Pulse wave propagation in a model human arterial network: Assessment of 1-D visco-elastic simulations against in vitro measurements

The accuracy of the nonlinear one-dimensional (1-D) equations of pressure and flow wave propagation in Voigt-type visco-elastic arteries was tested against measurements in a well-defined experimental 1:1 replica of the 37 largest conduit arteries in the human systemic circulation. The parameters required by the numerical algorithm were directly measured in the in vitro setup and no data fitting was involved. The inclusion of wall visco-elasticity in the numerical model reduced the underdamped high-frequency oscillations obtained using a purely elastic tube law, especially in peripheral vessels, which was previously reported in this paper [Matthys et al., 2007. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech. 40, 3476-3486]. In comparison to the purely elastic model, visco-elasticity significantly reduced the average relative root-mean-square errors between numerical and experimental waveforms over the 70 locations measured in the in vitro model: from 3.0% to 2.5% (p<0.012) for pressure and from 15.7% to 10.8% (p<0.002) for the flow rate. In the frequency domain, average relative errors between numerical and experimental amplitudes from the 5th to the 20th harmonic decreased from 0.7% to 0.5% (p<0.107) for pressure and from 7.0% to 3.3% (p<10(-6)) for the flow rate. These results provide additional support for the use of 1-D reduced modelling to accurately simulate clinically relevant problems at a reasonable computational cost.     

Wave propagation in a model of the arterial circulation

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

Validation of a patient-specific one-dimensional model of the systemic arterial tree

The aim of this study is to develop and validate a patient-specific distributed model of the systemic arterial tree. This model is built using geometric and hemodynamic data measured on a specific person and validated with noninvasive measurements of flow and pressure on the same person, providing thus a patient-specific model and validation. The systemic arterial tree geometry was obtained from MR angiographic measurements. A nonlinear viscoelastic constitutive law for the arterial wall is considered. Arterial wall distensibility is based on literature data and adapted to match the wave propagation velocity of the main arteries of the specific subject, which were estimated by pressure waves traveling time. The intimal shear stress is modeled using the Witzig-Womersley theory. Blood pressure is measured using applanation tonometry and flow rate using transcranial ultrasound and phase-contrast-MRI. The model predicts pressure and flow waveforms in good qualitative and quantitative agreement with the in vivo measurements, in terms of wave shape and specific wave features. Comparison with a generic one-dimensional model shows that the patient-specific model better predicts pressure and flow at specific arterial sites. These results obtained let us conclude that a patient-specific one-dimensional model of the arterial tree is able to predict well pressure and flow waveforms in the main systemic circulation, whereas this is not always the case for a generic one-dimensional model.     

Epicardial coronary blood flow including the presence of stenoses and aorto-coronary bypasses--I: Model and numerical method

A computer model and numerical method for calculating left epicardial coronary blood flow has been developed. This model employs a finite-branching geometry of the coronary vasculature and the one-dimensional, unsteady equations for flow with friction. The epicardial coronary geometry includes the left main and its bifurcation, the left anterior descending and left circumflex coronary arteries, and a selected number of small branches. Each of the latter terminate in an impedance, whose resistive component is related to intramyocardial compression through a linear dependence on left ventricular pressure. The elastic properties of the epicardial arteries are taken to be non-linear and are prescribed by specifying the local small-disturbance wave speed. The model allows for the incorporation of multiple stenoses as well as aorto-coronary bypasses. Calculations using this model predict pressure and flow waveform development and allow for the systematic investigation of the dependence of coronary flow on various parameters, e.g., peripheral resistance, wall properties, and branching pattern, as well as the presence of stenoses and bypass grafts. Reasonable comparison between calculations and earlier experiments in horses has been obtained.     

