Air Pollution, the Automobile, and Public Health (1988)

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Transport and Uptake of Inhaled Gases JAMES S. ULTMAN Pennsylvania State University Implications of Lung Anatomy / 324 Upper Airways / 325 Conducting Airways / 327 Respiratory Zone / 328 Nonuniformities in Ventilation and Perfusion / 329 Fundamentals of Mass Transport / 331 Thermodynamic Equilibria / 331 Diffusion and Reaction Rates / 332 Individual Mass Transfer Coefficients / 336 Overall Mass Transfer Coefficients / 337 Longitudinal Gas Transport / 340 Mathematical Models / 341 Compartment Models / 342 Distributed-Parameter Upper Air-way Models / 346 Distributed-Parameter Lower Airway Models / 349 Experiments / 352 In Vitro Methods / 352 In Vivo Animal Experiments / 354 In Vivo Human Subject Studies / 358 Summary / 361 Summary of Research Recommendations / 361 Air Pollution, the Automobile, and Public Health. (3 1988 by the Health Effects Institute. National Academy Press, Washington, D.C. 323

324 Transport and Uptake of Inhaled Gases When a pollutant gas is breathed, its trans- port through the highly branched tracheo- bronchial tree results in a unique internal distribution of concentration and uptake rates. A major factor complicating the study of these processes is the fact that gases with different molecular properties can exhibit different internal dose distribu- tions. Highly water-soluble gases, such as sulfur dioxide (SO2), are essentially re- moved by the upper airways, the primary site of respiratory defense. A highly reac- tive gas of only moderate solubility, such as ozone (03), can reach the tracheobronchial tree, where it reacts with the protective mucous layer and eventually damages un- derlying tissue in the small bronchioles. A gas that has limited aqueous activity but is highly reactive with hemoglobin, such as carbon monoxide (CO), is able to penetrate further to the respiratory zone and diffuse to the pulmonary circulation in quantity. The guiding philosophy presented in this chapter is that direct measurements on hu- man volunteers can be minimized by using mathematical models to anticipate uptake rates and identify target tissues of poten- tially hazardous species. An ideal model enables one to calculate dose distribution from basic anatomic and physicochemical parameters and to estimate the effects of pathological airway derangements and changes in ventilatory demand. But in . . . . . - many situations it Is more practical to use models that simply extrapolate measure- ments from laboratory animals to humans. In either case, it is necessary to validate proposed models with experiments, and for that reason both theoretical and labora- tory methodologies are discussed in this paper. The role of physicochemical factors in the absorption of gas species from a flow- ing airstream is of central concern to chem- ical engineers who design industrial gas separation equipment. Many of the same concepts used to develop gas absorption or chromatographic separation processes can be applied with some modification to the lung. Two widely read sources on this subject are Bird et al. (1960), a fundamental treatise on fluid flow and diffusion phe- nomena, and Treybal (1980), a textbook that describes conventional design meth- ods. Complementing this engineering knowledge is a wealth of physiology liter- ature concerning the influence of lung anat- omy and mechanics on the distribution of foreign gases (for example, Engel and Paiva 1985~. In this chapter, we will integrate these separate fields as Hills (1974) has done for normal respiratory gases. Although the  material here is intended to encompass all pollutant gases those already recognized as well as those still unrecognized an en- cyclopedic review has not been attempted. Rather, basic concepts and methods are stressed to provide a general overview, and concrete examples with respect to three species- S02, 03, and CO have been used to illustrate the spectrum of solubility and reactivity effects. This article begins with two sections devoted to issues at the heart of mathemat- ical model development: first, lung anat- omy and its influence on aerodynamics; and- second, fundamentals of diffusion and chemical reaction and their characterization in terms of mass transfer coefficients. The next two sections address the structure of alternative mathematical models and their validation by experimental measurements. O O ~ Implications of Lung Anatomy This section, in presenting an overview of lung anatomy and its influence on aerody- namics and hemodynamics, focuses on the human respiratory system. Although most of the concepts are also valid for animals (see Schlesinger, this volume), the numer- ical values used to illustrate specific ana- tomic and functional features are typical of normal adult humans. As a supplement to these data, table 1 has been included to indicate the difference in scale between the lungs of humans and animals. The pulmonary airways consist of three distinct functional units. The upper air- ways, extending from the nares and lips to the larynx, are the primary sites at which the temperature and humidity of inspired air are equilibrated to body conditions. The conducting airways, a branched-tube net

James S. Ultman 325 Table 1. Comparisons Among Anatomical Measurements Related to Breathing in Humans and Laboratory Animals Anatomical Large Small Small MeasurementHumanDog Dog Monkey Rabbit Rat Mouse . . . Body weight (kg) 74 22.8 11.2 3.71 3.6 0.14 0.023 Total lung volume 4,341 1,501 736 184 79 6.3 0.74 (ml) Total alveolar surface 143 90 41 13 5.9 0.39 0.068 (m2) Total capillary surface 126 72 33 12 4.7 0.41 0.059 (m2) Total capillary volume 213 119 50 15 7.2 0.48 0.084 (ml) Mean air-blood barrier 2.22 1.42 1.64 1.52 1.51 1.42 1.25 thickness (,um) SOURCE: Adapted with permission from Gil (1982, tables 1 and 2). Copyright CRC Press, Boca Raton, Fla. work originating at the trachea and extend- ing to include terminal bronchioles, pro- vide an orderly yet compact expansion of flow cross-section. The respiratory zone, composed of all the alveolated airways and airspaces, is responsible for oxygenation of and carbon dioxide removal from the pul- monary capillaries. Taken together, the conducting airways and respiratory zone are referred to as the lower respiratory tract, and since they contain no gas ex- change surface, the upper and conducting airways combined are often called the ana- tomic dead space. The respiratory zone, with a volume of about 3 liters compared to the 160 ml of anatomic dead space, contains most of the functional residual gas volume, and yet the total path length of 6 mm along alveolated airways is only a fraction of the 40 cm between the nose and the terminal conduct- ing airways. In other words, the total cross-sectional area in the respiratory zone is much greater than in the dead space. However, this total is the sum of the cross- sections of a large number of small passages in which the axial diffusion distance is very short. Therefore, longitudinal mixing of fresh inspired air with residual gas is fa- vored in the respiratory zone, whereas bulk flow with little mixing of fresh and residual air occurs in the dead space (Go mez 1965~. The walls of the upper and conducting airways are composed of an inner mucosal layer, a submucosal layer, and an outer adventitia. The mucosal layer, sometimes called the mucous membrane, contains cil- iated epithelium and mucus-producing goblet cells, with additional mucus-pro- ducing glands located in the submucosal layer. In the lumen of these airways, cilia are surrounded by a low-viscosity pericil- iary fluid, above which high-viscosity mu-  cus floats. The cilia beat in an organized unidirectional fashion, propelling this mu- cous blanket from the lower respiratory tract to the oropharynx with a daily clear- ance of about 100 ml. Blood is supplied to the lower respira- tory tract through the bronchial arteries, which arise from the aorta and deliver oxygen to conducting airways, and through the pulmonary circulation that perfuses the respiratory zone via the pul- monary artery. During rest, the bronchial circulation constitutes only 1 percent of the total 5 liter/min of systemic arterial blood flow, and it feeds sparsely distributed cap- illaries in the submucosal layer of bronchial tissue. In contrast, the entire output of the right heart enters the pulmonary circulation to supply the dense meshwork of capillaries in the walls of the alveoli. Upper Airways During quiet breathing, a typical tidal vol- ume is 500 ml at a frequency of 12 breaths/ min. corresponding to a minute volume of 6 liters/mint Under these conditions, air normally enters the nose and flows by way of the nasopharynx, pharynx, and larynx to

326 the trachea. In the nose, the convoluted well-perfused surface of the nasal turbinates with its large specific surface area of 8 cm2/ml provides an efficient site for ab- sorption of soluble gases. Alternatively, air may follow a path from the mouth directly to the pharynx, thereby depriving the in- spirate of purification by the nose. Most individuals breathe exclusively through their nose when minute volume is below 34 liters, but during vigorous exercise when ventilatory demand is larger, 45 to 60 per- cent of respired air is inspired orally (Nii- nimaa et al. 1980, 1981~. The presence of a nasal obstruction or even a disease of the lower respiratory tract such as asthma can also exaggerate oral breathing, making the lung more likely to be exposed to pollutant gases. The geometry of upper airways is irreg- ular and variable, depending on the point of air access, on the breathing pattern, and on the state of mucous membrane engorge- ment by nasal blood flow. Perhaps this explains why no mathematical model of its anatomy has been established. Roughly speaking, upper airway volume is 50 ml, with a path length of 17 cm from lips to glottis and 22 cm from nares to glottis. As respired gas flows through the upper air- ways, it encounters a series of sudden con- tractions and expansions in cross-section leading to corresponding accelerations and decelerations in gas velocity (figure 1~. This is quite different from the steady flow of gases in straight tubes of constant cross- section. The nature of flow in a tube is character- ized by the Reynolds number (Re), the dimensionless ratio of inertial force to fric . . tlona resistance: Re = Vd/A u (1) where d is tube diameter, V is volumetric flow rate, A is cross-sectional area available for flow, and v is kinematic viscosity. In sufficiently long straight tubes, flow is fully developed and the Reynolds number uniquely determines whether the velocity field is laminar (Re ~ 2100), turbulent (Re > 4000), or in transition between these two regimes (4000 > Re > 2100~. In laminar flow, material moves along fixed axial 30t 254 cue O 20 LO In oh 15 o  10 5 _V Transport and Uptake of Inhaled Gases 12 11 10 Bronchial / x generations\ | - ~ ,6 x ~9t 1 ~i, V 5 10 15 20 25 30 35 40 45 DISTANCE FROM NASAL TIP (cm) to Figure 1. Cross-sectional area available for gas flow. Axial velocity (m/see for flow of 200 ml/see) during quiet respiration is also shown at selected "V" (velocity) points. (Adapted with permission from Swift and Proctor 1977, p. 68, by courtesy of Marcel Dekker, Inc.) streamlines so that radial transport can only occur by molecular diffusion. In turbulent flow, the mixing action of turbulent eddies facilitates radial transport of gas species to the tube wall, a favorable condition for gas absorption. Because of disturbances created by geo- metric irregularities, flow in the upper air- ways is never fully developed. For exam- ple, the inspired airstream that is propelled through the glottic constriction into the trachea forms a confined jet immediately downstream of the larynx. Large velocity gradients within the jet generate turbulent eddies at Reynolds numbers well below the fully developed transition value of 2100 (Simone and Ultman 1982), and these ed- dies can increase radial mixing as gas flows downstream, even during quiet inspiration. Also, the rapid deceleration of air passing from the ostium internum to the turbinated structures can induce turbulence in the nose (Swift and Proctor 1977~. In addition to the promotion of turbu- lence, three other phenomena unique to the upper airways may enhance gas absorption.

James S. Ultman 327 First, fresh mucus swept into the tubinates from the paranasal sinuses supplements the capacity of the nasal passages for airborne pollutants. Second, the nasal mucosa and submucosa are well perfused with blood flowing countercurrent to the mucus flow, and this is a favorable configuration for gas absorption (Brain 1970~. Third, because increased blood flow can elicit an erectile response in surrounding tissue, vasodila- tion by cholinergic pollutants may increase gas absorption both by increasing capillary surface and by reducing the lateral diffusion distance in the nasal passages. Conducting Airways The conducting airways, also referred to as the tracheobronchial tree, are a series of more or less dichotomously branched tubes originating at the larynx and extending just proximal to the point where alveolated airways arise. Those branches containing cartilage are defined as bronchi and gener- ally occupy the first 10 or so generations. The more distal cartilage-free branches are defined as bronchioli. Although volume changes in the trachea and major bronchi are minimized by stiff cartilage rings em- bedded in the airway walls, the smaller bronchi and the bronchioli undergo isotropic volume changes in proportion to changes in lung volume (Hughes et al. 1972~. Raabe (1982) reviewed several numerical models of the tracheobronchial tree, which may be divided into two categories: the so-called symmetric models, in which a regular dichotomy of branching exists at every generation and all branches in a given generation have the same diameters and the same lengths, and the asymmetric models, which account for the unequal distribution of lung lobes three on the right and two on the left or for irregularities within the lobes themselves. Whereas a symmetric model is an idealization wherein the geom- etry of all transport paths from larynx to terminal bronchioles is identical, in the more realistic asymmetric model, airway diameters and lengths and even the number of branch divisions may differ along alter- native paths. For example, in Horsfield and Cumming's asymmetric model (1968), there are between 8 and 25 branchings with a corresponding distribution in path lengths from 7 to 23 cm between the trachea and the respiratory lobules. On the other hand, in Weibel's symmetric model A (1963, p. 136) there are 16 branchings with an equal length of 13 cm along all conducting airway paths. A symmetric model is a convenient start- ing point for the analysis of diffusion and flow because all transport paths are equiv- alent. Weibel's widely used model A por-  trays the conducting airways as an expand- ing network of dichotomously branching tubes, wherein generation number, z, in- creases from zero at the trachea to 16 at the terminal bronchioli and 23 at the terminal alveolar sacs. Each generation contains 2Z branches. Figure 2 summarizes the important geometric features of this model (top) and their impact on gas flow (bottom). Although the diameters of individual branches gener- ally decrease with increasing z, beyond the eighth generation the total surface available for gas absorption, S. is a strongly increasing function of longitudinal distance, y. Increases in the summed cross-section available for flow, A, parallel those for S so that axial velocity as well as branch Reynolds number fall dramatically as gas is transported distally. During quiet respiration, the velocity is suf- ficiently low that Re does not exceed the upper limit of 2100 for fully developed lami- nar flow in any branch, but during moderate exercise, Re values in excess of 2100 indicate the possibility of turbulent flow in the trachea and first four generations. The entry length fraction, L/Le, is the ratio of actual airway length, L, to length, Le' required for flow to become fully de- veloped (Olson et al. 1970~. Entry length fractions are less than unity proximal to the seventh generation, implying that velocity fields within the trachea and major bronchi do not reach fully developed behavior, even during quiet breathing. This explains how it is possible for random velocity fluctuations to be observed in bronchial airway casts at inspiratory flows well within the laminar Reynolds number range (Dekker 1961; Olson et al. 1973~. Such turbulence was undoubtedly due to the propagation of incompletely dissipated ed- dies from the upper airway segment of the

