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CARS: a centerline-anchored attention surrogate for coronary blood flow on skeleton-generated lumen geometries

Reza Akbarian Bafghi, Sukirt Thakur, Maziar Raissi

arXiv:2610.30498v1cs.CEphysics.flu-dyn

TL;DR

Patient-specific coronary flow surrogates must handle the flow domain itself as a variable, despite its branching and geometric complexity. CARS uses a centerline-anchored attention model to predict pressure and velocity fields from centerline radii and inlet flow. It reports strong held-out FFR decisions and outperforms the compared configurations at matched parameter counts.

  • Problem

    Coronary flow prediction requires solving the steady incompressible Navier–Stokes equations on patient-specific lumens whose branching topology and geometry vary, making the domain itself a high-dimensional parameter.

    Source: §1 · p. 2

  • Method

    CARS uses the centerline and its radius profile as an attention focal set, with inlet flow rate conditioning, to predict pressure and velocity at supplied evaluation points.

    Source: §3.6 · p. 18

  • Results

    97.7% (LCA) and 96.3% (RCA) of held-out FFR ≤0.8 decisions match the CFD reference, and CARS is most accurate among compared configurations at parameter counts matched within 3%.

    Source: §1 · p. 4

  • Takeaways & Limitations

    The results support a geometry-defined focal set for this surrogate setting, while retaining volumetric field prediction and an FFR functional derived from pressure.

    Source: §7 · p. 40

    The target is a steady, laminar, rigid-wall, Newtonian OpenFOAM pipeline, so pulsatile and transient effects, compliance, and microvascular resistance are outside the supervision.

    Source: §6.3 · p. 38

Abstract

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When the parameter of a PDE surrogate is the flow domain itself, the key/value set of an attention model must encode that domain. We present CARS (Coronary Artery Rapid Surrogate), a learned surrogate for the finite-volume solution map of the steady incompressible Navier-Stokes equations on coronary lumens built from image-derived centerlines. CARS maps a centerline with per-point lumen radius and a scalar inlet flow rate to pressure and velocity at arbitrary query points. The attention focal set is the case's own centerline, so geometry fusion costs $O(N n_f)$ rather than $O(N^2)$ in the number of evaluation points. Trained per territory on $2{,}700$ OpenFOAM solutions with synthetic focal lesions, CARS reaches relative $\ell_2$ errors of $0.120/0.149$ (pressure/velocity) on left-coronary and $0.063/0.119$ on right-coronary test sets, in $87$~ms per case against a solver wall-clock of $572$-$1{,}063$~s. It outperforms four parameter-matched neural operators, and accuracy saturates eightfold below the operating focal count. Fractional flow reserve derived from the predicted pressure matches the CFD label at Pearson $r = 0.941/0.993$ and reproduces the $\mathrm{FFR}\le 0.8$ decision in $97.7\%/96.3\%$ of held-out cases. The predicted velocity does not satisfy continuity closely enough for derived quantities, and the weights are territory-specific.

1. Introduction

Patient-specific coronary flow prediction requires solving on domains whose topology and geometry vary, while existing reduced-order and grid-based approaches do not fit this setting. CARS uses the centerline as a geometry-defined attention focal set and reports strong held-out FFR decisions alongside comparisons with neural-operator baselines.

1. Introduction

  • Variable branching topology, branch caliber, curvature, torsion, and focal narrowing make coronary geometry a high-dimensional domain parameter that lacks a small shape-parameter description.Projection-based models need a common reference configuration, while attention over all volume points costs O(N_v^2).Source: §1 · p. 2
  • Using the centerline and per-point radius as the focal set makes geometry fusion O(N_v n_f), while evaluation points remain an input to the map.The scalar flow rate enters through feature-wise linear modulation, consistent with the approximately linear pressure–flow scaling of this regime.Source: §1 · p. 2
  • CARS predicts pressure and velocity fields in 87 ms per case, compared with reference CFD solver medians of 572 s (RCA) and 948–1,063 s (LCA).The stated CFD times are for the reference solver stage on eight MPI ranks.Source: Figure 1 · p. 3
  • The coronary setting makes surrogate error interpretable through FFR, a pressure ratio used to decide whether a lesion is flow-limiting at the FFR ≤0.8 threshold.CT-derived Navier–Stokes solutions have been reported to agree closely with invasive measurement at this threshold.Source: §1 · p. 3
  • CARS is the most accurate of the compared configurations at parameter counts matched within 3%, and its predicted-pressure FFR reproduces the CFD FFR ≤0.8 decision in 97.7% (LCA) and 96.3% (RCA) of held-out cases.The evaluation uses identical splits and includes four neural-operator baselines and alternative focal representations.Source: §1 · p. 4
  • The evidence does not isolate the effect of centerline attachment because each ablation that moves the focal set also changes geometric channels or positional treatment.The authors identify this as a limitation on interpreting the ablation results.Source: §1 · p. 5
  • Centerline anchors outperform learned inducing points and surface-subsampled anchors; accuracy saturates eightfold below the operating focal count.A learned attention-level coordinate embedding lowers error by 27–30% relative to parameter-free 3D rotary encoding, with a frequency-scale confound noted.Source: §1 · p. 5

