chapter 4 respiratory mechanics and gas exchange · 4.4 mechanics of the human body fluid)...

4.1

CHAPTER 4

RESPIRATORY MECHANICS ANDGAS EXCHANGE

James B. GrotbergUniversity of Michigan, Ann Arbor, Michigan

4.1 ANATOMY 4.1 4.6 AIRWAY FLOW, DYNAMICS, AND4.2 MECHANICS OF BREATHING 4.3 STABILITY 4.114.3 VENTILATION 4.4 REFERENCES 4.124.4 ELASTICITY 4.7 BIBLIOGRAPHY 4.144.5 VENTILATION, PERFUSION, ANDLIMITS 4.9

As shown in Fig. 4.1, the pulmonary system consists of two lungs; each lung is conical in shape, withan inferior border (or base) that is concave as it overlies the diaphragm and abdominal structures anda superior border (or apex) that is convex and extends above the first rib. The anterior, lateral, andposterior lung surfaces are adjacent to the rib cage, while the medial surface is adjacent to themediastinum, which contains the heart, great vessels, and esophagus. All of the lung surface iscovered by the visceral pleural membrane, while the inside of the chest wall, mediastinum, anddiaphragm are covered by the parietal pleural membrane. The two pleural membranes are separatedby a thin liquid film (~20 to 40 µm thick) called the pleural fluid which occupies the pleural space,Fig. 4.1. This fluid lubricates the sliding motion between the lung and chest wall, and its pressuredistribution determines lung inflation and deflation. Because the air-filled lung is surrounded by thepleural liquid, it experiences a buoyancy force that contributes to the overall force including lungweight and fluid pressures.17

The right main bronchus branches from the end of the trachea to the right lung, while the left mainbronchus branches to the left. This division of a “parent” airway into two “daughter” airways is calleda bifurcation. Each level of branching is called a generation and is given an integer value n,according to the Weibel symmetric model35 shown in Table 4.1. The trachea is designated as n = 0, themain bronchi, n = 1, etc. For a symmetrically bifurcating geometry, then, there would be 2n airwaysat each generation level. Generations 0 � n � 16 constitute the conducting zone (trachea to terminal

4.1 ANATOMY

4.1.1 General

4.1.2 Airway Divisions

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4.2 MECHANICS OF THE HUMAN BODY

bronchiole) whose airways have no alveoli and, hence, do not participate in gas exchange with thepulmonary blood circulation. Each terminal bronchiole enters an acinus or respiratory zone unit for17 � n � 23; see Fig. 4.2. The respiratory bronchioles occupy generations 17 � n � 19 and are tubeswith partially alveolated surfaces. Generations 20 � n � 22 are the alveolar ducts, tubes with totallyalveolated surfaces. The alveoli are polyhedral in shape, but, viewed as nearly spherical caps, theyeach have a diameter of ~200 µm at ¾ inflation.

The pulmonary circulation consists of mixed venous blood entering the pulmonary artery from theright ventricle of the heart. After passing through the pulmonary capillary bed, the blood returns tothe left atrium via the pulmonary vein. Flow through portions of the capillary bed that are inventilated alveolar regions of the lung will discharge CO2 and absorb O2. Flows that go tounventilated alveoli will not exchange gas and are called the shunt flow. Typical values of pulmonaryarterial blood pressures are 20 mmHg systolic/12 mmHg diastolic, about one-sixth the systemicvascular pressures. Because the pulmonary system is a low-pressure system, the vessel walls are muchthinner than their systemic counterparts. The pulmonary arterial and venous systems have accompa-nying divisions, forming a triad with each airway division. The pulmonary capillaries, with a vesseldiameter of ~6 µm, form a multivessel mesh surrounding each alveolus and are contained in thealveolar walls. The blood-gas barrier is composed of (starting from the gas side): the alveolar liquidlining, alveolar epithelial cell, basement membrane, and capillary endothelial cell. These combinedlayers sum to approximately 0.5 to 1.0 µm, in thickness. This is the distance gas must diffuse to enteror leave the blood under normal conditions.

