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.     

Non-Newtonian Behavior of Blood in Oscillatory Flow

Sinusoidal oscillatory flow of blood and of aqueous glycerol solutions was produced in rigid cylindrical tubes. For aqueous glycerol, the amplitude of the measured pressure gradient wave form conformed closely to that predicted by Womersley's theory of oscillatory flow, up to Reynolds numbers approaching 2000. Blood differed significantly from aqueous glycerol solutions of comparable viscosity, especially at low frequencies and high hematocrits. As frequency increased, the hydraulic impedance of blood decreased to a minimum at a frequency of about 1-2 CPS, increasing monotonically at higher frequencies. The dynamic apparent viscosity of blood, calculated from Womersley's theory, decreased with increasing flow amplitude. The reactive component of the hydraulic impedance increased with frequency as predicted by theory; the resistive component decreased with increasing frequency, differing from the resistance of a Newtonian fluid which increased with frequency.     

Systems engineering analysis of aortic root blood pressure

This paper describes the aortic blood pressure as a function of aortic blood flow and the parameters of the blood and circulatory system. The method of performance involves the analogue of a multi-branched electrical to hydraulic transmission line applying graphical convolution to the blood flow-transform impedance relationship resulting in a theoretical pressure curve for the infinite aorta. The difference between the single pressure pulse and the computed adjusted infinite aorta pressure curve is described as the reflected wave. This reflected wave is then shown to be of reasonable configuration in time and velocity. The blood pressure is thus finally described completely by the physical parameters of the blood and the circulatory system and the blood flow.     

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.     

On the transmission of external pressure fluctuations to the cerebrospinal fluid

A theory has been formulated to explain the manner in which external pressure fluctuations are transmitted to the cerebrospinal fluid (CSF). The theory is based upon a three-compartment model which consists of the cerebral ventricles, the basal cisterns and spinal subarachnoid space, and the cortical subarachnoid space. The external pressure disturbance is represented by a Fourier series summed over the frequency ω. The mathematical analysis leads to a time constant τ which depends upon the compliances of the spinal region and sources of external pressure fluctuations, the rate of CSF absorption and the rate of fluid transfer between compartments. For arterial pulsations where ωτ ? 1, the theory is in accord with the experimental observations that (i) the arterial and CSF pulse waves are nearly identical in shape, and (ii) the amplitude of the CSF pulse wave increases with intracranial pressure. Moreover, it predicts that the amplitude of the wave will be larger in the spinal region than in the ventricles. The theory also accounts for the observation of one per minute pulse waves observed in hydrocephalic patients with decreased absorption rates.     

Mathematical model analysis of pressure pulse propagation of the aortic wall

A mathematical blood vessel mode is established of a semi-infinite, fluid filled, cylindrical, elastic tube where a pressure wave is propagated in the fluid. This model has assigned to it physical values of the aorta and is subject to a pressure pulse at the origin of the same configuration and magnitude as that observed in the human. The resulting mathematical solution demonstrates wall displacement in agreement with observed phenomena of (a) a high frequency component analogous to cardiovascular sounds; (b) a low frequency component resembling the pressure pulse analogous to the pulsatile movement of the vessel wall.     

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.     

Neoplastic transformation in vitro after exposure to low doses of mammographic-energy X rays: quantitative and mechanistic aspects

The induction of neoplastic transformation in vitro after exposure of HeLa x skin fibroblast hybrid cells to low doses of mammography-energy (28 kVp) X rays has been studied. The data indicate no evidence of an increase in transformation frequency over the range 0.05 to 22 cGy, and doses in the range 0.05 to 1.1 cGy may result in suppression of transformation frequencies to levels below that seen spontaneously. This finding is not consistent with a linear, no-threshold dose- response curve. The dose range at which possible suppression is evident includes doses typically experienced in mammographic examination of the human breast. Experiments are described that attempt to elucidate any possible role of bystander effects in modulating this low-dose radiation response. Not unexpectedly, inhibition of gap junction intercellular communication (GJIC) with the inhibitor lindane did not result in any significant alteration of transformation frequencies seen at doses of 0.27 or 5.4 cGy in these subconfluent cultures. Furthermore, no evidence of a bystander effect associated with factors secreted into the extracellular medium was seen in medium transfer experiments. Thus, in this system and under the experimental conditions used, bystander effects would not appear to be playing a major role in modulating the shape of the dose-response curve.     

Linear propagation of pulsatile waves in viscoelastic tubes

An experimental and theoretical analysis is made of pulsatile wave propagation in deformable latex tubes as a model of the propagation of pressure pulses in arteries. A quasi one-dimensional linear model is used in which, in particular, attention is paid to the viscous phenomena in fluid and tube wall. The agreement between experimental and theoretical results is satisfactory. It appeared that the viscoelastic behaviour of the tube wall dominates the damping of the pressure pulse. Several linear models are used to describe the wall behaviour. No significant differences between the results of these models were found.     

