
Mathematical Biosciences and Engineering, 2018, 15(5): 10551076. doi: 10.3934/mbe.2018047.
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Mathematical modelling of cardiac pulse wave reflections due to arterial irregularities
1. Imperial College London, South Kensington Campus, London SW72AZ, United Kingdom
2. Ecole normale supérieure ParisSaclay, 61 Avenue du Président Wilson, Cachan 94230, France
Received: , Accepted: , Published:
This research aims to model cardiac pulse wave reflections due to the presence of arterial irregularities such as bifurcations, stiff arteries, stenoses or aneurysms. When an arterial pressure wave encounters an irregularity, a backward reflected wave travels upstream in the artery and a forward wave is transmitted downstream. The same process occurs at each subsequent irregularity, leading to the generation of multiple waves. An iterative algorithm is developed and applied to pathological scenarios to predict the pressure waveform of the reflected wave due to the presence of successive arterial irregularities. For an isolated stenosis, analysing the reflected pressure waveform gives information on its severity. The presence of a bifurcation after a stenosis tends do diminish the amplitude of the reflected wave, as bifurcations' reflection coefficients are relatively small compared to the ones of stenoses or aneurysms. In the case of two stenoses in series, local extrema are observed in the reflected pressure waveform which appears to be a characteristic of stenoses in series along an individual artery. Finally, we model a progressive change in stiffness in the vessel's wall and observe that the less the gradient stiffness is important, the weaker is the reflected wave.
Keywords: Arterial irregularties; blood pressure; wave mechanics; mathematical modelling; iterative algorithm
Citation: Alexandre Cornet. Mathematical modelling of cardiac pulse wave reflections due to arterial irregularities. Mathematical Biosciences and Engineering, 2018, 15(5): 10551076. doi: 10.3934/mbe.2018047
References:
 [1] L. Augsburger,P. Reymon,E. Fonck,Z. Kulcar,M. Farhat,M. Ohta,N. Stergiopulos,D. A. Rufenacht, Methodologies to assess blood flow in cerebral aneurysms: Current state of research and perspectives, Journal of Neuroradiology, 36 (2009): 270277.
 [2] I. Bakirtas,A. Antar, Effect of stenosis on solitary waves in arteries, International Journal of Engineering Science, 43 (2005): 730743.
 [3] W.S. Duan,Y.R. Shi,X.R Hong,K.P. Lu,J.B. Zhao, The reflection of soliton at multiarterial bifurcations and the effect of the arterial inhomogeneity, Physics Letters A, 295 (2002): 133138.
 [4] L. Formaggia,F. Nobile,A. Quarteroni,A. Veneziani, Multiscale modelling of the circulatory system: A preliminary analysis, Comput. Visual. Sci., 2 (1999): 7583.
 [5] L. Formaggia,F. Nobile,A. Quarteroni,J.F. Gerbeau, On the coupling of 3D and 1D NavierStokes equations for flow problems in compliant vessels, Comp. Methods Appl. Mech. Engng., 191 (2001): 561582.
 [6] K. Hayashi, K. Handa, S. Nagasawa and A. Okumura, Stiffness and elastic behaviour of human intracranial and extracranial arteries, J. Biomech., 13 (1980), 175179,181184.
 [7] G. L. Langewouters,K. H. Wesseling,W. J. A. Goedhard, The static properties of 45 human thoracic and 20 abdominal aortas in vitro and the parameters of a new model, J. Biomech., 17 (1984): 425435.
 [8] C. A. D. Leguy,E. M. H. Bosboom,H. Gelderblom,A. P. G. Hoeks,F. N. Van de Vosse, Estimation of distributed arterial mechanical properties using a wave propagation model in a reverse way, Medical Engineering & Physics, 32 (2010): 957967.
 [9] K. S. Matthus,J. Alastruey,J. Peiro,A. W. Khir,P. Segers,R. P. Verdonck,K. H. Parker,S. J. Sherwin, Pulse wave propagation in a model human arterial network: Assessment of 1D viscoelastic simulations against in vitro measurements, J. Biomechanics, 44 (2011): 22502258.
 [10] H. G. Morales,I. Larrabide,A. J. Geers,M. L. Aguilar,A. F. Frangi, Newtonian and nonNewtonian blood flow in coiled cerebral aneurysms, J. Biomechanics, 46 (2013): 21582164.
 [11] W. W. Nichols, J. W. Petersen, S. J. Denardo and D. D. Christou Arterial stiffness, wave reflection amplitude and left ventricular afterload are increased in overweight individuals, Artery Research, 7 (2013), 222229.
 [12] Z. OvadiaBlechman,S. Einav,U. Zaretsky,D. Castel,E. Eldar, Characterization of arterial stenosis and elasticity by analysis of highfrequency pressure wave components, Computer in Biology and Medicine, 33 (2003): 375393.
 [13] C. S. Park,S. J. Payne, Nonlinear and viscous effects on wave propagation in an elastic axisymmetric vessel, J. of Fluids and Structures, 27 (2011): 134144.
 [14] K. H. Parker, An introduction to wave intensity analysis, Medical & Biological Engineering & Computing, 47 (2009): 175199.
 [15] T. J. Pedley, Nonlinear pulse wave reflection at an arterial stenosis, J. of Biomechanical Engineering, 105 (1983): 353359.
 [16] S. I. S. Pinto,E. Doutel,J. B. L. M. Campos,J. M. Miranda, Blood analog fluid flow in vessels with stenosis: Development of an Openfoam code to stimulate pulsatile flow and elasticity of the fluid, APCBEE Procedia, 7 (2013): 7379.
 [17] A. Quarteroni,L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, Handbook of Numerical Analysis, 12 (2004): 3127.
 [18] P. Segers,J. Kips,B. Trachet,A. Swillens,S. Vermeersch,D. Mahieu,E. Rietzschel,M. D. Buyzere,L. V. Bortel, Limitations and pitfalls of noninvasive measurement of arterial pressure wave reflections and pulse wave velocity, Artery Research, 3 (2009): 7988.
 [19] D. Shahmirzadi,E. E. Konofagou, Quantification of arterial wall inhomogeneity size, distribution, and modulus contrast using FSI numerical pulse wave propagation, Artery Research, 8 (2014): 5765.
 [20] N. Stergiopulos,D. F. Young,T. R. Rogge, Computer simulation of arterial flow with applications to arterial and aortic stenoses, J. Biomechanics, 25 (1992): 14771488.
 [21] N. Stergiopulos,F. Spiridon,F. Pythoud,J. J. Meister, On the wave transmission and reflection properties of stenoses, J. Biomechanics, 29 (1996): 3138.
 [22] A. Swillens,P. Segers, Assessment of arterial pressure wave reflection: Methodological considerations, Artery Research, 2 (2008): 122131.
 [23] A. Tozeren, Elastic properties of arteries and their influence on the cardiovascular system, J. Biomech. Eng., 106 (1984): 182185.
 [24] C. Tu,M. Deville,L. Dheur,L. Vanderschuren, Finite element simulation of pulsatile flow through arterial stenosis, J. Biomechanics, 25 (1992): 11411152.
 [25] J. J. Wang,K. H. Parker, Wave propagation in a model of the arterial circulation, J. Biomechanics, 37 (2004): 457470.
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