328 cast. Ordered flow disturbances consisting of a paired vortex pattern during inspira- tion and a quadruple vortex pattern during expiration have also been reported in iso- lated bronchi at Reynolds numbers within the laminar range (Schroter and Sudlow 1969~. The radial mixing brought about by these lateral flow circulations aids in trans- port of gas species toward the airway walls. Accompanying the decrease in branch diameters between trachea and terminal bronchioles, there is a progressive decrease in airway wall thickness and tissue perfu- sion rate. There are also fewer mucus- secreting elements and less ciliated epithe- lium, which reduces mucus velocity and thickness. Estimates of some relevant mu- cus, tissue, and blood flow parameters are presented in table 2. Transport and Uptake of Inhaled Gases AIRWAY GENERATION, z 0 1 2 4 6 8 1216 'T I ~'i I I I I I I I Trln E to 4 - UJ 3 1  - <~ 2.0 E N - °1.6 at; z - °1.2 up _ u' 0.8 O _ 00.4 _ En 0 v 800C - cr m 600C at in 400C o at ~ 200C Lid r in LlLe (exercise) / I LlLe (rest) ~ | '\\ 11 Re (rest) \\ / / ~? / _-~-- ~---~-- ~ --~-~ 0.12 0.14 0.16 0.18 0.20 0.22 0.24 ~ Re (exercise) 8 2 3) ~ 2.0 z z I O - r ID LONGITUDINAL DISTANCE, y(m) Figure 2. Geometric and aerodynamic characteristics of a symmetric tracheobron- chial model. All values were derived for Weibel's model A (Weibel 1963) scaled to a functional residual capacity of 3 liters. Entry length fraction was computed with the equation of Olson et al. (1973). Flows of 0.4 and 1.6 liters/sec have been used for rest and exercise conditions, respectively.  Respiratory Zone The respiratory zone is defined by the presence of regions containing membra- nous outpouchings the alveoli normally responsible for gas exchange. An individual alveolus has a characteristic diameter of about 0.2 mm, and its shape may be mod- eled in various ways: a truncated sphere, a truncated cone, or a cylindrical wedge (Weibel 1963, p. 60~. The fine structure of the alveolar wall consists of little more than a shell-like mesh of pulmonary capillaries enclosed in a layer of airway epithelium covered by a thin surfactant film, and therefore the blood-gas barrier imposed by the alveolar membrane is only about 2 ,um thick. Because the pulmonary capillaries are short, typically 10 ,um long, there is

James S. Ultman 329 Table 2. Conducting Airway Parametersa Airway Diameter/Lengthb Mucus ThicknessC Tissue Thickness Blood Flowe Mucus Velocity' (lo~2 m) (10-6 m) (10-4 m) (10-7 m3/sec) (10-4 m/see) Trachea 1.6/12 7 8 1.7 2.5 Main bronchi 1.0/6.0 7 5 0.6 1.7 Lobar bronchi 0.4/3.0 7 2 0.2 0.50 Segmental 0.2/1.5 7 1 0.09 0.067 bronchi Subsegmental 0.15/0.5 7 0.75 0.2 0.010 bronchi Terminal 0. 06/0.3 4.2 0.30 1.7 0.00033 bronchioles a Parameter values are based on a single airway branch. b Landahl (1950). c National Research Council (1977, table 7-3). Computed as 5% of airway diameter (DuBois and Rogers 1968). e DuBois and Rogers (1968, table 2). extensive branching in the capillary mesh, and the associated pulmonary blood flow resembles a moving sheath of fluid inter- rupted by regularly spaced posts of tissue (Rosenquist et al. 1973~. When packed together to form the lung parenchyma, the alveoli give a honeycomb appearance, with a specific surface area greater than 200 cm2/ml airspace volume. The edges of contact between adjacent al- veoli, called alveolar sepia, protrude into the airspace, thereby partitioning gas and impeding axial diffusion near the airway wall. Because their walls are so thin and have such a large specific surface, alveoli are well suited to normal gas exchange and probably to the absorption of air pollutants as well. To illustrate this, consider that an erythrocyte, which normally resides in the pulmonary capillaries for only 1 see, has sufficient time to come to equilibrium with alveolar oxygen (O2) and carbon dioxide (CO2) partial pressures. The fraction of airway wall occupied by alveoli increases with distal distance from the first alveolated airway to the blind- ended alveolar sacs that terminate every path of the tracheobronchial tree. As was the case for the conducting airways, both symmetric and asymmetric models have been proposed for the respiratory zone. In Weibel's model A, there are seven symmet- ric generations of respiratory airways, and along the 6-mm path leading from the first respiratory bronchioli to the alveolar sacs, the air-tissue surface available for gas ex- change increases from 0.16 to 39 m2 per generation. Simultaneously, the flow cross- section increases from 0.03 to 1.2 m2 with a concomitant drop in branch Reynolds number from 0.5 to 0.01 during quiet breathing. More recent work by Hansen  and Ampaya (1975) indicates that Weibel's model significantly underestimates both tissue surface and flow cross-section, but in either case it is reasonable to conclude that the influence of gas flow is negligible and transport in the respiratory zone occurs principally by diffusion. The distal airway model of Parker et al. (1971 ) is an example of an asymmetric model of the respiratory zone. It is a di- chotomously branching network in which the total number of branchings varies from three to eight along different airway paths. In addition, this model incorporates distri- butions of alveoli, from 7 to 25 alveoli per duct (mean 15.9) and from 8 to 15 alveoli per sac (mean 9.6), rather than the fixed values of 20 alveoli in each duct and 17 in each sac of Weibel's symmetric model. Nonuniformities in Ventilation and Perfusion If the lungs were perfectly symmetric and mechanical tissue properties and forces were uniform among generations, then air

330 Transport and Uptake of Inhaled Gases 20 ~ 15 z 10 c' cr: MEL 5 O ~.1- ~ ~ ~ ~ L 200 400 600 800 1000 1200 TRANSIT TIME TO ALVEOLAR DUCTS (arbitrary units) Figure 3. Frequency distribution of transit time from the carina to the alveolar ducts. (Adapted with permission from Horsfield and Cumming 1968, p. 379, and from the American Physiological Society.) flow and blood flow to all sibling branches would also be uniform. In this idealized situation, the local concentration and up- take rate of inhaled pollutants could vary only with respect to longitudinal position along the equivalent airway transport paths. In the real lung, however, both small-scale (intraregional) and large-scale (interregional) inhomogeneities exist, and they may impose a difference in dose be- tween equivalent structural units. For ex- ample, two terminal bronchioles located in different lobes of the lung might experience different pollutant exposure levels. 0.15 0.10 O c ~ J ~ Cal CC or 0.05 Two specific types of nonuniformities normally present in the lung have been well documented. First, because of anatomic asymmetries, there is a natural distribution of path lengths between the larynx and the respiratory zone, with shorter path lengths containing less gas volume than longer path lengths. If the rate of alveolar expansion distal to these paths is uniform, the transit times for fresh inspired gas to reach the alveoli by different paths will differ (figure 3), and there will be differences in time of exposure among equivalent structural units. Second, under the influence of grav- ity, when the body is upright, there is more ventilation and more blood flow near the bottom of the lungs than near the top (figure 41. In addition to the effects caused  individually by the distribution of ventila- tion, VA, and perfusion, Q. the VA/Q ratio has special importance regarding gas up- take in the respiratory zone. That is, if a pollutant is to continuously absorb into the respiratory zone, it must first reach an alveolus and ultimately be removed by the pulmonary circulation. Therefore, alveoli that receive little local ventilation (that is, V^/Q is small) are as incapable of continu- ously absorbing soluble gases as are ade- quately ventilated alveoli that receive an Q - Base ~ A/Q - f Apex 5 4 3 2 RIB NUMBER ~3 ~2 ] it: Figure 4. Ventilation/perfusion ratio, VA/Q, from base to apex of the normal upright human lung. (Adapted with permission from West 1977, p. 202, and from Academic Press, Orlando, Fla.)

lames S. Ultman 331 . . abnormally small blood flow (that is, VA/Q is large). This implies that gas uptake is maximized at some intermediate value of the ventilation/perfusion ratio. Fundamentals of Mass Transport A complete description of mass transport consists of three basic elements: the mass conservation equation, rate equations for diffusion and chemical reaction, and ther- modynamic equilibrium expressions. The mass conservation equation balances the rate at which a species may appear with the rate of its accumulation in a spatial region of interest often called the control volume. Given that the permissible modes of ap- pearance are convection, diffusion, and chemical reaction, the mass conservation equation has the form Rate of Accumulation Net Input Net Input Net Rate of Rate by + Rate by + Production Bulk Flow Diffusion by Reaction (2) The mathematical expression of mass con- servation is a differential field equation ap- plicable to any three-dimensional time- varying problem (Bird et al. 1960, p. 559), but in practice the general equation must be simplified by selecting a set of physically realistic assumptions. This will be consid- ered in detail in the next section, which is devoted to mathematical modeling. In this section, attention is focused on the fundamentals of thermodynamic equilibria and rate expressions for diffusion and chemical reaction. The formulation of mass transfer coefficients to describe transverse transport and the use of retention parame- ters and mixing coefficients to characterize longitudinal transport are also described. Thermodynamic Equilibria Phase equilibrium refers to the distribution of a species between adjacent immiscible phases. Suppose, for example, that a liquid phase containing a physically dissolved spe cies X is placed in contact with gas contain- ing an arbitrary partial pressure Px of the same solute species. Thereafter, molecules of X will continually redistribute between the liquid and gas phases, altering Ax until it reaches a stationary value p* referred to as the equilibrium partial pressure or the gas tension. In this equilibrium state, the ratio of the molar concentration Cx of species X in the liquid phase to the corresponding value of p* in the gas phase is a parameter,  cYX, called the Bunsen solubility coefficient: Ax = CXIPX (3) Generally, cut is a function of temperature only, its value being equivalent to the mo- lar volume of the liquid divided by Henry's Law constant. Therefore, when tempera- ture is fixed, the dissolved species concen- tration is a linear function of gas tension. The solubility coefficients of most foreign gases in mucus, tissue, and blood are not precisely known and must be approxi- mated by values measured in readily avail- able solvents such as water (table 3) and hydrocarbon oils. Chemical reaction equilibrium can im- pose additional constraints on solute con- centrations. During chemical reactions, molecular combinations and decomposi- tions can both occur such that a dissolved pollutant species is reversibly bound to another endogenous species. Typical of such reactions are the aqueous ionization of SO2 to form bisulfite; the ionization of CO2 to form bicarbonate; and the binding of 02, C02' or CO to hemoglobin. To exemplify the underlying reaction equilib- rium, consider the ionization of SO2 in water given by the stoichiometric equation SO2 + H2O = H+ + HSO3 (4) Although HSO3 also undergoes a weak dissociation to SO32-, equation 4 is a suffi- ciently accurate model of the overall proc- ess (Schroeter 1966, p. 17~. As the concentration of SO2 is depleted by the formation of HSO3, the law of mass action dictates that the forward reaction rate must slow down. Simultaneously, the concentration of bisulfite builds up, and this leads to an increase in reverse reaction rate. This progressively decreasing forward

