Fig. 4.1
The main components of an artery
From a mechanical point of view the two most important constituents of arteries are collagen and elastin. These are arranged in layers around the artery. The number of sheets increases during gestation but is fixed at birth. The molecules are arranged in a helical pattern around the artery, with a different pitch for different layers within the artery. The collagen molecules are present in a loose, wavy network in the adventitia. As the artery expands so the molecules unfurl to reach their straightened lengths at which point they become extremely stiff. This behaviour protects the smooth muscle cells from acute over-distension.
Elastin is a stable protein with a long half-life of around 50 years, so that the elastin sheets laid down in early life remain in place into later life. Collagen, on the other hand, has a half-life of 2 weeks, and thus is in a continuous state of turnover.
4.1.2 Stress–Strain Behaviour in Arteries
The stress–strain behaviour of an artery may be characterised by a plot of pressure versus diameter. Figure 4.2 shows pressure–diameter behaviour in an excised artery subject to an inflation pressure. The diameter-pressure behaviour is nonlinear, though over a restricted range (e.g. 80–120 mmHg) it is approximately linear. Within the physiologic range Fig. 4.2 shows a 10 % increase in diameter which is typical of the variation found in vivo during the cardiac cycle.
Fig. 4.2
Pressure-distension for an artery. The behaviour is nonlinear; the physiologic range of 80–120 mm Hg produces a 10 % variation in diameter
The contribution of elastin and collagen to the stress–strain behaviour was explored by Roach and Burton (1957). Figure 4.3 shows the tension-radius behaviour for 3 samples of artery. In the left curve the elastin has been removed by a chemical process so that the mechanical behaviour of the artery is governed by the collagen. The artery is quite stiff so that high stress has to be provided to stretch the vessel. In the right trace the collagen has been digested so that mechanical behaviour is governed by the elastin. The artery is now quite elastic so that that small changes in stress result in large changes in diameter. The elastic behaviour of the untreated artery can be explained by the different behaviours of collagen and elastin. At small distensions the collagen fibres are quite crimped and the behaviour is dominated by elastin. At high extensions the collagen fibres have straightened out; these now dominate mechanical behaviour and the artery becomes very stiff. This may be likened to a balloon in a net. Blowing up the balloon is easy until it is the same size as the net, at which point it is difficult to blow up the balloon much further.
Fig. 4.3
Effect of collagen and elastin on the tension-radius behaviour for an artery. Removal of elastin leads to stiffening; removal of collagen leads to more elastic behaviour. From: Roach MR, Burton AC; The reason for the shape of the distensibility curve of arteries; Can J Biochem Physiol. 1957;35:681–690; © Canadian Science Publishing or its licensors; redrawn with permission
As introduced in Chap. 1, the elastic modulus of elastin, collagen and arteries may be measured in a tensile testing system. The elastic modulus for elastin is low at 0.4–0.6 MPa but for collagen is much higher at 100 MPa. Arteries have intermediate values of 1–5 MPa (Ryan and Foster 1997). Note that, in the case of collagen and artery, incremental elastic modulus values are reported as the stress–strain behaviour is nonlinear. The Young’s modulus for an artery describes the composite behaviour.
The artery is a multi-layer structure and each layer has different mechanical properties. However to describe the overall stress–strain or diameter–pressure behaviour for an artery it is not necessary to know the elastic moduli of the different layers. This approach assumes that the artery is uniform and homogeneous; i.e. that the wall thickness and elastic composition are the same for different positions around the circumference. This assumption is valid in healthy arteries although it may not be true in disease due to 3D changes in geometry, wall thickness and wall composition (see later chapters).
4.2 Pressure and Flow Waveforms in Arteries
Figure 4.4 shows pressure–time waveforms from various arteries in the systemic circulation. The baseline or diastolic pressure in this example is about 80 mm Hg. The peak or systolic pressure increases with distance from the heart; in this example from 110 mm Hg at the aortic outflow to 160 mm Hg at the tibial arteries. There is also some change in shape of the pressure waveform with distance from the heart. Figure 4.4 also shows velocity–time waveforms taken using Doppler ultrasound. The waveforms are all similar in that they have a period of forward flow followed by a period of reverse flow. Figure 4.5 shows more time–velocity waveforms, taken from arteries supplying the brain and kidney. These organs have a low vascular resistance and the time–velocity waveforms have flow throughout the cardiac cycle. In this section we will be looking at the origins of the shape of the pressure and flow waveforms in different arteries.
