When air is flowing—that is, under dynamic conditions—one must not only exert the force necessary to maintain the lung and chest wall at a certain volume (i.e., static component of force), but also exert an extra force to overcome the inertia and resistance of the tissues and air molecules (i.e., dynamic component of force).
Airflow is proportional to the difference between alveolar and atmospheric pressure, but inversely proportional to airway resistance
The flow of air through tubes is governed by the same principles governing the flow of blood through blood vessels and the flow of electrical current through wires (see Equation 171). Airflow is proportional to driving pressure (ΔP) but inversely proportional to total airway resistance (R_{AW}):
(277)
(units: L/s) is airflow; the dot above the V indicates the time derivative of volume. For the lung, the driving pressure is the difference between alveolar pressure (PA) and barometric pressure (PB). Thus, for a fixed resistance, more airflow requires a greater ΔP (i.e., more effort). Viewed differently, to achieve a desired airflow, a greater resistance requires a greater ΔP.
When airflow is laminar—that is, when air molecules move smoothly in the same direction—we can apply Poiseuille's law, which states that the resistance (R) of a tube is proportional to the viscosity of the gas (η) and length of the tube (l) but inversely proportional to the fourth power of the radius:
(278)
This equation is the same as Equation 1711 for laminar blood flow. In general, changes in viscosity and length are not very important for the lung, although the resistance while breathing helium is greater than that while breathing nitrogen, the major component of air, because helium has a greater viscosity. However, the key aspect of Equation 278 is that airflow is extraordinarily sensitive to changes in airway radius. The fourthpower dependence of R on radius means that a 10% decrease in radius causes a 52% increase in R—that is, a 34% decrease in airflow. Although Poiseuille's law strictly applies only to laminar flow conditions, as discussed below, airflow is even more sensitive to changes in radius when airflow is not laminar.
In principle, it is possible to compute the total airway resistance of the tracheobronchial tree from anatomical measurements, applying Poiseuille's law when the flow is laminar and analogous expressions for airways in which the flow is not laminar. In 1915, Rohrer used this approach, along with painstaking measurements of the lengths and diameters of the airways of an autopsy specimen, to calculate the R_{AW} of the tracheobronchial tree. However, it is not practical to compute R_{AW} values, especially if we are interested in physiological or pathological changes in R_{AW}. Therefore, for both physiologists and physicians, it is important to measure R_{AW} directly. Rearrangement of Equation 277 yields an expression from which we can compute R_{AW}, provided that we know the driving pressure and the airflow that it produces:
(279)
We can measure airflow directly with a flowmeter (pneumotachometer) built into a tube through which the subject breathes. Measuring the driving pressure is more problematic because of the difficulty in measuring PA during breathing. In 1956, DuBois and colleagues met this challenge by cleverly using Boyle's law N268 and a plethysmograph to measure PA (Fig. 2711). For example, if the peak during a quiet inspiration is −0.5 L/s (by convention, a negative value denotes inflow) and PA at the same instant is −1 cm H_{2}O (from the plethysmograph), then
(2710)
FIGURE 2711 Measurement of PA during airflow. This plethysmograph is similar to the one in Figure 269 except that the spirometer is replaced by a sensitive device for measurement of the pressure inside the plethysmograph (P_{box}). The subject breathes plethysmograph air through a tube that has an electronically controlled shutter as well as meters for measurement of airflow and pressure at the mouth. In B, with the subject making an inspiratory effort against a closed shutter, pressure at the mouth equals PA. We obtain the calibration ratio ΔPA/ΔP_{box}, which allows us to convert future changes in the P_{box} to changes in PA. In C, the subject inspires through an open shutter. During the first moments of inspiration, the thorax expands before much air enters the lungs. Because alveoli expand without much of an increase in the number of gas molecules, PA must fall. Conversely, because the thorax encroaches on the plethysmograph air, which has hardly lost any gas molecules to the lungs, P_{box} must rise. From the calibration ratio ΔPA/ΔP_{box}, we calculate ΔPA from ΔP_{box} during inspiration. PA at any point during the respiratory cycle is the sum of the known PB and the measured ΔPA.
In normal individuals, R_{AW} is ~1.5 cm H_{2}O/(L/s) but can range from 0.6 to 2.3. Resistance values are higher in patients with respiratory disease and can exceed 10 cm H_{2}O/(L/s) in extreme cases.
The resistance that we measure in this way is the airway resistance, which represents ~80% of total pulmonary resistance. The remaining 20% represents tissue resistance—that is, the friction of pulmonary and thoracic tissues as they slide past one another as the lungs expand or contract.
In the lung, airflow is transitional in most of the tracheobronchial tree
We have seen that laminar airflow is governed by a relationship that is similar to Ohm's law. What happens when the airflow is not laminar? And can we predict whether the airflow is likely to be laminar? The flow of a fluid down a tube is laminar when particles passing any particular point always have the same speed and direction. Because of their viscosity, real fluids move fastest down the midline of the tube, and velocity falls to zero as we approach the wall of the tube (Fig. 2712A), as discussed for blood on page 416. If the average velocity of the fluid flowing down the tube passes a critical value, flow becomes turbulent; local irregular currents, called vortices, develop randomly, and they greatly increase resistance to flow. Under ideal laboratory conditions, airflow generally is laminar when the dimensionless Reynolds number (Re) is <2000 (see p. 416):
(2711)
Here r is the radius of the tube, is the velocity of the gas averaged over the cross section of the tube, ρ is the density of the gas, and η is its viscosity. When Re exceeds ~3000, flow tends to be turbulent. Between Re values of 2000 and 3000, flow is unstable and may switch between laminar and turbulent.
FIGURE 2712 Laminar, transitional, and turbulent flow. In A, laminar airflow () is proportional to the driving pressure (ΔP = P_{1} − P_{2}). In C, where turbulent airflow is proportional to the square root of the driving pressure, a greater ΔP (i.e., effort) is needed to produce the same as in A.
