The effect of shear stress on solitary waves in arteries

In the present work, we study the propagation of solitary waves in a prestressed thick walled elastic tube filled with an incompressible inviscid fluid. In order to include the geometric dispersion in the analysis the wall inertia and shear deformation effects are taken into account for the inner pressure-cross-sectional area relation. Using the reductive perturbation technique, the propagation of weakly non-linear waves in the long-wave approximation is examined. It is shown that, contrary to thin tube theories, the present approach makes it possible to have solitary waves even for a Mooney-Rivlin (M-R) material. Due to dependence of the coefficients of the governing Korteweg-deVries equation on initial deformation, the solution profile changes with inner pressure and the axial stretch. The variation of wave profiles for a class of elastic materals are depicted in graphical forms. As might be seen from these illustrations, with increasing thickness ratio, the profile of solitary wave is steepened for a M-R material but it is broadened for biological tissues.     

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.     

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.     

Effects of Viscosity and Constraints on the Dispersion and Dissipation of Waves in Large Blood Vessels: I. Theoretical Analysis

The propagation of sounds and pulse waves within the cardiovascular system is subject to strong dissipative mechanisms. To investigate the effects of blood viscosity on dissipation as well as dispersion of small waves in arteries and veins, a parametric study has been carried out. A linearized analysis of axisymmetric waves in a cylindrical membrane that contains a viscous fluid indicates that there are two families of waves: a family of slow waves and one of fast waves. The faster waves are shown to be more sensitive to variations in the elastic properties of the medium surrounding the blood vessels and at high values of the frequency parameter α defined by α = √ρωR²₀/μ the blood viscosity attenuates them more strongly over a length than the slow waves. At low values of α, the effects of viscosity on attenuation are reversed; that is, the family of slow waves is much more attenuated than the family of fast waves. For the slow waves the radial displacement component generally exceeds the axial component except at very low frequencies. Conversely the axial displacements are much larger than the radial displacement for the faster waves. The presence of external constraints, however, can modify these results. In the case of the slow waves the phase angle between pressure and radial wall displacement is virtually negligible in the presence of mild external constraints, while the phase angles between pressure and fluid mass flow are at most 45°. The corresponding phase angles for the fast waves exhibit much larger variations with changes in the elastic properties of the surrounding medium.     

Mechanical anisotropy of inflated elastic tissue from the pig aorta

Uniaxial and biaxial mechanical properties of purified elastic tissue from the proximal thoracic aorta were studied to understand physiological load distributions within the arterial wall. Stress–strain behaviour was non-linear in uniaxial and inflation tests. Elastic tissue was 40% stiffer in the circumferential direction compared to axial in uniaxial tests and～100% stiffer in vessels at an axial stretch ratio of 1.2 or 1.3 and inflated to physiological pressure. Poisson’s ratio v_θz averaged 0.2 and v_zθ increased with circumferential stretch from ～0.2 to ～0.4. Axial stretch had little impact on circumferential behaviour. In intact (unpurified) vessels at constant length, axial forces decreased with pressure at low axial stretches but remained constant at higher stretches. Such a constant axial force is characteristic of incrementally isotropic arteries at their in vivo dimensions. In purified elastic tissue, force decreased with pressure at all axial strains, showing no trend towards isotropy. Analysis of the force–length–pressure data indicated a vessel with v_θz≈0.2 would stretch axially 2–4% with the cardiac pulse yet maintain constant axial force. We compared the ability of 4 mathematical models to predict the pressure-circumferential stretch behaviour of tethered, purified elastic tissue. Models that assumed isotropy could not predict the stretch at zero pressure. The neo-Hookean model overestimated the non-linearity of the response and two non-linear models underestimated it. A model incorporating contributions from orthogonal fibres captured the non-linearity but not the zero-pressure response. Models incorporating anisotropy and non-linearity should better predict the mechanical behaviour of elastic tissue of the proximal thoracic aorta.     

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.     

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.     

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.     

Wave intensity amplification and attenuation in non-linear flow: Implications for the calculation of local reflection coefficients

Local reflection coefficients (R) provide important insights into the influence of wave reflection on vascular haemodynamics. Using the relatively new time-domain method of wave intensity analysis, R has been calculated as the ratio of the peak intensities (R_PI) or areas (R_CI) of incident and reflected waves, or as the ratio of the changes in pressure caused by these waves (R_ΔP). While these methods have not yet been compared, it is likely that elastic non-linearities present in large arteries will lead to changes in the size of waves as they propagate and thus errors in the calculation of R_PI and R_CI. To test this proposition, R_PI, R_CI and R_ΔP were calculated in a non-linear computer model of a single vessel with various degrees of elastic non-linearity, determined by wave speed and pulse amplitude (ΔP₊), and a terminal admittance to produce reflections. Results obtained from this model demonstrated that under linear flow conditions (i.e. as ΔP₊→0), R_ΔP is equivalent to the square-root of R_PI and R_CI (denoted by R_PI^p and R_CI^p). However for non-linear flow, pressure-increasing (compression) waves undergo amplification while pressure-reducing (expansion) waves undergo attenuation as they propagate. Consequently, significant errors related to the degree of elastic non-linearity arise in R_PI and R_CI, and also R_PI^p and R_CI^p, with greater errors associated with larger reflections. Conversely, R_ΔP is unaffected by the degree of non-linearity and is thus more accurate than R_PI and R_CI.     

