Blood flow is driven by a constant pressure head across variable resistances
To keep concepts simple, first think of the left heart as a constant pressure generator that maintains a steady mean arterial pressure at its exit (i.e., the aorta). In other words, assume that blood flow throughout the circulation is steady or nonpulsatile (below in the chapter, see the discussion of the consequences of normal cyclic variations in flow and pressure that occur as a result of the heartbeat). As a further simplification, assume that the entire systemic circulation is a single, straight tube.
To understand the steady flow of blood, driven by a constant pressure head, we can apply classical hydrodynamic laws. The most important law is analogous to Ohm's law of electricity:
That is, the pressure difference (ΔP) between an upstream point (pressure P1) and a downstream site (pressure P2) is equal to the product of the flow (F) and the resistance (R) between those two points (Fig. 17-2). Ohm's law of hydrodynamics holds at any instant in time, regardless of how simple or how complicated the circuit. This equation also does not require any assumptions about whether the vessels are rigid or compliant, as long as R is constant.
FIGURE 17-2 Flow through a straight tube. The flow (F) between a high-pressure point (P1) and a low-pressure point (P2) is proportional to the pressure difference (ΔP). A1 and A2 are cross-sectional areas at these two points. A cylindrical bolus of fluid—between the disks at P1 and P2—moves down the tube with a linear velocity v.
In reality, the pressure difference (ΔP) between the beginning and end points of the human systemic circulation—that is, between the high-pressure side (aorta) and the low-pressure side (vena cava)—turns out to be fairly constant over time. Thus, the heart behaves more like a generator of a constant pressure head than like a generator of constant flow, at least within physiological limits. Indeed, flow (F), the output of the left heart, is quite variable in time and depends greatly on the physiological circumstances (e.g., whether one is active or at rest). Like flow, resistance (R) varies with time; in addition, it varies with location within the body. The overall resistance of the circulation reflects the contributions of a complex network of vessels in both the systemic and pulmonary circuits.
Blood can take many different pathways from the left heart to the right heart (Fig. 17-3): (1) a single capillary bed (e.g., coronary capillaries), (2) two capillary beds in series (e.g., glomerular and peritubular capillaries in the kidney), or (3) two capillary beds in parallel that subsequently merge and feed into a single capillary bed in series (e.g., the parallel splenic and mesenteric circulations, which merge on entering the portal hepatic circulation). In contrast, blood flow from the right heart to the left heart can take only a single pathway, across a single capillary bed in the pulmonary circulation. Finally, some blood also courses from the left heart directly back to the left heart across shunt pathways, the most important of which is the bronchial circulation.
FIGURE 17-3 Circulatory beds.
The overall resistance (Rtotal) across a circulatory bed results from parallel and serial arrangements of branches and is governed by laws similar to those for the electrical resistance of DC circuits. For multiple resistance elements (R1, R2, R3, …) arranged in series,
For multiple elements arranged in parallel,
Blood pressure is always measured as a pressure difference between two points
Physicists measure pressure in the units of grams per square centimeter or dynes per square centimeter. However, physiologists most often gauge blood pressure by the height it can drive a column of liquid. This pressure is
where ρ is the density of the liquid in the column, g is the gravitational constant, and h is the height of the column. Therefore, if we neglect variations in g and know ρ for the fluid in the column (usually water or mercury), we can take the height of the liquid column as a measure of blood pressure. Physiologists usually express this pressure in millimeters of mercury or centimeters of water. Clinicians use the classical blood pressure gauge (sphygmomanometer) to report arterial blood pressure in millimeters of mercury.
Pressure is never expressed in absolute terms but as a pressure difference ΔP relative to some “reference” pressure. We can make this concept intuitively clear by considering pressure as a force applied to a surface area A.
If we apply a force to one side of a free-swinging door, we cannot predict the direction the door will move unless we know what force a colleague may be applying on the opposite side. In other words, we can define a movement or distortion of a mechanical system only by the difference between two forces. In electricity, we compare the difference between two voltages. In hemodynamics, we compare the difference between two pressures. When it is not explicitly stated, the reference pressure in human physiology is the atmospheric or barometric pressure (PB). Because PB on earth is never zero, a pressure reading obtained at some site within the circulation, and referred to PB, actually does not express the absolute pressure in that blood vessel but rather the difference between the pressure inside the vessel and PB.
