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.     

Effects of transmural pressure and wall shear stress on LDL accumulation in the arterial wall: a numerical study using a multilayered model

The accumulation of low-density lipoprotein (LDL) is recognized as one of the main contributors in atherogenesis. Mathematical models have been constructed to simulate mass transport in large arteries and the consequent lipid accumulation in the arterial wall. The objective of this study was to investigate the influences of wall shear stress and transmural pressure on LDL accumulation in the arterial wall by a multilayered, coupled lumen-wall model. The model employs the Navier-Stokes equations and Darcy's Law for fluid dynamics, convection-diffusion-reaction equations for mass balance, and Kedem-Katchalsky equations for interfacial coupling. To determine physiologically realistic model parameters, an optimization approach that searches optimal parameters based on experimental data was developed. Two sets of model parameters corresponding to different transmural pressures were found by the optimization approach using experimental data in the literature. Furthermore, a shear-dependent hydraulic conductivity relation reported previously was adopted. The integrated multilayered model was applied to an axisymmetric stenosis simulating an idealized, mildly stenosed coronary artery. The results show that low wall shear stress leads to focal LDL accumulation by weakening the convective clearance effect of transmural flow, whereas high transmural pressure, associated with hypertension, leads to global elevation of LDL concentration in the arterial wall by facilitating the passage of LDL through wall layers.     

Wave Propagation through a Viscous Incompressible Fluid Contained in an Initially Stressed Elastic Tube

To have a better understanding of the flow of blood in arteries a theoretical analysis of the pressure wave propagation through a viscous incompressible fluid contained in an initially stressed tube is considered. The fluid is assumed to be Newtonian. The tube is taken to be elastic and isotropic. The analysis is restricted to tubes with thin walls and to waves whose wavelengths are very large compared with the radius of the tube. It is further assumed that the amplitude of the pressure disturbance is sufficiently small so that nonlinear terms of the inertia of the fluid are negligible compared with linear ones. Both circumferential and longitudinal initial stresses are considered; however, their origins are not specified. Initial stresses enter equations as independent parameters. A frequency equation, which is quadratic in the square of the propagation velocity is obtained. Two out of four roots of this equation give the velocity of propagation of two distinct outgoing waves. The remaining two roots represent incoming waves corresponding to the first two waves. One of the waves propagates more slowly than the other. As the circumferential and/or longitudinal stress of the wall increases, the velocity of propagation and transmission per wavelength of the slower wave decreases. The response of the fast wave to a change in the initial stress is on the opposite direction.     

A potential role of smooth muscle tone in early hypertension: a theoretical study

A conspicuous long-term consequence of hypertension is a thickening of the arterial wall, which many suggest returns the circumferential wall stress toward its normal value. This thickening results from an increase in smooth muscle and extracellular matrix, with the associated growth and remodeling processes depending on a host of regulatory signals that likely include the altered mechanical environment. Although the precise mechanotransduction pathways remain unknown, we propose that vasoconstriction may be an early response of the arterial wall to a step-change in pressure. In particular, computations suggest that such a response can decrease the magnitude and transmural gradients of the pressure-induced wall stresses and return the mean wall shear stress toward its homeostatic value. Such an initial 'compensatory vasoconstriction' could also help set into motion subsequent growth and remodeling responses due to growth regulatory characteristics of the vasoactive molecules (e.g., nitric oxide, endothelin-1, angiotensin-II). Although the consequences of growth and remodeling have been the focus of prior biomechanical and histological studies, early responses dictate subsequent developments and therefore deserve increased attention in vascular biomechanics and mechanobiology.     

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.     

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.     

Wave Propagation through a Viscous Fluid Contained in a Tethered, Initially Stressed, Orthotropic Elastic Tube

To give a realistic representation of the pulse propagation in arteries a theoretical analysis of the wave propagation through a viscous incompressible fluid contained in an initially stressed elastic tube is considered. The tube is assumed to be orthotropic and its longitudinal motion is constrained by a uniformly distributed additional mass, a dashpot and a spring. The fluid is assumed to be Newtonian. The analysis is restricted to propagation of small amplitude harmonic waves whose wavelength is large compared to the radius of the vessel. Elimination of arbitrary constants from the general solutions of the equations of motion of the fluid and the wall gives a frequency equation to determine the velocity of propagation. Two roots of this equation give the velocity of propagation of two distinct outgoing waves. One of the waves propagates slower than the other. The propagation properties of s lower waves are very slightly affected by the degree of anisotropy of the wall. The velocity of propagation of faster waves decreases as the ratio of the longitudinal modulus of elasticity to the circumferential modulus decreases; transmission of these waves is very little affected. The influence of the tethering on the propagation velocity of slower waves is negligibly small; transmission of these waves is seriously affected. In tethered tubes faster waves are completely attenuated.     

