Wave intensity in the ascending aorta: effects of arterial occlusion

We examine the effects of arterial occlusion on the pressure, velocity and the reflected waves in the ascending aorta using wave intensity analysis. In 11 anaesthetised, open-chested dogs, snares were used to produce total arterial occlusion at 4 sites: the upper descending aorta at the level of the aortic valve (thoracic); the lower thoracic aorta at the level of the diaphragm (diaphragm); the abdominal aorta between the renal arteries (abdominal) and the left iliac artery, 2 cm downstream from the aorta iliac bifurcation (iliac). Pressure and flow in the ascending aorta were measured, and data were collected before and during the occlusion. During thoracic and diaphragm occlusions a significant increase in mean aortic pressure (46% and 23%) and in wave speed (25% and 10%) was observed, while mean flow rate decreased significantly (23% and 17%). Also, the reflected compression wave arrived significantly earlier (45% and 15%) and its peak intensity was significantly greater (257% and 125%), all compared with control. Aortic occlusion distal to the renal arteries, however, caused an indiscernible change in the pressure and velocity waveforms, and in the intensities and timing of the waves in the forward and backward directions. The measured pressure and velocity waveforms are the result of the interaction between the heart and the arterial system. The separated pressure, velocity and wave intensity are required to provide information about arterial hemodynamic such as the timing and magnitude of the forward and backward waves. The net wave intensity is simpler to calculate but provides information only about the predominant direction of the waves and can be misleading when forward and backward waves of comparable magnitudes are present simultaneously.     

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.     

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.     

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.     

Age-related peculiarities of dynamic structure of wave of the human arterial blood pressure

A decrease of elasticity of walls of aorta and large arteries in the process of human natural aging leads to a significant increment of pulse arterial pressure (AP) due to a reflected wave and accordingly to a rise of the load on myocardium and arterial walls. Therefore, an enhancement of the arterial wall stiffness is considered as a possible mechanism and at the same time as a prognostic factor of risk of development of diseases of the cardiovascular system (CVS). For detection and quantitative estimation of this factor, we applied method of arterial piezopulsometry that allows revealing characteristic age-related peculiarities of the dynamic structure of AP pulse waves, which were used at development of the way and algorithm of express-diagnostics of the functional state of the human CVS. The perspective use of this method for detection of signs of the age-related CVS readjustment as a factor of the organism aging is discussed.     

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.     

Aortic reflection coefficients and their association with global indexes of wave reflection in healthy controls and patients with Marfan's syndrome

Early return of reflected pressure waves increases the load on central arteries and may increase the risk of aortic rupture in patients with Marfan's syndrome (MFS). To assess whether wave reflection is elevated in MFS, we used ultrasound and MRI to measure central pressure and flow waveforms in 26 patients (13-54 yr of age) and 26 age- and gender-matched controls. Aortic systolic and diastolic cross-sectional areas were measured at the ascending and descending aorta (AA and DA), diaphragm (DIA), and lower abdominal aorta (AB). From these measurements, local characteristic impedance (Z(0-xx)) and local reflection coefficients (Gamma(xx-yy)) were calculated. Calculated global wave reflection indexes were the augmentation index (AIx) and the ratio of backward to forward pressure wave (P(b)/P(f)). The aorta was wider in MFS patients at AA (P < 0.01) and DA (P < 0.01). Aortic pulse wave velocity was 42 cm/s higher in MFS patients (P < 0.05). Z(0-xx) was not different between groups, except at DA, where it was lower in MFS patients. In controls, Gamma(AA-DA) was 0.31 +/- 0.08, Gamma(DA-DIA) was 0.00 +/- 0.11, and Gamma(DIA-AB) was 0.31 +/- 0.16. Mean values of Gamma(xx-yy) were not different between MFS patients and controls. In controls, aging diminished Gamma(AA-DA) but increased Gamma(DIA-AB). Clear age-related patterns were absent in MFS patients. AIx or P(b)/P(f) was not higher in MFS patients than in controls. There were indications for enhanced wave reflection in young MFS patients. Our data demonstrated that the major determinants of AIx were pulse wave velocity and the effective length of the arterial system and, to a lesser degree, HR and P(b)/P(f).     

