pith. sign in

arxiv: 2604.06287 · v1 · submitted 2026-04-07 · 💻 cs.LG · cs.NA· math.NA· physics.comp-ph· physics.flu-dyn

Asymptotic-Preserving Neural Networks for Viscoelastic Parameter Identification in Multiscale Blood Flow Modeling

Giulia Bertaglia , Raffaella Fiamma Cabini This is my paper

Pith reviewed 2026-05-10 19:18 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NAphysics.comp-phphysics.flu-dyn
keywords viscoelastic parametersasymptotic-preserving neural networksmultiscale blood flowparameter identificationDoppler ultrasoundarterial wallspressure estimationcardiovascular modeling
0
0 comments X

The pith

Asymptotic-preserving neural networks identify viscoelastic parameters of arteries from ultrasound area and velocity data while reconstructing pressure.

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

The paper demonstrates that embedding the equations of a one-dimensional multiscale blood flow model into asymptotic-preserving neural networks lets the system infer the unknown viscoelastic parameters that govern arterial wall deformation. At the same time the networks reconstruct the full time-dependent state variables and produce pressure estimates. This matters because it converts standard, non-invasive Doppler ultrasound measurements of vessel cross-section and blood speed into quantities that cannot be measured directly in patients, making the model usable in realistic clinical scenarios.

Core claim

By incorporating the governing physical principles of the one-dimensional multiscale viscoelastic blood flow model into asymptotic-preserving neural networks, the framework simultaneously infers the viscoelastic parameters controlling arterial deformation and reconstructs the time-dependent evolution of the state variables, thereby estimating pressure waveforms from cross-sectional area and velocity measurements obtained via Doppler ultrasound in vascular segments where direct pressure data are unavailable.

What carries the argument

Asymptotic-preserving neural networks that embed the governing equations of the one-dimensional multiscale viscoelastic blood flow model.

If this is right

  • Pressure waveforms become estimable from routine Doppler ultrasound without needing invasive pressure sensors.
  • The approach applies successfully to both synthetic test cases and real patient-specific data sets.
  • It removes the main practical barrier to using the viscoelastic blood flow model in personalized cardiovascular simulations.
  • State variables and parameters are recovered together rather than in separate steps.

Where Pith is reading between the lines

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

  • The same embedding technique could be tested on other multiscale physiological systems where parameters are difficult to measure directly.
  • If the method remains stable under realistic measurement noise, it could support real-time clinical tools that update arterial properties during an ultrasound exam.
  • Extending the framework to include additional data sources such as ECG timing might further constrain the parameter search.

Load-bearing premise

The one-dimensional multiscale viscoelastic blood flow model accurately represents arterial wall behavior and the neural network embeds its equations faithfully enough to identify the parameters stably and uniquely from area and velocity data alone.

What would settle it

Generate synthetic area and velocity time series from the exact model with known viscoelastic parameters, run the network, and check whether the recovered parameters match the ground-truth values within the expected numerical tolerance.

Figures

Figures reproduced from arXiv: 2604.06287 by Giulia Bertaglia, Raffaella Fiamma Cabini.

Figure 1
Figure 1. Figure 1: Asymptotic limits of the multiscale constitutive framework. By [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Feedforward Neural Network architecture. The network maps [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: PINN architecture. The network incorporates both data-driven and [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Schematic illustration of the concept of asymptotic-preservation [PITH_FULL_IMAGE:figures/full_fig_p013_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Schematic representation of the APNN framework. The neural [PITH_FULL_IMAGE:figures/full_fig_p014_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Synthetic TA test results at the vessel midpoint. The first panel [PITH_FULL_IMAGE:figures/full_fig_p020_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Synthetic TA test results over the full spatial domain. The first [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Training progress of the APNN for the synthetic dataset. The [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: In vivo test results at the vessel measurement point for three dif [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: In vivo test results on the full spatial domain for three carotid [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Training progress of the APNN for the three carotid arteries [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
read the original abstract

