QC

631 chest CT TotalSegmentator v2

volume, extent, centroid 631 → 425

joint 11-label warp 425 → 399

FOV QC

R-local repair

locked anatomical rules 399 → 383

same-structure neighbours chambers unchanged

383 corresponding meshes 11 blocks, 11 571 vertices frozen dev/test: 307/76

2 Frozen completion benchmark

Conditional-Gaussian SSM

study method

bu = µu + Cuo (Coo + λI)−1 (xo − µo ) x one fit, any observed vertex mask

Frozen evaluation only

Matched comparators graph β-VAE

mixture-PPCA

local Cond-G

identical dev split, masks, endpoint and held-out cases

internal missing non-chamber MPVED at k ∈ {1, 3, 5, 9}, n = 76

separate expert-surface validation: CARE2026 n = 58 + MM-WHS n = 20 (no fitting or tuning)

Fig. 1. End-to-end pipeline and frozen evaluation benchmark. After segmentation QC, joint-label registration, registration/FOV QC, and same-structure R-local repair, the 383 corresponding meshes are divided into a development set used for model selection and a 76-case evaluation set withheld from every fit reported here (the graph architecture predates this partition; Section V). The orange block carries the conditional-mean equation; all comparators use the same split, panels, endpoint, and evaluation cases. CARE2026 and MM-WHS provide separate expert-surface validation without fitting or tuning.

TABLE II T HE THREE COHORTS AND THE EXCLUSIONS THAT PRODUCE THEM . T HE RELEASED INTERNAL METADATA SUPPLIES AGE , SEX , MANUFACTURER , SCANNER MODEL , TUBE VOLTAGE , PATHOLOGY CLASS , AND STUDY TYPE ; “ N / A” ATTRIBUTES ARE ABSENT FROM THE RELEASED METADATA AND ARE NOT ESTIMATED , AND “ NONE ” MEANS NO FITTING OR TUNING . T HE 76- CASE EVALUATION LIST IS FROZEN ( SPLIT FILE RELEASED WITH THE ARTIFACTS ). B OTH CARE2026 EXCLUSIONS FALL IN SUBSET B; MM-WHS HAS NO EXCLUSIONS . Internal (TotalSegmentator)

CARE2026

Role SSM fit; internal eval. indep. validation Source cases 631 60 (A/B/G) Segmentation QC −206 (vol., extent, centr.) → 425 – Affine check – −2 (non-orthon.) Registration QC −26 (>25 mm) → 399 – FOV truncation −16 – Retained 383 58 (20/18/20) † Split/tuning 307 dev. (5 folds) / 76 withheld none Labels silver, all 11 silver 11, expert 7 Scanner models 11 distinct n/a Median age 64 years n/a Study mix 336 non-cardiac, 47 cardiac n/a Contrast phase n/a n/a ECG gating, phase n/a, likely mixed n/a Spacing analysed 1 mm isotropic 1 mm isotropic † The graph architecture predates the partition and may have seen cases now in the 76-case evaluation list.

MM-WHS CT indep. secondary 20 – – – – 20 none expert 7 n/a n/a n/a n/a n/a per case → 1 mm

Per-structure surface meshes were extracted via marching cubes [39] on Gaussian-smoothed label volumes, Laplacianfiltered [40], and decimated by quadric edge-collapse to the vertex counts in Table III (proportional to each structure’s mean surface area). The template mesh is extracted from a single selected medoid case (Fig. 2); its 11 571 vertices and fixed connectivity define the correspondence used throughout. Percase correspondence is obtained purely by warping this template into each patient’s space (below), so only the template’s own extraction parameters enter the SSM correspondence. The template is the case minimising the sum of Euclidean distances to all others in a 121-dimensional shape descriptor (per-structure centroid, volume, surface area, bounding-box

TABLE III E LEVEN CARDIAC STRUCTURES AND TEMPLATE VERTEX COUNTS . MYO IS THE LEFT- VENTRICULAR WALL ; THE OTHER TEN ARE BLOOD - POOL OR LUMINAL SURFACES . T HE MODEL THEREFORE CARRIES NO RIGHT- VENTRICULAR OR ATRIAL WALL AND DOES NOT REPRESENT TOTAL MYOCARDIAL MASS . ID

Structure

1 2 3 4 5 6 7 8 9 10 11

Left ventricle Myocardium Right ventricle Left atrium Right atrium Aorta Pulmonary artery Left atrial appendage Pulmonary veins Superior vena cava Inferior vena cava Total

Abbrev.

Vertices

LV MYO RV LA RA AO PA LAA PV SVC IVC

1 502 2 502 2 002 1 002 1 002 1 202 802 402 602 402 151 11 571

dimensions, principal component analysis (PCA) principalaxis lengths), each feature z-scored across the cohort. Vesselextent features enter with unit weight; because truncation is common in the cohort, they can pull the medoid toward a truncation-typical case and bake a short-vessel template into the correspondence, one route to the flat distal vessel caps. Heart centroids vary by up to 870 mm across scans due to scanner-origin differences. Centre-of-mass (COM) prealignment shifts each case to match the template centroid, reducing median displacement from 275 mm to 17 mm. No isotropicscale or generalised-Procrustes normalisation is applied, so the per-vertex displacement encodes size together with shape rather than shape alone. Symmetric diffeomorphic registration (ANTs SyN [41]) computes a deformation between each patient label image and

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

unchanged if Left aLeft candidate repair increases its longest Right Right edge. The five chamber blocks are copied bit-for-bit: detection thresholds and replacement values use neither chamber displacement nor any other structure. Two repair settings enter the evaluation: the primary R-local rule and the raw arm, which applies no repair and measures sensitivity to omitting it. R-local changes 868 vertices in 39 cases (mean 0.0196% of vertices per case; median zero; maximum 0.467%); by structure the changed counts (fraction of all case-vertex opportunities) are AO 259 (0.0563%), PA 245 (0.0798%), LAA 98 (0.0637%), PV 141 (0.0612%), SVC 104 (0.0675%), and IVC 21 (0.0363%), and chamber counts Left Left Right Right are exactly zero by construction, so no completion-input information enters the targets. LVLV MYO MYO RV RV LALA RA RA AO AO PAPA LAA LAA PV PV SVC SVC IVC IVC The IVC template is trimmed at the z-minimum of the descending aorta to remove the variable abdominal segment, reducing IVC vertices from 402 to 151 and the total mesh from 11 822 to 11 571. The trim plane is fixed once in template space and not recomputed per case, so the short-stub IVC seen on external CARE2026 is consistent with the fixed template extent rather than evidence of a prediction-specific truncation. Cases with mean chamber-vertex displacement (over LV, MYO, RV, LA, RA) exceeding 25 mm relative to the template are rejected, flagging scans the SyN cannot recover, typically LV MYO RV LA RA AO PA LAA PVpartialSVC IVC (FOV) anatomy. This extreme rotation or field-of-view leaves 399 of the 425 segmentation-QC survivors (93.9%); RA RALA AO AO RAPAPA AO LAA LAA PV PV SVCLAA IVC IVC PV PA SVC SVC IVC dropping a further 16 cases with FOV truncation gives the Fig. 2. The eleven-structure whole-heart template mesh (11 571 vertices) in clean cohort of n=383 (90.1% of the 425 survivors; 383 of anterior, posterior, left, and right views, each structure a colour-coded surface the 631 raw volumes). Shapes are represented as per-vertex in fixed vertex correspondence. This single template is warped into every displacements from the centre-of-mass-prealigned template. patient’s space to build the SSM. Per-structure warp-overlap Dice similarity coefficient (the moving segmentation pushed through the SyN transform and the template (cross-correlation similarity, radius 4, iterations compared to the template label) measures how well the 160/80/40). A single joint warp is estimated on the multi- diffeomorphic transform brings each patient into template label volume and propagated to all eleven structures. The correspondence, not anatomical accuracy against a manual cross-correlation objective is dominated by the large chambers, reference. It was recorded for the CARE2026 external cohort so great-vessel correspondence is a by-product of the joint (n=58) only. Table IV reports the per-structure distribution. warp rather than separately optimised; this motivates the The four chamber medians exceed 0.97 and MYO reaches representation audit below. Template vertices are propagated 0.962 (mean of the five per-chamber medians 0.974; per-case to patient space via a coordinate-map technique: three scalar pooled chamber mean 0.954); this is not comparable to the images encoding moving-space x, y, and z coordinates are segmentation-accuracy Dice of prior whole-heart methods on warped through the SyN transform and sampled trilinearly at CT [19], [24], which measure overlap against a reference label. template vertex locations. Aorta, SVC, and IVC are systematically lower (medians 0.82, Validity-mask propagation prevents undefined coordinate 0.73, 0.53) because external CT FOVs frequently truncate samples: a ones-image is warped with the same transform, and these vessels. This Dice certifies the forward label warp a template vertex with sampled validity below 0.5 is left at (moving→template), whereas the SSM correspondence uses its centre-of-mass-prealigned template position. The resulting the inverse (template→moving) field; the two agree closely for unsmoothed coordinate-map displacement is retained as the a well-behaved diffeomorphism, so the 0.97 figure is an upper raw sensitivity representation. proxy for the uncharacterised inverse-consistency residual. The primary R-local representation repairs only detached Because the registration target is a TotalSegmentator silver components of an individual non-chamber surface. Within label rather than a manual one, we separately checked agreeeach structure, an edge is cut when its patient-space length is ment against the seven expert-labelled CARE2026 structures both greater than 15 mm and greater than 3.5 times its template (Dice in patient space, no registration). Median overlap is length. If this disconnects the graph, the largest component is high for the chambers and pulmonary artery, and low for the the anchor and every other component is filled by inverse- aorta (0.64), where the expert label covers a shorter aortic distance weighting from up to 12 nearest anchor vertices in extent than the silver label, so the unshared segment is charged template space (all anchor vertices are used if fewer than as disagreement; the chamber distributions have a noisy low 12 remain). A no-regression guard restores the raw structure tail (per-case minima to 0.45). This check characterises one

