Wave Propagation in a Viscous Fluid Contained in an Orthotropic Elastic Tube

The problem of pressure wave propagation through a viscous fluid contained in an orthotropic elastic tube is considered in connection with arterial blood flow. Solutions to the fluid flow and elasticity equations are obtained for the presence of a reflected wave. Numerical results are presented for both isotropic and orthotropic elastic tubes. In particular, the pressure pulse, flow rate, axial fluid velocity, and wall displacements are plotted vs. time at various stations along the ascending aorta of man. The results indicate an increase in the peak value of the pressure pulse and a decrease in the flow rate as the pulse propagates away from the heart. Finally, the velocity of wave propagation depends mainly on the tangential modulus of elasticity of the arterial wall, and anisotropy of the wall accounts in part for the reduction of longitudinal movements and an increase in the hydraulic resistance.     

Theoretical analysis of pressure pulse propagation in arterial vessels.

An original mathematical model of viscous fluid motion in a tapered and distensible tube is presented. The model equations are deduced by assuming a two-dimensional flow and taking into account the nonlinear terms in the fluid motion equations, as well as the nonlinear deformation of the tube wall. One distinctive feature of the model is the formal integration with respect to the radial coordinate of the Navier-Stokes equations by power series expansion. The consequent computational frame allows an easy, accurate evaluation of the effects produced by changing the values of all physical and geometrical tube parameters. The model is employed to study the propagation along an arterial vessel of a pressure pulse produced by a single flow pulse applied at the proximal vessel extremity. In particular, the effects of the natural taper angle of the arterial wall on pulse propagation are investigated. The simulation results show that tapering considerably influences wave attenuation but not wave velocity. The substantially different behavior of pulse propagation, depending upon whether it travels towards the distal extremity or in the opposite direction, is observed: natural tapering causes a continuous increase in the pulse amplitude as it moves towards the distal extremity; on the contrary, the reflected pulse, running in the opposite direction, is greatly damped. For a vessel with physical and geometrical properties similar to those of a canine femoral artery and 0.1 degree taper angle, the forward amplification is about 0.9 m-1 and the backward attenuation is 1.4 m-1, so that the overall tapering effect gives a remarkably damped pressure response. For a natural taper angle of 0.14 degrees the perturbation is almost extinct when the pulse wave returns to the proximal extremity.     

Influence of the distensibility of large arteries on the longitudinal impedance: application for the development of non-invasive techniques to the diagnosis of arterial diseases

Background

This study shows that the arterial longitudinal impedance constitutes a hemodynamic parameter of interest for performance characterization of large arteries in normal condition as well as in pathological situations. For this purpose, we solved the Navier?CStokes equations for an incompressible flow using the finite element analysis method and the Arbitrary Lagrangian Eulerian (ALE) formulation. The mathematical model assumes a two-dimensional flow and takes into account the nonlinear terms in the equations of fluid motion that express the convective acceleration, as well as the nonlinear deformation of the arterial wall. Several numerical simulations of the blood flow in large vessels have been performed to study the propagation along an arterial vessel of a pressure gradient pulse and a rate flow pulse. These simulations include various deformations of the wall artery leading to parietal displacements ranging from 0 (rigid wall) to 15% (very elastic wall) in order to consider physiological and pathological cases.

Results

The results show significant changes of the rate flow and the pressure gradient wave as a function of aosc, the relative variation in the radius of the artery over a cardiac cycle. These changes are notable beyond a critical value of aosc equal to 0.05. This critical value is also found in the evolution of the longitudinal impedance. So, above a variation of radius of 5%, the convective acceleration, created by the fluid-wall interactions, have an influence on the flow detectable on the longitudinal impedance.

Conclusions

The interpretation of the evolution of the longitudinal impedance shows that it could be a mean to test the performance of large arteries and can contribute to the diagnosis of parietal lesions of large arteries. For a blood vessel with a wall displacement higher than 5% similar to those of large arteries like the aorta, the longitudinal impedance is substantially greater than that obtained in the absence of wall displacement. This study also explains the effects of convective acceleration, on the shape of the decline of the pressure gradient wave and shows that they should not be neglected when the variation in radius is greater than 5%.     

