Introduction

Vascular interventions have developed rapidly since the first aortic replacement with Dacron by Dubois in 1952. Understanding vascular haemodynamics and the biological response to implanted materials is essential for vascular surgeons and scientists developing new interventional technologies.^1,2

This chapter will summarise and discuss the following laws, equations and phenomena to give a basic understanding of the haemodynamic principles of the conduits and fluids with which we work:

• Laplace’s law of wall tension
• Newtonian Fluid
• Non-Newtonian fluid
• Poiseuille Flow
• Bernoulli’s equation
• Young’s modulus and pulsatile flow
• Mass conversation
• Reynolds’ number: laminar and turbulent flow
• Shear stress and pressure
• Forces on graft systems
• Computational modelling

For those who understand electrical circuit theory, there is much similarity with haemodynamics. Understanding the physiology and physics of blood flow is aided by the use of that recognition. 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, and hence degree of ischaemia of limbs and organs.

The great vessels, like the aorta, are with-out muscle and their walls are composed of collagen and elastin fibres. This allows them to behave as capacitors and store some of the energy in systole to be released to power flow in diastole, that is so important into vessels such as the coronary artery. The elastic arteries stiffen with age and explain the flow changes that occur with ageing and for progressive arterial disease due to this most important of all the risk factors.

Laplace’s Law of Wall Tension

Laplace’s law relates the tension in an arterial or venous wall with the pressure that the elastic tube can apply to material inside the tube. To assist in understanding this law we consider Figure 8.1. In this figure, w, represents the thickness of the arterial wall, r is the inner radius of the artery, P the inward pressure force due to the elastic nature of the artery and T is tensional stress within the wall of the vessel, where the tensional stress points in a direction that is tangential to the vessel wall. Due to mass conservation the wall thins as the vessel expands.

FIGURE 8.1

Cross section of an artery showing the various physical components that make up Laplace’s law.

The formula for Laplace’s law is given by the Eq.:

where it is usually assumed that the wall thickness, w is small relative to r. This law tells us that the inward pressure that is 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. Thus the smaller the vessel the larger the pressure it can apply on the blood.

Large thin-walled vessels are low pressure vessels. Increasing the pressure distends the vessel and increases the vessel volume which is a characteristic property of veins. For arteries to maintain pressure, the width of the wall must obviously be greater, so large veins are thin-walled and arteries are thick-walled.

One consequence of this behaviour is that, to a certain extent, an artery acts like a long cylindrical party balloon. When one attempts to blow up such a balloon, it is quite difficult to do at the first blow, however once the balloon reaches a particular radius, it usually becomes much easier to expand the balloon. That is you require less pressure to increase the size of the balloon. This phenomenon is known as instability. If this hap-pens to an artery, then we are dealing with an aneurysm and the relatively constant blood pressure will keep on increasing the size of the aneurysm.

The radius of the artery at which this instability occurs is difficult to compute accurately, but some fairly general arguments suggest that the following formula is a good guide: where r_c is the critical radius for the onset

of the instability and r₀ is the initial radius of the artery. The median diameter of the aorta is 23 mm and the thus aortic rupture is very rare when less than 50 mm in diameter, which is consistent with recent clinical data.^3,4 This guide also directs us to consider that the ratio of the diameters is probably more important than the absolute diameter and this should be taken into account when assessing aneurysms in the smaller diameter vessels of women. How arterial wall instability arises is illustrated in Figure 8.2, where in Figure 8.2(a) we show the stress structure within a small artery. Here the tensile stresses have a component in the radial direction, where the letter T labels this component. In Figure 8.2(b) the aneurysm/balloon has become very large, such that over a small segment of the wall the artery has hardly any curvature. This is an extreme case, but it does show that there is now no radial component to the tensile stresses. In such a case, the aneurysm can expand freely for just about any internal arterial pressure.

FIGURE 8.2

Cross sections of a small artery (a) and a very large artery (b) showing the stress distribution within the artery.