A novel porous media-based approach to outflow boundary resistances of 1D arterial blood flow models

In this paper we introduce a novel method for prescribing terminal boundary conditions in one-dimensional arterial flow networks. This is carried out by coupling the terminal arterial vessel with a poro-elastic tube, representing the flow resistance offered by microcirculation. The performance of the proposed porous media-based model has been investigated through several different numerical examples. First, we investigate model parameters that have a profound influence on the flow and pressure distributions of the system. The simulation results have been compared against the waveforms generated by three elements (RCR) Windkessel model. The proposed model is also integrated into a realistic arterial tree, and the results obtained have been compared against experimental data at different locations of the network. The accuracy and simplicity of the proposed model demonstrates that it can be an excellent alternative for the existing models.

     

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.     

Anharmonic analysis of arterial blood pressure and flow pulses

Fourier analysis is usually employed for the computation of blood flow in arteries. Although the orthogonality of Fourier eigenfunctions guarantees the accurate mathematical modeling of the blood pressure and flow waveforms, the physics behind this objective function is frequently missing. We propose a new method to account for the blood pressure and flow, single-cycle (systole-diastole) waveforms. It is based on the one dimensional hydrodynamic mass and momentum conservation equations for viscous flow. The similarity of the linear problem, under discussion, with related transmission line theory in electromagnetic wave propagation, permits expansion in anharmonic, non-separable eigenfunctions. In some cases one term in the expansion is adequate to fit the main peak of the observed waveforms. Analytical formulas are derived for the dependence of the pressure and flow main peaks on whole blood viscosity and distance from the heart, which interpret observations related to hypertension.     

Communication by substratum vibration in the New Zealand tree weta, Hemideina femorata (Stenopelmatidae: Orthoptera)

The propagation of vibrations along the trunk and branches of a manuka tree, generated in response to the impact of a steel ball-bearing on the trunk, was measured with an accelerometer. The impact generated bending waves which travelled along the trunk and into the branches. Close to the point of impact the waveform was dominated by a damped oscillation at 518 Hz; as the bending wave progressed away from the point of impact the frequency of the dominant waveform increased. Beyond 200 cm the waveform became increasingly complex and a smallamplitude, high-frequency component progressively preceded the main wave. Branching points also induced complex waveforms, particularly where branches lay at a large angle to the trunk. Stridulating wetas also generated bending waves in the tree at a frequency close to that generated by the ball-bearing, as well as at a higher frequency of 7.5 kHz. The acoustic frequency of stridulation peaked at 0.8 and 3.4 kHz. Records from nerves serving the vibration-sensitive subgenual organs showed that wetas can detect oscillations at 1 kHz at 0.015ms^-2. A stridulating weta placed on the same log as a preparation in which the nerve from the subgenual organ was monitored generated oscillatins well above the threshold for detection.     

Abnormal arterial flows by a distributed model of the fetal circulation

Modeling the propagation of blood pressure and flow along the fetoplacental arterial tree may improve interpretation of abnormal flow velocity waveforms in fetuses. The current models, however, either do not include a wide range of gestational ages or do not account for variation in anatomical, vascular, or rheological parameters. We developed a mathematical model of the pulsating fetoumbilical arterial circulation using Womersley's oscillatory flow theory and viscoelastic arterial wall properties. Arterial flow waves are calculated at different arterial locations from which the pulsatility index (PI) can be determined. We varied blood viscosity, placental and brain resistances, placental compliance, heart rate, stiffness of the arterial wall, and length of the umbilical arteries. The PI increases in the umbilical artery and decreases in the cerebral arteries, as a result of increasing placental resistance or decreasing brain resistance. Both changes in resistance decrease the flow through the placenta. An increased arterial stiffness increases the PIs in the entire fetoplacental circulation. Blood viscosity and peripheral bed compliance have limited influence on the flow profiles. Bradycardia and tachycardia increase and decrease the PI in all arteries, respectively. Umbilical arterial length has limited influence on the PI but affects the mean arterial pressure at the placental cord insertion. The model may improve the interpretation of arterial flow pulsations and thus may advance both the understanding of pathophysiological processes and clinical management.     

