Interaction of peristaltic flow with pulsatile flow in a circular cylindrical tube

The effect of pulsatile flow on peristaltic transport in a circular cylindrical tube is analysed. The flow of a Newtonian viscous incompressible fluid in a flexible circular cylindrical tube on which an axisymmetric travelling sinusoidal wave is imposed, is considered. The initial flow in the tube is induced by an arbitrary periodic pressure gradient. A perturbation solution with amplitude ratio (wave amplitude/tube radius) as a parameter is obtained when the frequency of the travelling wave and that of the imposed pressure gradient are equal. The interaction effects of periodic wall induced flow and periodic pressure imposed flow are visualized through the presence of substantially different components of steady and higher harmonic oscillating flow in the first order flow solution. Numerical results show a strong variation of steady state velocity profiles with boundary wave number and Reynolds number and a strong phase shift behaviour of the flow in the radial direction.     

Separation of arterial pressure into a nonlinear superposition of solitary waves and a windkessel flow

A simplified model of arterial blood pressure intended for use in model-based signal processing applications is presented. The main idea is to decompose the pressure into two components: a travelling wave which describes the fast propagation phenomena predominating during the systolic phase and a windkessel flow that represents the slow phenomena during the diastolic phase. Instead of decomposing the blood pressure pulse into a linear superposition of forward and backward harmonic waves, as in the linear wave theory, a nonlinear superposition of travelling waves matched to a reduced physical model of the pressure, is proposed. Very satisfactory experimental results are obtained by using forward waves, the N-soliton solutions of a Korteweg–de Vries equation in conjunction with a two-element windkessel model. The parameter identifiability in the practically important 3-soliton case is also studied. The proposed approach is briefly compared with the linear one and its possible clinical relevance is discussed.     

A Resonance Approach to Cochlear Mechanics

Background

How does the cochlea analyse sound into its component frequencies? In the 1850s Helmholtz thought it occurred by resonance, whereas a century later Békésy''s work indicated a travelling wave. The latter answer seemed to settle the question, but with the discovery in 1978 that the cochlea emits sound, the mechanics of the cochlea was back on the drawing board. Recent studies have raised questions about whether the travelling wave, as currently understood, is adequate to explain observations.

Approach

Applying basic resonance principles, this paper revisits the question. A graded bank of harmonic oscillators with cochlear-like frequencies and quality factors is simultaneously excited, and it is found that resonance gives rise to similar frequency responses, group delays, and travelling wave velocities as observed by experiment. The overall effect of the group delay gradient is to produce a decelerating wave of peak displacement moving from base to apex at characteristic travelling wave speeds. The extensive literature on chains of coupled oscillators is considered, and the occurrence of travelling waves, pseudowaves, phase plateaus, and forced resonance in such systems is noted.

Conclusion and significance

This alternative approach to cochlear mechanics shows that a travelling wave can simply arise as an apparently moving amplitude peak which passes along a bank of resonators without carrying energy. This highlights the possible role of the fast pressure wave and indicates how phase delays and group delays of a set of driven harmonic oscillators can generate an apparent travelling wave. It is possible to view the cochlea as a chain of globally forced coupled oscillators, and this model incorporates fundamental aspects of both the resonance and travelling wave theories.     

Wave propagation in a model of the arterial circulation

The propagation of the arterial pulse wave in the large systemic arteries has been calculated using a linearised method of characteristics analysis to follow the waves generated by the heart. The model includes anatomical and physiological data for the 55 largest arteries adjusted so that the bifurcating tree of arteries is well matched for forward travelling waves. The peripheral arteries in the model are terminated by resistance elements which are adjusted to produce a physiologically reasonable distribution of mean blood flow. In the model, the pressure and velocity wave generated by the contraction of the left ventricle propagates to the periphery where it is reflected. These reflected waves are re-reflected by each of the bifurcations that they encounter and a very complex pattern of waves is generated. The results of the calculations exhibit many of the features of the systemic arteries, including the increase of the pulse pressure with distance away from the heart as well as the initial decrease and then the large increase in the magnitude of back flow during late systole going from the ascending aorta to the abdominal aorta to the arteries of the leg. The model is then used to study the effects of the reflection or absorption of waves by the heart and the mechanisms leading to the incisura are investigated. Calculations are carried out with the total occlusion of different arterial segments in order to model experiments in which the effects of the occlusion of different arteries on pressure and flow in the ascending aorta were measured. Finally, the effects of changes in peripheral resistance on pressure and velocity waveforms are also studied. We conclude from these calculations that the complex pattern of wave propagation in the large arteries may be the most important determinant of arterial haemodynamics.     

