Abstract
The differences in the pressure and flow waveforms in the aortic root have not been explained so far in a satisfactory mathematical way. It is a generally accepted idea that the existence of the reflected wave causes the differences in shapes of pressure and flow. In this paper, a mathematical model is proposed that explains the blood pressure and flow waveforms based on changes in left ventricular volume during blood ejection into the aorta. According to the model, a change in volume of the left ventricle during contraction can be mathematically presented with solutions of differential equations that describe the behavior of a second-order system. The proposed mathematical equations of pressure and flow waveforms are derived from left ventricular volume change and basic equations of fluid dynamics. The position of the reflected wave depends on the age and elasticity of arteries, and has an effect on the flow and pressure waveforms. The model is in acceptable agreement with the experimental data available.
Similar content being viewed by others
References
Avolio A, Van Bortel L, Boutouyrie P, Cockcroft J, McEniery C, Protogerou A, Roman M, Safar M, Segers P, Smulyan H (2009) Role of pulse pressure amplification in arterial hypertension: experts’ opinion and review of the data. Hypertension 54:375–383
Bellhouse BJ (1969) Velocity and pressure distributions in the aortic valve. J Fluid Mech 37(3):587–600
Bowman AW, Frihauf PA, Kovács SJ (2004) Time-varying effective mitral valve area: prediction and validation using cardiac MRI and Doppler echocardiography in normal subjects. Am J Physiol Heart Circ Physiol 287:H1650–H1657
Feng J, Khir AW (2008) The compression and expansion waves of the forward and backward flows: an in vitro arterial model. Proc Inst Mech Eng 222(4):531–542
Garcia D, Barenbrug P, Pibarot P, Dekker A, Van der Veen F, Maessen J, Dumesnil J, Durand LG (2005) A ventricular-vascular coupling model in presence of aortic stenosis. Am J Physiol Heart Circ Physiol 288:H1874–H1884
Ghidaoui M, Zhao M, McInnis D, Axworthy D (2005) A review of water hammer theory and practice. Appl Mech Rev 58:49–76
Hill JA, Olson EN (2008) Cardiac plasticity. N Engl J Med 358(13):1370–1380
Hung TK, Schuessler GB (1977) An analysis of the hemodynamics of the opening of aortic valves. J Biomech 10(9):597–606
Jones C, Sugawara M, Kondoh Y, Uchida K, Parker K (2002) Compression and expansion wavefront travel in canine ascending aortic flow: wave intensity analysis. Heart Vessels 16(3):91–98
Leischik R, Beller K, Erbel R (1996) Comparison of a new intravenous echo contrast agent (BY 963) with Albunex for opacification of left ventricular cavity. Basic Res Cardiol 91(1):101–109
Leischik R, Rose J, Caspari G, Skyschally A, Heusch G, Erbel R (1997a) Contrast echocardiography for assessment of myocardial perfusion. Herz 22(1):40–50
Leischik R, Bartel T, Möhlenkamp S, Bruch C, Buck T, Haude M et al (1997b) Stress echocardiography: new techniques. Eur Heart J. 18(Suppl D):49–56
Little WC, Ohno M, Kitzman DW, Thomas JD, Cheng CP (1995) Determination of left ventricular chamber stiffness from the time for deceleration of early left ventricular filling. Circulation 92:1933–1939
Nichols WW, O’Rourke MF (2005) McDonald’s blood flow in arteries: theoretic, experimental, and clinical principles. Hodder Arnold Publication, London
O’Rourke MF (2009) Time domain analysis of the arterial pulse in clinical medicine. Med Biol Eng Comput 47(2):119–129
Parker KH (2009) An introduction to wave intensity analysis. Med Biol Eng Comput 47(2):175–188
Parker K, Jones C, Dawson J, Gibson D (1988) What stops the flow of blood from the heart? Heart Vessels 4(4):241–245
Sugawara M, Uchida K, Kondoh Y, Magosaki N, Niki K, Jones CJ, Sugimachi M, Sunagawa K (1997) Aortic blood momentum—the more the better for the ejecting heart in vivo? Cardiovasc Res 33:433–446
Tyberg J, Davies J, Wang Z, Whitelaw W, Flewitt J, Shrive N, Francis D, Hughes A, Parker K, Wang J (2009) Wave intensity analysis and the development of the reservoir-wave approach. Med Biol Eng Comput 47:221–232
Van Steenhoven AA, Van Dongen MEH (1978) Model studies of the closing behavior of the aortic valve. J Fluid Mech 90(1):21–32
Van Steenhoven AA, Van Dongen MEH (1986) Model studies of the aortic pressure rise just after valve closure. J Fluid Mech 166:93–113
Virag Z, Lulić F (2008) Modeling of aortic valve dynamics in a lumped parameter model of left ventricular-arterial coupling. Annali dell’Università di Ferrara 54:335–347
Wang JJ, Parker KH (2004) Wave propagation in a model of the arterial circulation. J Biomech 37(4):457–470
Wang JJ, 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(4):H1358–H1368
Zeidan Z, Erbel R, Barkhausen J, Hunold P, Bartel T, Buck T (2003) Analysis of global systolic and diastolic left ventricular performance using volume–time curves by real-time three-dimensional echocardiography. J Am Soc Echocardiogr 16(1):29–37
Zhang H, Li J (2009) A novel wave reflection model of the human arterial system. Cardiovasc Eng 9:39–48
Author information
Authors and Affiliations
Corresponding author
Appendix
Appendix
From the equation for energy conservation:
(The change in kinetic energy: \(E_{\text{k}} = \frac{1}{2}m{\text{d}}v^{2}\); Strain energy of the fluid (stored by a volume and pressure change):\(\frac{\Delta p \cdot \Delta V}{2}\); Strain energy of the elastic arterial wall: \(\frac{{\sigma^{2} }}{2E} \cdot V\); Stress for thin arterial wall (d 0 diameter, t thickness) \(\sigma = \frac{{\Delta pd_{0} }}{2t}\))
Volume of the arterial wall:
Pulse wave velocity:\(c^{2} = \frac{A}{\rho }\frac{{{\text{d}}A}}{{{\text{d}}P}}\)
After substitution in Eq. (1)
Equation (3) maybe re-written as follows (with substitutions \(\Delta p = {\text{d}}p\), \(\Delta A = {\text{d}}A\) and \(A = \frac{{\pi d_{0}^{2} }}{4}\))
From Hooke’s law σ = Eδ and after substitutions
The change of cross-sectional area
After differentiating Eq. (5) with respect to p =>
Substituting Eq. (6) in Eq. (4) and after some simplification
After rearranging Eq. (7)
and finding the square root
For arteries
Rights and permissions
About this article
Cite this article
Žikić, D. A mathematical model of pressure and flow waveforms in the aortic root. Eur Biophys J 46, 41–48 (2017). https://doi.org/10.1007/s00249-016-1133-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00249-016-1133-2