Newtonian Fluid

When we wish to describe the behaviour of a fluid it is necessary to know something about the frictional properties of the fluid. Consider the schematic depiction of a fluid shown in Figure 8.3. In this figure, fluid is flowing from left to right along the x direction. For purposes of illustration, we assume that the speed of the fluid, u, is increasing with increasing height (i.e., increasing y). This means that elements of fluid are sliding past each other and so generate some frictional stress τ. 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 (Eq (5)). The du/dy in equation (5) corresponds to the shear rate. To a reasonable approximation, one can assume that blood is a Newtonian fluid, at least for flow along the major arteries.

FIGURE 8.3

Elements of fluid slide past each other and generate a frictional shear stress.

Non-Newtonian Fluid

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. Examples of such fluids are shampoos and paints. This behaviour usually occurs, because, at rest, a shear thinning fluid typically has a tangled molecular structure, which makes the fluid relatively viscous. When force is applied, the molecules become ordered, the fluid viscosity decreases and the fluid begins to flow more easily. In Figure 8.4 we show the experimentally determined shear thinning behaviour of blood, where the hematocrit value for the blood is 45%. These data show that 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 one hundred times that of water.

FIGURE 8.4

Blood viscosity as a function of shear rate for 0% and 45% hematocrit.

This change in viscosity is mostly due to the collective behaviour of red blood cells. At low shear rates, red blood cells form aggregates where they stack one upon another, somewhat like a cylindrical pile of coins. These “stacks” of red blood cells are known as “ rouleaux ” (Figure 8.5). When the shear rate increases, these aggregates of blood cells are broken down and the blood viscosity decreases. For high shear rates, the blood cells tend to become elongated and line up with the flow of the liquid. This also tends to decrease the viscosity of the blood.

FIGURE 8.5

Rouleaux blood cell network.

Given that the viscosity of blood increases with decreasing shear, one would think that the viscosity of blood within the body should increase as blood travels from the arteries through the arterioles and into the capillaries. This is because the shear rate and velocity of the blood decreases as the blood travels from the arteries through to the capillaries. The viscosity of blood, however, may be approximately constant throughout much of the body. This effect arises due to separate physical flow phenomena:

First, the viscosity of blood is dependent on the hematocrit. If the hematocrit decreases then the blood viscosity decreases. For exam-ple in Figure 8.4 we show the viscosity of blood as a function of shear rate for 45% and 0% hematocrit. For 0% hematocrit line the viscosity of the blood is constant and has a value of approximately 1.6 cP.

Second, the hematocrit level is dependent on the diameter of the blood vessel. As the blood vessel decreases in diameter, the hematocrit level also decreases (Figure 8.6). This effect occurs because the blood cells tend to move away from the vessel walls and travel where the flow velocity is a maximum. This behaviour is known as the Fahraeus Effect and it has been shown to occur in tubes with a diameter as small as 29 μm.^6,7 Given that a blood cell has a diameter of around 8 μm it is possible that the Fahraeus Effect may occur in tubes with diameters less than 29 μm.

FIGURE 8.6

Hematocrit as a function of tube diameter. The initial hematocrit value for each line is shown in the inset box.

The combination of these two effects implies that the viscosity of blood is approximately constant throughout the body. Understanding these properties affects the thinking of shear stress between blood and vessel walls — or more relevantly between blood and atheroma.

Poiseuille Flow

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 a. If the pipe is long enough, the flow will develop a parabolic velocity profile, which is generally called a Poiseuille flow profile (Figure 8.7). The flow takes its name from Jean Louis Poiseuille, a physician with training in physics and mathematics, who first described the flow structure in 1846.

FIGURE 8.7

Parabolic velocity profile for fully developed Poiseuille flow.

The volumetric flow rate (q) for Poiseuille flow, i.e., the volume of fluid flowing along the tube per unit time is given by the formula

where p₁ – p₂ is the pressure difference between the two ends of the tube and µ is the viscosity of the fluid.

The physics of the flow is nicely described by this equation. That is, flow is driven by the pressure gradient in the tube or conversely, when there is flow in a tube then you must have a pressure gradient to drive the flow.

