Arteries and Veins

Physical properties of vessels closely follow the level of branching in the circuit

The aorta branches out into billions of capillaries that ultimately regroup into a single vena cava ( Fig. 19-1 , panel 1). At each level of arborization of the peripheral circulation, the values of several key parameters vary dramatically:

• 1

  Number of vessels at each level of arborization

• 2

  Radius of a typical individual vessel

• 3

  Aggregate cross-sectional area of all vessels at that level

• 4

  Mean linear velocity of blood flow within an individual vessel

• 5

  Flow (i.e., volume per second) through a single vessel

• 6

  Relative blood volume (i.e., the fraction of the body's total blood volume present in all vessels of a given level)

• 7

  Circulation (i.e., transit) time between two points of the circuit

• 8

  Pressure profile along that portion of the circuit

• 9

  Structure of the vascular walls

• 10

  Elastic properties of the vascular walls

We discuss parameters 1 to 5 in this section and parameters 6 to 8 in the following two sections. Parameters 9 and 10 are the subject of the second subchapter.

The number of vessels at a particular level of arborization ( Table 19-1 ) increases enormously from a single aorta to ~10 ⁴ small arteries, ~10 ⁷ arterioles, and finally ~4 × 10 ¹⁰ capillaries. However, only about one fourth of all capillaries are normally open to flow at rest. Finally, all of the blood returns to a single vessel where the superior and inferior venae cavae join.

TABLE 19-1

Key Systemic Vascular Parameters That Vary with Arborization

PARAMETER	AORTA	SMALL ARTERIES *	ARTERIOLES *	CAPILLARIES †	VENA CAVA
Number of units (N)	1	8000	2 × 10 7	1 × 10 10 open (4 × 10 10 total)	1
Internal radius ( r i )	1.13 cm	0.5 mm	15 µm	3 µm	1.38 cm
Cross-sectional area ( A i = π r i 2 )	4 cm 2	7.9 × 10 −3 cm 2	7.1 × 10 −7 cm 2	2.8 × 10 −7 cm 2	6 cm 2
Aggregate cross-sectional area ( A total = N π r i 2 )	4 cm 2	63 cm 2	141 cm 2	2827 cm 2 ‡	6 cm 2
Aggregate flow ( F total )	83 cm 3 /s (mL/s)	83 cm 3 /s	83 cm 3 /s	83 cm 3 /s	83 cm 3 /s
Mean linear velocity ( = F total / A total )	21 cm/s	1.3 cm/s	0.6 cm/s	0.03 cm/s ‡	14 cm/s
Single-unit flow ( F i = F total / N = A i · )	83 cm 3 /s (mL/s)	0.01 cm 3 /s	4 × 10 −6 cm 3 /s	8 × 10 −9 cm 3 /s ‡	83 cm 3 /s

* The values in this column are for a representative generation.

† The values in this column are for the smallest capillaries, at the highest level of branching.

‡ Assuming that only 25% of the capillaries are open.

The radius of an individual vessel ( r _i ; see Table 19-1 ) declines as a result of the arborization, decreasing from 1.1 cm in the aorta to a minimum of ~3 µm in the smallest capillaries. Because the cross-sectional area of an individual vessel is proportional to the square of the radius, this parameter decreases even more precipitously.

The aggregate cross-sectional area (see Table 19-1 ) at any level of branching is the sum of the single cross-sectional areas of all parallel vessels at that level of branching. That is, it is the area that you would see if you sliced through all of the vessels at the same level of branching in panel 1 of Figure 19-1 .

A fundamental law of vessel branching is that at each branch point, the combined cross-sectional area of daughter vessels exceeds the cross-sectional area of the parent vessel. In this process of bifurcation, the steepest increase in total cross-sectional area occurs in the microcirculation (see Fig. 19-1 , panel 2). A typical microcirculation in the smooth muscle and submucosa of the intestine ( Fig. 19-2 ) encompasses a first-order arteriole, several orders of progressively smaller arterioles, capillaries, several orders of venules into which the capillaries empty, and eventually a first-order venule. In humans, the maximum cross-sectional area occurs not at the level of the capillaries, but at the “postcapillary” (i.e., fourth-order) venules. Because of anastomoses among capillaries, capillaries only slightly outnumber fourth-order venules, whereas the cross-sectional unit area of each venule is appreciably greater than the area of a capillary. Assuming that only a quarter of the capillaries are usually open, the peak aggregate cross-sectional area of these postcapillary venules can be ~1000-fold greater than the cross section of the parent artery (e.g., aorta), as shown in panel 2 of Figure 19-1 .

