Very Slight Changes in Diameter of a Vessel Can Change Its Conductance Tremendously! Slight changes in the diameter of a vessel cause tremendous changes in the vessel's ability to conduct blood when the blood flow is streamlined. This is demonstrated by the experiment illustrated in Figure 14-9A, which shows three vessels with relative diameters of 1,2, and 4 but with the same pressure difference of 100 mm Hg between the two ends of the vessels. Although the diameters of these vessels

Figure 14-9

A, Demonstration of the effect of vessel diameter on blood flow.

B, Concentric rings of blood flowing at different velocities; the farther away from the vessel wall, the faster the flow.

increase only fourfold, the respective flows are 1,16, and 256 ml/mm, which is a 256-fold increase in flow. Thus, the conductance of the vessel increases in proportion to the fourth power of the diameter, in accordance with the following formula:

Conductance ~ Diameter4

Poiseuille's Law. The cause of this great increase in conductance when the diameter increases can be explained by referring to Figure 14-95, which shows cross sections of a large and a small vessel. The concentric rings inside the vessels indicate that the velocity of flow in each ring is different from that in the adjacent rings because of laminar flow, which was discussed earlier in the chapter. That is, the blood in the ring touching the wall of the vessel is barely flowing because of its adherence to the vascular endothelium. The next ring of blood toward the center of the vessel slips past the first ring and, therefore, flows more rapidly. The third, fourth, fifth, and sixth rings likewise flow at progressively increasing velocities. Thus, the blood that is near the wall of the vessel flows extremely slowly, whereas that in the middle of the vessel flows extremely rapidly.

In the small vessel, essentially all the blood is near the wall, so that the extremely rapidly flowing central stream of blood simply does not exist. By integrating the velocities of all the concentric rings of flowing blood and multiplying them by the areas of the rings, one can derive the following formula, known as Poiseuille's law:

pD Pr4 8hl in which F is the rate of blood flow, DP is the pressure difference between the ends of the vessel, r is the radius of the vessel, l is length of the vessel, and h is viscosity of the blood.

Note particularly in this equation that the rate of blood flow is directly proportional to the fourth power of the radius of the vessel, which demonstrates once again that the diameter of a blood vessel (which is equal to twice the radius) plays by far the greatest role of all factors in determining the rate of blood flow through a vessel.

Importance of the Vessel Diameter "Fourth Power Law" in Determining Arteriolar Resistance. In the systemic circulation, about two thirds of the total systemic resistance to blood flow is arteriolar resistance in the small arterioles. The internal diameters of the arterioles range from as little as 4 micrometers to as great as 25 micrometers. However, their strong vascular walls allow the internal diameters to change tremendously, often as much as fourfold. From the fourth power law discussed above that relates blood flow to diameter of the vessel, one can see that a fourfold increase in vessel diameter can increase the flow as much as 256-fold. Thus, this fourth power law makes it possible for the arterioles, responding with only small changes in diameter to nervous signals or local tissue chemical signals, either to turn off almost completely the blood flow to the tissue or at the other extreme to cause a vast increase in flow. Indeed, ranges of blood flow of more than 100-fold in separate tissue areas have been recorded between the limits of maximum arteriolar constriction and maximum arteriolar dilatation.

Resistance to Blood Flow in Series and Parallel Vascular Circuits. Blood pumped by the heart flows from the high pressure part of the systemic circulation (i.e., aorta) to the low pressure side (i.e., vena cava) through many miles of blood vessels arranged in series and in parallel. The arteries, arterioles, capillaries, venules, and veins are collectively arranged in series. When blood vessels are arranged in series, flow through each blood vessel is the same and the total resistance to blood flow (Rtotal) is equal to the sum of the resistances of each vessel:

The total peripheral vascular resistance is therefore equal to the sum of resistances of the arteries, arteri-oles, capillaries, venules, and veins. In the example shown in Figure 14-10A, the total vascular resistance is equal to the sum of R1 and R2.

Blood vessels branch extensively to form parallel circuits that supply blood to the many organs and tissues of the body. This parallel arrangement permits each tissue to regulate its own blood flow, to a great extent, independently of flow to other tissues.

For blood vessels arranged in parallel (Figure 14-105), the total resistance to blood flow is expressed as:

R total R1 R2 R3 R4

It is obvious that for a given pressure gradient, far greater amounts of blood will flow through this

Figure 14-10

Vascular resistances: A, in series and B, in parallel.

Figure 14-10

Vascular resistances: A, in series and B, in parallel.

parallel system than through any of the individual blood vessels. Therefore, the total resistance is far less than the resistance of any single blood vessel. Flow through each of the parallel vessels in Figure 14-105 is determined by the pressure gradient and its own resistance, not the resistance of the other parallel blood vessels. However, increasing the resistance of any of the blood vessels increases the total vascular resistance.

It may seem paradoxical that adding more blood vessels to a circuit reduces the total vascular resistance. Many parallel blood vessels, however, make it easier for blood to flow through the circuit because each parallel vessel provides another pathway, or conductance, for blood flow. The total conductance (Ctotal) for blood flow is the sum of the conductance of each parallel pathway:

For example, brain, kidney, muscle, gastrointestinal, skin, and coronary circulations are arranged in parallel, and each tissue contributes to the overall conductance of the systemic circulation. Blood flow through each tissue is a fraction of the total blood flow (cardiac output) and is determined by the resistance (the reciprocal of conductance) for blood flow in the tissue, as well as the pressure gradient. Therefore, amputation of a limb or surgical removal of a kidney also removes a parallel circuit and reduces the total vascular conductance and total blood flow (i.e., cardiac output) while increasing total peripheral vascular resistance.

Effect of Blood Hematocrit and Blood Viscosity on Vascular Resistance and Blood Flow

Note especially that another of the important factors in Poiseuille's equation is the viscosity of the blood. The greater the viscosity, the less the flow in a vessel if all other factors are constant. Furthermore, the viscosity of normal blood is about three times as great as the viscosity of water.

But what makes the blood so viscous? It is mainly the large numbers of suspended red cells in the blood,

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