Prostheses are subject to the intermit-tent forces of pulsation and flow. The large elastic vessels are capacitors and provide on-flow in diastole and the muscular peripheral vessels maintain pressure by altering resistance mediated via physiological feedback. Current prostheses are not able to do this and have to withstand the forces.

Note also the parameter of length. Flow is therefore also related to length. Patency, such as in femoro-popliteal synthetic conduits, maybe 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.

Bernoulli’s Equation

Johann Bernoulli (1667–1748) was a professor in Basel and taught physics, anatomy and physiology and his understanding lies at the heart of vascular physics and relates pressure to motion and energy. For a fluid that has no viscosity, one can write

where p is the pressure, ρ the mass density of the liquid, u the speed of the fluid, g the gravitational acceleration, and y the height. In other words, the Bernoulli equation states that the pressure plus the kinetic energy per unit volume, ρ₂^u2, 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, while for constant flow speed, an increase in height implies a decrease in pressure.

It should be understood that Eq. (7) is an approximation, as it ignores the loss of energy due to shearing friction between the flowing blood and the walls of the artery. Even so, it does provide us with an intuitive understanding of the physics of the arterial/venous system. For example, suppose we wish to measure the blood pressure of a person. Typically one places a sleeve or an external cuff around the upper arm. The upper arm is chosen 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 Eq. (7) 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 a maximum. We then know that the pressure will be a minimum and this is the diastolic pressure in the artery.

In practice, the arterial system has two sources of potential energy to drive the blood forward. The first is blood pressure and this 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 – its capacitance. Consider what might happen when the kinetic energy meets a resistive obstacle – some energy is dissipated as heat as with circuit theory and some is stored for use in diastole for onward flow in the period of heart filling by the elasticity of the great vessels acting as a capacitor. However, some energy is used up as a water hammer. The repetitive alterations in forward pressure and resistive back-pressure with pulsatile flow in a physiologically responding, pressurized system sets up the potential for the water hammer. The injury and healing cycle effect of these water hammers on atherogenesis and aneurysm behaviour at stress points has yet to be fully determined.

Young’s Modulus and Pulsatile Flow

Blood flows through the 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, c, at which blood flows along an artery is determined by the speed that a pulse of fluid can travel along an elastic tube. This speed is given, approximately, by the Moen-Korteweg formula:

where E is Young’s modulus for the wall of the artery, h is the thickness of the artery, d is the inner diameter of the artery and ρ is the density of blood. A schematic depiction of how a pulsatile wave propagates along an artery is given in Figure 8.8.

FIGURE 8.8

An exaggerated, schematic view of blood flow in an artery.

As can be seen from Eq. (8), the speed at which blood travels along an artery is partially dependent on the Young’s Modulus of the arterial wall. To illustrate the definition of Young’s Modulus it is useful to consider Figure 8.9, where a block of material is being stretched due to an applied force on one end of the block.

FIGURE 8.9

A block of material with a length, L, and side area A is subject to a force F. The applied force stretches the block a distance δL.

The block has a natural length denoted by L, when a force F is applied to one side of the block then the length of the block increases by δ L. This change in length is known as a strain, ε, and it is defined by the equation

The stress that the force applies to the block of material has the definition

Young’s Modulus is defined as the stress over the strain, i.e.,

Young’s modulus is a measure of how easy it is to stretch and compress a material. Thomas Young (1773 – 1829) was a medical physician who made significant contributions to fields of Physics (through his experiments which demonstrated the wave-like nature of light), linguistics (via his identification of the Rosetta Stone), medicine (with his studies of blood flow), and structural mechanics (e.g., Young’s Modulus). He was well aware of the elastic nature of arteries, but, somewhat ironically, does not appear to have used Young’s Modulus to describe their properties.

One consequence of aging is increasing stiffness in the arteries. This means that the Young’s modulus increases and this, as a consequence of Eq. (8), increases the speed of pulsatile flow within the arterial system.