332 Transport and Uptake of Inhaled Gases Table 3. Estimates of Physical Properties for Various Gases Diffusivity in Aira Aqueous DifFusivityb Aqueous Solubility' First-Order Reactivity Gas Dx (10-s m2/sec) Dx (10-9 m2/sec) ax (kg-mol/m3/Pa) kr (103/sec) SO2 1.15 2.14 9.13 x 10-6 0 O3 2.11 3.06 5.62 x 10-8 1.2(m):50(t):5(b) CO 2.17 3.06 8.09 x 10-9 0 a Chapman-Enskog equation (Bird et al. 1960, p. 510). b Wilke-Chang equation (Bird et al. 1960, p. 515). c Computed from Henry's Law coefficients (National Research Council 1977, table 7-1). ~ Miller et al. (1985, table 3) for mucus (m), tissue (t), and blood (b). . . . rate anc increasing reverse rate continues until the two rates are equal and the con- centrations of all species become stationary. In this equilibrium state, the concentrations of reactants and products possess a specific algebraic relationship. For the SO2 ioniza- tion reaction, (Ke) SO2 = CHSO3 CH/ CSO~ (5) where (Ke)So2, the reaction equilibrium constant specific to the ionization of SO2, is a function of temperature (Pearson et al. 1951). In pure water, where hydroxyl ion con- centration is low, electroneutrality dictates that bisulfite and hydrogen ion concentra- tions are approximately equal, and the summed concentration of SO2 in the unre- acted form CsO2 and in the reacted form CHSO3 can then be expressed as CSO2 + CHSO3 ~SO2PSO2 + t(Ke~sO2ctso2pso2]~/2 (6) This equation illustrates the ability of a reversible chemical reaction to increase the capacity of a solution for gaseous solute. To emphasize the importance of this, figure 5a compares the physical solubility of SO2 to the corresponding level of bisulfite ions. Clearly, the concentration of reacted spe- cies is far greater than the concentration of physically dissolved species, and the rela- tionship of bisulfite concentration to gas tension is nonlinear. In mucus and tissue, increases in hydrogen ion concentration occurring with SO2 ionization are probably suppressed by the buffering action of other solutes, and as equation 5 indicates, the bisulfite concentration would be even larger than in pure water. To provide a unified treatment of the dissolved and reversibly bound forms of a soluble gas, a reactive capacitance coeff~- cient, ,13x, analogous to the solubility coef- ficient, can be defined as the derivative of the reacted solute concentration, Cal. with respect to gas tension: - xr7  Ax = dCxr/dPx (7) Unlike ox, which is independent of Px*, the reactive capacitance coefficient for a revers- ibly bound solute species is generally a decreasing function of gas tension (figure 5b). Diffusion and Reaction Rates Transport of any species occurs by a com- bination of diffusion and convection. Whereas convection is the translation of molecules at the mean flow velocity, diffu- sion is the transport that occurs in response to a concentration gradient whether or not any net flow occurs. Fick's First Law is the general rate equation that accounts for con- vection and diffusion (Bird et al. 1960, p. 502), and for the special case of one-dimen- sional diffusion of species X in the absence of flow, it reduces to MX =-DxSd(Cx/dz) (8) where MX is the mass transport rate of species X; dCx/dz is the concentration gra- dient in the diffusion direction, z; S is the surface area perpendicular to z; Dx is the molecular diffusion coefficient; and the negative sign indicates that diffusion is along the path of decreasing concentration. Strictly speaking, equation 8 is valid only for solutions composed of two species, but Fick's Law may also be applied to multi

James S. Ultman 333 A 3 co E o ~ 2 A - z z UJ at o 0 1 o UJ B HSO3 / 4 _ E to 3 _ o UJ z > LL. 1 tar 2 SON \ 1, \ ~ ~HSO3 \ ' - -  0 0.5 1 .0 0 0.1 0.5 1 .0 GAS TENSION PSO2 (Pa) Figure 5. Solubility of sulfur dioxide at 37°C in pure water. A: The Physical solubility of SO2 and its equilibrium concentration as reacted SOL (that is, HSO3 ) are compared. B: In the graphical construction of average reactive capacitance at SO2 tensions between 0.1 and 0.5 Pa, a horizontal line is drawn so that the areas I and I' are equal. Note that 1 Pa is equivalent to 21.5 ppm of SO2 when total pressure is 1 atm. component solutions when the solute spe- cies of interest is present at sufficiently low concentration or partial pressure relative to the solvent species (Bird et al. 1960, p. 571). Chang et al. (1975) analyzed situa- tions in the airway lumen where multicom- ponent diffusion effects may be important, but the existence of similar phenomena in mucus and tissue have not been investigated. The lung consists of gas, tissue, and blood regions, and the customary assertion that phase equilibrium applies at the inter- faces between regions leads to correspond . . . . . . . . sing c .~scont~nu~t~es In species concentration. This mathematical inconvenience can be circumvented by substituting, in Fick's Law, the gas tension gradient for the con . . centratlon grac lent. MX =-aXDxS(dp */dZ) (9) The product, axDx, sometimes called Krogh's constant of diffusion (Dejours 1981), indicates that the diffusion rate of gases with a small diffusion coefficient can be significant when compensated by a suf- ficiently large solubility. A useful form of equation 9 results for the case of steady-state diffusion through a planar barrier of thickness 1. Then the transport rate MX is constant and equation 9 can be integrated with the result that MX = (~xDxIl)S~x~-Pxo) (10) where Phi and phi are the gas tensions at the two sides of the barrier. Thus, the tension of an inert gas is linearly distributed in the diffusion direction, and the proper driving force for diffusion is the gas tension differ- ence (figure 6a). If a gas species undergoes reversible

334 Transport and Uptake of Inhaled Gases A Mx e 1 Px1 1 ~ _ B 1 C Px, Px2 Mx, Mx - 1 ~ ·1 Pax 1 INERT GAS FACILITATED DIFFUSION WITH DIFFUSION DIFFUSION REACTION Px2 ROB Figure 6. Schematic representation of steady-state diffusion through a stationary barrier of thickness 1. A: The gas tension distribution is linear for an inert gas. B: When a species simultaneously diffuses in physically dissolved (Mx) and reversibly reacted forms (Mar), the distribution is steepest at the highest gas tensions where the reactive capacitance is smallest. C: When irreversible reaction occurs with a biological substrate B. the distribution becomes increasingly nonlinear as the reaction rate constant increases and the diffusion coefficient of pollutant X decreases. chemical reaction, its transport rate is al- tered because it simultaneously diffuses in physically dissolved and reacted forms. When the reaction rate is so fast that chem- ical equilibrium is closely approached throughout the diffusion barrier, the reac- tive capacitance coefficient can be utilized with Fick's Law for steady-state conditions  to obtain Mx = Xx + l3xDxr~llis~px,-Pi) (11) where ,13x is the integral average reactive capacitance as illustrated by the graphical construction in figure 5b, and DXr is the diffusion coefficient of the reacted species. The term (aXDx + ,l3XDxr) is an extended Krogh constant that accounts for the in- crease in diffusion brought about by revers- ible reaction. The existence of such facili- tated diffusion has been used to explain enhanced diffusion of oxygen in hemoglo- bin solutions (Keller and Friedlander 1966) and might also be an important factor in the transport of foreign gases through mucus and tissue (figure fib). When chemical reaction occurs at a finite rate or is not reversible, reactive capaci- tance can no longer relate the concentration of reacted species to the gas tension. In- stead, a reaction rate expression must be used in conjunction with Fick's Law. Sup pose, tor example, that each molecule of pollutant X absorbed from the airway lu- men reacts in mucus with b molecules of a biological substrate B that is supplied from underlying tissue (figure 6c). That is, X + bB = cC (12) where species C represents reaction prod- ucts and c represents the number of mole- cules of C. Then the net rate of depletion of species X per unit volume of mucus can be formulated as RX = kRf (CB) (CX Con) (13) where lerf Is the rate constant of the forward reaction, Cx and CB are the reactant con- centrations, and Cx represents the concen- tration of pollutant that would be present if the system was in reaction equilibrium at the prevailing concentrations of biological materials B and C. Unfortunately, few detailed kinetic data have been collected for inhaled pollutants and, therefore, the rate constants necessary to use reaction expressions such as equation 13 cannot be evaluated. In modeling bron- chial uptake, some investigators (Mclilton et al. 1972; Miller et al. 1978) have circum- vented this problem by assuming that re- action of pollutants with biological sub- strates is instantaneous and irreversible. In

James S. Ultman 335 A - 4 I ~ 3 LU of I 1 // /Ha = [(kr/DX)2 0 1 2 3 4 5 B - z o ~ 0.5 x lo 1.n ~ 0 1 DAMKOHLER NUMBER, [(k,lDX)2 Figure 7. Simultaneous diffusion and first-order reaction in a planar stationary barrier. A: The Hatta number is the ratio of uptake rate in the presence and the absence of chemical reaction. B: The flux fraction ~ is the ratio of edlux from the barrier to influx into the barrier. that case, chemical reaction occurs at a thin plane whose position within the mucous layer is determined by the relative diffu- sional influx of pollutant and substrate. A more general approach is to assume that biological substrate is so abundant that CB is essentially constant, and equation 13 can be approximated by first-order kinetics (Miller et al. 1985~: RX = kr~cx- CXC) (14) where kr is a modified reaction rate con- stant. The diffusion equation can then be formulated and solved for two alternative cases fast irreversible reaction (that is, Ox = 0) or slow reversible reaction. For hot] of these situations the uptake rate is given by (Hobler 1966)  Mx = Ha ~ c~xDx/ 1) S ~ p *I-p *id ( 1 5) where the Hatta number, Ha, is a dimen- sionless parameter that depends upon the dimensionless ratio of reaction to diffusion time, lfEr/DX]~/2' known as a Damkohler number (figure 7a). This equation is similar to the inert gas uptake rate with the addition of the Hatta number to provide a proportional correc- tion for the effect of chemical reaction. As Damkohler number approaches zero, cor- responding to inert gas behavior, Ha re- duces to unity and equation 15 becomes identical to equation 10. When Damkohler number is greater than zero, Ha is greater than unity, indicating an enhancement of diffusion by chemical reaction. Above a value of 3.5, Hatta and Damkohler num- bers are essentially equal and are propor- tional to the barrier thickness so that the diffusion rate is independent of 1. In other words, when Damkohler numbers are larger than 3.5, the relative reaction time is so short that most of species X is converted to product before it can diffuse through the barrier. In analyzing the transport of an air pol- lutant through a chemically reactive bar- rier, it is important to assess the fraction of absorbed gas that completely penetrates. For the special case of first-order kinetics, figure 7b indicates how the ratio of reactant species efflux to influx, ¢,, depends upon Damkohler number. The fraction of influx that is converted to reaction products is (1 - Hi. Using estimates of the reaction rate constant, aqueous diffusivity (table 3), and barrier thickness (table 2), a value of ~ = 0.025 was computed for O3 diffusion

336 Transport and Uptake of Inhaled Gases through the mucous blanket of the trachea and first four bronchial generations. This indicates that more than 97 percent of the O3 absorbed at the airway wall is scav- enged by mucus. On the other hand, be- cause the mucous layer on the terminal bronchioli is much thinner, the associated value of ¢, was 0.14, indicating that 14 percent of the absorbed O3 is expected to penetrate the mucus and reach underlying tissue. Throughout the tracheobronchial tree, the estimated value of ~ for tissue was nearly zero, implying that airway walls are sufficiently thick to deplete all the O3 that reaches them. ~ Recommendation 1. There is a critical need to quantify the chemical interaction of specific pollutants with mucus, tissue, and blood. Besides determining solubility and diffusion coefficients, it is essential to deter- mine the coefficients in reaction rate equa- tions. Individual Mass Transfer Coefficients The uptake formulations reveal that trans- port rate normal to a diffusion barrier is generally proportional to the product of the solubility, surface area, and gas tension difference across the diffusion barrier: Mx = ~xkms~[x,-P x2) where the proportionality constant, km, is the individual mass transfer coefficient of species X. Table 4 summarizes some useful formulas for km with entry 1 listing the stationary barrier models represented by equations 10, 11, and 15. It is also important to consider the uptake that occurs in the presence of flow, as in the airway lumen and the bloodstream. In these situations convection is transverse to the principal direction of absorption, and a diffusion barrier, known as the concentra- tion boundary layer, appears in the moving fluid. In the airway lumen, a boundary layer is formed where air is in contact with the mucous blanket, and in blood vessels a boundary layer is formed where blood is in contact with the endothelial cell lining. In general, as velocity increases, the bound- ary-layer thickness decreases and uptake rate Increases. Entry 2 in table 4 is a commonly used boundary-layer equation that correlates in- dividual mass transfer coefficients with fluid velocity appearing in the Reynolds number and with the dimensic~nless ratio of kinematic viscosity to diffusion coefficient, the so-called Schmidt number (Sc). To use this equation, it is necessary to substitute specific values of the correlation constants (c, m, n, I, depending on conduit geometry and flow conditions. A collection of corre-  lation constants pertaining to diffusion within the airway lumen is given in table 5. Using these constants, gas phase km values `16' have been estimated in each conducting airway generation of Weibel's model A. The resulting distributions (figure 8) gen erally exhibit a maximum value between the eighth and twelfth generation, indicat ing that the mucus lining these airways is Table 4. Useful Formulas for Individual Mass Transfer Coefficients, km Description Formula (1) Stationary barrier of thickness la (a) Inert gas (b) Reactive gas at equilibrium (c) Reactive gas, first-order kinetics (2) Boundary-layer diffusion in tube of diameter d and length Lb (3) Surface renewals (a) Inert gas, turnover time ~ (b) Reactive gas, first-order kinetics (kmllD.r) (kml/D.~) = 1 + (§XDxrlaxD.r) (kml/Dx) = Ha (kmdlDx) = cRemS?'(d/L)4 km = [DX/11 2 km = [krDX] I/2 Equations 10, 11, and 15 of text. b Hills (1974, p. 58). - Astarita (1967, pp. 7, 37).

