Solitary waves in prestressed elastic tubes

In this work, we studied the propagation of non-linear waves in a pre-stressed thin elastic tube filled with an inviscid fluid. In the analysis, analogous to the physiological conditions of the arteries, the tube is assumed to be subject to a uniform pressureP ₀ and a constant axial stretch ratio λ_z. In the course of blood flow it is assumed that a large dynamic displacement is superimposed on this static field. Furthermore, assuming that the displacement gradient in the axial direction is small, the non-linear equation of motion of the tube is obtained. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is investigated. It is shown that the governing equations reduce to the Korteweg-deVries equation which admits a solitary wave solution. The result is discussed for some elastic materials existing in the literature.     

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.     

Experimental study of finite amplitude pulsatile pressure waves in elastic tubes

An experimental study of the propagation of pulsatile pressure waves in an elastic tube was made and results were compared to a theoretical analysis by Lou. The pressure waves were sinusoidally varying acting in a horizontal, longitudinally constrained tube containing water. The independent experimental parameters varied were the pressure wave frequency, pressure wave volume per cycle, static tube pressure and steady flow rate. The wave propagation speeds, measured by non-intrusive techniques, were found to be functions of the wave frequency and the phase angles of the wave elements as theoretically predicted by Lou.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

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.     

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.     

Supersonic and near-sonic solitary ion-acoustic waves in magnetized plasma

The propagation of large-amplitude solitary ion-acoustic waves in magnetized plasma is analyzed. The problem is solved without assuming plasma quasineutrality within the pulse, and the wave potential is described by Poisson’s equation. Solutions in the form of supersonic and near-sonic solitary waves propagating obliquely to the magnetic field are found. The pulses have several peaks and exist for a discrete set of the wave parameters. The amplitude and oscillation frequency of a solitary wave are determined as functions of the Mach number and the propagation angle with respect to the magnetic field.     

Wave propagation in protein microtubules modeled as orthotropic elastic shells including transverse shear deformations

Wave propagation along the microtubules is one of the issues of major concern in various microtubule cellular functions. In this study, the general wave propagation behavior in protein microtubules is investigated based on a first-order shear deformation shell theory for orthotropic materials, with particular emphasis on the role of strongly anisotropic elastic properties of microtubules. According to experimental observation, the first-order shear deformation theory is used for the modeling of microtubule walls. A general displacement representation is introduced and a type of coupled polynomial eigenvalue problem is developed. Numerical examples describe the effects of shear deformation and rotary inertia on wave velocities in orthotropic microtubules. Finally, the influences of the microtubule shear modulus, axial external force, effective thickness and material temperature dependency on wave velocities along the microtubule protofilaments, helical pathway and radial directions are elucidated. Most results presented in the present investigation have been absent from the literature for the wave propagation in microtubules.     

Exact solutions for oblique solitary Alfvén waves in plasma

The properties of solitary Alfvén waves are studied for different ratios between the thermal plasma pressure and the magnetic pressure. It is shown that the wave propagation is accompanied by the generation of a nonlinear ion current along the magnetic field, the contribution of which to the Sagdeev potential was previously ignored. An expression for the quasi-potential of Alfvén waves with allowance for this effect is derived. It is found that Alfvén waves are compression waves in the inertial limit, whereas kinetic Alfvén waves are rarefaction waves. In a high-pressure plasma, a solitary wave has the form of either a well or a hump in the plasma density, depending on the relations between the Mach number, angle between the wave propagation direction and the magnetic field, and the value of the plasma beta.     

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.     

Cytoplasmic molecular delivery with shock waves: importance of impulse

Cell permeabilization using shock waves may be a way of introducing macromolecules and small polar molecules into the cytoplasm, and may have applications in gene therapy and anticancer drug delivery. The pressure profile of a shock wave indicates its energy content, and shock-wave propagation in tissue is associated with cellular displacement, leading to the development of cell deformation. In the present study, three different shock-wave sources were investigated; argon fluoride excimer laser, ruby laser, and shock tube. The duration of the pressure pulse of the shock tube was 100 times longer than the lasers. The uptake of two fluorophores, calcein (molecular weight: 622) and fluorescein isothiocyanate-dextran (molecular weight: 71,600), into HL-60 human promyelocytic leukemia cells was investigated. The intracellular fluorescence was measured by a spectrofluorometer, and the cells were examined by confocal fluorescence microscopy. A single shock wave generated by the shock tube delivered both fluorophores into approximately 50% of the cells (p < 0.01), whereas shock waves from the lasers did not. The cell survival fraction was >0.95. Confocal microscopy showed that, in the case of calcein, there was a uniform fluorescence throughout the cell, whereas, in the case of FITC-dextran, the fluorescence was sometimes in the nucleus and at other times not. We conclude that the impulse of the shock wave (i.e., the pressure integrated over time), rather than the peak pressure, was a dominant factor for causing fluorophore uptake into living cells, and that shock waves might have changed the permeability of the nuclear membrane and transferred molecules directly into the nucleus.     