The conducting airways are fed by a second blood supply, the bronchial circulation, which comesfrom the systemic circulation via the bronchial arteries. Blood returns via the bronchial veins, whichempty partly to the systemic veins and partly to the pulmonary veins. The latter case is a circuit thatbypasses the lungs entirely and makes the systemic cardiac output slightly larger than the pulmonarycardiac output or lung perfusion .

4.1.3 Pulmonary Circulation

FIGURE 4.1 Sketch of the pulmonary system. FIGURE 4.2 The respiratory zone of theairway network.



RESPIRATORY MECHANICS AND GAS EXCHANGE

RESPIRATORY MECHANICS AND GAS EXCHANGE 4.3

To maintain homeostasis, the continuous transudation of fluid and solutes from the pulmonary capillarybed into the surrounding interstitium and alveolar space is balanced by lymphatic drainage out of thelung. The lymphatic flow is directed toward the hilum from the pleural surfaces. From lymph nodes inthe hilum, the lymph travels to the paratracheal nodes and then eventually into the venous system viathe thoracic duct. The lung has nerve fibers from both the vagal nerves (parasympathetic) and thesympathetic nerves. The efferent fibers go to the bronchial musculature and the afferents come from thebronchi and alveoli.

The rib cage and its muscles form the chest wall, which protects the vital thoracic organs whilekeeping the lungs inflated. During inspiration, the ribs swing on an axis defined by their articulationwith the vertebrae,20 dashed lines in Fig. 4.3. The result is that upper ribs, such as rib 1, swingforward and up, like a pump handle, increasing the anterior-posterior diameter of the upper chestwall. However, lower ribs swing primarily laterally, like a bucket handle pinned at the spine andsternum. Their motion increases the lateral diameter of the thorax.

The major muscles involved in inspiration and expiration include the diaphragm and the intercostalmuscles. During inspiration, the diaphragm contracts, pulling the inferior lung surface (via the pleural

4.1.4 Lymphatics and Nerves

4.2 MECHANICS OF BREATHING

4.2.1 Chest Wall

4.2.2 Muscles of Inspiration and Expiration

FIGURE 4.3 Rib motion during inspiration.





fluid) downward, while the external intercostal muscles contract, lifting the ribs; see Fig. 4.4. Expira-tion is primarily a passive event; the elastic structures simply return to their original, less-stretched,state as the diaphragm and external intercostals relax. At a normal breathing rate of 15 breaths perminute (bpm), inspiration may occupy one-third of the 4-second breathing cycle, while passiveexpiration occupies the rest, an inspiratory to expiratory time ratio of 1:2. Forceful expiration isaccomplished by contraction of the internal intercostal muscles and the abdominal wall muscles thatsqueeze the abdominal contents hard enough to push them upward. The normal inspiratory toexpiratory time ratio can increase with forceful expiration, as may occur during exercise, or decreasewith prolonged expiration, as one finds in obstructive airways disease like asthma or emphysema.

There is common terminology for differentlung volume measurements, as shown in Fig.4.5. The maximum volume is total lung ca-pacity (TLC), which can be measured by di-lution of a known amount of inspired heliumgas whose insolubility in tissue and bloodprevents it from leaving the air spaces. Theminimum is residual volume (RV). Normalventilation occurs within an intermediaterange and has a local minimum called func-tional residual capacity (FRC). The volumeswing from FRC to the end of inspiration isthe tidal volume VT. The vital capacity (VC)is defined by VC = TLC - RV. Within thelung there are also important volume con-cepts. The anatomic dead space VD is thesummed volume of all conducting airways,measured by the Fowler method, while thephysiologic dead space is that portion of the

FIGURE 4.4 Diaphragm and abdominal muscles during inspiration andexpiration.