Wave Propagation through a Newtonian Fluid Contained within a Thick-Walled,Viscoelastic Tube

The propagation of harmonic pressure waves through a Newtonian fluid contained within a thick-walled, viscoelastic tube is considered as a model of arterial blood flow. The fluid is assumed to be homogeneous and Newtonian, and its motion to be laminar and axisymmetric. The wall is assumed to be isotropic, incompressible, linear, and viscoelastic. It is also assumed that the motion is such that the convective acceleration is negligible. The motion of the fluid is described by the linearized form of the Navier-Stokes equations and the motion of the wall by classical elasticity theory. The frequency dependence of the wall mechanical properties are represented by a three parameter, relaxation-type model. Using boundary conditions describing the continuity of stress and velocity components in the fluid and the wall, explicit solutions for the system of equations of the model have been obtained. The longitudinal fluid impedance has been expressed in terms of frequency and the system parameters. The frequency equation has been solved and the propagation constant also expressed in terms of frequency and system parameters. The results indicate that the fluid impedance is smaller than predicted by the rigid tube model or by Womersley''s constrained elastic tube model. Also, the velocity of propagation is generally slower and the transmission per wavelength less than predicted by Womersley''s elastic tube model. The propagation constant is very sensitive to changes in the degree of wall viscoelasticity.     

Pressure and flow in the systemic arterial system

The dynamic characteristics of the proximal arterial system are studied by solving the nonlinear momentum and mass conservation equations for pressure and flow. The equations are solved for a model systemic arterial system that includes the aorta, common iliacs, and the internal and external iliac arteries. The model includes geometric and elastic taper of the aorta, nonlinearly elastic arteries, side flows, and a complex distal impedance. The model pressure wave shape, inlet and outlet impedance, wave travel, and apparent wave velocity compare favorably with the values measured on humans. Calculations indicate that: (i) reflections are the major factor determining the shape and distal amplification of the pressure wave in the arterial tree; (ii) although important in attenuating the proximal transmission of reflecting waves, geometric taper is not the major cause of the distal pressure wave amplification; (iii) the dicrotic wave is a result of peripheral reflection and is not due to the sudden change in flow at the end of systole; (iv) the elastic taper and nonlinearity of the wall elasticity are of minor significance in determining the flow and pressure profiles; and (v) in spite of numerous nonlinearities, the system behaves in a somewhat linear fashion for the lower frequency components.     

The Influence of Drag on Nonlinear Oscillatory Flow through Concentric Annulus

A mathematical model has been developed to study the effect of particle drag parameter and frequency parameter on velocity and pressure gradient in nonlinear oscillatory two phase flow. The main purpose is to apply the model to study the combined effect of introduction of the catheter and elastic properties of the arterial wall on the pulsatile nature of the blood flow. We model the artery as an isotropic thin walled elastic tube and the catheter as a coaxial flexible tube. Blood is modeled as an incompressible particulate viscous Newtonian fluid. Perturbation technique has been applied to find the approximations for velocity and pressure gradient up to second order. Numerical solutions are investigated with graphical presentations to understand the effects of drag parameter, frequency parameter and phase angle on velocity along radial direction and pressure gradient along axial directions. As the drag parameter increases, mean pressure gradient and mean velocity will be decreased. As frequency parameter increases mean velocity profile bends near the outer wall. Due to elastic nature of artery wall, a thin catheter experience small oscillations and a thick catheter remains stationary inside the artery. Finally, the effect of catheterization on various physiologically important flow rate characteristics—mean velocity, mean pressure gradient are studied for a range of different catheter sizes, particle drag parameter and frequency parameters.     

Effects of elastic property of the wall on flow characteristics through arterial stenoses

Hemodynamic characteristics of blood flow through arterial stenoses are numerically investigated in this work. The blood is assumed as a Newtonian fluid and the pulsatile nature of flow is modeled by using measured values of the flowrate and pressure for the canine femoral artery. An isotropic elastic and incompressible material is assumed for the wall at each axial section, but a non-uniform distribution of the shear modulus in axial direction is used to model the high stiffness of the wall at the stenosis location. Full Navier equations for a thick wall are used as the governing equations for the wall displacements. A continuous grid extending over the flow field and the wall is considered and governing equations are transformed for use in the computational domain. Discretized forms of the transformed wall and flow equations, which are coupled through the boundary conditions at their interface, are obtained by control volume method and simultaneously solved using the well-known SIMPLER algorithm. To study the effects of wall deformability, solutions are obtained for both rigid and elastic walls. The results indicate that deformability of the wall causes an increase in the time average of pressure drop, but a decrease in the maximum wall shear stress. Displacement and stress distributions in the wall are presented.     

