N. Stergiopulos

Computer simulation of arterial flow with applications to arterial and aortic stenoses

A computer model for simulating pressure and flow propagation in the human arterial system is developed. The model is based on the one-dimensional flow equations and includes nonlinearities arising from geometry and material properties. Fifty-five arterial segments, representing the various major arteries, are combined to form the model of the arterial system. Particular attention is paid to the development of peripheral pressure and flow pulses under normal flow conditions and under conditions of arterial and aortic stenoses. Results show that the presence of severe arterial stenoses significantly affects the nature of the distal pressure and flow pulses. Aortic stenoses also have a profound effect on central and peripheral pressure pulse formation. Comparison with the published experimental data suggests that the model is capable of simulating arterial flow under normal flow conditions as well as conditions of stenotic obstructions in a satisfactory manner.

Simple and accurate way for estimating total and segmental arterial compliance: the pulse pressure method

We derived and tested a new, simple, and accurate method to estimate the compliance of the entire arterial tree and parts thereof. The method requires the measurements of pressure and flow and is based on fitting the pulse pressure (systolic minus diastolic pressure) predicted by the two-element windkessel model to the measured pulse pressure. We show that the two-element windkessel model accurately describes the modulus of the input impedance at low harmonics (0–4th) of the heart rate so that the gross features of the arterial pressure wave, including pulse pressure, are accounted for. The method was tested using a distributed nonlinear model of the human systemic arterial tree. Pressure and flow were calculated in the ascending aorta, thoracic aorta, common carotid, and iliac artery. In a linear version of the systemic model the estimated compliance was within 1% of the compliance at the first three locations. In the iliac artery an error of 7% was found. In a nonlinear version, we compared the estimates of compliance with the average compliance over the cardiac cycle and the compliance at the mean working pressure. At the first three locations we found the estimated and “actual” compliance to be within 12% of each other. In the iliac artery the error was larger. We also investigated an increase and decrease in heart rate, a decrease in wall elasticity and exercise conditions. In all cases the estimated total arterial compliance was within 10% of mean compliance. Thus, the errors result mainly from the nonlinearity of the arterial system. Segmental compliance can be obtained by subtraction of compliance determined at two locations.

Theoretical study of dynamics of arterial wall remodeling in response to changes in blood pressure

The dynamics of arterial wall remodeling was studied on the basis of a phenomenological mathematical model. Sustained hypertension was simulated by a step increase in blood pressure. Remodeling rate equations were postulated for the evolution of the geometrical dimensions that characterize the zero stress state of the artery. The driving stimuli are the deviations of the extreme values of the circumferential stretch ratios and the average stress from their values at the normotensive state. Arterial wall was considered to be a thick-walled tube made of nonlinear elastic incompressible material. Results showed that thickness increases montonically with time whereas the opening angle exhibits a biphasic pattern. Geometric characteristics reach asymptotically a new homeostatic steady state, in which the stress and strain distribution is practically identical with the distribution under normotensive conditions. The model predictions are in good agreement with published experimental findings.

A theoretical investigation of low frequency diameter oscillations of muscular arteries.

Spontaneous low frequency diameter oscillations have been observedin vivo in some muscular arteries. The aim of this paper is to propose a possible mechanism for their appearance. A lumped parameter mathematical model for the mechanical response of an artery perfused with constant flow is proposed, which takes into account the active behavior of the vascular smooth muscle. The system of governing equations is reduced into two nonlinear autonomous differential equations for the arterial circumferential stretch ratio, and the concentration of calcium ions, Ca2+, within the smooth muscle cells. Factors controlling the muscular tone are taken into account by assuming that the rate of change of Ca2+ depends on arterial pressure and on shear stress acting on the endothelium. Using the theory of dynamical systems, it was found that the stationary solution of the set of governing equations may become unstable and a periodic solution arises, yielding self-sustained diameter oscillations. It is found that a necessary condition for the appearance of diameter oscillations is the existence of a negative slope of the steady state pressure-diameter relationship, a phenomenon known to exist in arterioles. A numerical parametric study was performed and bifurcation diagrams were obtained for a typical muscular artery. Results show that low frequency diameter oscillations develop when the magnitude of the perfused inflow, the distal resistance, as well as the length of the artery are within a range of critical values.

On the wave transmission and reflection properties of stenoses

This study is concerned with the wave reflection properties of arterial stenoses. Two theoretical models have been developed for deriving the reflection coefficient: a linear model resulting from the linearization of the pressure drop-flow equation and an indirect, quasi-nonlinear model, based on the separation of pressure waves into their forward and backward running components proximal and distal to the stenosis. The linear method gave consistently lower values for the reflection coefficient when compared to the quasi-nonlinear model. In vitro experiments in elastic tubes showed that the reflection coefficient is strongly dependent on stenosis severity, mean flowrate, and the elastic properties of the proximal unobstructed artery. For critical stenoses the reflection coefficient is frequency and pulsatility independent. The results suggest that hemodynamically nonsevere stenoses may cause significant wave reflections.

Intraluminal pressure modulates the magnitude and the frequency of induced vasomotion in rat arteries.

Arterial vasomotion and its relation to intraluminal pressure were investigated in vitro in isolated rat arteries. Femoral arteries (mean diameter = 768.2 +/- 25 microns, n = 5) and mesenteric arteries (mean diameter = 393.4 +/- 32 microns, n = 5) were used in this study. Arterial segments were excised, mounted on microcannulas and perfused with Tyrodes solution at a constant flow (100 microliters/min). After equilibration, intraluminal pressure was stepwise changed from 0 to 120 mm Hg. The changes in the outer diameter of the vessels were measured continuously over a period of 4 h after the equilibration. Vasomotion was induced by constrictor agonists (norepinephrine 10(-6) M for mesenteric arteries and norepinephrine 10(-6) M + Bay K8644 10(-7) M for femoral arteries) and was maintained only in the presence of the above-mentioned drugs. Both vasomotion magnitude and frequency are modulated by pressure. Vasomotion frequency increases with pressure increase. When intraluminal pressure varied between 0 and 120 mm Hg, vasomotion frequency varied between 0.19 and 0.49 Hz for mesenteric arteries and between 0.04 and 0.23 Hz for femoral arteries. Thus, vasomotion frequency differed clearly between the two vessel types. Vasomotion amplitude shows a biphasic relationship with a maximum occurring at about 40 mm Hg for mesenteric arteries and 50 mm Hg for femoral arteries. Based on these findings, it is hypothesized that vasomotion amplitude relates to the active mechanical properties of the artery and, in particular, to its contractile capacity.

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.

A theoretical investigation of low frequency diameter oscillations of muscular arteries

Periodic diametral fluctuations have been measured in vivo on the radial arter. A theoretical investigation was performed on a possible mechanism of these low frequency phasic contractions. We used an one-dimensional lumped parameter model, which took into account both passive and active component of the arterial wall. The nonlinear stability analysis showed that, under certain conditions, the muscular arterial became mechanically unstable and periodic self-sustained oscillations could exist.