Wave propagation in a model of the arterial circulation
Introduction
The method of characteristics has previously been applied to the study of waves in the arteries by a number of authors (Skalak, 1972; Stettler et al., 1981; Stergiopulos et al., 1992). These studies can be grouped loosely into two approaches: highly idealised models which are concerned primarily with the propagation of waves in single vessels and very complex models which have attempted to predict the detailed behaviour of the arterial system. In the more complex models, it is difficult to ascribe cause to effect in the interpretation of the results of the numerical simulations. That is, it is impossible to say whether a particular feature of the results is due primarily to one or another of the several complexities of the model.
In this study, we have taken a slightly different approach to the modelling of waves in the arteries. In order to study the influence of the complex branching geometry of the arterial system on its haemodynamics, we have used a linearised form of the general solution of the one-dimensional flow equations with highly idealised cardiac function but a fairly realistic model of the anatomy of the largest arteries. By including only this one complexity into our model, we are confident that any complexities in the results of our calculations are due to the interaction of the simple input wave with the reflections and re-reflections arising from the complex geometry of the large arteries. We then alter the model parameters to explore the effects of individual features of our model on arterial pressure and velocity waveforms. We do not suggest that the results of our calculations provide a realistic model of arterial haemodynamics, but we do believe that the results demonstrate that many of the features of flow in the arteries can be ascribed to the effects of wave reflections.
Wave behaviour in the arteries has primarily been studied using the impedance method which also presumes linearity. The earliest comparable study was by Taylor (1966) who calculated the input impedance of two random networks of vessels with eight generations. There was no attempt to model a realistic arterial system. More recent studies have used more realistic models of the arterial system in their impedance calculations (Westerhof and Noordergraaf, 1970; Avolio, 1980; O'Rourke and Avolio, 1980; Stergiopulos et al., 1992). Indeed, we use the data collected by Westerhof and Noordergraaf and Stergiopulos et al. as the basis of our arterial model. The impedance method has also been applied to models of the pulmonary circulation (Milnor, 1989) and to a model based upon measurements of a cast of a right coronary artery (Zamir, 1998). All of these studies, while comparable to this work, are carried out in the frequency domain of Fourier analysis and thus lack the immediacy of the temporal analysis presented here. Also, impedance analysis is intrinsically linear whereas the method which we present can incorporate various non-linearities.
In the theory section, we give a very brief outline of the theoretical basis of our calculations and the algorithms we used. The details are available in Wang (1997). We then calculate the pressure and velocity waveforms at different locations in the arterial system in response to a simple, half-sinusoidal left ventricular contraction. The model is then used to study (i) the role of reflections from the heart, (ii) the effect of complete occlusion of the aorta at different locations and (iii) the effects of changes in peripheral resistance on the arterial pulse waveforms. Whenever possible, the results of our calculations are compared with relevant experimental observations.
Section snippets
The method of characteristics
The behaviour of blood in the arterial system is simulated as a one-dimensional, incompressible flow in elastic tubes. The governing equations for the pressure, P, and mean velocity, U, are hyperbolic and can be solved by the method of characteristics (Anliker et al., 1971; Skalak, 1972; Parker and Jones, 1990). The solution shows that any disturbance introduced into the artery will generate pressure and velocity waves that will propagate with the characteristic velocities U±c, where c is the
Results
The model has been used to calculate the pressure and velocity waveforms from the ascending aorta to the femoral artery in response to an idealised waveform generated by the left ventricle, a single half-sinusoid of duration. Since we are interested only in the resultant waveforms, all results are normalised with respect to the input wave using the diastolic pressure as the gauge pressure. The first results that we show are calculated assuming that the heart absorbs all waves which are
Conclusions
Before drawing conclusions about our results, we would like to reiterate that the goal of this study was not to develop a realistic model of arterial pulse waves but to develop a model with which we could explore the influence of wave reflections in the complex geometry of the arteries. With this goal in mind, we have used a realistic model of the large arteries adjusted to ensure that the model bifurcations are well matched for the transmission of forward waves. We have made simplifying
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