Blood vessels are elastic tubes
The walls of blood vessels consist of three layers: the intima, the media, and the adventitia. Capillaries, which have only an intimal layer of endothelial cells resting on a basal lamina, are the exception. Regardless of the organization of layers, one can distinguish four building blocks that make up the vascular wall: endothelial cells, elastic fibers, collagen fibers, and smoothmuscle cells. Figure 196 shows how the relative abundance of these components varies throughout the vascular circuit. In addition to these principal components, fibroblasts, nerve endings, and blood cells invade the intima; other extracellular components (e.g., proteoglycans) may also be present.
FIGURE 196 Structure of blood vessels. The values for medium arteries, arterioles, venules, and veins are merely illustrative because the dimensions can range widely. The drawings of the vessels are not to scale.
Endothelial cells form a single, continuous layer that lines all vascular segments. Junctional complexes keep the endothelial cells together in arteries but are less numerous in veins. The organization of the endothelial cell layer in capillaries varies greatly, depending on the organ (see p. 461). The glomus bodies in the skin and elsewhere are unusual in that their “endothelial cells” exist in multiple layers of cells called myoepithelioid cells. These glomus bodies control small arteriovenous shunts or anastomoses (see p. 570).
Elastic fibers are a rubberlike material that accounts for most of the stretch of vessels at normal pressures as well as the stretch of other tissues (e.g., lungs). Elastic fibers have two components: a core of elastin and a covering of microfibrils. The elastin core consists of a highly insoluble polymer of elastin, a protein rich in nonpolar amino acids (i.e., glycine, alanine, valine, proline). After being secreted into the extracellular space, the elastin molecules remain in a randomcoil configuration. They covalently crosslink and assemble into a highly elastic network of fibers, capable of stretching more than 100% under physiological conditions. The microfibrils, which are composed of glycoproteins and have a diameter of ~10 nm, are similar to those found in the extracellular matrix in other tissues. In arteries, elastic fibers are arranged as concentric, cylindrical lamellae. A network of elastic fibers is abundant everywhere except in the true capillaries, in the venules, and in the aforementioned arteriovenous anastomoses.
Collagen fibers constitute a jacket of far less extensible material than the elastic fibers, like the fabric woven inside the wall of a rubber hose. Collagen can be stretched only 3% to 4% under physiological conditions. The basic unit of the collagen fibers in blood vessels is composed of type I and type III collagen molecules, which are two of the fibrillar collagens. After being secreted, these triplehelical molecules assemble into fibrils that may be 10 to 300 nm in diameter; these in turn aggregate into collagen fibers that may be several microns in diameter and are visible with a light microscope. Collagen fibers are present throughout the circulation except in the capillaries. Collagen and elastic fibers form a network that is arranged more loosely toward the inner surface of the vessel than at the outer surface. Collagen fibers are usually attached to the other components of the vascular wall with some slack, so that they are normally not under tension. Stretching of these other components may take up the slack on the collagen fibers, which may then contribute to the overall tension.
Vascular smoothmuscle cells (VSMCs) are also present in all vascular segments except the capillaries. In elastic arteries, VSMCs are arranged in spirals with pitch varying from nearly longitudinal to nearly transversecircular; whereas in muscular arteries, they are arranged either in concentric rings or as helices with a low pitch. Relaxed VSMCs do not contribute appreciably to the elastic tension of the vascular wall, which is mainly determined by the elastin and collagen fibers. VSMCs exert tension primarily by means of active contraction (see pp. 238–240 and 459).
Because of the elastic properties of vessels, the pressureflow relationship of passive vascular beds is nonlinear
Because blood vessels are elastic, we must revise our concept of blood flow, which was based on Poiseuille's law for rigid tubes (see p. 415). Poiseuille's law predicts a linear pressureflow relationship (Fig. 197A, broken line). However, in reality, the pressureflow relationship is markedly nonlinear in an in vivo preparation of a vascular bed (see Fig. 197A, red curve). Starting at the foot of the red curve at a driving pressure of about 6 mm Hg, we see that the curve rises more steeply as the driving pressure increases. The reason is that increase of the driving pressure (i.e., axial pressure gradient) also increases the transmural pressure, causing the vessel to distend. Because radius increases, resistance falls and flow rises more than it would in a rigid tube. Thus, the plot curves upward. The elastic properties of vessels are the major cause of such nonlinear pressureflow relationships in vascular beds exhibiting little or no “active tension.”
FIGURE 197 Nonlinear relationships among pressure, flow, and resistance. N198
N198
Nonlinear Relationships Among Pressure, Flow, and Resistance
Contributed by Emile Boulpaep, and Walter Boron
In Figure 197A, the broken line shows the theoretical pressureflow relationship for a rigid tube—flow is directly proportional to driving pressure. For a real vascular bed (red curve), the relationship is nonlinear. First, the curve does not pass through the origin (i.e., flow is zero until driving pressure exceeds critical closing pressure). Second, the curve bends upward because increased transmural pressure widens the vessels. Increasing the vasomotor tone (blue and green curves) has two effects: shifting the critical closing pressure to the right and decreasing the slope (i.e., increasing the resistance once the vessel is open).
In Figure 197B, the three curves describe the resistance for the likecolored curves in Figure 197A. Under control conditions (no sympathetic stimulation—red curve), the resistance is infinitely high at pressures below the critical closing pressure because the baseline active tension causes the vessel to collapse. Higher driving pressures cause the vessel to dilate, lowering resistance with the fourth power of the radius. Thus, the resistance falls dramatically with increasing driving pressure. Increasing the vasomotor tone (blue and green curves) shifts the pressureresistance relationships to the right (reflecting the increased critical closing pressures) and up (reflecting the higher resistance throughout).
As driving pressure—and thus transmural pressure—increases, vessel radius increases as well, causing resistance (see p. 415) to fall (see Fig. 197B, red curve). Conversely, resistance increases toward infinity when driving pressure falls. In Poiseuille's case, of course, resistance would be constant (see Fig. 197B, broken line), regardless of the driving pressure. Although many vascular beds behave like the red curves in Figure 197A and B, we will see in Chapter 24 that some are highly regulated (i.e., active tension varies with pressure). These special circulations therefore exhibit a pressureflow relationship that differs substantially from that of a system of elastic tubes.
