Hemodynamics

Velocity of the Bloodstream Depends on Blood Flow and Vascular Area

In describing the variations in blood flow in different vessels, the terms velocity and flow must first be distinguished. The former term, sometimes designated as linear velocity , means the rate of displacement with respect to time and it has the dimensions of distance per unit time, for example, cm/s. The flow, often designated as volume flow , has the dimensions of volume per unit time, for example, cm ³ /s. In a conduit of varying cross-sectional dimensions, velocity, v, flow, Q, and cross-sectional area, A, are related by the following equation:

The relationships among velocity, flow, and area are portrayed in Fig. 6.1 . The flow of an incompressible fluid past successive cross sections of a rigid tube must be constant. For a given constant flow, the velocity varies inversely as the cross-sectional area (see Fig. 1.3 ). Thus, for the same volume of fluid per second passing from section a into section b, where the cross-sectional area is five times greater, the velocity diminishes to one fifth of its previous value. Conversely, when the fluid proceeds from section b to section c, where the cross-sectional area is one tenth as great, the velocity of each particle of fluid must increase tenfold.

The velocity at any point in the system depends not only on the cross-sectional area but also on the flow, Q . This in turn depends on the pressure gradient, properties of the fluid, and dimensions of the entire hydraulic system, as discussed in the next section. For any given flow, however, the ratio of the velocity past one cross section relative to that past a second cross section depends only on the inverse ratio of the respective areas; that is,

This rule pertains regardless of whether a given cross-sectional area applies to a system that consists of a single large tube or to a system made up of several smaller tubes in parallel.

As shown in Fig. 1.3 , velocity decreases progressively as the blood traverses the aorta, its larger primary branches, the smaller secondary branches, and the arterioles. Finally, a minimal value is reached in the capillaries. As the blood then passes through the venules and continues centrally toward the venae cavae, the velocity progressively increases again. The relative velocities in the various components of the circulatory system are related only to the respective cross-sectional areas. Thus each point on the cross-sectional area curve is inversely proportional to the corresponding point on the velocity curve (see Fig. 1.3 ).

Blood Flow Depends on the Pressure Gradient

In that portion of a hydraulic system in which the total energy remains virtually constant, changes in velocity may be accompanied by appreciable alterations in the measured pressure. Consider three sections (A, B, and C) of the hydraulic system depicted in Fig. 6.2 . Six pressure probes, or Pitot tubes , have been inserted. The openings of three of these (2, 4, and 6) are tangential to the direction of flow and hence measure the lateral , or static , pressure within the tube. The openings of the remaining three Pitot tubes (1, 3, and 5) face upstream. Therefore they detect the total pressure , which is the lateral pressure plus a dynamic pressure component that reflects the kinetic energy of the flowing fluid. This dynamic component, P _dyn , of the total pressure may be calculated from the following equation:

where ρ is the density of the fluid and v is the velocity. If the midpoints of segments A, B, and C are at the same hydrostatic level, the corresponding total pressures, P ₁ , P ₃ , and P ₅ , will be equal, provided that the energy loss from viscosity in these segments is negligible. However, because of the changes in cross-sectional area, the concomitant velocity changes alter the dynamic component.

In sections A and C, let ρ = 1 g/cm ³ and v = 100 cm/s. From Eq. 6.3 :

because 1330 dynes/cm ² = 1 mm Hg. In the narrow section, B, let the velocity be twice as great as in sections A and C. Therefore:

Hence in the wide sections of the conduit, the lateral pressures (P ₂ and P ₆ ) will be only 3.8 mm Hg less than the respective total pressures (P ₁ and P ₅ ); whereas in the narrow section, the lateral pressure (P ₄ ) is 15 mm Hg less than the total pressure (P ₃ ).

The peak velocity of flow in the ascending aorta of normal dogs is about 150 cm/second (s). Therefore the measured pressure at this site may vary significantly, depending on the orientation of the pressure probe. In the descending thoracic aorta the peak velocity is substantially less than that in the ascending aorta ( Fig. 6.3 ), and lesser velocities have been recorded in still more distal arterial sites. In most arterial locations, the dynamic component is a negligible fraction of the total pressure, and the orientation of the pressure probe does not materially influence the pressure recorded. At the site of a constriction, however, the dynamic pressure component may attain substantial values. In aortic stenosis , for example, the entire output of the left ventricle is ejected through a narrow aortic valve orifice. The high flow velocity is associated with a large kinetic energy, and therefore the lateral pressure is correspondingly reduced.

