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.     

In vivo vibrational wave propagation in human tibiae at different ages

Vibrational wave propagation was tested in vivo on the tibial bone of both legs of 56 female volunteers. The impact was produced by a hammer with a force strain gauge and the response was monitored by two accelerometers. The peak amplitude of the accelerations, the velocity of the acceleration wave propagation and damping were analysed for comparison among the different age groups. The results showed significant negative correlations between age and the peak amplitude of acceleration, and the velocity of acceleration wave propagation (p less than 0.01). The damping time of the acceleration wave also had a negative correlation with age. These findings suggested that age differences were related to the differences in the mechanical properties of bone. With reduction of bone mineral density, the velocity of the vibrational wave propagation would decrease, with simultaneous increase in impedance. In addition, wave absorption would be accelerated. It is suggested that this method could be used as an indicator of bone density. The method could also be developed to provide an index to monitor the progress of osteoporosis.     

Computation of aortic pulse wave velocity and aortic pulse wave velocity and aortic extensibility from pressure gradient measurements.

This study is concerned with the computation of aortic pulse wave velocity based on simultaneous recordings of the aortic pressure gradient and first-time derivative of aortic pressure. These variables were recorded by means of a double-lumen catheter introduced in the aorta of four anesthetized closed chest dogs, and connected to critically damped manometer systems. Results of aortic pulse wave velocity were then compared: (i) to the true phase velocity obtained from spectra of apparent phase velocity, and (ii) to the pulse wave velocity computed from the time shift between maximum slopes of the pressure wave. From the aortic valves to 37 cm down the aortic trunk, pulse wave velocity increased from 410-460 cm/s to approximately 600-800 cm/s. Based on the wave propagation equation presented of Bramwell and Hill (Bramwell, J.C., and Hill, A. V. 1922. Proc. R. Soc. 93, 298-306), volumetric extensibility coefficients were computed from pulse wave velocity data. Results indicated that, from the aortic valves to 37 cm down to the aorta, the mean volumetric extensibility decreased from 0.43-0.56% deltaV/cm H2O to 0.16-0.25% deltaV/cm H2O (1 cm H2O = 94.1 N/m2).     

Determination of mechanical properties and tone of blood vessels

Based on the equation of the propagation rate of pulse wave and on the presentation of the blood vessel as a hollow thin-wall cylinder a formula for estimation of the relative modulus of the vessel wall elasticity was obtained. The obtained formula makes it possible to improve the determination procedure of the mechanical properties and tonus of the blood vessels.     

Mechanical vulnerability of lower second premolar utilising visco-elastic dynamic stress analysis

Stress analysis determines vulnerability of dental tissues to external loads. Stress values depend on loading conditions, mechanical properties and constrains of structural components. The critical stress levels lead to tissue damage. The aim of this study is to analyse dynamic stress distribution of lower second premolar due to physiological cyclic loading, and dependency of pulsatile stress characteristics to visco-elastic property of dental components by finite element modelling. Results show that visco-elastic property markedly influences stress determinants in major anatomical sites including dentin, cementum–enamel and dentin–enamel junctions. Reduction of visco-elastic parameter leads to mechanical vulnerability through elevation of stress pulse amplitude, maximum stress value; and reduction of stress phase shift as a determinant of stress wave propagation. The results may be applied in situations in which visco-elasticity is reduced such as root canal therapy and post and core restoration in which teeth are more vulnerable to fracture.     

A laser-induced ultrasonic probe of the mechanical properties of aligned lipid multibilayers.

The recently developed laser-induced phonon spectroscopy (LIPS) technique is applied to the determination of dynamic mechanical properties of aligned dilauroylphosphatidylcholine (DLPC) multibilayer arrays containing 2 and 20% water by weight. Sample excitation by two crossed 100-ps laser pulses generates a longitudinal ultrasonic wave whose wavelength depends on the crossing angle. In these experiments, the acoustic wave propagates parallel to the bilayer planes. The ultrasonic velocity and attenuation are monitored through the diffraction of a variably delayed probe pulse by the acoustic grating. The velocity measures the lateral area compressibility of the bilayers, while the attenuation is related to the viscosity. Velocities obtained in the gel and liquid crystal phases are compared with those found previously using Brillouin scattering. The acoustic attenuation is shown to be an order of magnitude more sensitive to the gel-liquid crystal phase transition than the velocity. The lipid area compressibility and viscosity of DLPC-20% water multilayers with and without 100 mM CaCl2 are found to be identical within our experimental error.     

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.     

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.     

