3 Hemodynamics

Viscosity is the coefficient of internal fraction. It denotes the force that is needed to move adjacent fluid layers relative to one another. The viscosity of blood depends on hematocrit and other factors, and it remains more or less constant up to a hematocrit of 10; above that value it increases according to an empirically defined exponential function.¹² Since blood is a suspension, its viscosity is normally determined chiefly by the relative volumes of hemoglobin-filled erythrocytes and plasma. In small-diameter vessels like terminal capillaries (7–10 μm), blood acquires an apparent viscosity that is only slightly higher than that of pure plasma (Fahraeus-Lindqvist effect). This effect is caused by the axial migration of red cells, giving rise to a peripheral, hypocellular layer of plasma-like viscosity along the vessel walls.

Hydrodynamic parameters of major importance in diagnostic ultrasound include the dimensionless Reynolds number, which essentially determines whether flow is laminar or turbulent. In simplified terms, the Reynolds number is calculated from viscosity, flow velocity, and vessel diameter. Unlike blood, an ideal fluid does not have viscosity and its Reynolds number would be infinite. As the Reynolds number approaches a critical value, even very minor disturbances are sufficient to induce a transition from laminar to turbulent flow.

The Reynolds number indicates that flow in larger blood vessels like the aorta is turbulent, whereas laminar flow prevails in smaller-caliber branch vessels that do not contain zones of disturbed flow. This is an important diagnostic distinction, since the presence of turbulent flow in smaller-caliber vessels is indicative of a flow disturbance. These principles apply to vessels in the 0.5- to 16-mm range of diameters that are accessible to ultrasound scanning (Doppler, B-mode).

All modern sonographic techniques are echo techniques in which the quality of signal generation (intensity, signal-to-noise ratio) depends critically on the nature and quantity of reflectors. It is important that reflection can occur only at a tissue-to-tissue interface when the tissues are of different acoustic impedance (complex resistance Z = |Z| × e^jφ). The cellular constituents of blood act as ultrasound reflectors. The strength (intensity) and nature of the reflection depend not only on the number and size of the reflectors but also on the impedance mismatch between the cellular reflectors and the plasma that surrounds them. If the impedance mismatch is small, the resulting echo will be weak.

The size of the reflectors is particularly important for sonographic imaging because it determines the intensity of the echoes and whether diffraction and/or scattering will occur. Ultrasonography is successful or fails with frequency-dependent absorption in the medium; this determines the signal-to-noise ratio, which is crucial for image quality and often dictates whether a technically sound examination can even be performed.

The following parameters basically determine the fluid mechanics of blood in a given vascular segment:

•Vessel diameter or cross-sectional area and its local change (expansion as in the carotid sinus or constriction due to stenosis)

•Pressure gradient along the vascular segment

•Flow velocity as a function of pressure gradient, cross-sectional area, and volume flow

•Vessel curves and branches

•Viscosity or apparent viscosity (Fahraeus-Lindqvist effect)

•Reynolds number as the parameter for laminar or turbulent flow

•Pulsatility of the flow

•Elasticity of the vessel wall

•Shear rate of the flowing blood

•Density

Blood flow characteristics depend in the first line on the cardiac output, vessel size and arterial sclerosis, its course, and branching arteries. At 90 degrees, it is believed that color Doppler will miss flow information, as the parabolic flow character (Vmax in the center, lowest V at the vessel wall) will indicate a zero flow at 90 degrees on CFI. The physiologic blood flow can be characterized by a spiral flow pattern, first due to a reflection at the aortic arch, then mowing downstream. Nearly all arteries show a spiral flow at least during the mid-systolic flow even the portal vein have this flow pattern. Therefore, at 90 degrees CDI will show blue and red colors, which will disappear when the angle of insonation is optimized.

3.2 Flow Characteristics in Steady Volume Flow

Steady flow (flow that is constant or continuous over time) usually occurs only in man-made systems, while flow in blood vessels typically consists of pulsatile flow in elastic tubes due to cardiac contractions and respiration. But a knowledge of the physical principles of steady flow in rigid tubes is still helpful for understanding hemodynamics in general, the phenomena associated with pulsatile flow in blood vessels, and the imaging of blood flow and morphology by color and spectral Doppler (e.g., in patients with stenotic vascular lesions).^{27 , 155}

3.2.1 Shear Rate

Shear rate (shear velocity) describes the spatial change of flow velocity in the transverse direction. Real fluids are permeated by frictional forces which cause different flow velocities to arise at different locations, which in turn generate a transfer of forces. The shear rate, then, is a measure of the mechanical stress exerted by the fluid components.