Anterior terior

Anterior Anterior

5

Posterior Posterior

Posterior Left

Left Right

Right

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

TABLE IV P ER - STRUCTURE WARP - OVERLAP D ICE (S Y N- WARPED PATIENT SEGMENTATIONS VS TEMPLATE ), CARE2026 (n=58): A REGISTRATION - CONSISTENCY MEASURE , NOT ACCURACY AGAINST A MANUAL REFERENCE . S TRUCTURES MARKED † (LAA, PV, SVC, IVC) ARE SILVER - ONLY. C HAMBERS IS THE MEAN OF THE FIVE PER - CHAMBER MEDIANS . P 25 IS THE 25 TH PERCENTILE . Structure

Median

p25

Structure

Median

p25

LV MYO RV LA RA

0.981 0.962 0.974 0.978 0.975

0.979 0.944 0.965 0.977 0.971

AO PA LAA† PV† SVC† IVC†

0.824 0.969 0.941 0.918 0.725 0.528

0.787 0.910 0.938 0.891 0.619 0.461

Chambers: 0.974

All 11: 0.889

study contribution

Observed mesh blocks

LV

MYO

RV

LA

RA

AO

PV

whole blocks or truncated vertex subsets

Selection mask

Closed-form conditional operator

x o = Po x

bu = µu + Cuo (Coo + λI)−1 (xo − µo ) x

Po changes, model does not

Completed correspondence

Cu|o = Cuu − Cuo (Coo + λI)−1 Cou

observed coordinates copied; missing predicted C is fitted once on the development cohort; only the observed-index blocks are extracted at inference

Fig. 3. The completion operator. A selection matrix Po extracts whichever structure blocks or partial vertex sets are observed. The covariance is fitted once; inference extracts the corresponding covariance blocks and evaluates the conditional mean. Observed coordinates are copied to the output; missing coordinates are predicted. The figure writes the operator in covariance form for readability; the implementation evaluates the low-rank coefficient-space posterior of Eq. (1), whose ridge acts on the posterior precision rather than on Coo .

upstream layer and neither validates point correspondence nor replaces the reference-closeness rule. The headline external numbers are pooled over the three acquisition subsets. Subset B is the hardest domain (lowest, most variable chamber Dice) and is also where the two excluded non-orthonormal-affine cases fall, so the pooled result is mildly optimistic on the registration-consistency axis, though the silver-label faithfulness it rests on is not specific to subset B. IV. C OMPLETION M ODELS

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displacement d̂ for all eleven structures. The model extends CoMA (a convolutional mesh autoencoder) [33] with a maskconditioning channel: the encoder takes [dmasked ; m] ∈ RV ×4 , displacements at observed vertices (zero elsewhere) plus a binary mask, and stacks Chebyshev graph convolutions [42] of order K=3 with mesh pooling across four resolutions (11 571 → 2 909 → 741 → 210 vertices), widening from 16 to 128 channels, to a dz -dimensional Gaussian latent. The decoder mirrors the encoder; a differentiable Laplacian output smoothing and a Laplacian-coordinate loss term act only on the deep model. During training, each sample retains k ∼ Uniform[kmin , kmax ] complete structure blocks (uniformly one to ten); the rest are zeroed and the mask updated. This covers the named-block panels in the primary evaluation but not arbitrary within-structure surfaces, so the fixed axial partial-vertex masks are a separate out-of-training-distribution sensitivity. The loss combines masked-weighted reconstruction (weight 2 on completed vertices), a Kullback–Leibler (KL) term warmed up linearly to the candidate βmax , and Laplacian, edge-length, normal-alignment, and inter-structure repulsion regularisers with fixed weights; the released configuration carries the exact values. The matched search crosses dz ∈ {16, 32, 64, 128} with βmax ∈ {10−4 , 10−3 , 10−2 } on five development folds and training seeds 42/43, under one fixed Adam recipe with early stopping. The selected cell is refitted on all 307 development cases for the upper median of its ten best-checkpoint epoch indices plus one, with no validation-free checkpoint selection during refit. At evaluation, each refit uses the deterministic posterior mean, and coordinates from seeds 42–46 are averaged before the evaluation endpoint of Section V-A is computed. B. Conditional-Gaussian and Classical Shape Baselines We complete the shape with the closed-form Bayesian posterior over PCA coefficients, which, unlike an unregularised least-squares projection, conditions on the prior over those coefficients. Writing the normalised flattened displacement as x ∈ R3V with development mean µ, principal basis U, retained mode variances Λ, and residual variance σ 2 , and treating the coefficients as z ∼ N (0, Λ), the posterior mean given observed coordinates xo is
−1 −1 1 ⊤ (1) ẑ = σ12 U⊤ o Uo + Λ σ 2 Uo (xo − µo ),

The completion problem is the same in every model below. A case is a vector of per-vertex displacements from the template, an observed set of structures fixes some of those vertices, and the model must supply the rest; the models differ only in how the conditional distribution of the unobserved vertices is represented. The conditional-Gaussian posterior accepts any coordinate subset for which the regularised solve is defined, and the completed shape is µ+Uẑ, with observed coordinates so the same fitted model serves the named-block panels and copied to the final output. The implementation adds a relative axial partial-vertex masks evaluated here without retraining; diagonal ridge r diag(A)I to the posterior precision A in (1). this interface is broader than the tested mask distributions, and Residual variance is the mean of the discarded eigenvalues we do not infer performance outside them. The models below within the centred training rank N − 1, bounded below by −10 for numerical safety; it is a regulariser, not a calibrated are standard estimator families used to create a controlled 10 measurement-noise estimate. We call this estimator conditional completion benchmark, not claimed as new algorithms. Gaussian (Cond-G). Global Cond-G searches M ∈ {50, 100, 150, 200, 240} and A. Structure-Conditioned Graph β-VAE r ∈ {10−4 , 10−3 , 10−2 , 10−1 } within each representation arm. Let d ∈ RV ×3 denote the displacement field of a registered Within 0.01 mm of the lowest development completion error, cardiac mesh (V =11 571). Given a subset S ⊂ {1, . . . , 11} the rule chooses fewer modes and then the larger ridge. Both of observed structures, the task is to reconstruct the complete raw and R-local select M = 200 and r = 10−4 .