Dynamic measurement of the viscoelastic properties of skin

A wave propagation technique was used to measure the dynamic viscoelastic properties of excised skin when subjected to a low incremental strain. The propagation velocity, attenuation, and storage and loss moduli were determined from measured characteristics of a pulse propagating along a strip of skin. Experiments were conducted with the skin subjected to static stresses of 1500 Pa and 20,000 Pa. At low static stresses the skin response was viscoelastic with a loss tangent of approximately 0.6. In the frequency range of 0-1000 Hz, the wave velocity was relatively constant while the attenuation increased roughly linearly with frequency. However, results depended on the static stress. At the higher stress level the velocity was greater and the attenuation less than at the lower stress. At low stresses both the storage and loss moduli were relatively constant over the frequency range tested. The strong viscoelastic behavior of the tissue at higher frequencies is not predicted from models of the tissue determined from quasi-static test methods. In selecting a model to describe the behavior of skin, the test methods used for establishing the model must be consistent with its intended application.     

The speed of sound through trabecular bone predicted by Biot theory

Cancellous bone is a highly porous material filled with fluid. The mechanical properties of cancellous bone determine whether the bone is normal or osteoporotic. Wave propagation can be used to measure the elastic constants of cancellous bone. Recently, poroelasticity theory has been used to predict the elastic constants of cancellous bone from the wave velocities. In this study, it is shown that the fast wave, predicted by the Biot theory, corresponds to the wave penetrating the trabeculae, while the slow wave is determined by the interaction between the trabeculae and the fluid. The trabecular shape does not affect the wave velocity significantly when using the variable, which is determined by the microstructure, and the slow wave velocity decreases after the porosity reaches 80%.     

Experimental evaluation of local wave speed in the presence of reflected waves

Wave speed (also called pulse wave velocity) is the speed by which disturbance travels along the medium and it depends on the mechanical and geometrical properties of the vessel and on the density of the blood. Wave speed is a parameter of clinical relevance because it is an indicator of arterial stiffness and cardiovascular diseases.     

Arterial pulse attenuation prediction using the decaying rate of a pressure wave in a viscoelastic material model

The present study examines the possibility of attenuating blood pulses by means of introducing prosthetic viscoelastic materials able to absorb energy and damp such pulses. Vascular prostheses made of polymeric materials modify the mechanical properties of blood vessels. The effect of these materials on the blood pulse propagation remains to be fully understood. Several materials for medical applications, such as medical polydimethylsiloxane or polytetrafluoroethylene, show viscoelastic behavior, modifying the original vessel stiffness and affecting the propagation of blood pulses. This study focuses on the propagation of pressure waves along a pipe with viscoelastic materials using the Maxwell and the Zener models. An expression of exponential decay has been obtained for the Maxwell material model and also for low viscous coefficient values in the Zener model. For relatively high values of the viscous term in the Zener model, the steepest part of the pulse can be damped quickly, leaving a smooth, slowly decaying wave. These mathematical models are critical to tailor those materials used in cardiovascular implants to the mechanical environment they are confronted with to repair or improve blood vessel function.     

Instantaneous blood flow, impedance and elastic properties computed in man from aortic pulse waves

A mathematical model of the pressure-flow relationship in the arterial circulation and its possible use in routine hemodynamics in man are described. The instantaneous blood flow velocity in the ascending aorta can be calculated from two pressure curves simultaneously recorded 5 cm apart. The mechanical aortic input impedance is computed from the recorded pressure and the calculated blood flow velocity curves. Projection of the pulse waves on a time-length plane leads to the determination of the pulse wave velocity and then an estimation of the elastic modulus of the aortic wall.     

Stress wave propagation in bone

The traveling wave characteristics for a single compressive pulse were studied in fresh and embalmed human long bones. The stress wave was generated by the longitudinal impact of a steel ball on one end of a bone and was monitored by bonded strain gages. The dynamic properties, namely velocity, attenuation coefficient and dispersion were correlated with the mineral density, porosity, and cross-sectional area of the specimens. Statistically significant relationships were found between several of these parameters. These stress wave propagation characteristics are important for our understanding of the dynamic loading of bone and they may also provide a basis for the development of noninvasive techniques for studies of diseased or fractured bone.     

Validity and reproducibility of arterial pulse wave velocity measurement using new device with oscillometric technique: A pilot study

Background  

Availability of a range of techniques and devices allow measurement of many variables related to the stiffness of large or medium sized arteries. There is good evidence that, pulse wave velocity is a relatively simple measurement and is a good indicator of changes in arterial properties. The pulse wave velocity calculated from pulse wave recording by other methods like doppler or tonometry is tedious, time-consuming and above all their reproducibility depends on the operator skills. It requires intensive resource involvement. For epidemiological studies these methods are not suitable. The aim of our study was to clinically evaluate the validity and reproducibility of a new automatic device for measurement of pulse wave velocity that can be used in such studies.     