Wave dissipation in flexible tubes in the time domain: in vitro model of arterial waves

Earlier work of wave dissipation in flexible tubes and arteries has been carried out predominantly in the frequency domain and most of the studies used the measured pressure waveform for presenting the results. In this work we investigate the pattern of wave dissipation in the time domain using the separated forward and backward travelling waves in flexible tubes. We tested four sizes of latex tubes of 2m in length each, where a single semi-sinusoidal in shape, pressure wave, was produced at the inlet of each tube. Simultaneous measurements of pressure and flow waveforms were recorded every 5cm along the tubes and wave speed was determined using the pressure-velocity loop method (PU-loop). The measured data and wave speed were used to separate the pressure waveform and wave intensity, into their forward and backward directions, using wave intensity analysis (WIA). Also, the energy carried by the wave was calculated by integrating the relevant area under the wave intensity curve. The peak of the measured pressure waveform increased downstream, however, the peak of the separated forward pressure waveform decreased exponentially along the tube. Wave intensity and energy also dissipated exponentially along the travelling distance. The peaks of the separated pressure and wave intensity decreased in the forward in a similar exponential way to that in the backward direction in all four tube sizes. Also, the smaller the size of the tube the greater wave dissipation it caused. We conclude that wave separation is useful in studying wave dissipation in elastic tubes, and WIA provides a convenient method for determining the dissipation of the energy carried by the wave along the travelled distance. The separated pressure waveform, wave intensity and wave energy dissipate exponentially with the travelling distance, and wave dissipation varies conversely with the diameter of elastic tubes.     

Arterial pulsation-driven cerebrospinal fluid flow in the perivascular space: a computational model

This study was conducted to determine whether local arterial pulsations are sufficient to cause cerebrospinal fluid (CSF) flow along perivascular spaces (PVS) within the spinal cord. A theoretical model of the perivascular space surrounding a "typical" small artery was analysed using computational fluid dynamics. Systolic pulsations were modelled as travelling waves on the arterial wall. The effects of wave geometry and variable pressure conditions on fluid flow were investigated. Arterial pulsations induce fluid movement in the PVS in the direction of arterial wave travel. Perivascular flow continues even in the presence of adverse pressure gradients of a few kilopascals. Flow rates are greater with increasing pulse wave velocities and arterial deformation, as both an absolute amplitude and as a proportion of the PVS. The model suggests that arterial pulsations are sufficient to cause fluid flow in the perivascular space even against modest adverse pressure gradients. Local increases in flow in this perivascular pumping mechanism or reduction in outflow may be important in the etiology of syringomyelia.     

Integrodifference equations in the presence of climate change: persistence criterion,travelling waves and inside dynamics

To understand the effects that the climate change has on the evolution of species as well as the genetic consequences, we analyze an integrodifference equation (IDE) models for a reproducing and dispersing population in a spatio-temporal heterogeneous environment described by a shifting climate envelope. Our analysis on the IDE focuses on the persistence criterion, travelling wave solutions, and the inside dynamics. First, the persistence criterion, characterizing the global dynamics of the IDE, is established in terms of the basic reproduction number. In the case of persistence, a unique travelling wave is found to govern the global dynamics. The effects of the size and the shifting speed of the climate envelope on the basic reproduction number, and hence, on the persistence criterion, are also investigated. In particular, the critical domain size and the critical shifting speed are found in certain cases. Numerical simulations are performed to complement the theoretical results. In the case of persistence, we separate the travelling wave and general solutions into spatially distinct neutral fractions to study the inside dynamics. It is shown that each neutral genetic fraction rearranges itself spatially so as to asymptotically achieve the profile of the travelling wave. To measure the genetic diversity of the population density we calculate the Shannon diversity index and related indices, and use these to illustrate how diversity changes with underlying parameters.

