Computational model for the transition from peristaltic to pulsatile flow in the embryonic heart tube

Early in development, the heart is a single muscle-wrapped tube without formed valves. Yet survival of the embryo depends on the ability of this tube to pump blood at steadily increasing rates and pressures. Developmental biologists historically have speculated that the heart tube pumps via a peristaltic mechanism, with a wave of contraction propagating from the inflow to the outflow end. Physiological measurements, however, have shown that the flow becomes pulsatile in character quite early in development, before the valves form. Here, we use a computational model for flow though the embryonic heart to explore the pumping mechanism. Results from the model show that endocardial cushions, which are valve primordia arising near the ends of the tube, induce a transition from peristaltic to pulsatile flow. Comparison of numerical results with published experimental data shows reasonably good agreement for various pressure and flow parameters. This study illustrates the interrelationship between form and function in the early embryonic heart.     

A mathematical simulation of the ureter: effects of the model parameters on ureteral pressure/flow relations

Ureteral peristaltic mechanism facilitates urine transport from the kidney to the bladder. Numerical analysis of the peristaltic flow in the ureter aims to further our understanding of the reflux phenomenon and other ureteral abnormalities. Fluid-structure interaction (FSI) plays an important role in accuracy of this approach and the arbitrary Lagrangian-Eulerian (ALE) formulation is a strong method to analyze the coupled fluid-structure interaction between the compliant wall and the surrounding fluid. This formulation, however, was not used in previous studies of peristalsis in living organisms. In the present investigation, a numerical simulation is introduced and solved through ALE formulation to perform the ureteral flow and stress analysis. The incompressible Navier-Stokes equations are used as the governing equations for the fluid, and a linear elastic model is utilized for the compliant wall. The wall stimulation is modeled by nonlinear contact analysis using a rigid contact surface since an appropriate model for simulation of ureteral peristalsis needs to contain cell-to-cell wall stimulation. In contrast to previous studies, the wall displacements are not predetermined in the presented model of this finite-length compliant tube, neither the peristalsis needs to be periodic. Moreover, the temporal changes of ureteral wall intraluminal shear stress during peristalsis are included in our study. Iterative computing of two-way coupling is used to solve the governing equations. Two phases of nonperistaltic and peristaltic transport of urine in the ureter are discussed. Results are obtained following an analysis of the effects of the ureteral wall compliance, the pressure difference between the ureteral inlet and outlet, the maximum height of the contraction wave, the contraction wave velocity, and the number of contraction waves on the ureteral outlet flow. The results indicate that the proximal part of the ureter is prone to a higher shear stress during peristalsis compared with its middle and distal parts. It is also shown that the peristalsis is more efficient as the maximum height of the contraction wave increases. Finally, it is concluded that improper function of ureteropelvic junction results in the passage of part of urine back flow even in the case of slow start-up of the peristaltic contraction wave.     

Peristaltic transport in a channel with a porous peripheral layer: model of a flow in gastrointestinal tract

Peristaltic transport in a two dimensional channel, filled with a porous medium in the peripheral region and a Newtonian fluid in the core region, is studied under the assumptions of long wavelength and low Reynolds number. The fluid flow is investigated in the waveframe of reference moving with the velocity of the peristaltic wave. Brinkman extended Darcy equation is utilized to model the flow in the porous layer. The interface is determined as a part of the solution using the conservation of mass in both the porous and fluid regions independently. A shear-stress jump boundary condition is used at the interface. The physical quantities of importance in peristaltic transport like pumping, trapping, reflux and axial velocity are discussed for various parameters of interest governing the flow like Darcy number, porosity, permeability, effective viscosity etc. It is observed that the peristalsis works as a pump against greater pressure in two-layered model with a porous medium compared with a viscous fluid in the peripheral layer. Increasing Darcy number Da decreases the pumping and increasing shear stress jump constant beta results in increasing the pumping. The limits on the time averaged flux Q for trapping in the core layer are obtained. The discussion on pumping, trapping and reflux may be helpful in understanding some of the fluid dynamic aspects of the transport of chyme in gastrointestinal tract.     