Contraction of smooth muscle halts blood flow when driving pressure falls below the critical closing pressure
As we have already seen, at low values of driving pressure, resistance rises abruptly toward infinity (see Fig. 197B, red curve). Viewed differently, flow totally ceases when the pressure falls below about 6 mm Hg, the critical closing pressure (see Fig. 197A, red curve). The stoppage of flow occurs because of the combined action of elastic fibers and active tension from VSMCs. Graded increases in active tension—produced, for example, by sympathetic stimulation—shift the pressureflow relationship to the right and decrease the slope (see Fig. 197A, blue and green curves), which reflects an increase in resistance over the entire pressure range (see Fig. 197B, blue and green curves). The critical closing pressure also shifts upward with increasing degrees of vasomotor tone. This phenomenon is important in hypotensive shock, in which massive vasoconstriction occurs—in an attempt to raise arterial pressure—and critical closing pressures rise to 40 mm Hg or more. As a result, blood flow can stop completely in the limb of a patient in hypotensive shock, despite the persistence of a finite, albeit small, pressure difference between the main artery and vein.
Elastic and collagen fibers determine the distensibility and compliance of vessels
Arteries and veins must withstand very different transmural pressures in vivo (see Table 193). Moreover, their relative blood volumes respond in strikingly different ways to increases in transmural pressure (Fig. 198). Arteries have a low volume capacity but can withstand large transmural pressure differences. In contrast, veins have a large volume capacity (and are thus able to act as blood reservoirs) but can withstand only small transmural pressure differences.
FIGURE 198 Compliance of blood vessels. In A and B, a relative volume of 100% represents the volume in a fully relaxed vessel.
The abundance of structural elements in the vascular walls also differs between arteries and veins (see Fig. 196). These disparities contribute to differences in the elastic behavior of arteries and veins. We can study the elastic properties of an isolated blood vessel either by recording the “static” volume change produced by pressurizing the vessel to different levels or, conversely, by recording the “static” pressure change produced by filling the vessel with liquid to different volumes. After the pressure or volume is recorded experimentally, it is important to wait until the vessel has finished responding (i.e., it becomes static) before the measurements are recorded.
The volume distensibility expresses the elastic properties of blood vessels. Three measurements are useful for assessment of distensibility. First, the absolute distensibility is the change in volume for a macroscopic step change in pressure, ΔV/ΔP. Second, because the unstretched size varies among vessels, it is preferable to normalize the volume change to the initial, unstretched volume (V_{0}) and thus to use a normalized distensibility instead:
(194)
This ratio is a measurement of the fractional change in volume with a given step in pressure. Third, the most useful index of distensibility is compliance (C), which is the slope of the tangent to any point along the pressurevolume diagram in Figure 198:
(195)
Here, the ΔV and ΔP values are minute displacements. The steeper the slope of a pressurevolume diagram, the greater the compliance (i.e., the easier it is to increase volume). In Figure 198, the slope—and thus the compliance—decreases with increasing volumes. For such a nonlinear relationship, this variable compliance is the slope of the tangent to the curve at any point. Thus, a compliance reading should always include the transmural pressure or the volume at which it was made. Similar principles apply to the compliance of airways in the lung (see p. 610).
Differences in compliance cause arteries to act as resistors and veins to act as capacitors
As we increase transmural pressure, arteries increase in volume. As they accommodate the volume of blood ejected with every heartbeat, large, muscular systemic arteries (e.g., femoral artery) increase in radius by about 10%. This change for a typical muscular artery is much smaller than the distention that one would observe for elastic arteries (e.g., aorta), which have fewer layers of smooth muscle and are not under nervous control. The higher compliance of elastic arteries is evident in the pressurevolume diagram in Figure 198A—obtained after the smooth muscle is relaxed. The compliance of a relaxed elastic artery is substantial: increasing the transmural pressure from 0 to 100 mm Hg (near the normal mean arterial pressure) increases relative volume by ~180%. With further increases in pressure and diameter, compliance decreases only modestly. For example, increase of transmural pressure from 100 to 200 mm Hg increases relative volume by a further ~100 percentage points. Thus, arteries are properly constituted for development and withstanding of high transmural pressures. Because an increase in pressure raises the radius only modestly under physiological conditions, particularly in muscular arteries, the resistance of the artery (inversely proportional to r^{4}) does not fall markedly. Because muscular arteries have a rather stable resistance, they are sometimes referred to as resistance vessels.
Veins behave very differently. The pressurevolume diagram of a vein (see Fig. 198B) shows that compliance is extremely high—far higher than for “elastic arteries”—at least in the lowpressure range. For a relaxed vein, a relatively small increase of transmural pressure from 0 to 10 mm Hg increases volume by ~200%. This high compliance in the lowpressure range is not due to a property of the elastic fibers. Rather, it reflects a change in geometry. At pressures <6 to 9 mm Hg, the vein's cross section is ellipsoidal. A small rise in pressure causes the vein to become circular, without an increase in perimeter but with a greatly increased crosssectional area. Thus, in their normal pressure range (see Table 193), veins can accept relatively large volumes of blood with little buildup of pressure. Because they act as volume reservoirs, veins are sometimes referred to as capacitance vessels. The true distensibility or compliance of the venous wall—related to the increase in perimeter produced by pressures >10 mm Hg—is rather poor, as shown by the flat slope at higher pressures in Figure 198B.
How do veins and elastic arteries compare in response to a sudden increase in volume (ΔV)? When the heart ejects its stroke volume into the aorta, the intravascular pressure increases modestly, from 80 to 120 mm Hg. This change in intravascular pressure—also known as the pulse pressure (see p. 418)—is the same as the change in transmural pressure. We have already seen that venous compliance is low in this “arterial” pressure range. Thus, if we were to challenge a vein with a sudden increase in volume (ΔV) in the arterial pressure range, the increase in transmural pressure (ΔP) would be much higher than in an artery of equal size. This is indeed the case when a surgeon uses a saphenous vein segment in a coronary artery bypass graft and anastomoses the vein between two sites on a coronary artery, forming a bypass around an obstructed site.
