# American Institute of Mathematical Sciences

October  2018, 15(5): 1055-1076. doi: 10.3934/mbe.2018047

## 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 Paris-Saclay, 61 Avenue du Président Wilson, Cachan 94230, France

* Corresponding author: alexandre.cornet@ens-paris-saclay.fr

Received  December 24, 2016 Accepted  January 23, 2018 Published  May 2018

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.

Citation: Alexandre Cornet. Mathematical modelling of cardiac pulse wave reflections due to arterial irregularities. Mathematical Biosciences & Engineering, 2018, 15 (5) : 1055-1076. doi: 10.3934/mbe.2018047
##### References:
 [1] L. Augsburger, P. Reymon, E. Fonck, Z. Kulcar, M. Farhat, M. Ohta, N. Stergiopulos and D. A. Rufenacht, Methodologies to assess blood flow in cerebral aneurysms: Current state of research and perspectives, Journal of Neuroradiology, 36 (2009), 270-277.  doi: 10.1016/j.neurad.2009.03.001. [2] I. Bakirtas and A. Antar, Effect of stenosis on solitary waves in arteries, International Journal of Engineering Science, 43 (2005), 730-743.  doi: 10.1016/j.ijengsci.2004.12.014. [3] W.-S. Duan, Y.-R. Shi, X.-R Hong, K.-P. Lu and J.-B. Zhao, The reflection of soliton at multi-arterial bifurcations and the effect of the arterial inhomogeneity, Physics Letters A, 295 (2002), 133-138.  doi: 10.1016/S0375-9601(02)00078-6. [4] L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modelling of the circulatory system: A preliminary analysis, Comput. Visual. Sci., 2 (1999), 75-83.  doi: 10.1007/s007910050030. [5] L. Formaggia, F. Nobile, A. Quarteroni and J.-F. Gerbeau, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels, Comp. Methods Appl. Mech. Engng., 191 (2001), 561-582.  doi: 10.1016/S0045-7825(01)00302-4. [6] K. Hayashi, K. Handa, S. Nagasawa and A. Okumura, Stiffness and elastic behaviour of human intracranial and extracranial arteries, J. Biomech., 13 (1980), 175-179,181-184. doi: 10.1016/0021-9290(80)90191-8. [7] G. L. Langewouters, K. H. Wesseling and 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), 425-435.  doi: 10.1016/0021-9290(84)90034-4. [8] C. A. D. Leguy, E. M. H. Bosboom, H. Gelderblom, A. P. G. Hoeks and 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), 957-967.  doi: 10.1016/j.medengphy.2010.06.010. [9] K. S. Matthus, J. Alastruey, J. Peiro, A. W. Khir, P. Segers, R. P. Verdonck, K. H. Parker and S. J. Sherwin, Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements, J. Biomechanics, 44 (2011), 2250-2258.  doi: 10.1016/j.jbiomech.2011.05.041. [10] H. G. Morales, I. Larrabide, A. J. Geers, M. L. Aguilar and A. F. Frangi, Newtonian and non-Newtonian blood flow in coiled cerebral aneurysms, J. Biomechanics, 46 (2013), 2158-2164.  doi: 10.1016/j.jbiomech.2013.06.034. [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), 222-229. doi: 10.1016/j.artres.2013.08.001. [12] Z. Ovadia-Blechman, S. Einav, U. Zaretsky, D. Castel and E. Eldar, Characterization of arterial stenosis and elasticity by analysis of high-frequency pressure wave components, Computer in Biology and Medicine, 33 (2003), 375-393.  doi: 10.1016/S0010-4825(03)00004-0. [13] C. S. Park and S. J. Payne, Nonlinear and viscous effects on wave propagation in an elastic axisymmetric vessel, J. of Fluids and Structures, 27 (2011), 134-144.  doi: 10.1016/j.jfluidstructs.2010.10.003. [14] K. H. Parker, An introduction to wave intensity analysis, Medical & Biological Engineering & Computing, 47 (2009), 175-199.  doi: 10.1007/s11517-009-0439-y. [15] T. J. Pedley, Nonlinear pulse wave reflection at an arterial stenosis, J. of Biomechanical Engineering, 105 (1983), 353-359.  doi: 10.1115/1.3138432. [16] S. I. S. Pinto, E. Doutel, J. B. L. M. Campos and 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), 73-79.  doi: 10.1016/j.apcbee.2013.08.015. [17] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, Handbook of Numerical Analysis, 12 (2004), 3-127.  doi: 10.1016/S1570-8659(03)12001-7. [18] P. Segers, J. Kips, B. Trachet, A. Swillens, S. Vermeersch, D. Mahieu, E. Rietzschel, M. D. Buyzere and L. V. Bortel, Limitations and pitfalls of non-invasive measurement of arterial pressure wave reflections and pulse wave velocity, Artery Research, 3 (2009), 79-88.  doi: 10.1016/j.artres.2009.02.006. [19] D. Shahmirzadi and E. E. Konofagou, Quantification of arterial wall inhomogeneity size, distribution, and modulus contrast using FSI numerical pulse wave propagation, Artery Research, 8 (2014), 57-65.  doi: 10.1016/j.artres.2014.01.006. [20] N. Stergiopulos, D. F. Young and T. R. Rogge, Computer simulation of arterial flow with applications to arterial and aortic stenoses, J. Biomechanics, 25 (1992), 1477-1488.  doi: 10.1016/0021-9290(92)90060-E. [21] N. Stergiopulos, F. Spiridon, F. Pythoud and J. J. Meister, On the wave transmission and reflection properties of stenoses, J. Biomechanics, 29 (1996), 31-38.  doi: 10.1016/0021-9290(95)00023-2. [22] A. Swillens and P. Segers, Assessment of arterial pressure wave reflection: Methodological considerations, Artery Research, 2 (2008), 122-131.  doi: 10.1016/j.artres.2008.05.001. [23] A. Tozeren, Elastic properties of arteries and their influence on the cardiovascular system, J. Biomech. Eng., 106 (1984), 182-185.  doi: 10.1115/1.3138479. [24] C. Tu, M. Deville, L. Dheur and L. Vanderschuren, Finite element simulation of pulsatile flow through arterial stenosis, J. Biomechanics, 25 (1992), 1141-1152.  doi: 10.1016/0021-9290(92)90070-H. [25] J. J. Wang and K. H. Parker, Wave propagation in a model of the arterial circulation, J. Biomechanics, 37 (2004), 457-470.  doi: 10.1016/j.jbiomech.2003.09.007.