Mathematical models and numerical simulations offer a non-invasive way to explore cardiovascular phenomena, providing access to quantities that cannot be measured directly. In this study, we start with a one-dimensional multiscale blood flow model that describes the viscoelastic properties of arterial walls, and we focus on improving its practical applicability by addressing a major challenge: determining, in a reliable way, the viscoelastic parameters that control how arteries deform under pulsatile pressure. To achieve this, we employ Asymptotic-Preserving Neural Networks that embed the governing physical principles of the multiscale viscoelastic blood flow model within the learning procedure. This framework allows us to infer the viscoelastic parameters while simultaneously reconstructing the time-dependent evolution of the state variables of blood vessels. With this approach, pressure waveforms are estimated from readily accessible patient-specific data, i.e., cross-sectional area and velocity measurements from Doppler ultrasound, in vascular segments where direct pressure measurements are not available. Different numerical simulations, conducted in both synthetic and patient-specific scenarios, show the effectiveness of the proposed methodology.

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.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes Asymptotic-Preserving Neural Networks (APNNs) that embed the governing equations of a one-dimensional multiscale viscoelastic blood flow model to simultaneously identify viscoelastic parameters and reconstruct time-dependent state variables (cross-sectional area A(t) and velocity u(t)) from Doppler ultrasound measurements. This enables non-invasive estimation of pressure waveforms in vascular segments where direct pressure data are unavailable, with demonstrations claimed in both synthetic and patient-specific scenarios.

Significance. If the central claims hold with rigorous validation, the work could offer a practical physics-informed approach to parameter identification and state reconstruction in cardiovascular modeling, leveraging readily available ultrasound data to infer otherwise inaccessible pressures. The asymptotic-preserving embedding provides a potential advantage over standard PINNs for multiscale problems, and the focus on viscoelastic arterial wall properties addresses a relevant clinical modeling gap.

major comments (3)
  1. [Abstract and §4 (numerical results)] Abstract and numerical experiments section: The assertion that 'different numerical simulations... show the effectiveness of the proposed methodology' is unsupported by any reported quantitative metrics (e.g., parameter recovery errors, state reconstruction RMSE, pressure waveform correlations), error bars, or baseline comparisons against standard optimization or non-asymptotic-preserving networks, leaving the central claim of reliable identification without evidence.
  2. [§2–3 (model and APNN formulation)] Method and inverse problem formulation (likely §2–3): No analysis or numerical test is provided to establish identifiability or stability of the viscoelastic parameters (e.g., wall viscosity coefficients) from A(t) and u(t) data alone. The tube-law and momentum equations couple pressure to area via these parameters, and the APNN loss enforces the forward PDEs but supplies no regularization, observability analysis, or synthetic tests with noise to rule out non-uniqueness under pulsatile forcing.
  3. [§3 (APNN architecture and training)] Validation of asymptotic-preserving property: The manuscript does not detail how the AP property is enforced (e.g., specific terms in the loss function, scaling with the small parameter, or limit behavior tests) nor provide validation that the network recovers the correct asymptotic regime of the multiscale model as the relevant parameter approaches zero.
minor comments (2)
  1. [§2] Notation for the viscoelastic parameters and the tube law should be introduced with explicit definitions and units in the model section to improve readability for readers outside the immediate subfield.
  2. [§4 (figures)] Figure captions for synthetic and patient-specific results should include the specific error metrics or parameter values recovered, rather than relying solely on visual comparison.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments, which highlight important aspects for strengthening the manuscript. We address each major comment point by point below and will revise the paper accordingly to provide the requested evidence and clarifications.

read point-by-point responses

  1. Referee: Abstract and §4 (numerical results)] Abstract and numerical experiments section: The assertion that 'different numerical simulations... show the effectiveness of the proposed methodology' is unsupported by any reported quantitative metrics (e.g., parameter recovery errors, state reconstruction RMSE, pressure waveform correlations), error bars, or baseline comparisons against standard optimization or non-asymptotic-preserving networks, leaving the central claim of reliable identification without evidence.

