Introduction to vascular arterial haemodynamics
Understanding the haemodynamics of the arterial system is fundamental to vascular surgery. This involves a knowledge of the laws of physics that govern flow, pressure and resistance and understanding the flow characteristics of the complex fluid of blood. Included are references to the importance of capacitance, wall compliance, shear stresses, turbulence and the forces exerted in a pressurised system.
Understanding the physiology and physics of blood flow is aided by the recognition of the similarity between electrical circuit theory and haemodynamics. When considering fluid dynamics instead of:
where V is the voltage, I is the current, and R is the electrical resistance. This formula maybe substituted by:
with P the pressure, Q the volume flow rate, and R the flow resistance.
Resistors in series and parallel govern degrees of ischaemia, and the behaviour of blood flow and contribution of collaterals.
The great vessels, like the aorta, are principally composed of collagen and elastin fibres. This allows them to behave as capacitors and store some of the energy in systole as potential energy to be released in diastole.
The physics of arterial diseases
Key arterial disease types and scenarios include:
◆ Aneurysms: leading to rupture.
◆ Occlusive disease: leading to ischaemia and necrosis.
◆ Arterial dissection.
◆ Existence and development of collateral flow.
◆ Trauma: bleeding and flow disturbance.
◆ Iatrogenic changes and consequences: false aneurysm, hyperperfusion, steal, endoleaks, competing flows and watershed ischaemia, stenosis from intimal hyperplasia, and sudden occlusions.
Aneurysms are dilations of arteries including abdominal aortic aneurysms (AAA) and thoracic aortic aneurysms (TAA), and also cerebral, visceral, and peripheral aneurysms. When the size of an aneurysm increases, a point is reached where there is a significant risk of rupture. Although there are statistically-derived guidelines for some aneurysms there are no absolute sizes at which this occurs. Physics can help predict the ‘at risk aneurysm’ and provide some rationale for intervention.
Laplace’s law of wall tension
Laplace’s law of wall tension relates the tension in a vessel wall with the pressure that an elastic tube can apply to the lumen. In Fig. 1.13.1, r is the inner radius of the artery, w represents the thickness of the arterial wall that is usually assumed to be small relative to r, P is the inward pressure force due to the elastic nature of the artery, and T is tensional stress within the vessel wall, where the tensional stress points in a direction that is tangential to the vessel wall. Due to mass conservation (in a non-biological system) the wall thins as the vessel expands.
Laplace’s law is given by the eqn :
The inward pressure exerted by the vessel wall on the blood is directly proportional to the tensional stress in the wall and inversely proportional to the radius of the wall.
Large thin-walled vessels are low pressure vessels. Increasing the pressure distends the vessel and the volume, which is a characteristic property of veins. For arteries to maintain pressure, the width of the wall must be greater. So large veins are thin walled and arteries are thick walled.
An artery acts like a long cylindrical party balloon. When one attempts to blow up such a balloon it is quite difficult to do this initially. However, once the balloon reaches a particular radius, it usually becomes much easier to expand the balloon. At some point less pressure is required to increase the size of the balloon, since the radial component of the tensional stress decreases as the diameter increases. This phenomenon is known as instability. When this happens with an aneurysm the relatively constant blood pressure will keep on increasing the size of the aneurysm. Intermittent rises in systolic pressure, such as with exercise and physical work, will act as a puff on a balloon.
The critical radius for the onset of the instability, rc, is approximately twice the initial radius of the artery, ro.
For men, the median diameter of the aorta is 23 mm and aortic rupture is rare when less than 55 mm in diameter, which is consistent with recent clinical data.1,2 The ratio of the diameters is more important than the absolute diameter and this should be taken into account when assessing aneurysms in the smaller diameter vessels of women. Thus, we can use haemodynamics to predict whether action or inaction is needed considering the size of aneurysm versus growth rate and rupture risk. When a man’s normal aortic diameter is 23 mm and a woman’s is only 18 mm what are the relative predictive AAA instability diameters? Using the diameter ratio, for a man it is 46 mm and for a woman it is 36 mm.
