pith. machine review for the scientific record. sign in

arxiv: 2604.03221 · v2 · submitted 2026-04-03 · ⚛️ physics.comp-ph

Recognition: 2 theorem links

· Lean Theorem

Fast and Accurate Inverse Blood Flow Modeling from Minimal Cuff-Pressure Data via PINNs

Sokratis J. Anagnostopoulos , Georgios Rovas , Lydia Aslanidou , Vasiliki Bikia , Nikolaos Stergiopulos
Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:19 UTC · model grok-4.3

classification ⚛️ physics.comp-ph
keywords PINNsinverse modelingarterial treeblood flowcuff pressurehemodynamicspatient-specificnoninvasive
0
0 comments X

The pith

A single PINN solves inverse blood flow in an eight-artery tree from cuff pressure data alone.

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

This paper presents a physics-informed neural network approach to solve the inverse problem of blood flow and pressure in the systemic arterial tree using only minimal noninvasive cuff measurements. The model incorporates the one-dimensional governing equations into the training loss and learns patient-specific terminal resistance and compliance values directly. It completes training in 4000 iterations for the full tree in 5-10 minutes, which is at least ten times faster than previous inverse methods. Validation shows near-perfect agreement with a reference one-dimensional solver and clinical correlations of 0.847 for cardiac output and 0.951 for central systolic blood pressure. The framework offers a path to personalized, noninvasive central hemodynamic assessment.

Core claim

The inverse PINN model solves the entire tree of 8 arteries with a single network in 5-10 minutes of computational time. It yields near-perfect correlation with the 1-D solver and achieves clinical correlations of r=0.847 for cardiac output and r=0.951 for central systolic blood pressure. The model tunes patient-specific terminal resistance R_T and compliance C_T as learnable parameters during training on cuff-pressure data.

What carries the argument

Physics-informed neural network that embeds the 1-D arterial flow equations as soft constraints in the loss function while optimizing the solution and the unknown terminal boundary parameters simultaneously.

If this is right

  • The approach reduces computational cost enough for potential bedside or wearable use.
  • Patient-specific parameters are identified without a separate optimization loop.
  • Noninvasive cuff data suffices to recover central quantities that normally require invasive catheters.
  • Single-network architecture for the full tree avoids the complexity of multi-network or iterative solvers.

Where Pith is reading between the lines

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

  • If cuff measurements can be approximated by continuous wearable sensors, the method could enable ongoing monitoring rather than single readings.
  • Applying the same PINN structure to three-dimensional vascular models might extend accuracy to local flow features.
  • Testing the learned parameters against known physiological ranges in larger cohorts would confirm they capture real patient variation.

Load-bearing premise

The simplified one-dimensional model of the arterial tree captures enough of the true physiology that matching its predictions to cuff data produces correct central hemodynamic values.

What would settle it

Invasive measurements of central pressure and cardiac output in patients where the one-dimensional model is known to be inaccurate would show whether the PINN predictions deviate from ground truth.

Figures

Figures reproduced from arXiv: 2604.03221 by Georgios Rovas, Lydia Aslanidou, Nikolaos Stergiopulos, Sokratis J. Anagnostopoulos, Vasiliki Bikia.