2. Related Work

Prior work addresses coronary flow surrogates through reduced-order models, fixed-grid and graph-based learning, attention, and scalar FFR estimation, but each has constraints on geometry, topology, or output. CARS instead retains volumetric prediction with a topology-agnostic token set and a centerline-defined focal set.

2. Related Work

  • Prior ct-FFR pipelines were validated against invasive FFR in DISCOVER-FLOW, DeFACTO, and NXT, with per-vessel sensitivity broadly in the 80–90% band and variable specificity.The passage notes specificity as low as 54% per patient in DeFACTO and pipeline runtimes of hours per case even with cloud acceleration.Source: §2 · p. 6
  • Projection-based reduced-order models need a common reference domain and regular deformation family, which image-derived coronary trees with varying branch count and connectivity do not provide.One-dimensional models avoid topology constraints, but their empirical trans-stenotic pressure-loss closure sets accuracy for the FFR endpoint.Source: §2 · p. 7
  • Learned cardiovascular surrogates trade off geometry flexibility and spatial structure: CNNs need fixed discretizations, GNNs rely on local message passing, and standard DeepONet and Fourier neural operators assume structured inputs or adaptations.Physics-informed neural networks in basic form require per-case optimization, while metalearned variants amortize across a parameter family.Source: §2 · p. 7
  • Attention-based PDE surrogates reduce global attention cost with learned latents, inducing points, slices, or sampled points, but those focal sets are not geometry objects supplied by the problem.CARS uses the centerline itself for geometry conditioning; its volume anchors are sampled from evaluation points.Source: §2 · p. 8
  • Earlier machine-learning FFR estimators predict a scalar from lesion descriptors, whereas this work predicts volumetric fields and extracts FFR as a functional.Reported per-vessel AUC against invasive FFR across the cited lineage is approximately 0.84–0.93.Source: §2 · p. 9

3. Methodology

CARS approximates the steady coronary-flow solution map from centerline geometry, inlet flow rate, and requested evaluation points. It uses centerline-anchored attention to condition volume queries on geometry, then refines the predicted fields and decodes pressure and velocity.

3.1. Problem formulation

  • The surrogate maps a centerline with its radius profile, inlet flow rate, and evaluation points to pressure and velocity, supervised against the finite-volume solution map.The prescribed terminal fluxes prevent lesions from diverting flow to parallel branches, and Q is sampled independently of geometry.Source: §3.1 · pp. 10–11
  • Because the evaluation set is an input, predictions at a fixed point can depend on which other points are requested alongside it.The measured dependence is 0.019 relative ℓ2 on pressure and 0.058 on velocity; the model is also not equivariant to joint rigid motions.Source: §3.1 · p. 12

3.2. Inputs and normalization

  • The network receives ordered centerline coordinates and radii, volume coordinates, and scalar inlet flow rate, but no branch labels, edge list, or surface mesh.The geometry is represented by centerline and radius on this cohort; that representation ceases to be injective for lumens with eccentric cross-sections and asymmetric plaque.Source: §3.1 · pp. 12–13, §3.2 · p. 14

3.3.1. 3D rotary position embeddings (RoPE3D)

  • RoPE3D encodes relative coordinate displacement through axis-wise rotations, while its default frequency scale is acknowledged as a confound in comparisons with learned position encoding.The unscaled language-model default completes at most 0.16 cycles across the normalized domain; a radius-rescaled control was not run.Source: §3.3.1 · p. 15