Changes in the volume of vessels during the cardiac cycle are due to changes in radius rather than length. During ejection of blood during systole, the length of the thoracic aorta may increase by only about 1%. N193
N193
Changes in Vessel Volume
Contributed by Emile Boulpaep, and Walter Boron
The volume (V) of a vessel is the product of its mean crosssectional area and length (L). Thus, for a cylindrical vessel with a radius of r:
Changes in the volume (ΔV) of a vessel—changes such as those that occur during the cardiac cycle or respiratory cycle—generally are due to changes in radius, rather than in length. During the cardiac cycle, the ejection during systole of a stroke volume of 70 mL of blood causes a substantial increase in the radius of the thoracic aorta but causes the length of the thoracic aorta to increase by only ~1%. Moreover, a 1% decrease in the length of the abdominal aorta compensates for this increase in the length of the thoracic aorta.
In the case of coiled vessels, an increase in vessel length may make a larger contribution to the volume change. We will discuss two examples, vessels in the uterus and those in “apical” skin.
1. The arteries that supply the uterus form arcuate arteries inside the myometrium, from which arise radial arteries oriented toward the lumen of the uterus. As the radial arteries enter the endometrium, they become coiled spiral arteries. These spiral arteries undergo dramatic changes as the endometrium grows during the endometrial cycle (see pp. 1124–1126). During the proliferative phase of the cycle, the spiral arteries grow toward the lumen of the uterus together with the endometrium. During the luteal/secretory phase, the spiral arteries increase their coiling. On pages 1136–1137, we discuss the role of these spiral arteries in the maternal blood flow during pregnancy. See also Figure 566, which shows an unlabeled spiral artery that branches off from the maternal artery.
2. Another example of coiled vessels is the arteriovenous anastomoses (glomus bodies) in apical skin (see pp. 570–571). These vessels act as shunts between arterioles and venules in the dermis and play a role in the control of cutaneous blood flow, and thus heat flow from the “core” and the skin (see pp. 1200–1201).
In the case of pulmonary vessels, increases in lung volume stretch the blood vessels within the parenchyma of the lung, thereby increasing their length. Imagine that lung volume increases from residual volume (1.5 to 1.9 L as shown in Fig. 268) to total lung capacity (4.9 to 6.4 L as shown in Fig. 268), resulting in a 3.4fold increase in lung volume. If that volume increase represents a uniform increase in all three axes of the lung, it would correspond to a ~50% increase along each dimension.
Laplace's law describes how tension in the vessel wall increases with transmural pressure
Because compliance depends on the elastic properties of the vessel wall, the discussion here focuses on how an external force deforms elastic materials. When the force vanishes, the deformation vanishes, and the material returns to its original state. The simplest mechanical model of a linearly elastic solid is a spring (Fig. 199A). The elongation ΔL is proportional to the force (), according to Hooke's law:
(196)
where k is a constant. If an elastic body requires a larger force to achieve a certain deformation, it is stiffer or less compliant. The largest stress that the material can withstand while remaining elastic is the elastic limit. If it is deformed beyond this limit, the material reaches its yield point and, eventually, its breaking point.
FIGURE 199 Elastic properties of a spring.
Stress is the force per unit crosssectional area (). Thus, if we pull on an elastic band—stretching it from an initial length L_{0} to a final length L—the stress is the force we apply, divided by the area of the band in cross section. Strain is the fractional increase in length, that is, ΔL/L_{0} or (L − L_{0})/L_{0} (see Fig. 199B). An equation analogous to Hooke's law describes the relationship between stress and strain:
(197)
The proportionality factor, the Young elastic modulus (Y), N194 is the force per crosssectional area required to stretch the material to twice its initial length and also the slope of the strainstress diagram in Figure 199B. Thus, the stiffer a material is, the steeper the slope, and the greater the elastic modulus. For example, collagen is >1000fold stiffer than elastin (Table 195).
TABLE 195
Elastic Moduli (Stiffness) of Materials
MATERIAL 
ELASTIC MODULUS (dynes/cm^{2})* 
Smooth muscle 
10^{4} or 10^{5} 
Elastin 
4 × 10^{6} 
Rubber 
4 × 10^{7} 
Collagen 
1 × 10^{10} 
Wood 
10^{10} to 10^{12} 
^{*}1 mm Hg = 1334 dyne/cm^{2}.
N194
Young Modulus
Contributed by Emile Boulpaep, and Walter Boron
The elastic modulus used in Equation 197 in the text is the elastic modulus measured in an experiment in which one stretches an object. The tensile elastic modulus, also called the Young elastic modulus, is expressed by the term Y (for Young). Y is a measure of the force needed to stretch a material. If the material that we stretch has a uniform crosssectional area A, the stress is the force we apply, normalized to the crosssectional area (). Stress is given in units of force per area, such as dynes per square centimeter or newtons per square meter. One pascal (Pa) is 1 N/m^{2}. Because 1 N = 10^{5} dynes and 1 m^{2} = 10^{4}^{ }cm^{2}, 1 Pa = 10 dyne/cm^{2}.
The elongation that results from the stress that we apply is expressed as a strain or the change in length of the material normalized to its initial length ([L – L_{0}]/L_{0}). The relationship between stress and strain is given by Equation 197 (shown here as Equation NE 1911):
(NE 1911)
Because strain is dimensionless (i.e., length divided by length), the coefficient Y has the same units as stress: dynes per square centimeter, or newtons per square meter = pascals.
Table 195 in the text gives the elastic moduli in dynes per square centimeter, whereas in engineering elastic moduli are often given in pascals. For further comparison, the elastic modulus of cortical long bone, such that of the femur, is 2 × 10^{10} Pa or 2 × 10^{11} dyne/cm^{2}, and that of most types of steel is only an order of magnitude larger, ~2 × 10^{11} Pa or 2 × 10^{12} dyne/cm^{2}.
Finally, the elastic modulus of materials can also be obtained by measuring the degree of bending of a material under a given weight. For example, one could suspend a beam between two fixed points at either end of the beam, with the weight placed in the middle. The deflection of the beam is described by an equation analogous to Equation 197 (which is only valid for the tensile elastic modulus).
Because the elastic and collagen fibers in blood vessels are not arranged as simple linear springs, the stresses and strains that arise during the pressurization or filling of a vessel occur, at least in principle, along three axes (Fig. 1910): (1) an elongation of the circumference (θ), (2) an elongation of the axial length (x), and (3) a compression of the thickness of the vessel wall in the direction of the radius (r). In fact, blood vessels actually change little in length during distention, and the thinning of the wall is usually not a major factor. Thus, we can describe the elastic properties of a vessel by considering only what happens along the circumference.