Computational models of heart pumping efficiencies based on contraction waves in spiral elastic bands

We present a framework for modeling biological pumping organs based on coupled spiral elastic band geometries and active wave-propagating excitation mechanisms. Two pumping mechanisms are considered in detail by way of example: one of a simple tube, which represents a embryonic fish heart and another more complicated structure with the potential to model the adult human heart. Through finite element modeling different elastic contractions are induced in the band. For each version the pumping efficiency is measured and the dynamics are evaluated. We show that by combining helical shapes of muscle bands with a contraction wave it is possible not only to achieve efficient pumping, but also to create desired dynamics of the structure. As a result we match the function of the model pumps and their dynamics to physiological observations.     

An analysis of water flow in tube-living animals

Water flow has been studied in six tube-dwelling animals that have different pumping mechanisms and layout of their tube systems. The characteristics of the pumping mechanisms and the resistance of the tube systems have been found. Cilia act as impeller pumps and can produce large rates of flow when arranged in parallel, but may produce sufficient flow for small, long thin bodied animals when arranged in series. Larger vermiform animals must use piston mechanism to produce sufficient rates of flow and the high pressures they can produce do not seem to be of use to the animals during normal pumping. Animals with rigid limbs can use these to impel water and produce high rates of flow. Echinocardium, which has a globular body, must match the flow at the inflow and outflow points on the test to the flow over the circumference and has special ciliated spines at these sites to boost flow.     

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.     

The velocity of the arterial pulse wave: a viscous-fluid shock wave in an elastic tube

Background

The arterial pulse is a viscous-fluid shock wave that is initiated by blood ejected from the heart. This wave travels away from the heart at a speed termed the pulse wave velocity (PWV). The PWV increases during the course of a number of diseases, and this increase is often attributed to arterial stiffness. As the pulse wave approaches a point in an artery, the pressure rises as does the pressure gradient. This pressure gradient increases the rate of blood flow ahead of the wave. The rate of blood flow ahead of the wave decreases with distance because the pressure gradient also decreases with distance ahead of the wave. Consequently, the amount of blood per unit length in a segment of an artery increases ahead of the wave, and this increase stretches the wall of the artery. As a result, the tension in the wall increases, and this results in an increase in the pressure of blood in the artery.

Methods

An expression for the PWV is derived from an equation describing the flow-pressure coupling (FPC) for a pulse wave in an incompressible, viscous fluid in an elastic tube. The initial increase in force of the fluid in the tube is described by an increasing exponential function of time. The relationship between force gradient and fluid flow is approximated by an expression known to hold for a rigid tube.

Results

For large arteries, the PWV derived by this method agrees with the Korteweg-Moens equation for the PWV in a non-viscous fluid. For small arteries, the PWV is approximately proportional to the Korteweg-Moens velocity divided by the radius of the artery. The PWV in small arteries is also predicted to increase when the specific rate of increase in pressure as a function of time decreases. This rate decreases with increasing myocardial ischemia, suggesting an explanation for the observation that an increase in the PWV is a predictor of future myocardial infarction. The derivation of the equation for the PWV that has been used for more than fifty years is analyzed and shown to yield predictions that do not appear to be correct.

Conclusion

Contrary to the theory used for more than fifty years to predict the PWV, it speeds up as arteries become smaller and smaller. Furthermore, an increase in the PWV in some cases may be due to decreasing force of myocardial contraction rather than arterial stiffness.     

Density gradient system: II. A 50 channel programmable undulating diphragm peristaltic pump

A special peristaltic pump is described that has functioned as part of a system for density gradient formation and fractionation. Twenty-five pumping tubes are actuated on both the top and bottom of the pump. A Mylar diaphragm interposed between the rollers and the pumping tubes filters out the horizontal, stretching component of the forces imparted to the tubes. This greatly prolongs tube life, increases the allowable pressures that can be achieved with such a pump, and thus permits accurate delivery of viscous solutions.An explanation of the cause of the pulsations produced by peristaltic pumps is presented and the virtual elimination of these pulsations is demonstrated.Both velocity and direction of flow of the pump are controlled. By means of two independent bidirectional digital counters, preset volumes of fluid can be delivered and the total volume of liquid determined.Studies demonstrating the relative independence of fluid volume delivered at a preset count versus flow velocity and composition are presented.Other possibilities for use of the pump in automating density gradient analysis are discussed. Possibilities for employment of the pump for autoanalysis and in artificial organs are indicated.     

