The Hemodynamics of Vascular Disease

Potential and kinetic energy

The physical factors that govern how blood once ejected from the heart dissipates energy as it transits through the circulation are related to friction, resistance, and the influences of laminar and turbulent flow. The basic relationships between flow, pressure, and resistance as summarized by Bernoulli's equation, Poiseuille's law, and Poiseuille's equation are discussed. The connecting vascular channels have intrinsic resistances that connect either in parallel or in series. These interconnections modulate how blood flows through the circulation.

The main form of energy stored in flowing blood is the potential energy due to the pressure distending the vessels. This is created by the pumping action of the heart. However, some of the energy stored in blood is kinetic and a direct function of its velocity. Usually, the kinetic energy component is small compared with the pressure energy, and under normal resting conditions, it is equivalent to only a few millimeters of mercury or less. The kinetic energy of blood is proportional to its density (which is stable in normal circumstances) and to the square of its velocity. In essence, over relatively straight arterial segments, the sum of kinetic (blood flow) and potential (blood pressure) energy is constant. The equation that summarizes this relationship is Bernoulli's equation ( Fig. 1.2 ). If the artery lumen increases, kinetic energy is converted back into pressure (potential energy) as velocity decreases. Conversely, if the artery lumen narrows, the potential energy is converted into kinetic energy. Therefore within certain limits, important increases in kinetic energy occur in the systemic circulation when blood flow is high (e.g., during exercise) and in mildly stenotic lesions where luminal narrowing leads to increases in blood flood velocities. The effects of gravity due to differences in height of the blood vessel (the term ρ gz in the equation) are normally neglected over short arterial segments especially if they are oriented along the horizontal plane.

Energy differences related to differences in the levels of body parts

There is a large variation in the potential energy of blood due to differences in posture. For example, in the standing position the pressure in the feet is increased by an amount that is proportional to the height of the column of blood resting on the blood in the legs. This hydrostatic pressure increases the transmural pressure and the distention of the vessels. Gravitational potential energy (potential for doing work related to the effect of gravity on a free-falling body), however, is reduced in the dependent parts of the body by the same amount as the increase resulting from hydrostatic pressure. Therefore differences in the level of the body parts do not usually lead to changes in the driving pressure along the vascular tree unless the column of blood is interrupted, as may be the case when the venous valves close. Changes in energy and pressure associated with differences in height are noticeable with changes in posture or with activation of the calf pump muscles during walking .

Energy differences due to laminar flow

As a hemodynamic approximation, blood flow is said to be laminar because blood flows in concentric layers or laminae. Each infinitesimal layer flows with a different velocity. In theory, a thin layer of blood is held stationary next to the vessel wall at zero velocity because of an adhesive force between the blood and the inner surface of the vessel. The next layer flows with low velocity, but its movement is delayed by the stationary layer because of friction between the layers, generated by the viscous properties of the fluid. The second layer, in turn, delays the next layer, which flows at a greater velocity. The layers in the middle of the vessel flow with the highest velocity and the basic physics underlying this effect are such that the mean velocity averaged across the vessel is half of the maximal velocity measured in the center. Because the rate of change of velocity is greatest near the walls and decreases toward the center of the vessel, a velocity profile in the shape of a parabola exists along the vessel diameter, and this type of blood flow is typically referred to as laminar flow ( Fig. 1.3A ).

Loss of energy during blood flow occurs because of friction, and the amount of friction and energy loss is largely determined by the dimensions of the vessels. In a small diameter vessel, especially in the microcirculation, even the layers in the middle of the lumen are relatively close to the wall and are thus delayed considerably, resulting in a significant opposition or resistance to flow in that vessel segment. In contrast, the central core of blood is far from the walls in large arteries, and the frictional energy losses are less important. As indicated later, friction and energy losses increase if laminar flow is disturbed.

In a cylinder tube model of radius, R , the linear velocity, v , of blood flow in a given lamina located at a given radial distance, r , from the center is directly proportional to the pressure difference, Δ P , between the ends of the tube and the square of the radius and is inversely proportional to the length of the tube and the viscosity of the fluid. This is summarized by Poiseuille's equation for velocity:

For volume flow, the equation can be integrated showing that volume flow, Q , is proportional to the fourth power of the vessel radius:

where Q is the volume flow; Δ P is the pressure difference between the proximal and distal ends of the tube; r and L are the radius and length of the tube, respectively; and η is the viscosity of the fluid.

Because volume flow is proportional to the fourth power of the radius, even small changes in radius can result in large changes in volume flow. For example, a decrease in radius of 10% would decrease volume flow in a tube model by about 35% and a decrease of 50% would lead to a 95% decrease in volume flow. Because the length of the vessels and the viscosity of blood do not change much in the cardiovascular system, alterations in volume blood flow occur mainly as a result of changes in the radius of the vessels and in the difference in the pressure energy level available for blood flow.