3.3.2. Learned coordinate embedding

  • The learned coordinate embedding adds a learned coordinate-dependent bias to queries and keys, and is the default for the reported CARS results.It adds 12,736 parameters (+0.3%) and is not translation-invariant by construction.Source: §3.3.2 · p. 16

3.4. Tokenization of the vessel and the volume

  • Centerline tokens combine position and learned radius embeddings, while volume tokens carry spatial information without supplied field values.Consequently, hemodynamic information must reach volume tokens through attention.Source: §3.4 · p. 16

3.5. Tiered attention with centerline anchors

  • Tiered attention uses all tokens as queries but only centerline tokens as keys and values, reducing geometry-fusion cost from O(N^2) to O(N·nf).At the stated settings, geometry fusion is roughly 9× cheaper; the centerline focal tokens are supplied by each case, not learned global parameters.Source: §3.5 · pp. 16–17

3.6. Two-stage processing

  • The two-stage stack first fuses geometry through centerline anchors, then refines volume tokens through volume anchors; FiLM conditions the final features on inlet flow rate before decoding four field channels.FiLM’s multiplicative form is motivated by approximately linear pressure scaling with Q at fixed geometry over the sampled range.Source: §3.6 · p. 18, §3.7 · p. 18

4. Experimental Setup

The experiments use territory-specific coronary datasets generated from ImageCAS centerlines with synthetic radius narrowings and OpenFOAM reference fields. Matched baselines, held-out parameter-set cases, field and FFR metrics, and two training budgets define the evaluation.

4.1. Dataset

  • Each territory has 2,700 training, 300 validation, and 300 test cases sampled from 101 LCA and 131 RCA anatomies, with synthetic narrowings and fresh flow rates.Splits are over parameter-set samples, so centerline trajectories recur across splits; the reported field and endpoint results measure parameter-space, not clinical, generalization.Source: §4.1 · pp. 19–20

4.2. Baselines and comparisons

  • CARS is compared with four neural-operator baselines within 3% of its parameter count on identical splits, alongside ablations of encoding, attention, conditioning, anchors, and stage design.The ablation arms include RoPE3D versus learned bias, centerline versus learned inducing-point anchors, and two-stage versus single-stage processing.Source: §4.2 · pp. 20–21

4.3. Metrics

  • Mean relative ℓ2 pressure error on the held-out test split is the primary endpoint; velocity error, FFR statistics, and inference cost are secondary.Field errors are measured on denormalized predictions in the cell-center measure; the endpoint hierarchy was not registered in advance.Source: §4.3 · p. 21

4.4. Training

  • Training minimizes normalized mean squared error, with both 100-epoch matched-baseline and 300-epoch primary budgets reported.The 300-epoch configuration followed a hyperparameter sweep indicating that the model had not saturated at 100 epochs.Source: §4.4 · p. 21

5. Results

The results assess CARS’s pressure, velocity, and FFR predictions, including how architectural choices, focal-set density, geometric perturbations, and case complexity affect accuracy and its limitations. They also examine territory-specific performance and show that zero-shot transfer from LCA to RCA failed.

5.1. It reproduces the reference fields

  • CARS reaches mean test relative ℓ2 errors of 0.120/0.149 pressure/velocity on LCA and 0.063/0.119 on RCA, with 87 ms per case.LCA values are means ± s.d. across three seeds; RCA values are from one run. Reference-field discretization error is smaller than surrogate error on the median case by about a factor of three, in a different norm.Source: §5.1 · pp. 21–22

5.2.1. Component ablations

  • Learned coordinate embeddings reduce pressure/velocity errors by 30%/27% versus RoPE3D, while removing focal self-attention, using a Perceiver bottleneck, or replacing centerline anchors degrades performance.Removing FiLM gives 0.329/0.487; the learned-PE comparison is 0.140/0.196 versus 0.199/0.269 for RoPE3D. The learned-anchor arm also removes the radius profile and Stage-1 positional encoding, so it tests the whole geometry-attached focal representation, not geometric attachment alone. Separately, accuracy saturates at an eightfold reduction from 2,048 to 256 focal points, with only a material rise at 128.Source: §5.2 · p. 23, §5.2.1 · pp. 24–25