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

Mixture PPCA [43] is fitted in the lossless N − 1-dimensional fold-PCA coefficient space. A diagonalcovariance Gaussian mixture model (GMM) on the first 20 coefficients uses K ∈ {1, 2, 3, 5}, seeds 0–2, and PPCA ranks q ∈ {20, 40, 80, 160}. A cell is invalid if the GMM fails to converge, an effective component contains fewer than 30 cases, or, for K > 1, q > ⌊neff,min /3⌋. Invalid cells are reported and not rescued. The local Cond-G challenger selects the nearest kNN ∈ {25, 50, 100, 200, all} training shapes by RMS distance over observed coordinates, fits a local posterior with q ∈ {20, 50, 100, 200}, and rejects q > kNN − 1. Both local searches select the unrestricted all-neighbour, q = 200 limit; therefore no distinct local model is promoted to the heldout endpoint. V. R ESULTS

7

carries every paired method, panel, structure, representation, and training seed together. We report two-sided percentile 95% intervals and paired-bootstrap achieved-significance levels; the sole primary contrast is unadjusted and the seven supportive p values are Holm-adjusted. A prespecified escalation would add refit seeds 47–51 in both representation arms if the absolute primary difference fell below the five-refit seed SD. We audit uncertainty exploratively in the R-local domain on four fixed nested observation panels: LV alone; LV/MYO/RV; all five chambers; and those five blocks plus AO/PA/LAA/PV. Cond-G uses its exact conditional covariance under the selected finite-rank Gaussian model, residual variance, and ridge; for a fixed panel this covariance is homoscedastic across cases, not a learned case-adaptive confidence estimate. For the βVAE, five refits each provide 32 posterior draws; the primary matrix is the empirical second moment of all 160 decoded draws about the deterministic five-refit ensemble used for efficacy, with a separately labelled sensitivity recentred on the Monte Carlo mean. Coverage of 50%, 80%, and 95% threedimensional χ2 ellipsoids, Gaussian negative log likelihood, and the ratio of case-mean squared error to case-mean covariance trace are reported over each panel’s missing non-chamber vertices and over the common SVC/IVC intersection, with descriptive case-bootstrap intervals and no tests. A 10−6 mm2 eigenvalue floor is used only for inverses and log determinants, and its activation is reported. A separate descriptive support audit fits the selected Cond-G PCA subspace on the 307 complete development shapes and assigns each test case its full-shape Mahalanobis distance in that subspace. The tail is fixed once as distance strictly above the R-local development 95th percentile, reusing the same case set for the raw arm; typical/tail distributions, tail-minus-typical case-bootstrap intervals, rank association, and five prespecified worst cases are reported without p-values or exclusion. CARE2026 native prediction surfaces are compared with expert labels by area-weighted symmetric average surface distance (ASSD), the 95th-percentile Hausdorff distance (HD95), and the Chamfer root-mean-square (RMS) distance, beside the registered-reference diagnostic and a prespecified referencecloseness rule: a structure receives native metrics only where its registered reference lies within fixed distance limits of the expert surface. MM-WHS supplies a separate 20-case expert-labelled CT validation with no retuning, exhaustive seven-label panels at k = 1, 3, 5, and its own case bootstrap, released with the artifacts; its intervals and contrasts are descriptive and never pooled with CARE2026 or the internal endpoint. Geometry, CARE expert-surface, and fixed axial AO/IVC truncation experiments use two fixed predictions, the deterministic Cond-G fit and the coordinate mean of the five deterministic β-VAE refits, in raw and R-local coordinates.

A. Experimental Setup The primary internal endpoint uses the frozen 76-case list only after every development choice has been locked. We evaluate k ∈ {1, 3, 5, 9} observed structures with 20 fixed panels per k (panel seed 42). Five k = 9 panels observe every non-chamber structure and contain no defined primary target; the same five are omitted for every method, leaving panel counts 20/20/20/15. For a case and panel, the endpoint is mean per-vertex Euclidean distance (MPVED) over all missing non-chamber vertices; panels are averaged within case and k, and the four k-specific case values receive equal weight in the primary aggregate. An equal-structure endpoint prevents larger meshes from dominating a secondary descriptive analysis. Every compatible estimator is selected on the same five development folds, panels, normalisation, and four-k MPVED criterion, over the search grids of Section IV. Mixture PPCA and local Cond-G use prespecified validity rules rather than repairing invalid cells after inspection. Full grids, tie rules, and failure accounting are given with the released artifacts. The all-development mean shape and observed-coordinate nearestneighbour retrieval are descriptive floors, not selected models. R-local is the primary representation and the raw coordinatemap field is a prespecified sensitivity arm. Hyperparameters are selected separately within raw and R-local development folds. The primary predeclared contrast is R-local β-VAE minus R-local Cond-G, equally averaged over the four k values; positive values favour Cond-G. Because the graph architecture predates the evaluation partition, its paired bootstrap interval and resampling-floor p value are descriptive conditional on this shared-pool design rather than strict confirmatory inference. Exactly seven supportive contrasts share one Holm family: the analogous raw four-k contrast, the four R-local per-k contrasts, R-local PPCA minus Cond-G at the selected K = 1, q = 160 limit, and the R-local local-minus-global structural identity. The last is retained under the family map frozen before development selection resolved and is zero by construction, because selection chose the unrestricted global limit and the B. Internal Completion on the Frozen Evaluation Split Development selection chose Cond-G with M = 200 modes scorer aliases the same rank-correct prediction; retaining it is conservative for the remaining Holm-adjusted contrasts. All and relative ridge 10−4 in both representations. The raw βVAE winner was (dz , βmax ) = (64, 10−2 ), refit for 233 other comparisons are descriptive or exploratory. Inference resamples cases, not vertices or panels. A single epochs; the R-local winner was (128, 10−2 ), refit for 226 frozen 10,000 × 76 index matrix, released with the artifacts, epochs. The five deterministic refit predictions are averaged as

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

coordinate fields before the evaluation endpoint of Section V-A is computed. No K > 1 mixture met the prespecified validity rules of Section IV-B; the only eligible winner is the singlecomponent PPCA limit at q = 160. The local Cond-G search selected all development cases and q = 200, exactly the global prediction, so it is retained as an identity check rather than a distinct estimator. Table V gives all frozen-split means. In the primary Rlocal representation, four-k MPVED is 3.717 mm for Cond-G against 5.248 mm for the β-VAE, and Cond-G has the lowest value among the non-identical compatible estimators at every k; the separately selected raw arm gives the same ordering. The primary paired β-VAE-minus-Cond-G difference is +1.531 mm (95% CI 1.384–1.711; p < 0.001 at the pairedbootstrap floor). Every empirical supportive contrast remains positive after the prespecified Holm adjustment (Table VI). Individual R-local β-VAE refits give differences of 1.791– 1.918 mm (seed SD 0.053 mm). Averaging their coordinate fields before the nonlinear Euclidean endpoint reduces the ensemble difference by 0.314 mm relative to the mean individualrefit difference; the two summaries measure different quantities. The frozen seed-escalation rule does not trigger. The graph architecture itself is a fixed input to this comparison: its Chebyshev order, channel ladder, and depth come from an earlier 11-cell sweep on a previous partition and representation, and because cases later assigned to the evaluation list could have contributed to that sweep, it is disclosed only as architecture-design evidence. Before scoring the frozen evaluation list, we therefore froze a developmentonly R-local sensitivity at the locked dz = 128, βmax = 10−2 cell: ten configurations vary Chebyshev order, width, one additional encoder/decoder block, learning rate, mask weight, and Laplacian weight over the same five folds and seeds 42/43, entering no hypothesis family and never scoring an evaluation case; the two seeds match the developmentselection budget of every candidate, while seeds 44–46 belong only to the all-development refit ensemble. The base and eight alternative configurations with complete finite outputs span 5.983–6.433 mm development MPVED, with paired alternative-minus-base means from −0.011 to +0.439 mm; the K = 10, width-16 configuration is invalid after two non-finite runs (122 predictions), with no value imputed and no aggregate computed. This bounded sensitivity neither establishes architectural robustness nor changes the primary refits. A further development-only experiment addressed the operator axis. With the convolution replaced by EdgeConv [44] inside the same architecture, results proved strongly dependent on the optimisation settings: under the settings shared with the graph β-VAE the substituted operator was far behind, while under operator-specific retuning it reached 5.774 mm, below the 5.983–6.433 mm envelope of the β-VAE configurations tested here and still above the conditional-Gaussian model’s 4.055 mm on the same folds. These post-primary runs never score an evaluation case. What they establish is that the shared training settings are not neutral across architectures, recorded as a limitation of the matched-protocol design; our conclusions concern the models evaluated here, not non-linear completion methods in general.