An alternative ultrasonic method for measuring the elastic properties of cortical bone

We studied the elastic properties of bone to analyze its mechanical behavior. The basic principles of ultrasonic methods are now well established for varying isotropic media, particularly in the field of biomedical engineering. However, little progress has been made in its application to anisotropic materials. This is largely due to the complex nature of wave propagation in these media. In the present study, the theory of elastic waves is essential because it relates the elastic moduli of a material to the velocity of propagation of these waves along arbitrary directions in a solid. Transducers are generally placed in contact with the samples which are often cubes with parallel faces that are difficult to prepare. The ultrasonic method used here is original, a rough preparation of the bone is sufficient and the sample is rotated. Moreover, to analyze heterogeneity of the structure we measure velocities in different points on the sample. The aim of the present study was to determine in vitro the anisotropic elastic properties of cortical bones. For this purpose, our method allowed measurement of longitudinal and transverse velocities (C(L) and C(T)) in longitudinal (fiber direction) and the radial directions (orthogonal to the fiber direction) of compact bones. Young's modulus E and Poisson's ratio nu, were then deduced from the velocities measured considering the compact bone as transversely isotropic or orthotropic. The results are in line with those of other methods.     

Pressure wave propagation in fluid-filled co-axial elastic tubes. Part 1: Basic theory

Our work is motivated by ideas about the pathogenesis of syringomyelia. This is a serious disease characterized by the appearance of longitudinal cavities within the spinal cord. Its causes are unknown, but pressure propagation is probably implicated. We have developed an inviscid theory for the propagation of pressure waves in co-axial, fluid-filled, elastic tubes. This is intended as a simple model of the intraspinal cerebrospinal-fluid system. Our approach is based on the classic theory for the propagation of longitudinal waves in single, fluid-filled, elastic tubes. We show that for small-amplitude waves the governing equations reduce to the classic wave equation. The wave speed is found to be a strong function of the ratio of the tubes' cross-sectional areas. It is found that the leading edge of a transmural pressure pulse tends to generate compressive waves with converging wave fronts. Consequently, the leading edge of the pressure pulse steepens to form a shock-like elastic jump. A weakly nonlinear theory is developed for such an elastic jump.     

Scaling phloem transport: elasticity and pressure-concentration waves

Earlier theoretical analyses of the rate of propagation of pressure-concentration waves in the phloem were performed without adequate attention to the elastic expansion of sieve tube walls. Here, it is shown that the rate of propagation of pressure-concentration waves in phloem sieve tubes is not significantly impeded by wall elasticity, but rather, as previously implicated, by the ratio of sap osmotic pressure to the axial drop in sap hydrostatic pressure. It is also shown that pressure-concentration waves move equally well in both the upstream and downstream directions. These results permit future models to ignore elastic effects, and lend additional theoretical support to the "osmoregulatory flow" hypothesis, which argues that efficient molecular control of the phloem is permitted by maintaining sieve sap hydrostatic pressure at a value that is spatially nearly constant, which in turn permits changes in sieve tube state to be rapidly transmitted throughout the sieve tube via pressure-concentration waves.     

Ion-acoustic supersolitons in plasma

A new class of solitary waves??supernonlinear solitons (supersolitons)??the phase trajectories of which envelop one or several inner separatrices on the wave phase portrait has been revealed. It is shown that supersolitons of the ion-acoustic type can exist in an unmagnetized plasma that contains no less than four kinds of charged particles. The conditions for the existence of supersolitons are specified. The profile of the electrostatic potential in an ion-acoustic supersoliton is determined. It is shown that a supersoliton can be easily recognized experimentally among conventional solitons by using a differentiating circuit in the measuring channel.     