Reynolds developed Equation 2711 to predict turbulence when fluids flow through tubes that are long, straight, smooth, and unbranched. Pulmonary airways, however, are short, curved, bumpy, and bifurcated. The branches are especially problematic, because they set up small eddies (see Fig. 2712B). Although these eddies resolve farther along the airways, the air soon encounters yet other bifurcations, which establish new eddies. This sort of airflow is termed transitional. Because of the complex geometry of pulmonary airways, the critical Re in the lungs is far lower than the ideal value of 2000. In fact, Re must be less than ~1 for lung airflow to be laminar. Such low Re values and thus laminar flow are present only in the small airways that are distal to terminal bronchioles (see p. 597).
Airflow is transitional throughout most of the tracheobronchial tree. Only in the trachea, where the airway radius is large and linear air velocities may be extremely high (e.g., during exercise, coughing), is airflow truly turbulent (see Fig. 2712C).
The distinction among laminar, transitional, and turbulent airflow is important because these patterns influence how much energy one must invest to produce airflow. When flow is laminar (see Equation 277), airflow is proportional to ΔP and requires relatively little energy. When flow is transitional, one must apply more ΔP to produce the same airflow because producing vortices requires extra energy. Thus, the “effective resistance” increases. When flow is turbulent, airflow is proportional not to ΔP but to . Thus, we must apply an even greater ΔP to achieve a given flow (i.e., effective resistance is even greater).
The smallest airways contribute only slightly to total airway resistance in healthy lungs
As discussed above, airway resistance for healthy individuals is ~1.5 cm H_{2}O/(L/s). Because effective resistance can increase markedly with increases in airflow—owing to transitional and turbulent airflow—it is customary to measure resistances at a fixed, relatively low flow of ~0.5 L/s. The second column of Table 272 shows how R_{AW} normally varies with location as air moves from lips to alveoli during a quiet inspiration. A striking feature is that the greatest aggregate resistance is in the pharynxlarynx and large airways (diameter > 2 mm, or before about generation 8). Of the R_{AW} of 1.5 cm H_{2}O/(L/s) in a normal subject, 0.6 is in the upper air passages, 0.6 is in the large airways, and only 0.3 is in the small airways.
TABLE 272
Airway Resistance*
LOCUS 
NORMAL 
COPD 
Pharynxlarynx 
0.6 
0.6 
Airways > 2 mm diameter 
0.6 
0.9 
Airways < 2 mm diameter 
0.3 
3.5 
Total airway resistance 
1.5 
5.0 
^{*}Units of resistance are cm H_{2}O/(L/s).
Because R increases with the fourth power of airway radius (see Equation 278), it might seem counterintuitive that the small airways have the lowest aggregate resistance. However, although each small airway has a high individual resistance, so many are aligned in parallel that their aggregate resistance is very low. N276 We see this same pattern of resistance in the vascular system, where capillaries make a smaller contribution than arterioles to aggregate resistance.
N276
Aggregate Resistance of Small Airways
Contributed by Emile Boulpaep, Walter Boron
In the text we point out that, although each small airway has a high individual resistance compared to each large airway, there are so many of these small airways aligned in parallel that their aggregate resistance is very low.
In Chapter 17, we introduce the concept of how to compute the aggregate resistance of a group of blood vessels (or resistors) arranged in parallel. We used Equation 173, which we reproduce here:
(NE 276)
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
(NE 277)
Thus,
(NE 278)
As an example, let us compare the aggregate resistance at generations 10 and 16. At generation 10, the airway diameter may be ~1.2 mm (r = 0.6 mm), whereas at generation 16, the diameter might be 0.6 mm (r = 0.3 mm). Because the radius has fallen by a factor of 2 between generations 10 and 16, the fourthpower relationship between radius and resistance (see Equation 278) tells us that the unitary resistance of each airway must increase by a factor of 16. However, there are 2^{6}fold more airways at generation 16 than at generation 10. Thus, if we make the simplifying assumption that the airway length is the same for the two generations, we would conclude that the aggregate resistance falls to 16/64 or 25% of the initial value as we move from generation 10 to generation 16.
Table 272 also shows how R_{AW} varies for a typical patient with moderately severe chronic obstructive pulmonary disease (COPD), a condition in which emphysema or chronic bronchitis increases R_{AW} (Box 272). COPD is a common and debilitating consequence of cigarette smoking, far more common than the lung cancer that receives so much attention in the lay press. Notice where the disease strikes. Even though COPD increases total airway resistance to 5.0 cm H_{2}O/(L/s)—3.3fold greater than in our normal subject—pharynxlarynx resistance does not change at all, and largeairway resistance increases only modestly. Almost all of the increment in R_{AW} is due to a nearly 12fold increase in the resistance of the smallest airways! According to Equation 278, we could produce a 12fold increase in R_{AW} by decreasing radius by about half.
Box 272
Obstructive Pulmonary Disease
Two major categories of pulmonary disease can markedly reduce total ventilation: the restrictive pulmonary diseases (see Box 271) and the obstructive pulmonary diseases, in which the pathological process causes an increase in airway resistance—primarily a property of the conducting airways (see p. 597).
The condition can be acute, as with the aspiration of a foreign body, the buildup of mucus in an airway lumen, or the constriction of the airway lumen due to the contraction of smooth muscle in asthma (see Box 273).
COPD is defined as an increase in airway resistance caused by chronic bronchitis (longstanding inflammation of the bronchi or bronchioles), emphysema (destruction of alveolar walls, producing a smaller number of large alveoli), or a combination of the two. In the United States, COPD is the fourth leading cause of death. The major risk factor is cigarette smoking, although the inherited absence of α_{1}antitrypsin (see Table 181) also predisposes to COPD. Inflammation leads to the infiltration of the walls of conducting airways by macrophages, activated T lymphocytes, and neutrophils as well as the infiltration of alveolar walls by activated lymphocytes. The release of neutrophil elastase and other proteases overwhelms natural antiproteases, such as α_{1}antitrypsin. Bronchitis increases airway resistance by narrowing the lumen. With its destruction of alveolar walls, emphysema increases the static compliance, which, by itself, would make it easier to inhale. However, the destruction of parenchyma also reduces the mechanical tethering (see p. 620) of conducting airways, leading to an exaggerated collapse of these airways during expiration (see p. 626) and thus to an increase in airway resistance.