8

C. Additional Non-Linear Baselines To test whether the deep architecture is representative rather than a weak instance of its class, two descriptive post-lock arms were evaluated on the same frozen cases, panels, and endpoint (Tables VII and VIII): SpiralNet++ [45], a published mesh convolution operator substituted into our architecture at parameter parity (14.155 M against 14.2 M), and latent optimisation following Litany et al. [46], a published completion method run on our own locked β-VAE decoders, so that pair shares weights and differs only in inference procedure. Latent optimisation outperforms our deep model by 0.204 mm [0.169, 0.243] internally and 0.368 mm [0.338, 0.403] on MMWHS, yet trails Cond-G by 1.327 mm [1.201, 1.482] and 1.426 mm [1.272, 1.579]; on CARE2026 expert surfaces the ordering holds a third time, by 0.032 mm (LA) and 0.091 mm (RA) over the β-VAE, with AO and PA suppressed by the reference-closeness rule as for the primary methods. That a published completion method beats the β-VAE on both cohorts while losing to the closed-form estimator is the evidence that the deep arm is not a strawman of our construction: latent optimisation fixes the observed coordinates and asks what the fitted model implies for the remainder, structurally what the conditional mean does, and it still falls short. Fig. 4 shows these aggregates on the median case of each cohort under the six displayed panels, so neither is a best case. The completed anatomy closes into a coherent wholeheart mesh at every panel, observed blocks are reproduced exactly because their coordinates are copied, and the residual concentrates on the vessel and appendage surfaces furthest from the observed chambers. The single-structure panels (LV only, AO only) carry visibly the largest residual, the geometric statement behind the k-dependence in Table V; the external case shows the same spatial pattern with a larger residual, consistent with Section V-D.

D. External Validation CARE2026. A completion is scored against an expert surface only where the registered reference itself lies close to that surface (the reference-closeness rule of Section V-A); otherwise the distance would measure annotation mismatch rather than completion quality. The rule passes LV, MYO, RV, LA, and RA in both representations. Under R-local, their median/p90 reference ASSD ranges from 0.830/2.201 to 1.510/2.091 mm. AO fails under its prespecified longitudinalcoverage mismatch (18.895/21.694 mm), while PA misses the strict p90 threshold by 0.026 mm (3.398/6.954 mm versus 6.928 mm). Thus neither AO nor PA receives a native accuracy summary, and the successful 1.2× PA threshold sensitivity does not change that fixed decision. The designated 50% AO region is also technically non-evaluable in 46/58 R-local cases. Tables IX and X report the eligible R-local native metrics, pooled and by acquisition subset. With biventricular input, Cond-G has lower pooled descriptive ASSD, HD95, and Chamfer RMS means than the β-VAE for both LA and RA. The leave-one-out panel shows the same pooled ordering for the five eligible structures, but the complete A/B/G display

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

9

TABLE V F ROZEN 76- CASE INTERNAL EVALUATION COMPLETION MPVED ( MM ). E ACH k ENTRY IS THE CASE MEAN OVER ITS FIXED PANELS ; THE FOUR -k COLUMN EQUALLY WEIGHTS THE FOUR k VALUES , WITH A CASE - BOOTSTRAP 95% CI. PPCA IS THE SELECTED K=1, q=160 LIMIT; LOCAL C OND -G REUSES THE EXACT GLOBAL PREDICTION BY CONSTRUCTION . k=1 k=3 k=5 k=9

Representation Method

Four-k [95% CI]

Raw

Mean shape 11.218 11.278 11.189 10.909 11.149 [10.310, 12.040] Nearest neighbour 10.096 9.031 8.508 8.644 9.070 [8.621, 9.563] Cond-G 5.479 3.918 3.193 3.003 3.898 [3.629, 4.206] β-VAE 6.596 5.238 4.693 4.748 5.319 [4.991, 5.701] PPCA (K = 1) 5.524 3.958 3.248 3.064 3.948 [3.683, 4.251] † Local Cond-G 5.479 3.918 3.193 3.003 3.898 [3.629, 4.206]

R-local

Mean shape 11.157 11.215 11.130 10.858 11.090 [10.245, 11.990] Nearest neighbour 9.942 8.904 8.352 8.527 8.931 [8.503, 9.405] Cond-G 5.304 3.732 3.007 2.825 3.717 [3.480, 3.984] β-VAE 6.522 5.172 4.621 4.677 5.248 [4.935, 5.613] PPCA (K = 1) 5.348 3.767 3.061 2.882 3.764 [3.530, 4.027] † Local Cond-G 5.304 3.732 3.007 2.825 3.717 [3.480, 3.984] †

Structural identity, not an independently fitted held-out result.

TABLE VI F ROZEN PAIRED FAMILY. D IFFERENCES ARE CHALLENGER MINUS C OND -G, SO POSITIVE VALUES FAVOUR C OND -G. T HE R- LOCAL FOUR -k PRIMARY IS UNADJUSTED ; THE NEXT SEVEN ROWS ARE THE COMPLETE H OLM FAMILY (m = 7). T HE FINAL ROW IS AN IDENTITY CHECK RATHER THAN EMPIRICAL EVIDENCE . Contrast

Difference (mm) 95% CI (mm)

Multiplicity result

R-local β-VAE minus Cond-G, four-k primary

1.531

[1.384, 1.711] Unadjusted p < 0.001 (floor)

Raw β-VAE minus Cond-G, four-k R-local β-VAE minus Cond-G, k = 1 R-local β-VAE minus Cond-G, k = 3 R-local β-VAE minus Cond-G, k = 5 R-local β-VAE minus Cond-G, k = 9 R-local PPCA (K = 1) minus Cond-G, four-k R-local local minus global Cond-G, four-k

1.421 1.218 1.440 1.613 1.852 0.047 0.000

[1.264, 1.610] [1.023, 1.439] [1.298, 1.602] [1.466, 1.800] [1.689, 2.047] [0.038, 0.056] [0.000, 0.000]

Biventricular

Chambers

pHolm < 0.002 pHolm < 0.002 pHolm < 0.002 pHolm < 0.002 pHolm < 0.002 pHolm < 0.002 Structural identity

Seven-structure

8 TS

CARE2026

TS

CARE2026

TS

CARE2026

6 4

Vessels only

LV only

AO only

2

per-vertex error (mm)

10

0

TS

CARE2026

TS

CARE2026

TS

CARE2026

Fig. 4. Completion across six observed-structure panels, one internal TotalSegmentator case (TS, s0365) beside one external CARE2026 case (A Case1020), each the median case of its cohort by the displayed Cond-G estimator’s mean missing-vertex error over the six panels. Observed structures are grey; the conditional-Gaussian model completes the rest, coloured by per-vertex error (mm). The estimator is the selected configuration of Table V, fitted on the 307 development cases in the R-local domain.

