# 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
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##### References:
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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