James S. Ultman 337 Table 5. Correlation Constants for Mass Transfer Coefficients in the Airway Lumen, kmg Conduit Geometry, Flow Conditions (kmgd/Dx) = cRe"'Se"(d/L)~ c m n q (1) Upper airways, nasal breathinga (a) Inspiration 0.028 0.854 0.854 0 (b) Expiration, Re < 7800 0.0045 1.08 1.08 0 (c) Expiration, Re > 7800 0.310 0.585 0.585 0 (2) Upper airways, oral breathinga (a) Inspiration 0.035 0.804 0.804 0 (b) Expiration, Re < 12,000 0.0006 1.269 1.269 0 (c) Expiration, Re > 12,000 0.094 0.704 0.704 0 (3) Lower respiratory tracta (a) Inspiration 0.0777 0.726 0.726 0 (b) Expiration 0.0589 0.752 0.752 0 (4) Straight tubes, developed flown (a) L/d < 0.1 ReSc, Re < 4000 1.86 0.333 0.333 0.333 (b) L/d > 0.1 ReSc, Re < 4000 0.500 1.0 1.0 1.0 (c) L/d > 0.1 ReSc, Re > 4000 0.026 0.800 0.333 0 a Nuckols (1981). Re is evaluated at tracheal conditions. b Bird et al. (1960). particularly susceptible to attack by air- borne pollutants. The computations also indicate that the increased respiratory flow present during exercise triples km values, but the difference in mass transfer coeff~- cients among the three pollutants is rela . . Eve. y minor. Entry 3 in table 4 is an alternate formu- lation for km that might be useful in mucus. Many investigators believe that the mucous blanket is composed of a mosaic of 1- to 100-,um patches Jeffery and Reid 1977~. As these patches flow, it is possible for under- lying mucus to be swept to the surface, thereby exposing a fresh sink for pollutant gas. With gaps between patches spaced as close as 1 ,um, and at the highest mucus velocity of 1 cm/min, the characteristic time for mucus turnover, a, could be as short as 6 msec. Higbie's penetration the- ory and Danckwert's surface renewal model (Astarita 1967) have been developed for predicting mass transfer coefficients in such situations, but unfortunately, existing data on the structure and fluid mechanics of mucus are insufficient to perform reliable computations. On the other hand, because mucus velocity is at least four orders of magnitude less than the adjacent gas veloc- ity, it is reasonable to apply stationary barrier models (table 4, entry 1) as a first . . approximation.  The estimation of km values in the blood spaces is an exceptionally difficult problem. First, diffusion processes depend on geome- try that varies widely between the erectile vessels of the nasal passages, the sparsely distributed capillaries of the bronchial circu- lation, and the densely perfused plate-and- post structure of the pulmonary capillary beds. Second, quantitative anatomic models of vascular structure in the bronchial tissue and the upper airways do not exist. Last, blood is a two-phase material composed pri- marily of plasma and red blood cells, a fact that must be considered when analyzing dif- fusion within the small blood vessels. Overall Mass Transfe?. Coefficients In transport across a bronchial (or alveolar) wall, four major barriers to diffusion are encountered: the flowing gas, the mucus (or surfactant-hypophase), the tissue. and the blood layers (figure 9~. However, gas tensions at air/mucus, mucus/tissue, and tissue/blood interfaces are usually un- known, and it is necessary to compute uptake rate from the gas-phase partial pres- pressure, Px, and blood gas tension, Pxb 7

10 9 8 G) E CM lo _ 7 - z UJ L 6 11 o cr ~ s 11 in at cr co co 4 3 2 1 f ~ , 0 0.12 0.14 it' Ail //' 0111 ~ ~ ''/Jl/l| 1< _, - ~: SO2 ~ __- co , so2 / / ,/ /~ 338 Transport and Uptake of Inhaled Gases AIRWAY GENERATION, z o 1 2 4 6 8 12 16 l l ////~\\; ,- ~ . Exercise /-> ! ,,  - / \ ~Rest ~ \/ \ I , . . . ~ 0.16 0.18 0.20 0.22 0.24 LONGITUDINAL DISTANCE, y(m) Figure 8. Estimates of individual mass transfer coeffiaents in the conducting airways for three common pollutants, based on a straight tube correlation (table 5, entry 4). Flows of 0.4 and 1.6 liters/sec have been used to simulate quiet respiration (solid curves, closed circles) and exercise conditions (broken curves, open circles), respectively. Points on the ordinate are average values computed for the upper airways (table 5, entry 1). By separately formulating diffusion rate in each of the barriers and applying continuity of uptake to combine these four equations, an absorption rate based on the overall driving force (px -Pxbi is obtained: Mx Km S (` P XR P Xb) ~ 1 7) where Km is the overall mass transfer coef- ficient. In the sense that finite values of blood gas tension reduce the driving force for uptake, Pxb is often referred to as a diffusion backpressure. When the flux of species X is equal in all diffusion layers, the overall mass transfer coefficient has the well-known form (Treybal 1980, p. 109)

339 Six lames S. Ultman :~!. Pats = ~ L, GAS(g) MUCUS (m) TISSUE (t) BLOOD (b) -- ___ 1 ~ PXb l 1 Figure 9. Schematic representation of the four-layer model of diffusion through the bronchial wall. For convenience the blood layer has been drawn as a continuous planar diffusion barrier when, in fact, the bronchial capillaries are sparsely distrib- uted throughout the tissue layer. Km = [RTIkmg + 1/(c~xEm)m + 1/(a!xkm)t + 1/(c~xEm) b] (18) so that Km can be constructed from indi- vidual mass transfer coefficients and sol- ubility coefficients in gas (subscript g), mu- cus (subscript m), tissue (subscript I), and blood (subscript b). Since Km is a harmonic mean, its value will be controlled mainly by the layer having the smallest value of diffusion conductance aXkm or, in other words, the largest diffusion resistance,  1 l~xkm For species that undergo irreversible reaction, the flux is progressively reduced from layer to layer, and the flux fraction, ¢', must be introduced as a correction fac- tor in the formulation of Km. If the species reacts in both mucous and tissue layers, then Km = tRTlEmg + 1/(afxAm)m + 56m/(afxEm)t + ~m<;b~l~c~xEm)b]-i (19) where the presence of fractional ~ values increases Km relative to the prediction of equation 18. To examine the implications of these equations, mass transfer coefficients were estimated for the common pollutants, CO, SO2, and 03, using the anatomic constants in table 2. For all three species, individual gas-phase coefficients were computed at quiet breathing conditions, using entry 2 in table 4. Individual mucus and tissue coeff~- cients were calculated from the formulas in table 4 by assuming that CO transport occurs by pure diffusion (entry la); SO2 undergoes instantaneous and reversible re- action to bisulfite ion (entry lb); and O3 undergoes irreversible first-order reaction (entry lc). The overall mass transfer coef- ficients (with omission of the blood-layer contribution) were then determined by us- ing equation 18 for CO and SO2 and equa- tion 19 for O3. The resulting values of the overall mass transfer coefficients (table 6) decrease from the species SO2 to O3 to CO. This ordering is consistent with the National Research Council report (1977, p. 282) that SO2 is almost completely absorbed within the up- per airways; O3 is transported beyond the upper airways, its uptake being most sig- nificant in the distal conducting airways and proximal respiratory branches; and CO absorption occurs deeper within the respi- ratory zone. A comparison of the values of aXkm shows that the controlling diffusion resistance is in gas and tissue layers for SO2, in the mucous layer for 03, and in the tissue layer for CO. Thus, it is clear that mucus provides a protective function for an irreversibly reacting gas such as 03, but its

340 Transport and Uptake of Inhaled Gases Table 6. Estimates of the Mass Transfer Coefficient, exam (10-~° kg-mol/m2/Pa/sec), in a Conducting Airway Model SO2a CO O3b Airway Gas Mucus Tissue Overall Gas Mucus Tissue Overall Gas Mucus Tissue Overall Trachea 32 3,970 35 17 57 0.035 Main bronchi 54 3,970 56 27 91 0.035 Lobar 136 3,970 139 68 227 0.035 bronchi Segmental bronchi Subsegmental 415 3,970 371 187 692 bronchi Terminal 20 6,620 927 20 14 broncholi 272 3,970 278 133 454 0.00031 0.00031 52 1.1 28 1.04 0.00050 0.00049 89 1.1 28 1.05 0.0012 0.0012 223 1.1 28 1.05 0.035 0.0025 0.0023 445 1.1 28 1.06 0.035 0.0033 0.0030 679 1.1 28 1.06 0.059 0.0083 0.0073 13 1.1 4.9 0.84 a A ,`3SO2 value of 0.0013 kg-mol/m3/Pa was used. This corresponds to the capacitance of pure water in contact with a gas containing 1 ppm SO2 at 1 arm total pressure. It was also assumed that DSO, = DHSO?; b The tissue coefficient was divided by ¢,m to account for flux reduction by chemical reaction. barrier effects have little influence on the absorption of inert or soluble species. The relative diffusion resistances also indicate that increases in respiratory flow tending to diminish the gas-phase mass transfer coef- ficient should have little effect on O3 and CO uptake, but would probably increase SO2 absorption measurably. individual diffusion barriers, particularly the mucous layer, the bronchial wall, and the alveolar capillary network. Nx = ~XVCx - ~xS(dCx/dy) (20) Total Rate Rate of Rate of of Transport Bulk Flow Mixing where V is volumetric flow rate through the airway, y is longitudinal position, Ax is a retention coefficient, and ax is an overall longitudinal mixing; coefficient. _ ~ Recommendation 2. A serious effort is Bulk transport refers to the longitudinal needed to analyze mass transfer through motion of a gas species that occurs because the entire gas mixture has a finite volumet ric flow. To understand the need to use a retention coefficient in expressing bulk Longitudinal Gas Transport Friction imposes drag on the layer of gas moving next to the wall of an airway. Viscosity communicates some of this drag to adjacent layers. As a result, gas near the center of an airway moves downstream  faster than gas near the airway wall. This situation can give rise to both radial and . . . . . . axle varlatlons In species concentrations. To avoid considering variations in both . . . . . ~ . spatla c erections, it IS often convenient to utilize a concentration, Cx, that has been averaged over the airway cross- section. In that case, Fick's Law for the total rate of longitudinal transport, Nx, is expressed by the sum of a bulk transport and an axial mixing term as follows: 1 Cal · 1 . ~r transport, consider a concentration front ~ . that divides inspired air containing a for- eign gas species from residual air initially devoid of foreign gas (figure 10~. When the foreign gas is chemically inert and insoluble in tissue, it is confined to the airway lumen, and the concentration front is displaced at a volumetric rate identical to the bulk flow rate of the entire gas mixture. In that case, it is not necessary to use a retention coeffi- cient (that is, Ax = 1~. On the other hand, when a soluble gas is inspired, then revers- ible absorption into the airway wall causes a reduction in its mean axial displacement rate, and the retention coefficient is less than unity (Dayan and Levenspiel 1969~. Finally, when a gas species is chemically reactive, then slower-moving molecules near the airway walls are preferentially and

James S. Ultman 341 A INERT INSOLUBLE ('x = ~ ) BY Bulk | transport B SOWBLE(7X<1) Mixing Bulk l transport Mixing C IRREVERSIBLE REACTION (>x > 1) Bulk I I. transport Mixing Figure 10. Wash-in of foreign gas. Bulk transport refers to the longitudinal displacement of the concen- tration front in a flowing gas stream, and mixing is characterized by the increase in the width of the front. permanently absorbed from the airstream, leaving a greater fraction of faster-moving molecules in the vicinity of the airway centerline. In that case, the mean rate of translation of the concentration front in- creases above V, and the retention coeff~- cient has a value greater than unity (Sanka- rasubramanian and Gill 1973~. In contrast to bulk transport, which has been visualized in terms of the axial dis- placement of a concentration front, longi- tudinal mixing is characterized by the axial spreading or dispersion of transported spe- cies about the front. Three basic mecha- nisms may be responsible for longitudinal mixing: axial diffusion is due to the Brown- ian motion of individual molecules and is always present, whether or not there is flow; mixing by asymmetry is due to the nonuniform distribution of transport path lengths and to the unequal flow along these paths; and convective diffusion is due to the presence of a velocity profile which causes a progressive separation between the faster- moving molecules near the airway center- line and the slower-moving molecules near the airway wall. Details regarding the mathematical formulation of the mixing coefficients, ~x, corresponding to these mechanisms, can be found in a recent re- view paper (Ultman 1985~. Mathematical Models Modeling is the process by which the be- havior of a system of interest is represented by a simplified mathematical formulation to serve a specific purpose such as correla- tion of data collected under a variety of conditions; scale-up or extrapolation of performance from small to large systems;  or the prediction of behavior in completely new situations. An integral part of the modeling process is the selection of physi- cal and mathematical assumptions, approx- imations, and simplifications to make the . . . governing equations perspicuous anc trac- table. This can lead to a variety of alterna- t~ve models of varying complexity and utility (Himmelblau and Bishoff 1968~. Se- lection of a suitable model must be made and justified within the context of the pur- pose it is to serve and the data required for its validation and implementation. A common simplification is to ignore . . . . . . . spatla1 variation in species concentration in one or more dimensions, in the most ex- treme case assuming that, in designated regions, concentration is independent of position and is a function of time only. This corresponds to the physical assumption that materials are well mixed within each region. The governing material balances for such compartment models are ordinary differential equations. Although compart- ment models require few parameter val- ues an advantage in scale-up calcula- tions they cannot realistically incorporate detailed effects of geometry or fluid dynamics. To do so is a more ambitious task that requires a model in which concen- tration varies continuously along one or more principal directions instead of being represented by a single value in each of a few well-mixed regions. Such a distribut- ed-parameter model is governed by partial rather than ordinary differential equations. Because its parameters can be closely re- lated to basic material properties and ana- tomic features, distributed-parameter mod- els are capable of predicting system behavior. In modeling pulmonary gas uptake, it is expedient to make assumptions that uncou- ple the mass transport processes in the

