Fig. 5.1
Forces on an artery wall, blood pressure and wall shear stress
Fig. 5.2
Cross section of an artery wall. The pressure acts on the arterial wall. This is balanced by a circumferential stress within the wall
The cyclic variation in pressure gives rise to a cyclic variation in the diameter and circumference of the artery. Each cell within the wall therefore undergoes a cyclic stretching and un-stretching. An artery diameter typically varies by 10 % over the cardiac cycle so each cell will typically be stretched and un-stretched by the same amount in the circumferential direction. The expansion in the circumferential direction is accompanied by a decrease in thickness in the radial direction. This can be seen as a cyclic variation in the thickness of the intima-media layer measured using ultrasound imaging (Meinders et al. 2003).
The WSS is the viscous drag of blood on the wall and acts in the plane of the wall. The WSS changes in a cyclic manner with changes in blood flow during the cardiac cycle (Fig. 5.3). If overall flow is axial then the WSS vector will also be axial. However in regions of complex flow such as near bifurcations the WSS vector may have a large non-axial component (Fig. 5.4).
Fig. 5.3
Variation of WSS magnitude with time in the common carotid, common femoral and brachial arteries. Reprinted from Atherosclerosis, Vol. 191; Stroev PV, Hoskins PR, Easson WJ; Distribution of wall shear rate throughout the arterial tree: a case study; pp. 276–280, Copyright (2007), with permission from Elsevier
Fig. 5.4
WSS vector orientation in the carotid arteries at peak systole with large non-axial components close to the bifurcation. Image kindly provided by Prof Yun Xu, Imperial College
If an artery is excised from the body its length will decrease by about 40 % implying that in vivo there is longitudinal tension; that is the artery is pre-stressed in the longitudinal direction (Fig. 5.5). If the excised artery is sliced open longitudinally it will partially spring open implying that the artery wall is also pre-stressed from the inner to the outer lumen. Arteries do not exist an unloaded state in vivo. In vivo, arteries are always in a longitudinally stretched state and are always subject to pressure. In the unloaded state this pre-stressing gives rise to an uneven stress distribution from inner to outer wall. However, in vivo under blood pressure, the stress distribution becomes more uniform (Fig. 5.6).
Fig. 5.5
Arteries in vivo are in a pre-stressed state in the longitudinal direction
Fig. 5.6
Circumferential stress distribution in an artery with no blood pressure and with blood pressure, with and without pre-stressing. The stress distribution is uniform for the pressurised artery with pres-stressing
In the above analysis the artery is treated as a homogenous material. However, in reality, the presence of several layers (intima, media and adventitia) leads to a more complicated stress distribution due the different mechanical properties and the different amounts of pre-stressing in each layer (Holzapfel and Ogden 2010; Karsaj and Humphrey 2012).
5.1.3 Murray’s Law
The arterial system is a branching network; the number of vessels increases with each branching and the diameters decrease. An early attempt to investigate the design of the branches was performed by Murray (1926a, b). He hypothesised that the arterial system is designed so that the heart needs to expend the minimum amount of effort in order to pump the blood along the arteries. From this he derived that the diameter d 1 of the parent artery to the third power is equal to the sum of the diameters d 2 and d 3 of the daughter arteries to the third power (Eq. 5.1, Fig. 5.7).
(5.1)
This is known as Murray’s law and has been widely quoted in publications on the structure of the arterial system. Though Murray did not concern himself with WSS it follows from Murray’s law that the mean WSS is constant for all arteries from the largest to the smallest (Sherman 1981). It is also worth saying that if Murray’s law applies to all mammals, which have basically the same design of cardiovascular system, then the WSS in all arteries in the whale should be the same, and the WSS in all arteries in the mouse should be the same.
The work of Murray is concerned with the principles by which the arterial system is designed. It follows that there must then be a mechanism for controlling arterial diameter according to some design principle. The mechanism suggested here is that the arterial system is designed to maintain WSS at a constant value (Zamir 1977). This is further considered below.