The profile of the mean linear velocity of flow ( ) along a vascular circuit (see Fig. 19-1 , panel 3) is roughly a mirror image of the profile of the total cross-sectional area. According to the principle of continuity, which is an application of conservation of mass, total volume flow of blood must be the same at any level of arborization. Indeed, as we make multiple vertical slices along the x-axis of panel 1 in Figure 19-1 , the aggregate flow for each slice is the same:

As a consequence, must be minimal in the postcapillary venules (~0.03 cm/s), where A _total is maximal. Conversely, is maximal in the aorta (~20 to 50 cm/s). Thus, both A _total and values range ~1000-fold from the aorta to the capillaries but are inversely related to one another. The vena cava, with a cross-sectional area ~50% larger than that of the aorta, has a mean linear velocity that is about one third less.

Single-vessel flow, in contrast to total flow, varies by ~10 orders of magnitude. In the aorta, the flow is ~ 83 mL/s, the same as the cardiac output (~5 L/min). When about 25% of the capillaries are open, a typical capillary has a mean linear velocity of 0.03 cm/s and a flow of 8 × 10 ⁻⁹ mL/s (8 pL/s)—10 orders of magnitude less than the flow in the aorta. Within the microcirculation, single-vessel flow has considerable range. At one extreme, a first-order arteriole ( r _i ~ 30 µm) may have a flow of 20 × 10 ⁻⁶ mL/s. At the other, the capillaries that are closed at any given time have zero flow.

Most of the blood volume resides in the systemic veins

The body's total blood volume (V) of about 5 L (see Table 5-1 ) is not uniformly distributed along the x-axis of panel 1 in Figure 19-1 . At any level of branching, the total blood volume is the sum of the volumes of all parallel branches. Table 19-2 summarizes—for a hypothetical 70-kg woman—the distribution of total blood volume expressed as both absolute blood volumes and relative blood volumes (percentage of total blood volume). Panel 4 in Figure 19-1 summarizes four useful ways of grouping these volumes.

First, we can divide the blood volume into the systemic circulation (where ~85% of blood resides), the pulmonary circulation (~10%), and the heart chambers (~5%). The pulmonary blood volume is quite adjustable (i.e., it can be much higher than 10%) and is carefully regulated.

Second, as can be seen in the next section, we can divide blood volume into what is contained in the high-pressure system (~15%), the low-pressure system (~80%), and the heart chambers (~5%).

Third, we can group the blood volumes into those in the systemic venous system versus the remainder of the circulation. Of the 85% of the total blood volume that resides in the systemic circulation, about three fourths—or 65% of the total—is on the venous side, particularly in the smaller veins. Thus, the venous system acts as a volume reservoir. Changes in the diameters of veins have a major impact on the amount of blood they contain. For example, an abrupt increase in venous capacity causes pooling of blood in venous segments and may lead to syncope (i.e., fainting).

We can also use a fourth approach for grouping of blood volumes—divide the blood into the central blood volume (volumes of heart chambers and pulmonary circulation) versus the rest of the circulation. This central blood volume is very adjustable, and it constitutes the filling reservoir for the left heart. Left-sided heart failure can cause the normally careful regulation of the central blood volume to break down.

The circulation time is the time required for a bolus of blood to travel either across the entire length of the circulation or across a particular vascular bed. Total circulation time (the time to go from left to right across panel 1 in Fig. 19-1 ) is ~1 minute. Circulation time across a single vascular bed (e.g., coronary circulation) may be as short as 10 seconds. Circulation times may be obtained in humans by injection of a substance such as ether into an antecubital vein and measurement of the time to its appearance in the lung (4 to 8 seconds), or by injection of a bitter or sweet substance and measurement of the time to the perception of taste in the tongue (10 to 18 seconds). Although, in the past, circulation time was used clinically as an index of cardiac output, the measurement has little physiological significance. The rationale for determination of circulation time was that a shortening of circulation time could signify an improvement of cardiac output. However, the interpretation is more complicated, because circulation time is actually the ratio of blood volume to blood flow:

Changes in circulation time may thus reflect changes in volume as well as in flow. For instance, a patient in heart failure may have a decreased cardiac output (i.e., F ) or an increased blood volume (i.e., V ), either of which would contribute to an elevation of transit time.

The intravascular pressures along the systemic circuit are higher than those along the pulmonary circuit

Panel 5 in Figure 19-1 shows the profile of pressure along the systemic and pulmonary circulations. Pressures are far higher in the systemic than in the pulmonary circulation. Although the cardiac outputs of the left and right hearts are the same in the steady state, the total resistance of the systemic circulation is far higher than that of the pulmonary circulation (see pp. 683–684 ). This difference explains why the upstream driving pressure averages ~95 mm Hg in the systemic circulation but only ~15 mm Hg in the pulmonary circulation. Table 19-3 summarizes typical mean values at key locations in the circulation. In both the systemic and pulmonary circulations, the systolic and diastolic pressures decay downstream from the ventricles ( Fig. 19-3 ). The instantaneous pressures vary throughout each cardiac cycle for much of the circulatory system (see Fig. 19-1 , panel 5). In addition, the systemic venous and pulmonary pressures vary with the respiratory cycle, and venous pressure in the lower limbs varies with the contraction of skeletal muscle.

