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Comparison of stenosis models for usage in the estimation of pressure gradient across aortic coarctation


Non-invasive estimation of the pressure gradient in cardiovascular stenosis has much clinical importance in assisting the diagnosis and treatment of stenotic diseases. In this research, a systematic comparison is conducted to investigate the accuracy of a group of stenosis models against the MRI- and catheter-measured patient data under the aortic coarctation condition. Eight analytical stenosis models, including six from the literature and two proposed in this study, are investigated to examine their prediction accuracy against the clinical data. The two improved models proposed in this study consider comprehensively the Poiseuille loss, the Bernoulli loss in its exact form, and the entrance effect, of the blood flow. Comparison of the results shows that one of the proposed models demonstrates a cycle-averaged mean prediction error of −0.15 ± 3.03 mmHg, a peak-to-peak prediction error of −1.8 ± 6.89 mmHg, which is the best among the models studied.

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Fig. 1
Fig. 2
Fig. 3


A :

Sectional area of the vessel


Heart rate

k :


L :

Length of the vessel segment; Inertial effect of blood flow

Q :

Flow rate

P :


r :


R :

Resistance (frictional loss) of blood flow

T :

Heart period

v :

Flow velocity

ρ :

Density of the blood

μ :

Dynamic viscosity of the blood


Ascending aorta (aortic root position)

d :

Distal side of the vessel stenosis


Descending aorta (diaphragm position)

s :


u :

Proximal side of the vessel stenosis


  1. 1.

    Webb, G.: Treatment of coarctation and late complications in the adult. Semin. Thorac. Cardiovasc. Surg. 17, 139–142 (2005).

    Article  Google Scholar 

  2. 2.

    Kenny, D., Hijazi, Z.M.: Coarctation of the aorta: from fetal life to adulthood. Cardiol. J. 18, 487–495 (2011).

    Article  Google Scholar 

  3. 3.

    Jurcut, R., Daraban, A.M., Lorber, A., Deleanu, D., Amzulescu, M.S., Zara, C., Popescu, B.A., Ginghina, C.: Coarctation of the aorta in adults: what is the best treatment? Case report and literature review. J. Med. Life 4, 189–195 (2011). N/A

  4. 4.

    Scheer, B., Perel, A., Pfeiffer, U.J.: Clinical review: complications and risk factors of peripheral arterial catheters used for haemodynamic monitoring in anaesthesia and intensive care medicine. Crit. Care Lond. Engl. 6, 199–204 (2002).

    Article  Google Scholar 

  5. 5.

    Tan, R.H.H., Dart, A.J., Dowling, B.A.: Catheters: a review of the selection, utilisation and complications of catheters for peripheral venous access. Aust. Vet. J. 81, 136–139 (2003).

    Article  Google Scholar 

  6. 6.

    Lien, W.W., Lee, A.H., Kono, Y., Steinbach, G.C., Mattrey, R.F.: Noninvasive estimation of the pressure gradient across stenoses using sonographic contrast: in vitro validation. J. Ultrasound Med. 23, 683–691 (2004).

    Article  Google Scholar 

  7. 7.

    Hatle, L., Angelsen, B.A., Tromsdal, A.: Non-invasive assessment of aortic stenosis by Doppler ultrasound. Br. Heart J. 43, 284–292 (1980).

    Article  Google Scholar 

  8. 8.

    Hatle, L., Brubakk, A., Tromsdal, A., Angelsen, B.: Noninvasive assessment of pressure drop in mitral stenosis by Doppler ultrasound. Br. Heart J. 40, 131–140 (1978).

    Article  Google Scholar 

  9. 9.

    Dodds, S.R., Bourne, N.K., Chant, A.D.: Effect of flow on the resistance of modelled femoral artery stenoses. Br. J. Surg. 83, 957–961 (1996).

    Article  Google Scholar 

  10. 10.

    Teirstein, P.S., Yock, P.G., Popp, R.L.: The accuracy of Doppler ultrasound measurement of pressure gradients across irregular, dual, and tunnellike obstructions to blood flow. Circulation. 72, 577–584 (1985).

    Article  Google Scholar 

  11. 11.

    Marx, G.R., Allen, H.D.: Accuracy and pitfalls of Doppler evaluation of the pressure gradient in aortic coarctation. J. Am. Coll. Cardiol. 7, 1379–1385 (1986).

    Article  Google Scholar 

  12. 12.

    Teien, D., Karp, K., Eriksson, P.: Non-invasive estimation of the mean pressure difference in aortic stenosis by Doppler ultrasound. Br. Heart J. 56, 450–454 (1986).

    Article  Google Scholar 

  13. 13.

    Holen, J., Simonsen, S.: Determination of pressure gradient in mitral stenosis with Doppler echocardiography. Br. Heart J. 41, 529–535 (1979).

    Article  Google Scholar 

  14. 14.

    Seitz, W.S., Kashani, I.A.: Non-invasive determination of the aortic valve area in stenosis: hydraulic orifice formula for application to echocardiography and correlation with catheterization. Eur. Heart J. 4, 31–40 (1983).

    Article  Google Scholar 

  15. 15.

    Lima, C.O., Sahn, D.J., Valdes-Cruz, L.M., Goldberg, S.J., Barron, J.V., Allen, H.D., Grenadier, E.: Noninvasive prediction of transvalvular pressure gradient in patients with pulmonary stenosis by quantitative two-dimensional echocardiographic Doppler studies. Circulation. 67, 866–871 (1983).

    Article  Google Scholar 

  16. 16.

    Zhang, Y., Nitter-Hauge, S.: Determination of the mean pressure gradient in aortic stenosis by Doppler echocardiography. Eur. Heart J. 6, 999–1005 (1985).