This means, for example, that if the vessel radius is decreased by one-half, the resistance R is increased by a factor of 16. Most of the total flow resistance in the vascular system comes from the arterioles,⁷ i.e., the portion of the arterial vascular tree that has the greatest regulatory capacity (Fig. 3.1). As a result, the luminal diameter of the arterioles at any given time has a significant bearing on total flow resistance and also influences the flow phenomena that occur in vessels proximal to the arterioles. This means that peripheral vascular resistance affects both the spectral Doppler waveforms and the color flow image in duplex sonography.

3.2.3 Fahraeus-Lindqvist Effect, Apparent Viscosity, and Axial Migration

The viscosity of a Newtonian fluid is assumed to be constant across the cross section of the tube, but this is not true in a suspension such as blood. The shear forces depend on the velocity gradient of the streamlines, as described in the section Flow Profile at Constrictions and Expansions (p. 58). If the velocities of the individual cylindrical flow layers are similar, they will generate less shear. The gradient is largest at the vessel wall. As a result, the red cells are crowded toward the center of the stream, where they can move faster than at the wall; this phenomenon is called axial migration. Viscosity decreases toward the vessel wall, while shear forces increase. Zones with the highest shear forces become depleted of cellular constituents, so the viscosity in those areas approximates that of the fluid and is called the apparent viscosity. The wall layer, then, becomes a hypocellular layer composed mostly of blood plasma. This effect occurs in vessels of all sizes, from large to extremely small, and is highly advantageous in the capillary network, where vessel diameters are in the range of 7 to 10 μm, barely larger than the red cells themselves. As a result, the peripheral resistance in the capillary bed is significantly lower than it would be if axial migration did not occur. Another result is that the hematocrit has very little effect on the degree of peripheral resistance in the smallest vessels.

3.3 Flow Characteristics in Straight Vessels of Constant Cross Section

When the Reynolds number exceeds a certain critical value R_{e crit}, even very minor disturbances will be sufficient to cause turbulence (i.e., induce a transition from laminar to turbulent motion).¹⁴ The reason is as follows: A high mean velocity v_m implies that local, transient velocity changes will lead to high acceleration. If this acceleration is checked by a correspondingly high internal friction (viscosity), the changes cannot propagate through the fluid. But if internal friction is low, small changes will spread across the whole flow profile, leading to turbulence. The Reynolds number, then, describes the relationship between accelerational forces and frictional forces.

The critical Reynolds number is 2040 ± 10 according to Avila et al⁴ and 2000 according to McDonald,¹⁹ as measured at sharp-edged inlets or outlets from the main flow. If disturbances are present in the form of stenosis-like constrictions and they are hemodynamically irregular (not streamline-shaped), turbulence can occur even at Reynolds numbers considerably lower than 2000.

When velocity is increased past the critical Reynolds number, vortices (sites of localized turbulence) also form near the vessel wall. The implication for diagnostic ultrasound is that the insonation angles are no longer the same for different streamlines and may assume arbitrary values, appearing in the color duplex image as an inhomogeneous distribution of colors and brightness levels. The velocity profile in the turbulent flow region is flat, resulting in a high velocity gradient near the vessel wall. A knowledge of these factors that influence the development of turbulent flow is helpful in understanding hemodynamic phenomena.²⁷

When velocity is less than the Reynolds number and there is no spatial disturbance or change over the longitudinal section of a vessel, the result is laminar flow with a nearly parabolic velocity profile. Velocities are highest at the axial center of the vessel, while the fluid layer in contact with the vessel wall has a velocity close to zero. This zone, called the boundary layer, is defined as the region with less than 99% of the mainstream velocity (Fig. 3.2).

While the degree of shear (degree of layer movement per unit time) in a Newtonian or homogeneous fluid is linearly proportional to the applied shear stress, non-Newtonian fluids such as blood shows a nonlinear relationship to shear stress (Fig. 3.2). As a result, Newtonian and non-Newtonian fluids exhibit different behaviors in the same vessel (Fig. 3.2).