The reduction of lateral pressure in the region of the stenotic valve orifice influences coronary blood flow in patients with aortic stenosis. The orifices of the right and left coronary arteries are located in the sinuses of Valsalva, just behind the valve leaflets. The initial segments of these vessels are oriented at right angles to the direction of blood flow through the aortic valves. Therefore the lateral pressure is that component of total pressure that propels the blood through the two major coronary arteries. During the ejection phase of the cardiac cycle, the lateral pressure is diminished by the conversion of potential energy to kinetic energy. This process is greatly exaggerated in aortic stenosis because of the high flow velocities.

Clinical Box

The pressure tracings shown in Fig. 6.4 were obtained from two pressure transducers inserted into the left ventricle of a patient with aortic stenosis. The transducers were located on the same catheter and were 5 cm apart. When both transducers were well within the left ventricular cavity (see Fig. 6.4A ), they both recorded the same pressures. However, when the proximal transducer was positioned in the aortic valve orifice (see Fig. 6.4B ), the lateral pressure recorded during ejection was much less than that recorded by the transducer in the ventricular cavity. This pressure difference was associated almost entirely with the much greater velocity of flow in the narrowed valve orifice than in the ventricular cavity. The pressure difference reflects mainly the conversion of some potential energy to kinetic energy. When the catheter was withdrawn still farther, so that the proximal transducer was in the aorta (see Fig. 6.4C ), the pressure difference was even more pronounced, because substantial energy was lost through friction (viscosity) as blood flowed rapidly through the narrow aortic valve.

Relationship Between Pressure and Flow Depends on the Characteristics of the Conduits

The most fundamental law that governs the flow of fluids through cylindrical tubes was derived empirically by the French physiologist Jean Poiseuille. He was primarily interested in the physical determinants of blood flow, but he replaced blood with simpler liquids for his measurements of flow through glass capillary tubes. His work was so precise and important that his observations have been designated Poiseuille’s law . Subsequently, this same law has been derived theoretically.

Poiseuille’s law is applicable to the flow of fluids through cylindrical tubes only under special conditions, namely, in the case of steady, laminar flow of Newtonian fluids. The term steady flow signifies the absence of variations of flow in time, that is, a nonpulsatile flow. Laminar flow is the type of motion in which the fluid moves as a series of individual layers, with each stratum moving at a different velocity from its neighboring layers ( Fig. 6.5 ). In the case of flow through a tube, the fluid consists of a series of infinitesimally thin concentric tubes sliding past one another. Laminar flow is described in greater detail later, where it is distinguished from turbulent flow. Also, a Newtonian fluid is defined more precisely. For the present discussion, it may be considered to be a homogeneous fluid, such as water, in contradistinction to a suspension, such as blood.

Pressure is one of the principal determinants of the rate of flow. The pressure, P, in dynes/cm ² , at a distance h, in centimeters, below the surface of a liquid is:

where ρ is the density of the liquid in g/cm ³ and g is the acceleration of gravity in cm/s ² . For convenience, however, pressure is frequently expressed simply in terms of the height of the column of liquid above some arbitrary reference point.

Consider the tube connecting reservoirs R ₁ and R ₂ in Fig. 6.6A . Let reservoir R ₁ be filled with liquid to height h ₁ , and let reservoir R ₂ be empty, as in Fig. 6.6A . The outflow pressure, P _o , is therefore equal to the atmospheric pressure, which shall be designated as the zero, or reference, level. The inflow pressure, P _i , is then equal to the same reference level plus the height, h ₁ , of the column of liquid in reservoir R ₁ . Under these conditions, let the flow, Q, through the tube be 5 mL/s.