342 Transport and Uptake of Inhaled Gases lungs from those in other organ systems. For example, in treating the uptake of 03, it has been commonly assumed that rapid irreversible reaction occurs in the mucus, bronchial tissue, or both, so that the O3 has no opportunity to reach other organs. And in classic models applied to the uptake of CO, it has been assumed that the extraor- dinary affinity of hemoglobin for CO sup- presses its blood gas tension. Thus, the molar concentration of bound CO reaching the systemic circulation may be significant, but its backpressure on the pulmonary dif- fusion process is negligible. Both of these approximations make sense in the analysis of acute exposure, where lung tissue and blood do not have the opportunity to build up a significant store of pollutant and where chemical reaction is not limited by the availability of biological substrates. However, the modeling of chronic expo- sure should consider such effects, and in so doing, must also consider the influence of metabolism and excretion in other organs (see, for example, Haggard 1924; Kety 1951~. Compartment Models The oldest mathematical model of the lungs is attributed to Bohr (1891), who suggested that the airways be idealized by two serial compartments: a dead space representing the upper and conducting airways in which gas is transported by bulk flow with complete ab- sence of mixing, and an alveolar region in which mixing is so complete that composi- tion is uniform. Whereas the Bohr model predicts that expired gas is composed of a dead space of constant composition followed immediately by alveolar gas also of constant composition, modern gas monitoring has shown that in addition to pure dead-space gas (phase D, there is a finite transition in com- positicn (phase II) to the alveolar gas sample (phase III) which exhibits a gradually sloping plateau as expiration continues (see, for ex- ample, Fowler 1948~. To analyze gas uptake, the Bohr airway model must be coupled to a model of pulmonary circulation that simulates the progressive equilibration of capillary blood with alveolar gas. A realistic approach is to neglect backmixing in the direction of blood flow and to combine the uptake rate given by equation 17 with a steady-state material balance along a differential length, dy, of capillary (figure 1 la): KmS~ pxg-P*b) (dy/L) = Uptake Rate into Blood Q~dCx + dcxr) (21) Net Output Rate by Blood Flow  where L is capillary length, Q is volumetric rate of blood flow, Px is the (constant) alveolar partial pressure, Pxb is the blood gas tension, and Cx and Car are the molar concentrations in blood of species X in physically dissolved and chemically com- bined forms, respectively. To solve this equation, it is necessary to provide relation- ships between Cx, Cxr, and PXb. For inert gases, equation 3 constitutes this relation- ship, but the situation is more complicated for reactive gases. Consider, for example, the case of a reversibly reacting gas that undergoes a reaction sufficiently rapid that equilibrium occurs at all points in the capillary. Then equation 7 and a differential form of equa- tion 3 must both be applied to equation 21 with the result: (`KmS/Q) (y/L) = JPXb( r) Pxu ~ tYXb + AXE) / ~ PXg PXb) dpXb (22) where Px and PXb(L) are the equilibrium partial pressures of species X in venous and arterial blood, respectively. To perform this so-called Bohr integration (Hills 1974, p. 64), the exact dependence of reactive capacitance on gas tension must be known. For purposes of discussion, however, ,13x can be fixed at an appropriate mean value, fix, thereby allowing analytical integration of equation 22 (Piiper et al. 1971~. The resulting gas tension distributions (figure lib) depend upon a single dimen- sionless parameter, B 0 = Kin S. / ~ Q ~ ~Xb + Fob) ~ ~ ~ the ratio of characteristic convection time to diffusion time. For small values of this

James S. Ultman A J ~ Pxg region / Diffusion barrier t~ dl~lX ,: /Capillary PXV ~'ltr''''''_~' ~P*b(L) ol ~ Y ~ 343 B Pxg Pxv Figure 11. Alveolar uptake model. A: Schematic diagram of model. B: Longitudinal capillary gas tension distributions resulting from Bohr integration. (Adapted with permission from Piiper and Scheid 1980, p. 138, and from Academic Press, Orlando, Fla.) ratio, diffusion across the parenchymal tis- sue is slow, and there is insufficient time for end-capillary blood to become equilibrated with alveolar gas. For large values of the Bo ratio, however, the relative perfusion rate is small enough to ensure complete equilibra- tion. This opposition of convection and diffusion effects is also apparent in the overall uptake equation derived from the Bohr integration (Piiper and Scheid 1980~. Mx = KmSt(1-e B°~/Bol(PXR-Px',) (24) Equation 24 suggests that the overall driv- ing force for alveolar uptake is the differ- ence between alveolar partial pressure Px and venous tension Px and the quantity in square brackets corrects the overall mass transfer coefficient for any added resistance due to a perfusion limitation. When Bo is small, this correction factor approaches unity so that the uptake is determined by KmS alone (that is, it is diffusion limited), but when Bo is large, the correction term  approaches 1/Bo and uptake is proportional to Q(~Xb + Fib) (that is, perfusion limited). Although envisioned by physiologists as a means of evaluating pulmonary function, CO uptake should be regarded as the most extensively studied prototype of pollutant absorption. Carbon monoxide has a low aqueous solubility, but its reactive capaci- tance with hemoglobin is so large that the value of Bo is only 0.02, implying diffu- sion-limited uptake. Moreover, because of its strong affinity for hemoglobin, CO gas tension in pulmonary capillary blood is much below its alveolar partial pressure. In the extreme case where CO venous tension can be disregarded entirely, equation 24 indicates that KmS may be computed as the ratio of uptake rate to alveolar partial pres- sure, MX/PX, a quantity called the pulmo- nary diffusing capacity. And even though carboxyhemoglobin reaction may not be sufficiently rapid to justify this assumption of negligible CO backpressure, it is still possible to equate KmS with MX/PX when the reaction kinetics are first order with respect to CO concentration (Roughton and Forster 1957~. In that case, KmS is composed of a hemoglobin reaction capac- ity, (C~XkrV)b, in addition to a true alveolar membrane-diffusing capacity, (KmS)', as given by KmS = [1 /(KmS) r + 1 /(C~Xkr V)b] - ~ (25) where Ah is blood volume in the capillary bed, and krb is the reaction rate constant of species X in blood. For CO, the membrane diffusion resistance 1 I(KmS)' is typically

344 three times the reaction limitation 1/ (C[xEr VJb. A popular determination of diffusing ca- pacity is the single-breath method in which the decline in expired alveolar composition during a known breath-holding interval is measured. In this situation, it is appropriate to apply an unsteady-state differential ma- terial balance to the alveolar compartment. T [d( VgpX )/dt] + MX = 0 (26) Accumulation Uptake Rate Rate An important feature of the Bohr model is that inspiration and expiration are, in effect, occurring simultaneously and the alveolar volume, fig, is constant. After introducing the additional assumption that venous gas tension is negligible, equations 24 and 26 can be integrated to obtain PER (fib) /PXR(O) expel-[KmS(1-e-B°)R T/VgBoitb) (27) where Px (tb) is the alveolar partial pressure of species X after breath-holding time tb. This formula, which accounts for both perfusion and diffusion effects, is a gener- alization of the more traditional Krogh equation in which a diffusion limitation for CO is assumed. Three important limitations in the Bohr model have been identified in previous applications. First, the dynamics occurring during periods of inspiration and expiration are not included in the conventional Bohr model and, therefore, the effect of breath- ing pattern cannot be evaluated. This re- striction may be overcome by using a dy- namic Bohr model (see, for example, Murphy 1969; Graham et al. 1980) in which input and output by gas flow explicitly ap- pear as terms in the differential material bal- ance. Second, the Bohr model fails to antic- ipate imperfect mixing phenomena, such as those responsible for phase II and sloping phase III behavior of expired single-breath concentration curves. These mixing dynam- ics can be simulated by using multicompart- ment models in which parallel ventilation Transport and Uptake of Inhaled Gases paths are provided (Robertson et al. 1950~. Third, the Krogh equation predicts a linear semilogarithmic relationship between ex- pired alveolar concentration and breath- holding time, whereas the relationship for highly soluble gases has been shown to be curved (Cander and Forster 1959~. Models incorporating permeable bronchial compart- ments in addition to the alveolar region would be useful in simulating such data. The model of Saidel et al. (1973), which  overcomes the first two of these three limitations, exemplifies the potential flexi- bility of multicompartment analysis. Their five-compartment model (figure 12a) can simulate stratified and regional concentra- tion . inhomogeneities, nonuniformities in gas flow distribution, and distension of airway volumes, and it can incorporate any desired breathing pattern. The general mass balance equation for a compartment j in Saidel's model is similar to that of the Bohr model, equation 26, with addition of terms to account for input and output by gas flow: 1 R T [do Vipxj)Idt] Accumulation Rate (VIRT) ~ (fkjpX'-fjkPxj)- MXj (28) k Net Input Rate by Uptake Gas Flow Rate where V is the time-dependent tracheal flow; Vj is compartment volume; PXj is species partial pressure; Ok is the fraction of tracheal flow from compartment j to an adjacent compartment k; and Mx is uptake rate through the walls of compartmentj. In the two alveolar compartments, Mx was related to Px by equation 17, and in the three bronchial compartments, Mx was ne- glected. The five simultaneous differential equations that result cannot be solved ana- lytically, but must instead be integrated using well-established computer algorithms. In simulating the single-breath CO up- take test, Saidel et al. (1973) assumed that the CO backpressure was zero. They also

James S. Ultman 345 A COMPARTMENTS Upper airways Conducting airways Alveolar regions \ GO l ~G-5 ~ / ~ Pulmonary Venous _l ~ Arterial capillaries blood blood B G In ~ US C\S LL ~ ~ .m a c '( '- ~ o - c Ul ~ CC oD _ X 100 80 60 40 20 10 8 6 2 1 \ Normal 4 8 12 16 EXPIRATION TIME (see) Figure 12. Multicompartment model of CO uptake. A: Schematic diagram of the model. B: Simulation of CO partial pressure expired from compartment G-1. (Adapted with permission from Saidel et al. 1973, p. 484, and  from the National Center for Scientific Research, Paris.) assumed that CO uptake from the two alveolar compartments was governed by identical values of KmS. And instead of the conventional technique of changing breath- holding time to visualize the effect of up- take on expired alveolar composition, these investigators simulated expiration phases of varying times. Figure 12b illustrates the re- sulting CO uptake in the presence of chronic obstructive pulmonary disease (COPD) as compared to the more rapid uptake in a healthy lung. The dramatic difference be- tween the two simulations is merely the result of adjusting the fk parameters to ac- count for the inhomogeneity of alveolar ven- tilation in the presence of COPD. It would be useful to extend Saidel's model for general application to pollutant gases. The principal element required to complete the formulation is a rate expres- sion for bronchial absorption, analogous to equation 24 for alveolar absorption. Du- Bois and Rogers (1968) modeled bronchial uptake of nonreactive foreign gases as dif- fusion from a well-mixed airway compart- ment to a well-mixed capillary compart- ment across a homogeneous tissue barrier (figure 13, inset). The use of a well-mixed compartment implies that pulmonary cap- illaries are randomly oriented, thus provid ing extensive backmixing of blood; and the assignment of a diffusion resistance to the tissue alone implies that diffusion through the gas boundary layer and mucous blanket is relatively rapid. In addition, the analysis was restricted to acute exposure where recirculation of pollutant to afferent capil- lary blood is negligible. Under these conditions, a steady-state material balance around capillary compart- ment j results in the following relationship between uptake rate and airway partial pressure: Mxj = KmjSjp~j (29) where Sj is the contact area between tissue and capillary compartments, and KmjSj may be viewed as a bronchial diffusing capacity. According to the DuBois-Rogers model, Kntj = (1lax~kn~j + Sj/aXbQj) 1 (30) so that Kmj is analogous to an overall mass transfer coefficient that results from the sum of a tissue diffusion resistance charac- terized with an individual mass transfer coefficient, km' and a capillary perfusion re- sistance arising from the finite rate of blood flow, Qj. Using the stationary barrier model (table 4, entry la) to estimate values of k,nj, DuBois and Rogers predicted the

346 Transport and Uptake of Inhaled Gases Lo Q Q UJ cr: 0.8 In In cr tL 0.6 0.4 0.2 c, o m N2O ~/~Ac/etone // / Gas Px~ My,- O ,... 0 5 10 BRONCHIAL ZONE, j Capillary Tissue 0 Q 15 Figure 13. Bronchial uptake model. Partitioning of pollutant between capillary blood, p *j, and airway lumen, Px,, in the conducting airway generations. (Adapted with permission from DuBois and Rogers 1968, pp. 40, 42, and from Elsevier Science Publish- ers. ) partitioning of gas tension between ca- pillary blood and the airway lumen, Px,/Pxj' in each of 15 conducting airway genera- tions of Weibel's model A. When capillaries were assumed to envelop the outside of the bronchial wall, Px./Px values distal to the tenth generation are at least 0.95 (figure 13), indicating that blood gas tension is within 5 percent of airway gas partial pres- sure. In other words,- soluble gas penetra- tion from the airway lumen through the bronchial wall is nearly complete. How- ever, in more proximal generations where the tissue layer is thicker, there is a more significant departure from complete pene- trat~on. Recognizing that the spatial distribution of blood vessels can influence the penetra- tion of pollutants through tissue, DuBois and Rogers also investigated models in which perfusion occurred within the bron- chial wall. Generally speaking, they found that capillaries located near the airway lu- men are effective in removing absorbing  species before they can penetrate far into tissue, and deep tissue damage is thereby averted. For example, when perfusion was assumed to be uniform throughout the bronchial wall, Pxj/Px never exceeded 0.25. Hori and Suzuki (1984) presented a model of bronchial uptake of reactive gases wherein diffusion from an airway compart- ment to tissue and to capillary blood oc- curred in parallel. A physical basis for this model would be a bronchial surface com- posed of coexisting capillary and tissue regions. Although this may be a reasonable picture of the upper airways where a rich network of superficial capillaries exists, it would seem to be less appropriate in bron- chial airways. The overall mass transfer coefficient for the Hori-Suzuki model was formulated as Km = [RT/km~ + 1l(axbEm`~) (1 + ¢] ~ (31) where 6, the ratio of aXkm values between tissue and blood, reflects the relative af- finity of the pollutant species for these two competing absorbent phases. In estimating uptake in the conducting airways during inspiration, Hori and Su- zuki analyzed the steady flow of polluted air through a series of 16 well-mixed air- way compartments corresponding to the bronchial generations of Weibel's model A. Utilizing equation 31 in an appropriate steady-state material balance, uptake rates for each compartment were computed, and then summed to determine accumulated uptake. Since constant values of 0.0181 and 1.41 cm/see were specified for kmb and km, respectively, and ~ was fixed at 6.21 regardless of the diffusing species, uptake values were a function of blood solubility only. The results of these computations (figure 14a) indicate that accumulated up- take is negligible when axb < 10-9 kg- mol/m3/Pa, but when ax, > 10-s, absorp- tion into the conducting airways is essentially 100 percent. Distributed-Parameter Upper Airway Models The upper and lower respiratory tracts have often been modeled separately as dis- tributed-parameter models, but rarely to