These are the predicted instability diameters, not the diameters of rupture. Intervention is determined by the clinical ratio of iatrogenic risk versus natural history. Like any balloon, the physical quality and properties of the wall materials determine the limits of expansion. Practically, median and peak systolic blood pressure control would be expected to reduce the rate of aneurysm expansion. The mass conservation principle determines that the thickness of the wall decreases with expansion and suggests, given the same materials, that a thinner-walled variant may be weaker and more susceptible to expansion. Possibly, the best hint and adjunct guide to risk of rupture at smaller sizes is patient age with increased risk in younger patients due to the wall properties and the risk being the reciprocal of the age of instability onset.
The resistance effect of an arterial stenosis or constriction is related to flow rate, pressure gradient, percentage occlusion of the lumen, artery size, and demand. The outcomes of occlusive disease are most adverse when acute, and there is no anatomical or developed collateral flow. Understanding collateral flow is fundamental to clinical practice. Symptoms may be mild or unnoticed in progressive disease and a good collateral flow. This difference invokes the physics of resistors in series and parallel flows. Two stenoses in one arterial segment, such as the superficial femoral artery (SFA) are additive, but may have less adverse effect than the additive effect of one stenosis in each of two separate segments, e.g. with one in the SFA and the other in the iliac, or popliteal, or tibial, or in the related collateral segment in this case the profunda femoris—even though the sum of the resistors is the same. A resistor that affects the collateral flow acts to affect two segments not one. If a limb is ischaemic it is clinically practical to look for the second lesion. A complete occlusion cannot be made worse unless another segment is breached. Hence, angioplasty of a patent stenosis may be more risky than angioplasty attempts on a complete occlusion unless complications of embolization or extended dissection occur. A lesion that takes out primary and collateral flow segments at once is more adverse, e.g. complete occlusion of the common femoral artery versus complete occlusion of the proximal left subclavian artery.
The formula for resistors in parallel is
and for resistors in series
Suppose that you have a Newtonian fluid flowing in a steady, non-pulsatile manner, down a cylindrical non-elastic pipe of length, L, and radius, r. If the pipe is long enough, the flow will develop a parabolic velocity profile, which is generally called a Poiseuille flow profile (Fig. 1.13.2).
The volumetric flow rate (Q) for Poiseuille flow, is given by
where p1 – p2 is the pressure difference between the two ends of the tube and µ is the viscosity of the fluid. The equation shows the flow is driven by the pressure gradient in the tube.
Note the flow is also related to length. Patency, such as in femoro-popliteal synthetic conduits, may be as much, if not more, related to length of conduit as it is to angulation across bend points depending on the haematological factors depositing thrombus. This may also partly explain better patency in shorter bypass grafts.
For a fluid that has no viscosity,
where p is the pressure, ρ the mass density of the liquid, υ the speed of the fluid, g the gravitational acceleration, and y the height. The Bernoulli equation states that the pressure plus the kinetic energy per unit volume, ρ(υ2/2), plus the potential energy per unit volume, ρgy, is a constant at any point along the blood vessel. So for a constant height, an increase in flow speed implies a decrease in pressure at that location.
Eqn  provides an intuitive understanding of the physics of the circulation. For example, narrowing a vessel increases the speed and this is used to estimate blood pressure using an external cuff. Typically, one places a sleeve or an external cuff around the upper arm because it is at approximately the same level of the heart and so the pressure will not be affected by any difference in height. To measure the systolic pressure, the cuff pressure is increased until all blood flow ceases: from eqn  we know that this ‘cut-off’ pressure is the maximum pressure in the artery. The pressure in the external cuff is then decreased until the flow is laminar and there is no sound so we then know that the pressure will be at the minimum and an estimate of diastolic pressure.
Practically, a stenosis may allow maximum flow when the demand beyond is low and become significant with increased flow demand. The Ankle:Brachial Index (ABI) is the ratio of blood pressure at the ankle over the blood pressure in the arm, and is used to indicate stenosis (peripheral vascular disease). At rest, an ABI value less than normal indicates a stenosis affecting the lower limb. Based on eqn (2), if the flow is restricted and demand increases (during exercise in claudication) the pressure will fall, because the peripheral resistance goes down as the muscular arteries dilate and the flow cannot increase because of the stenosis. The expected response with exercise is a fall in ABI. If the ABI does not fall it suggests the collateral circulation (parallel flow) opens up in response to exercise. However, if the vessels in the leg are less compressible or incompressible (hardened arteries) then the test is unreliable.
