Journal of Engineering Mathematics

, Volume 77, Issue 1, pp 19–37 | Cite as

Physical determining factors of the arterial pulse waveform: theoretical analysis and calculation using the 1-D formulation

  • Jordi Alastruey
  • Tiziano Passerini
  • Luca Formaggia
  • Joaquim Peiró
Open Access


The shape of the arterial pulse waveform is intimately related to the physical properties of the cardiovascular system. It is clinically relevant to measure those properties that are related to cardiovascular function, such as the local elasticity and viscosity of the arterial wall, total compliance and net peripheral resistance of the systemic arterial tree. Most of these properties cannot be directly measured in vivo, but they can be calculated from pressure, flow and wall displacement measurements that can be obtained in vivo. We carry out a linear analysis of the one-dimensional (1-D) equations of blood flow in Voigt-type visco-elastic vessels to study the effects on pulse wave propagation of blood viscosity, flow inertia, wall visco-elasticity, total arterial compliance, net resistance, peripheral outflow pressure, and flow rate at the aortic root. Based on our analysis, we derive methods to calculate the local elastic and viscous moduli of the arterial wall, and the total arterial compliance, net resistance, time constant and peripheral outflow pressure of the systemic arterial tree from pressure, flow and wall displacement data that can be measured in vivo. Analysis of in vivo data is beyond the scope of this study, and therefore, we verify the results of our linear analysis and assess the accuracy of our estimation methods using pulse waveforms simulated in a nonlinear visco-elastic 1-D model of the larger conduit arteries of the upper body, which includes the circle of Willis in the cerebral circulation.


Circle of Willis Flow inertia Nonlinear one-dimensional modelling Peripheral outflow pressure Pulse wave propagation Systemic arteries Voigt-type visco-elasticity Wall compliance Windkessel pressure 