5.1.4 Brief Review of Wall Shear Stress
It was described in Sect. 5.1.3 that Murray’s law predicts that the mean WSS is constant in all arteries. This section looks at experimental data on WSS and whether Murray’s law has been confirmed to apply in vivo.
Early studies which estimated WSS in arteries assumed a parabolic velocity profile in the vessel. With this assumption an equation can be derived which relates wall shear rate to the flow rate and the diameter. In invasive studies flow rate can be measured using a variety of techniques including electromagnetic flow probes and thermodilution. Diameter can be measured using X-ray imaging or from calliper measurements in exposed vessels. These measurements need not even be made at the same time. Once the wall shear rate is calculated it is multiplied by viscosity to give the WSS. An early summary of WSS data is provided by Giddens et al. (1993) which demonstrated that mean WSS lies within the narrow range of 1–2 Pa in a range of arteries in the human and dog. This finding of a constant mean WSS is consistent with Murray’s law and seemed to provide good confirmation that Murray’s law was correct.
It is worth restating the situation in the early 1990s concerning WSS; Murray’s law suggested that WSS is constant within a species; experimental findings suggested that WSS was similar in several different arteries in the human in the dog.
The constancy of WSS was challenged by later studies which used in vivo techniques based on MRI and ultrasound. The spatial resolution and functionality of in vivo imaging gradually improved to the point where it was possible to obtain reliable measurements of WSS in vivo. Advances in computing power and the availability of computational fluid dynamics also meant that by the 1990s it was possible to estimate WSS in realistic geometries (rather than in simple straight tubes). This combination of imaging and modelling data came up with a slightly different story concerning WSS which is summarised by Cheng et al. (2007). Figure 5.8 illustrates findings in 3 arteries, the common carotid, brachial and femoral. There are clear differences, with the common carotid having a higher mean WSS than the brachial or femoral arteries. It was also found that there are differences in WSS between species with smaller species having high WSS. An MRI study by Greve et al. (2006) demonstrated a dependence of mean WSS on body-mass to the power of −0.38, which is in excellent agreement with the dependence of −0.375 predicted by Weinberg and Ethier (2007) from scaling laws.
Fig. 5.8
Variation of mean WSS in the common carotid, femoral and brachial arteries. The differences are due to differences in the pulsatility of the flow waveform, with higher values of mean WSS for the carotid and lower values for the femoral and brachial. Data from Cheng et al. (2007)
The evidence above points to a control mechanism which attempts to maintain WSS within a narrow bound (Langille 1996). The ‘set point’ for WSS in human arteries is around 1–2 Pa and deviation from this value will result in change in diameter; this is called ‘arterial remodelling’. If the mean WSS is greater than the set-point, the diameter increases to reduce WSS until its value lies at the set-point. Conversely if the mean WSS is less than the set-point the diameter decreases until the mean WSS value lies at the set-point. It is noted that temporary increases in mean WSS, such as occur during a few minutes exercise, will not result in long-term change in diameter. The control mechanism comes into play when the changes are longer term.
The discussion around Murray’s law above suggests that it was initially thought that the value of the set-point for WSS was the same in all arteries. It has become clear that the set-point is different for different arteries. One possible explanation for this was offered by Reneman et al. (2006). He noted that the flow waveform is not the same in all arteries. This is illustrated in Fig. 5.8; the common carotid has a high degree of flow throughout the cardiac cycle (and higher mean WSS) whereas the brachial and femoral (which supply blood to muscle) have high pulsatility (and low-mean WSS). However during exercise the flow waveforms in arteries supplying muscle change significantly; the waveforms have a much higher component of flow and a much higher mean WSS. Reneman’s observation was that during exercise the mean WSS is similar in different arteries, implying that the set-point may be relevant for conditions of high flow, not resting flow (for which most measurements are taken).
5.1.5 The Law of Laplace
It was noted above that blood pressure is opposed by a tension within the arterial wall. The relationship between pressure and tension was investigated by the French physicist Pierre de Laplace in the eighteenth century in the context of surface tension in water. Approximating an artery as a thin-walled cylinder gives the ‘law of Laplace’ as shown in Eq. 5.2.