show all references

##### References:
 [1] L. Augsburger, P. Reymon, E. Fonck, Z. Kulcar, M. Farhat, M. Ohta, N. Stergiopulos and D. A. Rufenacht, Methodologies to assess blood flow in cerebral aneurysms: Current state of research and perspectives, Journal of Neuroradiology, 36 (2009), 270-277.  doi: 10.1016/j.neurad.2009.03.001. [2] I. Bakirtas and A. Antar, Effect of stenosis on solitary waves in arteries, International Journal of Engineering Science, 43 (2005), 730-743.  doi: 10.1016/j.ijengsci.2004.12.014. [3] W.-S. Duan, Y.-R. Shi, X.-R Hong, K.-P. Lu and J.-B. Zhao, The reflection of soliton at multi-arterial bifurcations and the effect of the arterial inhomogeneity, Physics Letters A, 295 (2002), 133-138.  doi: 10.1016/S0375-9601(02)00078-6. [4] L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modelling of the circulatory system: A preliminary analysis, Comput. Visual. Sci., 2 (1999), 75-83.  doi: 10.1007/s007910050030. [5] L. Formaggia, F. Nobile, A. Quarteroni and J.-F. Gerbeau, On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels, Comp. Methods Appl. Mech. Engng., 191 (2001), 561-582.  doi: 10.1016/S0045-7825(01)00302-4. [6] K. Hayashi, K. Handa, S. Nagasawa and A. Okumura, Stiffness and elastic behaviour of human intracranial and extracranial arteries, J. Biomech., 13 (1980), 175-179,181-184. doi: 10.1016/0021-9290(80)90191-8. [7] G. L. Langewouters, K. H. Wesseling and 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), 425-435.  doi: 10.1016/0021-9290(84)90034-4. [8] C. A. D. Leguy, E. M. H. Bosboom, H. Gelderblom, A. P. G. Hoeks and 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), 957-967.  doi: 10.1016/j.medengphy.2010.06.010. [9] K. S. Matthus, J. Alastruey, J. Peiro, A. W. Khir, P. Segers, R. P. Verdonck, K. H. Parker and S. J. Sherwin, Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements, J. Biomechanics, 44 (2011), 2250-2258.  doi: 10.1016/j.jbiomech.2011.05.041. [10] H. G. Morales, I. Larrabide, A. J. Geers, M. L. Aguilar and A. F. Frangi, Newtonian and non-Newtonian blood flow in coiled cerebral aneurysms, J. Biomechanics, 46 (2013), 2158-2164.  doi: 10.1016/j.jbiomech.2013.06.034. [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), 222-229. doi: 10.1016/j.artres.2013.08.001. [12] Z. Ovadia-Blechman, S. Einav, U. Zaretsky, D. Castel and E. Eldar, Characterization of arterial stenosis and elasticity by analysis of high-frequency pressure wave components, Computer in Biology and Medicine, 33 (2003), 375-393.  doi: 10.1016/S0010-4825(03)00004-0. [13] C. S. Park and S. J. Payne, Nonlinear and viscous effects on wave propagation in an elastic axisymmetric vessel, J. of Fluids and Structures, 27 (2011), 134-144.  doi: 10.1016/j.jfluidstructs.2010.10.003. [14] K. H. Parker, An introduction to wave intensity analysis, Medical & Biological Engineering & Computing, 47 (2009), 175-199.  doi: 10.1007/s11517-009-0439-y. [15] T. J. Pedley, Nonlinear pulse wave reflection at an arterial stenosis, J. of Biomechanical Engineering, 105 (1983), 353-359.  doi: 10.1115/1.3138432. [16] S. I. S. Pinto, E. Doutel, J. B. L. M. Campos and 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), 73-79.  doi: 10.1016/j.apcbee.2013.08.015. [17] A. Quarteroni and L. Formaggia, Mathematical modelling and numerical simulation of the cardiovascular system, Handbook of Numerical Analysis, 12 (2004), 3-127.  doi: 10.1016/S1570-8659(03)12001-7. [18] P. Segers, J. Kips, B. Trachet, A. Swillens, S. Vermeersch, D. Mahieu, E. Rietzschel, M. D. Buyzere and L. V. Bortel, Limitations and pitfalls of non-invasive measurement of arterial pressure wave reflections and pulse wave velocity, Artery Research, 3 (2009), 79-88.  doi: 10.1016/j.artres.2009.02.006. [19] D. Shahmirzadi and E. E. Konofagou, Quantification of arterial wall inhomogeneity size, distribution, and modulus contrast using FSI numerical pulse wave propagation, Artery Research, 8 (2014), 57-65.  