Mathematical modeling of the response of a resistive vessel to pressure

The response of small arterial vessels to internal pressure makes an essential contribution to autoregulation in the vascular bed. It is believed that free cytosolic Ca²⁺ concentration plays a pivotal role in the regulation of smooth muscle contractility and hence of the vascular lumen. A simple mathematical model of blood flow in a resistive vessel is suggested. The model is based on the experimental data obtained for cerebral arteries, but may be used for any other resistive vessel. The model not only describes the regulation of the vascular lumen by transmural pressure but also shows realistic behavior of the vessel radius and cytosolic [Ca²⁺] at different rates of pressure change. Possible variations in the radius along the vessel due to the Bayliss effect are considered.     

The inverse Womersley problem for pulsatile flow in straight rigid tubes

In this study a numerical solution for the problem of pulsating flow in rigid tubes is described. The method applies to the case of known flow rate waveform, as opposed to Womersley solution where the pressure gradient was the known quantity. The solution provides the pressure gradient and wall shear stress waveforms as well as the instantaneous velocity profiles. Results show that the method can be used to study the blood flow characteristics in large arteries.     

Stomatal oscillations in orange trees under natural climatic conditions

BACKGROUND AND AIMS: Stomatal oscillations have been reported in many plant species, but they are usually induced by sudden step changes in the environment when plants are grown under constant conditions. This study shows that in navel orange trees (Citrus sinensis) pronounced stomatal oscillations occur and persist under natural climatic conditions. METHODS: Oscillations in stomatal conductance were measured, and related to simultaneous measurements of leaf water potential, and flow rate of sap in the stems of young, potted plants. Cycling was also observed in soil-grown, mature orchard trees, as indicated by sap flow in stem and branches. KEY RESULTS: Oscillations in stomatal conductance were caused by the rapid propagation and synchronization of changes in xylem water potential throughout the tree, without rapid changes in atmospheric conditions. CONCLUSIONS: The results show marked stomatal oscillations persisting under natural climatic conditions and underscore the need to discover why this phenomenon is so pronounced in orange trees.     

Pulsatile blood flow effects on temperature distribution and heat transfer in rigid vessels

The effect of blood velocity pulsations on bioheat transfer is studied. A simple model of a straight rigid blood vessel with unsteady periodic flow is considered. A numerical solution that considers the fully coupled Navier-Stokes and energy equations is used for the simulations. The influence of the pulsation rate on the temperature distribution and energy transport is studied for four typical vessel sizes: aorta, large arteries, terminal arterial branches, and arterioles. The results show that: the pulsating axial velocity produces a pulsating temperature distribution; reversal of flow occurs in the aorta and in large vessels, which produces significant time variation in the temperature profile. Change of the pulsation rate yields a change of the energy transport between the vessel wall and fluid for the large vessels. For the thermally important terminal arteries (0.04-1 mm), velocity pulsations have a small influence on temperature distribution and on the energy transport out of the vessels (8 percent for the Womersley number corresponding to a normal heart rate). Given that there is a small difference between the time-averaged unsteady heat flux due to a pulsating blood velocity and an assumed nonpulsating blood velocity, it is reasonable to assume a nonpulsating blood velocity for the purposes of estimating bioheat transfer.     

Computer simulation of blood flow in the human arm

This paper considers a finite element method to characterize blood flow in the human arm arteries. A set of different pressure waveforms, which represent normal and diseased heart pulses, is used for the proximal boundary conditions, and a modified Windkessel model is used for the distal arterial boundary conditions. A comparison of the distal pressure and flow waveforms, for each different proximal pressure, is made to determine whether such waveforms are significantly altered from normal waveforms. The results show that the distal pressure and/or flow waveforms in certain cases are sufficiently different to be possibly used as a diagnostic indicator of an abnormal heart condition. Also considered is the effect of stenosis, change of compliance, and dilatation of the distal beds on the pressure and flow waveforms. A stenosis which has an area reduction of greater than approximately 75% is found to significantly alter both the distal pressure and flow waveforms. Changes in arterial compliance, however, do not strongly influence the waveforms. Dilatation of distal vascular beds is simulated by reducing the lumped resistance of these beds, and this reduction increases mean flow and decreases mean distal pressure, but has little effect on the basic shape of either the pressure or flow waveform.     