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.     

Characterization of the separation properties of the beckman elutriator system

The role of fluid flow in the elutriation process was visualized by pumping dye solution through the Beckman JE-6 elutriator rotor. Three major fluid flow disturbances were observed in the separation chambers, namely; jet-streaming, ripple flow, and whirl flow. In order to evaluate the effects of these non-ideal fluid flow patterns on the separation of homogeneous populations of particles or cells, 12–35 μm diameter latex spheres and 9L rat brain tumor cells were fractionated with the Beckman elutriator system. The elutriator system was evaluated on the basis of: (1) recovery, (2) elution loss during loading, (3) homogeneity of the size distributions, and (4) the relationship of the median volume of eluted particles or cells to the rotor speed and the collection fluid velocity. Both a conventional collection method (two 40-mL fractions at each collection rotor speed) and a long collection method (10–15 40-mL fractions at several collection rotor speeds) were compared to determine if collection procedures could compensate for some of the difficulties caused by the non-ideal fluid flow patterns. Although more than 90% of the particles or cells were always recovered, about 5% eluted during the loading procedure. Neither collection method altered this phenomenon. The collected populations, but this was accompanied by a reduction in cell yield. The median particle or cell volume of each fraction agreed with that expected under ideal fluid flow conditions except at high and low rotor speeds when the conventional collection method was used.     

Characterization of the separation properties of the Beckman elutriator system

The role of fluid flow in the elutriation process was visualized by pumping dye solution through the Beckman JE-6 elutriator rotor. Three major fluid flow disturbances were observed in the separation chambers, namely; jet-streaming, ripple flow, and whirl flow. In order to evaluate the effects of these non-ideal fluid flow patterns on the separation of homogeneous populations of particles or cells, 12--35 micron diameter latex spheres and 9L rat brain tumor cells were fractionated with the Beckman elutriator system. The elutriator system was evaluated on the basis of: (1) recovery, (2) elution loss during loading, (3) homogeneity of the size distributions, and (4) the relationship of the median volume of eluted particles or cells to the rotor speed and the collection fluid velocity. Both a conventional collection method (two 40-mL fractions at ech collection rotor speed) and a long collection method (10--15 40-mL fractions at several collection rotor speeds) were compared to determine if collection procedures could compensate for some of the difficulties caused by the non-ideal fluid flow patterns. Although more than 90% of the particles or cells were always recovered, about 5% eluted during the loading procedure. Neither collection method altered this phenomenon. The long collection method significantly improved the homogeneity of the collected populations, but this was accompanied by a reduction in cell yield. The median particle or cell volume of each fraction agreed with that expected under ideal fluid flow conditions except at high and low rotor speeds when the conventional collection method was used.     

Pulse waves in prestressed arteries

In order to better understand the effect of initial stress in blood flow in arteries, a theoretical analysis of wave propagation in an initially inflated and axially stretched cylindrical thick shell is investigated. For simplicity in the mathematical analysis, the blood is assumed to be an incompressible inviscid fluid while the arterial wall is taken to be an isotropic, homogeneous and incompressible elastic material. Employing the theory of small deformations superimposed on a large initial field the governing differential equations of perturbed solid motions are obtained in cylindrical polar coordinates. Considering the difficulty in obtaining a closed form solution for the field equations, an approximate power series method is utilized. The dispersion relations for the most general case of this approximation and for the thin tube case are thoroughly discussed. The speeds of waves propagating in an unstressed tube are obtained as a special case of our general treatment. It is observed that the speeds of both waves increase with increasing inner pressure and axial stretch.     