Transmission characteristics of axial waves in blood vessels

The elastic behavior of blood vessels can be quantitatively examined by measuring the propagation characteristics of waves transmitted by them. In addition, specific information regarding the viscoelastic properties of the vessel wall can be deduced by comparing the observed wave transmission data with theoretical predictions. The relevance of these deductions is directly dependent on the validity of the mathematical model for the mechanical behavior of blood vessels used in the theoretical analysis. Previous experimental investigations of waves in blood vessels have been restricted to pressure waves even though theoretical studies predict three types of waves with distinctly different transmission characteristics. These waves can be distinguished by the dominant displacement component of the vessel wall and are accordingly referred to as radial, axial and circumferential waves. The radial waves are also referred to as pressure waves since they exhibit pronounced pressure fluctuations. For a thorough evaluation of the mathematical models used in the analysis it is necessary to measure also the dispersion and attenuation of the axial and circumferential (torsion) waves.

To this end a method has been developed to determine the phase velocities and damping of sinusoidal axial waves in the carotid artery of anesthetized dogs with the aid of an electro-optical tracking system. For frequencies between 25 and 150 Hz the speed of the axial waves was between 20 and 40 m/sec and generally increased with frequency, while the natural pressure wave travelled at a speed of about 10 m/sec. On the basis of an isotropic wall model the axial wave speed should however be approximately 5 times higher than the pressure wave speed. This discrepancy can be interpreted as an indication for an anisotropic behavior of the carotid wall. The carotid artery appears to be more elastic in the axial than in the circumferential direction.     

Nonlocal shear deformable shell model for postbuckling of axially compressed microtubules embedded in an elastic medium

Buckling and postbuckling analysis is presented for axially compressed microtubules (MTs) embedded in an elastic matrix of cytoplasm. The microtubule is modeled as a nonlocal shear deformable cylindrical shell which contains small scale effects. The surrounding elastic medium is modeled as a Pasternak foundation. The governing equations are based on higher order shear deformation shell theory with a von Kármán-Donnell-type of kinematic nonlinearity and include the extension-twist and flexural-twist couplings. The thermal effects are also included and the material properties are assumed to be temperature-dependent. The small scale parameter e ₀ a is estimated by matching the buckling load from their vibrational behavior of MTs with the numerical results obtained from the nonlocal shear deformable shell model. The numerical results show that buckling load and postbuckling behavior of MTs are very sensitive to the small scale parameter e ₀ a. The results reveal that the MTs under axial compressive loading condition have an unstable postbuckling path, and the lateral constraint has a significant effect on the postbuckling response of a microtubule when the foundation stiffness is sufficiently large.     

Mechanisms of very fast oscillations in networks of axons coupled by gap junctions

Because electrical coupling among the neurons of the brain is much faster than chemical synaptic coupling, it is natural to hypothesize that gap junctions may play a crucial role in mechanisms underlying very fast oscillations (VFOs), i.e., oscillations at more than 80 Hz. There is now a substantial body of experimental and modeling literature supporting this hypothesis. A series of modeling papers, starting with work by Roger Traub and collaborators, have suggested that VFOs may arise from expanding waves propagating through an “axonal plexus”, a large random network of electrically coupled axons. Traub et al. also proposed a cellular automaton (CA) model to study the mechanisms of VFOs in the axonal plexus. In this model, the expanding waves take the appearance of topologically circular “target patterns”. Random external stimuli initiate each wave. We therefore call this kind of VFO “externally driven”. Using a computational model, we show that an axonal plexus can also exhibit a second, distinctly different kind of VFO in a wide parameter range. These VFOs arise from activity propagating around cycles in the network. Once triggered, they persist without any source of excitation. With idealized, regular connectivity, they take the appearance of spiral waves. We call these VFOs “re-entrant”. The behavior of the axonal plexus depends on the reliability with which action potentials propagate from one axon to the next, which, in turn, depends on the somatic membrane potential V _s and the gap junction conductance g _gj . To study these dependencies, we impose a fixed value of V _s , then study the effects of varying V _s and g _gj . Not surprisingly, propagation becomes more reliable with rising V _s and g _gj . Externally driven VFOs occur when V _s and g _gj are so high that propagation never fails. For lower V _s or g _gj , propagation is nearly reliable, but fails in rare circumstances. Surprisingly, the parameter regime where this occurs is fairly large. Even a single propagation failure can trigger re-entrant VFOs in this regime. Lowering V _s and g _gj further, one finds a third parameter regime in which propagation is unreliable, and no VFOs arise. We analyze these three parameter regimes by means of computations using model networks adapted from Traub et al., as well as much smaller model networks.     

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.     

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.     