Mass Conversion

In Figure 8.10 we view a schematic depiction of an artery that is changing in shape as one travels along the artery. The blood flows in at one end with a speed u₁. The area at the inlet of the artery is given by A₁. In its simplest form, the mass conservation equation provides us with the relationship between the quantities at the proximal and distal ends of the artery:

FIGURE 8.10

A change in the diameter of an artery leads to a change in the blood flow speed.

We can see from Eq. (12) that if an artery becomes narrower, i.e., A₂ becomes smaller, then the flow speed, u₂, increases. This occurs because the mass flow is conserved and so if the tube becomes narrower then the flow rate has to increase.

Some diseased blood vessels develop a constriction or stenosis (Figure 8.11). This narrowing of the blood vessel wall may be caused by atherosclerosis or neo-intimal hyperplasia after an intervention. If we assume steady-state, Newtonian blood flow and ignore gravity then the pressure in a compromised blood vessel with a stenosis can be calculated by combining Bernoulli’s equation (equation 7) and the mass conservation (equation 12) to obtain

FIGURE 8.11

Blood vessel with a stenosis.

Since A₂ < A₁ the energy last term of equation 14 becomes negative so then the blood pressure is lower at the stenosed section of the blood vessel (the constriction) to ensure that the sum of the pressure and energy at each point along the blood vessel remains equal. 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. Stents or drug -eluting stents may be inserted in an artery that has a stenosis to keep the artery open after the blockage has been cleared using angioplasty. Mass conservation shows that the velocity is higher at the stenosis due to a smaller area at the stenosis.

Reynold’s Number

The Reynolds’ number (Re)is a dimensionless number, which provides an indication of how blood is flowing in an artery. The Reynolds’ number is given by:

where U is the speed of the flow, D is the diameter of the blood vessel, ρ the blood density and μ the blood viscosity. For an artery, the flow tends to change from laminar to turbulent at a Reynolds’ number of approximately 2000. This number should be treated as only a representative value, since the transition from laminar to turbulent flow may occur at higher Reynolds’ numbers.

To see a representative peak value of Reynolds’ number, we consider an abdominal aorta of diameter, D = 2.5 cm = 0.025 m, peak blood flow speed U = 60 cm/s = 0.6 m/s, blood density ρ = 1 gram/cc = 1000 kg/m³ and blood viscosity μ = 0.0036 P a s. These values give Re ≈ 4,200. So, in principle, it is possible for turbulent flow to occur in the aorta during the systolic phase.

Fluid flowing in a laminar fashion is dominated by the viscosity and at a high Reynolds ’ number by its inertia. A bruit is audible chaotic flow at high velocity with energy transformed to noise – inefficient flow that maybe disruptive as in a carotid stenosis – and blood needs to be able to flow fast in order to deliver its load at a cardiac output of up to 30L/min in an athlete.

Turbulent flow is less efficient relative to laminar flow. This means that more energy or a greater pressure drop is required to drive turbulent flow compared to laminar flow. A quantitative way of measuring this inefficiency is given by the formula for energy or “head” loss for flow along a pipe

where f is the loss coefficient, L the length of the artery or appropriate subsection of an artery and g the acceleration due to gravity. For laminar flow,

while for turbulent flow

One can show that f_turb > f_lam when Re > 1200, which implies that turbulence consumes more energy relative to laminar flow. This result is represented schematically in Figure 8.12, where we have plotted the ratio f_turb/If_lam as a function of Re. Here we see that at a Reynolds’ number of around 2000, turbulent flow loses 1.5 times more energy relative to laminar flow. As Re approaches 5000 turbulent flow tends to lose 3 times as much energy as laminar flow.

FIGURE 8.12

Plot of the ratio of turbulent to laminar energy loss coefficients.

It is interesting to speculate that the particulate nature of blood and plasma composition may act to discourage the formation of turbulent flow. Each red cell, being bi-concave, could change the local interactions between the cells and the blood plasma so that the flow tends to remain laminar. The shape of the red cell then may enhance the efficiency of blood flow, in addition to increasing surface area for oxygen delivery.