lames S. Ultman 347 100 AL 80 y cam J 3 60 40 20 o - ~O3 so2 10-8 10-7 10-6 10-5 LIQUID PHASE SOLUBILITY, c~xb(kg-mol/m3/Pa) Figure 14. Comparison of model predictions of total lung uptake. Based on data from a: Hori and Suzuki (1984); b: McJilton et al. (1972). "ether. There are at least two reasons why this separation is natural. First, the upper airways are adapted to perform some func- tions not shared with the lower respiratory tract: filtering out particulate material and warming and moistening inspired air be- fore it reaches the lower airways. Second, the geometry of the upper airways is com- plex, irregular, and asymmetric, making it difficult to represent concisely, but the tra- cheobronchial tree is regular and symmet- ric in a way that makes it possible to represent a vast number of interconnecting branches with a simple and tractable ideal 1zatlon. When air is inhaled simultaneously through the nose and mouth, analyzing uptake rate through the upper airways re- quires separate models for nasal and for oral absorption coupled according to the distri- bution of airflow through the two chan- nels. As applied to absorption during a steady inspiratory flow, this has been ex- pressed formally (Oulrey et al. 1983; Klein- man 1984) as: PxO/Pxi fn(PxOlPx`)n + (1 fn) (PxOlPxi~m (32) where Pxo/Px, sometimes known as tracheal penetration, is the ratio of pollutant partial pressures in the trachea and inhaled air; (PxO/Px,~n and (fix /Px~m are the separate tra-  cheal penetrations for air access through the nose and through the mouth. resoectivelv; fn are available from direct flow measure- ments (Niinimaa et al. 1981), the nasotra- cheal and orotracheal penetrations have been formulated with distributed-param- eter models. To explain the penetration of a variety of soluble vapors during steady inspiratory flow, Aharonson et al. (1974) modeled the nose as a tortuous conduit surrounded by a combined mucous/tissue layer that is en- veloped by a sheath of capillaries (figure 15a). Neglecting longitudinal mixing in the airstream, a steady-state material balance was written around a differential length, dy, of the conduit. v dpx _ _ R = Km~y~a~y)(px,~,,-pX6) Net Input Rate by Gas Flow Uptake Rate (33) where the overall mass transfer coefficient Kitty) and airway perimeter any) are both functions of longitudinal position, y. By assuming that gas tension of pollutant throughout the blood is negligible, this ordinary differential equation was inte- grated from the airflow inlet to the outlet with the result PxOlPxj = exp ~-KmSR T/V) (34) v , . . . end In is the fraction of trachea! flow that where K,nS is the value of Km~y~a~y) inte .1 ~ ~ Errs ~ r grated over the length of the conduit. A enters through the nose. Whereas values of

348 Transport and Uptake of Inhaled Gases A Blood ~ Mucus lined tissue ~' Air Pxj ~ B / pxb at._ I Pxg I dy 1: Bulk transport RT - Longitudinal _ diffusion ~ ~ _&XS ~ RT by / __; __ A(y) a(y)' Uptake Mx ~ PxO Figure 15. Geometry of distributed-parameter models. A: Upper airways (adapted with permission from Aharonson et al. 1974, p. 655, and from the American Physiological Society). B.: Symmetric tracheobronchial tree. similar model was utilized both by Oulrey et al. (1983) and by Kleinman (1984) to correlate SO2 absorption data obtained in the upper airways. They alternatively ap- plied equation 34 to the nose and to the mouth by using separate values of KmS. As Brain (1970) pointed out, the nose functions much as an industrial gas-liquid scrubber, and general design methods for such devices are common in chemical en- gineering literature (see, for example, Treybal 1980, pp. 30~313~. These meth- ods facilitate integration of the differential  gas-phase material balance (eq. 33) without the need to neglect the possible buildup of pollutant backpressure in the liquid absor- bent. For example, when a nonreactive soluble pollutant is absorbed with a parallel flow of initially pure liquid, then PXolpxl = {F + exp [(-KmSRT/V) (1 + F)~) /~1 + F) (35) where F is a dimensionless ratio defined in terms of the gas-liquid flow ratio V/Q as F = V/R T`~xbQ (36) On the basis of an estimated respiratory air-to-blood flow ratio of V/Q = 10 (La Belle et al. 1955), gases of low solubility (10-6 > axb > 10-8 kg-mol/m3/Pa) should have nasal F values of one or more, whereas pollutants of very high solubility (axb > 10-5 kg-mol/m3/Pa) may have F values considerably less than one. In the latter case, equation 35 reduces to the Aha- ronson model (eq. 34), implying that back- pressure created by absorbed pollutant is negligible. On the other hand, when F is

lames S. Ultman 349 greater than one, there is significant back- pressure, and equation 35 predicts penetra- tion ratios, PxOlPx., which are larger than those expected from the Aharonson model. Distributed-Parameter Lower Airway Models Whereas the development of upper airway models has been limited, distributed-pa- rameter modeling of gas transport in the tracheobronchial tree has received more attention. A primary motivation has been to understand airway mixing phenomena such as the slope of the phase III alveolar plateau or the dependence of anatomic dead space on respiratory flow and breath- holding time. Rauwerda (1946) was the first to combine a spatially varying ana- tomic geometry with the diffusion equation in order to analyze the longitudinal distri- bution of inert gas within the airways. He assumed that summed airway cross-section increased linearly with longitudinal dis- tance, and he ignored the contribution of individual airway dimensions and branch- ing characteristics. Major improvements in Rauwerda's model have included more re- alistic representation of summed airway cross-section (LaForce and Lewis 1970~; the incorporation of convective bulk transport due to breathing pattern (summing et al. 1971~; the use of a concentric two-zone model of respiratory airways to account for the stagnant gas pockets between alveolar septa (Scherer et al. 1972; Paiva 1973~; and the analysis of diffusion and convection be- tween parallel units (Paiva and Engel 1979) to simulate ventilation inhomogeneities. These analyses all share the philosophy that, first, the key variable along a given transport path is the longitudinal distribu- tion of partial pressure resulting from si- multaneous diffusion and convection, and, second, alternative transport paths are alike enough that the tracheobronchial tree can be represented as a single airway with summed cross-section that expands with distal dis- tance from the trachea (figure 15b). Mc~ilton et al. (1972) followed the same philosophy in developing the first distributed-parameter model of reactive gas uptake, beginning with a material balance on a differential section of the airway lumen given by A bpx RT bt Accumulation Rate V Pa, RT by Net Input Rate by Gas Flow  + ~ ~ {D A-~-atMX) RT by ~ by J As ~ Longitudinal Uptake DiEusion Rate Rate (37) -where Px represents partial pressure aver- aged over the cross-section A available for flow, a is the perimeter available for gas absorption, and both A and a are increasing functions of distal longitudinal distance, y. In solving this equation, Mc~ilton et al. used a finite-difference algorithm in which respiratory flow, V, varied with time in sinusoidal fashion, and in which the airway domain was divided into 25 longitudinal zones patterned after the 23 generations of Weibel's model A. Each airway zone was bounded by a tissue layer coated with an inner liquid film of either mucus in con- ducting zones or surfactant in respiratory zones. The principal assumptions in speci- fying absorption between the gas, liquid, and tissue layers were that the diffusion resistance of the gas boundary layer within the airway lumen is negligible; transport through the liquid film occurs by steady- state diffusion; and chemical reaction oc- curs exclusively at the liquid/tissue inter- face, where pollutant is instantaneously consumed. Since there is no accumulation or depletion of pollutant within the liquid film, the influx of pollutant from the air- way lumen is identical to tissue dose. And because pollutant concentration is negligi- ble at the liquid/tissue interface, the uptake flux, MxlS, could be expressed as the prod- uct of diffusion conductance in the liquid film, aXkm, with the pollutant partial pres- sure in the airway lumen, Px In perform- ing computations, cake values were esti- mated using the stationary barrier equation (table 4, entry la) in conjunction with

350 Transport and Uptake of Inhaled Gases appropriate values of aqueous-phase sol- ubility and diffusion coefficient and liquid film thickness of 10 Em in upper genera- tions, ~5 ,um in alveolar ducts, and 0.3 ,um in the alveoli. Predicted values of the local dose, ex- pressed as mass absorbed per unit bronchial surface area per breath, are shown in figure 16 for O3 and SO2. The O3 dose is uni- formly distributed within the conducting airway segments and then declines sharply in the terminal airspaces distal to the twen- tieth model segment. For SO2, a much more soluble gas, the uniform portion of the dose distribution is at a higher level but does not span as many airway generations as in the case of O3. Thus, the integrated doses of these two pollutants are similar. This explains why predicted values of ac- cumulated uptake are relatively insensitive to solubility, experiencing an increase of only 30 percent for a 100-fold increase in cat' (figure 14b). Compared to Hori and Suzuki's (1984) steady-state inhalation model (figure 14a), the Mc~ilton model predicts a greater total uptake for a low-solubility gas such as O3, but less uptake for a higher-solubility gas such as SO2. The former result is due to absorption in the respiratory zone, which was ignored in the Hori-Suzuki model and which is an important consideration for gases of low solubility. The latter result is probably due to mass transfer coefficient values: those used by Mc~ilton et al. were far smaller than those of Hori and Suzuki. A major shortcoming of the Mc.}ilton model is the localization of chemical reac- tion at the liquid/tissue interface. There- fore, Miller et al. (1978) developed a model of O3 transport in which the site of pollut- ant reaction could vary within the mucus. In particular, the mucous layer was mod- eled as a stagnant barrier into which O3 diffuses from the airway lumen and reactive biological substrates simultaneously diffuse from underlying tissue. At the reaction plane where the O3 and substrates meet, they are instantaneously consumed by chemical reaction. And the depth of pene- tration of this plane into the mucous layer is limited by the rate of O3 absorption relative to the rate of substrate supply. By defini ~o ' o2: To 3L 1o 4 -  10-5 L ~_ 10 6 0 7 10 8 10 9 L __ ~ - ~ so2 Lobar Trachea bronchi 1 ' 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 10 15 20 2s \ \ ~o3 1 Rbl Alv 0 5 MODEL SEGMENT Figure 16. Dose distributions predicted by the Mc- Jilton model. Environmental concentration is 1,000 ,uglm3, tidal volume is 500 ml, and breathing fre- quency is 15 breaths/mint Rbl refers to respiratory bronchioles; Alv to alveolus. (Adapted with permis- sion from McJilton et al. 1972.) lion, tissue exposure is nonzero only when O3 diffusion is sufficiently rapid that the reaction plane reaches the mucus/tissue in- terface, at which point all pollutant is con- sumed by the tissue. Other additions to this model include the use of a mass transfer coefficient, km' and a mixing coefficient, dx, to better describe lateral and longitudi- nal transport in the airway gas. The tissue doses predicted by Miller et al. in Weibel's geometric model generally in- crease from a value of zero in the trachea, where the reaction plane cannot completely penetrate the thick mucous layer, to a max- imum value in the 17th generation where the mucous layer is thinnest (figure 17~. In more distal generations, the dose declines because absorption is spread out over the rapidly increasing alveolar surface area and also because the surfactant layer is assumed to be unreactive with O3. In spite of the marked decline in tissue dose per unit area

lames S. Ultman 351 jo-4f ~n-5L . s Cal n N ~ 10-6 o 1n-9 0-7 ~ Tra h - t Oa concentratio - · 4000.0 \ ~ \ \ 10-8 · 10000 ~ %\~\\6 o 5000 \ ~ I . 250.0 in ~ 425.0 \ \ - 0 62.5 :~ ~ 9 lo '' 42 is 44 45 46 47 ,8 49 20 2' 22 23 _~/ AIRWAY GENERATION Figure 17. Ozone dose distribution predicted by the model of Miller et al. Tidal volume is 500 ml and breathing frequency is 15 breaths/mint (Adapted from Miller et al. 1978, p. 91.) trom conducting airways to respiratory air- spaces, the total mass of O3 absorbed into the respiratory zone was found to be five times that absorbed into conducting air- ways. The model of Mc~ilton et al. predicts a tissue dose strictly proportional to tracheal  concentration, but this relation in the Miller model is nonlinear because of the moving reaction plane. For example, con- sider the influence of tracheal concentration on dose distribution in conducting airways as predicted by the Miller model. Proximal to the sixteenth generation, there is no tissue exposure at tracheal concentrations below 60 ,ug/m3. But at progressively larger concentrations, more and more con- ducting airway tissue is exposed to pollut- ant, beginning with the more distal gener- ations and progressing toward the trachea. Thus, the total uptake in conducting air- way tissue increases with tracheal concen- tration in a disproportionate fashion. The most significant finding in the anal- yses of both Mc~ilton et al. and Miller et al. is that certain tissues receive a much higher . dose than others, indicating that O3 dam- age might be localized. The predictions of the two models are somewhat different, however. Mc~ilton's model exhibits a uni- form dose throughout the conducting air- ways that is considerably greater than O3 exposure in the respiratory zone. On the other hand, the distribution curves com- puted by Miller et al. have dosage peaks in the respiratory bronchioli that are many times larger than doses in adjacent conduct- ing airways and respiratory airspaces alike. Such sharply defined peaks indicate that the respiratory bronchioli are particularly sus- ceptible to O3 damage, and evidence from histological studies on primates (Dung- worth et al. 197.5) supports this conclusion. Miller et al. (1983) have applied the same model to NO2 uptake, and the shape of resulting dose distribution curves is similar to those of O3. Miller et al. (1985) sought to improve their previous model for O3 transport by accounting for the finite rate of O3 reaction in surfactant, tissue, and blood, as well as in mucus. The bronchial wall was modeled by adjacent layers of stagnant mucus and solid tissue, and the respiratory airway wall was modeled by a series of stagnant surfactant, solid tissue, and stationary blood layers. Quasi-steady transport by simultaneous diffusion and irreversible chemical reaction was assumed to occur through each layer. The use of first-order reaction kinetics im- plying a constant concentration of substrate precluded a limitation on pollutant flux by the supply of biological species. Rather, the O3 flux was restricted by boundary conditions at the edge of the (infinitely thick) bronchial tissue layer and at the centerline of the pulmonary blood layer. The equations that ultimately de- scribed this model were linear, and there- fore, when normalized by the tracheal con- centration, the predicted dose distributions were independent of tracheal concentra-  t~on. A primary objective in carrying out computations with this model was to de- termine the sensitivity of dose predictions to several parameters including the gas- phase mass transfer coefficient, the longitu- dinal mixing coefficient, the mucus reac