4.3 VENTILATION

4.3.1 Lung Volumes

FIGURE 4.5 Lung volumes and definitions.





lung that does not transfer CO2 from the capillaries, measured by the Bohr method. While themethods used to determine these two values are different, normally they yield essentially the sameresult, approximately 150 cm3 in an adult male. In some disease states, however, the Bohr methodmay be affected by abnormalities of the ventilation-perfusion relationship. The alveolar volume VA,where gas exchange occurs, is the lung volume minus the dead space.

A typical VT of 500 cm3 at a rate of 15 bpm yields a total ventilation of (0.5 liters × 15 breaths/min)= 7.5 liters/min. Assuming that air flows in the conducting airways like a plug, for each 500 cm3

breath inspired the tail 150 cm3 fills the dead space while the front 350 cm3 expands the alveoli,where it mixes with the alveolar gas previously retained at FRC. This makes the alveolar ventilation

liters × 15 bpm = 5.25 liters/min. The plug flow assumption is not correct, ofcourse, but for resting ventilation it is a useful simplification. It fails, for example, in high-frequencyventilation,3,24 where the mechanical ventilator can operate at 15 Hz with VT � 5 cm3; i.e., the tidalvolumes can be smaller than the dead space. Adequate gas exchange occurs under these circum-stances, because of the Taylor dispersion mechanism,33 which is a coupling of curvilinear, axialvelocity profiles with radial diffusion applied to a reversing flow.5 The properties of airway branch-ing, axial curvature, and flexibility can modify the mechanism considerably.6,8,14

The ventilatory flow rate results from the pressure boundary conditions imposed at the tracheaand alveoli. In lung physiology, the flow details are often neglected and simply lumped into aresistance to air flow defined as the ratio of the overall pressure drop ∆P to the flow rate; i.e., ∆P/= Res. A typical value is Res = 0.2 cmH2O-s/L for resting breathing conditions. The appeal of definingairway resistance this way is its analogy to electrical circuit theory and Ohm’s law, and other elementssuch as capacitance (see compliance, below) and inertance may be added.7,27,32 Then an airway atgeneration n has its own resistance, Resn, that is in series with its n - 1 and n + 1 connections but inparallel with the others at n.

The detailed fluid dynamics can be viewed as contributing to either the pressure drop due tokinetic energy changes, ∆PK, or that due to frictional, viscous, effects, ∆PF, so that

(4.1)

For a frictionless system, ∆PF = 0, Eq. (4.1) becomes the Bernoulli equation, from which we learnthat, in general, , where uout(uin) is the outlet (inlet) velocity. For a given flowrate , the average air velocity at generation n is un = /An, where An is the total cross-sectional areaat generation n. Because A0 A23, air velocities diminish significantly from the tracheal value to thedistal airways. This implies that ∆PK < 0 for inspiration while ∆PK > 0 for expiration.

The frictional pressure drop ∆PF is always positive, but its value depends on the flow regime in eachairway generation. For fully developed, laminar tube flow (Poiseuille flow) ,where µ is the fluid (gas) viscosity, d is the tube diameter, and L is the axial distance. This formula ishelpful in understanding some of the general trends for R, like its strong dependence on airway diameter,which can decrease in asthma. Airways are aerodynamically short tubes under many respiratory situations,i.e., not long enough for Poiseuille flow to develop. Then the viscous pressure drop is governed by energydissipation in a thin boundary layer region near the airway wall where the velocity profile is curvilinear.For this entrance flow, ∆PF ~ V3/2, so resistance now is flow dependent. The local Reynolds number isgiven by Ren = undn/ν, where ν ≈ 0.15 cm2/s is the kinematic viscosity of air. For Ren � 2300, (see Table4.1), the flow is turbulent and ∆PF in those airways has an even stronger dependence on ventilation,

smooth wall, ~ 2 rough wall. Expressing these three cases for a single tube (airway) in termsof the friction coefficient, CF = ∆PF/1/2ρu2. The comparison is shown in Eq. (4.2):

(4.2)

4.3.2 Air Flow and Resistance





where k depends on the wall roughness; a typical value is k = 0.04. Experiments of inspiratory flowwith a multigeneration cast of the central airways31 show that Poiseuille effects dominate for 350 �

Re0 � 500, entrance effects for 500 � Re0 � 4000, and rough-walled turbulent effects for 4000 �

Re0 � 30,000, approximately. CF tends to be higher during expiration than inspiration.As shown from measurements in airway models19 or theories using modifications of the friction

coefficients,28 the vast majority of the pressure drop across the entire lung occurs within the largeairways, say 0 � n � 8. Thus clinical evaluation of total airway resistance can miss diseases of thesmall airways whose diameters are less than 2 mm, the so-called silent zone of the network. Pressuredrop measurements and models are also important for the design of ventilators and other respiratoryassist or therapeutic devices that interface with lung mechanics.

Transport of O2, CO2, anesthetics, toxins, or any other soluble gas occurs by convection, diffusion,and their interplay. The ability of one gas species to diffuse through another is quantified by themolecular diffusivity D for the gas pair. For O2 diffusing through air, the value of D is

(at atmospheric pressure and 37°C), whereas for CO2 the value is less, . A characteristic time for diffusion to provide effective transport of a concentration front

over a distance L is Td = L2/D, where L can be the airway length Ln. By contrast, a characteristic timefor convective transport over the same distance is Tc = L/U, where U = un is the average flow speedof the bulk gas mixture in an airway. The ratio of the two time scales tells us which mechanism,diffusion, or convection, may dominate (i.e., occur in the shortest time) in a given airway situation.This ratio is the Peclet number, Pe = Td/Tc = UL/D, where Pe 1 indicates convection-dominatedtransport while Pe 1 indicates diffusion-dominated transport. Table 4.1 shows how Pen for O2

transport varies through the airway tree when . Convection dominates in the conduct-ing airways, where negligible O2 is lost, so the O2 concentration in these airways on inspiration isessentially the ambient inspired level, normally . At approximately generation 15,where Pe is close to unity and both convection and diffusion are important, the axial concentrationgradient in O2 develops over the next ~0.5 cm in path length to reach the alveolar level 100mmHg). CO2 has the reverse gradient during inspiration from its alveolar level to the

4.3.3 Gas Transport

TABLE 4.1 Airway Geometry

*Ren and Pen for O2 transport are evaluated for .Source: Data from Weibel (1963) (Ref. 35).





ambient level of near zero. During expiration the alveolar levels of both gases are forced out throughthe airway tree.

Once in the alveoli, gas transport through the al-veolar wall and endothelium into the blood (or viceversa) is diffusive transport, which is quantified by theconcept of the lung’s diffusing capacity. As shown inFig. 4.6, with wall thickness h, total alveolar mem-brane surface area Am, and a gas concentration or par-tial pressure difference of P1 (alveolar) - P2 (bloodserum), the mass flow rate across is

. Dgas-tiss isthe diffusivity of the gas (O2, CO2) in tissue and is pro-portional to solubility/(molecular weight)1/2, making

since CO2 has far greater solubilityin tissue. The details of the area, thickness, anddiffusivity are lumped into an overall coefficient calledthe diffusing capacity DL. The diffusion equation can besimplified for transport of carbon monoxide, CO, whichis so rapidly taken up by hemoglobin within the redblood cells that the serum P2 value is essentially zero. Then we find, after rearranging, that the diffusing

capacity is given by and there are both single-breath and steady-state methods employed in pulmonaryfunction testing to evaluate DL by breathing small concentrations of CO. Diseases that thicken themembrane wall, like pulmonary fibrosis, or reduce the available surface area, like emphysema, cancause abnormally low DL values.