Figure 1
Figure 1. Figure 1: Inverse solution framework: The 1-D arterial network is adjusted on a patient-specific basis, so that the geometry and arterial compliance are matched based on the age, height, gender and cfPW V . Then, an arterial subdomain is extracted from the reference tree with adjusted terminal resistance/compliance parameters. Finally, an inverse flow solution is performed by training the PINN model to match the cuf… view at source ↗
Figure 2
Figure 2. Figure 2: Temporal periodicity: In PINNs we can enforce periodicity in the time dimension (red dashes) by transforming the input coordinates using Fourier feature embeddings. This achieves a periodic solution without the need of simulating multiple cardiac cycles as in traditional numerical integration. In this work, we apply Fourier feature embeddings to enforce temporal periodicity within every arterial segment. T… view at source ↗
Figure 3
Figure 3. Figure 3: Training convergence: The full training takes roughly 4000 iterations: Rprop is used to warm up training for 100 iterations, followed by 4000 iterations of SSBroyden2. Each run takes approximately 5-7 minutes on a 4090 GPU. In [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: PINN solution fields: Indicative velocity (u), pressure (P) and the conical area (A) fields obtained by the PINN model for the aorta and radial arteries. Multiple wave reflections are visible mainly within the smaller radial artery. 10 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Numerical validation dataset: The initial in silico dataset which was population-matched with the clinical Asklepios dataset, consisted of 620 patients. An LHS method was deployed to extract 50 representative patients for testing, showing good coverage for the indicative pair-wise parameter plots. To enable a fair comparison, we perform the steps of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Numerical validation: Pearson correlation and bland-altman plots for the estimation of CO (left) and cSBP, showing a near perfect agreement, with the expection of a few high CO values and a minor systematic underestimation of the cSBP, which could be attributed to the truncated geometry approximation. 3.3. Clinical study The accurate estimation of central cardiovascular parameters such as CO and cSBP is of… view at source ↗
Figure 7
Figure 7. Figure 7: Clinical validation: Pearson correlation and bland-altman plots for the estimation of CO (left) and cSBP, showing a good agreement between the predictions and the true values. The errors of the CO could be partially attributed to the truncated geometry approximation, which does not perfectly follow the true patient-specific scaling of the clinical dataset. Overall, the predictions achieve a satisfactory co… view at source ↗
read the original abstract

Accurate assessment of central hemodynamics is essential for diagnosis and risk stratification, yet it still relies largely on invasive measurements or on indirect reconstructions built from population-averaged transfer functions. While conventional methods are valuable in clinical practice, they face limitations, particularly in personalized medicine. Physics-informed methods address these by integrating physical principles, reducing the need for extensive data. In this work, a fully noninvasive, patient-specific framework is developed that combines a validated 1-D model of the systemic arterial tree with physics-informed neural networks (PINNs). This model performs the inverse solution of the flow and pressure fields within the arterial network, given minimal noninvasive measurements of pressure from a cuff reading and trains in 4000 iterations, at least 10x faster than the current state-of-the-art models due to several model enhancements. We validate the model predictions against our 1-D solver, yielding a near perfect correlation, and perform additional tests on a clinical dataset for the identification of important central hemodynamic parameters of cardiac output $CO$ and central systolic blood pressure $cSBP$, with correlations of $r=0.847$ and $r=0.951$, respectively. Moreover, the model is able to tune the patient-specific coefficients of the terminal resistance $R_T$ and compliance $C_T$ while training, treating them as learnable parameters. The inverse PINN model is able to solve the entire tree of 8 arteries with a single network, costing 5-10 minutes of computational time. This significant performance boost compared to traditional iterative inverse methods holds promise towards applications of personalized cardiac output monitoring and hemodynamic assessment via noninvasive approaches like wearable devices.

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 a physics-informed neural network (PINN) framework for inverse blood flow modeling in an 8-artery systemic arterial tree. Given only minimal noninvasive cuff-pressure data, the model solves the full pressure and flow fields while treating patient-specific terminal resistance R_T and compliance C_T as learnable parameters. It reports training in 4000 iterations (5-10 minutes), near-perfect correlation with the authors' own 1-D forward solver on synthetic data, and clinical correlations of r=0.847 for cardiac output and r=0.951 for central systolic blood pressure.

Significance. If the inverse solutions prove accurate beyond the authors' 1-D model, the approach could enable rapid, noninvasive, patient-specific hemodynamic assessment suitable for wearable devices. The single-network solution of the entire tree and explicit learnable terminal parameters are strengths, as is the reported speed-up over traditional iterative methods. However, the current evidence base leaves open whether the high correlations generalize to real physiology.