    Authors: We agree that the current numerical results section and abstract do not include sufficient quantitative metrics or comparisons, which weakens the support for the effectiveness claims. In the revised manuscript, we will expand §4 with explicit metrics: relative errors and standard deviations for recovered viscoelastic parameters (e.g., wall viscosity coefficients), RMSE for reconstructed A(t) and u(t) across synthetic and patient-specific cases, and correlation coefficients for inferred pressure waveforms. We will also add error bars from multiple independent training runs and direct comparisons against standard PINNs (without asymptotic preservation) and conventional least-squares optimization methods applied to the same data. revision: yes

  2. Referee: [§2–3 (model and APNN formulation)] Method and inverse problem formulation (likely §2–3): No analysis or numerical test is provided to establish identifiability or stability of the viscoelastic parameters (e.g., wall viscosity coefficients) from A(t) and u(t) data alone. The tube-law and momentum equations couple pressure to area via these parameters, and the APNN loss enforces the forward PDEs but supplies no regularization, observability analysis, or synthetic tests with noise to rule out non-uniqueness under pulsatile forcing.

    Authors: The referee is correct that the manuscript currently lacks dedicated identifiability or stability analysis. Although the physics-informed loss couples the parameters to the governing equations, this alone does not rigorously establish uniqueness or robustness. In the revision, we will add a new subsection in §3 with synthetic experiments that inject controlled Gaussian noise (1–5% levels) into the A(t) and u(t) measurements under pulsatile conditions. These tests will quantify parameter recovery stability, include an observability discussion based on the tube-law and momentum balance, and explore simple regularization (e.g., parameter bounds or smoothness penalties) if non-uniqueness appears in certain regimes. Results will be reported transparently, including any limitations. revision: yes

  3. Referee: [§3 (APNN architecture and training)] Validation of asymptotic-preserving property: The manuscript does not detail how the AP property is enforced (e.g., specific terms in the loss function, scaling with the small parameter, or limit behavior tests) nor provide validation that the network recovers the correct asymptotic regime of the multiscale model as the relevant parameter approaches zero.

    Authors: We acknowledge that the description of the asymptotic-preserving enforcement is insufficiently detailed. In the revised §3, we will explicitly specify the loss terms that incorporate the small-parameter scaling (including how the network architecture is modified to respect the asymptotic limit) and provide the precise formulation used during training. We will also add dedicated validation experiments in §4 that systematically reduce the relevant small parameter (e.g., wall stiffness or viscosity scaling) to zero and compare the network outputs against the analytically known reduced-model solution, reporting quantitative convergence errors to confirm the AP property. revision: yes

Circularity Check

0 steps flagged

No significant circularity; method is a standard physics-informed inverse solver

full rationale

The paper presents an asymptotic-preserving neural network that embeds the 1D multiscale viscoelastic blood flow PDEs (tube law, momentum balance) into the loss to jointly reconstruct states and identify wall viscosity parameters from A(t) and u(t) data. This is a conventional PINN setup for inverse problems: the network is trained to minimize PDE residuals plus data mismatch, with parameters as trainable variables. Synthetic tests recover known ground-truth parameters (standard validation, not tautological), and patient-specific cases use independent Doppler measurements. No quoted step reduces the claimed inference to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The embedding supplies external physical constraints rather than assuming the target result. Identifiability concerns are a separate correctness issue, not circularity.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the starting 1D multiscale viscoelastic model and the capacity of the neural network architecture to preserve its asymptotic behavior while performing parameter inference from limited measurements.

free parameters (1)
  • viscoelastic parameters
    These are the target quantities identified by the network from data; their values are not known a priori and are learned during training.
axioms (1)
  • domain assumption The one-dimensional multiscale blood flow model with viscoelastic wall properties accurately captures arterial deformation under pulsatile pressure.
    The paper explicitly starts from this model and builds the neural network around its governing principles.

pith-pipeline@v0.9.0 · 5493 in / 1290 out tokens · 58631 ms · 2026-05-10T19:18:28.328403+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

What do these tags mean?
matches
The paper's claim is directly supported by a theorem in the formal canon.
supports
The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends
The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses
The paper appears to rely on the theorem as machinery.
contradicts
The paper's claim conflicts with a theorem or certificate in the canon.
unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