     

Experimental validation of a time-domain-based wave propagation model of blood flow in viscoelastic vessels

Time-domain-based one-dimensional wave propagation models of the arterial system are preferable over one-dimensional wave propagation models in the frequency domain since the latter neglect the non-linear convection forces present in the physiological situation, especially when the vessel is tapered. Moreover, one-dimensional wave propagation models of the arterial system can be used to provide boundary conditions for fully three-dimensional fluid-structure interaction computations that are usually defined in the time domain. In this study, a time-domain-based one-dimensional wave propagation model in a cross-sectional area, flow and pressure (A,q,p)-formulation is developed. Using this formulation, a constitutive law that includes viscoelasticity based on the mechanical behaviour of a Kelvin body, is introduced. The resulting pressure and flow waves travelling through a straight and tapered vessel are compared to experimental data obtained from measurements in an in vitro setup. The model presented shows to be well suited to predict wave propagation through these straight and tapered vessels with viscoelastic wall properties and hereto can serve as a time-domain-based method to model wave propagation in the human arterial system.     

Travelling Waves in Hybrid Chemotaxis Models

Hybrid models of chemotaxis combine agent-based models of cells with partial differential equation models of extracellular chemical signals. In this paper, travelling wave properties of hybrid models of bacterial chemotaxis are investigated. Bacteria are modelled using an agent-based (individual-based) approach with internal dynamics describing signal transduction. In addition to the chemotactic behaviour of the bacteria, the individual-based model also includes cell proliferation and death. Cells consume the extracellular nutrient field (chemoattractant), which is modelled using a partial differential equation. Mesoscopic and macroscopic equations representing the behaviour of the hybrid model are derived and the existence of travelling wave solutions for these models is established. It is shown that cell proliferation is necessary for the existence of non-transient (stationary) travelling waves in hybrid models. Additionally, a numerical comparison between the wave speeds of the continuum models and the hybrid models shows good agreement in the case of weak chemotaxis and qualitative agreement for the strong chemotaxis case. In the case of slow cell adaptation, we detect oscillating behaviour of the wave, which cannot be explained by mean-field approximations.     

Arterial Pulsation-driven Cerebrospinal Fluid Flow in the Perivascular Space: A Computational Model

This study was conducted to determine whether local arterial pulsations are sufficient to cause cerebrospinal fluid (CSF) flow along perivascular spaces (PVS) within the spinal cord. A theoretical model of the perivascular space surrounding a "typical" small artery was analysed using computational fluid dynamics. Systolic pulsations were modelled as travelling waves on the arterial wall. The effects of wave geometry and variable pressure conditions on fluid flow were investigated. Arterial pulsations induce fluid movement in the PVS in the direction of arterial wave travel. Perivascular flow continues even in the presence of adverse pressure gradients of a few kilopascals. Flow rates are greater with increasing pulse wave velocities and arterial deformation, as both an absolute amplitude and as a proportion of the PVS. The model suggests that arterial pulsations are sufficient to cause fluid flow in the perivascular space even against modest adverse pressure gradients. Local increases in flow in this perivascular pumping mechanism or reduction in outflow may be important in the etiology of syringomyelia.     

Non-linear bioconvection in a deep suspension of gyrotactic swimming micro-organisms

 The non-linear structure of deep, stochastic, gyrotactic bioconvection is explored. A linear analysis is reviewed and a weakly non-linear analysis justifies its application by revealing the supercritical nature of the bifurcation. An asymptotic expansion is used to derive systems of partial differential equations for long plume structures which vary slowly with depth. Steady state and travelling wave solutions are found for the first order system of partial differential equations and the second order system is manipulated to calculate the speed of vertically travelling pulses. Implications of the results and possibilities of experimental validation are discussed. Received: 26 May 1997 / Revised version: 10 May 1998     

Detection without deflection? A hypothesis for direct sensing of sound pressure by hair cells

It is widely thought that organisms detect sound by sensing the deflection of hair-like projections, the stereocilia, at the apex of hair cells. In the case of mammals, the standard interpretation is that hair cells in the cochlea respond to deflection of stereocilia induced by motion generated by a hydrodynamic travelling wave. But in the light of persistent anomalies, an alternative hypothesis seems to have some merit: that sensing cells (in particular the outer hair cells) may, at least at low intensities, be reacting to a different stimulus — the rapid pressure wave that sweeps through the cochlear fluids at the speed of sound in water. This would explain why fast responses are sometimes seen before the peak of the travelling wave. Yet how could cells directly sense fluid pressure? Here, a model is constructed of the outer hair cell as a pressure vessel able to sense pressure variations across its cuticular pore, and this ‘fontanelle’ model, based on the sensing action of the basal body at this compliant spot, could explain the observed anomalies. Moreover, the fontanelle model can be applied to a wide range of other organisms, suggesting that direct pressure detection is a general mode of sensing complementary to stereociliar displacement.     