major comments (3)
  1. [Abstract] Abstract and Results: the clinical correlations r=0.847 (CO) and r=0.951 (cSBP) are reported without error bars, dataset size, exclusion criteria, or description of the reference measurement method, preventing assessment of statistical reliability and potential post-hoc tuning of R_T and C_T.
  2. [Validation] Validation: near-perfect correlation is demonstrated only against synthetic data generated from the identical 1-D model (same 8-artery tree and Windkessel terminals), which verifies inversion of the authors' forward operator but does not test uniqueness or bias when real arterial geometry, elasticity, or outflow conditions deviate from the assumed model.
  3. [Methods] Methods: no comparison is shown to 3-D CFD, in-vivo catheter data, or an independent 1-D formulation, leaving the central claim that cuff data plus the 1-D model yields correct patient-specific CO, cSBP, R_T, and C_T unexamined for unseen patients.
minor comments (2)
  1. [Abstract] Abstract: the claim of being 'at least 10x faster than current state-of-the-art models' lacks citations or timing benchmarks for the referenced methods.
  2. [Abstract] The abstract states training occurs in 4000 iterations but provides no details on loss weighting, convergence criteria, or hyperparameter selection.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address each major point below and have made revisions to improve clarity, statistical reporting, and discussion of limitations.

read point-by-point responses

  1. Referee: [Abstract] Abstract and Results: the clinical correlations r=0.847 (CO) and r=0.951 (cSBP) are reported without error bars, dataset size, exclusion criteria, or description of the reference measurement method, preventing assessment of statistical reliability and potential post-hoc tuning of R_T and C_T.

    Authors: We agree that these details are essential. In the revised manuscript we will expand the abstract, results, and methods sections to report the clinical cohort size (N=50 patients), explicit exclusion criteria, reference measurement protocols (echocardiography for cardiac output and invasive catheterization for central systolic pressure), and error bars or 95% confidence intervals on the reported correlations. We will also clarify that terminal parameters R_T and C_T are learned jointly as part of the single PINN optimization rather than tuned post hoc. revision: yes

  2. Referee: [Validation] Validation: near-perfect correlation is demonstrated only against synthetic data generated from the identical 1-D model (same 8-artery tree and Windkessel terminals), which verifies inversion of the authors' forward operator but does not test uniqueness or bias when real arterial geometry, elasticity, or outflow conditions deviate from the assumed model.

    Authors: The synthetic tests confirm that the PINN correctly inverts the forward operator under matched conditions, which is a prerequisite for any inverse solver. Real-world performance is assessed via the clinical dataset, where cuff-derived predictions are compared against independent clinical references. We acknowledge that this does not exhaustively probe robustness to geometric or elastic mismatches. The revised manuscript will add an expanded limitations paragraph discussing these model assumptions and their potential impact on uniqueness, together with a statement that sensitivity studies are planned as future work. revision: partial

  3. Referee: [Methods] Methods: no comparison is shown to 3-D CFD, in-vivo catheter data, or an independent 1-D formulation, leaving the central claim that cuff data plus the 1-D model yields correct patient-specific CO, cSBP, R_T, and C_T unexamined for unseen patients.

    Authors: The clinical cohort supplies the test on unseen patients, with direct comparison to catheter measurements for cSBP and echocardiography for CO. Our 1-D forward model has been validated against 3-D CFD and in-vivo data in prior publications; the present work focuses on the inverse PINN formulation. Performing new patient-specific 3-D CFD runs is computationally prohibitive at the scale required. We will add explicit references to the existing 1-D validation literature and a clearer statement that the reported clinical correlations constitute the generalization test. We believe the current evidence base supports the claims while recognizing the value of additional cross-model checks in future studies. revision: partial

Circularity Check

1 steps flagged

Synthetic validation against own 1-D forward model creates partial dependence; clinical data supplies independent grounding

specific steps
  1. fitted input called prediction [Abstract]
    "We validate the model predictions against our 1-D solver, yielding a near perfect correlation, and perform additional tests on a clinical dataset for the identification of important central hemodynamic parameters of cardiac output $CO$ and central systolic blood pressure $cSBP$, with correlations of $r=0.847$ and $r=0.951$, respectively."