doi: 10.1016/j.artres.2014.01.006. [20] N. Stergiopulos, D. F. Young and T. R. Rogge, Computer simulation of arterial flow with applications to arterial and aortic stenoses, J. Biomechanics, 25 (1992), 1477-1488.  doi: 10.1016/0021-9290(92)90060-E. [21] N. Stergiopulos, F. Spiridon, F. Pythoud and J. J. Meister, On the wave transmission and reflection properties of stenoses, J. Biomechanics, 29 (1996), 31-38.  doi: 10.1016/0021-9290(95)00023-2. [22] A. Swillens and P. Segers, Assessment of arterial pressure wave reflection: Methodological considerations, Artery Research, 2 (2008), 122-131.  doi: 10.1016/j.artres.2008.05.001. [23] A. Tozeren, Elastic properties of arteries and their influence on the cardiovascular system, J. Biomech. Eng., 106 (1984), 182-185.  doi: 10.1115/1.3138479. [24] C. Tu, M. Deville, L. Dheur and L. Vanderschuren, Finite element simulation of pulsatile flow through arterial stenosis, J. Biomechanics, 25 (1992), 1141-1152.  doi: 10.1016/0021-9290(92)90070-H. [25] J. J. Wang and K. H. Parker, Wave propagation in a model of the arterial circulation, J. Biomechanics, 37 (2004), 457-470.  doi: 10.1016/j.jbiomech.2003.09.007.
Graphical representation of the system with n discontinuities, $S_{n}$ with $n \in \mathbb N$
$(x, t)$-diagram of the system with 3 discontinuities, $S_{3}$
Graphical representation of the system with 4 discontinuities, $S_{4}$
Normalized reflected pressure versus time at the entrance of a 5 mm long stenosis modelled with S2, with different values of reflection coefficients
Computational solution absolutely converging towards the analytical solution as $\epsilon \rightarrow 0$ in the case of two discontinuities $(n = 2)$ and $\gamma_{01} = - \gamma_{12} = 0.8$
Normalized reflected pressure versus time at the entrance of a 5 mm long stenosis, 6 cm before a bifurcation modelled with $S_3$, with different values of reflection coefficients for the upstream stenosis and a constant reflection coefficient for the bifurcation
Normalized reflected pressure versus time at the entrance of two 5 mm stenoses in series modelled with $S_4$, with different value of reflection coefficients for the upstream stenosis and constant reflection coefficients for the downstream stenosis
Normalized reflected pressure versus time at the entrance of a vessel with a smooth change in stiffness modelled with $S_4$ for different gradients in stiffness
Numerical analysis of the effet of the control parameter $m$ on the computational solution in the case of two discontinuities $(n = 2)$ with $\gamma_{01} = - \gamma_{12} = 0.8$ as in Section 4.2
Numerical analysis of the effet of the control parameter $m$ on the computational solution in the case of three discontinuities $(n = 3)$ with $\gamma_{01} = - \gamma_{12} = 0.8$ and $\gamma_{23} = 0.05$ as in Section 4.3
Numerical analysis of the effet of the control parameter $m$ on the computational solution in the case of four discontinuities $(n = 4)$ with $\gamma_{01} = - \gamma_{12} = 0.8$ and $\gamma_{23} = -\gamma_{34} = 0.2$ as in Section 4.4
Absolute convergence of the computational solution as a function of number of iterations in the case of two discontinuities $(n = 2)$ and for different values of reflection coefficients
(Top) computational solution absolutely converging towards the analytical solution as $\epsilon \rightarrow 0$ for (left) n = 3 with $\gamma_{01} = - \gamma_{12} = 0.8$ and $\gamma_{23} = 0.05$ and for (right) n = 4 with $\gamma_{01} = - \gamma_{12} = 0.8$ and $\gamma_{23} = -\gamma_{34} = 0.2$ and (bottom) $\Delta_r(F_N)$ convergence as the number of iterations increases for (left) $n = 3$ and (right) $n = 4$ for different values of reflection coefficients
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