5.2.2. Comparison with neural-operator baselines

  • At 300 epochs, CARS reduces test error versus GNOT by 66%/71% at 100 epochs and beats AB-UPT on both fields at every matched seed at 300 epochs.The 300-epoch matched-seed CARS–AB-UPT margins are 0.039 ± 0.014 pressure and 0.023 ± 0.012 velocity; the comparison is fixed-budget, and the velocity margin misses α = 0.05.Source: §5.2.2 · p. 26, Table 4 · p. 27

5.3. What it costs

  • Increasing training from 100 to 300 epochs reduces validation pressure and velocity errors by 23% and 28%, whereas widening to dmodel=256 does not pay off within the 100-epoch budget.At fixed evaluation density, pressure error varies by at most 0.004 across a 16-fold range from 4,096 to 65,536 points; an eightfold reduction in anchor count from 16,384 to 2,048 costs 0.008 pressure error and 0.073 velocity error. bfloat16 raises pressure error by 21% on LCA and 37% on RCA and degrades velocity by about 3.6×.Source: §5.3 · pp. 27–28, Table 5 · p. 29

5.4.1. Fractional flow reserve endpoint

  • On the full test split, predicted and reference FFR agree at Pearson r = 0.941/0.993 and MAE 0.026/0.010 on LCA/RCA, with FFR ≤0.8 decisions matching in 97.7%/96.3%.The primary analysis includes all 300 cases per territory and uses exact-coordinate queries; its ROC AUC is 0.996/0.996 on the full cohort.Source: §5.4.1 · pp. 29–30, Table 6 · p. 31

5.4.2. Parametric sensitivity to the geometric inputs

  • Radius perturbations are the binding geometric sensitivity: 5% relative noise leaves both territories intact, 10% moderately degrades LCA, and 20% substantially degrades both.Under independent position noise, LCA is unchanged up to σ = 1 mm, while RCA loses three threshold-accuracy points at 0.5 mm; correlated position biases up to 2 mm leave both at their unperturbed operating point. MC-dropout uncertainty ranks cases but is overconfident as an interval.Source: §5.4.2 · pp. 30–32

5.5.1. Error versus geometric and flow complexity

  • No covariate produces a large or reliable error dependence, and the reproducible high-error tail is case-specific rather than explained by flow rate, vessel size, or branching.Branch count is weakly associated with error after correction, while pressure error spans factors of 52 on LCA and 38 on RCA; per-case errors correlate across seeds at Pearson 0.89–0.93.Source: §5.5.1 · pp. 32–33

5.5.2. What the construction does not enforce

  • CARS does not enforce governing equations pointwise: median |∇·u|D/U is 0.26/0.30 on LCA/RCA, and inlet flux differs from prescribed Q by 12.2%/5.5%.The supervised field norms and pressure-derived endpoint can be accurate without conservation; quantities differentiating velocity, such as wall shear stress, vorticity, and dissipation, should not be read from predictions without physics-constrained training or a divergence-free projection.Source: §5.5.2 · pp. 34–35

5.5.3. Cross-territory generalization: RCA

  • Zero-shot LCA→RCA inference produced relative ℓ2 errors of order 8 on both fields, roughly two orders worse than within-territory results, with pooled pressure correlation only +0.02 to +0.06.RCA geometries typically have a single dominant trunk, and their mean inlet flow rate is 3,515 ± 571 mm3/s, about 55% of the LCA mean, leaving flow-rate distributions separated by roughly 3 standard deviations; shifted geometry and flow-rate distributions drive extrapolation.Source: §5.5.3 · pp. 35–36
  • RCA-statistic position/radius normalization reduced pooled pressure error from 15.7 to 7.0, but also normalizing Q raised it to 22.5; the best global scale remained indistinguishable from zero.Per-territory conditioning did not repair the uncorrelated zero-shot output, indicating representational failure under extrapolation rather than a calibration problem.Source: §5.5.3 · p. 35
  • A fresh RCA-trained CARS reached relative ℓ2 errors of 0.063/0.119 and FFR Pearson r = 0.994, MAE 0.008, and threshold accuracy 96.2% on the physiologic-range subset.Field errors, r, and MAE improved on the LCA result under matched architecture, but threshold accuracy and specificity did not; inference cost was indistinguishable between territories.Source: §5.5.3 · p. 36
  • The RCA results show within-territory performance, not cross-territory transfer: one checkpoint cannot distinguish the regimes without cross-territory training or an explicit territory token.Training in the non-dimensional variables of Eq. (7) remained untested and was identified as a promising candidate.Source: §5.5.3 · p. 36

6. Discussion

CARS's FFR disagreements are usually threshold crossings near 0.8, while its forward pass replaces only the flow solve, not the upstream geometry pipeline or meshing. The discussion bounds the evidence to the modeled solver and sampled geometries, and identifies further validation and broader supervision as steps toward use.