A mathematical model of transmural transport of oxygen to the retina

A mathematical model of transmural transport of oxygen to a metabolizing retina is presented based on the equations of fluid dynamics. The equations of oxygen transfer are derived and then solved subject to the condition that the capillaries begin to transport oxygen at an initial time. The resulting transient analysis gives us insight into how diffusive and filtrative processes lead to the oxygen distributions both inside and outside capillaries. On the other hand, the steady state solution allows us to predict the cutoff intraocular pressure above which no oxygen is transferred to retinal tissue. It also gives quantitative relationships which allow us to postulate how intracapillary hypertension counterbalances elevated intraocular pressures and how low pressure glaucoma may arise from ineffective diffusive and filtrative processes of oxygen transport.     

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.     

A thick walled viscoelastic model for the mechanics of arteries

A knowledge of the mechanics of arteries is of importance in the determination of vessel rheological properties and in the studies of blood flow and certain arterial diseases. Most existing arterial models treat only wave motions; however, other types of motion, in particular those associated with flow development and other end effects, occur in the vascular system. Thus, a model is needed which can be applied to a variety of possible types of motion.

An arterial model is described which includes the effects of thick walls, linear viscoelasticity, and wall tethering. The forms of the displacements and stresses are found independently of the exact form of the applied fluid stresses; thus, the results are applicable to a range of possible dynamical conditions. Displacements and stress states can then be found from experimental or theoretical knowledge of the blood pressure and flow. The results are applied to flow development and wave propagation regions in the arteries.     

Theory of longitudinal plasma waves with allowance for ion motion

A study is made of the propagation of steady-state large-amplitude longitudinal plasma waves in a cold collisionless plasma with allowance for both electron and ion motion. Conditions for the existence of periodic potential waves are determined. The electric field, potential, frequency, and wavelength are obtained as functions of the wave phase velocity and ion-to-electron mass ratio. Taking into account the ion motion results in the nonmonotonic dependence of the frequency of the waves with the maximum possible amplitudes on the wave phase velocity. Specifically, at low phase velocities, the frequency is equal to the electron plasma frequency for linear waves. As the phase velocity increases, the frequency first decreases insignificantly, reaches its minimum value, and then increases. As the phase velocity increases further, the frequency continues to increase and, at relativistic phase velocities, again becomes equal to the plasma frequency. Finally, as the phase velocity approaches the speed of light, the frequency increases without bound.     

Elastic expulsion from a long tube

The paper presents a theoretical analysis of elastic expulsion from a long pressurized tube following instantaneous severance. The governing equations of motion and continuity are used, together with an equation representing radial equilibrium of the tube wall, to construct a differential equation for the inner radius of the tube as a function of axial position and time. Similarity solutions for this equation are presented for a Newtonian fluid. The effect of the relaxed/fully-extended radius ratio on wall dilatation and expulsion rate is discussed.     

Myogenic tone,reactivity, and forced dilatation: a three-phase model of in vitro arterial myogenic behavior

Myogenic behavior, prevalent in resistance arteries and arterioles, involves arterial constriction in response to intravascular pressure. This process is often studied in vitro by using cannulated, pressurized arterial segments from different regional circulations. We propose a comprehensive model for myogenicity that consists of three interrelated but dissociable phases: 1) the initial development of myogenic tone (MT), 2) myogenic reactivity to subsequent changes in pressure (MR), and 3) forced dilatation at high transmural pressures (FD). The three phases span the physiological range of transmural pressures (e.g., MT, 40-60 mmHg; MR, 60-140 mmHg; FD, >140 mmHg in cerebral arteries) and are characterized by distinct changes in cytosolic calcium ([Ca(2+)](i)), which do not parallel arterial diameter or wall tension, and therefore suggest the existence of additional regulatory mechanisms. Specifically, the development of MT is accompanied by a substantial (200%) elevation in [Ca(2+)](i) and a reduction in lumen diameter and wall tension, whereas MR is associated with relatively small [Ca(2+)](i) increments (<20% over the entire pressure range) despite considerable increases in wall tension and force production but little or no change in diameter. FD is characterized by a significant additional elevation in [Ca(2+)](i) (>50%), complete loss of force production, and a rapid increase in wall tension. The utility of this model is that it provides a framework for comparing myogenic behavior of vessels of different size and anatomic origin and for investigating the underlying cellular mechanisms that govern vascular smooth muscle mechanotransduction and contribute to the regulation of peripheral resistance.     

