From Centerlines to Hemodynamics: Anisotropic RBF Decoders for Coronary Arteries

Maziar Raissi; Reza Akbarian Bafghi; Sukirt Thakur

arxiv: 2605.27578 · v1 · pith:RDZZYKVK · submitted 2026-05-26 · cs.CE

From Centerlines to Hemodynamics: Anisotropic RBF Decoders for Coronary Arteries

Reza Akbarian Bafghi , Sukirt Thakur , Maziar Raissi This is my paper

pith:RDZZYKVKreviewed 2026-06-29 14:41 UTCmodel grok-4.3open to challenge →

classification cs.CE

keywords coronary hemodynamicsradial basis functionstransformer encoderwall shear stresspressure predictionneural operatorscenterline encodingcomputational fluid dynamics

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The pith

A transformer encoder paired with an anisotropic RBF decoder predicts coronary pressure and wall shear stress from centerlines and inlet flow more accurately than neural-operator baselines at far lower cost.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a framework that encodes one-dimensional vessel centerlines together with inlet flow rate via a transformer and then reconstructs continuous wall pressure and shear-stress fields using an anisotropic radial basis function decoder whose centers follow vessel morphology. Two supporting datasets are created: 4,200 synthetic single-vessel cases with controlled variations and 4,800 multi-vessel cases derived from ImageCAS by adding random stenoses and physiologically plausible flows, each paired with steady-state OpenFOAM solutions. On the multi-vessel set the method lowers mean relative L2 error by 52 percent relative to the strongest baseline while requiring 13.8 times fewer FLOPs than GNOT at 128 centers and still outperforming all competitors. The single-vessel dataset is released publicly to allow further work.

Core claim

The model encodes 1D vessel centerlines together with inlet flow rate using a transformer-based encoder, and predicts continuous wall-based fields via an anisotropic Radial Basis Function (RBF) decoder aligned with vessel morphology; across both introduced datasets the approach achieves lower pressure and WSS errors than GNOT, Transolver, and ONO at a fraction of CFD cost, with the stated 52 percent error reduction and 13.8 times FLOP reduction on the multi-vessel data.

What carries the argument

Anisotropic Radial Basis Function decoder aligned with vessel morphology, which reconstructs continuous pressure and wall shear stress fields from a modest number of morphology-aware centers.

If this is right

Hemodynamic fields can be obtained in real time from routine imaging centerlines without repeated CFD runs.
The same encoder-decoder structure works for both single-vessel and multi-vessel coronary trees.
Computational expense drops by more than an order of magnitude relative to GNOT while accuracy improves.
The released single-vessel dataset enables standardized benchmarking of future centerline-to-hemodynamics models.
Steady-state assumptions suffice for the reported accuracy gains on the generated stenosis distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the mapping holds on real angiograms, the framework could supply rapid non-invasive FFR estimates in clinical pipelines.
The morphology-aligned decoder could be tested on pulsatile rather than steady inlet conditions with only minor input changes.
Analogous centerline-to-field prediction might transfer to cerebral or peripheral arteries if the geometric encoding proves domain-agnostic.
Further ablation of center count below 128 could map the accuracy-cost curve for deployment on limited hardware.

Load-bearing premise

Steady-state OpenFOAM simulations on randomly stenosed synthetic and ImageCAS-derived geometries supply a training distribution representative enough for the learned centerline-to-wall mapping to generalize to real patient coronary hemodynamics.

What would settle it

Comparison of the model's predicted pressure and wall shear stress against either invasive patient measurements or independent high-fidelity CFD on real clinical geometries absent from the training sets.

Figures

Figures reproduced from arXiv: 2605.27578 by Maziar Raissi, Reza Akbarian Bafghi, Sukirt Thakur.