6.1. Structure of the endpoint error

  • 7/286 LCA and 11/293 RCA cases were discordant, with most within ±0.05 of FFR = 0.8; an LCA false positive reflected a near-uniform pressure offset, not a focal geometry miss.Velocity also degrades faster than pressure under anchor thinning and reduced precision, and does not satisfy continuity closely enough for differentiation.Source: §6.1 · p. 36

6.2. Position of the surrogate in the end-to-end cost

  • 87 ms is the forward-pass time, but image-to-result time remains dominated by segmentation, centerline extraction, lesion localization, and volume meshing.The reported 6,600–12,000× speed-up compares the flow-solution stage; CARS replaces the solve on a given mesh, not meshing.Source: §6.2 · pp. 36–37

6.3. Limitations

  • The target is a steady, laminar, rigid-wall, Newtonian OpenFOAM pipeline with prescribed outlet flow splits, not patient physiology.Pulsatility, transitional jets, compliance, microvascular resistance, and supervised wall shear stress are outside the stated supervision.Source: §6.3 · p. 38
  • Reported errors measure interpolation over the sampled parameter set, not transfer to unseen anatomies or acquisition settings; trajectories recur across splits, and some test cases differ from training only in flow rate.An anatomy-disjoint LCA split sharply degrades both CARS and AB-UPT; CARS remains more accurate on most of eleven test anatomies, with a patient-clustered interval spanning a factor of two.Source: §6.3 · p. 38
  • The cohort's tubular construction and injective geometry encoding do not establish how much geometry a surrogate needs; independently segmented anatomy may introduce an unmeasured fiber-spread error floor.Stenoses are radial changes to the radius profile, so modeled trans-lesional loss reflects caliber rather than plaque morphology.Source: §6.3 · pp. 38–39
  • Inputs must be approximately aligned to the training cohort's frame, and fixed outlet fluxes prevent lesions from diverting flow into parallel branches.The prescribed split simplifies the map and creates a severity-dependent pressure-drop bias; baseline comparisons also retain five stated asymmetries.Source: §6.3 · p. 39

6.4. Extensions

  • The input contract may fit other branching or tubular domains with a deterministic skeleton and caliber profile, but this reach is proposed rather than demonstrated.Airway trees, duct networks, fractured media, and heat-exchanger manifolds would each need reference data and a training run; extensions to richer physics likewise require new reference data.Source: §6.4 · pp. 39–40

7. Conclusion

CARS uses centerline-anchored attention to predict pressure and velocity on coronary lumens, and its pressure-derived FFR reproduces the solver's threshold decision in most held-out cases. The conclusion supports geometry-defined focal sets for the evaluated coronary task while identifying validation and transfer boundaries.

7. Conclusion

  • CARS runs in 87 ms per case and is the most accurate compared configuration at parameter counts matched within 3%, while reproducing the solver's FFR ≤0.8 decision in 97.7% of LCA and 96.3% of RCA held-out cases.The comparison includes four neural-operator baselines and alternative focal representations.Source: §7 · p. 40
  • The results support a geometry-defined focal set for this surrogate, but the ablations do not isolate focal-set provenance from the geometric information its tokens carry.The conclusion says the margin is neither a density effect nor a parameter-count effect.Source: §7 · p. 40
  • Other branching or tubular domains are candidate applications, but coronary flow is the only evaluated domain and meeting the input contract does not guarantee cross-cohort transfer.The conclusion identifies invasive FFR calibration and broader geometry distributions as extensions toward clinical use.Source: §7 · p. 40

CRediT authorship contribution statement

The contribution statement credits Reza Akbarian Bafghi and Sukirt Thakur with core technical and research work, and Maziar Raissi with conceptual and supervisory roles.

CRediT authorship contribution statement

Declaration of competing interest

The authors disclose that Sukirt Thakur is employed by AngioInsight, Inc., and report no other competing interests or relationships.

