Journal of Theoretical
and Applied Mechanics

0, 0, pp. , Warsaw 0


Nataliya Kizilova, Jeremi Mizerski, Helen Solovyova
The problem of the pulse wave propagation and reflection along human aorta and the diagnostic importance of the pressure and flow signals are studied. The geometrical model is based on the detailed postmortem measurements on the systemic arterial trees. The aortic model consists of 36 aortic segments and 57 side branches of the aorta including the larger and medium vessels. It was shown most of the branches have zero wave reflection coefficients but the large branches like celiac, renal and iliac arteries could produce noticeable wave reflections. The smaller branches possess negative wave reflection coefficient and, thus, contribute to the blood suction effect and lower aortic resistance to the blood flow. Mathematical model is based on the axisymmetric incompressible Navier-Stokes equations for blood and the momentum equations for incompressible viscoelastic arterial wall. The Windkessel and structured tree outflow boundary conditions at the outlets of the branches have been used. The solution has been found as superposition of the forward and backward running waves. The blood flow curves measured in vivo by Doppler ultrasound in the larger systemic arteries of healthy volunteers have been used for identification of the model parameters. It is shown, the individual geometry plays an essential role in the location of the positive and negative wave reflection sites along the aorta and, thus, in the pressure and flow patterns and blood distribution into the branches. The influence of occlusion of the iliac arteries, low/high wall rigidity, and total length of aorta are studied on different individual geometries. The model can be used for determination of the individual parameters for patient-specific cardiovascular models and further modeling of the outcomes of the surgical and therapeutic procedures.
Keywords: Biomechanics, Fluid mechanics


Nichols W., O’Rourke M., 2005, McDonald’s, blood flow in arteries. Theoretical, experimental and clinical principles, Oxford: Oxford Univ. Press.

Nguyen K.T., Clark Ch.D., Chancellor Th.J.,2008, Papavassiliou D.V. Carotid geometry effects on blood flow and on risk for vascular disease, Journal of Biomechanics, 41, 11−19.

Frazin L.J., Lanza G., Vonesh M., et al., 1990, Functional chiral asymmetry in descending thoracic aorta, Circulation, 82, 1985–1994.

Kilner P.J., Yang G.Z., Mohiaddin R.H., et al. 1993, Helical and retrograde secondary flow patterns in the aortic arch studied by three-directional magnetic resonance velocity mapping, Circulation, 88, 2235–2247.

Kvitting P., Hessevik I., Matre K., et al.. 1996, Three-dimensional cross-sectional velocity distribution in the ascending aorta in cardiac patients, Clinical Physiology, 16, 239–258.

Tenenbaum A., M. Motro, M.S. Feinberg, et al., 2000, Retrograde flow in the thoracic aorta in patients with systemic emboli, Chest, 118, 1703–1708.

Mitchell G.F., 2009, Arterial Stiffness and Wave Reflection: Biomarkers of Cardiovascular Risk,

Artery Research, 3, 56–64.

Latham R.D., Westerhof N., Sipkema P., et al., 1985, Regional wave travel and reflections along the human aorta: a study with six simultaneous micromanometric pressures, Circulation, 72, 1257−1269.

Westerhof B.E., Guelen I., Westerhof N., et al., 2006, Quantification of wave reflection in the human aorta from pressure alone: a proof of principle, Hypertension, 48, 595−601.

Epstein S., WillemetM., Chowienczyk P.J., Alastruey J., 2015, Reducing the number of parameters in 1D arterial blood flow modeling: less is more for patient-specific simulations, American Journal of Physiology, 309, H222–H234.

Karamanoglu M., Feneley M.P., 1997, On-line synthesis of the human ascending aortic pressure pulse from the finger pulse, Hypertension, 30, 1416–1424.

Frank O., 1926, Die Theorie der Pulswellen, Zeitschrift fur Biologie, 85, 91−130.

Alastruey J., ParkerK.H., PeiroJ., Sherwin S.J., 2008, Lumped parameter outflow models for 1-d blood flow simulations: effect on pulse waves and parameter estimation, Communications in Computational Physics, 4, 317−336.

Karamanoglu M., Gallagher D.E., Avolio A.P., O’Rourke M.F., 1995, Pressure wave propagation in a multibranched model of the human upper limb, American Journal of Physiology, 269, H1363–H1369.

Womersley J.R., 1957, An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries, Technical Report TR-56-614.

Atabek H.B., 1968, Wave propagation through a viscous fluid contained in a tethered, initially stressed, orthotropic elastic tube, Biophysical Journal, 8,626−649.

Shoenberg M., 1968, Pulse wave propagation in elastic tubes having longitudinal changes in area and stiffness, Biophysical Journal, 8,991−1008.

Kizilova N.N., 2006, Pressure wave propagation in liquid-filled tubes of viscoelastic material, Fluid Dymanics, 41,444−456.