A model of herd grazing as a travelling wave,chemotaxis and stability

Possible constitutive models are examined for the formation of a herd, under the assumption that a herd forms a travelling wave while grazing. Under quite general conditions, it is found that the only possibility for a travelling wave is a balance between food seeking and natural dispersion, such as in chemotaxis. The stability of the travelling wave previously conjectured, is shown both for one- and two-dimensional perturbations.     

Mathematical modelling of ciliary propulsion of an electrically-conducting Johnson-Segalman physiological fluid in a channel with slip

Bionic systems frequently feature electromagnetic pumping and offer significant advantages over conventional designs via intelligent bio-inspired properties. Complex wall features observed in nature also provide efficient mechanisms which can be utilized in biomimetic designs. The characteristics of biological fluids are frequently non-Newtonian in nature. In many natural systems super-hydrophobic slip is witnessed. Motivated by these phenomena, in this paper, we discussed a mathematical model for the cilia-generated propulsion of an electrically-conducting viscoelastic physiological fluid in a ciliated channel under the action of magnetic field. The rheological behavior of the fluid is simulated with the Johnson-Segalman constitutive model which allows internal wall slip. The regular or coordinated movement of the ciliated edges (which line the internal walls of the channel) is represented by a metachronal wave motion in the horizontal direction which generates a two-dimensional velocity profile. This mechanism is imposed by a periodic boundary condition which generates propulsion in the channel flow. Under the classical lubrication approximation, the boundary value problem is non-dimensionalized and solved analytically with a perturbation technique. The influence of the geometric, rheological (slip and Weissenberg number) and magnetic parameters on velocity, pressure gradient and the pressure rise (evaluated via the stream function in symbolic software) are presented graphically and interpreted at length.     

Computer modeling of fluid dynamics related to a myocardial bridge in a coronary artery

Fluid mechanics associated with blood flows induced by the so-called myocardial bridge (MB) has been studied systematically using a computational fluid dynamic modeling of the Newtonian, incompressible, two-dimensional, unsteady flow in a channel with a time-dependently flushing in/out indentation. During each cycle, a train of vortex wave flow was observed downstream of the phasic stenosis and both upper and lower walls suffer severely from consistently high, oscillating wall shear stresses (WSS). Extensive studies were conducted on the influence of the Reynolds number, the geometry and the Strouhal number of the MB movement on the nature of the vortex flow and the time-dependent wall shear stress distribution. Special attention was drawn to the relationship between the vortex wave and the pressure distribution. It was found that the pressure gradient changed markedly during one cycle, which was apparently dominated by the dynamics of the indentation. A steep, adverse pressure gradient was observed when the indentation was flushing out, which corresponded to the existence of the most developing vortices. It implies the possibility that the MB in a coronary artery can produce an extremely low pressure region immediately downstream of the phasic stenosis, where elastic choking or collapse of the coronary artery might occur.     

Mathematica numerical simulation of peristaltic biophysical transport of a fractional viscoelastic fluid through an inclined cylindrical tube

This paper studies the peristaltic transport of a viscoelastic fluid (with the fractional second-grade model) through an inclined cylindrical tube. The wall of the tube is modelled as a sinusoidal wave. The flow analysis is presented under the assumptions of long wave length and low Reynolds number. Caputo's definition of fractional derivative is used to formulate the fractional differentiation. Analytical solutions are developed for the normalized momentum equations. Expressions are also derived for the pressure, frictional force, and the relationship between the flow rate and pressure gradient. Mathematica numerical computations are then performed. The results are plotted and analysed for different values of fractional parameter, material constant, inclination angle, Reynolds number, Froude number and peristaltic wave amplitude. It is found that fractional parameter and Froude number resist the flow pattern while material constant, Reynolds number, inclination of angle and amplitude aid the peristaltic flow. Furthermore, frictional force and pressure demonstrate the opposite behaviour under the influence of the relevant parameters emerging in the equations of motion. The study has applications in uretral biophysics, and also potential use in peristaltic pumping of petroleum viscoelastic bio-surfactants in chemical engineering and astronautical applications involving conveyance of non-Newtonian fluids (e.g. lubricants) against gravity and in conduits with deformable walls.     