Although small airways normally have a very low aggregate resistance, it is within these small airways that COPD has its greatest and earliest effects. Even a doubling of smallairway resistance from 0.3 to 0.6 cm H_{2}O/(L/s) in the early stages of COPD would produce such a small increment in R_{AW} that it would be impossible to identify the COPD patient in a screening test based on resistance measurements. As discussed on pages 687–689, approaches that detect the nonuniformity of ventilation are more sensitive for detection of early airway disease. In addition to COPD, the other common cause of increased R_{AW} is asthma (Box 273).
Box 273
Asthma
Asthma is a very common condition, occurring in 5% to 10% of the American population. Asthma is primarily an inflammatory disorder; the familiar bronchospasm is secondary to the underlying inflammation. One hypothesis is that asthma represents the inappropriate activation of immune responses designed to combat parasites in the airways. When a susceptible person inhales a trigger (e.g., pollen), inflammatory cells rush into the airways, releasing a multitude of cytokines, leukotrienes, and other humoral substances (e.g., histamine) that induce bronchospasm.
The patient experiencing an acute asthma attack is usually easy to recognize. The classic presentation includes shortness of breath, wheezing, and coughing. Triggers include allergens, heat or cold, a host of occupational irritants, and exercise. The patient can often identify the specific trigger. Spirometry can confirm the diagnosis; the most characteristic feature is a decreased FEV_{1}. Many asthmatic patients use peak flowmeters at home because the severity of symptoms does not always correlate with objective measurements of the disease's severity.
The type of treatment depends on the frequency and severity of the attacks. For patients with infrequent attacks that are not particularly severe, it is often possible to manage the disease with an inhaled β_{2}adrenergic agonist, used only when needed. These medications, easily delivered by a metereddose inhaler that can be carried around in one's pocket or purse, act on β_{2}adrenergic receptors to oppose bronchoconstriction. A patient who requires such an agent more than one or two times a week should receive an inhaled corticosteroid on a regular basis to suppress inflammation. Inhaled corticosteroids generally lack the side effects of oral corticosteroids, but oral corticosteroids may be required in a patient with sustained and severe asthma. Many patients rely on regular dosing of longacting βagonist inhalers and inhaled corticosteroids to keep their asthma under control. Theophylline (a phosphodiesterase inhibitor that raises [cAMP]_{i}; see p. 51), once a mainstay of asthma therapy, is now used far less commonly. Inhaled anticholinergic agents are more useful in COPD patients than in asthmatic patients but can be beneficial in some individuals who cannot tolerate the side effects of βadrenergic agonists (notably tachycardia). Smoothmuscle relaxants (e.g., cromakalimrelated drugs; see p. 198) and other antiinflammatory agents (e.g., leukotriene inhibitors) also play a role in asthma therapy.
Regardless of its cause, increased R_{AW} greatly increases the energy required to move air into and out of the lungs. If the increase in R_{AW} is severe enough, it can markedly limit exercise. In extreme cases, even walking may be more exercise than the patient can manage. The reason is obvious from the pulmonary version of Ohm's law (see Equation 277). The maximal ΔP (ΔP_{max}) that we can generate between atmosphere and alveoli during inspiration, for example, depends on how low we can drive alveolar pressure using our muscles of inspiration. For a given ΔP_{max}, a 3.3fold increase in R_{AW} (COPD column in Table 272) translates to a 3.3fold reduction in maximal airflow (). N277
N277
Effect of Increased Airway Resistance on Maximal Airflow
Contributed by Emile Boulpaep, Walter Boron
Beginning on page 619 of the text, we presented an example in which COPD increased airway resistance (R_{AW}) by a factor of 3.3 (from 1.5 to 5.0 respiratory resistance units of cm H_{2}O/[L/s]; see Table 272). If the maximal ΔP that the subject can generate between atmosphere and alveoli (ΔP_{max}) were to remain normal, then the maximal airflow (; units: L/s) would also decrease by a factor of 3.3:
(NE 279)
Although the relationship between and the maximal amount of O_{2} taken up per minute (; see pp. 1213–1214) is not directly proportional, it is intuitively obvious that as falls, must also fall … and with it, the maximal amount of exercise that the individual can perform. Thus, if a healthy individual—running with maximal effort—were capable of jogging a kilometer in, say, 5 minutes, then a COPD patient might at best be able to walk that distance over a much longer time. With further increases in R_{AW}, the maximal exercise capacity of the COPD patient would gradually decline, and with sufficiently severe disease, the patient might have difficulty even walking along a perfectly flat surface without aid of supplemental oxygen.
Vagal tone, histamine, and reduced lung volume all increase airway resistance
Several factors can modulate R_{AW}, including the autonomic nervous system (ANS), humoral factors, and changes in the volume of the lungs themselves. The vagus nerve, part of the parasympathetic division of the ANS, releases acetylcholine, which acts on an M_{3} muscarinic receptor on bronchial smooth muscle (see p. 341). The result is bronchoconstriction and therefore an increase in R_{AW}. The muscarinic antagonist atropine blocks this action. Irritants such as cigarette smoke cause a reflex bronchoconstriction (see pp. 717–718) in which the vagus nerve is the efferent limb.
Opposing the action of the vagus nerve is the sympathetic division of the ANS, which releases norepinephrine and dilates the bronchi and bronchioles, but reduces glandular secretions. However, these effects are weak because norepinephrine is a poor agonist of the β_{2}adrenergic receptors that mediate this effect via cAMP (see pp. 342–343).
Humoral factors include epinephrine, released by the adrenal medulla. Circulating epinephrine is a far better β_{2} agonist than is norepinephrine and therefore a more potent bronchodilator. Histamine constricts bronchioles and alveolar ducts and thus increases R_{AW}. Far more potent is the bronchoconstrictor effect of the leukotrienes LTC_{4} and LTD_{4}.