352 Transport and Uptake of Inhaled Gases 10-4 _ D c`i ~ 10-5 CO o o UJ o 10-6 C] 3 en - o 10-7 I o m 10-8 G En a Humans (first generation of respiratory bronchioles) Rabbit 0 Guinea pig I I I ~ I I Ill 1 1 1 1 1 1lil 1 1 1 1 1 I til 10 1o2 103 104 TRACHEAL O3 CONCENTRATION ~g/m3) Figure 18. Comparison of respiratory bronchial tis- sue dose in humans and laboratory animals. (Adapted from Miller et al. 1978, p. 95. ) lion coefficient, and the minute volume of breathing. As has been established in table 6, the gas-phase diffusion resistance of O3 is relatively small because of its low solubil- ity. Therefore, it is not surprising that dose .. . . . . predictions were Insensitive to t he gas- phase mass transfer coefficient. Similarly, the magnitude of the longitudinal mixing coefficient had little effect on O3 uptake throughout the respiratory zone, which is consistent with previous mathematical analyses of O2 and CO2 exchange (Pack et al. 19771. On the other hand. dose oredic t~ons were quite sensitive to the mucus reaction rate constant, he. As the value of kr was increased, the dose to bronchial mucus became larger and the bronchial tissue dose decreased dramatically. This demonstrates the effectiveness of mucus in removing O3 before it can damage underlying tissue. Simulations performed at progressively in- creasing tidal volumes and breathing fre- quencies implied that exposure of respira tory airway tissue to O3 is aggravated by increasing levels of exercise.  An important application of these mod- els is the comparison of dose distributions in laboratory animals to those in humans. Miller et al. (1978, 1983) utilized their model with morphometric and physico- chemical data available for guinea pigs and rabbits as well as for humans. From the viewpoint of O3 dose absorbed into respi- ratory bronchioles (figure 18), the model predicts that humans are more like rabbits than guinea pigs, at least at tracheal con- centrations above 100 ,ug/m3. It must be borne in mind, however, that the validity of such predictions depends strongly on the accuracy of the many anatomic, equilib- rium, diffusion, and reaction parameters used in the model. Recommendation 3. To analyze total uptake of pollutants and to predict dose distribution, mathematical models that ac- count for ventilation and perfusion limita- tions, including their regional distribution, should be developed and validated. Experiments In this section, experimental methods that can be used to validate proposed transport models and estimate relevant thermody- namic, diffusion, and reaction parameters are discussed. Where published data exist, they are compared with the predictions of models already presented in the Mathemat- ical Models section. Three categories of experimental methods are described: in vi- tro methods that use physical models or isolated tissue preparations; in viva meth- ods that. because of a high degree of risk or Invasiveness, use animals as subjects of study; and in viva techniques that are suit- able for use on human subjects. In Vitro Methods Physical models fabricated from stock ma- terials or from casts of actual lung airways offer the opportunity to evaluate mathe- matical models in the absence of undesired ..

lames S. Ultman 353 biological variables. Physical models were used to study gas transport as early as 1915, when Henderson et al. (1915) introduced smoke into a tube in order to visualize airflow patterns and concluded that in- dividual gas species can penetrate well beyond their tidal volume because of non- uniform longitudinal convection. A con . . . ~ . . . . tmulng senes ot sue. ~ Investigations, using inert tracer gases in both upper airway casts (Simone and Ultman 1982) and tracheo- bronchial network models (see, for exam- ple, Scherer et al. 1975; Ultman and Blat- man 1977), have added considerably to the understanding of longitudinal convective- diffusion processes and to the evaluation of longitudinal mixing coefficients. Of greater importance than longitudinal mixing is the lateral diffusion of absorbable species in the vicinity of airway walls. Nuckols (1981) applied the analogy be- tween heat and mass transfer (Bird et al. 1960, p. 642) to infer gas-phase mass trans- fer coefficients from thermal measurements in an adult upper airway cast as well as in a symmetric tracheobronchial model. Mea- surements were made in both steady inspi- ratory and steady expiratory flows, and the resulting correlations of the Sherwood number, km d/DX, show a strong depen- dence on both Reynolds and Schmidt numbers (table 5~. Using the napthalene wall sublimation technique, Hanna and Scherer (1986) deter- mined local mass transfer coefficients in an adult upper airway cast supplied with a fixed inspiratory airflow of 12 liters/mint The distribution in km values from the external nares to the larynx was highly nonuniform, with the largest value occur- ring at the entrance of the nasopharynx where there is an abrupt convergence of airflow from the turbinates. Neither Nuckols nor Hannah and Scherer attempted to simulate absorption processes within the airway wall. Ichioka (1972), however, measured SO2 absorption from a flowing gas stream into moistened filter paper intended to represent the upper- airway mucosa. In each experiment, a straight tube, 60 cm long x 0.9 cm diam- eter, was exposed to a steady flow of a mixture of 5 ppm SO2 in nitrogen for 15 min. and the cylindrical sections of filter paper lining the tube were moistened with either distilled water or with 1 percent bovine serum albumin (BSA) solution. Ichioka's data are presented in terms of the inlet-to-outlet partial pressure ratio. / 11 ' · . .1 A 1 Px,lPxO, allowing comparison to tne ~na- ronson model of uptake into the upper  airways (figure 19~. According to equation 34, a chord drawn from the origin to any data point on a plot of lntpx/px) versus S/V has a slope equal to RTKm, and there- fore the plot is linear when the overall mass transfer coefficient is constant. Clearly this is not the case for Ichioka's measurements. The slopes of the curves progressively de- crease, implying that Km decreases as the transit time, S/V, increases. To interpret these results, consider the overall mass transfer coefficient as being composed of diffusion resistances due to the gas boundary layer (subscript g) and to the filter paper (subscript 11: A , RTKm = [l/km + 1/RT(C~xkm~l]-1 (38) According to this equation, the largest Km values occur when the resistance of the filter paper is negligible, and RTKm is well approximated by km . The Px,/Px curve for this gas-phase limited diffusion has been constructed in figure 19 using standard correlations for km (table 5, entry 4~. Be- cause the experimental Px./Px values are consistently smaller than the gas-limited predictions, the wet filter paper must con- stitute a significant impediment to diffu- sion. This is especially true for long transit times, when absorption is so great that a significant backpressure of SO2 is built up at the gas liquid interface. Judging from typical values of km and R TKm calcu- lated from the slopes of the dotted chords in figure 19, equation 38 indicates that the filter paper contributes 33 percent of the overall diffusion resistance. Unlike the gas-limited curve, the data correlations in figure 19 depend upon flow rate, but curiously this effect is different for distilled water and for BSA solution. At any given S/V value, the BSA data suggest that Km is larger at a gas flow of 3 liters/min than it is at 1 liter/min, whereas the oppo- site is true for distilled water. This differ

354 5 r 4 _ ~ _ UJ 3 3 _ Oh UJ a: Q E ji 2 J At _- Gas-phase limitation ~ _ my' .. c, . , ~ / o · / ~ / ~ . / : ~- · ~ ~ \AI_.A~ \` 0 BSA 3 lito s/minWate { BSA 1 L..-, , , , , , , , , , 0 5 10 TRANSIT TIME, S/V (see/cm) Figure 19. Measurements of Ichioka (1972) for SO2 absorption from a nitrogen stream flowing through a straight-tube model of the upper respiratory tract. Dotted chords illustrate the graphical determination of RTKm, and the solid curves have been drawn through the data by eye. The broken curve representing gas- phase limited diffusion has been computed using entry 4 in table 5. ence is most likely due to a variation in SO2 chemistry between the two liquids. For example, BSA solution may have a buff- ering capacity for hydrogen ions produced in the bisulfite reaction (eq. 4), thereby increasing the reaction affinity for SO2. Lacking this buffering capacity, distilled water creates higher SO2 backpressures, which reduce the driving force for absorp- t~on. Beyond certain limits of minimum scale and maximum geometric complexity, it is  impractical to construct physical models. Rather, an isolated organ preparation should be used, which more accurately represents the biochemistry and mechanics. Postlethwait and Mustafa (1981) studied NO2 uptake by an isolated perfused rat lung that was supplied with a reciprocating flow of the pollutant-in-air mixture to the trachea. These investigators were able to measure both total uptake and the parti- tioning of reaction products between pa Transport and Uptake of Inhaled Gases renchymal tissue and perfusate. They dem- onstrated that NO2 is absorbed into tissue primarily as NO2, which is then converted to NO3- by erythrocytes in the vascular space. Although not pursued in this partic- ular study, the effects of perfusion and ventilation rate on absorption could be measured in an isolated lung preparation. By analyzing such data in conjunction with a suitable mathematical model such as equation 24, global values of mass transfer coefficients could be estimated and the rel- ative importance of perfusion and ventila- tion limitations could be determined. It might also be desirable to quantify reaction dynamics between pollutants and biological substrates by using isolated tis- sue samples. In such small-scale systems, the confounding effects of diffusion resis- tance can be eliminated by using high con- vection rates and thin samples. Rasmussen (1984) reviewed various systems for cultur- ing lung cells for this purpose. As he points out, the unique problem in achieving a realistic model is that the tissue sample must simultaneously be exposed to aque- ous nutrient medium on the endothelial side and pollutant gas flow on the epithelial side. One possible solution to this problem is to culture cells in growth medium on the outside of permeable hollow fiber tube bundles (Knazek et al. 1972) and then pass the pollutant gas through the lumen of the bundles. Such a bioreactor has the added advantage of a very high surface-to-volume ratio for pollutant uptake. Ire Vivo Animal Experiments The most popular and fruitful approach in studying absorption dynamics has been to ventilate an anesthetized animal through a specially designed face mask or cannula from which a pollutant air mixture can be directed to either the nose or the mouth. Then, by measuring pollutant concentra- tion at the gas inlet as well as in the trachea, it is possible to infer pollutant penetration along a path from nose to trachea or from mouth to trachea. Figure 20 shows tracheal concentration monitored in response to the onset of various O3 levels in a steady air- flow entering the nose (solid lines) or the

James S. Ultman 355 0.6 Inhaled concentration (ppm) - z ~ 0.4 at LL o c: cat o c: ~ 0.2 at //~/~\,__' 1 ~ ~^__~___ 0.29 10 15 20 TIME (min) Figure 20. Time course of tracheal O3 concentration in response to the introduction of a fixed concentra- tion into the nose (solid curves) or the mouth (broken curves) of a dog. O3 entered the upper airways at a steady airflow of 3.5 liters/min and at the concentra- tion labeled on the curves. (Adapted from Yokoyama and Frank 1972, p. 134, with permission of the Helen Dwight Reed Educational Foundation. Published by Heldref Publications. @) 1972.) mouth (broken lines). In this experiment, the trachea was surgically isolated from the lower respiratory tract, which was sepa- rately ventilated with clean air (Yokoyama and Frank 1972~. For each curve, there is a finite time during which O3 concentration rises from its initial value of zero toward a steady-state level, and this rise time appears to be longer for the nose than for the mouth. In principle, a dynamic model of these data would yield useful information about the underlying transport processes. However, the slow-responding devices that are typically used to measure pollutant concentration are incapable of monitoring the true transient behavior. Therefore, all previous studies have been restricted to the analysis of steady-state PxilPxO values In figure 21, PxlPX data for airflow between the nose and the trachea of dogs has been plotted for several noxious gases. The use of transit time for expressing the observed variation with flow is consist ent with the treatment of in vitro data in figure 19. Because SO2 absorption by the nose is so great, it has to be graphed separately using a compressed ordinate. At the other extreme, CO is so poorly ab  sorbed by the upper airways (Vaughan et al. 1969) that it does not appear in figure 21. n 75 It is obvious that none of the data curves ~ 0.78 in figure 21 are linear. Instead there is a consistent trend of increasing slope with increasing airflow rate. It has been sug aested that a higher airflow stimulates greater blood flow, causing distension of blood vessels and thereby increasing the effective area available for absorption. 0.28 ~ ~ ~ ~mu~taneous~y, increased blood flow would more effectively wash out the capil lary bed and diminish vascular backpres sure (Aharonson et al. 1974~. Then again, similar curvature was exhibited by the in vitro data (figure 19), where no physiolog ical response was possible. An explanation that conforms to both in vivo and in vitro data is that the gas-phase mass transfer coefficient, which is directly related to air flow, makes a substantial contribution to the overall absorption process. To explore this hypothesis further, tra cheal penetration data obtained in dogs at two steady inspiratory flows and along nasotracheal as well as orotracheal paths have been assembled in table 7. By apply ing equation 34 to these data, R TKmS values have been computed for O3 and for SO2 at every experimental condition. For 03, these values are of the order of 100 cm3/sec and, assuming an upper airway surface of 100 cm2, the order of magnitude of RTKn7 is 1 cm/sec. But the range of individual km values predicted by the up per airway correlations in table 5 is from 1 to 3.5 cm/sec. Therefore, gas-phase resis tance is indeed a significant factor in the overall absorption of O3 and in the uptake of the even more soluble SO2. In characterizing the flow behavior of the data in table 7, RTKmS is assumed to vary as Am, and the flow sensitivity parameter m was computed from each data pair. For SO2, the orotracheal mass transfer coeff~-