(5.2)
This states that, for a fixed pressure P, as the radius r increases the tension T in the wall increases. An alternative formulation of the law of Laplace is in the context of stress (force per unit area). The circumferential stress or hoop stress H is shown in Eq. 5.3 where w is wall thickness.
(5.3)
Figure 5.9 shows wall stress as a function of age in the aorta and carotid artery (from Åstrand et al. 2005). Values range from 3000 to 16,000 Pa.
Fig. 5.9
Plots of circumferential stress as a function of age in the aorta and common carotid arteries. Closed circles are made, open circles female. Reprinted from Journal of Vascular Surgery, Vol. 42; Åstrand H, Rydén-Ahlgren A, Sandgren T, Länne T; Age-related increase in wall stress of the human abdominal aorta: an in vivo study; pp. 926–931, Copyright (2005), with permission from The Society for Vascular Surgery
5.1.6 Brief Review of Circumferential and Longitudinal Wall Stress
The mechanical strength of an artery is governed primarily by the elastin and collagen fibres which are laid down in a concentric fashion. The idea of a lamellar unit of structure consisting of muscle, elastin and collagen was formulated by Wolinsky and Glagov (1967). During fetal development the number of lamellar units increases and remains constant after birth. The number of lamellar units in the adult aorta of different species was studied by Wolinsky and Glagov (1967). In decreasing order of animal size there are around 70 units in the pig, 60 in the human, 45 in the dog, 13 in the guinea pig and 5 in the mouse. The circumferential tension in the wall of the aorta increases with animal size (5 N m−1 for the mouse increasing to 190 N m−1 for the pig). However the significant finding reported in the Wolinsky paper was that the tension per lamellar unit was similar across all species at 1–3 N m−1. Wall thickness therefore adjusts in order to maintain circumferential stress within a narrow range. Thickening of the wall during embryonic development occurs by increase in the number of lamellar units. Post-birth increases in wall thickness arise through thickening of each lamellar unit. In healthy individuals this is achieved by an increase in smooth muscle content, however in disease such as in hypertension thickening is accompanied by increasing collagen deposition and the changes become irreversible (Hayashi and Naiki 2009). Conversely, where there is decrease in circumferential stress, possibly as a result of disease, this leads to atrophy of the wall (Bomberger et al. 1980).
It was noted above that arteries are pre-stretched; that is there is longitudinal wall stress. Arteries are surrounded by a fibrous adventitia which is attached to tissues surrounding the other organs of the body. These organs will exert forces on the arteries via the adventitia which can be referred to as ‘tethering forces’. The main determinant of structure within the body is the skeleton. As the skeleton grows the arteries will be subject to stretching which will increase the longitudinal stress. The response of the artery is to lengthen in order to normalise the longitudinal stress. However, where an artery is decreased in length it does not shorten to normalise the longitudinal stress and the result is a tortuous artery (Wagenseil and Mecham 2009).
5.2 Arterial System Growth and Adaptability
This section examines the role of forces in the growth and remodelling of arteries from the embryo through birth into old age. It is noted that there are gaps in the evidence base and the literature comes from several different mammals including chick, sheep and human. Further detail is provided in review articles (Langille 1996; Pries et al. 2005; Humphrey 2008; Hayashi and Naiki 2009; Wagenseil and Mecham 2009). This section is mostly not concerned with disease. The underpinning process governing growth and remodelling is termed ‘mechanotransduction’ and is described in detail in Sects. 5.3 and 5.4.
