Pressure and flow in the systemic arterial system

The dynamic characteristics of the proximal arterial system are studied by solving the nonlinear momentum and mass conservation equations for pressure and flow. The equations are solved for a model systemic arterial system that includes the aorta, common iliacs, and the internal and external iliac arteries. The model includes geometric and elastic taper of the aorta, nonlinearly elastic arteries, side flows, and a complex distal impedance. The model pressure wave shape, inlet and outlet impedance, wave travel, and apparent wave velocity compare favorably with the values measured on humans. Calculations indicate that: (i) reflections are the major factor determining the shape and distal amplification of the pressure wave in the arterial tree; (ii) although important in attenuating the proximal transmission of reflecting waves, geometric taper is not the major cause of the distal pressure wave amplification; (iii) the dicrotic wave is a result of peripheral reflection and is not due to the sudden change in flow at the end of systole; (iv) the elastic taper and nonlinearity of the wall elasticity are of minor significance in determining the flow and pressure profiles; and (v) in spite of numerous nonlinearities, the system behaves in a somewhat linear fashion for the lower frequency components.     

Determination of viscoelastic moduli of arteries

A model for the arterial wall, motivated by wave propagation findings, is developed. The wall is taken to be a viscoelastic, orthotropic, prestressed shell which is materially characterized at any prestress level by ten incremental moduli. By using the momentum equations and the wave propagation characteristics for three axisymmetric modes, the ten moduli are found in terms of the three wave speeds, the three attenuation coefficients and the prestresses.     

Dynamic measurement of the viscoelastic properties of skin

A wave propagation technique was used to measure the dynamic viscoelastic properties of excised skin when subjected to a low incremental strain. The propagation velocity, attenuation, and storage and loss moduli were determined from measured characteristics of a pulse propagating along a strip of skin. Experiments were conducted with the skin subjected to static stresses of 1500 Pa and 20,000 Pa. At low static stresses the skin response was viscoelastic with a loss tangent of approximately 0.6. In the frequency range of 0-1000 Hz, the wave velocity was relatively constant while the attenuation increased roughly linearly with frequency. However, results depended on the static stress. At the higher stress level the velocity was greater and the attenuation less than at the lower stress. At low stresses both the storage and loss moduli were relatively constant over the frequency range tested. The strong viscoelastic behavior of the tissue at higher frequencies is not predicted from models of the tissue determined from quasi-static test methods. In selecting a model to describe the behavior of skin, the test methods used for establishing the model must be consistent with its intended application.     

Pulmonary vascular impedance and wave reflections in the hypoxic calf.

The alterations in pulsatile hemodynamics that occur during hypoxic pulmonary vasoconstriction have not been well characterized. Changes in oscillatory hemodynamics, however, may affect right ventricular-pulmonary vascular coupling and the dissipation of energy within the lung vasculature. To better define hypoxic pulsatile hemodynamics, we measured main pulmonary artery proximal and distal micromanometric pressures and ultrasonic flow in four open-chest calves during progressive hypoxia. Main pulmonary artery impedance and pressure transmission spectra were calculated using spectral analysis methods. Measured pressure and flow signals were separated in the time domain into forward and backward components. Hypoxia increased pulmonary blood pressure and resistance and produced multiple modifications in the impedance and pressure transmission spectra that indicated increased wave reflections and elasticity. The impedance and apparent phase velocity first-harmonic values were increased in amplitude, and the pressure transmission modulus plot showed an increased peak value. In addition, the impedance modulus plot demonstrated a rightward shift and increased oscillation in the mid- to high-frequency range. The time domain analysis also confirmed increased wave reflections and elasticity. Hypoxia produced large backward-traveling (reflected) pressure and flow waves. The initial portions of these waves arrived at the heart during systole, producing characteristic changes in the measured pressure and flow waveforms. With prolonged hypoxia, main pulmonary artery pulse wave velocity increased by 30%. Thus, hypoxia is associated with complex alterations in pulmonary artery elasticity and wave reflections that act to increase the oscillatory afterload of the right ventricle.     