The flow velocity gradient between adjacent fluid layers (laminae) depends on the internal friction of the fluid.⁶ The shape of this flow profile is influenced by:

•Internal friction (viscosity)

•Cohesion (the “forces of attraction” between adjacent similar molecules)

•Adhesion of the fluid to the vessel wall (external friction)

Laminar Flow

Flow is described as laminar if all the particles or concentric fluid cylinders (“layered” flow) are moving in the same direction and there is no mixing of streamlines, i.e., no exchange of particles at right angles to the flow direction. In the color duplex imaging of laminar flow, the brightest color (relative to the color bar) is displayed at the center of the vessel, indicating the relatively highest flow velocity, while darker color shades indicating lower flow velocities are displayed near the vessel wall (Fig. 3.2).

Turbulent Flow

Turbulent flow is present if the streamlines exhibit (usually) three-dimensional, random-appearing, unsteady movements over the whole cross section of the vessel, with a resultant mixing of all the streamlines. Unlike simple eddy formation at sites of disturbed flow (branch points, plaques, stenoses), turbulent flow is characterized by a large spatial extent, a disorganized and largely unpredictable spatiotemporal structure, and a strong dependence on initial and boundary conditions.

In pulsatile flow as in steady flow, the transition from laminar to turbulent flow is marked by the occurrence of transitional phenomena. However, pulsatile flow differs from laminar flow in that the Reynolds number in pulsatile flow changes during the cardiac cycle due to changes in flow velocity, flow profile (different velocities with different streamline gradients at different times), and possible changes in vessel diameter. Even under physiologic conditions, this can lead to the development of periodic turbulence (local turbulent zones) in certain phases of the cardiac cycle.³⁰

3.4 Flow Characteristics in Vessels of Variable Cross Section

3.4.1 Bernoulli Principle

The law of energy conservation also applies to fluid flow and is described by the Bernoulli principle, an important equation in fluid mechanics. Basically this principle states that all energy components in a flowing fluid are equal at all sites. This presumes that there is no friction or loss of fluid.

The Bernoulli principle is also an important basis for the sonographic detection of stenoses by the measurement of flow velocities and accelerations. The equation is as follows:

The product of pressure and mean velocity, therefore, will be the same in measurements taken at all sites. It follows that the velocity at sites of narrowing⁵ must increase at the expense of hydrostatic pressure so that the volume flow rate will remain equal at all points in the vessel (1, 2, and 3).

An ideal fluid is frictionless. In this case the velocity measured at point 3 would precisely equal that measured at point 1. This does not apply to nonideal fluids like blood, in which case the flow velocity will steadily decline over the length of the vessel due to friction losses.

3.4.2 Flow Profile at Constrictions and Expansions

When there is a sudden decrease in vessel caliber, pressure and velocity changes are accompanied by a change in the velocity profile, which is usually flat downstream from the constriction (entrance effect). The fluid must travel a certain distance before it reestablishes a nearly parabolic profile. Conversely, an increase in vessel diameter causes the flow profile to become steeper (exit effect), with an increased velocity gradient between the central and peripheral wall-adjacent layers.

3.4.3 Flow Separation, Separation Zones, and Turbulent Zones

The separation of streamlines from the main flow into secondary flows with possible vortices and turbulence is called flow separation. This may occur at stenoses or branch points (flow dividers). Physiologic flow separation with a local turbulent zone (resident turbulence) is normally present in the carotid sinus, for example, just past the division of the common carotid artery into the internal and external carotid arteries (Fig. 3.3).

Turbulent zones are usually areas of low shear. Along with oscillating and time-varying shear forces, published studies include these zones among the pathologic flow disturbances that are most active in promoting atherosclerosis. Clinically, these factors create sites of predilection for plaque formation along the inner curve of vessels and at bifurcations.¹

The vessel diameter distal to a stenosis is generally larger than within the stenosis, creating an expansion effect like that described above. If the stenosis does not have a streamlined shape, it leads to poststenotic flow separation and local turbulence. If, on the other hand, a stenosis or branch point happens to be favorably streamlined, flow separation and turbulence may not occur. This is typically seen with lower-grade “flat” stenoses.