Arterial Dissection, Collateral Circulation and Competing Flows

To this point we have essentially discussed flow in series. Much of the normal flow and some pathological flow occurs in parallel. For example, the collateral circulation in each segment of the body; the profunda system in the thigh, the geniculate system around the knee and the tibial systems in the leg. Another good example is the carotid and vertebral systems combining to form the cerebral circulation. In parallel circulation, 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 ongoing flow from the two systems is determined by the resistance of each system. These two therefore compete for the proportion of on-flow. 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 or the muscles during exercise. The branches of the great vessels and arteries to the tissues are resistance vessels and they have muscular walls for this purpose. The formula for resistors in parallel circuits is

where n is the number of parallel circuits.

These 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 artificially 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. Similarly, with femoro-popliteal bypass after long-standing superficial femoral artery occlusion when the profunda collateral flow has been well developed.

In aortic dissection, the outflow from the false lumen is met with greater resistance than the outflow from the true lumen. The flows compete where the intima has been torn off the origin of a branch vessel which therefore comes off the false lumen and leaves a hole in the membrane at that point. The pressure is higher in the false lumen at any time in the cardiac cycle other than peak systole.

Figure 8.13 shows the trace from true and false lumens of a dissected aorta. Note the systolic pressure is same in each lumen at 138 mmHg. The diastolic is higher in the false lumen at 93 mmHg compared to the diastolic in the true lumen of 82 mmHg. The area under the curve is the same and so the pulse wave in the false lumen is wider. The mean pressure in the false lumen is higher at 109 mmHg than the true lumen where the mean is 91 mmHg.

FIGURE 8.13

Pressure readings from the true and false lumens of a dissected abdominal aorta (courtesy of Dr John Anderson).

This means that the false lumen is almost always 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. The membrane that is the remnant of the intima oscillates as the pressure ratio between the true and false lumen changes during the cardiac cycle. This dynamic also applies for a Type 1 endoleak into the residual sac of an aortic aneurysm treated by an endovascular graft.

Shear Stress and Pressure

All vascular clinicians are familiar with the ultimate shearing force injury of high velocity impact when the mobile arch of the aorta and heart continue to move forward while the descending aorta, held by the intercostals and posterior mediastinum, is held to the vertebral bodies. What of subtle persistent long-term shear stresses and the relationship with the greatest risk factor for arterial disease – age? There are known common sites for occlusive atheromatous plaques e.g. the carotid bifurcation, aortic bifurcation, origins of branches of the aorta and coronary arteries and shear stress points such as the adductor canal.

Atheroma is an arterial lesion. Occlusive and dilating diseases of the arteries progressively 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 “ arterialised”. 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. 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, τ_w, due to Poiseuille fluid flow is given by the formula

where a 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 at specific areas where low and oscillatory endothelial shear stress occur. High risk plaques have a large lipid core, thin and inflamed fibrous cap and excessive expansive remodelling⁹. Wall shear stress may rupture the established plaque. Plaque rupture and intraplaque haemorrhage are recognized causes of cardiac events. Computational modelling of carotid bifurcations with atherosclerotic plaques that had patient-specific geometries obtained from Magnetic Resonance Imaging (MRI) scans and modelled the fluid-structure interactions have shown that stresses in the fibrous cap and around the plaque shoulders affect plaque rupture risk, with higher stress and plaque rupture risk for thinner caps.^10-13