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.     

Evaluation of a three point pressure method for the determination of arterial transmission characteristics

The mathematical expressions required to analyse wave transmission characteristics (frequency dependent phase velocity, attenuation, and reflection) in arteries, under in vivo conditions, are summarized. In addition, a three point pressure method, which theoretically permits experimental determination of the propagation constant (phase velocity and attenuation), was tested under in vitro conditions. It is found to be a potentially powerful tool for in vivo studies if used with the appropriate constraints.     

Smooth muscle relaxation and local hydraulic impedance properties of the aorta.

Smooth muscle relaxation is expected to yield beneficial effects on hydraulic impedance properties of large vessels. We investigated the effects of intravenous diltiazem infusion on aortic wall stiffness and local hydraulic impedance properties. In seven anesthetized, closed-chest dogs, instantaneous cross-sectional area and pressure of the descending thoracic aorta were measured using transesophageal echocardiography combined with acoustic quantification and a micromanometer, respectively. Data were acquired during a vena caval balloon inflation, both at the control condition and with diltiazem infusion. At the operating point, diltiazem reduced blood pressure in all dogs but did not alter aortic dimensions or wall stiffness. Over the observed pressure range, aortic area-pressure relationships were linear. Whereas diltiazem affected the slope of this relationship variably (no change in 3 dogs, increase in 1 dog, decrease in 3 dogs), the zero-pressure area intercept was significantly increased in every case such that higher area was observed at any given pressure. When comparisons were made at a common level of wall stress, wall stiffness was either increased or unchanged during diltiazem infusion. In contrast, diltiazem decreased wall stiffness in every case when comparisons were made at a common level of aortic midwall radius. Aortic characteristic impedance and pulse wave velocity, components of left ventricular hydraulic load that are determined by aortic elastic and geometric properties, were affected variably. A comparison of wall stiffness at matched wall stress appears inappropriate for assessing changes in smooth muscle tone. Because of the competing effects of changes in vessel diameter and wall stiffness, smooth muscle relaxation is not necessarily accompanied by the expected beneficial changes in local aortic hydraulic impedance. These results can be reconciled by recognizing that components other than vascular smooth muscle (e.g., elastin, collagen) contribute to aortic wall stiffness.     

Examination of elastic non-uniformity in the arterial system using a hydraulic model

Pressure wave propagation has been examined in a model artery with spatially varying compliance. Although results were affected by viscous losses, appropriate allowance for such losses produced agreement between experimental findings and predictions of linear wave transmission theory. Particularly, the ability of non-uniformity of the tube wall to generate amplification of the pressure wave was confirmed. However, extrapolation to the physiological situation suggests that reflections from discrete sites in peripheral beds have a greater effect on pressure wave propagation than does elastic non-uniformity of major vessels. A theoretical analysis has demonstrated that the effects of elastic non-uniformity can be interpreted as the integrated effects of infinitesimal reflections from each progressive increment in wall stiffness.     

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.     

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.     

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.     

Near-wall micro-PIV reveals a hydrodynamically relevant endothelial surface layer in venules in vivo

High-resolution near-wall fluorescent microparticle image velocimetry (micro-PIV) was used in mouse cremaster muscle venules in vivo to measure velocity profiles in the red cell-depleted plasma layer near the endothelial lining. micro-PIV data of the instantaneous translational speeds and radial positions of fluorescently labeled microspheres (0.47 microm) in an optical section through the midsagittal plane of each vessel were used to determine fluid particle translational speeds. Regression of a linear velocity distribution based on near-wall fluid-particle speeds consistently revealed a negative intercept when extrapolated to the vessel wall. Based on a detailed three-dimensional analysis of the local fluid dynamics, we estimate a mean effective thickness of approximately 0.33 micro m for an impermeable endothelial surface layer or approximately 0.44 micro m assuming the lowest hydraulic resistivity of the layer that is consistent with the observed particle motions. The extent of plasma flow retardation through the layer required to be consistent with our micro-PIV data results in near complete attenuation of fluid shear stress on the endothelial-cell surface. These findings confirm the presence of a hydrodynamically effective endothelial surface layer, and emphasize the need to revise previous concepts of leukocyte adhesion, stress transmission to vascular endothelium, permeability, and mechanotransduction mechanisms.     