Poiseuille's equation can be rewritten as follows:

The resistance term, R , depends on the viscous properties of the blood and on the dimensions of the vessels. Although these parameters cannot be measured in a complex system, the pressure difference, Δ P , and the volume blood flow, Q , can be measured, and the resistance can thus be calculated as a systemic vascular resistance. Because resistance is equal to the pressure difference divided by the volume flow (the pressure difference per unit flow), it can be thought of as the pressure difference needed to produce one unit of flow and therefore can be considered as an index of the difficulty in forcing blood through vessels.

Loss of Laminar Flow Patterns.

Various degrees of deviation from orderly laminar flow occur in the human circulation. Minor factors responsible for these deviations include changes in blood flow velocity during the cardiac cycle as a result of acceleration during early systole and deceleration in late systole and diastole and alterations of the lines of flow due to small changes in the diameter of the vessel. Alterations in the blood flow profiles occur at curves ( Fig. 1.4 ), at bifurcations ( Fig. 1.5 ), at any branch point, and at stenotic lesions. Once altered, the laminar (parabolic) velocity profile is often not reestablished for a considerable distance. Instead, the velocity distributions can remain flattened within and just distal to stenotic lesions as plug flow ( Fig. 1.3B ), be skewed after curves and branches ( Fig. 1.4 ), or create a jet at stenotic lesions.

In certain circumstances, laminar flow can evolve into a blood flow pattern that is mixed: a flow profile that has both forward and backward flow velocity components across the diameter of the artery. The transition zone where the lamina has zero velocity is then referred to as the site of boundary layer separation. This phenomenon can occur at branch points and is classically described at the carotid artery bifurcation ( Fig. 1.5 ). At the carotid bifurcation, the blood flow profile in the distal common carotid artery tends to migrate toward the internal carotid artery while a zone of boundary layer separation evolves in the proximal internal carotid artery ( Fig. 1.5 ) opposite to the flow divider. This zone of flow separation can also occur on the opposite side, near the origin of the external carotid artery.

Laminar flow may be altered, become disturbed or fully turbulent, even in a uniform tube. Turbulence occurs when the velocity of blood flow reaches a certain velocity within a tube of a given diameter. This transition point is expressed by the dimensionless Reynolds number (Re ):

$Re=\frac{2rv\rho }{\eta }$

where r is the radius of the tube , v is the velocity, ρ is the density of the fluid, and η is the viscosity of the fluid. Because the density ( ρ ) and viscosity ( η ) of blood are relatively constant at 1.04 to 1.05 g/cm ³ and 0.03 to 0.05 poise (g/cm⋅s), respectively, the development of turbulence depends mainly on the diameter of the vessel and on the blood flow velocity . In a tube model, laminar flow tends to be present if the Reynolds number is less than 2000, is considered in transition between 2000 and 4000, and is replaced by turbulent flow at values above 4000. However, in the circulatory system, disturbances and various degrees of turbulence are likely to occur at lower values because of body movements, the pulsatile nature of blood flow, changes in vessel dimensions, roughness of the endothelial surface, and other factors. Turbulence develops more readily in large vessels under conditions of high flow and can be detected clinically by the finding of bruits or thrills. This would typically be seen in dialysis access fistulae ( Fig. 1.6 ). Bruits may sometimes be heard over the ascending aorta during systolic acceleration in normal individuals at rest and are frequently heard in states of high cardiac output and blood flow, even in more distal arteries, such as the femoral artery. Distortion of the laminar flow velocity profiles as seen on Doppler waveforms is the “bread and butter” of vascular ultrasound. For example, in arteries with severe stenosis, pronounced turbulence is a diagnostic feature observed in the poststenotic zone and is typically associated with a bruit ( Fig. 1.7 ) that corresponds to soft tissue vibrations in the range of 100 to 300 Hz.

Turbulence occurs because a jet of blood with high velocity and high kinetic energy suddenly encounters a normal diameter lumen or a lumen of increased diameter (because of poststenotic dilatation), where both the velocity and energy level are lower than in the stenotic region. During turbulent flow, the loss of pressure energy between two points in a vessel is greater than that which would be expected from the factors in Poiseuille's and Bernoulli's equations ( Fig. 1.2 ), and the parabolic velocity profile is flattened ( Fig. 1.3B ).

Practical Tips

• According to Bernoulli's principle, when blood flow velocity increases, the pressure of the moving blood decreases.

• Laminar blood flow is normally described by a parabolic distribution of moving red cells where the highest velocity is in the middle of the artery and the slowest is near the wall (Poiseuille's equation).

• Resistance to blood flow increases as the radius of the artery decreases (to the fourth power).

• Boundary layer separation refers to the loss of laminar flow at a branch point such as the carotid bifurcation. It is typically associated with the formation of eddies and reversed blood flow on one side and normal direction of blood flow on the other. It can also occur at the margin of the stenotic jet.

• Turbulence occurs distal to a stenosis in a region where blood flow velocities are still elevated enough so that the product of velocity and effective diameter is greater than the Reynolds number.