As noted above, the circulation can be divided into a high-pressure and a low-pressure system. The high-pressure system extends from the left ventricle in the contracted state all the way to the systemic arterioles. The low-pressure system extends from the systemic capillaries, through the rest of the systemic circuit, into the right heart, and then through the pulmonary circuit into the left heart in the relaxed state. The pulmonary circuit, unlike the systemic circuit, is entirely a low-pressure system; mean arterial pressures normally do not exceed 15 mm Hg, and the capillary pressures do not rise above 10 mm Hg.

Under normal conditions, the steepest pressure drop in the systemic circulation occurs in arterioles, the site of greatest vascular resistance

If we assume for simplicity's sake that the left heart behaves like a constant pressure generator of 95 mm Hg and the right heart behaves like a constant pressure generator of 15 mm Hg, it is the resistance of each vascular segment that determines the profile of pressure fall between the upstream arterial and downstream venous ends of the circulation. In particular, the pressure difference between two points along the axis of the vessel (i.e., the driving pressure difference, Δ P ) depends on flow and resistance: Δ P = F · R (see p. 412 ). According to Poiseuille's law (see p. 415 ), the resistance ( R _i ) of an individual, unbranched vascular segment is inversely proportional to the fourth power of the radius. Thus, the pressure drop between any two points along the circuit depends critically on the diameter of the vessels between these two points. However, the steepest pressure drop (Δ P /Δ x ) does not occur along the capillaries, where vessel diameters are smallest, but rather along the precapillary arterioles. Why? The aggregate resistance contributed by vessels of a particular order of arborization depends not only on their average radius but also on the number of vessels in parallel. N19-1 The more vessels in parallel, the smaller the aggregate resistance (see Equation 19-3 ). Although the resistance of a single capillary exceeds that of a single arteriole, capillaries far outnumber arterioles ( Table 19-4 ). The result is that the aggregate resistance is larger in the arterioles, and this is where the steepest Δ P occurs.

N19-1

Relative Resistance of Arterioles Versus Capillaries

In Chapter 17 , we introduced the concept of how to compute the aggregate resistance of a group of blood vessels (or resistors) arranged in parallel. We used Equation 17-3 , shown here as Equation NE 19-1 :

If we assume that each of N parallel branches has the same resistance R _i (i.e., R _i = R ₁ = R ₂ = R ₃ = R ₄ = …), then the overall resistance is

Thus,

If the aorta gives rise to 10 ⁷ arterioles (the N in the above equations), each with a resistance ( R _i ) of 15 × 10 ⁷ dyne s/cm ⁵ , then the total resistance of all the arterioles would be 15 dyne s/cm ⁵ :

If the same aorta gives rise to 3000 × 10 ⁷ capillaries (N), each with a resistance ( R _i ) of 1 × 10 ¹⁰ dyne s/cm ⁵ , then the total resistance of all the capillaries would be 3 dyne s/cm ⁵ :

Thus, because the capillaries vastly outnumber the arterioles, their aggregate resistance is less, even though their unit resistance is considerably greater.

Local intravascular pressure depends on the distribution of vascular resistance

We have just seen that the Δ P between an upstream checkpoint and a downstream checkpoint depends on the resistance between these points. What determines the absolute pressure at some location between the two checkpoints? To answer this question, we need to know not only the upstream arterial and downstream venous pressures but also the distribution of resistance between the two checkpoints. A good example to explain this concept is the distribution of resistance and pressure in the systemic microvasculature. Just how the upstream pressure in the arteriole and the downstream pressure in the venule affect the pressure at the midpoint of the capillary ( P _c ) depends on the relative size of the upstream and downstream resistances. For the simple circuit illustrated in Figure 19-4 A , N19-2

N19-2

Derivation of Equation for Midcapillary Pressure

On page 451 of the text, we present Equation 19-3 (shown here as Equation NE 19-6 )

The capillary pressure ( P _c ) depends on arteriolar pressure ( P _a ), venular pressure ( P _v ), the precapillary resistance upstream of the capillary bed ( R _pre ), and the postcapillary resistance downstream of the capillary ( R _post ). This equation describes the hydrodynamic equivalent of the electrical voltage drop (“voltage divider”) across each of two resistances in series, such as the pair of resistors shown in Figure 19-4 A .

