Computational and Mathematical Methods in Medicine

Volume 2017, Article ID 4198095, 7 pages

https://doi.org/10.1155/2017/4198095

## Numerical Investigation of Pulse Wave Propagation in Arteries Using Fluid Structure Interaction Capabilities

^{1}Department of Mechanical Engineering, King Saud University, P.O. Box 800, Riyadh 11421, Saudi Arabia^{2}Laboratoire de Mécanique de Lille, UMR CNRS 8107, Villeneuve-d’Ascq, France

Correspondence should be addressed to Essam Al-Bahkali; as.ude.usk@ilakhabe

Received 5 May 2017; Revised 26 July 2017; Accepted 2 August 2017; Published 24 September 2017

Academic Editor: Michele Migliore

Copyright © 2017 Hisham Elkenani et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The aim of this study is to present a reliable computational scheme to serve in pulse wave velocity (PWV) assessment in large arteries. Clinicians considered it as an indication of human blood vessels’ stiffness. The simulation of PWV was conducted using a 3D elastic tube representing an artery. The constitutive material model specific for vascular applications was applied to the tube material. The fluid was defined with an equation of state representing the blood material. The onset of a velocity pulse was applied at the tube inlet to produce wave propagation. The Coupled Eulerian-Lagrangian (CEL) modeling technique with fluid structure interaction (FSI) was implemented. The scaling of sound speed and its effect on results and computing time is discussed and concluded that a value of 60 m/s was suitable for simulating vascular biomechanical problems. Two methods were used: foot-to-foot measurement of velocity waveforms and slope of the regression line of the wall radial deflection wave peaks throughout a contour plot. Both methods showed coincident results. Results were approximately 6% less than those calculated from the Moens-Korteweg equation. The proposed method was able to describe the increase in the stiffness of the walls of large human arteries via the PWV estimates.

#### 1. Introduction

Computational analysis of cardiovascular problems incorporating FSI is a challenging problem. Detailed analysis of the blood flow field and artery wall behavior can assist in clinicians’ assessment of vascular diseases [1]. The first person to investigate a formula for the velocity of pressure waves in a thin elastic tube was Young [2] in 1808. Womersley [3] investigated the dynamic response of a tube with a sinusoidal flow and defined an analytical solution for the flow domain.

In this study, we investigated the propagation of a pulse wave through an elastic vessel. This application is of clinical relevance as PWV measurements are currently considered to be the clinical gold-standard measure of arterial stiffness [4]. PWV is typically a disturbance’s propagation speed through a vessel resulting from the flow pressure. As blood is an almost-incompressible fluid [5–7], the finite PWV is mainly the result of the FSI between the local pressure of the blood on the vessel wall and the resultant wall deformation it causes.

To validate the obtained results, we used the same model used by Kuntz et al. [8] and Penrose and Staples [9] who validated their simulation, conducted with ANSYS-CFX commercial software, with the theoretical results obtained by the Moens-Korteweg equation [10]. Moatamedi et al. [11] and Souli et al. [12] used the same model in their study and validated their simulation, conducted with LS-DYNA commercial package, with the results obtained by the Moens-Korteweg equation.

Shahmirzadi et al. [13] validated their work conducted with ABAQUS explicit solver with the Moens-Korteweg equation, but they used a different model. Dias et al. [14] implemented their model using the commercial code RADIOSS (Altair Engineering) to investigate the effect of both elasticity and wall thickness on PWV through a long elastic tube.

#### 2. Flow in an Elastic Vessel

The transient progression of a pressure pulse through a tube has been investigated by many researchers over the years. A good review of this research is available [17]. The first work on wave propagation in an elastic tube was presented by Moens and Korteweg at the end of the 19th century [10]. It is based on Newton’s work on the speed of sound in air. Taking as the wall’s Young’s Modulus, as the thickness of the wall, as the inner radius, and as the fluid density and relating the change in radius to the applied pressure, the wave speed () can be written as Errors resulting from the thin tube assumption can be compensated by using the Bergel correction [18], which accounts for the thickness through Poisson’s ratio (). The difference between the modified wave speed () and the original one () is calculated by the following equation:where is the ratio of the wall thickness and tube outer radius. Simplifying this expression and incorporating it into (1), the modified wave speed () becomes

#### 3. Numerical Modeling

The numerical setup used for this three-dimensional fluid structure interaction study was based on a tube with an internal diameter of 4 mm and a wall thickness of 0.12 mm as shown in Figure 1. The length of the model was set as 100 times as long as the internal radius of the tube to be long enough as per the Moens-Korteweg equation’s condition. The vessel wall was considered as elastic with a density of 1075 kg/m^{3}, a Poisson’s ratio of 0.45, and an initial Young’s modulus () of 3 MPa. These values are well representative of a blood vessel’s physiological state. The blood material was modeled with a Newtonian incompressible equation of state (EOS) which related the density of the blood () to the external pressure and the sound speed () according to the following equation:where is the initial blood density. This constitutive material law gives slight compressibility to the blood, which improves the stability and convergence of our computation. Thus, the higher the sound speed the higher the incompressibility of the fluid. The initial blood density was set to 1000 kg/m^{3} and dynamic viscosity was 0.001 Pa·s. The sound speed was set to 60 m/s according to a separate study, where a range of sound speeds were tested, ranging from 15 m/s to the real value of 1460 m/s. It was found that a sound speed of 60 m/s provided reasonable results and incurred less computational cost. Higher values yielded similar accuracy but with relatively high computational cost. Lower values such as 15 m/s led to underestimation in the results. We thus concluded that a value of 60 m/s was practical for simulating vascular biomechanical problems.