Measurement of oscillatory flow pressure gradient in an elastic artery model

In vitro experiments were conducted to measure the oscillatory flow pressure gradient along an elastic tube in order to assess the recent nonlinear theory of Wang and Tarbell. According to this theory, in an elastic tube with oscillatory flow, the mean (time-averaged) pressure gradient cannot be calculated using Poiseuille's law. The effect of wall motion creates a nonlinear convective acceleration, and an induced mean pressure gradient is required to balance the convective acceleration. The induced mean pressure gradient depends on the diameter variation over a cycle, the pulsatility and unsteadiness of the flow, and the phase difference between the pressure wave form and the flow wave form. The amplitude of the pressure gradient also depends on these parameters and may deviate significantly from Womersley's rigid tube theory. A flow loop was constructed to produce oscillatory flow in an elastic tube. Flow wave forms were measured with an ultrasonic flow probe, and ultrasonic diameter crystals were used to measure wall movement. A special device for pressure drop measurement was constructed using Millar catheter tip transducers to obtain both forward and backward pressure drops that were then averaged. This averaging method eliminated the static error of the pressure transducers. The pressure-flow phase angle was varied by clamping a distal elastic section at various locations. Pressure gradients were obtained for a range of phase angles between −55 ° and +49 °. The mean and amplitude of the measured pressure gradient were compared to theoretical values. Both positive and negative induced mean pressure gradients were measured over the range of phase angles. The measured pressure gradient amplitudes were always lower than predicted by Womersley's rigid tube theory. The experimental means and amplitudes are in good agreement with the elastic tube theoretical values. Thus, the experiments verify the theory of Wang and Tarbell.     

Arterial remodeling in response to increased blood flow using a constituent-based model

Previous theoretical models of arterial remodeling in response to changes in blood flow were based on the assumption that material properties of the arterial wall remain unchanged during the remodeling process. According to experimental findings, however, remodeling due to increased flow is accompanied by alteration in the structural properties of elastin, which results in a decrease in its effective elastic stiffness. To account for these effects, we propose a predictive model of arterial remodeling hypothesizing that the variation in mechanical properties of elastin is initiated and driven by the deviation of the intimal shear stress from its baseline value. Geometrical remodeling restores the wall stress distribution as it was under normal flow conditions. A constrained mixture approach is followed. Artery is modeled as a thick-walled cylindrical tube made of non-linear, elastic, anisotropic and incompressible material. Data for a rabbit thoracic aorta have been employed. At the final adapted state, the model predicts a non-monotonic dependence of arterial compliance on the magnitude of flow. This result is in agreement with available experimental data in the literature.     

Optimal modeling of an arterial system

The aortic blood flow is described by a set of nonlinear hyperbolic partial differential equations that account for mass and momentum conservation, and nonlinear models for the mechanical properties of the artery. Identification is used for determining the wave speed, arterial taper, and cross section: these parameters reflect the elastic characteristics of the aorta wall and control the pulsatile response. The differential equations were numerically integrated by the Lax-Wendroff scheme of Abarbanel and Goldberg [J. Comput. Phys. 10:1–21 (1972)] that avoids nonlinear oscillations. The Gauss-Newton technique was used for the parameter identification. By reference to reported elocity and pressure input-output pairs, a parameter vector is found such that the distance in the L₂ norm between the predicted outputs and the measured functions is minimal. Calculations of the velocity and pressure waves show excellent compatibility of the model with reported experimental data: starting from arbitrary parameter estimates, which yield grossly distorted waveforms, the error is typically reduced to 7–8%. Introduction of viscoelastic behaviour for the arterial wall in the form of a Volterra integral for the cross section does not lead to significant improvement. Numerical examples are presented which prove the convergence, accuracy, and stability of the algorithm. Emphasis is placed on the computational feasibility of the proposed system identification.     

