Hemodynamics for the Vascular Surgeon

Bernoulli's Principle

When fluid flows from one point to another, its total energy (E) along any given streamline is constant provided that flow is steady and there are no frictional energy losses. This is in accordance with the law of conservation of energy and constitutes Bernoulli's principle:

This equation expresses the relationships among pressure, gravitational potential energy, and kinetic energy in an idealized fluid system. In the horizontal diverging tube shown in Fig. 5.1 , steady flow between point 1 and point 2 is accompanied by an increase in cross-sectional area and a decrease in flow velocity. Although the fluid moves against a pressure gradient of 2.5 mm Hg and therefore gains potential energy, the total fluid energy remains constant because of the lower velocity and a proportional loss of kinetic energy. In other words, the widening of the tube results in the conversion of kinetic energy to potential energy in the form of pressure. In a converging tube, the opposite would occur; a pressure drop and increase in velocity would result in potential energy being converted to kinetic energy.

The situation depicted in the preceding example is not observed in human arteries because the ideal flow conditions specified in the Bernoulli relationship are not present. The fluid energy lost in moving blood through the arterial circulation is dissipated mainly in the form of heat. When this source of energy loss is accounted for, Eq. 5.5 becomes the following:

Viscous Energy Losses and Poiseuille's Law

Energy losses in flowing blood occur either as viscous losses resulting from friction or as inertial losses related to changes in the velocity or direction of flow. The term viscosity describes the resistance to flow that arises because of the intermolecular attractions between fluid layers. The coefficient of viscosity (η) is defined as the ratio of shear stress (τ) to shear rate (D) :

Shear stress is proportional to the energy loss resulting from friction between adjacent fluid layers, whereas shear rate is the relative velocity of adjacent fluid layers. Fluids with particularly strong intermolecular attractions offer a high resistance to flow and have high coefficients of viscosity. For example, motor oil has a higher coefficient of viscosity than water. The unit of viscosity is the poise, which equals 1 dyne-s/cm ² . Because it is difficult to measure viscosity directly, relative viscosity is often used to relate the viscosity of a fluid to that of water. The relative viscosity of plasma is approximately 1.8, whereas the relative viscosity of whole blood is in the range of 3 to 4.

Because viscosity increases exponentially with increases in hematocrit, the concentration of red blood cells is the most important factor affecting the viscosity of whole blood. The viscosity of plasma is determined largely by the concentration of plasma proteins. These constituents of blood are also responsible for its non-Newtonian character. In a Newtonian fluid, viscosity is independent of shear rate or flow velocity. Because blood is a suspension of cells and large protein molecules, its viscosity can vary greatly with shear rate ( Fig. 5.2 ). Blood viscosity increases rapidly at low shear rates but approaches a constant value at higher shear rates. In most of the arterial circulation, the prevailing shear rates place the blood viscosity on the asymptotic portion of the curve. Thus, for arteries with diameters greater than approximately 1 mm, human blood resembles a constant-viscosity or Newtonian fluid.

Poiseuille's law describes the viscous energy losses that occur in an idealized flow model. This law states that the pressure gradient along a tube ( P ₁ − P ₂ , in dynes per square centimeter) is directly proportional to the mean flow velocity ( , in centimeters per second) or volume flow ( Q, in cubic centimeters per second), the tube length ( L , in centimeters), and the fluid viscosity ( η , in poise) and is inversely proportional to either the second or fourth power of the radius ( r , in centimeters):

When this equation is simplified to Pressure = flow × resistance, it is analogous to Ohm's law of electrical circuits.

The strict application of Poiseuille's law requires the steady, laminar flow of a Newtonian fluid in a straight, rigid, cylindrical tube. Because these conditions seldom exist in the arterial circulation, Poiseuille's law can only estimate the minimum pressure gradient or viscous energy losses that may be expected in arterial flow. Energy losses owing to inertial effects often exceed viscous energy losses, particularly in the presence of arterial disease.

Inertial Energy Losses

Energy losses related to inertia (Δ E ) are proportional to a constant (K) , the specific gravity of blood, and the square of blood velocity:

Because velocity is the only independent variable in this equation, inertial energy losses result from the acceleration and deceleration of pulsatile flow, variations in lumen diameter, and changes in the direction of flow at points of curvature and branching. The combined effects of viscous and inertial energy losses are illustrated in Fig. 5.3 . When the pressure drop across an arterial segment is measured at varying flow rates, the experimental data fit a line with both linear (viscous) and squared (inertial) terms. The viscous energy losses predicted by Poiseuille's law are considerably less than the total energy loss actually observed.