The arterial system has two sources of potential energy to drive the blood forward. The first is blood pressure, which is transformed into kinetic energy of flow during the period between systole and diastole, and the second is stored energy in the wall of the artery—the capacitance. The effect of height as a potential energy is evident, for example, with elevating and lowering ischaemic limbs, and lowering the head after syncope. The effect of height is much more evident in the venous system because of the low average pressure. The venous system uses valves to reduce the height of each inter-valve segment, muscle contraction forces blood up through veins from one chamber to the next. The arterial system is much more dependent on stored potential energy, especially for heart perfusion during diastole, and potential energy is stored in the great arteries by controlling resistance using muscular arteries. The venous system has little or no resistance system in health, and responds to demand with every muscle in the body acting as a heart. Imagine the instantaneous venous return in a sprinter emerging from the starting blocks. If every muscle is a ‘heart’ there must be multiple valves (there are no valves in the central veins that are not encased in muscle). Flow is almost unlimited in the venous system, which is essentially open-ended and they need only be thin walled. When resistance occurs and flow is limited after deep venous thrombosis (DVT), and the deep valves fail the effects are clearly evident in leg ulceration. The dynamic variation between arteries and veins is again defined by eqn .
In its simplest form, the mass conservation equation defines the relationship between flow speed (υ), and area at the proximal (A1) and distal (A2) ends of the artery:
We can see from eqn  that if an artery becomes narrower, i.e. A2 becomes smaller, then the flow speed, υ2, increases. This occurs because the mass flow is conserved and so if the tube becomes narrower then the velocity has to increase.
If in a stenosis (Fig. 1.13.2b) steady-state Newtonian blood flow is assumed then the pressure in a compromised blood vessel can be calculated by combining Bernoulli’s equation (eqn ) and the mass conservation (eqn ) to obtain:
Since A2 < A1 the energy term, which is the last term of eqn  becomes negative so then the blood pressure is lower at the stenotic section of the blood vessel (the constriction). The lower pressure at the stenosis makes the blood vessel with a stenosis more prone to collapse if an external pressure were applied to the blood vessel. Mass conservation confirms that the velocity is higher at the stenosis due to a smaller area. This increased flow velocity can cause turbulence which accentuates the pressure drop.
Young’s modulus and pulsatile flow
Young’s modulus is a measure of how easy it is to stretch and compress a material, and is defined as the ratio of stress to strain. Blood flows through arteries in a pulsatile fashion. Arteries are semi-elastic tubes and the arteries expand and contract as the pulse of blood flows along the artery. The speed at which blood flows along an artery is determined by the speed that a pulse of fluid can travel along an elastic tube. This wave speed, c, is given, approximately, by the Moen–Korteweg formula:
where E is Young’s modulus for the wall of the artery, w is the thickness of the artery, D is the inner diameter of the artery and ρ is the density of blood. The elastic arteries stiffen with age, and explain the flow changes that occur with ageing and for progressive arterial disease. Hence, Young’s modulus increases with age, and as a consequence increases the speed of pulsatile flow within the arterial system. The loss of compliance limits the pulse rate because the valve cannot close until the stroke volume has flowed on. Young people respond to effort with increase in pulse rate and older people with a rise in pulse pressure.
Stress sites and arterial damage leading to atherosclerosis
Vascular clinicians are familiar with the shearing force injury of high velocity impact when mobile arterial segments move on the junction with more fixed segments, e.g. the arch on the descending aorta. What of subtle persistent long-term shear stresses and the relationship with the greatest risk factor for arterial disease—age? Typical sites for stenoses and occlusive atheroma plaques to develop include bifurcations, apices of curves, moving-static junctions, and tethered shear stress points like the adductor canal. The sites are predictable, known, and coincide with stress points
Atheroma is an arterial lesion. Occlusive diseases of the arteries occur with ageing, and obviously age is the greatest risk factor. It is not seen in children and only seen in veins subject to long-term pulsatile pressure when they are said to be ‘arterialized’, for example, when a vein is used for an arterial bypass or for a dialysis fistula. Pressure and pulsatility are the forces involved. Persistent raised blood pressure above the norm causes progressive wall damage, as well as increased base load on the heart. With age there is degeneration of the wall of the artery and loss of compliance. Pulse pressure and peak systolic pressures rise because of the loss of compliance. Peaks of pressure occur with exertion and acute damage may occur at such times. Age will eventually affect all, but some are more genetically predisposed to arterial lesions, and other risk factors such as poor diet and smoking accelerate any genetic predisposition.