Modelling the circle of Willis to assess the effects of anatomical variations and occlusions on cerebral flows

Blood flow in the circle of Willis (CoW) is modelled using the 1-D equations of pressure and flow wave propagation in compliant vessels. The model starts at the left ventricle and includes the largest arteries that supply the CoW. Based on published physiological data, it is able to capture the main features of pulse wave propagation along the aorta, at the brachiocephalic bifurcation and throughout the cerebral arteries. The collateral ability of the complete CoW and its most frequent anatomical variations is studied in normal conditions and after occlusion of a carotid or vertebral artery (VA). Our results suggest that the system does not require collateral pathways through the communicating arteries to adequately perfuse the brain of normal subjects. The communicating arteries become important in cases of missing or occluded vessels, the anterior communicating artery (ACoA) being a more critical collateral pathway than the posterior communicating arteries (PCoAs) if an internal carotid artery (ICA) is occluded. Occlusions of the VAs proved to be far less critical than occlusions of the ICAs. The worst scenario in terms of reduction in the mean cerebral outflows is a CoW without the first segment of an anterior cerebral artery combined with an occlusion of the contralateral ICA. Furthermore, in patients without any severe occlusion of a carotid or VA, the direction of flow measured at the communicating arteries corresponds to the side of the CoW with an absent or occluded artery. Finally, we study the effect of partial occlusions of the communicating arteries on the cerebral flows, which again confirms that the ACoA is a more important collateral pathway than the PCoAs if an ICA is occluded.     

Ascending pathways mediating somatoautonomic reflexes in exercising dogs

The ascending spinal pathways mediating somatocardiovascular reflexes during exercise were studied in unanesthetized dogs by placing lesions in the lumbar spinal cord. After training to run on a treadmill with hindlimbs only, 20 dogs were anesthetized and instrumented using sterile surgical techniques. To chronically record heart rate and arterial blood pressure, the aorta was cannulated via the omocervical artery. To test the intactness of descending spinal sympathetic pathways, reflex pressor responses to baroreceptor hypotension were produced by bilateral carotid arterial occlusion using pneumatic vessel occluders placed around the common carotid arteries. To generate transient ischemic exercise (120 s), a pneumatic occluder was placed around the left iliac artery. Eight to 10 days after instrumentation, blood pressure and heart rate were monitored at rest and during hindlimb running with and without simultaneous iliac arterial occlusion. The modest pressor response and tachycardia elicited by hindlimb exercise were markedly augmented by simultaneous hindlimb ischemia (i.e., iliac arterial occlusion). Lesion placement in the dorsolateral sulcus area and the dorsolateral funiculus at L2 significantly reduced the blood pressure and heart rate responses to simultaneous exercise occlusion. The cardiovascular responses to nonischemic exercise and bilateral carotid arterial occlusion were not altered by such spinal sections. It is concluded that in the dog the ascending spinal pathways mediating cardiovascular responses to ischemic exercise are located in the lateral funiculus, including the dorsolateral sulcus area and dorsolateral funiculus.     

Modelling pulse wave propagation in the rabbit systemic circulation to assess the effects of altered nitric oxide synthesis

Pulse wave propagation in the mature rabbit systemic circulation was simulated using the one-dimensional equations of blood flow in compliant vessels. A corrosion cast of the rabbit circulation was manufactured to obtain arterial lengths and diameters. Pulse wave speeds and inflow and outflow boundary conditions were derived from in vivo data. Numerical results captured the main features of in vivo pressure and velocity pulse waveforms in the aorta, brachiocephalic artery and central ear artery. This model was used to elucidate haemodynamic mechanisms underlying changes in peripheral pulse waveforms observed in vivo after administering drugs that alter nitric oxide synthesis in the endothelial cells lining blood vessels. According to our model, these changes can be explained by single or combined alterations of blood viscosity, peripheral resistance and compliance, and the elasticity of conduit arteries.     