Vascular Resistance

Hemodynamic resistance (R) can be defined as the ratio of the energy drop between two points along an artery ( E ₁ − E ₂ ) to the mean blood flow (Q):

Because the kinetic energy term ( ) is typically a small component of the total fluid energy and the artery is usually assumed to be horizontal so that the gravitational potential energy terms ( ρgh ) cancel, Eq. 5.4 can be used to express resistance as the simple ratio of pressure drop ( P ₁ − P ₂ ) to flow. Thus Eq. 5.10 becomes a rearranged version of Poiseuille's law (see Eq. 5.8 ), and the minimum resistance or viscous energy losses are given by the resistance term

The hemodynamic resistance of an arterial segment increases as the flow velocity increases, provided that the lumen size remains constant ( Fig. 5.4 ). These additional energy losses are related to inertial effects and are proportional to .

According to Eq. 5.11 , the predominant factor influencing hemodynamic resistance is the fourth power of the radius. The relationship between radius and pressure drop for various flow rates along a 10-cm vessel segment is shown in Fig. 5.5 . For a wide range of flow rates, the pressure drop is negligible until the radius is reduced to approximately 0.3 cm; for radii less than 0.2 cm, the pressure drop increases rapidly. These observations may explain the frequent failure of femoropopliteal autogenous vein bypass grafts less than 4 mm in diameter.

In the human circulation, approximately 90% of the total vascular resistance results from flow through the arteries and capillaries, whereas the remaining 10% results from venous flow. The arterioles and capillaries are responsible for more than 60% of the total resistance, whereas the large- and medium-sized arteries account for only about 15%. Thus, the arteries that are most commonly affected by atherosclerotic occlusive disease are normally vessels with low resistance.

Laminar Flow

In the steady-state conditions specified by Poiseuille's law, the flow pattern is laminar. All motion is parallel to the walls of the tube, and the fluid is arranged in a series of concentric layers, or laminae, like those shown in Fig. 5.6 . While the velocity within each lamina remains constant, the velocity is lowest adjacent to the tube wall and increases toward the center of the tube. This results in a velocity profile that is parabolic in shape ( Fig. 5.7 ). As discussed previously, the energy expended in moving one lamina of fluid over another is proportional to viscosity.

Turbulent Flow

In contrast to the linear streamlines of laminar flow, turbulence is an irregular flow state in which velocity varies rapidly with respect to space and time. These random velocity changes result in the dissipation of fluid energy as heat. Although turbulent flow is uncommon in normal arteries, the arterial flow pattern is often disturbed. The condition of disturbed flow is an intermediate state between stable laminar flow and fully developed turbulence. It is a transient perturbation in the laminar streamlines that disappears as the flow proceeds downstream. Arterial flow may become disturbed at points of branching and curvature.

When turbulence is the result of a stenotic arterial lesion, it generally occurs immediately downstream from the stenosis and may be present only over the systolic portion of the cardiac cycle when the velocities are highest. Under conditions of turbulent flow, the velocity profile changes from the parabolic shape of laminar flow to a rectangular or blunt shape (see Fig. 5.7 ). Because of the random velocity changes, energy losses are greater for a turbulent or disturbed flow state than for a laminar flow state. Consequently the linear relationship between pressure and flow expressed by Poiseuille's law cannot be applied. This deviation from Poiseuille's law in arterial flow is shown in Fig. 5.3 .

Boundary Layer Separation

In fluid flowing through a tube, the portion of fluid adjacent to the tube wall is referred to as the boundary layer . This layer is subject to both frictional interactions with the tube wall and viscous forces generated by the more rapidly moving fluid toward the center of the tube. When the tube geometry changes suddenly, as at points of curvature, branching, or variations in lumen diameter, small pressure gradients are created that cause the boundary layer to stop or reverse direction. This change results in a complex, localized flow pattern known as an area of boundary layer separation or flow separation .

Areas of boundary layer separation have been observed in models of arterial anastomoses and bifurcations. In the carotid artery bifurcation shown in Fig. 5.8 , the central rapid flow stream of the common carotid artery is compressed along the inner wall of the carotid bulb, producing a region of high shear stress. An area of flow separation has formed along the outer wall of the carotid bulb that includes helical flow patterns and flow reversal. The region of the carotid bulb adjacent to the separation zone is subject to relatively low shear stresses. Distal to the bulb, in the internal carotid artery, flow reattachment occurs and a more laminar flow pattern is present.

The complex flow patterns described in models of the carotid bifurcation have also been documented in human subjects by pulsed Doppler studies. As shown in Fig. 5.9 , the Doppler spectral waveform obtained near the inner wall of the carotid bulb is typical of the forward flow pattern found in the internal carotid artery. However, sampling of flow along the outer wall of the bulb demonstrates lower velocities with periods of both forward and reverse flow. These spectral characteristics are consistent with the presence of flow separation and are considered to be a normal finding, particularly in young individuals. Alterations in arterial distensibility with increasing age make flow separation less prominent in older individuals.