One of the most powerful determinants of R_{AW} is lung volume. R_{AW} is extremely high at residual volume (RV) but decreases steeply as V_{L} increases (Fig. 2713A). One reason for this effect is obvious: all pulmonary airways—including the conducting airways, which account for virtually all of R_{AW}—expand at high V_{L}, and resistance falls steeply as radius increases (see Equation 278). A second reason is the principle of interdependence (see p. 613)—alveoli tend to hold open their neighbors by exerting radial traction or mechanical tethering (see Fig. 2713B). This principle is especially important for conducting airways, which have thicker walls than alveoli and thus a lower compliance. At high V_{L}, alveoli dilate more than the adjacent bronchioles, pulling the bronchioles farther open by mechanical tethering. Patients with obstructive lung disease, by definition, have an increased R_{AW} at a given V_{L} (see Fig. 2713A). However, because these patients tend to have a higherthannormal FRC, they breathe at a higher V_{L}, where airway resistance is—for them—relatively low.
FIGURE 2713 Airway resistance.
Intrapleural pressure has a static component (−P_{TP}) that determines lung volume and a dynamic component (PA) that determines airflow
In Equation 272, we defined transpulmonary pressure as the difference between alveolar and intrapleural pressure (P_{TP} = PA − P_{IP}). What is the physiological significance of these three pressures, and how do we control them?
P_{IP} is the parameter that the brain—through the muscles of respiration—directly controls. Rearranging the definition of P_{TP} in Equation 272 yields
(2712)
Thus, P_{IP} has two components, −P_{TP} and PA, as summarized in Figure 2714 and Table 273. As we will see in the next section, P_{TP} and PA literally flow from P_{IP}.
FIGURE 2714 Static and dynamic components of P_{IP}.
TABLE 273
Static Versus Dynamic Properties of the Lungs
PROPERTY 
STATIC 
DYNAMIC 
Anatomical correlate 
Alveoli 
Conducting airways 
Key “constant” 
Static compliance, C 
Airway resistance, R_{AW} 
Key pressure 
P_{TP} (P_{TP} = PA – P_{IP}) 
PA 
Key parameter 
V_{L} (C = ΔV_{L}/ΔP_{TP}) 

Pathology 
Restrictive disease (e.g., fibrosis), caused by ↓ C 
Obstructive disease (e.g., COPD), caused by ↑ R_{AW} 
Transpulmonary Pressure
P_{TP} is a static parameter. It does not cause airflow. Rather, along with static compliance, P_{TP} determines V_{L}. The curve in the lower left part of Figure 2714—like the middle plot of Figure 275—describes how V_{L}depends on P_{TP}. That is, this curve describes the P_{TP} required to overcome the elastic (i.e., static) forces that oppose lung expansion but makes no statement about . We have already seen that the slope of this curve is static compliance (see p. 610), a property mainly of the alveoli, and that a decrease in C can produce restrictive lung disease (see p. 610). Note that P_{TP} not only determines V_{L} under static conditions, when there is no airflow, but also under dynamic conditions (i.e., during inspiration and expiration). However, the brain does not directly control P_{TP}.
Alveolar Pressure
PA is a dynamic parameter. It does not determine V_{L} directly. Instead, along with airway resistance, PA determines airflow. The curve in the lower right part of Figure 2714 describes how depends on PA. That is, this curve describes the PA required to overcome inertial and resistive (i.e., dynamic) forces that oppose airflow but makes no statement about V_{L}. The slope of this plot is airway conductance, the reciprocal of R_{AW}, N278 which is mainly a property of the conducting airways. A decrease in R_{AW} can produce obstructive lung disease. When PA is zero, must be zero, regardless of whether V_{L} is at RV or TLC or anywhere in between. If the PA is positive and the glottis is open, air flows from alveoli to atmosphere, regardless of V_{L}. If PA is negative, air flows in the opposite direction. As is the case with P_{TP}, the brain does not directly control PA.
N278
Airway Resistance
Contributed by Emile Boulpaep, Walter Boron
In the lower right panel of Figure 2714, we have assumed that airway resistance (R_{AW}) is constant during a respiratory cycle, so that the curve is a straight line. However, R_{AW} changes with lung volume (see red and teal curves in Fig. 2713A) and, as discussed on pages 626–627, airway collapse during expiration causes R_{AW} to be greater during expiration than during inspiration.
During inspiration, a sustained negative shift in P_{IP} causes PA to become transiently more negative
During a quiet respiratory cycle—an inspiration of 500 mL, followed by an expiration—the body first generates negative and then positive values of PA. The four large gray panels of Figure 2715 show the idealized time courses of five key parameters. N279 The uppermost panel is a record of V_{L}. The next panel is a pair of plots, −P_{TP} and P_{IP}. The third shows the record of PA. The bottom panel shows a simultaneous record of .
FIGURE 2715 The respiratory cycle. PB, P_{IP}, P_{TP}, and PA are all in centimeters of H_{2}O. The colored points (labeled a, b, c, and d) in each of the central panels correspond to the illustrations on the left with the samecolored backgrounds. The two panels on the far right are taken from Figure 2714.
N279
Model of the Respiratory Cycle
Contributed by Emile Boulpaep, Walter Boron
The four large graphs with gray backgrounds in Figure 2715 represent a hypothetical respiratory cycle. We generated the tracings using a rather simple computer model, making the following assumptions:
1. Throughout the respiratory cycle, the static compliance (C) was a constant 0.2 L/cm H_{2}O.
2. Throughout the respiratory cycle, the resistance (R) was a constant 1 cm H_{2}O/(L/s).
3. The initial lung volume (V_{L}) was 3.2 L, corresponding to time zero in the graphs (i.e., point a in the four panels).
4. During inspiration, the subject shifted P_{IP} from –5 cm H_{2}O to –7.5 cm H_{2}O with an exponential time course (time constant = 0.01 second). This value of τ for P_{IP} results in a τ for V_{L} of ~0.2 seconds, consistent with the theoretical requirement that = R × C = 1 × 0.2 = 0.2 second. Thus, P_{IP} is the prime mover, the parameter that our hypothetical subject controlled. A second consequence is that because (a) P_{IP}changes by 2.5 cm H_{2}O and (b) the compliance is 0.2 L/cm H_{2}O, the total change in lung volume (ΔV_{L}) was 0.5 L.