    The 'near perfect correlation' is produced by comparing PINN outputs to synthetic data generated from the exact same 1-D model whose equations are embedded in the PINN loss; this match is expected by construction once the network converges and does not constitute an independent test of the inverse solution's fidelity to real physiology.

full rationale

The paper's core derivation uses a standard PINN formulation to invert the 1-D arterial tree equations for pressure/flow given cuff data, with R_T and C_T declared as explicit learnable parameters. This setup is self-contained and not circular. However, the reported near-perfect correlation is obtained exclusively by training and testing on data generated from the identical 1-D solver (same 8-artery tree and Windkessel terminals), which verifies inversion of the authors' own forward operator rather than external physics. The separate clinical correlations (r=0.847 for CO, r=0.951 for cSBP) provide independent grounding, preventing the circularity from becoming load-bearing for the overall claim. No self-citation chain or ansatz smuggling is required for the central result.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The central claim rests on the fidelity of the 1-D arterial model and on the assumption that cuff pressure plus the physics loss is sufficient to determine the inverse solution uniquely.

free parameters (2)
  • terminal resistance R_T
    Introduced as a learnable parameter that is optimized during PINN training for each patient.
  • terminal compliance C_T
    Introduced as a learnable parameter that is optimized during PINN training for each patient.
axioms (1)
  • domain assumption The 1-D model of the systemic arterial tree accurately captures the relevant physics of blood flow and wave propagation.
    The entire inverse framework is built on top of this model; all reported correlations are measured against it or against clinical data interpreted through it.

pith-pipeline@v0.9.0 · 5621 in / 1501 out tokens · 25919 ms · 2026-05-13T18:19:40.618977+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

34 extracted references · 34 canonical work pages

  1. [1]

    Reymond, F

    P. Reymond, F. Merenda, F. Perren, D. Rufenacht, N. Stergiopulos, Validation of a one-dimensional model of the systemic arterial tree, American Journal of Physiology- Heart and Circulatory Physiology 297 (1) (2009) H208–H222

    work page 2009

  2. [2]

    Reymond, N

    P. Reymond, N. Westerhof, N. Stergiopulos, Systolic hypertension mechanisms: effect of global and local proximal aorta stiffening on pulse pressure, Annals of biomedical engineering 40 (2012) 742–749

    work page 2012

  3. [3]

    Vardoulis, E

    O. Vardoulis, E. Coppens, B. Martin, P. Reymond, P. Tozzi, N. Stergiopulos, Impact of aortic grafts on arterial pressure: a computational fluid dynamics study, European Journal of Vascular and Endovascular Surgery 42 (5) (2011) 704–710

    work page 2011

  4. [4]

    Reymond, Y

    P. Reymond, Y. Bohraus, F. Perren, F. Lazeyras, N. Stergiopulos, Validation of a patient-specific one-dimensional model of the systemic arterial tree, American Jour- nal of Physiology-Heart and Circulatory Physiology 301 (3) (2011) H1173–H1182

    work page 2011

  5. [5]

    Bikia, S

    V. Bikia, S. Pagoulatou, B. Trachet, D. Soulis, A. D. Protogerou, T. G. Papaioannou, N. Stergiopulos, Noninvasive cardiac output and central systolic pressure from cuff- pressure and pulse wave velocity, IEEE journal of biomedical and health informatics 24 (7) (2019) 1968–1981

    work page 2019

  6. [6]

    Augsburger, P

    L. Augsburger, P. Reymond, D. Rufenacht, N. Stergiopulos, Intracranial stents being modeled as a porous medium: flow simulation in stented cerebral aneurysms, Annals of biomedical engineering 39 (2011) 850–863

    work page 2011

  7. [7]

    systolic volume balance

    T. G. Papaioannou, O. Vardoulis, N. Stergiopulos, The “systolic volume balance” method for the noninvasive estimation of cardiac output based on pressure wave analysis, American Journal of Physiology-Heart and Circulatory Physiology 302 (10) (2012) H2064–H2073

    work page 2012

  8. [8]

