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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.

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 [

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 [

To validate the obtained results, we used the same model used by Kuntz et al. [

Shahmirzadi et al. [

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 [

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 ^{3}, a Poisson’s ratio of 0.45, and an initial Young’s modulus (^{3} and dynamic viscosity was 0.001 Pa·s. The sound speed

The long elastic tube model and boundary conditions.

From (^{3} more than the arterial wall’s Young’s modulus (^{3} or more are not practicable for the FSI study with the current available modeling techniques as noted by Moatamedi et al. [

Meshed parts of model.

Inlet boundary conditions; a velocity pulse in the axial direction.

Grote and Keller [

The simulation ran with a 20 ms explicit step time for a Young’s moduli range of 3, 2, 1, 0.5, and 0.1 MPa assigned to the tube material. Using a Dell computer with an Intel® Xeon® CPU running at 2.8 GHz with 12 processors and 24 GB RAM, it took approximately 13 h to complete each calculation. Figure

Three different time frames of radial disturbance of tube wall because of wave propagation. The displacement of the structure has been magnified by factor of 5. The unit of the scale is mm.

Figure

Axial velocity waveforms at different positions along the tube center line for an artery with Young’s modulus (

Two different methods were used to assist the PWV: foot-to-foot measuring of velocity waveforms and slope of the regression line of the wall radial deflection wave peaks throughout a contour plot.

The PWV was calculated from the time delay of each waveform relative to its preceding waveform. The waveforms were plotted for equal-spaced intervals along the tube center line. To measure the time interval between two sites, we used the foot-to-foot method [

Arrival time for two progressive waveforms indicated by foot-to-foot method.

We take the sites at distances 70, 100, and 130 mm from the tube inlet to be far away from the tube ends, where wave reflections are expected due to fixed boundary conditions at those ends.

The wall radial deflection at equally spaced locations along the tube center line was plotted against time and is shown as a 3D mesh plot in Figure

Model with Young’s modulus,

3D mesh plot

2D contour plot

The obtained results were plotted with the values calculated from the modified Moens-Korteweg equation, (

PWV as a function of Young’s modulus: comparison of numerical results conducted with different commercial packages and values calculated from the Moens-Korteweg equation.

Figure

Effect of ageing on elastic properties of arteries; incremental Young’s modulus plotted against age for normal human aorta at a pressure of 100 mm Hg [

Normal values for PWV (average according to age (1455 subjects); boxes contain 50% of the data and bars contain the remainder; horizontal lines indicate medians) [

Figure

In this study, a basic computational scheme for a strongly coupled FSI in an elastic artery was developed and validated with basic theory. Qualitative agreement was obtained, indicating that this computational method for PWV analysis is accurate enough to evaluate its value with accepted accuracy. The scaling down of sound speed has a significant effect on results convergence and computation cost, and we conclude that a value of 60 m/s is reasonable enough for solving vascular biomechanical problems. On the other side, the PWV values obtained from our new approach corresponded well with in vivo reference values published in literatures [

The authors declare that there are no conflicts of interest regarding the publication of this paper.

The authors are grateful to the College of Engineering Research Center at King Saud University for their support of this work.