is not uniformly ordered in every metric–subset cell: subset- completed structures. Under R-local, median/p90 reference A RV HD95 is 8.073 mm for the β-VAE and 8.349 mm for ASSD is 0.628/0.687 mm for LA, 0.620/0.838 mm for RA, Cond-G. The subsets are descriptive; no subgroup robustness and 2.884/4.724 mm for PA, all within the fixed 3.464/6.928claim or hypothesis test is made. mm limits. AO fails at 17.060/19.024 mm, and the directional decomposition identifies annotation scope rather than registraMM-WHS. MM-WHS supplies a second independent CT tion error as the cause: where MM-WHS labels the aorta, the cohort (20 cases), with no exclusions, pooling, or retuning. registered reference agrees with the expert surface to 1.075 mm As on CARE2026, the reference-closeness rule decides which averaged over the 20 cases, comparable to LA and RA, while structures receive native summaries: it admits three of the four

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

TABLE VII A DDITIONAL NON - LINEAR BASELINES , R- LOCAL REPRESENTATION , MPVED ( MM ) WITH CASE - BOOTSTRAP 95% INTERVALS . T HE INTERNAL COLUMN IS THE FROZEN 76- CASE SPLIT AT FOUR OBSERVED - STRUCTURE COUNTS ; THE MM-WHS COLUMN IS EXHAUSTIVE PANELS AT k = 1, 3, 5 OVER THE SEVEN EXPERT- OVERLAP STRUCTURES , SO ONLY THE ORDERING WITHIN A COLUMN IS COMPARABLE . B OTH ARMS LIE OUTSIDE THE PRESPECIFIED H OLM FAMILY AND ARE DESCRIPTIVE ; THE LATENT- OPTIMISATION SETTING IS NOT RETUNED EXTERNALLY. C OND -G, THE β-VAE AND THE FLOORS REPEAT TABLE V FOR REFERENCE ; THE SAME ARMS ON CARE2026 EXPERT SURFACES , A DIFFERENT ENDPOINT, ARE IN TABLE VIII. Method

Type

Cond-G Latent optimisation‡ β-VAE SpiralNet++§

Closed-form linear Published method Feed-forward deep Published operator

Nearest neighbour Mean shape

Retrieval floor Population floor

10

TABLE X R- LOCAL CARE LEAVE - ONE - EXPERT- STRUCTURE - OUT COMPLETION , POOLED (n = 58) AND BY ACQUISITION SUBSET (A/B/G CONTAIN 20/18/20 CASES ). VALUES ARE MEAN ASSD / HD95 / C HAMFER RMS IN MM . AO AND PA ARE OMITTED BECAUSE THEIR REFERENCES ARE TOO FAR . Missing structure Population

Cond-G

β-VAE

LV

Pooled A B G

2.362 / 6.407 / 3.119 3.899 / 9.623 / 4.942 1.818 / 4.744 / 2.358 3.367 / 8.177 / 4.254 3.697 / 10.361 / 4.969 4.841 / 12.575 / 6.258 1.704 / 4.510 / 2.215 3.582 / 8.411 / 4.446

Internal 76

MM-WHS (n = 20)

MYO

3.717 [3.480, 3.984] 5.044 [4.746, 5.389] 5.248 [4.935, 5.613] 6.710 [6.283, 7.218]

4.823 [4.446, 5.226] 6.248 [5.755, 6.777] 6.616 [6.109, 7.152] 8.916 [8.105, 9.741]

Pooled A B G

2.056 / 5.667 / 2.729 1.695 / 4.541 / 2.212 2.947 / 8.627 / 4.036 1.616 / 4.129 / 2.068

RV

Pooled A B G

2.717 / 8.098 / 3.770 3.377 / 8.555 / 4.335 2.868 / 8.349 / 3.969 3.016 / 8.073 / 3.994 3.247 / 10.314 / 4.619 4.148 / 10.495 / 5.305 2.088 / 5.852 / 2.807 3.045 / 7.290 / 3.801

LA

Pooled A B G

2.417 / 6.228 / 3.192 2.169 / 5.919 / 2.871 3.023 / 7.642 / 3.919 2.120 / 5.264 / 2.860

3.393 / 7.837 / 4.207 2.577 / 6.451 / 3.266 4.445 / 10.407 / 5.495 3.262 / 6.909 / 3.989

RA

Pooled A B G

2.800 / 7.434 / 3.763 2.577 / 7.013 / 3.458 3.246 / 9.006 / 4.441 2.621 / 6.439 / 3.456

3.878 / 9.530 / 4.988 3.474 / 8.729 / 4.448 4.372 / 10.762 / 5.607 3.836 / 9.224 / 4.972

8.931 [8.503, 9.405] 10.907 [10.145, 11.684] 11.090 [10.245, 11.990] 13.174 [12.045, 14.419]

‡ Litany et al., run on our frozen β-VAE decoders; the autoencoder is ours, the completion algorithm is theirs. § Operator substituted into our architecture at parameter parity, not a reproduction of a published end-to-end system.

TABLE VIII A DDITIONAL NON - LINEAR BASELINES ON CARE2026 EXPERT SURFACES , R- LOCAL , BIVENTRICULAR PANEL , MEAN ASSD / HD95 / C HAMFER RMS ( MM ) OVER 58 CASES . T HE ENDPOINT IS DISTANCE TO EXPERT MANUAL SEGMENTATIONS AND IS NOT COMPARABLE TO TABLE VII. AO AND PA ARE SUPPRESSED FOR EVERY METHOD BY THE PRESPECIFIED REFERENCE - CLOSENESS RULE . C OND -G AND THE β-VAE REPEAT TABLE IX FOR REFERENCE .

2.943 / 7.583 / 3.769 2.526 / 6.257 / 3.171 3.847 / 10.332 / 5.046 2.547 / 6.434 / 3.219

For the three eligible structures, Cond-G has lower mean ASSD than the β-VAE on LA (2.854 versus 3.989 mm), RA (2.733 versus 3.435 mm), and PA (4.895 versus 5.441 mm), Method Type LA RA and the same ordering holds for HD95 (7.570/7.647/15.041 Cond-G Closed-form linear 3.275 / 7.846 / 4.138 3.479 / 9.007 / 4.586 versus 9.742/9.147/16.579 mm) and Chamfer RMS Latent optimisation Published method 3.902 / 8.941 / 4.822 4.243 / 10.265 / 5.413 β-VAE Feed-forward deep 3.934 / 8.990 / 4.860 4.333 / 10.372 / 5.495 (3.704/3.655/6.963 versus 4.995/4.467/7.652 mm). SingleSpiralNet++ Published operator 4.299 / 9.701 / 5.266 4.605 / 10.868 / 5.779 component PPCA is not separable from Cond-G at this cohort size (2.896/2.753/4.928 mm); both remain well below the TABLE IX nearest-neighbour (6.474/5.263/8.092 mm) and populationR- LOCAL CARE BIVENTRICULAR NATIVE PREDICTION - VERSUS - EXPERT mean (6.673/6.628/8.064 mm) floors. The structure-local SURFACE METRICS , POOLED (n = 58) AND BY ACQUISITION SUBSET Cond-G ablation is numerically identical to Cond-G here, (A/B/G CONTAIN 20/18/20 CASES ). T HE PANEL OBSERVES LV/MYO/RV; VALUES ARE MEAN ASSD / HD95 / C HAMFER RMS IN MM . AO AND PA with a paired difference of zero, because the observed panel ARE OMITTED AFTER FAILING THE REFERENCE - CLOSENESS RULE . is exactly LV, MYO, and RV; it separates only on the internal eleven-block panels. These are descriptive means over 20 Population Method LA RA cases from a segmentation challenge’s training split: they Pooled Cond-G 3.275 / 7.846 / 4.138 3.479 / 9.007 / 4.586 corroborate the internal ordering on an independent cohort Pooled β-VAE 3.934 / 8.990 / 4.860 4.333 / 10.372 / 5.495 without supporting a confirmatory claim. A Cond-G 2.634 / 6.643 / 3.353 3.035 / 8.085 / 3.989 A β-VAE 3.103 / 7.553 / 3.911 3.780 / 9.373 / 4.811 MM-WHS also transfers the internal representation-space B Cond-G 4.010 / 9.642 / 5.032 4.028 / 10.598 / 5.351 endpoint. Against the seven-label registered correspondence B β-VAE 4.878 / 11.412 / 6.034 4.706 / 11.572 / 6.011 target, R-local Cond-G gives 4.823 mm MPVED (95% G Cond-G 3.253 / 7.432 / 4.119 3.427 / 8.496 / 4.494 G β-VAE 3.915 / 8.247 / 4.752 4.549 / 10.290 / 5.715 descriptive case-bootstrap interval 4.446–5.226), the βVAE 6.616 mm (6.109–7.152), and single-component PPCA 4.863 mm (4.487–5.265), averaged equally over k = 1, 3, 5; the opposite direction reaches 32.558 mm because the refer- the nearest-neighbour and population-mean floors give 10.907 ence extends beyond the annotated ascending segment into the and 13.174 mm, the corresponding raw values are 5.502, 6.963, arch and descending aorta, and a symmetric surface distance and 5.512 mm, and the paired β-VAE minus Cond-G contrast charges that unannotated extent as error. The same mismatch is 1.794 mm (1.623–1.961). These are representation-space fails AO on CARE2026 (18.895 mm), so AO receives no transfer results, not native expert-surface accuracy. Evidence by structure. native accuracy summary in either external cohort. The rule is Table XI, fixed before frozen-split selected-model scoring, symmetric by design and was not modified for this structure; the favourable one-directional value is a diagnostic, not partial answers three questions for every structure separately: whether validation. PA passes only in the R-local representation; in its reference is an expert manual label or an automatic silver the raw arm it reaches 9.999 mm and fails, so its eligibility is label, whether that reference is close enough to score against, and what evidence exists. The internal endpoint scores only representation-dependent and reported as such.