Shear stress on an arterial wall, τw, due to Poiseuille fluid flow is given by the formula
where r is the radius of the artery and Q is the volume flow rate of blood through the artery. From this formula it can be seen that shear stress increases with the increase of blood flow through the artery and tends to increase as the artery becomes smaller in diameter—provided that the volume flow rate and the viscosity are approximately constant.
Atherosclerotic lesions form partly as a result of healing-injury cycles at specific areas where low and oscillatory wall shear stresses occur. High risk plaques have lipid cores, thin and inflamed fibrous caps and excessive expansive remodelling.3,4,5,6,7 High wall stress may rupture the established plaques, especially for thinner caps.4 Plaque rupture and intraplaque haemorrhage are recognized causes of cardiac and carotid related cerebral events.
Collateral circulation, competing flows, and arterial dissection
Much of normal flow occurs in parallel, for example, the collateral circulation in each segment of the body. Another good example is the carotid and vertebral systems combining to form the cerebral circulation. In parallel circulations, the pressure at the separation of the two systems is theoretically the same for each, and the pressure at the re-union is also the same for each. The proportion of flow in the two systems is determined by the resistance of each system. This works well to direct or redirect the flow to the target tissues. The body may select priorities for flow, for example, the brain and heart in shock because the branches of the great vessels and arteries to the tissues are resistance vessels.
Parallel circuits also provide alternative channels should the dynamics change due to injury or disease. Not all parallel circuits are beneficial. Detrimental competing flows may occur with artificial created channels, for example, aorto-bifemoral bypass, when one iliac system is normal and the other occluded. The competing flows on the normal side predispose for either that limb of the graft or part of the iliac system on that side to occlude. In bypass surgery treatment consideration of parallel flow through collaterals should be taken into account.
In a dissected aorta, the outflow from the false lumen is met with greater resistance than the outflow from the true lumen because there is less run-off. The pressure is higher in the false lumen at any time in the cardiac cycle other than peak systole. Furthermore the flows compete at re-entry sites and where the intima has been torn off at the origin of a branch vessel and a hole is left in the membrane at that point. Fig. 1.13.3 shows the trace from true and false lumens of a dissected aorta. The systolic pressure is the same in each lumen at 138 mmHg and the branch vessels are perfused from both true and false lumens. The competing flows and higher median false lumen pressure merely prevent contrast flow imaging. If the intimal connection is intact as an intimal tube then it can be compressed and flow affected as per the physics above. The diastolic is higher in the false lumen at 93 mmHg compared with 82 mmHg in the true lumen. The mean pressure in the false lumen is higher at 109 mmHg than the true lumen where the mean is 91 mmHg. This means that the false lumen is invariably the larger of the two and is more likely to dilate. Flow of contrast injected into the true lumen is not seen to flow out to the false lumen through the holes in the membrane unless the pressure of the injection and the pressure of the lumen together exceed the pressure of the false lumen.
Endovascular forces on aortic prostheses
The performance of endovascular grafts was found to be different to open repair with a sewn replacement of the artery because of unsuspected influences that relate to sustained physical forces.8,9 The diameters of the grafts used for the same AAA differ markedly between the open and endograft methods. The common diameters used for tube replacement surgically of infrarenal AAA are 18 or 20 mm. The most common diameter for an endograft is 26 or 28 mm and 30+ mm is not uncommon. This discrepancy is due to the different types of attachment and haemostasis. In open surgery, there are sutures through the graft and the full thickness of the aortic wall. The aortic diameter at that point is permanently fixed to the diameter of the graft in its pressurized state. Endovascular grafts do not bind the adventitia to the prosthesis—they merely attach and a residual radial force is required for seal. The oversize allowance must accommodate elasticity and compliance, while maintaining the seal between pulsations for the whole of the length of the sealing zone. Endoleaks, which are blood flow between graft and the AAA wall, can also cause problems in AAA, such as elevated sac pressure and high stresses, which may lead to rupture.10,11 The haemodynamics applying to an endovascular graft with an endoleak are similar to a dissected aorta—the diastolic phase is important.