FIGURE 1910 Three axes of vessel deformation. N199
The transmural pressure (ΔP) is the distending force that tends to increase the circumference of the vessel. Opposing this elongation is a force inside the vessel wall. It is convenient to express this wall tension(T) as the force that must be applied to bring together the two edges of an imaginary slit, of unit length L, cut in the wall along the longitudinal axis of the vessel (Fig. 1911). Note that in using tension (units: dynes per centimeter) rather than stress (units: dynes per square centimeter), we assume that the thickness of the vessel wall is constant.
FIGURE 1911 Laplace's law. The circumferential arrows that attempt to bring together the edges of an imaginary slit along the length of the vessel represent the constricting force of the tension in the wall. The radial arrows that push the wall outward represent the distending force. At equilibrium, the two sets of forces balance.
The equilibrium between ΔP and T depends on the vessel radius and is expressed by a law derived independently by Thomas Young and the Marquis de Laplace in the early 1800s. N195 For a cylinder,
N195
Laplace's Law
Contributed by Emile Boulpaep, and Walter Boron
In N199, we point out that, although one could consider as many as three axes of stressstrain relationships for a blood vessel, the most important is the circumferential stressstrain relationship. Laplace's law illustrates how the circumferential geometry of an elastic tube depends on the balance of the distending force, which is generated by the transmural pressure (ΔP) that pushes the wall outward, and the constricting force from elastic components within the wall that pulls the wall inward. We can compute the distending force acting on a vessel (vessel length = L) from ΔP. For a blood vessel, ΔP is the transmural pressure, which is the difference between the intravascular pressure and the tissue pressure (see Fig. 174). The pressure is simply the force () divided by the inner surface area of the vessel wall, which is (A_{surface}), 2π · r · L. Because we disregard the changes in the thickness of the wall, the present treatment assumes a single radius (r), which is roughly equivalent to the midwall radius. Thus,
(NE 1912)
If we normalize this distending force per unit length of vessel, (in dynes per centimeter), we have the following:
(NE 1913)
The constricting force that balances the distending force is the circumferential stress in the wall of the vessel. This stress is the force that must be applied to bring together the two edges of an imaginary slit of length L that has been cut through the vessel wall, along the longitudinal axis of the vessel wall (see Fig. 1911). You can imagine that, as we cut through the vessel wall, we see that the crosssectional area cut through the wall (A_{slit}) is A_{slit} = h · x, where h is the thickness of the wall. This crosssectional area is highlighted in yellow in Figure 1910 and is labeled “A_{slit}.” The circumferential stress is the force acting over the area A_{slit}. However, because we lack information about thickness of the wall, and thus about A_{slit}, we will ignore the dimension of thickness and instead will focus on the constricting force per unit length of the slit rather than per unit area. Thus, instead of working with the circumferential stress () of the vessel, we will instead work with the circumferential tension (, in dynes per centimeter):
(NE 1914)
In order for the system to be in equilibrium, the work done by the distending force must balance the work done by the constricting force. Work is force times displacement. The work done by the distending force (see Equation NE 1913) is done over a displacement in the radial axis (Δr), as shown in Figure 1910:
(NE 1915)
The work done by the constricting force (see Equation NE 1914) is done over a displacement in the circumferential axis and has the magnitude 2π · Δr:
(NE 1916)
Based on the principles of virtual work and conservation of energy, we conclude that—at equilibrium—the distending work (see Equation NE 1915) is the same as the constricting work (see Equation NE 1916):
(NE 1917)
Simplifying this equation yields the law of Young/Laplace for a cylinder:
(NE 1918)
or
As we shall see on page 613, Equation 276 describes Laplace's law for a sphere, such as a balloon. In this case, the equation of Young and Laplace becomes:
(NE 1919)
Here, ΔP is the pressure distending the balloon (i.e., the difference between the internal and external pressures), T is the tension within the wall of the balloon, and r is the radius of the balloon. If the sphere were instead a soap bubble, T would be the surface tension.
N199
Axes of Vessel Deformation
Contributed by Emile Boulpaep, and Walter Boron
Figure 1910 in the text shows the three axes along which deformation can occur during the filling of a blood vessel: circumferential (θ), longitudinal (x), and radial (r). Therefore, as a blood vessel is distended by blood filling its lumen, we can consider three stresses and three strains, each with a different orientation. The three strains are as follows:
1. An elongation of the circumference (Δθ = Δ2πr), where r is the average of the vessel's inner radius (r_{1}) and outer radius (r_{2}) as shown in Fig. 174)
2. An elongation of the length of the vessel (Δx)
3. A compression of the thickness of the vessel wall (Δh) in the direction of the radius; that is, a change in the difference between outer and inner radii, Δ(r_{2} – r_{1})
As noted in N193, blood vessels actually change little in length during distention, and therefore we can disregard the strainstress along the axis x of Figure 1910. Although thinning of the wall does occur during blood vessel distention, the radial stress from wall compression is usually far less than the circumferential stress; therefore, we will also ignore the strainstress along the axis r. Thus, we are left to consider only the strainstress relationship along the circumferential axis (θ), which is given by Laplace's law in Equation 198:
Note that in the above equation, stress (i.e., force per unit area) is replaced by tension (i.e., force per unit length).
(198)
Thus, for a given transmural pressure, the wall tension in the vessel gets larger as the radius increases.
The vascular wall is adapted to withstand wall tension, not transmural pressure
In Figure 198, we plotted the volume of a vessel against transmural pressure, finding that the slope of this relationship is compliance. However, this sort of analysis does not provide us with direct information on the springlike properties of the materials that make up the vessel wall. With Laplace's law (T = ΔP · r), we can transform the pressurevolume relationship into the same kind of “elastic diagram” that we used in Figure 199B to understand the properties of a rubber band.
Figure 1912A is a replot of the pressurevolume diagrams in Figure 198A and B. In Figure 1912B, we reverse the coordinate system so that volume is now on the xaxis and pressure on the yaxis. Finally, in Figure 1912C, we convert volume to radius on the xaxis and use Laplace's law to convert pressure into wall tension on the yaxis. The plots in Figure 1912C are analogous to a stressstrain diagram for a rubber band (see Fig. 199B). Thus, stretching the radius of the aorta (solid red curve) causes a considerable rise in wall tension, which reflects the aorta's moderate compliance. The vena cava, on the other hand, fills over a wide range of radii before any wall tension develops, which reflects the shape change that makes it appear to be very compliant at the low pressures that are physiological for a vein. However, further stretching of the vein's radius results in a steep rise in wall tension, which reflects the inherently limited compliance of veins. N196
FIGURE 1912 Elastic diagram (tension versus radius) of blood vessels. In A, the curves are replots of the curves in Figure 198, A and B. In B, the plot is the result of reversing the two axes in A. In C, the solid red and blue curves are transformations of the curves in B. If we solve for r, the xaxis of B (volume) becomes the radius in C; if we use Laplace's law (T = ΔP · r) to solve for T, the yaxis of B (transmural pressure) becomes tension in C.