Influence of the distensibility of large arteries on the longitudinal impedance: application for the development of non-invasive techniques to the diagnosis of arterial diseases

Background

This study shows that the arterial longitudinal impedance constitutes a hemodynamic parameter of interest for performance characterization of large arteries in normal condition as well as in pathological situations. For this purpose, we solved the Navier?CStokes equations for an incompressible flow using the finite element analysis method and the Arbitrary Lagrangian Eulerian (ALE) formulation. The mathematical model assumes a two-dimensional flow and takes into account the nonlinear terms in the equations of fluid motion that express the convective acceleration, as well as the nonlinear deformation of the arterial wall. Several numerical simulations of the blood flow in large vessels have been performed to study the propagation along an arterial vessel of a pressure gradient pulse and a rate flow pulse. These simulations include various deformations of the wall artery leading to parietal displacements ranging from 0 (rigid wall) to 15% (very elastic wall) in order to consider physiological and pathological cases.

Results

The results show significant changes of the rate flow and the pressure gradient wave as a function of aosc, the relative variation in the radius of the artery over a cardiac cycle. These changes are notable beyond a critical value of aosc equal to 0.05. This critical value is also found in the evolution of the longitudinal impedance. So, above a variation of radius of 5%, the convective acceleration, created by the fluid-wall interactions, have an influence on the flow detectable on the longitudinal impedance.

Conclusions

The interpretation of the evolution of the longitudinal impedance shows that it could be a mean to test the performance of large arteries and can contribute to the diagnosis of parietal lesions of large arteries. For a blood vessel with a wall displacement higher than 5% similar to those of large arteries like the aorta, the longitudinal impedance is substantially greater than that obtained in the absence of wall displacement. This study also explains the effects of convective acceleration, on the shape of the decline of the pressure gradient wave and shows that they should not be neglected when the variation in radius is greater than 5%.     

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.     

A fast computational model for circulatory dynamics: effects of left ventricle–aorta coupling

The course of diseases such as hypertension, systolic heart failure and heart failure with a preserved ejection fraction is affected by interactions between the left ventricle (LV) and the vasculature. To study these interactions, a computationally efficient, biophysically based mathematical model for the circulatory system is presented. In a four-chamber model of the heart, the LV is represented by a previously described low-order, wall volume-preserving model that includes torsion and base-to-apex and circumferential wall shortening and lengthening, and the other chambers are represented using spherical geometries. Active and passive myocardial mechanics of all four chambers are included. The cardiac model is coupled with a wave propagation model for the aorta and a closed lumped-parameter circulation model. Parameters for the normal heart and aorta are determined by fitting to experimental data. Changes in the timing and magnitude of pulse wave reflections by the aorta are demonstrated with changes in compliance and taper of the aorta as seen in aging (decreased compliance, increased diameter and length), and resulting effects on LV pressure–volume loops and LV fiber stress and sarcomere shortening are predicted. Effects of aging of the aorta combined with reduced LV contractile force (failing heart) are examined. In the failing heart, changes in aortic properties with aging affect stroke volume and sarcomere shortening without appreciable augmentation of aortic pressure, and the reflected pressure wave contributes an increased proportion of aortic pressure.

     

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.     

Wave reflection effects in the central circulation of American alligators (Alligator mississippiensis): what the heart sees