This work was supported by a British–Italian partnership programme for young researchers (British Council/MIUR). Jordi Alastruey was also funded by a British Heart Foundation Intermediate Basic Science Research Fellowship (FS/09/030/27812) and the Centre of Excellence in Medical Engineering funded by the Wellcome Trust and EPSRC under grant number WT 088641/Z/09/Z. Luca Formaggia gratefully acknowledges the support of MIUR through a PRIN07 grant.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. 1.
    Blacher J, Asmar R, Djane S, London GM, Safar ME (1999) Aortic pulse wave velocity as a marker of cardiovascular risk in hypertensive patients. Hypertension 33: 1111–1117CrossRefGoogle Scholar
  2. 2.
    Laurent S, Boutouyrie P, Asmar R, Gautier I, Laloux B, Guize L, Ducimetiere P, Benetos A (2001) Aortic stiffness is an independent predictor of all-cause and cardiovascular mortality in hypertensive patients. Hypertension 37: 1236–1241CrossRefGoogle Scholar
  3. 3.
    Meaume S, Rudnichi A, Lynch A, Bussy C, Sebban C, Benetos A, Safar ME (2001) Aortic pulse wave velocity as a marker of cardiovascular disease in subjects over 70 years old. J Hypertens 19: 871–877CrossRefGoogle Scholar
  4. 4.
    Randall SO, Esler MD, Calfee RV, Bulloch GF, Maisel AS, Culp B (1976) Arterial compliance in hypertension. Aust N Z J Med 6: 49–59CrossRefGoogle Scholar
  5. 5.
    Simon AC, Safar ME, Levenson JA, London GM, Levy BI, Chau NP (1979) An evaluation of large arteries compliance in man. Am J Physiol 237: H550–H554Google Scholar
  6. 6.
    Armentano RL, Barra JG, Pessana FM, Craiem DO, Graf S, Santana DB, Sanchez RA (2007) Smart smooth muscle spring-dampers. Smooth muscle smart filtering helps to more efficiently protect the arterial wall. IEEE Eng Med Biol Mag 26: 62–70CrossRefGoogle Scholar
  7. 7.
    Peiró J, Veneziani A (2009) Reduced models of the cardiovascular system. In: Formaggia L, Quarteroni A, Veneziani A (eds) Cardiovascular mathematics. Modeling and simulation of the circulatory system. Springer, Milano, pp 347–394Google Scholar
  8. 8.
    van de Vosse FN, Stergiopulos N (2011) Pulse wave propagation in the arterial tree. Annu Rev Fluid Mech 43: 467–499ADSCrossRefGoogle Scholar
  9. 9.
    Alastruey J, Khir AW, Matthys KS, Segers P, Sherwin SJ, Verdonck P, Parker KH, Peiró J (2011) Pulse wave propagation in a model human arterial network: assessment of 1-D visco-elastic simulations against in vitro measurements. J Biomech 44: 2250–2258CrossRefGoogle Scholar
  10. 10.
    Devault K, Gremaud P, Novak V, Olufsen M, Vernières G, Zhao P (2008) Blood flow in the circle of Willis: modeling and calibration. Multiscale Model Simul 7: 888–909MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Martin V, Clément F, Decoene A, Gerbeau JF (2005) Parameter identification for a one-dimensional blood flow model. ESAIM Proc 14: 174–200zbMATHGoogle Scholar
  12. 12.
    Leguy CAD, Bosboom EMH, Gelderblom H, Hoeks APG, van de Vosse FN (2010) Estimation of distributed arterial mechanical properties using a wave propagation model in a reverse way. Med Eng Phys 32: 957–967CrossRefGoogle Scholar
  13. 13.
    Khir AW, O’Brien A, Gibbs JSR, Parker KH (2001) Determination of wave speed and wave separation in the arteries. J Biomech 34: 1145–1155CrossRefGoogle Scholar
  14. 14.
    Davies JE, Whinnett ZI, Francis DP, Willson K, Foale RA, Malik IS, Hughes AD, Parker KH, Mayet J (2006) Use of simultaneous pressure and velocity measurements to estimate arterial wave speed at a single site in humans. Am J Physiol Heart Circ Physiol 290: H878–H885CrossRefGoogle Scholar
  15. 15.
    Alastruey J, Parker KH, Peiró J, Byrd SM, Sherwin SJ (2007) Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows. J Biomech 40: 1794–1805CrossRefGoogle Scholar
  16. 16.
    Armentano R, Megnien JL, Simon A, Bellenfant F, Barra J, Levenson J (1995) Effects of hypertension on viscoelasticity of carotid and femoral arteries in humans. Hypertension 26: 48–54CrossRefGoogle Scholar
  17. 17.
    Gariepy J, Massonneau M, Levenson J, Heudes D, Simon A (1993) Evidence for in vivo carotid and femoral wall thickening in human hypertension. Groupe de Prévention Cardio-vasculaire en Médecine du Travail. Hypertension 22: 111–118CrossRefGoogle Scholar
  18. 18.
    Smith NP, Pullan AJ, Hunter PJ (2001) An anatomically based model of transient coronary blood flow in the heart. SIAM J Appl Math 62: 990–1018MathSciNetCrossRefGoogle Scholar
  19. 19.
    Brook BS, Falle SAEG, Pedley TJ (1999) Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state. J Fluid Mech 396: 223–256MathSciNetADSzbMATHCrossRefGoogle Scholar
  20. 20.
    Armentano R, Barra J, Levenson J, Simon A, Pichel RH (1995) Arterial wall mechanics in conscious dogs: assessment of viscous, inertial, and elastic moduli to characterize aortic wall behavior. Circ Res 76: 468–478CrossRefGoogle Scholar
  21. 21.
    Craiem D, Graf S, Pessana F, Grignola J, Bia D, Gines F, Armentano R (2005) Cardiovascular engineering: modelization of ventricular arterial interaction in systemic and pulmonary circulation. Lat Am Appl Res 35: 111–114Google Scholar
  22. 22.
    Čanić S, Tambača J, Guidoboni G, Mikelić A, Hartley C, Rosenstrauch D (2006) Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow. SIAM J Appl Math 67: 164–193MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Bessems D, Giannopapa CG, Rutten MCM, van de Vosse FN (2008) Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels. J Biomech 41: 284–291CrossRefGoogle Scholar
  24. 24.
    Valdez-Jasso D, Haider MA, Banks HT, Santana DB, German YZ, Armentano RL, Olufsen MS (2009) Analysis of viscoelastic wall properties in ovine arteries. IEEE Trans Biomed Eng 56: 210–219CrossRefGoogle Scholar