In addition to the hemodynamic effects described above, the separation and reattachment of streamlines occur in areas of caliber change and at branch points. These phenomena, also called secondary flow, occur in the main vessel just proximal to the origin of a side branch and particularly in the side branch itself just distal to its origin (Fig. 3.4).

Key factors that determine the degree of flow separation are as follows:

The flow in a “dead-water zone” is characterized by flow reversal and vortex formation with low velocities and multidirectional flow ( Fig. 3.5 ).

A key implication for ultrasound imaging is that the boundary layers between different flow components (vortices) are relatively echogenic and thus generate high-intensity echoes of low Doppler frequency, which are clearly apparent both in the frequency spectrum and in the color flow image. Platelets may remain suspended for some time in these turbulent zones, where shear forces can promote their aggregation, adhesion, and release of procoagulant agents. This stimulates local atherogenic processes in the vessel wall (Fig. 3.5).

3.5 Characteristics of Pulsatile Volume Flow

The description of a time-varying or discontinuous pulsatile flow is considerably more complex than in the case of steady flow.¹⁸ Other references (e.g., McDonald¹⁹) may be consulted for a more detailed exploration, but a somewhat simplified discussion will suffice for clinical purposes.

Blood flow in the arterial vascular system is characterized by alternating phases of acceleration and deceleration caused by the pulsatile pumping action of the heart, which performs both pressure and volume work. Anatomic and physical conditions tend to modify the original intraventricular pressure and volume curves. Although a large pressure amplitude is generated in the left ventricle during one cardiac cycle, it is moderated by the compliance of the aorta and large vessels (“windkessel effect”) in the presence of a competent aortic valve and elastic vascular system (Fig. 3.6). The large vessels function as a reservoir whose compliance enables them to store some of the pulsatile energy of the heart. This tends to make the flow more uniform over time, especially in the arteries more distant from the left side of the heart. The pulsatility (ratio of systole to diastole) of the flow is also strongly influenced by peripheral vascular resistance, which in turn reflects the tonicity of the arterioles. It also determines the amplitude of reflected pulse waves, which affects the movement patterns of individual streamlines at arbitrary sites of measurement.

Pulsatile flow is basically laminar as well when compared with steady flow, but its velocity profile is more variable due to the inertia and internal friction of blood. The velocity profile in the acceleration phase is flat and approaches a parabolic pattern only if forward flow still predominates after the acceleration phase. If transient reverse flow occurs (especially in the limb arteries), it often begins in the central arteries, i.e., the regions in which the highest maximum velocity formerly prevailed (Fig. 3.7).

3.5.2 Approach to the Flow Complexity

Color flow imaging and spectral Doppler analyze blood flow angle-dependent. The estimated velocity should be adjusted to the streamlines. Because of limited frame rate and low spatial resolution, both Doppler techniques provide reduced information about the flow complexity in different parts of blood vessels, especially close to bifurcations or bending of vessels. At plane wave imaging techniques, frame rates of about 400-1000 fps provide an insonation of at least three angles, thus allowing the visualizing of blood flow vectors at a high spatial and temporal resolution. It supports the spiral and helical flow pattern in nearly all arteries during mid to end-systole except the small-sized vessels. Larger veins may also show a spiral flow character. A great deal of work is needed to establish the rheologic parameters for the physiologic values (see Chapter 21).^{156 , 157}

3.5.3 Waveforms of Pulsatile Flow/Helical Flow

As a result of the different boundary conditions, two different waveforms are seen in association with pulsatile flow:

•High-resistance flow

•Low-resistance flow

The classic waveform produced by high peripheral resistance is found in the limb arteries at rest (Fig. 3.7a). The time-velocity waveform or Doppler spectrum is characterized by a steep systolic upstroke and a rapid decline of flow velocity. Significant reverse flow occurs in early and mid-diastole, while a second, small forward flow component may be observed in late diastole, creating a triphasic waveform. The development of this waveform is at least partly explained by noting that the primary pulse wave is reflected in the periphery, travels back up the arterial system, is reflected against the now-closed aortic valve, then travels back in the peripheral direction where it causes the second forward-flow peak (“ping-pong” phenomenon).⁸ The amplitude dwindles distally because the flow velocity is decreased by the viscosity of the blood and by the greatly increased total vascular cross-sectional area in the periphery (see Chapter 7.1.1 and Chapter 7.2.1).

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