Theoretical study of flow,pressure and solute concentration in double membrane systems

A model system consisting of two rigidly held membranes in series was investigated through the application of the Kedem and Katchalsky thermodynamic single membrane flow equations. This analysis results in predictions of the steady state flow properties as well as values for the solute concentration and pressure of the internal compartment when the system is under the influence of a constant solute concentration or hydrostatic pressure gradient. It is demonstrated that although the flow properties and internal compartment pressure are complicated functions of the membrane permeability coefficients and driving gradient across the system, the relationships are greatly simplified by the explicit appearance of the internal compartment steady state solute concentration in the equations. It is shown that the steady state volume flow rate depends on the absolute value of the solute concentration in the external compartments, as well as the solute concentration gradient across the system. The properties of non-linear dependence of volume flow on concentration gradient, and rectification of volume flow are discussed and shown to be independent properties of the system. For the system under the influence of a solute concentration gradient, the internal compartment pressure can be greater or less than the ambient pressure, and depends mainly on the order in which the membranes are encountered by the volume flow. These properties are qualitatively correlated with certain available experimental observations in biological systems.     

RV instantaneous intraventricular diastolic pressure and velocity distributions in normal and volume overload awake dog disease models

Intraventricular diastolic right ventricular (RV) flow field dynamics were studied by functional imaging using three-dimensional (3D) real-time echocardiography with sonomicrometry and computational fluid dynamics in seven awake dogs at control with normal wall motion (NWM) and RV volume overload with diastolic paradoxical septal motion. Burgeoning flow cross section between inflow anulus and chamber walls induces a convective pressure rise, which represents a "convective deceleration load" (CDL). High spatiotemporal resolution dynamic pressure and velocity distributions of the intraventricular RV flow field revealed time-dependent, subtle interactions between intraventricular local acceleration and convective pressure gradients. During the E-wave upstroke, the total pressure gradient along intraventricular flow is the algebraic sum of a pressure decrease contributed by local acceleration and a pressure rise contributed by a convective deceleration that partially counterbalances the local acceleration gradient. This underlies the smallness of early diastolic intraventricular gradients. At peak volumetric inflow, local acceleration vanishes and the total adverse intraventricular gradient is convective. During the E-wave downstroke, the strongly adverse gradient embodies the streamwise pressure augmentations from both local and convective decelerations. It induces flow separation and large-scale vortical motions, stronger in NWM. Their dynamic corollaries on intraventricular pressure and velocity distributions were ascertained. In the NWM pattern, the strong ring-like vortex surrounding the central core encroaches on the area available for flow toward the apex. This results in higher linear velocities later in the downstroke of the E wave than at peak inflow rate. The augmentation of CDL by ventriculoannular disproportion may contribute to E wave and E-to-A ratio depression with chamber dilatation.     

On the propagation of a wave front in viscoelastic arteries

In formulating a mathematical model of the arterial system, the one-dimensional flow approximation yields realistic pressure and flow pulses in the proximal as well as in the distal regions of a simulated arterial conduit, provided that the viscoelastic damping induced by the vessel wall is properly taken into account. Models which are based on a purely elastic formulation of the arterial wall properties are known to produce shocklike transitions in the propagating pulses which are not observed in man under physiological conditions. The viscoelastic damping characteristics are such that they are expected to reduce the tendency of shock formation in the model. In order to analyze this phenomenon, the propagation of first and second-order pressure waves is calculated with the aid of a wave front expansion, and criteria for the formation of shocks are derived. The application of the results to the human arterial system show that shock waves are not to be expected under normal conditions, while in case of a pathologically increased pressure rise at the root of the aorta, shocklike transitions may develop in the periphery. In particular, it is shown that second-order waves never lead to shock formation in finite time for the class of initial conditions and mechanical wave guides which are of interest in the mammalian circulation.     

Bio-medico-mechanical behavior of natural artery blood vessel under constant and variable internal pulsatile pressure flow test in vitro

Studies have been carried out on the bio-medico-mechanical behavior in vitro of natural blood vessel (dogs) under constant and variable internal pulsatile pressure flow. The apparatus designed by us well simulated the arterial system. The studies were made for the case of pressure amplitude kept as constant, of the two-step-multi-duplicated pulsatile pressure and of the fluctuating pressure. For the case of the fluctuating pressure, the strength of the artery becomes considerably lower than those under constant amplitude and two-step-multi-duplicated pulsatile pressure. SEM observations of the inner walls of the artery shows that collagen fibers are more elongated under fluctuating pulsatile pressure flow. In conclusion, in order to avoid the mechanical deterioration of the artery strength, it is useful to keep the pulsatile blood pressure at constant amplitude. Even for the case of the blood pressure fluctuation, it is necessary to manage to keep the blood pressure as near a regular wave as possible, the total number of repeated pulse being equal.