Forces on Graft Systems

The performance of endoluminal grafts (ELG) was found to be different to open repair with a sewn replacement of the artery because of unsuspected influences, as mentioned above, that relate to sustained physical forces.¹ The openly-sewn prosthesis binds the wall of the artery to the prosthesis with a transmural suture. The artery may expand above or below the prosthesis. However, at the point of attachment the artery wall is held to the fixed diameter by the through-wall suture for as long as the suture holds. ELG’s to date do not bind the adventitia to the prosthesis – they merely attach. The ELG must continue to act to bridge the gap between normal artery above and below until, if ever, the aneurysm’s cavity shrinks right down. In open surgery, the suture is binding and the tissues around supportive. The diameters of the grafts used for the same abdominal aortic aneurysm (AAA) differ markedly between the open and ELG methods. The common diameters used for tube replacement surgically of infrarenal AAA is 18 or 20 mm. The commonest diameter for an endoluminal graft is 26 or 28 mm and 30+ mm is not uncommon. Why such a discrepancy when the surgeon judges the diameter to suitable fit? This discrepancy is due to the different types of attachment of an open graft and an endoluminal graft. With the former, there are sutures through the graft and the full thickness of the aortic wall. This means that the aortic diameter at that point is permanently fixed to the diameter of the graft in its pressurised state. The diameter of a crimped vascular graft is, by definition, the minimum internal distance between the crimps in the non-pressurised state. It is increased by approximately 10% when pressurized. With the ELG, a residual radial force is required for seal and the attachment may or may not be enhanced by latching barbs. The oversize allowance must accommodate elasticity and compliance while maintaining the seal between pulsations for the whole of the length of the sealing zone. These latching barbs sometimes cross the renal arteries but have been shown to have a minor effect of 1 % on the renal artery flow rate for 3 mm diameter artery.¹⁴

With an endoluminal graft the device must bridge a gap for an indeterminate time before the body reabsorbs the contents of the aneurysm and encases the graft in foreign body fibrous tissue support. Therefore the long term function and durability demands are different and more demanding.¹ Under-standing the forces involved is basic to design and use of new technology and the weaknesses that lead to aneurysmal disease provides a challenge.^1,15

A mistaken clinical impression is that the forces on a thoracic ELG should be greater than those on an abdominal ELG. The flow and diameter of the thoracic aorta are greater and the haemodynamic forces potentially much larger. However, because the diameter of the graft changes little, if at all, the down-ward displacement force in the thoracic ELG is small as the resistance in the graft is low – except on the aortic arch. The resistance of any graft that extends into the iliac vessels is much greater because of the significant change in diameter and high resistance within the graft acting like a windsock or sea anchor.¹⁵ an aorto uni-iliac device affords greater resistance than a bifurcated graft and detachment at the neck and migration is a common problem due to high displacement forces. In contrast with the thoracic aorta there is little drag because there is little or no change in diameter. In contrast, there is little drag on a graft in the thoracic aorta because there is little or no change in diameter along the graft. The force applied to the graft is on the curve and centrifugal forces apply. Since every action has an equal and opposite reaction (Newton’s third law), one must ask where is the reaction. The reaction is to pull the graft out from the top and the bottom almost equally. When endoluminal grafts were first used in the thorax, unexpected upward migration of the distal end emerged as the problem, especially when there was a significant curve on the graft. For the same reason, this ‘lift out ’ may also be seen from the iliacs when the graft fixation is weak because of ectasia and/or short length of distal attachment. Type 1B endoleak can be more dangerous than Type 1A if this factor is ignored.

An important issue in vascular intervention is the durability of endoluminal grafts. Such grafts are often used to protect aneurysms from the effects of arterial pressure. Unfortunately, hemodynamic forces can displace a graft and thereby, potentially, interrupt the seal between the graft and the neck of the aneurysm. It is important, therefore to have an understanding of the possible forces that may be exerted on a graft.

To illustrate the steps used in determining the forces on a graft system, via analytic equations, we consider the steady flow of blood through a bent pipe (Figure 8.14). In this figure, the proximal inlet entrance is labelled by 1 and the distal exit by 2. D₁, A₁ and D₂, A₂ are the diameters and crosssectional areas, respectively, of the graft at the points 1 and 2. The vector normals of the cross-sectional areas are, respectively, at angles of θ₁ and θ ₂ to the vertical. Similarly p and v refer to the pressures and velocities at these points. R_x and R_y are the x and y components of the restoring force. The external pressure on the graft system is denoted by p_ex.

FIGURE 8.14

The characteristic velocity, pressure, area and force vectors required to compute the restraining forces on a bent, single-tube graft system.

In our analysis, we assume steady-state, i.e., non-pulsatile, flow. We do this as it gives us a basic idea of how the system is behaving.

The first equation is the steady-state mass conservation equation, which we rewrite in the form

One should note that v₁ and v₂ are average flow speeds, where the average is taken over the areas of A₁ and A₂ respectively.