Cox R.H., 1968, Wave propagation through the Newtonian fluid contained within thick-walled viscoelastic tube, Biophysical Journal, 8,691−709.

Hollander E.H., Wang J.J., Dobson G.M., et al., 2001, Negative wave reflections in pulmonary arteries, American Journal of Physiology, 281, 895−902.

Sun Y.-H., Anderson T.J., Parker K.H., Tyberg J.V., 2000, Wave-intensity analysis: a new approach to coronary hemodynamic, Journal of Applied Physiology, 89, 1636–1644 .

Zenin O.K., Кizilova N.N., Philippova E.N., 2007, Studies on the structure of human coronary vasculature, Biophysics,52, 924−930.

Kizilova N.N., 2007, Modeling of intraorgan arterial vasculature. II. Propagation of pressure waves,Biophysics, 52, 77–82.

Kizilova N., Philippova H., Zenin O., 2010, A realistic model of human arterial system: blood flow distribution, pulse wave propagation and modeling of pathology, Mechanika w Medycynie, Korzynski M. and Cwanka J. (eds.), vol.10, Rzeszow University Press, 103−118.

Lighthill M J 1978 Waves in fluids (Cambridge: Cambridge Univ. Press)

Taylor C.A., Hughes T.H., ZarinsC., 1996, Computational investigations in vascular disease, Computational Physics, 10, 224–232.

Taylor C.A., Draney M.T., KuJ.P., et al., 1999, Predictive medicine: Computational techniques in therapeutic decision-making, Computer Aided Surgery, 4, 231–247.

Taylor C.A., Hughes Th.J.R., Zarins C.K., 1998, Finite element modeling of blood flow in arteries, Computational Methods in Applied Mechanical Engineering, 158, 155–196.

Shipkowitz T., Rodgers V.G.J., FrazinL.J., Chandran K.B., 2000, Numerical study on the effect of secondary flow in the human aorta on local shear stresses in abdominal aortic branches, Journal of Biomechanics, 33, 717−728.

Xiao N., Alastruey J., Figueroa C.A., 2014, A Systematic Comparison between 1-D and 3-D hemodynamics in compliant arterial models, International Journal on Numerical Methods in Biomedical Engineering, 30, 204–231.

Quarteroni A., TuveriM., VenezianiA., 2000, Computational vascular fluid dynamics: Problems, models and methods, Computer Visualization Sciences, 2, 163–197.

Quarteroni A., 2001, Modeling the Cardiovascular System: A Mathematical Challenge, Mathematics Unlimited — 2001 and Beyond, Engquist B., Schmid W. (eds), Springer, Berlin, Heidelberg, 961−970.

Díaz-Zuccarini V., NarracottA.J., BurriesciG., et al., 2006, Adaptation and development of software simulation methodologies for cardiovascular engineering: present and future challenges from an end-user perspective, Philosophical Transactions of the Royal Society A, 367, 2655−2666.

Westerhof N., Bosman F., de Vries C.J., Noordegraaf A., 1969, Analog studies of the human systemic arterial tree, Journal of Biomechanics, 2, 121−143.

Kizilova N., Solovyova H., Mizerski J., 2019, Modeling of pulse wave propagation and reflection along human aorta, Biomechanics in Medicine and Biology, K. Arkusz, R.Będziński, T. Klekiel, S. Piszczatowski (eds.),SpringerSeries “Advances in Intelligent Systems and Computing”, Vol. 831, 23−35.

KizilovaN., MizerskiJ., 2018,Validation of numerical models for flow simulation and wave propagation along human aorta, Journal of Physics: Conference Series, 1101, 012014.

Kassab Gh.S., 2006, Biomechanics of the cardiovascular system: the aorta as an illustratory example, Journal of the Royal Society Interface, 3, 719–740.

Wang J.J., Parker K.H., 2004, Wave propagation in a model of the arterial circulation,Journal of Biomechanics, 37, 457–470.

Willemet M., ChowienczykP., Alastruey J., 2015, A database of virtual healthy subjects to assess the accuracy of foot-to-foot pulse wave velocities for estimation of aortic stiffness, American Journal of Physiology, 309, H663–H675.

Blanco P J, Watanabe S M, Passos M A, et al., 2015, An anatomically detailed arterial network model for one-dimensional computational hemodynamics, IEEE Transactions on. Biomedical Engineering, 62,736–753.

Alastruey J., Hunt A.A.E., Weinberg P.D., 2014, Novel wave intensity analysis of arterial pulse wave propagation accounting for peripheral reflections, International Journal for Numerical Methods in Biomedical Engineering, 30, 249−279.

Olufsen M., 1999, Structured tree outflow condition for blood in larger systemic arteries, American journal of Physiology, 276, H257−268.