Effects of muscle contraction on pulsatile pressure-flow relations in femoral bed

Femoral arterial pressure-flow relations and vascular impedance were studied during isometric contraction of the gastrocnemius-plantaris muscle group in anesthetized dogs. Contractions were synchronized with the electrocardiogram to occur in the first or second half of the cardiac cycle and included twitches as well as low-, intermediate-, and high-frequency tetanuses. The effects of fatigue and recovery were also documented. Marked changes in pressure and flow waveforms and corresponding femoral arterial input impedance spectra were seen for all contraction modes. Impedance moduli and estimated characteristic impedance were elevated regardless of contraction mode and were associated with fluctuations in impedance phase. All tetanuses placed in the first half of the cardiac cycle produced a striking and consistent reversal of impedance phase for the fundamental harmonic from negative to positive values which decreased with progressive fatigue. During recovery, impedance spectra were unchanged from control spectra. We have demonstrated marked alterations in pressure and flow waveforms and impedance spectral patterns during isometric contraction in the canine hindlimb. These changes may be explained by 1) markedly increased wave reflection as a result of muscle contraction and/or 2) the generation of a retrograde pulse by contracting muscle that fuses with the antegrade pulse of cardiac origin.     

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

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

Electric field-induced orientation of L- and DL-phosphatidylcholine bilayers

Membrane orientation induced by an alternating electric field has been examined for the L-enantiomer and racemic dipalmitoylphosphatidylcholine (DPPC) bilayers. The orientation effect was measured by bending curvature of hairpin-like deformation of the multilamellar cylindrical tubes with varying field-strength, frequency and tube size. It has been observed that both L- and DL-DPPC tubes are similar in the profiles of field-strength dependence and frequency dependence on the curvature deformation, but different in the deformed curvatures. DL-DPPC tubes deform largely as compared with L-DPPC tubes. The square of the deformed curvature of DL-DPPC tubes is larger than that of L-DPPC by about 37% on average. The result indicates that the racemic membrane is responsive to the electric field as compared with the L-enantiomer membrane. This suggests that a hybrid arrangement of head groups of the racemic lipid leads an effective response of the membrane due to the head group orientation.     

Electric field-induced orientation of l- and dl-phosphatidylcholine bilayers

Membrane orientation induced by an alternating electric field has been examined for the l-enantiomer and racemic dipalmitoylphosphatidylcholine (DPPC) bilayers. The orientation effect was measured by bending curvature of hairpin-like deformation of the multilamellar cylindrical tubes with varying field-strength, frequency and tube size. It has been observed that both l- and dl-DPPC tubes are similar in the profiles of field-strength dependence and frequency dependence on the curvature deformation, but different in the deformed curvatures. dl-DPPC tubes deform largely as compared with l-DPPC tubes. The square of the deformed curvature of dl-DPPC tubes is larger than that of l-DPPC by about 37% on average. The result indicates that the racemic membrane is responsive to the electric field as compared with the l-enantiomer membrane. This suggests that a hybrid arrangement of head groups of the racemic lipid leads an effective response of the membrane due to the head group orientation.     

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.     

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.     

Effects of diabetes and gender on mechanical properties of the arterial system in rats: aortic impedance analysis

We determined the effects of diabetes and gender on the physical properties of the vasculature in streptozotocin (STZ)-treated rats based on the aortic input impedance analysis. Rats given STZ 65 mg/kg i.v. were compared with untreated age-matched controls. Pulsatile aortic pressure and flow signals were measured and were then subjected to Fourier transformation for the analysis of aortic input impedance. Wave transit time was determined using the impulse response function of the filtered aortic input impedance spectra. Male but not female diabetic rats exhibited an increase in cardiac output in the absence of any significant changes in arterial blood pressure, resulting in a decline in total peripheral resistance. However, in each gender group, diabetes contributed to an increase in wave reflection factor, from 0.47 +/- 0.04 to 0.84 +/- 0.03 in males and from 0.46 +/- 0.03 to 0.81 +/- 0.03 in females. Diabetic rats had reduced wave transit time, at 18.82 +/- 0.60 vs 21.34 +/- 0.51 msec in males and at 19.63 +/- 0.37 vs 22.74 +/- 0.57 msec in females. Changes in wave transit time and reflection factor indicate that diabetes can modify the timing and magnitude of the wave reflection in the rat arterial system. Meanwhile, diabetes produced a fall in aortic characteristic impedance from 0.023 +/- 0.002 to 0.009 +/- 0.001 mmHg/min/kg/ml in males and from 0.028 +/- 0.002 to 0.014 +/- 0.001 mmHg/min/kg/ml in females. With unaltered aortic pressure, both the diminished aortic characteristic impedance and wave transit time suggest that the muscle inactivation in diabetes may occur in aortas and large arteries and may cause a detriment to the aortic distensibility in rats with either sex. We conclude that only rats with male gender diabetes produce a detriment to the physical properties of the resistance arterioles. In spite of male or female gender, diabetes decreases the aortic distensibility and impairs the wave reflection phenomenon in the rat arterial system.     