A large central compliance is thought to dominate the hemodynamics of all vertebrates except birds and mammals. Yet large crocodilians may adumbrate the avian and mammalian condition and set the stage for significant wave transmission (reflection) effects, with potentially detrimental impacts on cardiac performance. To investigate whether crocodilians exhibit wave reflection effects, pressures and flows were recorded from the right aorta, carotid artery, and femoral artery of six adult, anesthetized American alligators (Alligator mississippiensis) during control conditions and after experimentally induced vasodilation and constriction. Hallmarks of wave reflection phenomena were observed, including marked differences between the measured profiles for flow and pressure, peaking of the femoral pressure pulse, and a diastolic wave in the right aortic pressure profile. Pulse wave velocity and peripheral input impedance increased with progressive constriction, and thus changes in both the timing and magnitude of reflections accounted for the altered reflection effects. Resolution of pressure and flow waves into incident and reflected components showed substantial reflection effects within the right aorta, with reflection coefficients at the first harmonic approaching 0.3 when constricted. Material properties measured from isolated segments of blood vessels revealed a major reflection site at the periphery and, surprisingly, at the junction of the truncus and right aorta. Thus, while our results clearly show that significant wave reflection phenomena are not restricted to birds and mammals, they also suggest that rather than cope with potential negative impacts of reflections, the crocodilian heart simply avoids them because of a large impedance mismatch at the truncus.     

Pulse wave propagation in a model human arterial network: assessment of 1-D numerical simulations against in vitro measurements

A numerical model based on the nonlinear, one-dimensional (1-D) equations of pressure and flow wave propagation in conduit arteries is tested against a well-defined experimental 1:1 replica of the human arterial tree. The tree consists of 37 silicone branches representing the largest central systemic arteries in the human, including the aorta, carotid arteries and arteries that perfuse the upper and lower limbs and the main abdominal organs. The set-up is mounted horizontally and connected to a pulsatile pump delivering a periodic output similar to the aortic flow. Terminal branches end in simple resistance models, consisting of stiff capillary tubes leading to an overflow reservoir that reflects a constant venous pressure. The parameters required by the numerical algorithm are directly measured in the in vitro set-up and no data fitting is involved. Comparison of experimental and numerical pressure and flow waveforms shows the ability of the 1-D time-domain formulation to capture the main features of pulse wave propagation measured throughout the system test. As a consequence of the simple resistive boundary conditions used to reduce the uncertainty of the parameters involved in the simulation, the experimental set-up generates waveforms at terminal branches with additional non-physiological oscillations. The frequencies of these oscillations are well captured by the 1-D model, even though amplitudes are overestimated. Adding energy losses in bifurcations and including fluid inertia and compliance to the purely resistive terminal models does not reduce the underdamped effect, suggesting that wall visco-elasticity might play an important role in the experimental results. Nevertheless, average relative root-mean-square errors between simulations and experimental waveforms are smaller than 4% for pressure and 19% for the flow at all 70 locations studied.     

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.     

Beyond the aorta: partial transmission of reflected waves from aortic coarctation into supra-aortic branches modulates cerebral hemodynamics and left ventricular load

Wave reflection from the site of aortic coarctation produces a reflected backward compression wave (BCW) that raises left ventricular (LV) afterload. However, not all reflected wave power will propagate back to the LV. This study investigated the hypothesis that the BCW is partially transmitted into supra-aortic vessels as a forward wave and explored the consequences of this phenomenon for cerebral and LV haemodynamic load. In eight sheep, high fidelity pressure and flow were measured in the aortic trunk (AoT) and brachiocephalic trunk (BCT, the single supra-aortic vessel present in sheep) at baseline and during two levels of proximal descending aortic constriction. Wave power analysis showed that aortic constriction produced not only a BCW in the AoT, but also a second forward compression wave (\(\mathrm{FCW}_{2})\) in the BCT that augmented pressure and flow after the initial forward compression wave (\(\mathrm{FCW}_{1})\). Mathematical analysis and a one-dimensional model of the human systemic arteries and aortic coarctation suggested that the relative transmission of waves into supra-aortic vessels versus the aorta was determined by the relative admittances of these vessels. Reducing supra-aortic admittance (1) increased pressure and flow pulsatility in cerebral arteries, (2) produced carotid and middle cerebral arterial flow waveforms with an older adult phenotype, (3) promoted transmission of reflected wave power towards the LV and (4) substantially increased mid- to late-systolic myocardial stress, which may promote LV hypertrophy. These findings suggest that wave transmission into supra-aortic branches has an important impact on both cerebral hemodynamics and LV load in aortic coarctation.