A mathematical model of umbilical venous pulsation

Pulsations in the fetal heart propagate through the precordial vein and the ductus venosus but are normally not transmitted into the umbilical vein. Pulsations in the umbilical vein do occur, however, in early pregnancy and in pathological conditions. Such transmission into the umbilical vein is poorly understood. In this paper we hypothesize that the mechanical properties and the dimensions of the vessels do influence the umbilical venous pulsations, in addition to the magnitude of the pressure and flow waves generated in the fetal atria. To support this hypothesis we established a mathematical model of the umbilical vein/ductus venosus bifurcation. The umbilical vein was modeled as a compliant reservoir and the umbilical vein pressure was assumed to be equal to the stagnation pressure at the ductus venosus inlet. We calculated the index of pulsation of the umbilical vein pressure ((max-min)/mean), the reflection and transmission factors at the ductus venosus inlet, numerically and with estimates. Typical dimensions in the physiological range for the human fetus were used, while stiffness parameters were taken from fetal sheep. We found that wave transmission and reflection in the umbilical vein ductus venosus bifurcation depend on the impedance ratio between the umbilical vein and the ductus venosus, as well as the ratio of the mean velocity and the pulse wave velocity in the ductus venosus. Accordingly, the pulsations initiated by the fetal heart are transmitted upstream and may arrive in the umbilical vein with amplitudes depending on the impedance ratio and the ratio between the mean velocity and the pulse wave velocity in the ductus venosus.     

Role of tapering in aortic wave reflection: hydraulic and mathematical model study

Pressure and flow have been measured simultaneously at six locations along the aorta of an anatomically correct 1:1 scale hydraulic elastic tube model of the arterial tree. Our results suggest a discrete reflection point at the level of the renal arteries based on (i) the quarter-wavelength formula and (ii) the comparison of foot-to-foot (c(ff)) and apparent phase velocity (c(app)). However, separation of the pressure wave into an incident and reflected wave at all six locations indicates continuous reflection: a reflected wave is generated at each location as the forward wave passes by. We did a further analysis using a mathematical transmission line model with a simple tapering geometry (length 50 cm, 31 and 11 mm proximal and distal diameter, respectively) for a low (0.32 ml/mmHg), normal (1.6 ml mmHg) and high (8 ml/mmHg) value of total arterial compliance. Using the quarter-wavelength formula, a discrete reflection point is found at x = 33 cm, the level of the renal arteries, independent of the value of total compliance. However, local analysis comparing c(ff) and c(app) does not reveal a marked reflection site, and the analysis of incident and reflected waves merely suggests a continuous reflection. We therefore conclude that the measured in vivo aortic wave reflection indices are the result of at least two interacting phenomena: a continuous wave reflection due to tapering, and local reflections arising from branches at the level of the diaphragm. The continuous reflection is hidden in the input impedance pattern. Using the quarter-wavelength formula or the classical wave separation theory, it appears as a reflection coming from a single discrete site, confusingly also located at the level of the diaphragm. Therefore, the quarter-wavelength formula and the linear wave separation theory should be used with caution to identify wave reflection zones in the presence of tapering, i.e., in most mammalian arteries.     

The mechanism of propagation of variation potentials in wheat leaves

Here we examined the mechanism of propagation of variation potential (VP) induced by burning in wheat leaves. Participation of hydraulic and chemical mechanisms in VP transmission was analyzed by optical coherent tomography and a radioactive tracer method, respectively. The speed of the hydraulic signal considerably exceeded the VP velocity. Investigation of a chemical substance spreading from the zone of local wounding was based on experimental data for radioactive marker transmission derived with a one-dimensional diffusion equation. The speed of the marker transmission was in accordance with VP velocity. The elimination of the potential transmission of a chemical signal by a timed severing of the leaf between the burn site and the recorded site blocked VP propagation. We suggest that a VP is formed by the transmission of a wound substance, the velocity of which is likely increased by hydraulic wave propagation.     