The overall mechanical properties of the isolated lungare demonstrated by its pressure-volume (P-V) curve.Lung inflation involves both inflating open alveoliand recruiting those which are initially closed becauseof liquid plugs or local collapse. Deflation from TLCstarts with a relatively homogenous population of in-flated alveoli. This irreversible cycling process ispartly responsible for the inflation and deflation limbsof the curve being different. Inflation requires higherpressures than deflation for any given volume; see theair-cycled (solid line) curve of Fig. 4.7 adapted fromRef. 26. Irreversible cycling of a system exhibits itshysteresis, and the hatched area within the loop cre-ated by the two limbs is called the hysteresis area. Themagnitude of the area is the work per cycle requiredto cycle the lung. The overall slope of the curve, ob-tained by connecting the endpoints with a straight line, is the total compliance Ctot = ∆V/∆P, whichmeasures the total volume change obtained for the total pressure increase. Stiffer lungs have lowercompliance, for example. Often one wants a local measure of compliance, which is the local slope of

(4.3)

4.4 ELASTICITY

4.4.1 Pressure-Volume Relationship of Isolated Lung

FIGURE 4.6 Diffusion across the alveolar/endothe-lial membrane.

FIGURE 4.7 Air cycling of isolated dog lungs.





the V(P) curve, defined as the dynamic compliance Cdyn = dV/dP. Other useful parameters are thespecific compliance V-1 dV/dP, the elastance 1/C, and the specific elastance V dP/dV.

Shown in Fig. 4.8 is the curve for a lung inflated and deflated with saline rather than air. Salinecycling was first noted by Von Neergaard34 to reduce inflation pressures, as can be seen in theresulting curve that is shifted to the left, because of increased compliance, and has lost almost all ofits hysteresis area. Because it removes the air-liquid interface that alveoli normally possess, salinecycling reveals that the surface tension forces exerted by this interface have very significant effects onthe lung’s mechanical response. Saline cycling allows us to understand the elasticity and hysteresisproperties of the lung tissue by removing the surface tension forces and their contributions tohysteresis and compliance.

The P-V curve for an isolated lung is only part of the total picture. The intact pulmonary system iscontained within the chest cavity, which has its own P-V characteristics. In Fig. 4.9 adapted from Ref.22, the static P-V curves for an isolated lung (PL), a passive chest with no lung contents (PW), and thetotal (PT = PL + PW) are shown. When PT = 0, the system is in equilibrium with the surroundingatmospheric pressure and the graph indicates that the lung is stretched from its near-zero volume atPL = 0 while the chest cavity is compressed from its higher equilibrium volume at PW = 0. Hence theelastic recoil of the chest wall is pulling outward while the lung tissue is pulling inward to balance theforces, which are in series.

The internal surface of the lung is coated with a thin liquid film, which is in contact with the residentair. The surface tension arising at this air-liquid interface has significant effects on the overall lungmechanical response, as shown in Fig. 4.8, when it is removed. Alveolar Type II cells produceimportant surface-active substances called surfactants, which reduce the interfacial tension. A majorcomponent of lung surfactant is dipalmitoylphosphatidylcholine (DPPC). Lung surfactants reducealveolar surface tension from the surfactant-free value for air-water, 70 dyn/cm, to 1 to 5 dyn/cm inthe alveoli depending on the concentration and state of lung inflation. Figure 4.10 shows the relation-ship between surface tension and surface area for cycling of an air-liquid interface containing lung

4.4.2 Pressure-Volume Relationship of the Respiratory System

4.4.3 Surface Tension Versus Surface Area

FIGURE 4.8 Saline cycling compared to air cycling.FIGURE 4.9 Pressure-volume curves forlung (PL), chest wall (Pw), and total respiratorysystem (PT).





surfactants. Note that there is a hysteresis area and averageslope, or compliance, that characterize the loop and havesignificant effects on their counterparts in Fig. 4.7.