58 extracted references · 58 canonical work pages · 1 internal anchor

  1. [1]

    Aghaee and M

    A. Aghaee and M. O. Khan. Pinning down the accuracy of physics- informed neural networks under laminar and turbulent-like aortic blood flow conditions.Computers in Biology and Medicine, 185:109528, 2025

    work page 2025

  2. [2]

    Alastruey, A

    J. Alastruey, A. W. Khir, K. S. Matthys, P. Segers, S. J. Sherwin, P. R. Verdonck, K. H. Parker, and J. Peir´ o. Pulse wave propagation in a model human arterial network: assessment of 1-D visco-elastic 29 simulations against in vitro measurements.Journal of biomechanics, 44(12):2250–2258, 2011

    work page 2011

  3. [3]

    Alastruey, K

    J. Alastruey, K. Parker, J. Peir´ o, S. Byrd, and S. Sherwin. Modelling the circle of willis to assess the effects of anatomical variations and occlusions on cerebral flows.Journal of biomechanics, 40(8):1794–1805, 2007

    work page 2007

  4. [4]

    Alastruey, K

    J. Alastruey, K. H. Parker, and S. J. Sherwin. Arterial pulse wave haemodynamics. In11th international conference on pressure surges, pages 401–443. Virtual PiE Led t/a BHR Group, 2012

    work page 2012

  5. [5]

    Alastruey, T

    J. Alastruey, T. Passerini, L. Formaggia, and J. Peir´ o. Physical deter- mining factors of the arterial pulse waveform: theoretical analysis and calculation using the 1-D formulation.Journal of Engineering Mathe- matics, 77(1):19–37, 2012

    work page 2012

  6. [6]

    Barzaghi, F

    L. Barzaghi, F. Brero, R. F. Cabini, M. Paoletti, M. Monforte, F. Lizzi, F. Santini, X. Deligianni, N. Bergsland, S. Ravaglia, et al. Myo-regressor deep informed neural network (Myo-DINO) for fast mr parameters map- ping in neuromuscular disorders.Computer Methods and Programs in Biomedicine, 256:108399, 2024

    work page 2024

  7. [7]

    Bertaglia

    G. Bertaglia. Asymptotic-preserving neural networks for hyperbolic sys- tems with diffusive scaling. InAdvances in Numerical Methods for Hy- perbolic Balance Laws and Related Problems, pages 23–48. SEMA SIMAI Springer Series, 2023

    work page 2023

  8. [8]

    Bertaglia, V

    G. Bertaglia, V. Caleffi, and A. Valiani. Modeling blood flow in vis- coelastic vessels: the 1D augmented fluid–structure interaction system. Computer Methods in Applied Mechanics and Engineering, 360:112772, 2020

    work page 2020

  9. [9]

    Bertaglia, C

    G. Bertaglia, C. Lu, L. Pareschi, and X. Zhu. Asymptotic-preserving neural networks for multiscale hyperbolic models of epidemic spread. Mathematical Models and Methods in Applied Sciences, 32(10):1949– 1985, 2022

    work page 1949

  10. [10]

    Bertaglia, A

    G. Bertaglia, A. Navas-Montilla, A. Valiani, M. I. M. Garc´ ıa, J. Murillo, and V. Caleffi. Computational hemodynamics in arteries with the one- dimensional augmented fluid-structure interaction system: viscoelastic parameters estimation and comparison with in-vivo data.Journal of Biomechanics, 100:109595, 2020. 30

    work page 2020

  11. [11]

    Bertaglia and L

    G. Bertaglia and L. Pareschi. Multiscale constitutive framework of one-dimensional blood flow modeling: Asymptotic limits and numeri- cal methods.Multiscale Modeling & Simulation, 21(3):1237–1267, 2023

    work page 2023

  12. [12]