The clinical importance of boundary layer separation is that these localized flow disturbances may contribute to the formation of atherosclerotic plaques. Examination of human carotid bifurcations, both at autopsy and during surgery, indicates that intimal thickening and atherosclerosis tend to occur along the outer wall of the carotid bulb, whereas the inner wall is relatively spared. These findings suggest that atherosclerotic lesions form near areas of flow separation and low shear stress.

Pulsatile Flow

In a pulsatile system, pressure and flow vary continuously with time and the velocity profile changes throughout the cardiac cycle. The hemodynamic principles already discussed are based on steady flow, and they are not adequate for a precise description of pulsatile flow in the arterial circulation; however, as previously stated, they can be used to determine the minimum energy losses occurring in a specific flow system. The resistance term of Poiseuille's law (see Eq. 5.11 ) estimates viscous energy losses in steady flow, but it does not account for the inertial effects, arterial wall elasticity, and wave reflections that influence pulsatile flow. The term vascular impedance is used to describe the resistance or opposition offered by a peripheral vascular bed to pulsatile blood flow.

Pulsatile flow appears to be important for optimal organ function. For example, when a kidney is perfused by steady flow instead of pulsatile flow, a reduction in urine volume and sodium excretion occurs. The critical effect of pulsatile flow is probably exerted on the microcirculation. Although the exact mechanism is unknown, transcapillary exchange, arteriolar tone, and lymphatic flow are all influenced by the pulsatile nature of blood flow.

Bifurcations and Branches

The branches of the arterial system produce sudden changes in the flow pattern that are potential sources of energy loss. However, the effect of branching on the total pressure drop in normal arterial flow is relatively small. Arterial branches commonly take the form of bifurcations. Flow patterns in a bifurcation are determined mainly by the area ratio and the branch angle. The area ratio is defined as the combined area of the secondary branches divided by the area of the primary artery.

Bifurcation flow can be analyzed in terms of pressure gradient, velocity, and transmission of pulsatile energy. According to Poiseuille's law, an area ratio of 1.41 would allow the pressure gradient to remain constant along a bifurcation. If the combined area of the branches equals the area of the primary artery, the area ratio is 1.0, and there is no change in the velocity of flow. For efficient transmission of pulsatile energy across a bifurcation, the vascular impedance of the primary artery should equal that of the branches, a situation that occurs with an area ratio of 1.15 for larger arteries and 1.35 for smaller arteries. Human infants have a favorable area ratio of 1.11 at the aortic bifurcation, but there is a gradual decrease in the ratio with age. In the teenage years, the average area ratio is less than 1.0; in the 20s, it is less than 0.9; and by the 40s, it drops below 0.8. This decline in the area ratio of the aortic bifurcation leads to an increase in both the velocity of flow in the secondary branches and the amount of reflected pulsatile energy. For example, with an area ratio of 0.8, approximately 22% of the incident pulsatile energy is reflected in the infrarenal aorta. This mechanism may play a role in the localization of atherosclerosis and aneurysms in this arterial segment.

The curvature and angulation of an arterial bifurcation can also contribute to the development of flow disturbances. As blood flows around a curve, the high-velocity portion of the stream is subjected to the greatest centrifugal force; rapidly moving fluid in the center of the vessel tends to flow outward and be replaced by the slower fluid originally located near the arterial wall. This can result in complex helical flow patterns, such as those observed in the carotid bifurcation. As the angle between the secondary branches of a bifurcation is increased, the tendency to develop turbulent or disturbed flow also increases. The average angle between the human iliac arteries is 54 degrees; however, with diseased or tortuous iliac arteries, this angle can approach 180 degrees. In the latter situation, flow disturbances are particularly likely to develop.

Tangential Stress and Tension

The tangential stress (τ) within the wall of a fluid-filled cylindrical tube can be expressed as follows:

where P is the pressure exerted by the fluid (in dynes per square centimeter), r is the internal radius (in centimeters), and δ is the thickness of the tube wall (in centimeters). Stress (τ) has the dimensions of force per unit area of tube wall (dynes per square centimeter). Thus tangential stress is directly proportional to pressure and radius but inversely proportional to wall thickness.

Eq. 5.12 is similar to Laplace's law, which defines tangential tension (T) as the product of pressure and radius:

Tension is given in units of force per tube length (dynes per centimeter). The terms stress and tension have different dimensions and describe the forces acting on the tube wall in different ways. Laplace's law can be used to characterize thin-walled structures such as soap bubbles; however, it is not suitable for describing the stresses in arterial walls.

The Properties of Arterial Walls in Specific Conditions