The Input Impedance of an Assembly of Randomly Branching Elastic Tubes

Computations are presented of the input impedance of assemblies of randomly bifurcating elastic tubes, as a generalized model of the arterial system. Account is taken of the viscosity of the fluid, the viscoelastic properties of the walls, the variation of elasticity in the different orders of branches, and the variation in cross-sectional area at the bifurcations. The results show that the distributed and scattered nature of the terminations of such an assembly greatly reduces the influence of reflections upon the behavior of the input impedance. The variation of impedance with frequency is very similar in form to that found in animal experiments for the input impedance of the aorta. The architecture of the arterial system may thus be considered to play an important part in determining the favorably low impedance presented to the heart by the aorta.     

Fluid pressure relaxation depends upon osteonal microstructure: modeling an oscillatory bending experiment.

When bone is mechanically loaded, bone fluid flow induces shear stresses on bone cells that have been proposed to be involved in bone's mechanosensory system. To investigate bone fluid flow and strain-generated potentials, several theoretical models have been proposed to mimic oscillatory four-point bending experiments performed on thin bone specimens. While these previous models assume that the bone fluid relaxes across the specimen thickness, we hypothesize that the bone fluid relaxes primarily through the vascular porosity (osteonal canals) instead and develop a new poroelastic model that integrates the microstructural details of the lacunar-canalicular porosity, osteonal canals, and the osteonal cement lines. Local fluid pressure profiles are obtained from the model, and we find two different fluid relaxation behaviors in the bone specimen, depending on its microstructure: one associated with the connected osteonal canal system, through which bone fluid relaxes to the nearby osteonal canals; and one associated with the thickness of a homogeneous porous bone specimen (approximately 1 mm in our model), through which bone fluid relaxes between the external surfaces of the bone specimen at relatively lower loading frequencies. Our results suggest that in osteonal bone specimens the fluid pressure response to cyclic loading is not sensitive to the permeability of the osteonal cement lines, while it is sensitive to the applied loading frequency. Our results also reveal that the fluid pressure gradients near the osteonal canals (and thus the fluid shear stresses acting on the nearby osteocytes) are significantly amplified at higher loading frequencies.     

Pulsatile blood flow in the entire coronary arterial tree: theory and experiment

The pulsatility of coronary circulation can be accurately simulated on the basis of the measured branching pattern, vascular geometry, and material properties of the coronary vasculature. A Womersley-type mathematical model is developed to analyze pulsatile blood flow in diastole in the absence of vessel tone in the entire coronary arterial tree on the basis of previously measured morphometric data. The model incorporates a constitutive equation of pressure and cross-section area relation based on our previous experimental data. The formulation enables the prediction of the impedance, the pressure distribution, and the pulsatile flow distribution throughout the entire coronary arterial tree. The model is validated by experimental measurements in six diastolic arrested, vasodilated porcine hearts. The agreement between theory and experiment is excellent. Furthermore, the present pulse wave results at low frequency agree very well with previously published steady-state model. Finally, the phase angle of flow is seen to decrease along the trunk of the major coronary artery and primary branches toward the capillary vessels. This study represents the first, most extensive validated analysis of Womersley-type pulse wave transmission in the entire coronary arterial tree down to the first segment of capillaries. The present model will serve to quantitatively test various hypotheses in the coronary circulation under pulsatile flow conditions.     

Evaluation by fluid/structure-interaction spinal-cord simulation of the effects of subarachnoid-space stenosis on an adjacent syrinx

A finite-element numerical model was constructed of the spinal cord, pia mater, filum terminale, cerebrospinal fluid in the spinal subarachnoid space (SSS), and dura mater. The cord was hollowed out by a thoracic syrinx of length 140 mm, and the SSS included a stenosis of length 30 mm opposite this syrinx. The stenosis severity was varied from 0% to 90% by area. Pressure pulse excitation was applied to the model either at the cranial end of the SSS, simulating the effect of cranial arterial pulsation, or externally to the abdominal dura mater, simulating the effect of cough. A very short pulse was used to examine wave propagation; a pulse emulating cardiac systole was used to examine the effects of fluid displacement. Additionally, repetitive sinusoidal excitation was applied cranially. Bulk fluid flow past the stenosis gave rise to prominent longitudinal pressure dissociation ("suck") in the SSS adjacent to the syrinx. However, this did not proportionally increase the longitudinal motion of fluid in the syrinx. The inertia of the fluid in the SSS, together with the compliance of this space, gave a resonance capable of being excited constructively or destructively by cardiac or coughing impulses. The main effect of mild stenosis was to lower the frequency of this resonance; severe stenosis damped out to-and-fro motions after the end of the applied excitation. Syrinx fluid motion indicated the fluid momentum and thus the pressure developed when the fluid was stopped by the end of the syrinx; however, the tearing stress in the local cord material depended also on the instantaneous local SSS pressure and was therefore not well predicted by syrinx fluid motion. Stenosis was also shown to give rise to a one-way valve effect causing raised SSS pressure caudally and slight average cord displacement cranially. The investigation showed that previous qualitative predictions of the effects of suck neglected factors that reduced the extent of the resulting syrinx fluid motion and of the cord tearing stress, which ultimately determines whether the syrinx lengthens.