If reservoir R ₁ is filled to height h ₂ , which is twice h ₁ , and reservoir R ₂ is again empty (as in panel B), the flow is twice as great, that is, 10 mL/s. Thus with reservoir R ₂ empty, the flow is directly proportional to the inflow pressure, P _i .

If reservoir R ₂ is now allowed to fill to height h ₁ , and the fluid level in R ₁ is maintained at h ₂ (as in panel C), the flow again becomes 5 mL/s. Thus flow is directly proportional to the difference between inflow and outflow pressures:

If the fluid level in R ₂ attains the same height as in R ₁ , flow ceases (panel D).

For any given pressure difference between the two ends of a tube, the flow depends on the dimensions of the tube. Consider the tube connected to the reservoir in Fig. 6.7A . With length l ₁ and radius r ₁ , the flow Q ₁ is observed to be 10 mL/s.

The tube connected to the reservoir in panel B has the same radius but is twice as long. Under those conditions the flow Q ₂ is found to be 5 mL/s, or only half as great as Q ₁ . Conversely, for a tube half as long as l ₁ , the flow would be twice as great as Q ₁ . In other words, flow is inversely proportional to the length of the tube:

The tube connected to the reservoir in Fig. 6.7C is the same length as l ₁ , but the radius is twice as great. Under these conditions, the flow Q ₃ is found to increase to a value of 160 mL/s, which is 16 times greater than Q ₁ . The precise measurements of Poiseuille revealed that flow varies directly as the fourth power of the radius:

Because r ₃ = 2r ₁ in the previous example (see Fig. 6.7C ), Q ₃ will be proportional to (2r ₁ ) ⁴ , or $16{\text{r}}_1^4$ ; therefore Q ₃ will equal 16Q ₁ .

Finally, for a given pressure difference and for a cylindrical tube of given dimensions, the flow varies as a function of the nature of the fluid itself. This flow-determining property of fluids is termed viscosity , η, which Newton defined as the ratio of shear stress to the shear rate of the fluid. Those fluids for which the shear rate is proportional to the shear stress are known as Newtonian fluids . If the shear rate is not proportional to the shear stress, the fluid is non-newtonian .

These terms may be comprehended more clearly if one considers the flow of a homogeneous fluid between parallel plates. In Fig. 6.8 , let the bottom plate (the bottom of a large basin) be stationary, and let the upper plate move at a constant velocity along the upper surface of the fluid. The shear stress , τ, is defined as the ratio of F:A, where F is the force applied to the upper plate in the direction of its motion along the upper surface of the fluid, and A is the area of the upper plate in contact with the fluid. The shear rate is du/dy, where u is the velocity of a minute element of the fluid in the direction parallel to the motion of the upper plate, and y is the distance of that fluid element above the bottom, stationary plate.

For a movable plate traveling at constant velocity across the surface of a homogeneous fluid, the velocity profile of the fluid will be linear. The fluid layer in contact with the upper plate will adhere to it and therefore will move at the same velocity, U, as the plate. Each minute element of fluid between the plates will move at a velocity, u, proportional to its distance, y, from the lower plate. Therefore the shear rate will be U/Y, where Y is the total distance between the two plates. Because viscosity, η, is defined as the ratio of shear stress, τ, to the shear rate, du/dy, in the example illustrated in Fig. 6.8 ,

Thus the dimensions of viscosity are dyn/cm ² divided by (cm/s)/cm, or dyn•s/cm ² . In honor of Poiseuille, 1 dyn•s/cm ² has been termed a poise . The viscosity of water at 20°C is approximately 0.01 poise, or 1 centipoise.

With regard to the flow of Newtonian fluids through cylindrical tubes, the flow varies inversely as the viscosity. Thus in the example of flow from the reservoir in Fig. 6.7D , if the viscosity of the fluid in the reservoir were doubled, the flow would be halved (5 mL/s instead of 10 mL/s).

In summary, for the steady, laminar flow of a Newtonian fluid through a cylindrical tube, the flow, Q, varies directly as the pressure difference, P _i − P _o , and the fourth power of the radius, r, of the tube; whereas it varies inversely as the length, l, of the tube and the viscosity, η, of the fluid. The full statement of Poiseuille’s law is

where π/ 8 is the constant of proportionality.