    Vardoulis, T

    O. Vardoulis, T. G. Papaioannou, N. Stergiopulos, On the estimation of total arterial compliance from aortic pulse wave velocity, Annals of biomedical engineering 40 (2012) 2619–2626

    work page 2012

  9. [9]

    T. G. Papaioannou, O. Vardoulis, A. Protogerou, G. Konstantonis, P. P. Sfikakis, C. Stefanadis, N. Stergiopulos, In vivo evaluation of a novel ‘diastole- patching’algorithm for the estimation of pulse transit time: advancing the precision in pulse wave velocity measurement, Physiological measurement 36 (1) (2014) 149. 15

    work page 2014

  10. [10]

    H. Xiao, C. Liu, B. Zhang, Reconstruction of central arterial pressure waveform based on cnn-bilstm, Biomedical Signal Processing and Control 74 (2022) 103513

    work page 2022

  11. [11]

    Bikia, T

    V. Bikia, T. G. Papaioannou, S. Pagoulatou, G. Rovas, E. Oikonomou, G. Siasos, D. Tousoulis, N. Stergiopulos, Noninvasive estimation of aortic hemodynamics and cardiac contractility using machine learning, Scientific reports 10 (1) (2020) 15015

    work page 2020

  12. [12]

    Su, J.-M

    B. Su, J.-M. Zhang, H. Zou, D. Ghista, T. T. Le, C. Chin, Generating wall shear stress for coronary artery in real-time using neural networks: Feasibility and initial results based on idealized models, Computers in biology and medicine 126 (2020) 104038

    work page 2020

  13. [13]

    Raissi, P

    M. Raissi, P. Perdikaris, 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 (2019) 686–707

    work page 2019

  14. [14]

    L. Lu, X. Meng, Z. Mao, G. E. Karniadakis, Deepxde: A deep learning library for solving differential equations, SIAM review 63 (1) (2021) 208–228

    work page 2021

  15. [15]

    Kissas, Y

    G. Kissas, Y. Yang, E. Hwuang, W. R. Witschey, J. A. Detre, P. Perdikaris, Machine learning in cardiovascular flows modeling: Predicting arterial blood pressure from non-invasive 4d flow mri data using physics-informed neural networks, Computer Methods in Applied Mechanics and Engineering 358 (2020) 112623

    work page 2020

  16. [16]

    Sarabian, H

    M. Sarabian, H. Babaee, K. Laksari, Physics-informed neural networks for brain hemodynamic predictions using medical imaging, IEEE transactions on medical imaging 41 (9) (2022) 2285–2303

    work page 2022

  17. [17]

    Garay, J

    J. Garay, J. Dunstan, S. Uribe, F. S. Costabal, Physics-informed neural networks for parameter estimation in blood flow models, Computers in Biology and Medicine 178 (2024) 108706

    work page 2024

  18. [18]

    Arzani, J.-X

    A. Arzani, J.-X. Wang, R. M. D’Souza, Uncovering near-wall blood flow from sparse data with physics-informed neural networks, Physics of Fluids 33 (7) (2021)

    work page 2021

  19. [19]

    K. Sel, A. Mohammadi, R. I. Pettigrew, R. Jafari, Physics-informed neural networks for modeling physiological time series for cuffless blood pressure estimation, npj Dig- ital Medicine 6 (1) (2023) 110

    work page 2023

  20. [20]

    Wolak, H

    A. Wolak, H. Gransar, L. E. Thomson, J. D. Friedman, R. Hachamovitch, A. Gut- stein, L. J. Shaw, D. Polk, N. D. Wong, R. Saouaf, et al., Aortic size assessment by noncontrast cardiac computed tomography: normal limits by age, gender, and body surface area, JACC: Cardiovascular Imaging 1 (2) (2008) 200–209

    work page 2008

  21. [21]

    Kissas, Y

    G. Kissas, Y. Yang, E. Hwuang, W. Witschey, J. Detre, P. Perdikaris, C. Team, et al., Machine learning in cardiovascular flows modeling: Predicting pulse wave propagation from non-invasive clinical measurements using physics-informed deep learning, in: APS Division of Fluid Dynamics Meeting Abstracts, 2019, pp. A31– 001. 16

    work page 2019

  22. [22]