    Boileau, P

    E. Boileau, P. Nithiarasu, P. J. Blanco, L. O. M¨ uller, F. E. Fossan, L. R. Hellevik, W. P. Donders, W. Huberts, M. Willemet, and J. Alastruey. A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling.International journal for numerical methods in biomedical engineering, 31(10):e02732, 2015

    work page 2015

  13. [13]

    Boscarino, L

    S. Boscarino, L. Pareschi, and G. Russo. A unified imex runge–kutta ap- proach for hyperbolic systems with multiscale relaxation.SIAM Journal on Numerical Analysis, 55(4):2085–2109, 2017

    work page 2085

  14. [14]

    Boscarino, L

    S. Boscarino, L. Pareschi, and G. Russo.Implicit-Explicit Methods for Evolutionary Partial Differential Equations. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2024

    work page 2024

  15. [15]

    R. F. Cabini, L. Barzaghi, D. Cicolari, P. Arosio, S. Carrazza, S. Figini, M. Filibian, A. Gazzano, R. Krause, M. Mariani, M. Peviani, A. Pichiec- chio, D. U. Pizzagalli, and A. Lascialfari. Fast deep learning reconstruc- tion techniques for preclinical magnetic resonance fingerprinting.NMR in Biomedicine, 37(1):e5028, 2024

    work page 2024

  16. [16]

    C. G. Caro.The mechanics of the circulation. Cambridge University Press, 2012

    work page 2012

  17. [17]

    Chan and L

    T. Chan and L. Vese. An active contour model without edges. InIn- ternational conference on scale-space theories in computer vision, pages 141–151. Springer, 1999

    work page 1999

  18. [18]

    F. A. Choudhry, J. T. Grantham, A. T. Rai, and J. P. Hogg. Vascular ge- ometry of the extracranial carotid arteries: an analysis of length, diam- eter, and tortuosity.Journal of neurointerventional surgery, 8(5):536– 540, 2016

    work page 2016

  19. [19]

    Coccarelli, J

    A. Coccarelli, J. M. Carson, A. Aggarwal, and S. Pant. A framework for incorporating 3d hyperelastic vascular wall models in 1d blood flow simulations.Biomechanics and Modeling in Mechanobiology, 20:1231– 1249, 2021

    work page 2021

  20. [20]

    M. J. Colebank and N. C. Chesler. Efficient uncertainty quantification in a spatially multiscale model of pulmonary arterial and venous hemo- dynamics.Biomechanics and Modeling in Mechanobiology, 2024. 31

    work page 2024

  21. [21]

    Dumbser and E

    M. Dumbser and E. F. Toro. On universal Osher-type schemes for gen- eral nonlinear hyperbolic conservation laws.Communications in Com- putational Physics, 10:635–671, 2011

    work page 2011

  22. [22]

    Dumbser and E

    M. Dumbser and E. F. Toro. A simple extension of the Osher Rie- mann solver to non-conservative hyperbolic systems.Journal of Scien- tific Computing, 48:70–88, 2011

    work page 2011

  23. [23]

    Formaggia, A

    L. Formaggia, A. Quarteroni, and A. Veneziani.Cardiovascular Math- ematics: Modeling and simulation of the circulatory system, volume 1. Springer Science & Business Media, 2010

    work page 2010

  24. [24]

    H. Gao, X. Zhu, and J. X. Wang. A bi-fidelity surrogate modeling ap- proach for uncertainty propagation in three-dimensional hemodynamic simulations.Computer Methods in Applied Mechanics and Engineering, 366:113047, 2020

    work page 2020

  25. [25]

    Garay, J

    J. Garay, J. Dunstan, S. Uribe, and F. Sahli Costabal. Physics-informed neural networks for parameter estimation in blood flow models.Com- puters in Biology and Medicine, 178:108706, 2024

    work page 2024

  26. [26]

    Goodfellow, Y

    I. Goodfellow, Y. Bengio, A. Courville, and Y. Bengio.Deep learning, volume 1. MIT press Cambridge, 2016

    work page 2016

  27. [27]

    Holenstein, P

    R. Holenstein, P. Niederer, and M. Anliker. A viscoelastic model for use in predicting arterial pulse waves.Journal of Biomechanical Engineer- ing, 102(4):318–325, 11 1980

    work page 1980

  28. [28]