Declaration of competing interest

  • Sukirt Thakur is employed by AngioInsight, Inc., a company active in coronary hemodynamics; the authors declare no other competing interests or relationships.The authors also state that no invasive measurements, identifiable patient data, or proprietary clinical pipeline entered the work.Source: Declaration of competing interest · p. 41

Funding

The paper reports that it received no specific grant funding.

Funding

  • The research received no specific grant from funding agencies in the public, commercial, or not-for-profit sectors.Source: Funding · p. 41

Appendix A.1. Architecture hyperparameters

This appendix specifies CARS’s defaults, training objective, and optimization setup, then examines computational scaling and design choices. It also reports configuration comparisons and variability across runs.

Appendix A.1. Architecture hyperparameters

Appendix A.2. Loss and optimization

  • The normalized-space loss weights pressure once and velocity three times, while pressure remains the primary endpoint; batch-size-1 training instability is unresolved.The authors note that the pressure–velocity trade-off may partly reflect this weighting and that the learning rate was not jointly tuned with batch size.Source: Appendix A.2 · pp. 50–51

Appendix A.3. Computational complexity, analytic and measured

  • Measured scaling is near-linear when Stage-2 anchors are fixed, but approaches quadratic when every evaluation point is also an anchor.The accuracy configuration uses 16,384 Stage-2 anchors; the appendix reports no approximation bound in focal count.Source: Appendix A.3 · pp. 51–52, Table A.2 · p. 52

Appendix A.4. 3D rotary position embeddings

  • RoPE3D encodes relative 3D coordinate displacement without learnable parameters, but its inherited frequency scale is poorly matched to normalized coordinates.Most rotation channels remain at small angles across the domain, confounding the learned-encoding comparison with frequency-scale choice.Source: Appendix A.4 · pp. 52–53

Appendix A.5. Hyperparameter sweep

  • Table A.4 varies training length, depth, and width one at a time from the 100-epoch learned-position-encoding reference.The corresponding 300-epoch row has test relative ℓ2 errors of 0.116 / 0.146 for pressure / velocity.Source: Appendix A.5 · p. 53, Table A.4 · p. 54

Appendix B.1. What each ablation arm receives

  • The learned-anchor ablation removes centerline coordinates and radii and makes Stage-1 positional encoding inactive, so it changes multiple inputs and design choices.The reported 0.205 → 0.385 pressure delta bounds their joint effect from above and attributes none of it to anchors alone.Source: Appendix B.1 · p. 54

Appendix B.2. Reference configuration

  • The paired positional-encoding run reduces relative errors from 0.199/0.269 to 0.140/0.196, or 30%/27%, for pressure/velocity.The component-ablation reference uses a separate RoPE3D run at 0.205/0.287.Source: Appendix B.2 · pp. 54–55

Appendix B.3. Run-to-run variance

  • Across three 300-epoch seeds, the primary configuration gives test errors of 0.1198 ± 0.0032 and 0.1491 ± 0.0047 for pressure and velocity.AB-UPT has greater pressure spread, and both single-run rows happen to be their arm’s best seed.Source: Appendix B.3 · p. 55, Table B.2 · p. 56

Appendix B.4. Distribution of the per-case error and focal count

This appendix examines per-case error distributions, sensitivity to focal count, and the effect of duplicate geometries in the test split.

Appendix B.4. Distribution of the per-case error and focal count

  • Table B.3 reports the per-case relative ℓ2 error distribution, with the mean as the headline figure because the distribution is right-skewed.Source: Table B.3 · p. 56, Appendix B.4 · p. 56
  • Table B.4 tests focal-count sensitivity at fixed trained weights by thinning the 2,048-point centerline while keeping the total evaluation budget at 16,384 points.It measures sensitivity to the queried focal density, not accuracy after retraining at each count.Source: Appendix B.4 · p. 56, Table B.4 · p. 57

Appendix B.5. Field accuracy excluding duplicate geometries

  • Removing duplicate geometries raises mean pressure error by 0.010 on LCA and 0.002 on RCA, without changing the paper’s conclusions.The 34 LCA and 22 RCA duplicates are easier cases that differ from training cases only in inlet flow rate.Source: Appendix B.5 · pp. 56–57, Table B.5 · p. 57

Appendix C. Neural-operator baselines: configurations

The appendix documents how neural-operator baselines were configured and details the protocol and comparisons used to assess FFR predictions. It also evaluates a one-dimensional reduced-order comparator, which performs unevenly across territories.