N196
LengthTension Versus StrainStress Diagrams
Contributed by Emile Boulpaep, and Walter Boron
In Figure 1912C of the text, we plot the wall tension of a cylindrical blood vessel against its radius (r, in millimeters). This wall tension is the force per unit length (T, in dynes per centimeter) of an imaginary slit in the wall of the blood vessel (see Fig. 1910). The radius in Figure 1912C is related by a constant (2π) to the length of the circumference of the vessel (2πr). Thus, the xaxis is really a length, so that Figure 1912C is really a lengthtension diagram similar to that shown for passive tension in muscle (see Fig. 99C). Note, however, that in Figure 99C, what we plotted on the yaxis was the percent of maximum active force, not the wall tension (in which the force is normalized to the length of an imaginary slit in the vessel).
In contrast, an engineer studying the elastic properties of a solid would use a strainstress diagram, as shown in Figure 199B, which differs from a lengthtension diagram on both the y and xaxes. On the yaxis of a strainstress diagram, we plot the force, normalized to the crosssectional area of the material—we define this parameter as stress (dynes per square centimeter). On the xaxis, we plot the fractional change in the length of the material—we define this dimensionless parameter as the strain. The following equation (which reproduces Equation 197) shows that the relationship between stress and strain is linear:
(NE 1920)
Note that the proportionality constant in this equation is the Young elastic modulus (Y), which is a characteristic of a particular material, as shown in Table 195.
It would be advantageous to convert the lengthtension diagram in Figure 1912C into a true strainstress diagram because that would allow us to estimate the Young elastic modulus for the blood vessel wall, which in turn would give us insight into the material that makes up the wall. This conversion requires that we (1) convert the yaxis from wall tension (dynes per centimeter) to stress (dynes per square centimeter), and (2) convert the xaxis from absolute length (centimeters) to fractional change in length (dimensionless). Starting from the definition of stress, we get
(NE 1921)
We see that we can obtain the stress by dividing wall tension (yaxis in Fig. 1912C) by the wall thickness. Note that the experimental data of Figure 1912C were obtained on vessels of an experimental animal with vessel diameters that were about half those of the human aorta or vena cava. Assuming that wall thickness of both the aorta and vena cava in the experimental animal were 0.1 cm, we can convert the yaxis of Figure 1912C to units of stress (dynes per square centimeter) simply by dividing the wall tension by 0.1 cm. We will perform this conversion separately for the aorta and vena cava and, in the process, compute the Young elastic modulus for the materials that constitute the wall of the aorta and vena cava.
Aorta
Because the lengthtension diagram in Figure 1912C was not linear, the resulting strainstress diagram will also be nonlinear. Therefore, we will analyze the relationship between strain and stress at two different points along the curve (i.e., at two different degrees of elongation of the circumference of the vessel wall).
For an elongation of the vesselwall circumference from a radius of 3.15 mm (the intersection of the curve with the xaxis) to a radius of 4 mm, the fractional change in length is (L – L_{0})/L_{0} = (4 – 3.15)/3.15 = 0.27, or a change of 27%. Achieving this elongation requires that we increase the wall tension from zero to 21,500 dyne/cm. Viewed another way, assuming a wall thickness of 0.1 cm, we must increase the stress from zero to (21,500 dyne/cm)/(0.1 cm) or 215,000 dyne/cm^{2}. Inserting the preceding values into Equation NE 1920 yields an elastic modulus (Y) of ~800,000 dyne/cm^{2}. Referring to Table 195, we see that this value is intermediate between the elastic modulus of smooth muscle and that of elastin, suggesting that both smooth muscle and elastin contribute to the elastic properties of the vessel wall in this low range of elongation of the vesselwall circumference.
For an elongation of the vesselwall circumference from a radius of 6 mm (the red point on the red curve in Fig. 1912C) to a radius of 6.1 mm, the fractional change in length is (6.1 – 6.0)/6.0 = 0.017, or a change of 1.7%. Achieving this elongation requires that we increase the wall tension from 120,000 dyne/cm to 140,000 dyne/cm, an increase in wall tension of 20,000 dyne/cm. Viewed another way, and assuming a wall thickness of 0.1 cm, we must increase the stress by (20,000 dyne/cm)/(0.1 cm) or 200,000 dyne/cm^{2}. Inserting the preceding values into Equation NE 1920 yields an elastic modulus (Y) of ~ 12,000,000 dyne/cm^{2}. This estimated elastic modulus somewhat exceeds the typical elastic modulus of elastin, indicating the likely recruitment—in this range of elongation—of some material with a higher elastic modulus (i.e., collagen).
This analysis confirms the conclusion that we reached on page 458. Thus, depending on the degree of stretch, different materials of different elastic modulus contribute to the overall elastic diagram of the aorta.
Vena Cava
For an elongation of the vesselwall circumference from a radius of 3 mm to a radius of 6 mm, the fractional change in length is (L – L_{0})/L_{0} = (6 – 3)/3 = 1, or a change of 100%. Achieving this elongation requires that we increase the wall tension from zero to ~1000 dyne/cm. Assuming a wall thickness of 0.1 cm, we must increase the stress from zero to (1000 dyne/cm)/(0.1 cm) or 10,000 dyne/cm^{2}. Inserting the preceding values into Equation NE 1920 yields an elastic modulus (Y) of ~100,000 dyne/cm^{2}, which is in the range of the elastic modulus for pure smooth muscle (see Table 195). However, because smooth muscle is arranged in parallel with elastin, it is impossible to pull on one without pulling on the other. Thus, the elastic modulus of ~100,000 dyne/cm^{2} is unrealistically low. In fact, the stress that we are applying in this range of elongation does not place any increased strain on the smooth muscle and elastic fibers of the vessel wall. Rather, the small increase in wall stress is merely that required to change the geometry of the vena cava from ellipsoidal (i.e., collapsed) to circular (i.e., fully rounded), as discussed on page 455.