    Article  Google Scholar 

  17. 17.

    van Laar, P.J., van der Grond, J., Mali, W.P.T.M., Hendrikse, J.: Magnetic resonance evaluation of the cerebral circulation in obstructive arterial disease. Cerebrovasc. Dis. Basel Switz. 21, 297–306 (2006).

    Article  Google Scholar 

  18. 18.

    Mustert, B.R., Williams, D.M., Prince, M.R.: In vitro model of arterial stenosis: correlation of MR signal dephasing and trans-stenotic pressure gradients. Magn. Reson. Imaging 16, 301–310 (1998).

    Article  Google Scholar 

  19. 19.

    Giardini, A., Tacy, T.A.: Pressure recovery explains Doppler overestimation of invasive pressure gradient across segmental vascular stenosis. Echocardiography 27, 21–31 (2010).

    Article  Google Scholar 

  20. 20.

    Valdes-Cruz, L.M., Yoganathan, A.P., Tamura, T., Tomizuka, F., Woo, Y.R., Sahn, D.J.: Studies in vitro of the relationship between ultrasound and laser Doppler velocimetry and applicability to the simplified Bernoulli relationship. Circulation 73, 300–308 (1986).

  21. 21.

    Popp, R.L., Teplitsky, I.: Lessons from in vitro models of small, irregular, multiple and tunnel-like stenoses relevant to clinical stenoses of valves and small vessels. J. Am. Coll. Cardiol. 13, 716–722 (1989).

    Article  Google Scholar 

  22. 22.

    Young, D.F., Cholvin, N.R., Kirkeeide, R.L., Roth, A.C.: Hemodynamics of arterial stenoses at elevated flow rates. Circ. Res. 41, 99–107 (1977).

    Article  Google Scholar 

  23. 23.

    Young, D.F., Cholvin, N.R., Roth, A.C.: Pressure drop across artificially induced stenoses in the femoral arteries of dogs. Circ. Res. 36, 735–743 (1975).

    Article  Google Scholar 

  24. 24.

    Santamore, W.P., Bove, A.A.: A theoretical model of a compliant arterial stenosis. Am. J. Physiol. 248, H274–H285 (1985).

  25. 25.

    Gould, K.L.: Quantification of coronary artery stenosis in vivo. Circ. Res. 57, 341–353 (1985).

    Article  Google Scholar 

  26. 26.

    May, A.G., De Weese, J.A., Rob, C.G.: Hemodynamic effects of arterial stenosis. Surgery 53, 513–524 (1963). N/A

  27. 27.

    Huo, Y., Svendsen, M., Choy, J.S., Zhang, Z.-D., Kassab, G.S.: A validated predictive model of coronary fractional flow reserve. J. R. Soc. Interface 9, 1325–1338 (2012).

    Article  Google Scholar 

  28. 28.

    Shi, Y., Valverde, I., Lawford, P.V., Beerbaum, P., Hose, D.R.: Patient-specific non-invasive estimation of pressure gradient across aortic coarctation using magnetic resonance imaging. J. Cardiol. 73, 544–552 (2019).

    Article  Google Scholar 

  29. 29.

    Avrahami, I., Kersh, D., Liberzon, A.: Pulsatility index as a diagnostic parameter of reciprocating wall shear stress parameters in physiological pulsating waveforms. PLoS One 11, e0166426 (2016).

    Article  Google Scholar 

  30. 30.

    Murgo, J.P., Westerhof, N., Giolma, J.P., Altobelli, S.A.: Aortic input impedance in normal man: relationship to pressure wave forms. Circulation 62, 105–116 (1980).

    Article  Google Scholar 

  31. 31.

    Wilkinson, I.B., MacCallum, H., Flint, L., Cockcroft, J.R., Newby, D.E., Webb, D.J.: The influence of heart rate on augmentation index and central arterial pressure in humans. J. Physiol. 525(Pt 1), 263–270 (2000).

    Article  Google Scholar 

  32. 32.

    Ninos, G., Bartzis, V., Merlemis, N., Sarris, I.E.: Uncertainty quantification implementations in human hemodynamic flows. Comput. Methods Prog. Biomed. 203, 106021 (2021).

    Article  Google Scholar 

  33. 33.

    Working Group 1 of the Joint Committee for Guides in Metrology: Evaluation of measurement data — Guide to the expression of uncertainty in measurement. Bureau International des Poids et Mesures (2008)

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This research was funded by the European Community’s Seventh Framework Programme (FP7/2007–2013) under the grant agreement number 224495 (euHeart project).

Author information




YS and DRH contributed to conception and design; IV, HBG, and PB provided the clinical data; YS, IV, PVL, and DRH analysed and interpreted the data; YS drafted the manuscript; and YS, PVL, and DRH critically revised the manuscript. All authors read and gave final approval of the final manuscript.

Corresponding author

Correspondence to D. Rodney Hose.

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All procedures performed in the study involving human participants were in accordance with the ethical standards of the Ethical Committee of King’s College London on human experimentation and with the Helsinki Declaration of 1975, as revised in 2000. This article does not contain any study with animals performed by any of the authors.

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Informed consent was obtained from all individual participants included in the study.

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The authors declare no competing interests.

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Shi, Y., Valverde, I., Lawford, P.V. et al. Comparison of stenosis models for usage in the estimation of pressure gradient across aortic coarctation. J Biol Phys 47, 171–190 (2021).

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  • Pressure gradient
  • Aortic coarctation
  • Stenosis model
  • Poiseuille loss
  • Bernoulli loss
  • Entrance effect