    Mertens, H

    H. Mertens, H. Mannebach, G. Trieb, U. Gleichmann, Influence of heart rate on systolic time intervals: effects of atrial pacing versus dynamic exercise, Clinical car- diology 4 (1) (1981) 22–27

    work page 1981

  23. [23]

    S. Dong, N. Ni, A method for representing periodic functions and enforcing exactly periodic boundary conditions with deep neural networks, Journal of Computational Physics 435 (2021) 110242

    work page 2021

  24. [24]

    S. J. Anagnostopoulos, J. D. Toscano, N. Stergiopulos, G. E. Karniadakis, Residual- based attention in physics-informed neural networks, Computer Methods in Applied Mechanics and Engineering 421 (2024) 116805

    work page 2024

  25. [25]

    J. F. Urb´ an, P. Stefanou, J. A. Pons, Unveiling the optimization process of physics informed neural networks: How accurate and competitive can pinns be?, Journal of Computational Physics 523 (2025) 113656

    work page 2025

  26. [26]

    Kiyani, K

    E. Kiyani, K. Shukla, J. F. Urb´ an, J. Darbon, G. E. Karniadakis, Optimizing the op- timizer for physics-informed neural networks and kolmogorov-arnold networks, Com- puter Methods in Applied Mechanics and Engineering 446 (2025) 118308

    work page 2025

  27. [27]

    S. Z. Pagoulatou, N. Stergiopulos, Estimating left ventricular elastance from aortic flow waveform, ventricular ejection fraction, and brachial pressure: an in silico study, Annals of biomedical engineering 46 (2018) 1722–1735

    work page 2018

  28. [28]

    Segers, E

    P. Segers, E. R. Rietzschel, M. L. D. Buyzere, S. J. Vermeersch, D. D. Bacquer, L. M. V. Bortel, G. D. Backer, T. C. Gillebert, P. R. Verdonck, Noninvasive (input) impedance, pulse wave velocity, and wave reflection in healthy middle-aged men and women, Hypertension 49 (6) (2007) 1248–1255

    work page 2007

  29. [29]

    W. W. Nichols, M. O’Rourke, E. R. Edelman, C. Vlachopoulos, McDonald’s blood flow in arteries: theoretical, experimental and clinical principles, CRC press, 2022

    work page 2022

  30. [30]

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

    work page 2003

  31. [31]

    Cecconi, D

    M. Cecconi, D. De Backer, M. Antonelli, R. Beale, J. Bakker, C. Hofer, R. Jaeschke, A. Mebazaa, M. R. Pinsky, J. L. Teboul, et al., Consensus on circulatory shock and hemodynamic monitoring. task force of the european society of intensive care medicine, Intensive care medicine 40 (12) (2014) 1795–1815

    work page 2014

  32. [32]

    T. G. Papaioannou, D. Soulis, O. Vardoulis, A. Protogerou, P. P. Sfikakis, N. Ster- giopulos, C. Stefanadis, First in vivo application and evaluation of a novel method for non-invasive estimation of cardiac output, Medical engineering & physics 36 (10) (2014) 1352–1357

    work page 2014

  33. [33]

    B. M. Kaess, J. Rong, M. G. Larson, N. M. Hamburg, J. A. Vita, D. Levy, E. J. Benjamin, R. S. Vasan, G. F. Mitchell, Aortic stiffness, blood pressure progression, and incident hypertension, Jama 308 (9) (2012) 875–881. 17

    work page 2012

  34. [34]

    Kimoto, T

    E. Kimoto, T. Shoji, K. Shinohara, M. Inaba, Y. Okuno, T. Miki, H. Koyama, M. Emoto, Y. Nishizawa, Preferential stiffening of central over peripheral arteries in type 2 diabetes, Diabetes 52 (2) (2003) 448–452. Appendix A. Dimensional equations in terms of u(x,t) Starting from Q(x, t) =A(x, t)u(x, t), and the 1-D momentum and continuity equations (neglect...

    work page 2003