For an elongation of the vesselwall circumference from a radius of 6.8 mm (the blue point on the blue curve in Fig. 1912C) to a radius of 6.9 mm, the fractional change in length is (6.9 – 6.8)/6.8 = 0.015, or a change of 1.5%. Achieving this elongation requires that we increase the wall tension from 20,000 dyne/cm to 40,000 dyne/cm, an increase in wall tension of 20,000 dyne/cm. Assuming a wall thickness of 0.1 cm, we must increase the stress by (20,000 dyne/cm)/(0.1 cm) or 200,000 dyne/cm^{2}. Inserting the preceding values into Equation NE 1920 yields an elastic modulus (Y) of ~13,600,000 dyne/cm^{2}. The estimated elastic modulus of the vena cava in this range of elongation of the circumference somewhat exceeds the typical elastic modulus of elastin, which indicates—as in the case of the aorta—the recruitment of a material with a higher elastic modulus (i.e., collagen).
What does Laplace's law (see Equation 198) tell us about the wall tension necessary to withstand the pressure inside a blood vessel? On the plot of tension versus radius for the aorta in Figure 1912C, we have chosen a single point at a radius of 6 mm; because the aorta is relatively stiff, stretching it to this radius produces a wall tension of 120,000 dyne/cm. According to Laplace's law, exactly one pressure—200,000 dyne/cm^{2} or 150 mm Hg—satisfies this combination of r and T. In Figure 1912C, this pressure is indicated by the slope of the red broken line that connects the origin with our point. In a similar exercise for the vena cava, we assume a radius of ~6.7 mm, which is similar to the one we chose for the aorta; because the vena cava is readily deformed in this range, expanding it to this radius produces a wall tension of only 12,000 dyne/cm (blue broken line). This combination of r and T yields a much lower pressure—18,000 dyne/cm^{2}, or 13.5 mm Hg. Thus, comparing vessels of similar size, Laplace's law tells us that a high wall tension is required to withstand a high pressure.
Comparing two vessels of very different size reveals a disparity between wall tension and pressure. A large vein, such as the vena cava, must resist only 10 mm Hg in transmural pressure but is equipped with a fair amount of elastic tissue. A capillary, on the other hand, which must resist a transmural pressure of 25 mm Hg, does not have any elastic tissue at all. Why? The key concept is that what the vessel really has to withstand is not pressure but wall tension. According to Laplace's law (see Equation 198), wall tension is the product of transmural pressure and radius (ΔP · r). Hence, wall tension in a capillary (10 dyne/cm) is much smaller than that in the vena cava (18,000 dyne/cm), even though the capillary is at a higher pressure. Table 196 shows that the amount of elastic tissue correlates extremely well with wall tension but very poorly with transmural pressure. The higher the tension that the vessel wall must bear, the greater is its complement of elastic tissue.
TABLE 196
Comparison of Wall Tensions and ElasticTissue Content in Various Vessels
MEAN TRANSMURAL PRESSURE ΔP (mm Hg) 
RADIUS (r) 
WALL TENSION T (dynes/cm) 
ELASTIC TISSUE* 

Aorta 
95 
1.13 cm 
140,000 
++++ 
Small arteries 
90 
0.5 cm 
60,000 
+++ 
Arterioles 
60 
15 µm 
1200 
+ 
Capillaries 
25 
3 µm 
10 
0 
Venules 
15 
10 µm 
20 
0 
Veins 
12 
>0.02 cm 
320 
+ 
Vena cava 
10 
1.38 cm 
18,000 
++ 
^{*}The number of plus signs is a relative index of the amount of elastic tissue.
Elastin and collagen separately contribute to the wall tension of vessels
The solid red and blue curves in Figure 1912C are quite different from the linear behavior predicted by Hooke's law (see Fig. 199B). With increasing stretch, the vessel wall resists additional deformation more; that is, the slope of the relationship becomes steeper. The increasing slope (i.e., increasing elastic modulus) of the radiustension diagram of a blood vessel is due to the heterogeneity of the elastic material of the vascular wall. Elastic and collagen fibers have different elastic moduli (see Table 195). We can quantitate the separate contributions of elastic and collagen fibers by “chemical dissection.” After selective digestion of elastin with elastase, which unmasks the behavior of collagen fibers, the lengthtension relationship is very steep and closer to the linear relationship expected from Hooke's law (Fig. 1913, orange curve). After selective digestion of collagen fibers with formic acid, which unmasks the behavior of elastic fibers, the lengthtension relationship is fairly flat (see Fig. 1913, violet curve). The orange curve (collagen) is steeper than the violet curve (elastin) because collagen is stiffer than elastin (see Table 195). In a normal vessel (see Fig. 1913, red curve), modest degrees of stretch elongate primarily the elastin fibers along a relatively flat slope. Progressively greater degrees of stretch recruit collagen fibers, resulting in a steeper slope.
FIGURE 1913 Chemical dissection of elastic moduli of collagen and elastin.
Aging reduces the distensibility of arteries
With aging, important changes occur in the elastic properties of blood vessels, primarily arteries. We can look at these age relationships for arteries from two perspectives, the pressurevolume curve and the radiustension curve.
The most obvious difference in aortic pressurevolume curves with increasing age is that the curves shift to progressively higher volumes (Fig. 1914A), which reflects an increase in diameter. In addition, the compliance of the aorta first rises during growth and development to early adulthood and then falls during later life. After early adulthood, two unfavorable changes occur. First, arteriosclerotic changes reduce the vessel's compliance per se. Thus, during ventricular ejection, a normalsized increase in aortic volume (ΔV) in a young adult produces a relatively small pulse pressure (ΔP) in the aorta. In contrast, the same change in aortic volume in an elderly individual produces a much larger pulse pressure. Second, because blood pressure frequently rises with age, the older person operates on a flatter portion of the pressurevolume curve, where the compliance is even lower than at lower pressures. Thus, a normalsized increase in aortic volume produces an even larger pulse pressure.
FIGURE 1914 Effect of aging on arteries. In A and B, a relative radius of 100% represents the fully relaxed value. In A, the white triangles show the expected change in transmural pressure (blue line) for the increase in aortic blood volume (red line) produced by each heartbeat—assuming a diastolic pressure of 80 mm Hg.