Because a pressure difference is always between two points—and these two points are separated by some distance (Δx) and have a spatial orientation to one another—we can define a pressure gradient (ΔP/Δx) with a spatial orientation. Considering orientation, we can define three different kinds of pressure differences in the circulation:
1. Driving pressure. In Figure 17-4, the ΔP between points x1 and x2 inside the vessel—along the axis of the vessel—is the axial pressure difference. Because this ΔP causes blood to flow from x1 to x2, it is also known as the driving pressure. In the circulation, the driving pressure is the ΔP between the arterial and venous ends of the systemic (or pulmonary) circulation, and it governs blood flow. Indeed, this is the only ΔP we need to consider to understand flow in horizontal rigid tubes (see Fig. 17-2).
FIGURE 17-4 Three kinds of pressure differences, and their axes, in a blood vessel.
2. Transmural pressure. The ΔP in Figure 17-4 between point r1 (inside the vessel) and r2 (just outside the vessel)—along the radial axis—is an example of a radial pressure difference. Although there is normally no pressure difference through the blood along the radial axis, the pressure drops steeply across the vessel wall itself. The ΔP between r1 and r2 is the transmural pressure; that is, the difference between the intravascular pressure and the tissue pressure. Because blood vessels are distensible, transmural pressure governs vessel diameter, which is in turn the major determinant of resistance.
3. Hydrostatic pressure. Because of the density of blood and gravitational forces, a third pressure difference arises if the vessel does not lie in a horizontal plane, as was the case in Figure 17-2. The ΔP in Figure 17-4 between point h1 (bottom of a liquid column) and h2 (top of the column)—along the height axis—is the hydrostatic pressure difference P1 − P2. This ΔP is similar to the P in Equation 17-4 (here, ρ is the density of blood), and it exists even in the absence of any blood flow. If we express increasing altitude in positive units of h, then hydrostatic ΔP = −ρg(h1 − h2).
Total blood flow, or cardiac output, is the product (heart rate) × (stroke volume)
The flow of blood delivered by the heart, or the total mean flow in the circulation, is the cardiac output (CO). The output during a single heartbeat, from either the left or the right ventricle, is the stroke volume (SV). For a given heart rate (HR),
The cardiac output is usually expressed in liters per minute; at rest, it is 5 L/min in a 70-kg human. Cardiac output depends on body size and is best normalized to body surface area. The cardiac index (units: liters per minute per square meter) is the cardiac output per square meter of body surface area. The normal adult cardiac index at rest is about 3.0 L/(min m2).
The principle of continuity of flow is the principle of conservation of mass applied to flowing fluids. It requires that the volume entering the systemic or pulmonary circuit per unit time be equal to the volume leaving the circuit per unit time, assuming that no fluid has been added or subtracted in either circuit. Therefore, the flow of the right and left hearts (i.e., right and left cardiac outputs) must be equal in the steady state. N17-1
Cardiac Output of the Left and Right Hearts
Contributed by Emile Boulpaep
As shown on the right side of Figure 17-3 the bronchial circulation—which carries ~2% of the cardiac output or ~100 mL/min at rest—originates from the aorta (i.e., the output of the left heart). After passing through bronchial capillaries, about half of this bronchial blood empties into the azygos vein (see p. 693) and returns to the right atrium, and about half enters pulmonary venules (i.e., the input to the left heart). In other words, ~1% or ~50 mL/min of the blood leaving the left ventricle reenters the left atrium, thus bypassing the right heart (i.e., a right-to-left shunt). Thus, although we generally say that the outputs of the left and right hearts are identical in the steady state, in fact the cardiac output of the left heart exceeds the cardiac output of the right heart by about 1% or 50 mL/min at rest.
Flow in an idealized vessel increases with the fourth power of radius (Poiseuille equation)
Flow (F) is the displacement of volume (ΔV) per unit time (Δt):
In Figure 17-2, we could be watching a bolus (the blue cylinder)—with an area A and a length L—move along the tube with a mean velocity . During a time interval Δt, the cylinder advances by Δx, so that the volume passing some checkpoint (e.g., at P2) is (A · Δx). Thus,
This equation holds at any point along the circulation, regardless of how complicated the circulation is or how irregular the cross-sectional area.