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Both selected models degrade outside development support. Among the 12 cases above the frozen R-local development 95th-percentile full-shape Mahalanobis threshold, Cond-G inREGISTERED REFERENCE LIES WITHIN THE FIXED DISTANCE LIMITS OF THE EXPERT SURFACE , SO A NATIVE COMPARISON IS MEANINGFUL ; THE creases from 3.335 to 5.758 mm (tail penalty 2.423 mm; 95% INTERNAL BENCHMARK SCORES ONLY STRUCTURES THAT ARE descriptive interval 1.693–3.193) and the β-VAE from 4.774 to COMPLETION TARGETS THERE . “C OND -G BETTER ” IS THE LOWER 7.778 mm (3.004 mm; 1.967–4.329). Continuous distance/error DESCRIPTIVE MEAN ERROR AGAINST THE β-VAE AT EVERY k. AO’ S REFERENCE IS TOO FAR BECAUSE THE EXPERT ANNOTATIONS COVER A Spearman correlations are 0.883 and 0.790, respectively. No diSHORTER AORTIC EXTENT THAN THE WHOLE - AORTA REFERENCE , NOT rect interval or test was prespecified for the difference between BECAUSE OF MISREGISTRATION (S ECTION V-D). penalties, so no differential robustness claim is made. The distance uses the complete target, is unavailable at inference, Structures Internal benchmark CARE2026 expert MM-WHS expert and is not a marker of pathology. check check True triangle tests contradict a collision-free interpretaLV, MYO, RV never a completion reference close expert label, but obtarget enough (leave-one- served as input tion. Across 76 R-local biventricular completions, Cond-G/βout panel) VAE/target cases with self-intersection include LA 6/17/10, LA, RA never a completion reference close reference close target enough (both panels) enough; Cond-G RA 4/31/14, AO 8/11/17, and PA 7/9/6. Any true triangle better contact with observed LV, MYO, or RV occurs in 34/63/3 AO Cond-G better at reference too far reference too far cases for AO and 45/56/6 for PA; frequent LA–LV and RA– every k PA Cond-G better at reference too far reference close RV contacts also occur in the targets because these structures every k enough under Rhave shared anatomical interfaces. Predicted degenerate-face local; Cond-G better LAA, PV, Cond-G better at no expert label no expert label counts are zero, but triangle contacts are not overlap volume SVC, IVC every k and zero degeneracy does not imply anatomical validity. Finally, fixed axial AO/IVC retention at 0, 25, 50, 75, and 100% evaluates only completion after template registration; missing non-chamber blocks, so LV, MYO, RV, LA, and RA hidden-set composition changes with the cut and several raw receive no internal completion score. For each of the six nonand R-local error curves are non-monotone, so the experichamber structures (AO, PA, LAA, PV, SVC, IVC) at each ment establishes neither monotonic truncation robustness nor observed-structure count k ∈ {1, 3, 5, 9}, 24 combinations in modality-specific partial-surface registration performance. all, Cond-G has a lower descriptive mean error than the β-VAE, and the same holds in all 24 raw-representation combinations. The four k values are kept separate in the released artifacts, F. Downstream Measurement, Robustness, and Uncertainty and no cross-k average, reliability tier, or composite rank is Three further analyses, each with its plan fixed before the formed, so no summary can hide a weak structure. numbers were seen, address what a user would take from a CARE thus provides expert-surface evidence with a closecompleted mesh, whether the fitted variation looks like cardiac enough reference for five structures, MM-WHS for three of its phase, and what happens when an observed structure is only four completed targets, and four structures have no external partly present. The plans and result files are released with the expert reference at all; AO fails the reference-closeness rule reproducibility artifacts. in both external cohorts, PA is eligible on MM-WHS only Morphometric retention. From the biventricular panel we under R-local, and the subset-A RV HD95 reversal prevents complete the remaining structures and compute per-structure a uniform per-metric claim. Fixed-topology and FOV caveats enclosed volume for the seven structures with usable volumes are strongest for AO extent, PA distal branching, variable PV (the trimmed IVC stub is excluded), comparing the completion branching, and the capped SVC/IVC stumps. These absences against the case’s own registered mesh (Table XII). Because and qualifications are results, not values to be averaged away. a completion is a deterministic function of the observed structures, the question is how much of a measurement survives, E. Repair Ablation and Validity Audits so each structure is also compared against three alternatives R-local modifies only 868 of 4,431,693 internal case–vertex available without a shape model: the development-set median opportunities, in 39/383 cases, and no chamber coordinate. volume, the mean shape, and a development-only ridge reOmitting it raises the four-k Cond-G mean from 3.717 to gression of the missing log volume on the three observed 3.898 mm and the β-VAE mean from 5.248 to 5.319 mm; chamber volumes. Improvement is claimed only when the Cond-G remains favoured in both arms, which preserves the paired case-bootstrap interval for the difference in absolute selected model ordering without establishing anatomical truth. error excludes zero. Internally, five of seven structures beat The same fixed transform intervenes more on both external all three alternatives; pulmonary-vein volume is not sepacohorts: on CARE2026 it modifies 2,174 of 671,118 opportu- rated from any of them (−0.01 mL, [−0.31, +0.28] against nities in all 58 cases (per-case mean 0.324%, median 0.251%, the regression) and caval volume is not separated from the maximum 1.193%), and on MM-WHS 640 of 231,420 oppor- regression. Externally the atria retain their advantage over the tunities in all 20 cases (per-case mean 0.277%, median 0.216%, regression while AO, PA, and PV are not separated from any maximum 1.219%), versus a 0.0196% internal per-case mean; baseline, and caval volume is worse than a constant (+2.49 mL, we treat that disparity as preprocessing-domain sensitivity, not [+1.65, +3.35]). Atrial sphericity improves on the mean shape successful external correction. internally for both atria (−0.021 and −0.028) but degrades TABLE XI T HE EVIDENCE BEHIND EACH STRUCTURE ; STRUCTURES WITH IDENTICAL STATUS SHARE A ROW. “R EFERENCE CLOSE ENOUGH ” MEANS THE