  25. 25.
    Reymond P, Merenda F, Perren F, Rüfenacht D, Stergiopulos N (2009) Validation of a one-dimensional model of the systemic arterial tree. Am J Physiol Heart Circ Physiol 297: H208–H222CrossRefGoogle Scholar
  26. 26.
    Valdez-Jasso D, Bia D, Zócalo Y, Armentano RL, Haider MA, Olufsen MS (2011) Linear and nonlinear viscoelastic modeling of aorta and carotid pressure–area dynamics under in vivo and ex vivo conditions. Ann Biomed Eng 39: 1438–1456CrossRefGoogle Scholar
  27. 27.
    Formaggia L, Lamponi D, Quarteroni A (2003) One-dimensional models for blood flow in arteries. J Eng Math 47: 251–276MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Passerini T (2009) Computational hemodynamics of the cerebral circulation: multiscale modeling from the circle of Willis to cerebral aneurysms. Ph.D. thesis, Politecnico di Milano, ItalyGoogle Scholar
  29. 29.
    Giannopapa CG (2004) Fluid–structure interaction in flexible vessels. Ph.D. thesis, University of London, UKGoogle Scholar
  30. 30.
    Caro CG, Pedley TJ, Schroter RC, Seed WA (2011) The mechanics of the circulation, 2nd edn. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  31. 31.
    Olufsen MS (1999) Structured tree outflow condition for blood flow in larger systemic arteries. Am J Physiol 276: H257–H268Google Scholar
  32. 32.
    Westerhof N, Bosman F, de Vries CJ, Noordergraaf A (1969) Analog studies of the human systemic arterial tree. J Biomech 2: 121–143CrossRefGoogle Scholar
  33. 33.
    Alastruey J, Parker KH, Peiró J, Sherwin SJ (2008) Lumped parameter outflow models for 1-D blood flow simulations: effect on pulse waves and parameter estimation. Commun Comput Phys 4: 317–336MathSciNetGoogle Scholar
  34. 34.
    Anliker M, Histand MB, Ogden E (1968) Dispersion and attenuation of small artificial pressure waves in the canine aorta. Circ Res 23: 539–551CrossRefGoogle Scholar
  35. 35.
    Lynn PA, Fuerst W (1989) Introductory digital signal processing with computer applications. Wiley, New YorkGoogle Scholar
  36. 36.
    Matthys KS, Alastruey J, Peiró J, Khir AW, Segers P, Verdonck PR, Parker KH, Sherwin SJ (2007) Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in vitro measurements. J Biomech 40: 3476–3486CrossRefGoogle Scholar
  37. 37.
    Alastruey J (2010) On the mechanics underlying the reservoir-excess separation in systemic arteries and their implications for pulse wave analysis. Cardiovasc Eng 10: 176–189CrossRefGoogle Scholar
  38. 38.
    Frank O (1899) Die Grundform des arteriellen Pulses. Erste AbhandlungMathematische Analyse Z Biol 37: 483–526Google Scholar
  39. 39.
    Aguado-Sierra J, Alastruey J, Wang J-J, Hadjiloizou N, Davies JE, Parker KH (2008) Separation of the reservoir and wave pressure and velocity from measurements at an arbitrary location in arteries. Proc Inst Mech Eng H 222: 403–416Google Scholar
  40. 40.
    Wang J-J, O’Brien AB, Shrive NG, Parker KH, Tyberg JV (2003) Time-domain representation of ventricular-arterial coupling as a windkessel and wave system. Am J Physiol Heart Circ Physiol 284: H1358–H1368Google Scholar
  41. 41.
    Roy C (1880) The elastic properties of the arterial wall. J Physiol (Lond) 3: 125–159Google Scholar
  42. 42.
    Parker KH (2009) An introduction to wave intensity analysis. Med Biol Eng Comput 47: 175–188CrossRefGoogle Scholar
  43. 43.
    Olufsen MS, Peskin CS, Kim WY, Pedersen EM, Nadim A, Larsen J (2000) Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann Biomed Eng 28: 1281–1299CrossRefGoogle Scholar
  44. 44.
    Oates C (2001) Cardiovascular haemodynamics and Doppler waveforms explained. Greenwich Medical Media, LondonGoogle Scholar
  45. 45.
    Ibrahim E-SH, Johnson KR, Miller AB, Shaffer JM, White RD (2010) Measuring aortic pulse wave velocity using high-field cardiovascular magnetic resonance: comparison of techniques. J Cardiovasc Magn Reson 12: 26–39CrossRefGoogle Scholar
  46. 46.
    Zambanini A, Cunningham SL, Parker KH, Khir AW, Thom SA, Hughes AD (2005) Wave-energy patterns in carotid, brachial, and radial arteries: a noninvasive approach using wave-intensity analysis. Am J Physiol Heart Circ Physiol 289: H270–H276CrossRefGoogle Scholar
  47. 47.
    Levenson JA, Peronneau PP, Simon AC, Safar ME (1981) Pulsed Doppler: determination of diameter, blood flow velocity and volumic flow of brachial artery in man. Cardiovasc Res 15: 164–170CrossRefGoogle Scholar
  48. 48.
    Cebral JR, Castro MA, Soto O, L"ohner R, Alperin N (2003) Blood-flow models of the circle of Willis from magnetic resonance data. J Eng Math 47: 369–386MathSciNetzbMATHCrossRefGoogle Scholar
  49. 49.
    Moore SM, David T, Chase JG, Arnold J, Fink J (2006) 3D models of blood flow in the cerebral vasculature. J Biomech 39: 1454–1463CrossRefGoogle Scholar
  50. 50.
    Alastruey J, Nagel SR, Nier B, Hunt AAE, Weinberg PD, Peiró J (2009) Modelling pulse wave propagation in the rabbit systemic circulation to assess the effects of altered nitric oxide synthesis. J Biomech 42: 2116–2123CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  • Jordi Alastruey
    • 1
  • Tiziano Passerini
    • 2
    • 3
  • Luca Formaggia
    • 2
  • Joaquim Peiró
    • 4
  1. 1.Department of Biomedical Engineering, Division of Imaging Sciences and Biomedical EngineeringKing’s College London, King’s Health Partners, St. Thomas’ HospitalLondonUK
  2. 2.Dipartimento di Matematica F. BrioschiPolitecnico di MilanoMilanItaly
  3. 3.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA
  4. 4.Department of AeronauticsImperial CollegeLondonUK

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