Pulse waves in prestressed arteries

In order to better understand the effect of initial stress in blood flow in arteries, a theoretical analysis of wave propagation in an initially inflated and axially stretched cylindrical thick shell is investigated. For simplicity in the mathematical analysis, the blood is assumed to be an incompressible inviscid fluid while the arterial wall is taken to be an isotropic, homogeneous and incompressible elastic material. Employing the theory of small deformations superimposed on a large initial field the governing differential equations of perturbed solid motions are obtained in cylindrical polar coordinates. Considering the difficulty in obtaining a closed form solution for the field equations, an approximate power series method is utilized. The dispersion relations for the most general case of this approximation and for the thin tube case are thoroughly discussed. The speeds of waves propagating in an unstressed tube are obtained as a special case of our general treatment. It is observed that the speeds of both waves increase with increasing inner pressure and axial stretch.     

Effects of velocity jumps on blood flow in large arteries

When a human being experiences a sudden velocity change, the blood flow is disturbed. A theoretical analysis to predict the effects of sudden velocity changes on blood flow in large arteries is presented. The situations is modelled as a one-dimensional flow problem in a viscoelastic tube where the fluid viscosity convective term in the equation of motion and nonlinearity in the elastic modulus of the tube wall are neglected. The governing equations of the model are solved by Laplace transformation. The computed results show that relatively high blood pressures, capable of harming circulation, are produced even by relatively moderate velocity jumps.     

Large deforming buoyant embolus passing through a stenotic common carotid artery: a computational simulation

Arterial embolism is responsible for the death of lots of people who suffers from heart diseases. The major risk of embolism in upper limbs is that the ruptured particles are brought into the brain, thus stimulating neurological symptoms or causing the stroke. We presented a computational model using fluid-structure interactions (FSI) to investigate the physical motion of a blood clot inside the human common carotid artery. We simulated transportation of a buoyant embolus in an unsteady flow within a finite length tube having stenosis. Effects of stenosis severity and embolus size on arterial hemodynamics were investigated. To fulfill realistic nonlinear property of a blood clot, a rubber/foam model was used. The arbitrary Lagrangian-Eulerian formulation (ALE) and adaptive mesh method were used inside fluid domain to capture the large structural interfacial movements. The problem was solved by simultaneous solution of the fluid and the structure equations. Stress distribution and deformation of the clot were analyzed and hence, the regions of the embolus prone to lysis were localized. The maximum magnitude of arterial wall shear stress during embolism occurred at a short distance proximal to the throat of the stenosis. Through embolism, arterial maximum wall shear stress is more sensitive to stenosis severity than the embolus size whereas role of embolus size is more significant than the effect of stenosis severity on spatial and temporal gradients of wall shear stress downstream of the stenosis and on probability of clot lysis due to clot stresses while passing through the stenosis.     

Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements

The accuracy of the nonlinear one-dimensional (1-D) equations of pressure and flow wave propagation in Voigt-type visco-elastic arteries was tested against measurements in a well-defined experimental 1:1 replica of the 37 largest conduit arteries in the human systemic circulation. The parameters required by the numerical algorithm were directly measured in the in vitro setup and no data fitting was involved. The inclusion of wall visco-elasticity in the numerical model reduced the underdamped high-frequency oscillations obtained using a purely elastic tube law, especially in peripheral vessels, which was previously reported in this paper [Matthys et al., 2007. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech. 40, 3476-3486]. In comparison to the purely elastic model, visco-elasticity significantly reduced the average relative root-mean-square errors between numerical and experimental waveforms over the 70 locations measured in the in vitro model: from 3.0% to 2.5% (p<0.012) for pressure and from 15.7% to 10.8% (p<0.002) for the flow rate. In the frequency domain, average relative errors between numerical and experimental amplitudes from the 5th to the 20th harmonic decreased from 0.7% to 0.5% (p<0.107) for pressure and from 7.0% to 3.3% (p<10(-6)) for the flow rate. These results provide additional support for the use of 1-D reduced modelling to accurately simulate clinically relevant problems at a reasonable computational cost.     