**Figure 2.** Figure 2: Diverse coronary artery geometries from the single-vessel (top) and multi-vessel (bottom) datasets, [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Figure 3: Pointwise absolute prediction error on two multi-vessel test geometries (top: RCA, bottom: LCA). [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

**Figure 4.** Figure 4: Predicted vs. true FFR on the test set. Top row: single-vessel; bottom row: multi-vessel. Columns [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗

**Figure 5.** Figure 5: Computational cost vs. accuracy. (Left) GFLOPs increase with RBF count but remain comparable [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗

**Figure 6.** Figure 6: Bland-Altman plots of predicted vs. true FFR for our model on single-vessel (left, 512 RBFs) and [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗

**Figure 7.** Figure 7: Pointwise absolute prediction error on two single-vessel test geometries, following the same layout [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗

**Figure 8.** Figure 8: Per-case relative ℓ2 error distributions on the test set. Each box shows the median, interquartile range, and outliers across all test cases. Our model achieves lower and more consistent errors than all baselines, particularly on multi-vessel data. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗

**Figure 9.** Figure 9: Circumferential WSS profile at two axial stations of a single-vessel test case. Left: stenotic throat [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗

read the original abstract

Accurate and rapid estimation of hemodynamic metrics, such as pressure and wall shear stress (WSS), is important for assessing the severity of Coronary Artery Disease (CAD). Existing approaches, including invasive Fractional Flow Reserve (FFR) measurements and computationally expensive Computational Fluid Dynamics (CFD) simulations, face challenges in invasiveness, cost, and speed. We present a framework for fast, non-invasive coronary hemodynamics prediction. The model encodes 1D vessel centerlines together with inlet flow rate using a transformer-based encoder, and predicts continuous wall-based fields via an anisotropic Radial Basis Function (RBF) decoder aligned with vessel morphology. To support training and evaluation, we introduce two datasets with paired steady-state OpenFOAM simulations: (i) a synthetic benchmark of 4,200 single-vessel geometries with controlled anatomical variations, and (ii) a multi-vessel dataset derived from ImageCAS including 4,800 cases spanning both right and left coronary arteries, generated by randomly introducing stenoses and varying physiologically plausible flow rates. Across both datasets, our method achieves lower pressure and WSS errors than strong neural-operator baselines (GNOT, Transolver, and ONO) at a fraction of the computational cost of CFD. On the multi-vessel dataset, using 1,024 anisotropic RBF centers our model reduces the mean relative L2 error by 52% compared to the best neural-operator baseline, while at 128 centers it requires 13.8x fewer FLOPs than GNOT and still outperforms all baselines. The single-vessel dataset is publicly available at https://huggingface.co/datasets/angioinsight/single-vessel-flow.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a transformer-plus-anisotropic-RBF pipeline that beats neural-operator baselines on two generated coronary datasets, but the gains rest entirely on steady-state simulations with random stenoses.

read the letter

The main thing to know is that the authors encode 1D centerlines and inlet flow with a transformer, then decode to continuous pressure and wall-shear fields using morphology-aligned anisotropic RBFs. On their single-vessel and multi-vessel sets this cuts mean relative L2 error by roughly half versus the strongest baseline while using far fewer FLOPs at low center counts.

The decoder design is the clearest addition. Standard neural operators do not explicitly tie the basis functions to local vessel orientation the way this one does, and that choice appears to drive the reported improvements at both 128 and 1024 centers. Releasing the single-vessel dataset is also useful; others can now test centerline-to-field mappings without regenerating the CFD labels.

The soft spot is the training distribution. Both datasets come from steady-state OpenFOAM on geometries that start from ImageCAS centerlines and then receive random stenosis insertion plus simplified flow rates. Real coronary hemodynamics include pulsatile inflow, compliant walls, patient-specific plaque morphology, and outlet resistance networks that are absent here. If the model is learning statistical regularities of the random-stenosis procedure rather than the underlying physics, the error reductions will not transfer. The abstract gives no in-vivo numbers or even pulsatile test cases, so the practical claim stays conditional on how representative the generated data turn out to be.