5. Expiration was the reverse of inspiration, with P_{IP} shifting from –7.5 to –5.0 cm H_{2}O with a time constant, once again, of 0.01 seconds. Again, the τ for V_{L} was ~0.2 seconds.
In the actual modeling, we divided the inspiration and expiration into multiple 10ms intervals. Each interval had two phases, one in which we allowed the change in P_{IP} to change the lung volume—but with no airflow—and a second in which we allowed air to flow as dictated by the alveolar pressure (PA) and airway resistance.
First phase. At time zero (point a in the four panels), we imagined that we plugged the trachea with a cork and then allowed the subject to begin the inspiratory effort for a duration of 10 ms. P_{IP} became slightly more negative according to the aforementioned exponential time course. At the end of the 10ms period, we computed the new value of V_{L}, knowing the current value of P_{IP} and the compliance and also assuming that the air molecules inside the lung obeyed the ideal gas law (PA ⋅ V_{L} = nRT). Keep in mind that because we plugged the trachea with a cork, the number of air molecules in the lung was fixed during the 10ms period of inspiratory effort. Thus, we could compute the new value of PA. From this new value of PA and the current value of P_{IP}, we could compute the P_{TP} after 10 ms. Thus, at the end of the first phase, we have new values for P_{IP}, V_{L}, PA, and P_{TP}.
Second phase. Once we knew the new value of PA, we removed our hypothetical cork for 10 ms and allowed air to flow into the lung, governed by the equation = ΔP/R. Note that ΔP is the difference between PB and PA. At the end of 10 ms, we replaced the cork (the flow of air is far from complete!). Knowing how many air molecules entered during the 10ms interval, and assuming that P_{IP} has not changed, we can now recompute all the relevant parameters and start the first phase of the second cycle.
Thus, although we assumed the time course of the green P_{IP}versustime curve in the second gray panel of Figure 2715, all of the other curves follow directly from the calculations and the assumed parameters. The time constant (τ) for V_{L} that falls out of our computer simulation is ~0.2 second. As we will see in N2710 the theoretical time constant for a change in lung volume is the product R × C. In the model we have been discussing, R × C evaluates to a time constant of 0.2 seconds, which agrees well with our simulation. Note that the τ we assumed for the change in P_{IP} was 0.01 second, which is considerably faster than the τ for the change in V_{L} and thus is not rate limiting.
N2710
Calculating the Time Constant for a Change in Lung Volume
Contributed by Emile Boulpaep, Walter Boron
It should not be very surprising that the τ for the time course of V_{L} is 0.2 second. Just as for electrical circuits, where the time constant of a resistorcapacitor network is R × C, where R is resistance and C is capacitance, the time constant for inflating a lung is the product of the airway resistance and the static compliance:
(NE 2710)
For healthy human lungs, we might assume that R is ~1 cm H_{2}O/(L/s)—on page 617 we pointed out that normal values for R might range from 0.6 to 2.3 cm H_{2}O/(L/s). We also might assume that C is ~0.2 L/cm H_{2}O (see Box 261). Thus, our predicted time constant is
(NE 2711)
If R increased 5fold from 1 to 5 cm H_{2}O/(L/s), the time constant would also quintuple:
(NE 2712)
Note that the time constant that we have computed is a theoretical minimum. The actual τ can only be as small as R × C if the individual making an inspiratory or expiratory effort decides to change P_{IP} (or is capable of changing P_{IP}) very rapidly compared to R × C. If the person decides to inhale or exhale slowly, then of course the actual time course with which V_{L} changes can be far larger than R × C. As an extreme example, if an oboe player decides to exhale over a period of many tens of seconds, the time constant no longer depends on the mechanical properties of the lung, but on the activity of the cerebral cortex.
On the right side of Figure 2715 are the static P_{TP}V_{L} curve and the dynamic relationship (both copied from Fig. 2714). On the left side is a series of four cartoons that represent snapshots of the key pressures (i.e., P_{IP}, P_{TP}, and PA) at four points during the respiratory cycle:
a. Before inspiration begins. The lungs are under static conditions at a volume of FRC.
b. Halfway through inspiration. The lungs are under dynamic conditions at a volume of FRC + 250 mL.
c. At the completion of inspiration. The lungs are once again under static conditions but at a volume of FRC + 500 mL.
d. Halfway through expiration. The lungs are under dynamic conditions at a volume of FRC + 250 mL.
a. At the end of expiration/ready for the next inspiration. The lungs are once again under static conditions at a volume of FRC.
The V_{L} record in the top gray panel of Figure 2715 shows that V_{L} rises more or less exponentially during inspiration and similarly falls during expiration.
Knowing the time course of V_{L}, we obtained the P_{TP} values in the second gray panel of Figure 2715 by reading them off the static P_{TP}V_{L} diagram to the right and plotted them as −P_{TP} (for consistency with Equation 2712). As V_{L} increases during inspiration, P_{TP} increases (i.e., −P_{TP} becomes more negative). The opposite is true during expiration. Remember that P_{TP} (along with static compliance) determines V_{L} at any time.
The P_{IP} record in the second gray panel of Figure 2715 shows that P_{IP} is the same as −P_{TP} whenever the lungs are under static conditions (points a, c, and a). However, during inspiration, P_{IP} rapidly becomes more negative than −P_{TP} but then merges with −P_{TP} by the end of inspiration. The difference between P_{IP} and −P_{TP} is PA, which must be negative to produce airflow into the lungs. During expiration, P_{IP} is more positive than −P_{TP}.
The PA record in the third gray panel of Figure 2715 shows that PA is zero under static conditions (points a, c, and a). During inspiration, PA rapidly becomes negative but then relaxes to zero by the end of inspiration. The opposite is true during expiration. The PA values in this plot represent the differences between the P_{IP} and −P_{TP} plots in the preceding panel.
We computed (bottom gray panel) from the relationship in Equation 277: = (PA − PB)/R_{AW}. Remember that PA (along with R_{AW}) determines at any time. Here, we assume that R_{AW} is fixed during the respiratory cycle at 1 cm H_{2}O/(L/s). Thus, the record has the same time course as PA.