Periodic flow of a viscous liquid in a thick-walled elastic tube

Arterial blood flow is analyzed on the basis of a realistic model consisting of a viscous liquid contained in a thick-walled viscoelastic tube. Approximate forms of the Navier-Stokes and continuity equations are derived for this model and solved in conjunction with the equations of motion of an elastic solid. Expressions are found for the displacement of the tube wall, velocity distribution, volume flow rate and phase velocity of the pressure wave. Changes in the shape of the pressure wave caused by damping and dispersion are determined, and the effect of viscoelasticity is assessed. Numerical results are presented which correspond to observed parameters of the circulatory systems of living animals.     

Wall shear rate measurements in an elastic curved artery model

Since atherosclerotic lesions tend to be localized at bends and branching points, knowledge of wall shear rate patterns in models of these geometries may help elucidate the mechanism of atherogenesis. This study uses the photochromic method of flow visualization to determine both the mean and amplitude of the wall shear rate waveform in straight and curved elastic arterial models to demonstrate the effects of curvature, elasticity, and the phase angle between the flow and pressure waveforms (impedance phase angle). Under sinusoidal flow conditions characteristic of large arteries, the mean shear rate at the inner wall of the curved tube is reduced 40–56% from its steady flow value, depending on the phase angle. Wall shear rate amplitudes in the curved tube are significantly reduced by wall motion (36–55% of the Womersley amplitude for a straight rigid tube). The shear rate amplitude at the outer wall decreases 30% as the phase angle is reduced from −20° to −66°, while the shear rate amplitude at the inner wall increases 45%. As a result, the oscillatory nature of flow at the outer wall decreases with decreasing negative phase angle, but flow at the inner wall becomes much more oscillatory. At large negative phase angles, characteristic of hypertension or vasoactive agents, the shear rate at the inner wall has a small mean and cycles through positive and negative values; the shear rate at the outer wall remains positive throughout the flow cycle. Thus, the impedance phase angle could affect atherogenesis along the inner wall if temporal and directional changes in wall shear rate play a role.     

Flow in arteries in the presence of stenosis

Of concern in the paper is a study of blood flow in an arterial segment having a stenosis. The artery is modelled as an initially stressed orthotropic elastic tube filled with a viscous incompressible fluid. The analysis is based on the assumption of the presence of a mild stenosis in the artery. Blood is treated as a Newtonian fluid. The effect of the surrounding connective tissues on the motion of the wall has been incorporated. Propagation of small amplitude harmonic waves, generated due to the flow of blood where the wave length is large compared to the radius of the arterial segment, is is considered in detail.     

Effects of vessel compliance on flow pattern in porcine epicardial right coronary arterial tree

The compliance of the vessel wall affects hemodynamic parameters which may alter the permeability of the vessel wall. Based on experimental measurements, the present study established a finite element (FE) model in the proximal elastic vessel segments of epicardial right coronary arterial (RCA) tree obtained from computed tomography. The motion of elastic vessel wall was measured by an impedance catheter and the inlet boundary condition was measured by an ultrasound flow probe. The Galerkin FE method was used to solve the Navier–Stokes and Continuity equations, where the convective term in the Navier–Stokes equation was changed in the arbitrary Lagrangian–Eulerian (ALE) framework to incorporate the motion due to vessel compliance. Various hemodynamic parameters (e.g., wall shear stress—WSS, WSS spatial gradient—WSSG, oscillatory shear index—OSI) were analyzed in the model. The motion due to vessel compliance affects the time-averaged WSSG more strongly than WSS at bifurcations. The decrease of WSSG at flow divider in elastic bifurcations, as compared to rigid bifurcations, implies that the vessel compliance decreases the permeability of vessel wall and may be atheroprotective. The model can be used to predict coronary flow pattern in subject-specific anatomy as determined by noninvasive imaging.     

Numerical 3D-simulation of pulsatile wall shear stress in an arterial T-bifurcation model

The structure of pulsatile blood flow and wall shear stress in a 90° T-bifurcation model is analysed numerically. The nonlinear Navier-Stokes equations for time-dependent incompressible Newtonian fluid flow are approximated using a newly developed pressure correction, finite element method. The wall shear stress is calculated from the finite element velocity field. The investigation shows viscous flow phenomena such as flow separation and stagnation and the distribution of high and low wall shear stress during the pulse cycle. Furthermore, the effect of a sharp corner the bifurcation edge on the wall shear stress is analysed. Detailed local flow investigation is required to examine fluid dynamic contribution to the development of arterial diseases such as atherosclerosis and thrombosis.