Insufficient surfactant levels can occur in prematureneonates whose alveolar cells are not mature enough toproduce sufficient quantities, a condition leading torespiratory distress syndrome, also called hyaline membranedisease.2 Instilling liquid mixtures, containing either naturalor manmade surfactants, directly into airways via the tracheahas developed from early work in animal models9 into animportant clinical tool called surfactant replacement therapy.The movement of these liquid boluses through the networkrelies on several mechanisms, including air-blown liquidplug flow dynamics, gravity, and surface tension and itsgradients.4,10,18

The lung differs from many other organs in its combina-tion of gas-phase and liquid-phase constituency. Becausethe densities of these two phases are so different, gravityplays an important role in determining regional lung be-havior, both for gas ventilation and for blood perfusion. Inan upright adult, the lower lung is compressed by theweight of the upper lung and this puts the lower lung’salveoli on a more compliant portion of their regional P-Vcurve; see Fig. 4.7. Thus inhaled gas tends to be directedpreferentially to the lower lung regions, and the regionalalveolar ventilation decreases in a graded fashion mov-ing upward in the gravity field.

Blood flow is also preferentially directed toward thelower lung, but for different reasons. The blood pressure inpulmonary arteries, Pa, and veins, P?, sees a hydrostaticpressure gradient in the upright lung. These vessels areimbedded within the lung’s structure, so they are sur-rounded by alveolar gas pressure, PA, which is essentiallyuniform in the gravity field since its density is negligible.Thus there is a region called Zone III in the upper lungwhere Pa < PA. This difference in pressures squeezes thecapillaries essentially shut and there is relatively little pul-monary blood flow there. In the lower lung called Zone I,the hydrostatic effect is large enough to keep Pa > Pv > PA

and blood flow is proportional to the arterial-venous pres-sure difference, Pa - Pv. In between these two zones is ZoneII where Pa > PA > Pv. Here there will be some length of thevessel that is neither fully closed nor fully opened. It ispartially collapsed into more of a flattened, oval shape. Thephysics of the flow, called the vascular waterfall29 or chokedflow30 or flow limitation23, dictates that Q is no longer depen-dent on the downstream pressure, but is primarily deter-mined by the pressure difference Pa - PA.

Figure 4.11a shows this interesting type of flexible tube

4.5 VENTILATION, PERFUSION, AND LIMITS

FIGURE 4.10 Surface area As versus sur-face tension s for a cycled interface con-taining lung surfactant. There is a hyster-esis area and mean slope or compliance tothe curve; compare to Fig. 4.7.

FIGURE 4.11 Choked flow through a flexibletube, (a) upstream pressure Pu, downstream pres-sure Pd, external pressure Pext, flow F, pressure P,cross-sectional area A. (b) A/A0 versus transmuralpressure with shapes indicated, (c) Flow versuspressure drop, assuming Pd is decreased and Pu isfixed.





configuration and flow limitation phenomena where Pa = Pu, PA = Pext, P? = Pd and . Asdownstream pressure Pd is decreased and upstream pressure Pu is kept fixed, the flow increases untilthe internal pressure of the tube drops somewhat below the external pressure Pext. Then the tubepartially collapses, decreasing in cross-sectional area according to its pressure-area or “tube law”relationship A(P), shown in Fig. 4.11b. As Pd is further decreased, the tube reduces its cross-sectionalarea while the average velocity of the flow increases. However, their product, the volumetric flowrate F, does not increase as shown in Fig. 4.11c. A simplified understanding of this behavior may beseen from the conservation of momentum equation for the flow, a Bernoulli equation, where theaverage fluid velocity U is defined as the ratio of flow to cross-sectional area, U = F/A, all of theterms pressure dependent.

(4.4)

where Pres

= pressure in a far upstream reservoir where fluid velocity is negligibly small ρ = fluid density

Pres is the alveolar air pressure, for example, when applied to limitation of air flow discussed below.Taking the derivative of Eq. (4.4) with respect to P and setting the criterion for flow limitation asdF/dP = 0 gives

(4.5)

where E is the specific elastance of the tube. The quantity (E/ρ)1/2 is the “wave speed” of smallpressure disturbances in a fluid-filled flexible tube, and flow limitation occurs when the local fluidspeed equals the local wave speed. At that point, pressure information can no longer propagateupstream, since waves carrying the new pressure information are all swept downstream.