The second approach for assessment of the effects of age on the elastic properties of blood vessels is to examine the radiustension diagram (see Fig. 1914B). Because of the progressive, diffuse fibrosis of vessel walls with age, and because of an increase in the amount of collagen, the maximal slope of the radiustension diagram increases with age. In addition, with age, these curves start to bend upward at lower radii, as the same degree of stretch recruits a larger number of collagen fibers. Underlying this phenomenon is an increased crosslinking among collagen fibers (see p. 1239) and thus less slack in their connections to other elements in the arterial wall. Thus, even modest elongations challenge the stiffer collagen fibers to stretch.
Active tension from smoothmuscle activity adds to the elastic tension of vessels
Although we have been treating blood vessels as though their walls are purely elastic, the active tension (see pp. 238–240) from VSMCs also contributes to wall tension. Stimulation of VSMCs can reduce the internal radius of muscular feed arteries by 20% to 50%. Laplace's law (T = ΔP · r) tells us that as the VSMC shortens—thereby reducing vessel radius against a constant transmural pressure—there is a decrease in the tension that the muscle must exert to maintain that new, smaller radius.
For a blood vessel in which both passive elastic components and active smoothmuscle components contribute to the total tension, the radiustension relationship reflects the contributions of each. N197 The red curve in Figure 1915 shows such a compounded (passive + active) radiustension diagram for an artery in which the sympathetic neurotransmitter norepinephrine (see p. 342) has maximally stimulated the VSMCs. The green curve shows a passive radiustension relationship for just the elastic component of the tension (after the active component is eliminated by poisoning of the VSMCs with potassium cyanide). Of course, the green curve is the radiustension diagram on which we have focused the previous figures (e.g., see Fig. 1912C). Subtraction of the green from the red curve in Figure 1915 yields the active lengthtension diagram (blue curve) for the vascular smooth muscle.
FIGURE 1915 Active versus passive tension. The relative radius of 100% is that of an excised vessel maximally stimulated by norepinephrine (red curve, total tension) but not “stressed” (i.e., transmural pressure is 0). When the vessel is maximally poisoned with potassium cyanide, the baseline radius is ~150% (green curve, passive tension), which reflects its relaxed state. The blue curve is just the active (i.e., smoothmuscle) component of tension.
N197
Stability of Vessels with Combined Active and Passive Tension
Contributed by Ridder Emile Boulpaep
Figure 1915 in the text shows that total vascular wall tension from active and passive elements (red curve) is a rising function of relative radius (i.e., length). We can compare the active, passive, and total tensions of a blood vessel wall with those of skeletal muscle, which are shown in Figure 99C. Active tensions in both skeletal muscle and vascular smoothmuscle cells (VSMCs) plotted against length exhibit a maximum, as shown for skeletal muscle in Figure 99D and for VSMCs by the blue curve in Figure 1915. However, note that the plot of the total tension of skeletal muscle against length (orange curve in Fig. 99C) exhibits a plateau or a maximum, whereas the plot of total tension of VSMCs against length rises monotonically throughout (red curve in Fig. 1915). The latter phenomenon is due to the increasingly large passive tension component (green curve in Fig. 1915), which rises very steeply as one stretches the wall of the blood vessel.
The absence of a plateau in the totaltension versus radius curve for a blood vessel wall is of great benefit for the stability of a blood vessel over a wide range of transmural pressures. We can make this point more clearly by considering a lengthtension diagram similar to the red curve in Figure 1915, but in which we plot the absolute radius on the xaxis. The solid red curve in Figure 1912C is just such a plot. Here, the dashed red line represents the particular transmural pressure of 150 mm Hg that satisfies the physical equilibrium of that vessel at a wall tension of 120,000 dyne/cm and a radius of 6 mm. We could draw a family of dashed lines in Figure 1915—each one representing one transmural pressure and passing through the origin (zero tension and zero radius) and intersecting various points on the rising red curve. Each of these dashed lines would represent one of a wide range of transmural pressures that can be in physical equilibrium, each with its own wall tension and vessel radius.
We now return to the red curve in Figure 1915, which represents a vessel whose VSMCs have been maximally stimulated by norepinephrine. Note that in this plot, the xaxis starts with 100%, rather than zero. This 100% represents the “unloaded” diameter in which the transmural pressure is zero. This is also the minimum radius that the vessel can achieve under the current conditions of norepinephrine stimulation. Now let us assume that we extend the xaxis of Figure 1915 to the left to 0% to obtain the true origin of a diagram such as that in Figure 1912 (tension = 0, radius = 0). Because we have drawn Figure 1915 for an artery of an experimental animal, we will assume that the 100% radius corresponds to 3 mm = 0.3 cm.
Just as we introduced the concept of drawing a family of dashed lines in Figure 1912C, we can draw a family of dashed lines in Figure 1915, with its xaxis artificially extended to the left to 0%. The table below shows the coordinates of the end points of just such a family of dashed lines. In each row we show the coordinates of the origin (columns 1 and 2), the coordinates of the intersection of the dashed line with the red curve (columns 3 and 4), and finally the appropriate equilibrium pressure in two different units (columns 5 and 6) computed from Laplace's law (i.e., P = T/r). For example, the first row of data pertains to the unloaded condition. Here, the dashed line is horizontal and connects the new origin at 0% with 100% (i.e., the old origin). Note that the slope is the transmural pressure, which is zero. Another example is the last row of data, which corresponds to a radius of 200%, or twice the initial radius of 100%. Here, the dashed line has a slope of 275 mm Hg, which is the transmural pressure necessary for the physical equilibrium between the radius of 200% and wall tension of 220,000 dyne/cm. Thus, the table shows that the system can be in equilibrium for transmural pressures ranging from zero to 275 mm Hg, and even beyond.
Coordinates of the Origin 
Coordinates of Intersection of Dashed Line with Red Curve 
Equilibrium Pressure = Slope of Dashed Line = T/r 

Tension (dynes/cm) 
Radius 
T (dynes/cm) 
Radius 
P (dynes/cm^{2}) 
P (mm Hg) 
0 
0% = 0 cm 
0 
100% = 0.3 cm 
0 
0 
0 
0% = 0 cm 
10,000 
120% = 0.36 cm 
28,000 
20 
0 
0% = 0 cm 
30,000 
140% = 0.42 cm 
71,000 
54 
0 
0% = 0 cm 
68,000 
160% = 0.48 cm 
142,000 
106 
0 
0% = 0 cm 
120,000 
180% = 0.54 cm 
222,000 
167 
0 
0% = 0 cm 
220,000 
200% = 0.6 cm 
367,000 
275 
We will now use two examples to illustrate why it is important for blood vessels to contain elastic material with a reasonably high elastic modulus. That is, the green curve in Figure 1915 must be reasonably steep.