A hybrid one-dimensional/Womersley model of pulsatile blood flow in the entire coronary arterial tree

Using a frequency-domain Womersley-type model, we previously simulated pulsatile blood flow throughout the coronary arterial tree. Although this model represents a good approximation for the smaller vessels, it does not take into account the nonlinear convective energy losses in larger vessels. Here, using Womersley's theory, we present a hybrid model that considers the nonlinear effects for the larger epicardial arteries while simulating the distal vessels (down to the 1st capillary segments) with the use of Womersley's Theory. The main trunk and primary branches were discretized and modeled with one-dimensional Navier-Stokes equations, while the smaller-diameter vessels were treated as Womersley-type vessels. Energy losses associated with vessel bifurcations were incorporated in the present analysis. The formulation enables prediction of impedance and pressure and pulsatile flow distribution throughout the entire coronary arterial tree down to the first capillary segments in the arrested, vasodilated state. We found that the nonlinear convective term is negligible and the loss of energy at a bifurcation is small in the larger epicardial vessels of an arrested heart. Furthermore, we found that the flow waves along the trunk or at the primary branches tend to scale (normalized with respect to their mean values) to a single curve, except for a small phase angle difference. Finally, the model predictions for the inlet pressure and flow waves are in excellent agreement with previously published experimental results. This hybrid one-dimensional/Womersley model is an efficient approach that captures the essence of the hemodynamics of a complex large-scale vascular network. The present model has numerous applications to understanding the dynamics of coronary circulation.     

Model analysis of respiratory responses to inspiratory resistive loads

Based on experimental inspiratory driving pressure waveforms and active respiratory impedance data of anesthetized cats, we made model predictions of the factors that determine the immediate (first loaded breath) intrinsic (i.e., nonneural) tidal volume compensation to added inspiratory resistive loads. The time course of driving pressure (P) was given by P = atb, where a is the pressure at 1 s from onset of inspiration and represents the intensity of neuromuscular drive, t is time, and b is a dimensionless index of the shape of the driving pressure wave. For a given value of active respiratory impedance, tidal volume compensation to added resistive loads increases with increasing inspiratory duration and decreasing value of b but is independent of a. Model predictions of load compensation are compared to experimental results.     

The subscapular arterial tree as a source of microvascular arterial grafts

The subscapular arterial tree may be used as a source of microvascular grafts to replace damaged or diseased portions of arteries, particularly in the hand and forearm. By studying cadaver dissections, it is possible to estimate the number of branches that may be found at different arterial segment lengths from the origin of the subscapular artery. Fifty-five preserved cadaver subscapular arterial trees were dissected, and the branching patterns were documented. Three major arterial branching patterns of the subscapular artery were observed with one, two, and three major branches to the serratus anterior in 60 percent, 29 percent, and 9 percent of the cases, respectively. The authors determined the number of 1-mm-diameter, 1-cm-long branches arising from each of six 3-cm regions of the arterial tree measured from the origin of the subscapular artery to the end of the longest terminal branch. The probability of finding at least one usable terminal branch that is at least 12.0 cm in length was found to be 98 percent. Typically, there are two to five useful branches at this distance. Such information may help surgeons fine tune their process of selecting an appropriate arterial donor site for a particular arterial defect and supports the use of the subscapular arterial tree as a donor site for microvascular arterial grafts.     

Reverse flow in the major infrarenal vessels--a capacitive phenomenon

The arterial blood flow waveform is shown to change abruptly when passing from the thoracic aorta into the abdominal aorta in humans. Although this change has been accurately predicted by numerical solution of complicated pulse propagation equations, this paper demonstrates the ability of a simple lumped parameter model to explain this change in the waveforms using easily understood physical terms. The model correctly predicts changes in flow waveform under conditions of exercise and peripheral vascular disease. This analysis is useful in understanding abdominal artery physiology and explains the basis for clinical ultrasound Doppler examination of the legs.