In a physically well defined system, it is also possible to predict the flow from the geometry of the vessel and the properties of the fluid. In 1840 and 1841, Jean Poiseuille observed the flow of liquids in tubes of small diameter and derived the law associated with his name. In a straight, rigid, cylindrical tube,
This is the Hagen-Poiseuille equation, N17-2 where F is the flow, ΔP is the driving pressure, r is the inner radius of the tube, l is its length, and η is the viscosity. The Poiseuille equation requires that both driving pressure and the resulting flow be constant.
Jean Léonard Marie Poiseuille (1797–1869) was a French physician and experimentalist whose studies in hemodynamics in 1838, later published in 1840, are now known as the Hagen-Poiseuille law. (For more information, see http://www.cartage.org.lb/en/themes/Biographies/MainBiographies/P/Poiseuille/1.html.)
What is widely known as Poiseuille's equation was in fact derived by Gotthilf Heinrich Ludwig Hagen (1797–1884), a German physicist, who in 1839 independently confirmed the findings of Poiseuille's experiments. (For more information, see http://www.wikipedia.org/wiki/Gotthilf_Heinrich_Ludwig_Hagen.)
The Hagen-Poiseuille law describes the laminar flow of a viscous liquid through a cylindrical tube (see Fig. 17-5B). The viscous resistance reflects the frictional interaction between adjacent layers of fluid, each of which moves at a different velocity. This resistance does not reflect the friction between the outermost layer of fluid—which is stationary—and the wall of the tube.
In the Hagen-Poiseuille law (Equation 17-9, shown here as Equation NE 17-1),
F is the flow (in milliliters · second–1), ΔP is the pressure difference (in dynes · centimeter–2), r is the inner radius of the tube (in centimeters), l is the length of the tube (in centimeters), and η is the dynamic viscosity (in dynes · second · centimeter–2 = poise). The unit of dynamic viscosity, the poise, is named after Poiseuille.
The discoveries of Hagen and Poiseuille are often described as the Hagen-Poiseuille equation, Poiseuille's law, or the Poiseuille equation. These terms are synonymous in meaning and are used interchangeably.
Pappenheimer JR. Contributions to microvascular research of Jean Léonard Marie Poiseuille. American Physiological Society: Bethesda, MD; 1984:1–10. Handbook of Physiology, Section 2: The Cardiovascular System. vol 4 [parts 1 and 2].
Sutera SP, Skalak R. The history of Poiseuille's law. Annu Rev Fluid Mech. 1993;25:1–19.
Wikipedia. s.v. Hagen–Poiseuille equation. http://en.wikipedia.org/wiki/Hagen%E2%80%93Poiseuille_equation.
Three implications of Poiseuille's law are as follows:
1. Flow is directly proportional to the axial pressure difference, ΔP. The proportionality constant—(πr4)/(8ηl)—is the reciprocal of resistance (R), as is presented below.
2. Flow is directly proportional to the fourth power of vessel radius.
3. Flow is inversely proportional to both the length of the vessel and the viscosity of the fluid.
Unlike Ohm's law of hydrodynamics (F = ΔP/R), which applies to all vessels, no matter how complicated, the Poiseuille equation applies only to rigid, cylindrical tubes. Moreover, discussion below in this chapter reveals that the fluid flowing through the tube must satisfy certain conditions.
Viscous resistance to flow is proportional to the viscosity of blood but does not depend on properties of the blood vessel walls
The simplest approach for expressing vascular resistance is to rearrange Ohm's law of hydrodynamics (see Equation 17-1):
This approach is independent of geometry and is even applicable to very complex circuits, such as the entire peripheral circulation. Moreover, we can conveniently express resistance in units used by physicians for pressure (millimeters of mercury) and flow (milliliters per second). Thus, the units of total peripheral resistance are millimeters of mercury/(milliliters per second)—also known as peripheral resistance units (PRUs).