The key message in Figure 2715 is that during inspiration, the negative shift in P_{IP} has two effects. The body invests some of the energy represented by ΔP_{IP} into transiently making PA more negative (dynamic component). The result is that air flows into the lungs and V_{L} increases; but this investment in PA is only transient. Throughout inspiration, the body invests an increasingly greater fraction of its energy in making P_{TP} more positive (static component). The result is that the body maintains the new, higher V_{L}. By the end of inspiration, the body invests all of the energy represented by ΔP_{IP} in maintaining V_{L} and none in further expansion. The situation is not unlike that faced by Julius Caesar as he, with finite resources, conquered Gaul. At first, he invested all of his resources in expanding his territory at the expense of the feisty Belgians; but as the conquered territory grew, he was forced to invest an increasingly greater fraction of his resources in maintaining the newly conquered territory. In the end, he necessarily invested all of his resources in maintaining his territory and was unable to expand further.
Dynamic compliance falls as respiratory frequency rises
In the preceding section, we examined pressure, volume, and flow changes during an idealized respiratory cycle of 5 seconds, which corresponds to a respiratory frequency of 12 breaths/min. The top curve in Figure 2716A shows a normal V_{L} time course during an inspiration. As for any exponential process, the time constant (τ) N2710 is the interval required for ΔV_{L} to be ~63% complete. For healthy lungs, τ is ~0.2 second. Thus, for inspiration, the increase in V_{L} is 63% complete after 0.2 second, 86% complete after 0.4 second, 95% complete after 0.6 second, and so on. We will make the simplifying assumption that the time available for inspiration is half this time or 2.5 seconds, which represents >12 time constants! Thus, if the V_{T} after infinite time were 500 mL, the ΔV_{L} measured 2.5 seconds after initiation of inspiration would also be ~500 mL (see Fig. 2716A, green point on top curve). In Figure 2716B, we replot this value as the green point at a frequency of 12 breaths/min on the top curve (i.e., normal lungs).
FIGURE 2716 Dynamic compliance. In A and B, the colored points represent changes in volume at respiratory frequencies of 12, 24, and 48 breaths/min.
For a respiratory frequency of 24 breaths/min, 1.25 seconds is available for inspiration. At the end of this time, the ΔV_{L} is ~499 mL (see Fig. 2716B, blue point on the top curve).
If we further increase the respiratory frequency to 48 breaths/min, only 0.625 second is available for inspiration. At the end of this period, only slightly more than three time constants, the ΔV_{L} is ~478 mL (see Fig. 2716B, red point on top curve). Thus, over a wide range of frequencies, ΔV_{L} is largely unchanged. Only when respiratory frequency approaches extremely high values does ΔV_{L} begin to fall off.
The situation is very different for a subject with substantially increased airway resistance. If R_{AW} increased 5fold, τ would also increase 5fold to 1 second. N2710 As a result, the trajectory of V_{L} also would slow by a factor of 5 (see Fig. 2716A, bottom curve). If we could wait long enough during an inspiration, these unhealthy lungs would eventually achieve a ΔV_{L} of 500 mL. The problem is that we cannot wait that long. Therefore, if the frequency is 12 breaths/min and inspiration terminates after only 2.5 seconds (i.e., only 2.5 time constants), the ΔV_{L} would be only 459 mL (see Fig. 2716B, green point on bottom curve). Thus, even at a relatively low frequency, the patient with increased R_{AW} achieves a ΔV_{L} that is well below normal.
For a respiratory frequency of 24 breaths/min, when only 1.25 seconds is available for inspiration, the ΔV_{L} is only 357 mL (see Fig. 2716B, blue point on bottom curve). At a respiratory frequency of 48 breaths/min, when only 0.625 second is available for inspiration, the ΔV_{L} is only 232 mL (see Fig. 2716B, red point on bottom curve). Thus, the ΔV_{L} for the subject with increased R_{AW} falls rapidly as frequency increases.
We can represent the change in V_{L} during cyclic breathing by a parameter called dynamic compliance:
(2713)
Note that C_{dynamic} is proportional to ΔV_{L} in Figure 2716B. Under truly static conditions (i.e., frequency of 0), −ΔP_{IP} equals ΔP_{TP}, and dynamic compliance is the same as static compliance (C or C_{static}) that we introduced in Equation 274. As frequency increases, C_{dynamic} falls below C_{static}. The degree of divergence increases with resistance. For the normal lung in Figure 2716B, ΔV_{L} and thus C_{dynamic} fall by only ~5% as frequency rises from 0 to 48 breaths/min. However, for the lung with a 5fold increased airway resistance, C_{dynamic} falls by >50% as frequency rises from 0 to 48 breaths/min.
The above pathological pattern is typical of asthma; R_{AW} is elevated, but C_{static} is relatively normal. In emphysema, both R_{AW} and C_{static} are elevated (see Fig. 275, upper curve). Thus, a plot of C_{dynamic} versus frequency would show that C_{dynamic} is initially greater than C_{static} at low respiratory frequencies, but it falls below C_{static} as frequency increases. What do these frequencydependent decreases in C_{dynamic} mean for a patient? The greater the respiratory frequency, the less time is available for inspiration or expiration, and the smaller the V_{T} (see Fig. 2716C). N2711
N2711
Implications of HighFrequency Breathing in a Patient with a High Time Constant
Contributed by Emile Boulpaep, Walter Boron
As discussed in the text, a patient with increased airway resistance will require longer than normal to inflate the lungs by the expected V_{T} during an inspiratory effort (i.e., τ is increased, as in Fig. 2716A) and will require longer than normal to deflate the lungs to the FRC during an expiratory effort. As a consequence, as such a patient breathes with everincreasing respiratory frequencies, the effective V_{T} will fall (see Fig. 2716C). The effect on the ventilation of alveoli is even more severe.
As we shall see in Chapter 31, even in healthy lungs, not all of the V_{T} reaches the alveoli—a fixed amount (~150 mL per breath) is wasted ventilating the conducting airways (see pp. 675–676), which have no alveoli. Only the part of the V_{T} in excess of, say, 150 mL actually reaches the alveoli. Thus, as the V_{T} gradually falls at increasing respiratory frequencies, the first volume to fall is the air ventilating the alveoli. The fall in V_{T} in a patient with increased airway resistance (i.e., an increased τ) can therefore cause alveolar ventilation to fall far more steeply than the total ventilation.