Fig. 4.4
Blood pressure–time and velocity–time waveforms at increasing distance from the heart. There is a baseline (diastolic) pressure of 80 mm Hg; the peak (systolic) pressure increases with distance and there are changes in the overall shape. The flow waveforms are taken from Doppler ultrasound. The waveforms are pulsatile with a period of forward flow followed by a period of reverse flow then more forward flow. Circulation figure reproduced from Wikipedia; https://commons.wikimedia.org/wiki/File:Circulatory_System_en.svg. This image is in the public domain and was authored by Mariana Ruiz Villarreal
Fig. 4.5
Velocity–time waveforms from arteries supplying organs (brain and kidney) with high metabolic demand. Waveforms demonstrate a high baseline of flow. Circulation figure reproduced from Wikipedia; https://commons.wikimedia.org/wiki/File:Circulatory_System_en.svg. This image is in the public domain and was authored by Mariana Ruiz Villarreal
4.2.1 Windkessel Model
Models of pressure and flow in the arterial system generally only consider the arteries. The pressure at the capillaries (as noted in Chap. 2) is a fixed value close to zero so the venous system does not need to be taken into account when modelling the arterial system. An early model which tried to explain the pressure–time and flow-time waveforms is the Windkessel model. This comes from the German word ‘Windkessel’ meaning ‘air chamber’. The Windkessel model of the systemic circulation consists of three elements (Fig. 4.6); a pump representing the heart, an elastic chamber representing the arteries, and an outflow resistance representing flow through the arterioles. The process can be broken into 3 phases:
Fig. 4.6
Windkessel model consisting of a pump (heart), an elastic chamber (artery) and an outflow resistance (arteriolar bed). a Pre-ejection phase where the heart is full of blood. b In the ejection phase the heart contracts ejecting blood into the compliant chamber. The ejection takes about 100 ms. The tension in the wall of the chamber causes a pressure on the blood, which is forced out through the resistance vessels. Pressure and flow reach a maximum at the end of the ejection phase. c The ejection of blood from the heart stops. This is the relaxation phase. As the chamber gradually returns to its resting position the pressure on the blood decreases. As a result of the decrease in pressure, the flow also reduces. d The chamber returns to its resting position. The pressure reduces to its resting value and the flow reduces to zero
Ejection phase. The heart contracts ejecting blood into the compliant chamber. The ejection takes about 100 ms. The tension in the wall of the chamber causes a pressure on the blood, which is forced out through the resistance vessels. Pressure and flow reach a maximum at the end of the ejection phase.
Relaxation phase. The ejection of blood from the heart stops. This is the relaxation phase. As the chamber gradually returns to its resting position the pressure on the blood decreases. As a result of the decrease in pressure, the flow also reduces.
Resting phase. The chamber returns to its resting position. This is the completion of the relaxation phase. The pressure reduces to its resting value and the flow reduces to zero.
There are some similarities between the pressure waveforms from the Windkessel model and those seen in Fig. 4.4, in that pressure reaches a peak and gradually reduces. The Windkessel flow waveforms also have a peak and gradually reduce in value thereafter, similar to those in Fig. 4.5 for arteries supplying brain and kidney. However, the Windkessel flow waveforms bear little resemblance to the flow waveforms seen in Fig. 4.4 which have a period of reverse flow.
4.2.2 Wave Propagation Model
The Windkessel model is useful in basic understanding but it omits a key feature of the arterial system which is wave propagation. Ejection of blood from the left ventricle leads to expansion of the aorta in order to accommodate the ejected volume. The increase in circumference leads to an increase in tension within the arterial wall and hence an increase in pressure within the blood. The pressure passes down the artery in the form of a wave. This phenomenon can be demonstrated by recording the pressure with time at various locations from the heart. Figure 4.7 shows the pressure–time waveform at different distances from the heart up to 60 cm. The increase in pressure at the beginning of the waveform occurs at later times. This is consistent with the idea that the pressure pulse propagates down the artery as a wave. The pressure wave propagation speed is usually called the ‘pulse wave velocity’ or PWV. The typical PWV in the thoracic aorta is about 5 m s−1 and increases, with distance from the heart, to about 15 m s−1 in the arteries of the lower leg. An important point is that pressure wave speed is not the same as blood speed. The pressure wave speed is the speed at which the pressure wave propagates and is typically 5–15 m s−1. The blood speed is the speed at which the blood moves which is typically 0–1 m s−1 in health.
Fig. 4.7
Aortic pressure measured 10, 20, 30, 40, 50 and 60 cm downstream from the aortic valve plotted on as a function of time at different distances. The increase in pressure at the beginning of the waveform occurs at increasingly later times with distance demonstrating that the pressure propagates as a wave down the arterial tree. From Medical and Biological Engineering and Computing, An introduction to wave intensity analysis, Vol. 47, 2009, pp. 175–180, Parker KH, © International Federation for Medical and Biological Engineering 2009, with permission of Springer
An early attempt to formulate an equation for PWV was made in 1878 by Moens and Korteweg. This states that PWV is related to the elastic modulus E of the artery, the wall thickness h, the diameter d and the density ρ of blood. This simple equation in practice predicts the correct PWV to within about 10 % in healthy arteries.
(4.1)
4.2.3 Propagation Model with Reflected Waves
For propagation of any wave, in any medium, the wave will be scattered or reflected where there is a change in local impedance and this is also true for pressure waves in arteries. A change in impedance occurs as a result of change in diameter, shape or elastic properties. Common sites of reflected waves are at bifurcations; that is when a single parent artery splits into 2 or more daughter arteries. However, by far the largest contribution to reflected waves comes from the change in impedance which occurs between arteries and arterioles. The pressure–time waveform at a particular location in an artery is therefore a composite of the forward-going pressure wave, and a reverse-going pressure wave arising from downstream reflections at the level of the arterioles (Fig. 4.8a). The flow-time waveform can be considered in the same manner as a combination of a forward-going flow wave and a reverse-going flow wave (see e.g. Murgo et al. 1981). However, the flow waves combine in a subtractive manner (Fig. 4.8b). In the example shown in Fig. 4.8b the composite flow waveform has a period of reverse flow, similar to that seen in Fig. 4.4.