Alternatively, if the flow through the tube fulfills Poiseuille's requirements, we can express “viscous” resistance in terms of the dimensions of the vessel and the viscous properties of the circulating fluid. N17-3 Combining Equation 17-9 and Equation 17-10, we get
Contributed by Emile Boulpaep
The Hagen-Poiseuille equation describes the laminar flow of a viscous liquid through a cylindrical tube (see Fig. 17-5B). The viscous resistance reflects the frictional interaction between adjacent layers of fluid, each of which moves at a different velocity. This resistance does not reflect the friction between the outermost layer of fluid—which is stationary—and the wall of the tube. (In other words, Hagen and Poiseuille assumed that the outer edge of fluid does not move. It sticks to the wall!) Rather, viscous resistance depends on the fluid's viscosity and shape.
In Equation 17-11 (shown here as Equation NE 17-2), we define the viscous resistance as
Here, the resistance term R has the fundamental dimensions (mass) · (length)–4 · (time)–4. If the length (l) and the radius (r) are given in centimeters, and if the dynamic viscosity (η) is given in poise (or dynes · second · centimeter–2), then resistance is in the units dynes · centimeter–5 · second–1.
If one instead expresses the dynamic viscosity (η) not in poise but in the units grams centimeter–1 second–1 (remembering that, because force = mass × acceleration, the dyne has the units gram · centimeter second–2), then the units of resistance become gram centimeter–4 second–1.
Note that if the vessel is not straight, rigid, cylindrical (which implies a smooth internal surface), and unbranched, other nonviscous parameters will sum with the viscous resistance to make up the total resistance of the system (R) that appears in Ohm's law of hydrodynamics (see Equation 17-1). Such nonviscous resistances can arise from contributions from rough vessel walls and obstructions in the path of fluid flow—qualities of the container.
Thus, viscous resistance is proportional to the viscosity of the fluid and the length of the tube but inversely proportional to the fourth power of the radius of the blood vessel. Note that this equation makes no statement regarding the properties of the vessel wall per se. The resistance to flow results from the geometry of the fluid—as described by l and r—and the internal friction of the fluid, the viscosity (η). Viscosity is a property of the content (i.e., the fluid), N17-3 unrelated to any property of the container (i.e., the vessel).
The viscosity of blood is a measure of the internal slipperiness between layers of fluid
Viscosity expresses the degree of lack of slipperiness between two layers of fluid. Isaac Newton described the interaction as illustrated in Figure 17-5A. Imagine that two parallel planes of fluid, each with an area A, are moving past one another. The velocity of the first is v, and the velocity of the slightly faster moving second plane is + Δv. The difference in velocity between the moving planes is Δv and the separation between the two planes is Δx. Thus, the velocity gradient in a direction perpendicular to the plane of shear, Δv/Δx (units: [centimeters/second]/centimeter = second–1), is the shear rate. The additional force that we must apply to the second sheet to make it move faster than the first is the shear stress. The greater the area of the sheets, the greater the force needed to overcome the friction between them. Thus, shear stress is expressed as force per unit area (). The shear stress required to produce a particular shear rate Newton defined as the viscosity:
FIGURE 17-5 Viscosity.
Viscosity measures the resistance to sliding when layers of fluid are shearing against each other. The unit of viscosity is the poise (P). Whole blood has a viscosity of ~3 centipoise (cP).
If we apply Newton's definition of viscosity to a cylindrical blood vessel, the shearing laminae of the blood are not planar but concentric cylinders (see Fig. 17-5B). If we apply a pressure head to the blood in the vessel, each lamina will move parallel to the long axis of the tube. Because of cohesive forces between the inner surface of the vessel wall and the blood, we can assume that an infinitesimally thin layer of blood close to the wall of the tube (see Fig. 17-5B, layer #0) cannot move. However, the next concentric cylindrical layer, layer #1, moves in relation to the stationary outer layer #0, but slower than the next inner concentric cylinder, layer #2, and so on. Thus, the velocities increase from the wall to the center of the cylinder. The resulting velocity profile is a parabola with a maximum velocity, vmax, at the central axis. The lower the viscosity, the sharper the point of the bullet-shaped velocity profile.