This analysis greatly oversimplifies what happens in the lungs of real people. Although we have treated the lungs as if there were one value for R_{AW} and one for C_{static}, each conducting airway has its own airway resistance and each alveolar unit has its own static compliance, and these values vary with parameters such as V_{L}, posture, and hormonal status. As a result, some alveolar units have greater time constants than others. Airway disease may make some of these time constants substantially higher. As respiratory frequency increases, alveoli with relatively high time constants will have less time to undergo volume changes. As a result, these “slow” airways—compared with the “faster” airways—will make progressively smaller contributions to the overall ventilation of the lungs. At sufficiently high frequencies, very slow airways may drop out of the picture entirely.
Transmural pressure differences cause airways to dilate during inspiration and to compress during expiration
We have noted three factors that modulate airway caliber: (1) the ANS, (2) humoral substances, and (3) V_{L} (see Fig. 2713). A fourth factor that modulates R_{AW} is flow of air through the conducting airway itself. Airflow alters the pressure difference across the walls of an airway, and this change in transmural pressure (P_{TM}) can cause the airway to dilate or to collapse. Figure 2717A through C depicts the pressures along a single hypothetical airway, extending from the level of the alveolus to the lips, under three conditions: during inspiration (Fig. 2717A), at rest (Fig. 2717B), and during expiration (Fig. 2717C). In all three cases, the lung is at the same volume, FRC; the only difference is whether air is flowing into the lung, not flowing at all, or flowing out of the lung. Because V_{L} is at FRC, P_{TP} (the P_{TM} for the alveoli) is 5 cm H_{2}O in all three cases.
FIGURE 2717 Dilation and collapse of airways with airflow. In all four panels, V_{L} is FRC. P_{TM} is the transmural pressure across conducting airways. P_{AW} and values in pale blue balloons represent pressures inside conducting airways (all in centimeters of H_{2}O). Graphs at bottom show P_{AW} profiles along airways (blue curve) from alveolus to mouth. P_{IP} is constant throughout the thorax (green curve). The upward (i.e., positive) P_{TM} arrows tend to expand the airways, whereas the downward (i.e., negative) P_{TM} arrows tend to squeeze them.
Static Conditions
First consider what happens under static conditions (see Fig. 2717B). In the absence of airflow, the pressures inside all airways must be zero. Considering first the alveoli, P_{TP} is 5 cm H_{2}O and the PA is zero, and thus P_{IP} is −5 cm H_{2}O. We will ignore the effects of gravity on P_{IP} and thus assume that P_{IP} is uniform throughout the chest cavity. The P_{IP} of −5 cm H_{2}O acts not only on alveoli but on all conducting airways within the thoracic cavity. For these, P_{TM} at any point is the difference between the pressure inside the airway (P_{AW}) and P_{IP} (see Equation 271):
(2714)
In other words, a transmural pressure of +5 cm H_{2}O acts on all thoracic airways (but not the trachea in the neck, for example), tending to expand them to the extent that their compliance permits.
Inspiration
Now consider what happens during a vigorous inspiration (see Fig. 2717A). We first exhale to a V_{L} below FRC and then vigorously inhale, so that the PA is −15 cm H_{2}O at the instant that V_{L} passes through FRC. Because the lung is at FRC, P_{TP} is +5 cm H_{2}O. The P_{IP} needed to produce a PA of −15 cm H_{2}O is
(2715)
This “inspiring” P_{IP} of −20 cm H_{2}O is just enough to produce the desired airflow and also to maintain the alveoli at precisely the same volume that they had under static conditions. But how does this exceptionally negative P_{IP} affect airways upstream from the alveoli? P_{AW} gradually decays from −15 cm H_{2}O in the alveoli to zero at the lips. The farther we move from the alveoli, the less negative is P_{AW}, and thus the greater is P_{TM}.
As an illustration, consider a point about halfway up the airway's resistance profile, where P_{AW} is −8 cm H_{2}O:
(2716)
Thus, the transmural pressure opposing the elastic recoil at this point has increased from +5 cm H_{2}O at rest (see Fig. 2717B) to +12 cm H_{2}O during this vigorous inspiration (see Fig. 2717A). Because P_{TM} has increased, the airway will dilate. The tendency to dilate increases as we move from the alveoli to larger airways. As shown in the graph in the lower part of Figure 2717A, P_{AW} (and thus P_{TM}, as indicated by the upward arrows) gradually increases. Note that the very positive P_{TM} values that develop in the larger airways determine only the tendency to dilate. The extent to which an airway actually dilates also depends on its compliance. The amount of cartilage supporting the airways gradually increases from none for 11thgeneration airways to a substantial amount for the mainstem bronchi. Because the increasing amount of cartilage in the larger airways decreases their compliance, they have an increasing ability to resist changes in caliber produced by a given change in P_{TM}.
Expiration
As might be expected, conducting airways tend to collapse during expiration (see Fig. 2717C). We first inhale to a V_{L} above FRC and then exhale vigorously, so that PA is +15 cm H_{2}O at the instant that V_{L} passes through FRC. Because the lung is at FRC, P_{TP} is +5 cm H_{2}O. The P_{IP} needed to produce a PA of +15 cm H_{2}O is
(2717)
This P_{IP} of +10 cm H_{2}O is 5 less than the PA and thus maintains the alveoli at the same volume that prevailed under static conditions and during inspiration. What is the effect of this very positive P_{IP} on the upstream airways? P_{AW} must decrease gradually from +15 cm H_{2}O in the alveoli to zero at the lips. The farther we move from the alveolus, the lower the P_{AW} and thus the lower the P_{TM}. At a point about halfway up the airway's resistance profile, where P_{AW} is +8 cm H_{2}O,
(2718)
Thus, at this point during a vigorous expiration, the transmural pressure opposing elastic recoil has fallen sharply from +5 cm H_{2}O at rest (which tends to mildly inflate the airway) to −2 cm H_{2}O during expiration (which actually tends to squeeze the airway). As we move from the alveoli to larger airways, P_{AW} gradually decreases. That is, P_{TM} gradually shifts from an everdecreasing inflating force (positive values) to an everincreasing squeezing force (negative values), as indicated by the change in the orientation of the arrows in the lower panel of Figure 2717C. Fortunately, these larger airways—with the greatest collapsing tendency—have the most cartilage and thus some resistance to the natural collapsing tendency that develops during expiration. In addition, mechanical tethering (see p. 620) helps all conducting airways surrounded by alveoli to resist collapse. Nevertheless, R_{AW} is greater during expiration than it is during inspiration.