A mathematical model for the peristaltic flow of chyme movement in small intestine

A mathematical model based on viscoelastic fluid (fractional Oldroyd-B model) flow is considered for the peristaltic flow of chyme in small intestine, which is assumed to be in the form of an inclined cylindrical tube. The peristaltic flow of chyme is modeled more realistically by assuming that the peristaltic rush wave is a sinusoidal wave, which propagates along the tube. The governing equations are simplified by making the assumptions of long wavelength and low Reynolds number. Analytical approximate solutions of problem are obtained by using homotopy analysis method and convergence of the obtained series solution is properly checked. For the realistic values of the emerging parameters such as fractional parameters, relaxation time, retardation time, Reynolds number, Froude number and inclination of tube, the numerical results for the pressure difference and the frictional force across one wavelength are computed and discussed the roles played by these parameters during the peristaltic flow. On the basis of this study, it is found that the first fractional parameter, relaxation time and Froude number resist the movement of chyme, while, the second fractional parameter, retardation time, Reynolds number and inclination of tube favour the movement of chyme through the small intestine during pumping. It is further revealed that size of trapped bolus reduces with increasing the amplitude ratio whereas it is unaltered with other parameters.     

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.     

A Mathematical Model of Valveless Pumping: A Lumped Model with Time-Dependent Compliance,Resistance, and Inertia

A new lumped model of flow driven by pumping without valves is presented, motivated by biomedical applications: the circulation of the human fetus before the development of the heart valves and mechanism of blood flow during the external cardiopulmonary resuscitation (CPR). The phenomenon of existence of a unidirectional net flow around a loop of tubing that consists of two different compliances is called valveless pumping. The lumped parameter model of valveless pumping in this paper is governed by the ordinary differential equations for pressure and flow, with time-dependent compliance, resistance, and inertia. This simple model can represent the essential features of valveless pumping we observed in earlier mathematical models and physical experiments of valveless pumping. We demonstrate that not only parameters of the driving function, such as frequency or amplitude, but also physical parameters, such as wall thickness and tube stiffness, are important in determining the direction and magnitude of a net flow. In this system, we report two new and interesting phenomena of valveless pumping: One is that the shifted peak frequency can be predicted by the pulsewave speed and the other is that time-dependent resistance is a crucial factor in generating valveless pumping. We also demonstrate that this lumped model can be extended to a one-dimensional flow model of valveless pumping and explain why a linear case, the case of the constant compliance, resistance, and inertia, generates almost zero net flow. This emphasizes that the nonlinearity of valveless pumping is also an important factor to generate a net flow in a closed loop model of valveless pumping.     

A capillary-based microfluidic instrument suitable for immunoaffinity chromatography

The analysis of biological samples to produce clinical or research data often requires measurement of analytes from complex biological matrices and limited volumes. Miniaturized analytical systems capable of minimal sample consumption and reduced analysis times have been employed to meet this need. The small footprint of this technology offers the potential for portability and patient point-of-care testing. A prototype microfluidic system has been developed and is presented for potential rapid assessment of clinical samples. The system has been designed for immunoaffinity chromatography as a means of separating analytes of interest from biological matrices. The instrument is capable of sub-microliter sample injection and detection of labeled antigens by long wavelength laser-induced fluorescence (LIF). The laboratory-constructed device is assembled from an array of components including two syringe pumps, a nano-gradient mixing chip, a micro-injector, a diode laser, and a separation capillary column made from a polymer/silica (PEEKsil) tube. An in-house program written with LabVIEW software controls the syringe pumps to perform step gradient elution and collects the LIF signal as a chromatogram. Initial columns were packed with silica beads to evaluate the system. Optimization of the device has been achieved by measuring flow accuracy with respect to column length and particle size. Syringe size and pressure effects have also been used to characterize the capability of the pumps. Based on test results, a 200-microm x 25-mm column packed with 1-microm silica beads was chosen for use with a 500-microL syringe. The system was tested for mixer proportioning by pumping different compositions of buffer and fluorescent dye solutions in a stepwise fashion. A linear response was achieved for increasing concentrations of fluorescent dye by online mixing (R2=0.9998). The effectiveness of an acidic gradient was confirmed by monitoring pH post-column and measuring premixed solutions offline. Finally, assessment of detectability was achieved by injecting fluorescent dye solutions and measuring the signal from the LIF detector. The limit of detection for the system with these solutions was 10.0 pM or 10.0 amol on-column. As proof-of-principle, immunoaffinity chromatography was demonstrated with immobilized rabbit anti-goat IgG and a fluorescent dye-goat IgG conjugate as a model antigen.     