5.2.1 Embryogenesis
Following fertilisation of the mother’s egg with the father’s sperm the offspring begins to develop. The process starts with a single cell which quickly subdivides into many cells. During the initial few weeks of this development the offspring is called an ‘embryo’ and the formation of the embryo is called ‘embryogenesis’. Over time, a few days and weeks, organs form and a vascular system is laid down. The initial laying down of the vascular system, though fascinating, does not concern us. However once the earliest vessels are formed then these grow and develop into the vascular system; arteries, veins, capillaries etc. These immature simple vessels consist of a tube of endothelial cells without any surrounding smooth muscle, very similar to capillaries. The rudimentary heart pumps blood along the early vascular system and as the flow rate increases the arteries increase in diameter. Studies on the chick embryo (le Noble et al. 2005) estimated ‘reduced-velocity’ which is the ratio of blood velocity to diameter, as an index which was easy to calculate and proportional to WSS. They showed that reduced-velocity was similar for embryonic arteries greater than 40 μm in diameter (Fig. 5.10).
Fig. 5.10
Reduced-velocity (a measure of wall shear stress) versus diameter in embryonic arteries of the chick. Reduced-velocity is constant for arteries from 60–160 microns indicating the presence of a control mechanism. From; le Noble F, Fleury V, Pries A, Corvol P, Eichmann A, Reneman RS; Control of arterial branching morphogenesis in embryogenesis: go with the flow; Cardiovasc Res. 2005;65(3);619–28; by permission of the European Society of Cardiology
Even at this extremely early stage in the development of the arterial system 2 things emerge. First the constancy of reduced-velocity with diameter suggests a control mechanism in which diameter is adjusted to maintain reduced-velocity (i.e. shear stress) within a narrow bound; secondly that the endothelium (in the absence of any other cell types) is responsible for this control.
5.2.2 Fetal Growth
By the end of 8 weeks most of the major organs in the human embryo are in place and the remaining 32 weeks in the womb are spent growing. The embryo is now referred to as a fetus. Over this period the fetus grows from 3 g to 3 kg in weight, and the arterial system also grows. The diameter of the aortic root (where the aorta emerges from the left ventricle) is 1 mm at 11 weeks increasing linearly with time to 9 mm at 40 weeks (Cartier et al. 1987; Achiron et al. 1998; Haak et al. 2002). There is very little literature on WSS in the fetus. Struijk et al. (2005) used ultrasound imaging in the descending aorta in fetuses with gestational age from 18 to 39 weeks, demonstrating that there was no change in mean WSS which had an average value of 2.2 Pa.
During the period from 8 weeks to birth the development of the arterial system can be described as follows; as the fetus grows, the cardiac output increases and the blood pressure increases. The arteries will increase in diameter in an attempt to maintain WSS at a constant value. The increase in pressure will result in an increase in wall thickness in an attempt to maintain circumferential stress at a constant value. This increase in thickness is associated with an increase in the number of lamellar units. The general enlargement of the skeleton and organs gives rise to an increase in longitudinal stress and the arteries will lengthen in an attempt to maintain longitudinal stress at a constant value.
5.2.3 Birth
The fetus is supplied with oxygen from the mother via the placenta and the umbilical cord. The left and right sides of the heart are connected via a hole (the foramen ovale) between the left and right atria, and the pulmonary and systemic circulations are connected via a bridging vessel (the ductus arteriosis) which links the pulmonary artery and the aorta. After birth the umbilical supply is cut and there are a number of plumbing changes which occur. The foramen ovale closes separating the left and right sides of the heart and there is closure of the ductus arteriosis separating the pulmonary and systemic circulations. The loss of the umbilical supply leads to an increase in pressure in the aorta and the reduction in pulmonary resistance leads to a decrease in pressure in the pulmonary artery. Before birth the aorta and pulmonary artery carry similar flow rate, have similar diameter and similar wall thickness. After birth the flow rate and diameter remain similar. However, there is increase in wall thickness of the aorta and decrease in wall thickness in the pulmonary artery. These changes in wall thickness are consistent with the renormalisation of circumferential stress following changes in pressure immediately following birth (Leung et al. 1977).
5.2.4 Childhood
Human life post-birth may be divided into two phases; birth to adulthood (childhood) at around 20 years when height no longer increases, then 20 years to death at age up to around 100 years.
Stay updated, free articles. Join our Telegram channel
Full access? Get Clinical Tree
