One-Dimensional Model for Propagation of a Pressure Wave in a Model of the Human Arterial Network: Comparison of Theoretical and Experimental Results

Abstract

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.

Comparing different numerical methods for solving arterial 1D flows in networks

“…We are now able to plan to present a more realistic arterial network and we want to couple it with a venous network such as the venous network mentioned in Fullana and Zaleski (2009). Saito et al (2011). A pump produces a pulse of water which propagates in a nine-artery network.…”

“…Alastruey et al (2011) and Saito et al (2011) successfully compared an experimentally realistic visco-elastic model of tubes and such 1D models. Here, we emphasise the numerical methods and we will simulate the results of Saito et al (2011) as a test case.…”

“…Many authors (Zagzoule and Marc-Vergnes 1986;Olufsen et al 2000;Alastruey et al 2011;Saito et al 2011) have succeeded in simulating an arterial tree with a 1D or integral approach. Alastruey et al (2011) and Saito et al (2011) successfully compared an experimentally realistic visco-elastic model of tubes and such 1D models. Here, we emphasise the numerical methods and we will simulate the results of Saito et al (2011) as a test case.…”

Euler, Father of Hemodynamics

“…The attenuation is primarily caused by the viscosity of the walls. Furthermore, the stress-strain relation for flexible tubes such as arteries is nonlinear (Saito et al, 2011). However, as a first approximation, we can assume that the artery is a linearly elastic material that obeys Hooke's law, where the stress is related to the strain in both the longitudinal and circumferential direction through the Young's modulus E. From these we get that in a Kelvin-Voigt material, stress σ, strain ε, and their rates of change with respect to time t are governed by an equation of the form…”

Effects of aortic valve diseases on pressure profiles in selected locations of the human arterial system

“…The tube law is determined by the viscoelastic properties of the vessel walls [26]. Voigt-type visco-elastic laws reproduce, in the first approximation, the main effects of the walls' properties on the blood flow in large arteries, including hysteresis and creep [27][28][29][30][31][32]. An example of this type of law that neglects the effects of wall inertia and longitudinal prestress [36] is given by [32]:…”