A critical issue in vascular intervention is the durability of endoluminal grafts. Unfortunately, haemodynamic forces can displace a graft and interrupt the seal between the graft and the aneurysm neck. It is important therefore to have an understanding of the possible forces that may be exerted on an endograft. If we assume steady-state, i.e. non-pulsatile, flow, we can obtain an equation for the resultant force for a horizontal graft by considering the momentum equation, together with mass and energy conservation.12,13,14
The resultant force on a horizontal graft is the force due to pressure and flow of blood through the graft.
A mistaken clinical impression is that the forces on a thoracic endograft should be greater than those on an abdominal endograft since the flow and diameter of the thoracic aorta are greater. However, because the diameter of the cylindrical graft changes little (A1 = A2) the downward displacement force is small as the resistance in the graft is low—except on the curve. The resistance of any graft that extends into the iliac vessels is much greater because of the significant change in diameter (A1 >> A2) and the graft acts like a sea anchor or windsock14 with a larger drag force. An aorto-uni-iliac device affords greater resistance than a bifurcated graft for the same reason, and detachment at the neck and migration is a common problem due to high displacement forces. A symmetric bifurcated graft has a horizontal restraint force that is strongly dependent on inlet area, the pressure, and on the bifurcation angle (especially > 15°). The inlet flow rate has negligible effect on the horizontal restraint force.14,15 Pressure is more important than flow rate for migration and displacement. For a curved graft both the pressure and velocity components add together to produce a greater total force on the graft as shown in Fig. 1.13.4.
Properties of blood that influence flow
The properties that affect viscosity, turbulence, and behaviour of a non-Newtonian fluid act to give blood its unique flow characteristics.
Newtonian fluid and non-Newtonian fluid
In a Newtonian fluid, the frictional stress is proportional to the rate at which the speed changes as a function of distance, where µ is the viscosity and dυ/dr corresponds to the shear rate (eqn ). In an artery, r is the radial direction. To a reasonable approximation, one can assume that blood is a Newtonian fluid only for flow along the major arteries.
Non-Newtonian fluids have a viscosity that depends on the strain rate. A shear thinning fluid is a fluid that changes from ‘thick’ to ‘thin’ when force is applied to the fluid. Shear thinning behaviour of blood is observed at a haematocrit value of 45%. For high shear rates, which may occur in the large arteries of the body, the viscosity of blood is about four times that of water (where the viscosity of water is approximately one centipoise (cP)). However, for lower shear rates, the viscosity of blood can be over 100 times that of water.16 Despite this, the viscosity of blood is kept approximately constant throughout the body. The viscosity of blood is dependent on the haematocrit and the haematocrit level depends on blood vessel diameter.17 If the haematocrit decreases, then the blood viscosity decreases. As the blood vessel decreases in diameter, the haematocrit level also decreases, because the blood cells tend to move away from the vessel walls and travel where the flow velocity is a maximum (known as the Fahraeus effect).18,19 Understanding these properties affects the thinking of blood flow in small vessels and shear stress between blood and vessel walls in atherogenesis.
The Reynolds’ number provides an indication of how blood is flowing in an artery and is given by:
Where ρ the blood density, υ is the speed of the flow, D is the diameter of the blood vessel and μ the blood viscosity. For an artery, the flow tends to change from laminar to turbulent at a Reynolds’ number greater than 2500. Blood may have properties that allow laminar flow at higher numbers and this requires further research. Flow in arteries are typically laminar, but can be turbulent in a stenosis. Turbulent flow is less efficient relative to laminar flow and more energy or a greater pressure is required to drive turbulent flow compared with laminar flow.
A bruit is audible turbulent flow at high velocity with energy transformed to noise—inefficient flow that maybe disruptive as in a carotid stenosis or may be innocent with high flow murmurs in children or associated with anaemia. Blood still needs to be able to flow fast in order to deliver its load at a cardiac output of up to 30 L/min in an athlete. Turbulent flow is less efficient relative to laminar flow and more energy or a greater pressure is required to drive turbulent flow compared with laminar flow.
Knowledge of the physics of the vascular system in health and disease will influence vascular management. Understanding the vessels and the flow properties of the blood within them should assist vascular intervention, planning, and decision making.
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