Because of conservation of flow along the vascular circuit, the flow (F) across R _pre and R _post is, by definition, the same. For each of the two resistive elements, Ohm's law of hydrodynamics (see Equation 17-1 ) states that

$P_a-P_c=F\cdot R_{\text{pre}}$

$P_c-P_v=F\cdot R_{\text{post}}$

Thus, because F is identical in Equations NE 19-7 and NE 19-8 ,

$\left(P_a-P_c\right)/R_{\text{pre}}=(P_c-P_v)/R_{\text{post}}$

Solving for P _c in the above equation, we get the following:

$\begin{array}{c}\left(R_{\text{post}}/R_{\text{pre}}\right)(P_a-P_c)=(P_c-P_v)\\ \left(R_{\text{post}}/R_{\text{pre}}\right)P_a-\left(R_{\text{post}}/R_{\text{pre}}\right)P_c+P_v=P_c\\ \left(R_{\text{post}}/R_{\text{pre}}\right)P_a+P_v=P_c+\left(R_{\text{post}}/R_{\text{pre}}\right)P_c\\ \left(R_{\text{post}}/R_{\text{pre}}\right)P_a+P_v=P_c[1+(R_{\text{post}}/R_{\text{pre}}\left)\right]\\ \left[\right(R_{\text{post}}/R_{\text{pre}})P_a+P_v]/[1+(R_{\text{post}}/R_{\text{pre}}\left)\right]=P_c\end{array}$

The last in the preceding set of equations is a simple rearrangement of Equation NE 19-6 above.

where P _a is arteriolar pressure, P _v is venular pressure, R _pre is precapillary resistance upstream of the capillary bed, and R _post is postcapillary resistance downstream of the capillary.

From this equation, we can draw three conclusions about local microvascular pressure. First, even though arteriolar pressure is 60 mm Hg and venular pressure is 15 mm Hg in our example (see Table 19-3 ), capillary pressure is not necessarily 37.5 mm Hg, the arithmetic mean. According to Equation 19-3 , P _c would be the arithmetic mean only if the precapillary and postcapillary resistances were identical (i.e., R _post / R _pre = 1). The finding that P _c is ~25 mm Hg in most vascular beds (see Fig. 19-4 A ) implies that R _pre exceeds R _post (i.e., R _post / R _pre < 1). Under normal conditions in most microvascular beds, R _post / R _pre ranges from 0.2 to 0.4.

The second implication of Equation 19-3 is that as long as the sum ( R _pre + R _post ) is constant, reciprocal changes in R _pre and R _post would not alter the total resistance of the circuit and would therefore leave both P _a and P _v constant. However, as R _post / R _pre increases, P _c would increase, thereby approaching P _a (see Fig. 19-4 B ). Conversely, as R _post / R _pre decreases, P _c would decrease, approaching P _v (see Fig. 19-4 C ). This conclusion is also intuitive. If we reduced R _pre to zero (but increased R _post to keep R _post + R _pre constant), no pressure drop would occur along the precapillary vessels, and P _c would be the same as P _a . Conversely, if R _post were zero, P _c would be the same as P _v .

The third conclusion from Equation 19-3 is that, depending on the value of R _post / R _pre , P _c may be more sensitive to changes in arteriolar than in venular pressure, or vice versa. For example, when the ratio R _post / R _pre is low (i.e., R _pre > R _post , as in Fig. 19-4 A or C ), capillary pressure tends to follow the downstream pressure in large veins. This phenomenon explains why standing may cause the ankles to swell. The elevated pressure in large leg veins translates to an increased P _c , which, as discussed below, leads to increased transudation of fluid from the capillaries into the interstitial spaces (see pp. 467–468 ). It also explains why elevation of the feet, which lowers the pressure in the large veins, reverses ankle edema.

Vascular resistance varies in time and depends critically on the action of vascular smooth-muscle cells. The major site of control of vascular resistance in the systemic circulation is the terminal small arteries (or feed arteries) and arterioles. Figure 19-5 illustrates the effect of vasoconstriction or vasodilation on the pressure profile. Whereas the overall Δ P between source and end point may not vary appreciably during a change in vascular resistance, the shape of the local pressure profile may change appreciably. Thus, during arteriolar constriction, the pressure drop between two points along the circuit (i.e., axial pressure gradient, Δ P /Δ x ) is steep and concentrated at the arteriolar site (see Fig. 19-5 , green curve). During arteriolar dilation, the gradient is shallow and more spread out (see Fig. 19-5 , violet curve).

Although the effects of vasoconstriction and vasodilation are greatest in the local arterioles (lighter red band in Fig. 19-5 ), the effects extend both upstream (darker red band) and downstream (purple and blue bands) from the site of vasomotion because the pressure profile along the vessel depends on the resistance profile. Thus, if one vascular element contributes a greater fraction of the total resistance, a greater fraction of the pressure drop will occur along that element.