In vivo repeatability of the pulse wave inverse problem in human carotid arteries

Accurate arterial stiffness measurement would improve diagnosis and monitoring for many diseases. Atherosclerotic plaques and aneurysms are expected to involve focal changes in vessel wall properties; therefore, a method to image the stiffness variation would be a valuable clinical tool. The pulse wave inverse problem (PWIP) fits unknown parameters from a computational model of arterial pulse wave propagation to ultrasound-based measurements of vessel wall displacements by minimizing the difference between the model and measured displacements. The PWIP has been validated in phantoms, and this study presents the first in vivo demonstration. The common carotid arteries of five healthy volunteers were imaged five times in a single session with repositioning of the probe and subject between each scan. The 1D finite difference computational model used in the PWIP spanned from the start of the transducer to the carotid bifurcation, where a resistance outlet boundary condition was applied to approximately model the downstream reflection of the pulse wave. Unknown parameters that were estimated by the PWIP included a 10-segment linear piecewise compliance distribution and 16 discrete cosine transformation coefficients for each of the inlet boundary conditions. Input data was selected to include pulse waves resulting from the primary pulse and dicrotic notch. The recovered compliance maps indicate that the compliance increases close to the bifurcation, and the variability of the average pulse wave velocity estimated through the PWIP is on the order of 11%, which is similar to that of the conventional processing technique which tracks the wavefront arrival time (13%).     

Optimal Velocity and Safety of Discontinuous Conduction through the Heterogeneous Purkinje-Ventricular Junction

Slow and discontinuous wave conduction through nonuniform junctions in cardiac tissues is generally considered unsafe and proarrythmogenic. However, the relationships between tissue structure, wave conduction velocity, and safety at such junctions are unknown. We have developed a structurally and electrophysiologically detailed model of the canine Purkinje-ventricular junction (PVJ) and varied its heterogeneity parameters to determine such relationships. We show that neither very fast nor very slow conduction is safe, and there exists an optimal velocity that provides the maximum safety factor for conduction through the junction. The resultant conduction time delay across the PVJ is a natural consequence of the electrophysiological and morphological differences between the Purkinje fiber and ventricular tissue. The delay allows the PVJ to accumulate and pass sufficient charge to excite the adjacent ventricular tissue, but is not long enough for the source-to-load mismatch at the junction to be enhanced over time. The observed relationships between the conduction velocity and safety factor can provide new insights into optimal conditions for wave propagation through nonuniform junctions between various cardiac tissues.     

Gap Junction Channels and Cardiac Impulse Propagation

The role of gap junction channels on cardiac impulse propagation is complex. This review focuses on the differential expression of connexins in the heart and the biophysical properties of gap junction channels under normal and disease conditions. Structural determinants of impulse propagation have been gained from biochemical and immunocytochemical studies performed on tissue extracts and intact cardiac tissue. These have defined the distinctive connexin coexpression patterns and relative levels in different cardiac tissues. Functional determinants of impulse propagation have emerged from electrophysiological experiments carried out on cell pairs. The static properties (channel number and conductance) limit the current flow between adjacent cardiomyocytes and thus set the basic conduction velocity. The dynamic properties (voltage-sensitive gating and kinetics of channels) are responsible for a modulation of the conduction velocity during propagated action potentials. The effect is moderate and depends on the type of Cx and channel. For homomeric-homotypic channels, the influence is small to medium; for homomeric-heterotypic channels, it is medium to strong. Since no data are currently available on heteromeric channels, their influence on impulse propagation is speculative. The modulation by gap junction channels is most prominent in tissues at the boundaries between cardiac tissues such as sinoatrial node-atrial muscle, atrioventricular node-His bundle, His bundle-bundle branch and Purkinje fibers-ventricular muscle. The data predict facilitation of orthodromic propagation.     

Effect of initial stresses on the wave propagation in arteries

A theoretical analysis for the problem of wave propagation in arteries is presented. Blood is treated as a Newtonian, viscous incompressible fluid. On the basis of information derived from experimental investigations on the mechanical properties of wall tissues, the arterial wall is considered to be nonlinearly viscoelastic and orthotropic. The analysis is carried out for a cylindrical artery, under the purview of membrane theory, by taking account the effect of initial stresses. The motion of the wall and that of the fluid are assumed to be axisymmetric. Particular emphasis has been paid to the propagation of small amplitude harmonic waves whose wavelength is large compared to the radius of the vessel. By employing the equations of motion of the fluid and those for the wall, together with the equations of continuity, a frequency equation is derived by exploiting the conditions of continuity of the velocity of the arterial wall and that of blood on the endosteal surface of the wall. In order to illustrate the validity of the derived analytical expressions a quantitative analysis is made for the variations of the phase velocities as well as the transmission coefficient with frequency corresponding to different transmural pressures which relate to different initial stresses. Computational results indicate that phase velocities increase with the increase of transmural pressures.