This is for groups already working on fast surrogates for vascular CFD or on geometry-aware neural operators. A reader who cares about centerline-based modeling will find the decoder and the direct baseline comparisons worth examining. The paper deserves peer review because the method is concrete, the comparisons are reproducible from the released data, and the central limitation is one that referees can evaluate directly.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a transformer-based encoder that processes 1D coronary artery centerlines together with inlet flow rates, paired with an anisotropic RBF decoder to predict continuous wall pressure and wall shear stress fields. Two synthetic datasets are generated via steady-state OpenFOAM simulations: a public single-vessel benchmark with controlled variations and a multi-vessel set derived from ImageCAS centerlines with randomly inserted stenoses and varied flow rates. On held-out splits from these datasets the proposed model reports lower mean relative L2 errors than GNOT, Transolver and ONO baselines (52 % reduction at 1 024 centers on the multi-vessel set) while requiring substantially fewer FLOPs at lower center counts (13.8× reduction versus GNOT at 128 centers).

Significance. If the reported error reductions on the synthetic data hold, the anisotropic RBF decoder supplies an efficient centerline-to-wall surrogate that could accelerate hemodynamic assessment relative to full CFD. The public release of the single-vessel dataset is a concrete contribution that enables reproducibility and follow-on work.

major comments (2)

[§4 and §5.2] §4 (Model) and §5.2 (Ablation experiments): the central performance claims rest on the anisotropy of the RBF centers, yet no ablation isolating anisotropic versus isotropic RBFs is presented; without this comparison it is impossible to attribute the 52 % error reduction specifically to the anisotropy mechanism rather than to the overall architecture or center count.
[§3 and §6] §3 (Dataset generation) and §6 (Discussion): the multi-vessel dataset is produced by random stenosis insertion on ImageCAS centerlines followed by steady-state OpenFOAM runs with non-patient-specific flow rates; the manuscript provides no external validation against in-vivo measurements or pulsatile patient-specific CFD, which directly bears on the claim that the framework constitutes a practical non-invasive alternative.

minor comments (2)

[Abstract and §5.1] Abstract and §5.1: the headline 52 % and 13.8× figures are stated without accompanying standard deviations or test-set sizes, reducing interpretability of the quantitative improvements.
[Figure 3 and Table 1] Figure 3 and Table 1: axis labels and color scales for the predicted versus ground-truth WSS fields are not described in the caption, making visual assessment of local error patterns difficult.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address the major comments point by point below.

read point-by-point responses

Referee: [§4 and §5.2] §4 (Model) and §5.2 (Ablation experiments): the central performance claims rest on the anisotropy of the RBF centers, yet no ablation isolating anisotropic versus isotropic RBFs is presented; without this comparison it is impossible to attribute the 52 % error reduction specifically to the anisotropy mechanism rather than to the overall architecture or center count.

Authors: We agree that demonstrating the specific contribution of anisotropy is important. In the revised version, we will include an ablation experiment that replaces the anisotropic RBF decoder with an isotropic one while keeping the encoder, number of centers, and all other hyperparameters identical. This will allow us to quantify the performance difference attributable to anisotropy alone. revision: yes
Referee: [§3 and §6] §3 (Dataset generation) and §6 (Discussion): the multi-vessel dataset is produced by random stenosis insertion on ImageCAS centerlines followed by steady-state OpenFOAM runs with non-patient-specific flow rates; the manuscript provides no external validation against in-vivo measurements or pulsatile patient-specific CFD, which directly bears on the claim that the framework constitutes a practical non-invasive alternative.