The next analysis tool at our disposal is the momentum conservation equation, which can be expressed in the form

and

where in these formulae, we have ignored the weight of the graft and the weight of blood in the graft. These terms are easily included into the equations, if required.

Energy is the final conserved quantity that we can use in our analysis. The energy conservation equation has the form:

where g is the gravitational acceleration, γ = ρg is the weight density of blood, z₁ and z₂ are the vertical heights of the proximal and distal ends of the graft, respectively, and h_L is the ‘head loss ’ in the pipe, i.e., the amount of pressure or energy that is lost due to frictional viscous effects as the fluid travels through the pipe. Head loss is usually given by the equation

where K_L is a constant, the value of which is usually dependent on the shape, length and diameter of the pipe. The coefficients α₁ and α₂ are kinetic energy correction factors that have different values depending on the type of flow. For example, for uniform flow α = 1, turbulent flow has α ≈ 1, and laminar flow gives α = 2.

By combining Eqs (21), (24) and (25), one obtains

So, by using Eqs (26) and (21), we can express p₂ and v₂ in terms of quantities at the entrance of the graft. This then allows us to compute the restraining forces on the graft system by then using Eqs (22) and (23).

Case 2 – The windsock graft

Suppose now we consider a graft in the shape of a windsock, such as in Figure 8.16.

FIGURE 8.16

An endoluminal graft in the shape of a wind-sock.

For this case, the inlet area is now larger than and the outlet area, so, by Eq. (21), the outlet flow speed is greater than the inlet flow speed as given by

$${\nu }_2=\left(\frac{A_1}{A_2}\right){\nu }_1$$

(29)

As in the previous case, the angles θ₁ and θ₂ are equal and have a value of 90° and the inlet and outlet pressures are not equal due to the frictional, shear interaction between the blood and the graft.

The restraint forces on the graft are from Eqs (22) and (23):

$$R_x=p_2{A}_2-p_1{A}_1+p{\nu }_2^2{A}_2-p{\nu }_1^2{A}_1,$$

(30)

and

$$R_y=0.$$

(31)

When you put in the appropriate numbers into Eq. (30), it is found that the dominant term in this equation is the p₁A₁ term. Many endoluminal grafts have this ‘wind sock’ shape with a distal exit area, which is smaller than the proximal, inlet area. This shape has a much larger drag force than for a cylindrical graft.

FIGURE 8.15

Cylindrical graft.

Case 4 – The symmetric bifurcated graft

Suppose that we consider a symmetric bifurcated graft, such as shown in Figure 8.18, where the two outlet distal legs of the graft are at an angle α to the horizontal, the two distal ends are equal and gravity is ignored. The proximal end of the graft is labelled by the number 1, the symmetric distal ends by 2 and 3. B y satisfying momentum conversation the horizontal restraint force is given by:

$$R_x=-p_1{A}_1-2p_2{A}_2\cos \alpha +p{\nu }_1^2{A}_1-2p{\nu }_2^2{A}_2\cos \alpha .$$

(33)

FIGURE 8.18

Symmetric bifurcated graft.

The more general, non-symmetric case with gravity is described elsewhere.¹⁵

FIGURE 8.17

Curved graft.

We also know to satisfy mass conversation that the flow in has to equal the sum of the outflows

$${\nu }_1{A}_1={\nu }_2{A}_2+{\nu }_3{A}_3,$$

(34)

and since it is a symmetric bifurcated graft then

$${\nu }_1{A}_1=2{\nu }_2{A}_2.$$

(35)

By applying Bernoulli’s equation 7 and mass conversation equation 35 then p₂ and ν₂ can be eliminated from equation 33.

This equation shows that the horizon-tal restraint force is strongly dependent on inlet area, pressure and on the bifurcation angle (especially > 15°). But the blood inlet velocity or flow rate has negligible effect on the horizontal restraint force.^16-18 Naturally, a steady-state assumption is questionable, since pulsatile flow occurs in the human body. However, it was shown experimentally that a steady-state analytical model can be used, with variable pressure and flow rate inputs, to predict forces on a symmetric, bifurcated graft in pulsatile flow with reason-able approximation within design limits.^17-18 This steady-state analytical force model is now used in the design of grafts.