The overall effect of nonuniform ventilation and perfusion is that both decrease as one progressesvertically upward in the upright lung. But perfusion decreases more rapidly so that the dimensionlessratio of ventilation to perfusion, , decreases upward, and can vary from approximately 0.5 atthe lung’s bottom to 3 or more at the lung’s top.36 Extremes of this ratio are ventilated regions withno blood flow, called dead space, where , and perfused regions with no ventilation, calledshunt, where .

The steady-state gas concentrations within analveolus reflect the balance of inflow to outflow,as shown in the control volumes (dashed lines) ofFig. 4.12. For CO2 in the alveolar space, net in-flow by perfusion must equal the net outflow byventilation, where C indicates concentration. In the tissuecompartment, the CO2 production rate from cellu-lar metabolism, , is balanced by net inflowversus outflow in the tissues; i.e.,

. Noting that and combining the two equations, while convert-ing concentrations to partial pressures, leads to thealveolar ventilation equation

(4.6)

where the constant 8.63 comes from the units con-version. The inverse relationship between alveolar

FIGURE 4.12 Schematic for ventilation and perfusion inthe lung and tissues. Dashed line indicates control volumefor mass balance. Assume instantaneous and homogenousmixing. Subscripts indicate gas concentrations for sys-temic (s) or pulmonary (p) end capillary Cc, venous Cc, al-veolar CA, and arterial Ca.





ventilation and alveolar CO2 is well known clinically. Hyperventilation drops alveolar, and hencearterial, CO2 levels whereas hypoventilation raises them. Using a similar approach for oxygen con-sumption and using the results of the CO2 balance yields the ventilation-perfusion equation

(4.7)

Equation 4.7 uses the definition of the respiratory exchange ratio, , which usually hasa value of R ≈ 0.8 for . It also replaces the end capillary concentration with the systemicarterial value, , assuming equilibration. From Eq. 4.7, the extreme limits of ,mentioned earlier, may be recognized. Intermediate solutions are more complicated, however, sincethere are nonlinear relationships between gas partial pressure and gas content or concentration in theblood. Equation 4.7 also demonstrates that higher , as occurs in the upper lung, is consistentwith a higher end capillary and alveolar oxygen level. It is often thought that tuberculosis favors theupper lung for this reason.

The variation leads to pulmonary venous blood having a mix of contributions from differentlung regions. Consequently, there is a difference between the lung-average alveolar and theaverage or systemic arterial , sometimes called the A-a gradient of O2. An average can bederived from the alveolar gas equation,

(4.8)

which derives from the mass balance for O2 in Fig. 4.12. Here is the inspired value and f is asmall correction normally ignored. Clinically, an arterial sample yields , which can be substitutedfor in Eq. 4.8. The A-a gradient becomes abnormally large in several lung diseases that causeincreased mismatching of ventilation and perfusion.

A common test of lung function consists of measuringflow rate by a spirometer apparatus. When the flow sig-nal is integrated with time, the lung volume is found.Important information is contained in the volume versustime curves. The amount of volume forcefully exhaledwith maximum effort in 1 second, FEV1, divided by themaximal volume exhaled or forced vital capacity, FVC,is a dimensionless ratio used to separate restrictive andobstructive lung disease from normal lungs. FEV1/FVCis normally 80 percent or higher, but in obstructed lungs(asthma, emphysema) the patient cannot exhale verymuch volume in 1 second, so FEV1/FVC drops to diag-nostically low levels, say 40 percent. The restricted lung(fibrosis) has smaller than normal FVC, though theFEV1/FVC ratio may fall in the normal range because ofthe geometric scaling as a smaller lung.Flow and volume are often plotted against one anotheras in Fig. 4.13. The flow-volume curves shown are forincreasing levels of effort during expiration. The

4.6 AIRWAY FLOW, DYNAMICS, AND STABILITY

4.6.1 Forced Expiration and Flow Limitation

FIGURE 4.13 Flow-volume curves for in-creasing effort level during expiration, in-cluding maximal effort. Note effort-inde-pendent portion of curves.