Nonlinear theory of ion-acoustic waves in an electron-positron-ion plasma

An analytical nonlinear gasdynamic theory of ion-acoustic waves in an e-p-i plasma is developed for the case in which all the plasma components in the wave undergo polytropic compression and rarefaction. An exact solution to the basic equations is found and analyzed by the Bernoulli pseudopotential method. The parameter range in which periodic waves can propagate and the range in which solitary waves (solitons) exist are determined. It is shown that the propagation velocity of a solitary is always higher than the linear ion sound velocity. The profiles of all the physical quantities in both subsonic and supersonic waves are calculated. The results obtained agree well with both the data from other papers and particular limiting cases.     

Pulse Wave Propagation in Elastic Tubes Having Longitudinal Changes in Area and Stiffness

The behavior of both step waves and sinusoidal waves in fluid-filled elastic vessels whose area and distensibility vary with distance is explored theoretically. It is shown that the behavior of these waves may be explained, to a large extent, by considering the effect of the continuous stream of infinitesimal reflections that is set up whenever any wave travels in a region of vessel where the local impedance, (that is, the ratio of elastic wavespeed to tube area) is not constant. It is found that in such vessels the behavior of sinusoidal waves over distances which are a fraction of a wavelength can be quite different from their average behavior over several wavelengths. Both behaviors are described analytically. The results are applied to the mammalian circulatory system, one of the most interesting results being that a longitudinal variation in the pressure and velocity amplitudes which has a wavelength roughly one-half that of standing waves is predicted. The treatment is essentially a linearized quasi-one-dimensional one, the major assumptions being that the fluid is inviscid, the mean flow is zero, and the vessel is perfectly elastic and constrained from motion in the longitudinal direction. As in the physiological situation, the ratio of fluid velocity to pulse propagation speed is assumed small. For comparison with the analytical results, the linearized equations are also solved numerically by computer.     

A multiscale approach for modelling wave propagation in an arterial segment

A mathematical model of blood flow through an arterial vessel is presented and the wave propagation in it is studied numerically. Based on the assumption of long wavelength and small amplitude of the pressure waves, a quasi-one-dimensional (1D) differential model is adopted. It describes the non-linear fluid-wall interaction and includes wall deformation in both radial and axial directions. The 1D model is coupled with a six compartment lumped parameter model, which accounts for the global circulatory features and provides boundary conditions. The differential equations are first linearized to investigate the nature of the propagation phenomena. The full non-linear equations are then approximated with a numerical finite difference method on a staggered grid. Some numerical simulations show the characteristics of the wave propagation. The dependence of the flow, of the wall deformation and of the wave velocity on the elasticity parameter has been highlighted. The importance of the axial deformation is evidenced by its variation in correspondence of the pressure peaks. The wave disturbances consequent to a local stiffening of the vessel and to a compliance jump due to prosthetic implantations are finally studied.     

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.     

A one-dimensional model of the spinal cerebrospinal-fluid compartment

Modeling of the cerebrospinal fluid (CSF) system in the spine is strongly motivated by the need to understand the origins of pathological conditions such as the emergence and growth of fluid-filled cysts in the spinal cord. In this study, a one-dimensional (1D) approximation for the flow in elastic conduits was used to formulate a model of the spinal CSF compartment. The modeling was based around a coaxial geometry in which the inner elastic cylinder represented the spinal cord, middle elastic tube represented the dura, and the outermost tube represented the vertebral column. The fluid-filled annuli between the cord and dura, and the dura and vertebral column, represented the subarachnoid and epidural spaces, respectively. The system of governing equations was constructed by applying a 1D form of mass and momentum conservation to all segments of the model. The developed 1D model was used to simulate CSF pulse excited by pressure disturbances in the subarachnoid and epidural spaces. The results were compared to those obtained from an equivalent two-dimensional finite element (FE) model which was implemented using a commercial software package. The analysis of linearized governing equations revealed the existence of three types of waves, of which the two slower waves can be clearly related to the wave modes identified in previous similar studies. The third, much faster, wave emanates directly from the vertebral column and has little effect on the deformation of the spinal cord. The results obtained from the 1D model and its FE counterpart were found to be in good general agreement even when sharp spatial gradients of the spinal cord stiffness were included; both models predicted large radial displacements of the cord at the location of an initial cyst. This study suggests that 1D modeling, which is computationally inexpensive and amenable to coupling with the models of the cranial CSF system, should be a useful approach for the analysis of some aspects of the CSF dynamics in the spine. The simulation of the CSF pulse excited by a pressure disturbance in the epidural space, points to the possibility that regions of the spinal cord with abnormally low stiffness may be prone to experiencing large strains due to coughing and sneezing.