Heterogeneous mechanics of the mouse pulmonary arterial network

Individualized modeling and simulation of blood flow mechanics find applications in both animal research and patient care. Individual animal or patient models for blood vessel mechanics are based on combining measured vascular geometry with a fluid structure model coupling formulations describing dynamics of the fluid and mechanics of the wall. For example, one-dimensional fluid flow modeling requires a constitutive law relating vessel cross-sectional deformation to pressure in the lumen. To investigate means of identifying appropriate constitutive relationships, an automated segmentation algorithm was applied to micro-computerized tomography images from a mouse lung obtained at four different static pressures to identify the static pressure–radius relationship for four generations of vessels in the pulmonary arterial network. A shape-fitting function was parameterized for each vessel in the network to characterize the nonlinear and heterogeneous nature of vessel distensibility in the pulmonary arteries. These data on morphometric and mechanical properties were used to simulate pressure and flow velocity propagation in the network using one-dimensional representations of fluid and vessel wall mechanics. Moreover, wave intensity analysis was used to study effects of wall mechanics on generation and propagation of pressure wave reflections. Simulations were conducted to investigate the role of linear versus nonlinear formulations of wall elasticity and homogeneous versus heterogeneous treatments of vessel wall properties. Accounting for heterogeneity, by parameterizing the pressure/distention equation of state individually for each vessel segment, was found to have little effect on the predicted pressure profiles and wave propagation compared to a homogeneous parameterization based on average behavior. However, substantially different results were obtained using a linear elastic thin-shell model than were obtained using a nonlinear model that has a more physiologically realistic pressure versus radius relationship.     

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

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

Wave Transmission through an Assembly of Randomly Branching Elastic Tubes

Calculations are presented of the transmission of oscillations through an assembly of randomly branching elastic tubes, as a model of not only the major arteries, but also a peripheral vascular bed. It appears that the viscosity of the arterial wall must be the major source of attenuation in the larger arteries, while the viscosity of the blood plays a significant role only in the smaller vessels. In all situations, variations of cross-sectional area have a considerable effect on wave transmission, causing a general decrease in amplitude and an accentuation of reflection from the terminations. The effects of variation in cross-sectional area are sufficiently great to indicate that they should be included in future models of the arterial system. Finally, it is argued that because of the presence of random branching and elastic nonuniformity, the determination of the reflection coefficient for a system such as the arterial tree may be quite misleading.     

Right ventricular adaptation to pulmonary hypertension: an interspecies comparison

Right ventricular (RV) adaptation is an important prognostic factor in acute and chronic pulmonary hypertension. Pulmonary vascular basal tone and hypoxic reactivity are known to vary widely between species. We investigated how RV adaptation to acute pulmonary hypertension is preserved in species with low, intermediate, and high pulmonary vascular resistance and reactivity. Acute pulmonary hypertension was induced by hypoxia, distal embolism, and proximal constriction in anesthetized dogs (n = 10), goats (n = 8), and pigs (n = 8). Pulmonary vessels were assessed by flow-pressure curves and by impedance to quantify distal resistance, proximal elastance, and wave reflections. RV function was assessed by pressure-volume curves to quantify afterload, contractility, and ventricular-arterial coupling efficiency. First, hypoxia was associated with a progressive increase of resistance, elastance, and wave reflection from dogs to goats and from goats to pigs. RV contractility increased proportionally to RV afterload, and optimal coupling was preserved in all species. Second, embolism increased resistance and wave reflection but not elastance. The increase in RV contractility matched the increase in RV afterload and optimal coupling was preserved. Finally, proximal pulmonary artery constriction increased resistance, increased and accelerated wave reflection, and markedly increased elastance. RV contractility increased markedly and coupling showed a nonsignificant trend to decrease. We conclude that optimal or near-optimal ventricular-arterial coupling is maintained in acute pulmonary hypertension, whether in absence or presence of chronic species-induced pulmonary hypertension.