Appendix C. Neural-operator baselines: configurations

  • Baseline models are parameter-matched within 3% and use the same splits and preprocessing, but no per-baseline learning-rate or schedule search was performed.Batch size 1 may disadvantage architectures intended for larger batches; the reported margins over GNOT, ONO, and Transolver are described as upper bounds on architectural differences.Source: Appendix C · pp. 57–58, Table C.1 · p. 58

Appendix D.1. Lesion localization and measurement station

  • The FFR measurement station is placed 20 mm distal to the lesion on the same branch, or at its distal end when less vessel remains.The lesion is selected from eligible focal narrowings using healthy shoulders on both sides, with branch boundaries handled separately.Source: Appendix D.1 · p. 61

Appendix D.2. Exact-coordinate against nearest-point reading

  • Exact-coordinate pressure lookup improves full-cohort LCA correlation from 0.921 to 0.941, while changing no LCA threshold decisions and one RCA decision.The comparison holds the measurement station, lesion localization, and case counts fixed.Source: Appendix D.2 · p. 62, Table D.2 · p. 63

Appendix D.3. Physiologic-range subset and agreement analyses

  • The physiologic-range analysis includes 286 LCA and 293 RCA cases and reports patient-clustered confidence intervals because test cases share segmentations.Excluding cases outside reference FFR 0.3–1.0 does not change the paper’s conclusions.Source: Appendix D.3 · p. 63

Appendix D.4. One-dimensional blood-flow reduced-order model

  • The 1D ROM reaches Pearson r = 0.19/0.56 and threshold accuracy 49.0%/77.1% on LCA/RCA, versus majority-class baselines of 78.7%/68.3%.It is presented as a physics sanity check, not a comparison against modern 1D solvers; the authors identify a stenosis-loss closure as a major missing comparator.Source: Appendix D.4 · pp. 65–66

Appendix D.5. Same-data scalar-regression baseline

The appendix compares same-data scalar regressors with CARS, reports MC-dropout triage and input-perturbation protocols, and examines equation consistency and cross-territory transfer.

Appendix D.5. Same-data scalar-regression baseline

  • The Itu-style scalar MLP and Scalar XGB use 15 hand-engineered features and the same CFD-derived labels and splits as CARS, but their AUCs are not comparable to clinical-cohort literature.The literature evaluates against invasive FFR, whereas these baselines use CFD labels on synthetic geometry and flow-rate samples.Source: Appendix D.5 · p. 66, Table D.3 · p. 67

Appendix D.6. Sensitivity and triage tables

  • MC-dropout intervals are overconfident: 1σ coverage is 52% on LCA and 44% on RCA, versus calibrated 68%, while 2σ coverage is 0.68/0.69 versus nominal 0.95.The passage says calibration would require a different uncertainty formulation, not rescaling; RCA cutoff transfer is less cleanly supported by its heavier-tailed validation σ distribution.Source: Appendix D.6 · pp. 66–67, Table D.4 · p. 67
  • Input-noise tests perturb centerline positions and radii independently, while segment-level tests apply one bias across a random contiguous 10-mm arc-length window.The σ = 0 row is the within-table baseline for both protocols; the segment perturbation seed is fixed across σ values.Source: Table D.5 · p. 68, Table D.6 · p. 69

Appendix E. Error structure, invariances and transfer

  • Appendix E lists binned error, frame-sensitivity, PDE-consistency, and zero-shot LCA-to-RCA transfer measurements, but the supplied passages give no corresponding outcome values.The table captions identify the scope of these measurements rather than their numerical results.Source: Appendix E · p. 70
  • PDE-consistency measurements use coordinate derivatives, with a 0.05 mm finite-difference stencil for the Laplacian; points near the wall can place stencil arms outside the supervised lumen.The reported all-points momentum residual is inflated by roughly 1.7× by such points, while divergence uses exact first derivatives.Source: Appendix E · p. 70

Appendix E.1. An anatomy-disjoint partition of the same cases

The anatomy-disjoint experiment tests generalization to unseen vessel trees, a harder question than parameter interpolation on familiar trees. Its small anatomy count leaves pressure uncertainty broad, while the CFD labels also reflect a specified steady-flow setup.