Feasibility Study of Ex Ovo Chick Chorioallantoic Artery Model for Investigating Pulsatile Variation of Arterial Geometry

Despite considerable research efforts on the relationship between arterial geometry and cardiovascular pathology, information is lacking on the pulsatile geometrical variation caused by arterial distensibility and cardiomotility because of the lack of suitable in vivo experimental models and the methodological difficulties in examining the arterial dynamics. We aimed to investigate the feasibility of using a chick embryo system as an experimental model for basic research on the pulsatile variation of arterial geometry. Optical microscope video images of various arterial shapes in chick chorioallantoic circulation were recorded from different locations and different embryo samples. The high optical transparency of the chorioallantoic membrane (CAM) allowed clear observation of tiny vessels and their movements. Systolic and diastolic changes in arterial geometry were visualized by detecting the wall boundaries from binary images. Several to hundreds of microns of wall displacement variations were recognized during a pulsatile cycle. The spatial maps of the wall motion harmonics and magnitude ratio of harmonic components were obtained by analyzing the temporal brightness variation at each pixel in sequential grayscale images using spectral analysis techniques. The local variations in the spectral characteristics of the arterial wall motion were reflected well in the analysis results. In addition, mapping the phase angle of the fundamental frequency identified the regional variations in the wall motion directivity and phase shift. Regional variations in wall motion phase angle and fundamental-to-second harmonic ratio were remarkable near the bifurcation area. In summary, wall motion in various arterial geometry including straight, curved and bifurcated shapes was well observed in the CAM artery model, and their local and cyclic variations could be characterized by Fourier and wavelet transforms of the acquired video images. The CAM artery model with the spectral analysis method is a useful in vivo experimental model for studying pulsatile variation in arterial geometry.     

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.     

Breaking symmetry in non-planar bifurcations: distribution of flow and wall shear stress

Non-planarity in blood vessels is known to influence arterial flows and wall shear stress. To gain insight, computational fluid dynamics (CFD) has been used to investigate effects of curvature and out-of-plane geometry on the distribution of fluid flows and wall shear stresses in a hypothetical non-planar bifurcation. Three-dimensional Navier-Stokes equations for a steady state Newtonian fluid were solved numerically using a finite element method. Non-planarity in one of the two daughter vessels is found to deflect flow from the inner wall of the vessel to the outer wall and to cause changes in the distribution of wall shear stresses. Results from this study agree to experimental observations and CFD simulations in the literature, and support the view that non-planarity in blood vessels is a factor with important haemodynamic significance and may play a key role in vascular biology and pathophysiology.     

Unsteady and three-dimensional simulation of blood flow in the human aortic arch

A three-dimensional and pulsatile blood flow in a human aortic arch and its three major branches has been studied numerically for a peak Reynolds number of 2500 and a frequency (or Womersley) parameter of 10. The simulation geometry was derived from the three-dimensional reconstruction of a series of two-dimensional slices obtained in vivo using CAT scan imaging on a human aorta. The numerical simulations were obtained using a projection method, and a finite-volume formulation of the Navier-Stokes equations was used on a system of overset grids. Our results demonstrate that the primary flow velocity is skewed towards the inner aortic wall in the ascending aorta, but this skewness shifts to the outer wall in the descending thoracic aorta. Within the arch branches, the flow velocities were skewed to the distal walls with flow reversal along the proximal walls. Extensive secondary flow motion was observed in the aorta, and the structure of these secondary flows was influenced considerably by the presence of the branches. Within the aorta, wall shear stresses were highly dynamic, but were generally high along the outer wall in the vicinity of the branches and low along the inner wall, particularly in the descending thoracic aorta. Within the branches, the shear stresses were considerably higher along the distal walls than along the proximal walls. Wall pressure was low along the inner aortic wall and high around the branches and along the outer wall in the ascending thoracic aorta. Comparison of our numerical results with the localization of early atherosclerotic lesions broadly suggests preferential development of these lesions in regions of extrema (either maxima or minima) in wall shear stress and pressure.     