Tissue Aspiration and Inverse Finite Element Characterisation of Soft Tissues

In this work we examined the determination of soft tissue parameters via tissue aspiration experiments and inverse finite element characterisation. An aspiration tube was put against the target tissue. The deformation of the tissue inside the tube caused by weak suction was tracked with a video based system. A strain energy function was employed to model the elastic behaviour of soft tissue and viscoelasticity was accounted for by means of a quasi-linear viscoelastic formulation. Friction between the aspiration tube and the aspirated tissue was included in the model. Based on the assumed material model, an optimal set of material parameters was found, in order to best fit the experimental data obtained from ex-vivo experiments on pig kidney cortex. The inverse method resulted in robust determination of the unknown material parameters.     

Morphogenetic movements during axolotl neural tube formation tracked by digital imaging

During neurulation in vertebrate embryos, epithelial cells of the neural plate undergo complex morphogenetic movements that culminate in rolling of the plate into a tube. Resolution of the determinants of this process requires an understanding of the precise movements of cells within the epithelial sheet. A computer algorithm that allows automated tracking of epithelial cells visible in digitized video images is presented. It is used to quantify the displacement field associated with morphogenetic movements in the axolotl (Ambystoma mexicanum) neural plate during normal neural tube formation. Movements from lateral to medial, axial elongations and area changes are calculated from the displacement field data and plotted as functions of time. Regional and temporal differences are identified. The approach presented is suitable for analyzing a wide variety of morphogenetic movements.     

Tissue-Specific Mathematical Models of Slow Wave Entrainment in Wild-Type and 5-HT2B Knockout Mice with Altered Interstitial Cells of Cajal Networks

Gastrointestinal slow waves are generated within networks of interstitial cells of Cajal (ICCs). In the intact tissue, slow waves are entrained to neighboring ICCs with higher intrinsic frequencies, leading to active propagation of slow waves. Degradation of ICC networks in humans is associated with motility disorders; however, the pathophysiological mechanisms of this relationship are uncertain. A recently developed biophysically based mathematical model of ICC was adopted and updated to simulate entrainment of slow waves. Simulated slow wave propagation was successfully entrained in a one-dimensional model, which contained a gradient of intrinsic frequencies. Slow wave propagation was then simulated in tissue models which contained a realistic two-dimensional microstructure of the myenteric ICC networks translated from wild-type (WT) and 5-HT_2B knockout (degraded) mouse jejunum. The results showed that the peak current density in the WT model was 0.49 μA mm⁻² higher than the 5-HT_2B knockout model, and the intracellular Ca²⁺ density after 400 ms was 0.26 mM mm⁻² higher in the WT model. In conclusion, tissue-specific models of slow waves are presented, and simulations quantitatively demonstrated physiological differences between WT and 5-HT_2B knockout models. This study provides a framework for evaluating how ICC network degradation may impair slow wave propagation and ultimately motility and transit.     

Subcellular propagation of calcium waves in Müller glia does not require autocrine/paracrine purinergic signaling

The polarized morphology of radial glia allows them to functionally interconnect different layers of CNS tissues including the retina, cerebellum, and cortex. A likely mechanism involves propagation of transcellular Ca²⁺ waves which were proposed to involve purinergic signaling. Because it is not known whether ATP release is required for astroglial Ca²⁺ wave propagation we investigated this in mouse Müller cells, radial astroglia-like retinal cells in which in which waves can be induced and supported by Orai/TRPC1 (transient receptor potential isoform 1) channels. We found that depletion of endoplasmic reticulum (ER) stores triggers regenerative propagation of transcellular Ca²⁺ waves that is independent of ATP release and activation of P₂X and P₂Y receptors. Both the amplitude and kinetics of transcellular, depletion-induced waves were resistant to non-selective purinergic P₂ antagonists such as pyridoxalphosphate-6-azophenyl-2′,4′-disulfonic acid (PPADS). Thus, store-operated calcium entry (SOCE) is itself sufficient for the initiation and subcellular propagation of calcium waves in radial glia.     

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.     

Bacterial transformation using micro-shock waves

Shock waves are one of the most competent mechanisms of energy dissipation observed in nature. We have developed a novel device to generate controlled micro-shock waves using an explosive-coated polymer tube. In this study, we harnessed these controlled micro-shock waves to develop a unique bacterial transformation method. The conditions were optimized for the maximum transformation efficiency in Escherichia coli. The maximum transformation efficiency was obtained when we used a 30 cm length polymer tube, 100 μm thick metal foil, 200 mM CaCl₂, 1 ng/μl plasmid DNA concentration, and 1 × 10⁹ cell density. The highest transformation efficiency achieved (1 × 10⁻⁵ transformants/cell) was at least 10 times greater than the previously reported ultrasound-mediated transformation (1 × 10⁻⁶ transformants/cell). This method was also successfully employed for the efficient and reproducible transformation of Pseudomonas aeruginosa and Salmonellatyphimurium. This novel method of transformation was shown to be as efficient as electroporation with the added advantage of better recovery of cells, reduced cost (40 times cheaper than a commercial electroporator), and growth phase independent transformation.