Appendix E.1. An anatomy-disjoint partition of the same cases

  • The paper says this cohort cannot settle unseen-anatomy generalization with useful precision, making a larger segmentation cohort necessary to resolve the pressure comparison.The anatomy-disjoint experiment reuses existing CFD cases rather than rerunning simulations.Source: Appendix E.1 · pp. 72, 74
  • The partition assigns all cases from each ImageCAS segmentation to one split, leaving 83 training and 11 test anatomies; the 373 test cases are therefore not independent observations.Resampling the 11 anatomies gives a 95% interval of [0.44, 0.93] for CARS pressure error, versus the misleadingly narrower case-level interval [0.61, 0.71].Source: Appendix E.1 · pp. 72–74, Table E.6 · p. 74
  • On the anatomy-disjoint split, both models degrade sharply, and neither predicts unseen-tree flow well; CARS has lower error on both fields, winning on 8 of 11 anatomies for pressure and 10 of 11 for velocity.The paired margin over AB-UPT is resolved for velocity but not pressure, and CARS’s pressure-error interval spans a factor of 2.1.Source: Appendix E.1 · pp. 73–74, Table E.6 · p. 74

Appendix F. Upstream CFD pipeline

  • Reference fields come from steady, rigid-wall, no-slip simulations with Newtonian blood, excluding pulsatile and transient phenomena such as wave reflection and instantaneous wave-free ratio.The simulations use prescribed terminal fluxes and a fixed-value inlet pressure condition.Source: Appendix F · p. 75
  • The CFD cases sample inlet flow rate independently of geometry, so FFR ≤0.8 prevalence reflects the sampling design rather than a clinical population.The imposed flow spans resting and hyperemic ranges, and CARS receives Q through its FiLM pathway.Source: Appendix F · pp. 75–76

Appendix F.1. Grid convergence of the reference solutions

The grid study refines production geometries across three levels to estimate reference-solution discretization error. Pressure convergence is near the formal second order, whereas velocity estimates are less reliable because observed orders exceed that regime.

Appendix F.1. Grid convergence of the reference solutions

  • The three refinement levels use 160, 324, and 544 cells per cross-section, with unequal realized ratios; the study therefore uses an iterative non-constant-ratio GCI formulation.The domain changes slightly with resolution because circular sections are represented by inscribed polygons, whose volume deficit converges at second order.Source: Appendix F.1 · pp. 77–78
  • The pressure observed-order median is 2.51, near formal second order, while the velocity median is 3.28 and makes its GCI an order-of-magnitude indication rather than a reliable asymptotic estimate.Two cases with observed orders near 0.48 were retained rather than excluded based on outcome.Source: Appendix F.1 · p. 79
  • GCI and surrogate errors use different norms, and the global volume-weighted GCI may understate pointwise error through cancellation; peri-stenotic restricted norms were not computed.The surrogate errors are unweighted over cell centers, which overweights near-wall regions relative to the GCI norm.Source: Appendix F.1 · p. 79
  • LCA has 1.5 times RCA’s pressure-norm error and 3.2 times its FFR error, with a heavier upper tail.The comparison is reported as the substantive territory difference in the grid study.Source: Appendix F.1 · p. 80
  • At the recorded distal endpoint, median FFR GCI is 0.055%, compared with 0.41% for the domain-minimum proxy, though the endpoint analysis includes only 12 cases.The endpoint is unavailable in three cases, and one additional case has no usable refinement triple.Source: Appendix F.1 · p. 80
  • Production-resolution FFR is low by 0.005 at the median against the Richardson extrapolate, and the worst 0.114 case is non-asymptotic, making it an upper envelope on the tail.The paper attributes the systematic direction to numerical dissipation and the inscribed-polygon lumen deficit; the surrogate inherits the common label bias.Source: Appendix F.1 · pp. 80–81

Appendix G. Further limitations of the evaluation

The appendix collects further evaluation limitations and identifies measurements that would remove them.

Appendix G. Further limitations of the evaluation

  • Table G.1 organizes further evaluation limitations by the claims they bound and the measurements needed to address them.The supplied passages do not specify the individual limitations or measurements.Source: Appendix G · p. 81, Table G.1 · p. 82

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