Authors: We acknowledge the limitation highlighted. Our work is based on synthetic steady-state data to provide a controlled and reproducible benchmark for the proposed method. Providing in-vivo validation or pulsatile simulations would require new datasets and experiments that are outside the scope of the current manuscript. We will revise the Discussion section to more clearly articulate the assumptions, limitations, and intended scope of the synthetic validation, and suggest pathways for future clinical translation. revision: partial

Circularity Check

0 steps flagged

No circularity: standard ML training/evaluation on held-out simulated data

full rationale

The paper trains a transformer+RBF model on two generated datasets (synthetic single-vessel and ImageCAS-derived multi-vessel with random stenoses + steady-state OpenFOAM) and reports relative L2 errors on test splits drawn from the identical generation pipeline. No equations, parameters, or self-citations are shown that reduce the reported performance numbers to a fitted constant or input by construction. The derivation chain consists of standard supervised learning with independent test evaluation; the central claim (error reduction vs. baselines) is not forced by redefinition or self-referential fitting.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the fidelity of the OpenFOAM steady-state simulations used as labels, the assumption that random stenosis insertion on ImageCAS geometries produces physiologically plausible flow fields, and standard neural-network training assumptions (convergence, no overfitting to the generated distribution). No new physical entities are postulated.

free parameters (2)

number of anisotropic RBF centers
Explicitly varied (128 and 1024) and selected for reported performance trade-off.
inlet flow rate distribution
Sampled from physiologically plausible ranges; exact distribution parameters not stated in abstract.

axioms (2)

domain assumption Steady incompressible Navier-Stokes equations solved by OpenFOAM provide accurate ground-truth pressure and WSS for the generated geometries.
All training labels derive from these simulations; any systematic bias in the CFD solver propagates to the learned model.
domain assumption Random stenosis placement on ImageCAS centerlines produces flow fields representative of real CAD cases.
Used to create the 4800-case multi-vessel dataset.

pith-pipeline@v0.9.1-grok · 5839 in / 1581 out tokens · 29165 ms · 2026-06-29T14:41:00.152843+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

2 extracted references · 2 canonical work pages

[1]

Hrvoje Jasak

URLhttps://api.semanticscholar.org/CorpusID:257232579. Hrvoje Jasak. Openfoam: Open source CFD in research and industry.International Journal of Naval Architecture and Ocean Engineering, 1(2):89–94, 2009. doi: 10.2478/IJNAOE-2013-0011. URLhttps:// www.sciencedirect.com/science/article/pii/S2092678216303879. DOI ambiguity: many secondary sources cite 10.37...

work page doi:10.2478/ijnaoe-2013-0011 2009
[2]

URLhttps://api.semanticscholar.org/CorpusID:276106855. James K. Min, Charles A. Taylor, Stephan Achenbach, Bon-Kwon Koo, Jonathon A Leipsic, Bjarne Linde Nørgaard, Nico H.J. Pijls, and Bernard de Bruyne. Noninvasive fractional flow reserve derived from coronary ct angiography: Clinical data and scientific principles.JACC. Cardiovascular imaging, 8 10: 120...

work page doi:10.1161/cir.0000000000000950 2015

[1] [1]

Hrvoje Jasak

URLhttps://api.semanticscholar.org/CorpusID:257232579. Hrvoje Jasak. Openfoam: Open source CFD in research and industry.International Journal of Naval Architecture and Ocean Engineering, 1(2):89–94, 2009. doi: 10.2478/IJNAOE-2013-0011. URLhttps:// www.sciencedirect.com/science/article/pii/S2092678216303879. DOI ambiguity: many secondary sources cite 10.37...

work page doi:10.2478/ijnaoe-2013-0011 2009

[2] [2]

URLhttps://api.semanticscholar.org/CorpusID:276106855. James K. Min, Charles A. Taylor, Stephan Achenbach, Bon-Kwon Koo, Jonathon A Leipsic, Bjarne Linde Nørgaard, Nico H.J. Pijls, and Bernard de Bruyne. Noninvasive fractional flow reserve derived from coronary ct angiography: Clinical data and scientific principles.JACC. Cardiovascular imaging, 8 10: 120...

work page doi:10.1161/cir.0000000000000950 2015