    S. Jin. Asymptotic-preserving schemes for multiscale physical problems. Acta Numerica, 31:415–489, 12 2021

    work page 2021

  29. [29]

    S. Jin, Z. Ma, and K. Wu. Asymptotic-preserving neural networks for multiscale time-dependent linear transport equations.Journal of Scien- tific Computing, 94(3):57, 2023

    work page 2023

  30. [30]

    G. E. Karniadakis, I. G. Kevrekidis, L. Lu, P. Perdikaris, S. Wang, and L. Yang. Physics-informed machine learning.Nature Reviews Physics, 3(6):422–440, 2021

    work page 2021

  31. [31]

    D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014

    work page internal anchor Pith review Pith/arXiv arXiv 2014

  32. [32]

    Kissas, Y

    G. Kissas, Y. Yang, E. Hwuang, W. R. Witschey, J. A. Detre, and P. Perdikaris. Machine learning in cardiovascular flows modeling: Pre- dicting arterial blood pressure from non-invasive 4D flow MRI data using 32 physics-informed neural networks.Computer Methods in Applied Me- chanics and Engineering, 358:112623, 2020

    work page 2020

  33. [33]

    R. S. Lakes.Viscoelastic Materials. Cambridge University Press, 2009

    work page 2009

  34. [34]

    T. Lee, R. Kashyap, and C. Chu. Building skeleton models via 3-D medial surface axis thinning algorithms.CVGIP: Graphical Models and Image Processing, 56(6):462–478, 1994

    work page 1994

  35. [35]

    Leibinger, M

    J. Leibinger, M. Dumbser, U. Iben, and I. Wayand. A path-conservative osher-type scheme for axially symmetric compressible flows in flexible visco-elastic tubes.Applied Numerical Mathematics, 105:47–63, 2016

    work page 2016

  36. [36]

    J. R. Levick.An introduction to cardiovascular physiology. Butterworth- Heinemann, 2013

    work page 2013

  37. [37]

    Liang, S

    F. Liang, S. Takagi, R. Himeno, and H. Liu. Biomechanical characteri- zation of ventricular–arterial coupling during aging: a multi-scale model study.Journal of biomechanics, 42(6):692–704, 2009

    work page 2009

  38. [38]

    Markl, A

    M. Markl, A. Frydrychowicz, S. Kozerke, M. Hope, and O. Wieben. 4D flow MRI.Journal of Magnetic Resonance Imaging, 36(5):1015–1036, 2012

    work page 2012

  39. [39]

    C. R. Maurer, R. Qi, and V. Raghavan. A linear time algorithm for computing exact euclidean distance transforms of binary images in arbi- trary dimensions.IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(2):265–270, 2003

    work page 2003

  40. [40]

    A. S. Meidert and B. Saugel. Techniques for non-invasive monitoring of arterial blood pressure.Frontiers in medicine, 4:231, 2018

    work page 2018

  41. [41]

    A. G. Morgan, M. J. Thrippleton, J. M. Wardlaw, and I. Marshall. 4D flow MRI for non-invasive measurement of blood flow in the brain: a systematic review.Journal of Cerebral Blood Flow & Metabolism, 41(2):206–218, 2021

    work page 2021

  42. [42]

    L. O. M¨ uller and E. F. Toro. A global multiscale mathematical model for the human circulation with emphasis on the venous system.In- ternational journal for numerical methods in biomedical engineering, 30(7):681–725, 2014

    work page 2014

  43. [43]

    A. A. Oglat, M. Matjafri, N. Suardi, M. A. Oqlat, M. A. Abdelrahman, and A. A. Oqlat. A review of medical doppler ultrasonography of blood 33 flow in general and especially in common carotid artery.Journal of medical ultrasound, 26(1):3–13, 2018

    work page 2018

  44. [44]