maximal effort curve sets a portion common to all of the curves, the effort-independent region.Expiratory flow limitation is another example of the choked flow phenomenon discussed earlier inthe context of blood flow. Similar flow versus driving pressure curves shown in Fig. 4.11 can beextracted from Fig. 4.13 by choosing flows at the same lung volume and measuring the pressure dropat that instant, forming the isovolume pressure-flow curve. Since airway properties and E vary alongthe network, the most susceptible place for choked flow seems to be the proximal airways. Forexample, we expect gas speeds prior to choke to be largest at generation n = 3 where the total cross-sectional area of the network is minimal; see Table 4.1. So criticality is likely near that generation. Aninteresting feature during choked flow in airways is the production of flutter oscillations, which areheard as wheezing breath sounds,13,15,16 so prevalent in asthma and emphysema patients whose maxi-mal flow rates are significantly reduced, in part because E and Uc are reduced.

Most of the dynamics during expiration, so far, have been concerned with the larger airways. Towardthe very end of expiration, smaller airways can close off as a result of plug formation from the liquidlining, a capillary instability,11,21 or from the capillary forces pulling shut the collapsible airway,25 orfrom some combination of these mechanisms. Surfactants in the airways help to keep them open byboth static and dynamic means. The lung volume at which airway closure occurs is called the closingvolume, and in young healthy adults it is ~10 percent of VC as measured from a nitrogen washouttest. It increases with aging and with some small airway diseases. Reopening of closed airways wasmentioned earlier as affecting the shape of the P-V curve in early inspiration as the airways arerecruited. When the liquid plugs break and the airway snaps open, a crackle sound, or cascade ofsounds from multiple airways, can be generated and heard with a stethoscope.1,12 Diseases that lead toincreased intra-airway liquid, such as congestive heart failure, are followed clinically by the extent oflung crackles, as are certain fibrotic conditions that affect airway walls and alveolar tissues.

1. Alencar, A. M., Z. Hantos, F. Petak, J. Tolnai, T. Asztalos, S. Zapperi, J. S. Andrade, S. V. Buldyrev, H. E. Stanley, and B. Suki,“Scaling behavior in crackle sound during lung inflation,” Phys. Rev. E, 60:4659–4663, 1999.

2. Avery, M. E., and J. Mead, “Surface properties in relation to atelectasis and hyaline membrane disease,” Am. J. Dis. Child.,97:517–523, 1959.

3. Bohn, D. J., K. Miyasaka, E. B. Marchak, W. K. Thompson, A. B. Froese, and A. C. Bryan, “Ventilation by high-frequencyoscillation,” J. Appl. Physiol., 48:710–16, 1980.

4. Cassidy, K. J., J. L. Bull, M. R. Glucksberg, C. A. Dawson, S. T. Haworth, R. B. Hirschl, N. Gavriely, and J. B. Grotberg, “A ratlung model of instilled liquid transport in the pulmonary airways,” J. Appl. Physiol., 90:1955–1967, 2001.

5. Chatwin, P. C., “On the longitudinal dispersion of passive contaminant in oscillatory flows in tubes,” J. Fluid Mech.,71:513–527, 1975.

6. Dragon, C. A., and J. B. Grotberg, “Oscillatory flow and dispersion in a flexible tube,” J. Fluid Mech., 231:135–155, 1991.

7. Dubois, A. B., A. W. Brody, D. H. Lewis, and B. F. Burgess, “Oscillation mechanics of lungs and chest in man.” J. Appl. Physiol.8:587–594, 1956.

8. Eckmann, D. M., and J. B. Grotberg, “Oscillatory flow and mass transport in a curved tube,” J. Fluid Mech., 188:509–527,1988.

9. Enhorning, G., and B. Robertson, “Expansion patterns in premature rabbit lung after tracheal deposition of surfactant,” ActaPath. Micro. Scand. Sect. A—Pathology, A79:682, 1971.

4.6.2 Airway Closure and Reopening

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