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.     

Pulse wave velocity as a diagnostic index: the pitfalls of tethering versus stiffening of the arterial wall

Pulse wave velocity (PWV) is often used as a clinical index of aging, vascular disease, or age related hypertension. This practice is based on the assumption that a higher wave speed indicates vascular stiffening. This assumption is well grounded in the physics of pulsatile flow of an incompressible fluid where it is fully established that a pulse wave travels faster in a tube of stiffer wall, the wave speed becoming infinite in the mathematical limit of a rigid wall. However, in this paper we point out that the physical principal of higher pulse wave velocity in a stiffer tube is strictly valid only when the wall is free from outside constraints, which in the physiological setting is present in the form of tethering of the vessel wall. The use of PWV as an index of arterial stiffening may thus lose its validity if tethering is involved. A solution of the problem of vessel wall mechanics as they arise from the physiological pulsatile flow problem is presented for the purpose of resolving this issue. The vessel wall is considered to have finite thickness with or without tethering and with a range of mechanical properties ranging from viscoelastic to stiff. The results show that, indeed, while the wave speed becomes infinite in the mathematical limit of a rigid free wall, the opposite actually happens if the vessel wall is tethered. Here the wave speed actually diminishes as the degree of tethering increases. This dichotomy in the effects of tethering versus stiffening of the arterial wall may clearly lead to error in the interpretation of PWV as an index of vessel wall stiffness. In particular, a normal value of PWV may lead to the conclusion that vessel wall stiffening is absent while this value may in fact have been lowered by tethering. In other words, the diagnostic test may lead to a false negative diagnosis. Our results indicate that the reason for which PWV is lower in a tethered wall compared with that in a free wall of the same stiffness is that the radial movements of the wall are greatly reduced by tethering. More precisely, the results show that PWV depends strongly on the ratio of radial to axial displacements and that this ratio is much lower in a tethered wall than it is in a free wall of the same stiffness.     

Predator–prey systems in streams and rivers

Many predator–prey systems are found in environments with a predominantly unidirectional flow such as streams and rivers. Alterations of natural flow regimes (e.g., due to human management or global warming) put biological populations at risk. The aim of this paper is to devise a simple method that links flow speeds (currents) with population retention (persistence) and wash-out (extinction). We consider systems of prey and specialist, as well as generalist, predators, for which we distinguish the following flow speed scenarios: (a) coexistence, (b) persistence of prey only or (c) predators only (provided they are generalists), and (d) extinction of both populations. The method is based on a reaction–advection–diffusion model and traveling wave speed approximations. We show that this approach matches well spread rates observed in numerical simulations. The results from this paper can provide a useful tool in the assessment of instream flow needs, estimating the flow speed necessary for preserving riverine populations.     

Flutter in collapsible tubes: a theoretical model of wheezes

A mathematical analysis of flow through a flexible channel is examined as a model of flow-induced flutter oscillations that pertain to the production of wheezing breath sounds. The model provides predictions for the critical fluid speed that will initiate flutter waves of the wall, as well as their frequency and wavelength. The mathematical results are separated into linear theory (small oscillations) and nonlinear theory (larger oscillations). Linear theory determines the onset of the flutter, whereas nonlinear theory determines the relationships between the fluid speed and both the wave amplitudes and frequencies. The linear theory predictions correlate well with data taken at the onset of flutter and flow limitation during experiments of airflow in thick-walled collapsible tubes. The nonlinear theory predictions correlate well with data taken as these flows are forced to higher velocities while keeping the flow rate constant. Particular ranges of the parameters are selected to investigate and discuss the applications to airway flows. According to this theory, the mechanism of generation of wheezes is based in the interactions of fluid forces and friction and wall elastic-restoring forces and damping. In particular, a phase delay between the fluid pressure and wall motion is necessary. The wave speed theory of flow limitation is discussed with respect to the specific data and the flutter model.