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externally for LA (+0.011, [+0.004, +0.018]). Multiplicative TABLE XII M ORPHOMETRIC RETENTION FROM THE BIVENTRICULAR PANEL : limits of agreement run ×[0.75, 1.36] for LA and widen to PER - STRUCTURE VOLUME AGREEMENT BETWEEN THE COMPLETION AND ×[0.67, 1.59] for SVC, and calibration slopes fall to 0.64 THE CASE ’ S OWN REGISTERED MESH . MAE IS MEAN ABSOLUTE ERROR ; (PV) and 0.73 (SVC), that is, reversion towards the mean for MEDAPE IS MEDIAN ABSOLUTE PERCENTAGE ERROR ; ∆ IS THE PAIRED DIFFERENCE IN ABSOLUTE ERROR AGAINST THE DEVELOPMENT- ONLY the smallest structures. Volume is one scalar summary of a REGRESSION ON THE THREE OBSERVED CHAMBER VOLUMES , WITH ITS surface and no clinical tolerance is defined here, so these are 95% CASE - BOOTSTRAP INTERVAL ( NEGATIVE FAVOURS COMPLETION ). measurement-agreement results, not clinical validation. V ESSEL AND APPENDAGE ROWS ARE SEGMENT VOLUMES UNDER THE FIXED TEMPLATE CROP. Anatomy and phase. Phase and anatomy are not identifiable from one static scan per subject, so we do not decompose Internal (n=76) CARE2026 (n=58) them; two model-side observations bound the concern. Within Structure MAE (mL) medAPE ∆ [95% CI] MAE (mL), medAPE the first 20 principal-component coefficients (a fixed analysis LA 7.99 8.3% −6.84 [−9.43, −4.43] 18.47, 18.5% subspace, distinct from the M =200 completion model), log RA 9.61 8.9% −7.29 [−10.36, −4.35] 20.89, 25.7% ventricular and log atrial volume correlate positively (r = AO 13.81 5.6% −7.99 [−12.75, −3.33] 23.08, 13.1% 6.74 8.5% −4.51 [−6.35, −2.68] 13.39, 17.5% 0.549 [0.439, 0.630] under R-local, 0.528 [0.425, 0.610] raw; PA LAA 0.93 10.7% −0.49 [−0.71, −0.27] 1.41, 14.8% 400 model-refitting bootstrap resamples), the signature of over- PV 1.41 13.1% −0.01 [−0.31, +0.28] 1.55, 13.7% SVC 3.18 12.3% −0.60 [−1.30, +0.21] 5.52, 22.6% all chamber size rather than the reciprocal covariance expected from phasic opposition, and the leading mode tracks patient age (Spearman 0.493). Completion error varies with a leftventricular cavity-fraction proxy (−13.8 mm per unit [−26.1, −2.3]); the proxy conflates phase with dilation, hypertrophy, and segmentation quality, so the association is reported without actually cover 0.927–0.988 and 0.991–0.999 of the missing a cause. non-chamber vertices, with error-to-covariance-trace ratios of Partial observation of an input structure. The fixed axial 0.097–0.293. The β-VAE is overconfident: its 160-draw emAO/IVC retention curves above truncate structures that are pirical second moment covers only 0.026–0.052 and 0.124– themselves completion targets, so we added a like-for-like 0.203, with ratios 4.448–10.054. The direction persists on the experiment in which one observed chamber is axially truncated fixed SVC/IVC population, the alternate Monte Carlo centre at 100, 75, 50, 25, and 0% of its extent while the hidden is nearly identical, and no covariance eigenvalue reaches the set is held fixed, from the inferior end as primary and the numerical floor. These are cohort-level diagnostics under two superior end as sensitivity. Non-inferiority was declared in non-equivalent constructions; they provide neither a causal exadvance at a 1.531 mm margin, the paired margin by which planation of miscoverage nor per-patient or clinical confidence. Cond-G beats the graph β-VAE. Against a fully observed The two constructions nevertheless fail in opposite direcpanel at 4.599 mm, truncating LV changes nothing measurable even when it is removed entirely (−0.004 mm, one-sided upper tions, and the direction matters more than the magnitude for bound 0.018 mm), MYO removal costs 0.037 mm, and RV, the any reliability claim. Crossing the audit with the frozen outonly arm with a visible trend, costs 0.295 mm (upper bound of-support strata of Section V-E separates the 76 evaluation 0.392 mm). All six primary cells (three truncated chambers, on cases into 64 typical and 12 atypical; both artifacts were the internal and CARE cohorts) are non-inferior at the margin, fixed before any result was seen. Table XIII reports nominal in both cut directions and in the raw arm; the regularised 95% coverage by stratum. Cond-G covers 1.000 on typical condition number rises only from 2.5×102 to 4.3×102 across anatomy in every panel and still 0.944 to 0.999 on the atypical the masks and a ridge sensitivity at 10−5 and 10−3 changes stratum, while its nominal 50% coverage falls from 0.978– nothing, so this is insensitivity rather than the regulariser 0.996 down to 0.651–0.949 and its error-to-trace ratio rises dominating. Under the tested masks and margin, the left- from 0.072–0.199 up to 0.229–0.863, toward one rather than ventricular blood pool contributes nothing measurable once its past it: its conservatism is consumed, not idle, on the cases wall is observed. The truncation is applied to meshes already furthest from the development distribution. The β-VAE covers in correspondence, built from complete anatomy, so this is 0.128 to 0.205 on typical and 0.105 to 0.194 on atypical cases, completion from partial input rather than registration of a with the panel-1 ratio rising from 9.177 to 12.163, so it is overconfident everywhere and most overconfident where the cropped image. anatomy is unusual. Root-mean-square error roughly doubles Model and uncertainty boundaries. No K > 1 PPCA mixture met the prespecified validity across the strata for both models, from 5.609 to 11.105 mm for rules, and the local search selects its all-neighbour global Cond-G and 6.206 to 10.638 mm for the β-VAE on panel 1, limit; neither supplies evidence for a distinct nonlinear or local and the same directions hold on the anatomy-fixed SVC/IVC advantage. The bounded graph sensitivity cannot exclude other population. This exploratory, post-hoc stratification rests on 12 operators, pooling hierarchies, losses, or larger cohorts, and the atypical cases and does not make either construction calibrated. axial partial-vertex masks above are the only within-structure What it supports is narrower and still useful: the target-derived Mahalanobis distance acts as a retrospective reliability flag masks evaluated. Neither uncertainty construction is calibrated on the 76 for where the reported regions are least trustworthy, and the evaluation cases. Cond-G is overconservative: across the four linear estimator fails in the conservative direction while the observation panels, its nominal 50% and 95% ellipsoids deep ensemble fails in the overconfident one.