Effects of pulsation frequency and endothelial integrity on enhanced arterial transmural filtration produced by pulsatile pressure

The role of the endothelium in regulating transmural fluid filtration into the artery wall under pulsatile pressure and the effects of changes in pulsatile frequency on filtration have received little attention. Previous experiments (Alberding JP, Baldwin AL, Barton JK, and Wiley E. Am J Physiol Heart Circ Physiol 286: H1827-H1835, 2004) demonstrated significantly increased filtration after initial onset of pulsatile pressure compared with that predicted by using parameters measured under steady pressure. To determine the role of the endothelium in this phenomenon, the following experiments were performed on five New Zealand White rabbits (anesthetized with 30 mg/kg pentobarbital sodium). One of each pair of carotid arteries was deendothelialized, and filtration measurements under steady and pulsatile pressure were compared with those made in intact vessels (Alberding JP, Baldwin AL, Barton JK, and Wiley E. Am J Physiol Heart Circ Physiol 286: H1827-H1835, 2004). To determine the effect of increasing pulsatile frequency on arterial filtration, transmural filtration was measured by using pulsatile pressure frequencies of 1 Hz, followed by 2 Hz, in another set of intact arteries (6 arteries and 3 animals). For deendothelialized vessels, the initial increase in filtration after onset of pulsatility was similar to that observed in intact vessels, but the subsequent reduction in filtration was less abrupt. When pulsatile frequency was increased from 1 to 2 Hz in intact arteries, an initial increase in filtration was observed, similar to that obtained after onset of pulsatile pressure subsequent to a steady pressure. The observed responses of arteries to pulsatile pressure, with and without endothelium, or undergoing a frequency change, suggest a dynamic role for the endothelium in regulating transvascular transport in vivo.     

Computational analysis of coupled blood-wall arterial LDL transport

The transport of macromolecules, such as low density lipoproteins (LDLs), across the artery wall and their accumulation in the wall is a key step in atherogenesis. Our objective was to model fluid flow within both the lumen and wall of a constricted, axisymmetric tube simulating a stenosed artery, and to then use this flow pattern to study LDL mass transport from the blood to the artery wall. Coupled analysis of lumenal blood flow and transmural fluid flow was achieved through the solution of Brinkman's model, which is an extension of the Navier-Stokes equations for porous media. This coupled approach offers advantages over traditional analyses of this problem, which have used possibly unrealistic boundary conditions at the blood-wall interface; instead, we prescribe a more natural pressure boundary condition at the adventitial vasa vasorum, and allow variations in wall permeability due to the occurrence of plaque. Numerical complications due to the convection dominated mass transport process (low LDL diffusivity) are handled by the streamline upwind/Petrov-Galerkin (SUPG) finite element method. This new fluid-plus-porous-wall method was implemented for conditions typical of LDL transport in a stenosed artery with a 75 percent area reduction (Peclet number=2 x 10(8)). The results show an elevated LDL concentration at the downstream side of the stenosis. For the higher Darcian wall permeability thought to occur in regions containing atheromatous lesions, this leads to an increased transendothelial LDL flux downstream of the stenosis. Increased transmural filtration in such regions, when coupled with a concentration-dependent endothelial permeability to LDL, could be an important contributor to LDL infiltration into the arterial wall. Experimental work is needed to confirm these results.     

Experimental determination of mechanical shear stress about an anastomotic junction

The present study is undertaken to determine whether the elastic tube model originally developed by Kuchar and Ostrach (Biomedical Fluid Mechanics Symposium, pp. 45-69, 1966) accurately provides a first approximation of the biomechanics of the anastomotic junction. The experimental protocol involves the use of canine carotid arteries as the host vessel and several graft materials including autogenous and prosthetic substitutes. The host artery-graft combinations are perfused in vitro in a pulsatile perfusion apparatus which simulates the natural hemodynamic environment. This apparatus provides accurate dynamic measurements of radial wall motion (measured at various longitudinal increments), associated pressures and rates of fluid flow. These data are then applied to the theoretical model for calculation of anastomotic induced bending stresses. The results indicate that the predictions derived from the elastic model consistently overestimate the measured radial change adjacent to the anastomotic junction. As a result shear stresses based on elastic theory deviate from values derived from a numerical curve fit to the experimental data.