The problem of airway compression during expiration is exaggerated in patients with emphysema, a condition in which the alveolar walls break down. This process results in fewer and larger air spaces with fewer points of attachment and less mutual buttressing of air spaces. Although the affected alveoli have an increased compliance and thus a larger diameter at the end of an inspiration, they are flimsy and exert less mechanical tethering on the conducting airways they surround. Thus, patients with emphysema have great difficulty exhaling because their conducting airways are less able to resist the tendency to collapse. However, these patients make their expirations easier in three ways. We could predict them all from our knowledge of dynamic respiratory mechanics:
1. They exhale slowly. A low during expiration translates to a less positive PA and thus a less positive P_{IP}, which minimizes the tendency to collapse.
2. They breathe at higher V_{L}. A high V_{L} maximizes the mechanical tethering that opposes airway collapse during expiration and thus minimizes R_{AW} (see Fig. 2713A).
3. They exhale through pursed lips. This maneuver—known as puffing—creates an artificial high resistance at the lips. Because the greatest pressure drop occurs at the location of the greatest resistance, puffing causes a greater share of the P_{AW} drop to occur across the lips than along collapsible, cartilagefree airways. Thus, puffing maintains relatively high P_{AW} values farther along the tracheobronchial tree (see Fig. 2717D) and reduces collapsing tendencies throughout. The greatest collapsing tendencies are reserved for the largest airways that have the most cartilage (see p. 597). Of course, the patient pays a price for puffing: and thus the ventilation of the alveoli are low.
Because of airway collapse, expiratory flow rates become independent of effort at low lung volumes
Cartilage and mechanical tethering (see p. 620) oppose the tendency of conducting airways to collapse during expiration. Because tethering increases as V_{L} increases, we expect airways to better resist collapse when V_{L} is high. To see if this is true, we will examine how expiratory airflow varies with effort (i.e., PA) at different V_{L}.
Imagine that we make a maximal inspiration and then hold our breath with glottis open (Fig. 2718A). Thus, PA is zero. In addition, P_{TP} is +30 cm H_{2}O to maintain TLC. From Equation 2712, P_{IP} = (−P_{TP}) + PA = −30 cm H_{2}O. Now, starting from TLC, we make a maximal expiratory effort. Figure 2718B summarizes the pressures in the alveoli and thorax at the instant we begin exhaling but before V_{L} has had time to change. P_{TP} is still +30 cm H_{2}O, but PA is now +40 cm H_{2}O (to produce a maximal expiration) and P_{IP} is therefore +10 cm H_{2}O. As V_{L} decreases during the course of the expiration, we will monitor , V_{L}, and PA.
FIGURE 2718 Dependence of airflow on effort. In D, note that if airway resistance were fixed, increased effort (i.e., increased PA) would yield a proportionate increase in , as indicated by the red line. However, increased effort tends to narrow airways, raising resistance and tending to flatten the curves. As V_{L} decreases (e.g., 4 or 3 L), airflow becomes independent of effort with increasingly low effort. PB, P_{IP}, P_{TP}, and PA are all in centimeters of H_{2}O.
The top curve in Figure 2718C shows how changes as a function of V_{L} when, starting from TLC, we make a maximal expiratory effort. Notice that rises to its maximal value at a V_{L} that is somewhat less than TLC and then gradually falls to zero as V_{L} approaches RV. The data in this top curve, obtained with maximal expiratory effort (i.e., at maximal PA), will help us determine how expiratory flow varies with effort. To get the rest of the necessary data, we repeat our experiment by again inhaling to TLC and again exhaling to RV. However, with each trial, we exhale with less effort (i.e., at smaller PA)—efforts labeled as high, medium, and low in Figure 2718C.
If we draw a vertical line upward from a V_{L} of 5 L in Figure 2718C, we see that, at this very high V_{L}, gets larger and larger as the effort increases from low to medium to high to maximal. Because these efforts correspond to increasing PA values, we can plot these four data points (all obtained at a V_{L} of 5 L) versus PA in the top curve of Figure 2718D. Because increases continuously with the PA, flow is effort dependent at a V_{L} of 5 L. If the airways were made of steel, this plot of versus PA would be a straight line. Because the actual plot bends downward at higher values of PA (i.e., greater efforts), R_{AW} must have increased with effort (i.e., the airways collapsed somewhat).
Returning to Figure 2718C, we see that an important characteristic of these data is that the curve for maximal expiratory effort defines an envelope that none of the other three curves could penetrate. Thus, at a V_{L} of 4 L, is ~7 L/s, regardless of whether the expiratory effort is high or maximal (i.e., the two points overlie one another on the graph). At a V_{L} of 3 L, the four curves have practically merged, so that is ~5 L/s regardless of effort. Thus, at lung volumes that are <3 L, it does not matter how much effort we make; the expiratory flow can never exceed a certain value defined by the envelope.
Shifting back to Figure 2718D, we see that for the lower two plots, increases with PA—up to a point. Further increases in effort (i.e., PA) are to no avail because they produce a proportional increase in R_{AW}—expirationinduced airway collapse. Thus, the more positive values of P_{IP} not only produce more positive values of PA but also increase R_{AW}, so that PA/R_{AW} and thus remain constant.
At low lung volumes, flow becomes effort independent because the reduced mechanical tethering cannot oppose the tendency toward airway collapse that always exists during expiration. Moreover, at progressively lower V_{L}, flow becomes effort independent earlier. In other words, particularly at low lung volumes, it simply does not pay to try any harder.