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and native expert distances prevents one favourable layer from certifying another. The concrete downstream use demonstrated here is cohort unification: the fixed biventricular CARE panel completes structures that an external cohort never labelled, putting cases with different label inventories into one eleven-block correCond-G β-VAE spondence space where per-structure morphometry can be Observed panel typical atypical typical atypical computed on a common vertex set. The example is deliberate LV 1.000 0.946 [0.868, 0.998] 0.128 0.105 [0.071, 0.138] and bounded. LV, MYO, RV 1.000 0.944 [0.878, 0.991] 0.129 0.140 [0.104, 0.174] Five chambers 1.000 0.974 [0.953, 0.993] 0.141 0.139 [0.103, 0.175] Finally, coordinate accuracy is not geometric or clinical Nine structures 1.000 0.999 [0.996, 1.000] 0.205 0.194 [0.110, 0.281] validity. Self-intersections and cross-structure triangle contacts remain, ostial distortion and volumetric overlap between completed and observed structures are not quantified, measurement VI. D ISCUSSION agreement degrades for the smallest blocks (pulmonary-vein volume is not separated from a constant, and external caval The main finding is deliberately narrow: in this registered volume is worse than one), uncertainty ellipsoids are miscaliCT displacement space, with matched development selection brated in opposite directions, and the axial truncation curves do and a frozen 76-case evaluation split drawn from the same not establish robustness. The released meshes and completion source pool as development (not an independent cohort), Cond- operator are research resources for controlled correspondenceG improves the four-k non-chamber MPVED over the selected space experiments; this study does not validate individual graph β-VAE by 1.531 mm. The raw-coordinate sensitivity, anatomical measurements, abnormality detection, clinical deeach individual k, five refits, and the eligible CARE native cisions, haemodynamic or electrophysiological simulation, or summaries support the same overall ordering but answer dif- completion from unregistered clinical images. ferent questions and are not pooled. The much smaller 0.047mm internal difference from the eligible single-component PPCA limit shows that much of the advantage is shared A. Limitations by regularised single-component Gaussian conditioning rather First, the internal 307/76 partition is a frozen evaluation than unique to one implementation. The result positions Cond- split drawn from the same source pool used to develop the G against matched alternatives rather than the wider literature representation and model families, not an independent cohort. (Section V-A). The reported fits are isolated from those 76 outcomes, but One plausible explanation is that centre-of-mass alignment, the graph architecture predates the partition and was selected joint SyN registration, and a fixed template absorb enough in a sweep whose training pool included cases now in the pose and correspondence variation that a regularised Gaussian evaluation set, so the paired bootstrap summaries are descripconditional mean is a strong estimator at this cohort size; this is tive conditional on this design rather than strict confirmatory a hypothesis about the constructed representation, not evidence inference; only the development-only bounded sensitivity is that cardiac anatomy is Gaussian. The invalid multi-component clean of that issue. CARE and MM-WHS are independent but cells, the global limit selected by the local search, and the small (n = 58 and n = 20) and their analyses are descriptive. bounded architecture sensitivity cannot distinguish insufficient We are not aware of a larger public CT cohort carrying expert sample size from model misspecification, nor exclude a better labels for all eleven structures, so the external benchmarks nonlinear model, registration strategy, or loss. answer the independence half of validation, not the scale half. The external cohorts sharpen rather than erase the boundary. Second, the internal endpoint is error to a TotalSegmentatorCARE provides eligible native expert-surface evidence for derived, SyN-registered silver target, not error to anatomical five structures and shows lower pooled Cond-G metrics in truth. CARE supplies expert surfaces for seven structures but the registered biventricular and leave-one-out settings, but the reference-closeness rule of Section V-D admits five, MMone subset–metric cell reverses and AO/PA fail the reference- WHS admits three (one only under R-local), and LAA, PV, closeness rule. MM-WHS preserves the descriptive registered- SVC, and IVC have no external expert surface (Table XI). target ordering without retuning and admits LA, RA, and PA The fixed-topology and FOV caveats of Section V-D apply, natively, where Cond-G is again lower on all three metrics; AO warp-overlap Dice is insensitive to tangential vertex sliding, fails there as on CARE, and PA is eligible only under R-local. and single-template propagation may introduce template bias. MM-WHS corroborates the internal ordering while showing Third, the cohort is CT-only, a convenience sample with that measurability is decided per structure and representation unknown health, scanner, ancestry, and geographic composirather than per dataset; at 20 cases it is corroboration, not tion, and ungated: a single static scan cannot separate withinconfirmatory validation. cycle motion from inter-subject variation, so the released The raw/R-local comparison carries its own caution: R-local mean and modes are not a normative healthy reference, sameis sparse internally and its omission does not reverse model instant input and target do not remove phase mixture from ordering, but the higher intervention prevalence on MM-WHS the population distribution, and cross-phase conditioning (such reveals domain sensitivity. Separating silver-target MPVED, as predicting an end-systolic atrium from an end-diastolic warp-overlap diagnostics, registered-reference measurability, ventricle) is outside the evaluated distribution and untested. TABLE XIII N OMINAL 95% ELLIPSOID COVERAGE BY OUT- OF - SUPPORT STRATUM , R- LOCAL , 76 EVALUATION CASES (64 TYPICAL , 12 ATYPICAL ). S TRATA COME FROM THE FROZEN DEVELOPMENT 95 TH - PERCENTILE M AHALANOBIS THRESHOLD . E XPLORATORY AND POST- HOC ; BRACKETS ARE DESCRIPTIVE CASE - BOOTSTRAP INTERVALS .

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The proxy analyses of Section V-F (concordant chamber co- structure template mesh, the core CT-to-correspondence variance, an age-linked leading mode) are consistent with size- pipeline, and model code under the MIT licence. A versioned driven variation but measure geometry, not timing, and cannot reproducibility archive accompanying this article provides exclude a phase contribution; the cavity-fraction association is the exact analysis scripts, frozen split and bootstrap files, reported without an identified cause. manifests, and machine-readable result tables. The per-case Fourth, the model comparison is finite and contains no R-local displacement fields and reconstructed meshes will published external system, because none emits the eleven be deposited in a versioned public archive at publication. blocks the endpoint scores (Section II). The graph search All source image data are public and de-identified; no new matches latent width and KL weight, the development-only data were collected and no image data are redistributed. The sensitivity varies nearby architecture choices and one alterna- reported results were produced with TotalSegmentator 2.13.0 tive operator, and that experiment showed a shared training (silver labels), ANTsPy 0.6.3 (registration), and a CUDA 12.8 protocol is not neutral across architectures: the comparison build of PyTorch (graph models). is controlled rather than architecture-optimal, and a matched ACKNOWLEDGEMENTS protocol is no guarantee of fairness. The pooling hierarchy, alternative objectives, extensive per-architecture optimisation, This work is funded by the EU NextGenerationEU through and substantially larger datasets remain untested, and invalid the Recovery and Resilience Plan for Slovakia under the K > 1 mixture cells are missing evidence, not evidence project No. 09I03-03-V04-00394. The authors acknowledge that multimodal models fail. Health status and diagnosis are the Technical University of Košice for providing highunavailable in all three cohorts, so no pathology stratum can performance computing (HPC) resources on the Perun cluster. be formed; the target-derived Mahalanobis tail is the feasible R EFERENCES substitute, measures degradation away from development support rather than on disease, and cannot serve at inference. [1] M. Salvador, M. Strocchi, F. Regazzoni, C. M. Augustin, L. Dedè, S. A. Niederer, and A. Quarteroni, “Whole-heart electromechanical Fifth, the outputs remain point completions. Neither tested simulations using Latent Neural ODEs,” npj Digital Medicine, vol. 7, uncertainty construction is calibrated and no conformal or no. 1, p. 90, 2024. external calibration layer is evaluated: a layer fitted on the [2] C. Rodero, M. Strocchi, M. Marciniak, S. Longobardi, J. Whitaker, M. D. O’Neill, K. Gillette, C. Augustin, G. Plank, E. J. Vigmond, P. 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Matej Gazda received the B.Sc. and M.Sc. degrees in computer modelling from the Technical University of Košice, Slovakia, in 2017 and 2019, respectively, and the Ph.D. degree in informatics from the same university in 2023. He is currently with the Department of Mathematics and Theoretical Informatics, Faculty of Electrical Engineering and Informatics, Technical University of Košice. His research interests include the application of artificial intelligence to medical imaging.

GAZDA ET AL.: ELEVEN-STRUCTURE CARDIAC CT SSM AND SHAPE COMPLETION

Jakub Gazda is a physician and researcher with the Faculty of Medicine, Pavol Jozef Šafárik University and L. Pasteur University Hospital, Košice, Slovakia. His research interests are in hepatology and gastroenterology, including the epidemiology and prognosis of primary biliary cholangitis.

Juraj Gazda received the Ph.D. degree from the Technical University of Košice, Slovakia, in 2010. He is currently a Professor with the Department of Computers and Informatics, Technical University of Košice. His research interests include 5G/6G networks, artificial intelligence, machine learning, and agent-based modelling of complex networks.

Peter Drotár received the M.Sc. and Ph.D. degrees in electronics from the Technical University of Košice, Slovakia, in 2007 and 2010, respectively. He is currently an Associate Professor with the Department of Computers and Informatics, Technical University of Košice, where he leads the Intelligent Information Systems Laboratory. His research interests include machine learning, medical imaging, biomedical signal processing, handwriting analysis, and feature selection. He is a Member of IEEE and EurAI.

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