First, consider the hypothetical case in which the active tension of smooth muscle (analogous to the blue curve in Fig. 1915) peaks at a wall tension of 100,000 dyne/cm at a radius of 190% = 0.57 cm and then falls very steeply to a wall tension of zero at a radius of 195%. If we now summed this new hypothetical blue curve and the existing green curve in Figure 1915, we would obtain a red curve that reaches a maximum total wall tension of ~150,000 dyne/cm at a radius of 190% and then falls off at higher radii. At the maximum wall tension, the transmural pressure (P) would be 197 mm Hg. Any transmural pressure in excess of 197 mm Hg (i.e., a pressure of P + ΔP) would further dilate the vessel to a radius in excess of 190% (i.e., a radius of r + Δr). The total wall tension required for equilibrium, according to Laplace's law, would be T_{required} = (P + ΔP) × (r + Δr), which would have to be >150,000 dyne/cm. In fact, the actual tension (T_{actual}) has already passed its peak of 150,000 dyne/cm. Accordingly, T_{required} = (P + ΔP) × (r + Δr) > T_{actual} and a blowout would result (see Fig. 1916B, panel 2).
FIGURE 1916 Mechanical stability of vessels.
Second, consider another hypothetical case in which smoothmuscle tension is perfectly constant. That is, the blue curve for active tension (analogous to the blue curve in Fig. 1915) is a flat horizontal line that might have a wall tension, for example, of 100,000 dyne/cm, irrespective the size of the vessel. If we now summed this new hypothetical blue curve and the existing green curve in Figure 1915, we would obtain a red curve that would be the same as the green curve, but upwardly displaced by 100,000 dyne/cm. Thus, this new red curve would be flat between a radius of 0% and a radius of 140%. Beyond a radius of 140%, this new red curve would rise with the same slope as the green curve. Let us now assume a starting radius of 140% (0.42 cm) and a wall tension of 100,000 dyne/cm, which would be in physical equilibrium—according to Laplace's law—at a transmural pressure of 178 mm Hg. If the transmural pressure then fell to <178 mm Hg (i.e., a pressure of P – ΔP), the vessel would tend to narrow below a radius of 140% (i.e., a radius of r – Δr). The total wall tension required for equilibrium, according to Laplace's law, would be T_{required} = (P – ΔP) × (r – Δr), which would have to be <100,000 dyne/cm. In fact, the actual tension (T_{actual}) is 100,000 dyne/cm. Accordingly, T_{required} = (P – ΔP) × (r – Δr) < T_{actual} and a complete vessel collapse would result (see Fig. 1916B, panel 3).
Elastic tension helps stabilize vessels under vasomotor control
Consider a vessel initially stretched to a radius that is 80% greater than the unstressed radius (i.e., 180 versus 100 on the xaxis of Fig. 1915). As we inject more fluid into the vessel, the radius increases (i.e., the vessel is more “stressed”). As a result, total wall tension must increase (see Fig. 1915, red curve). The opposite, of course, would happen if we were to withdraw some fluid from the vessel. During such changes in total wall tension, what are the individual contributions of vascular smoothmuscle tension and the passive connective tissue tension? In Figure 1916, we consider two extreme examples, one in which the vessel has only elastic elements and another in which it has only a fixed (i.e., isotonic) active tension.
First, consider a vessel lacking smooth muscle, so that only elastic elements contribute to total wall tension (see Fig. 1916A). If we increase transmural pressure from P (panel 1) to P + ΔP (panel 2), thereby increasing the radius from r to r + Δr, wall tension will automatically increase from T to T + ΔT. This increase in tension allows the vessel to reach a new equilibrium, according to Laplace's law:
(199)
Conversely, a decrease in pressure must lead to a decrease in radius and a decrease in tension (panel 3). To achieve either the stable inflation of panel 2 or the stable deflation of panel 3, ΔT cannot be zero. In other words, the passive radiustension diagram (see Fig. 1915, green curve) cannot have a slope of zero. The vessel wall must have some elasticity.
Second, consider a hypothetical case in which no elastic fibers are present and in which the total tension T is kept constant by active elements (i.e., VSMCs) alone. For example, this would be approximately the case for the blue curve in Figure 1915, between relative radii of 180% and 200%. At the start (see Fig. 1916B, panel 1), we will assume that the smoothmuscle tension exactly balances the P · r product. However, when we now increase pressure (panel 2), the automatic adaptation of tension that we saw in the previous example would not occur. Indeed any increase in pressure would make the product (P + ΔP) · (r + Δr) exceed the fixed T of the VSMCs. As a result, the vessel would blow out. Conversely, any decrease in pressure would make the product (P − ΔP) · (r − Δr) fall below the fixed T and lead to full collapse of the vessel (panel 3). This type of instability does in fact occur in arteriovenous anastomoses, which are characterized by poor elastic tissue and abundant myoepithelioid cells.
A real vessel, whose wall contains both elastic and smoothmuscle elements, can be stable (i.e., neither blown out nor collapsed) over a wide range of radii. The role of elastic tissue in vessels is therefore not only to withstand high transmural pressures, but also to stabilize the vessel. Thus, elastic tissue ensures a graded response when smoothmuscle tone changes. If vessels had only smooth muscle—and no elastic fibers—they would be like those in panels 2 and 3 of Figure 1916B, and tend to be either completely open or completely closed.
An imbalance between passive and active elastic components in blood vessels can be important in disease. For example, when the passive, elastic fibers of a blood vessel are reduced or damaged, vessels tend to become larger, as in aneurysms and varicosities. If the radius exceeds the value compatible with the physical equilibrium governed by Laplace's law, a blowout may occur.
A second pathological example is seen when smooth muscle has undergone maximal stimulation and, conversely, passive elastic elements have regressed. This is the case in Raynaud disease, in which exposure to cold leads to extreme constriction of arterioles in the extremities, particularly the fingers. Vessel closure occurs because active smoothmuscle tension dominates the radiustension diagram.