Computational Modelling

Computational fluid dynamics (CFD) and finite element modeling can assist in our understanding of vascular haemodynamics. CFD uses numerical methods to discretize and mesh the geometry and algorithms to solve the equations of motion (for example, the Navier-Stokes equation) and other relevant equations. In the last several years with advances in computing the computational modeling capability has greatly improved. It is now possible to incorporate patient-specific geometry from MRI, CT or magnetic resonance angiography (MRA) data. The computational models also now include fluid-structure interactions (fluid flow and wall deformation interaction), pulsatile flow and non-Newtonian flow. Some computational model-ling of vascular haemodynamic systems include AAA grafts,^19-22 fluid-structure interactions with cere bral aneurysms,^23-25 patient-specific cerebral aneurysms with coils,^26-28 and patient-specific circle of

FIGURE 8.19

Effect of graft placement on blood flow and AAA wall at peak systole pressure level (courtesy of Professor Clement Kleinstreuer).

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. Li and Kleinstreuer³⁴ modelling analysis indicated the sac pressure caused by type II endoleaks (leakage via collateral arteries) depends on the inlet branch pressure; thus, type II endoleaks may increase sac pressure to near the systemic pressure levels, which could cause more clinical concern. Other studies have shown that intrasac pressure measurements and haemodynamic analysis of the graft-aortic wall interactions can be used to detect type II endoleaks.^35,36

Recent Developments and Future Directions

Mathematics, principles of physics and computational modelling of vascular haemodynamics have been useful in verifying or modifying intuitive engineering of endovascular stent grafts; and towards better understanding of failure modes of the cardiovascular system and its prostheses. For example, based on vascular haemodynamics analysis relating a vascular geometric ratio to the likelihood of aneurysm rupture, plaque rupture and stent graft migration. CFD modelling of vascular haemodynamics is currently being used in the design and evaluation of implanted medical devices such as grafts. It is still a complex process to create patient-specific computational models and difficult for clinicians to interpret the results. In the future, computational modelling of vascular haemodynamics may be used as a tool for patient-specific blood flow quantification relevant to clinical practice to assist intervention planning, decisionmaking and optimization.

However, for computational model-ling to be more easily translated to clinical relevance there needs to be more extensive comparisons of in vitro and in vivo clinical studies to validate the codes, with the uncertainties quantified, so they can be used with confidence. Patient-specific geometries and measured flow distribution boundary conditions should be included. There needs to be better models of arterial mechanical proper-ties during each stage of a disease state (e.g. aneurysm formation), more accurate imaging techniques that can provide better information of the wall thicknesses, details on perivascular environment and much better coupling with vascular biology, mass transport and cellular biophysics. With grafts now being used in high curvature areas it is more important to understand the forces required to keep the grafts in place to mitigate their migration.

The intersect of clinical arterial pathology, feedback systems in physiology and computational fluid dynamics leads us to potentially the most exciting time in advances for arterial disease ever. The vascular system is dynamic in its function, its response to demand, its injury and repair cycle and its aging. The arterial system is intricately designed so that each arterial division is specific for its function and the demands placed upon it for up to one hundred years. To the empirical, statistical, biochemical, genetic and molecular biology knowledge of the cardiovascular system must be added the central role of haemodynamic physics and the pathology that results from the relentless forces of blood pressure and pulse wave. Modelling of arteries opens the door to much better understanding of why atheroma occurs at the known predictable sites such as the carotid bifurcation, the origins of branch vessels of the aorta and sites of stress for example the adductor canal.

Conclusion

Understanding the physics of the vascular system in health and disease will influence vascular management. This is a rich field for further research. Further clues to atherogenesis may lie in the differences of the fluid dynamics and stresses applied to the arterial systems. Computational modelling will be of increasing importance, as the science evolves, to our understanding of vascular haemodynamics.

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