    Orera, J

    J. Orera, J. Mairal, L. S´ anchez-Fuster, and J. Murillo. Reconstructing in-vitro and in-vivo signals and parameters in networks of elastic ves- sels using physics-informed neural networks.Computers in Biology and Medicine, 203:111472, 2026

    work page 2026

  45. [45]

    Orera, J

    J. Orera, J. Ram´ ırez, P. Garc´ ıa-Navarro, and J. Murillo. Roepinns: An integration of advanced cfd solvers with physics-informed neural net- works and application in arterial flow modeling.Computer Methods in Applied Mechanics and Engineering, 440:117933, 2025

    work page 2025

  46. [46]

    M. R. Pfaller, J. Pham, A. Verma, L. Pegolotti, N. M. Wilson, D. W. Parker, W. Yang, and A. L. Marsden. Automated generation of 0d and 1d reduced-order models of patient-specific blood flow.Interna- tional Journal for Numerical Methods in Biomedical Engineering, 38:0– 25, 2022

    work page 2022

  47. [47]

    Piccioli, G

    F. Piccioli, G. Bertaglia, A. Valiani, and V. Caleffi. Modeling blood flow in networks of viscoelastic vessels with the 1-D augmented fluid–structure interaction system.Journal of Computational Physics, 464:111364, 9 2022

    work page 2022

  48. [48]

    Quarteroni and L

    A. Quarteroni and L. Formaggia. Mathematical modelling and numerical simulation of the cardiovascular system.Handbook of numerical analysis, 12:3–127, 2004

    work page 2004

  49. [49]

    Raissi, P

    M. Raissi, P. Perdikaris, and G. E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations.Journal of Computational physics, 378:686–707, 2019

    work page 2019

  50. [50]

    M. E. Safar. Systolic blood pressure, pulse pressure and arterial stiff- ness as cardiovascular risk factors.Current opinion in nephrology and hypertension, 10(2):257–261, 2001

    work page 2001

  51. [51]

    M. E. Safar, B. I. Levy, and H. Struijker-Boudier. Current perspectives on arterial stiffness and pulse pressure in hypertension and cardiovascu- lar diseases.Circulation, 107(22):2864–2869, 2003

    work page 2003

  52. [52]

    Salvi, G

    P. Salvi, G. Lio, C. Labat, E. Ricci, B. Pannier, and A. Benetos. Valida- tion of a new non-invasive portable tonometer for determining arterial 34 pressure wave and pulse wave velocity: the pulsepen device.Journal of hypertension, 22(12):2285–2293, 2004

    work page 2004

  53. [53]

    B. V. Scheer, A. Perel, and U. J. Pfeiffer. Clinical review: complications and risk factors of peripheral arterial catheters used for haemodynamic monitoring in anaesthesia and intensive care medicine.Critical care, 6(3):199, 2002

    work page 2002

  54. [54]

    C.-W. Shu. Essentially non-oscillatory and weighted essentially non- oscillatory schemes for hyperbolic conservation laws operated by uni- versities space research association.ICASE Report, pages 1–78, 1997

    work page 1997

  55. [55]

    Valdez-Jasso, M

    D. Valdez-Jasso, M. A. Haider, H. Banks, D. B. Santana, Y. Z. German, R. L. Armentano, and M. S. Olufsen. Analysis of viscoelastic wall prop- erties in ovine arteries.IEEE transactions on biomedical engineering, 56(2):210–219, 2008

    work page 2008

  56. [56]

    Willemet and J

    M. Willemet and J. Alastruey. Arterial pressure and flow wave analysis using time-domain 1-D hemodynamics.Annals of biomedical engineer- ing, 43(1):190–206, 2015

    work page 2015

  57. [57]

    C. Wu, M. Zhu, Q. Tan, Y. Kartha, and L. Lu. A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks.Computer Methods in Applied Mechanics and Engi- neering, 403:115671, 2023

    work page 2023

  58. [58]

    N. Xiao, J. Alastruey, and C. Alberto Figueroa. A systematic compar- ison between 1-D and 3-D hemodynamics in compliant arterial models. International journal for numerical methods in biomedical engineering, 30(2):204–231, 2014. 35

    work page 2014