356 Transport and Uptake of Inhaled Gases ~0 - G 6.0 C,' 5.0 G 6 .L 6 I G . . LU z z 40 10 5 3 2 1.5 FLOW RATE (liters/min) 4.0 3.0 2.0 1 000 100 dig, 0.31 + 0.003 ppm - ~ O3 / ,~ 0.80 ~ 0.001 ppm i/ / '' 10 1 f ~ I ~ 0 0.3 TRANSIT TIME (m i n /l iter) _ calm_ · .v 0.2 ~_ 0.4 0.6 0.8 1.0 13.0 14.0 TRANSIT TIME (min/liter) Figure 21. Effect of airflow on the absorption of various foreign gases along the nasotracheal path of the upper airways. (Adapted with permission from Aharonson et al. 1974, p. 656, and from the American Physiological Society.) cient is much less sensitive to flow than is the nasotracheal coefficient. For O3, how- ever, there is an astonishing similarity of flow sensitivity for the nasotracheal and the orotracheal paths, and at both inlet concen- trations reported. With the exception of the orotracheal value for SO2, the values of m are all the same order of magnitude but somewhat less than the 0.854 value re-  ported for km in physical models (table 5, entry la). The smaller flow dependence of the in viva data is probably due to diffusion resistances in tissue and mucous layers that are not directly affected by gas flow. Animal data also suggest that both O3 and SO2 uptake are sensitive to atmospheric concentration. This effect is portrayed by the Px/Pxj tracheal penetration values in fig- ure 22. The SO2 data (Strandberg 1964) were obtained from free-breathing rabbits who inspired a pollutant mixture of known composition from a head chamber. Tra- cheal samples were obtained at peak inspi- ratory flow (I) and peak expiratory flow (E), the former sample being representative of upper airway absorption and the latter indicative of lower airway uptake. The O3 data were obtained on dogs by applying subatmospheric pressure to a tracheos- tomy, thereby withdrawing pollutant mix Table 7. Comparison of Absorption Characteristics of the Nose and Mouth Nasotracheal Steady Flow Inspired Concentration (liters/min) Px,,/Px; 0.001 0.032 0.283 0.631 0.408 0.733 R TKmS (10 4m3/sec) Orotracheal m pX`~/p.~' R TKmS (10 4m3/sec) m l ppm SO2a 3.5 35 26-34 ppm O3 3.5-6.5 35-45 3.5-6.5 35-45 78-80 ppm O3 4.03 20.1 1.05 3.07 0.747 2.07 0.70 0.0044 0.660 0.52 0.665 0.884 0.49 0.732 0.902 3.17 2.42 0.340 0.822  0.260 0.687 a Frank et al. (1969). b Yokoyama and Frank (1972).

James S. Ultman Lit 1.0 cr 6 ~ To I O 0.50 cr _ ~ 6 Al I S02(1) \ ~' 03(1) \ // ~ SO2(E) . o 10-2 10-1 1 10 1o2 INLET CONCENTRATION (ppm) 1n2 Figure 22. The concentration dependence of SO2 penetration (Strandberg 1964) in rabbits and O3 penetration (Vaughan et al. 1969) in dogs. Both inspiratory (I) and expiratory (E) samples were obtained in the trachea. lures from a reservoir into the nose and through the upper airways (Vaughan et al. 1969~. Tracheal concentration was deter- mined by sampling from a tracheostomy tube while the dog breathed spontaneously through the caudal portion of the tracheos- tomy. These data reveal that penetration of SO2 through the upper airways is inversely re- lated to its inlet concentration, but penetra- tion of O3 is directly related to concentra- tion. It is possible to attribute the behavior of O3 to a saturation limitation in chemical reaction rate as concentration increases. Such reaction kinetics are common for a variety of biochemical reactions and are often described by the Michaelis-Menten equation (Mahler and Cordes 1968, p. 153~. It is difficult to conceive of a purely phys- ical explanation for the concentration de- pendence of SO2 penetration. Rather, it has been postulated that short exposures to high levels of SO2 stimulate mucus secre- tion, thereby reducing penetration as com- pared to exposures at low concentrations (Brain 1970~. Far less information is available for ab- sorption in the lower airways than for uptake in the upper airways. In experi- ments where the lower airways of dogs were surgically isolated from the upper airways, penetration of O3 to the respira- tory airspaces estimated as the ratio of expired-to-inspired partial pressures was 357 ~n3 from 0.15 to 0.20, depending on the me- chanical ventilation rate and the inlet con- centration (Yokoyama and Frank 1972~. And in free-breathing rabbits (figure 22),  the analogous ratio for SO2 was from 0.2 to 0.4. Because of the naturally reversing res- piratory flow in the latter experiments, some Resorption of pollutant may have occurred during expiration, as pollutant- depleted air from distal airways passed over the pollutant-rich tissue and mucus in more proximal airways. Therefore, the expira . . . . . . . t1on-to-1nsp1rat1on partial pressure ratios may be somewhat larger than the actual lower airway penetrations. An important factor to consider in the design of uptake experiments is exposure time of the animal to the pollutant. Whereas an acute exposure results in data relevant to transport in a healthy animal lung, chronic exposure can result in ana- tomic and functional derangements that further affect the absorption process. All the investigations cited above utilized short exposures, usually less than an hour. How- ever, Moorman et al. (1973) measured O3 absorption into the upper airways of dogs that were chronically exposed for 8 to 24 hr. and compared the results with data from acutely exposed dogs. Their results show that in dogs, penetration through the nasotracheal path is generally greater dur- ing chronic exposure than during acute exposure. These investigators hypothesized

358 Transport and Uptake of Inhaled Gases that decreased mucus flow due to chronic exposure was responsible for the increased penetration. · Recommendation 4. Using a consistent experimental protocol, total uptake of se- lected pollutants should be measured in different animal species and then used to develop basic rules of extrapolation. In Vivo Human Subject Studies Extensive research into the transport of foreign gases in the human lung has been directed toward the development of non- invasive tests of pulmonary function. To a large extent, progress in this area has been stimulated by the development of reliable fast-responding gas analyzers. Literature on the use of inert insoluble gases for the characterization of gas mixing and distribu- tion is extensive (Engel and Paiva 1985~. One of the most widely used lung func- tion tests employing inert insoluble gas is the multibreath wash-out (Fowler et al. 1952~. In this measurement, the end-ex- pired nitrogen fraction is recorded for a series of regular breaths following a change . . . . ,~ . in 1nsplrec . gas mixture trom room air to pure oxygen. If the lungs behaved in ac- cordance with the static Bohr model, then a semilogarithmic plot of expired nitrogen fraction versus breath number would be a straight line having a slope and intercept from which dead space and alveolar vol- ume could be determined. However, for patients with diseased lungs, this plot is far from linear, and even in normal subjects, expired nitrogen fraction exhibits multiex- ponential rather than single-exponential decay. Such nonideal behavior has been successfully simulated using multicompart- ment models with ventilation inhomoge- neities between parallel regions (Robertson et al. 1950~. Most research on reactive foreign gases has been devoted to the uptake of CO. Because the absorption of this species is limited by diffusion through the alveolar membranes, decreased CO uptake can serve as an indicator of parenchymal tissue abnormalities. In the single-breath method, CO diffusing capacity is computed from the ratio of final-to-initial alveolar con- centration measured during a series of breath-holding periods of known duration (Apthorp and Marshall 1961~. The diffu- sion-limited form of equation 27 (that is, Bo = 0) is appropriate for this calculation, and it predicts that a semilogarithmic plot of alveolar concentration ratio versus breath- holding time has a slope proportional to KmS. Roughton and Forster (1957) recog-  nized that pulmonary diffusing capacity is composed of a true membrane-diffusing capacity and a hemoglobin reaction capac- ity that is proportional to capillary blood volume (eq. 25~. Moreover, they devel- oped a method of estimating the capillary blood volume by measuring the change in reaction capacity at different levels of in- spired O2. Although the method is fraught with difficulties, primarily because a unique value of alveolar concentration must be in- ferred from expired gas analysis, the single- breath CO uptake procedure as standardized by Ogilvie et al. (1957) is still in use. There has also been considerable interest in soluble nonreactive gases such as acety- lene. Since the absorption of acetylene is perfusion-limited, it can be used as an indicator of pulmonary blood flow, Q. Acetylene uptake has been measured by the same breath-holding procedure developed for CO, and the data were then analyzed with equation 27. For a moderately soluble gas, the Bo parameter is large enough that a semilogarithmic plot of expired alveolar concentration versus breath-holding time should have a slope proportional to Q. To study in detail the influence of solubility on uptake, Cander and Forster (1959) per- formed single-breath experiments using five different nonreactive gases including acetylene. Their results depart from the theory in two important ways, particularly for the most soluble gases, ethyl ether and acetone (figure 23~. First, the "percent of initial alveolar con- centration" does not extrapolate to the ex- pected value of 100 percent at zero breath- holding time. This was attributed to an initially rapid absorption of the foreign gas into parenchymal tissue, thereby causing an instantaneous drop in alveolar partial pres- sure. By extending the mathematical model

James S. Ultman 359 ~ ' 30 u, oh ~5 :~ ._ ~ 0 F ID A: ~ ~ c ~ g A: ~ ~ ~ 3 100 ~ h. 50 _ 10 5 b _~ W~ _ 04, _ 1 _ ~v -~ N2`J - - o . _ 1 0 1 0 20 30 40 50 BREATH-HOLDING TIME (see) Figure 23. Breath-holding uptake data for five inert gases of increasing solubility obtained in a series of single breaths. (Adapted with permission from Cander and Forster 1959, p. 544, and from the Amer ican Physiological Society.) to account for this, Cander and Forster were able to estimate reasonable values of the tissue volume. Second, the semiloga- rithmic plots are not linear. Instead, as breath-holding time increases, the alveolar concentration data become progressively larger than expected from a linear extrapo- lation of the initial data. Cander and Forster attributed this result to the contamination of expired alveolar gas with foreign gas that was initially absorbed into conducting air- way tissues. Although no formal model was proposed, this behavior could un- doubtedly be predicted by a multicompart- ment simulation that accommodates de- sorption processes during expiration. By combining individual measurements into one multiple-gas test, it is possible to simultaneously estimate several transport  parameters for the same subject. Moreover, since the parameters are measured under a single set of conditions, their values and the relations among their values may be more reliable than if each was measured under conditions that must necessarily differ, even if only slightly. An example of such an approach is the work of Sackner et al. (1975) who performed a series of breath- holding experiments on subjects inspiring a gas mixture containing helium (to assess alveolar volume), CO (to evaluate mem brane-diffusing capacity and capillary blood volume), and acetylene (to determine blood flow). Whereas Sackner's study used a single-compartment static model to ana- lyze the uptake data, Saidel et al. (1973) used a more sophisticated multicompart- ment model to elucidate both ventilation distribution and uptake dynamics. These investigators carried out parame . . . . ter estimation experiments in two stages. They first performed multibreath wash-out measurements in which uptake of the ni- trogen test gas is negligible. By matching simulations (eq. 28) to these data, flow fraction parameters, Ok, governing distri- bution of volume and ventilation among the four distensible lower-airway compart- ments (figure 12a) could be evaluated for each subject. Then, the subjects were ad- ministered a steady-state uptake test in which a dilute CO-air mixture was in- spired and the end-tidal concentration and uptake of CO was measured during con- secutive breaths. By using the fkj ventila- tion parameters already established from the nitrogen wash-out data, simulations of CO uptake data could be performed to estimate the parenchymal diffusion param- eters, KmS. In humans as in animals, it is also possi- ble to study uptake in the upper airways independent of the lower airways. For ex- ample, Speizer and Frank (1966) described an experiment in which cooperating sub- jects inhaled a 15-ppm mixture of SO2-air into the nose, while inspiratory and expi- ratory samples were automatically with- drawn through a nasal sampling tube just inside the nares and a pharyngeal sampling tube was inserted through the mouth. By comparing concentrations between the nose and pharynx, it was clear that SO2 penetration beyond the upper airways was only 1 percent during inspiration. And be- cause the pharyngeal concentration was on the order of 0.4 ppm during both inspiration and expiration, it appears that the lower airways neither absorbed nor desorbed a de- tectable quantity of SO2. However, the ex- pired nasal concentration was 2 ppm, five times larger than the pharyngeal value, indi- cating that Resorption from the nasal pas- sages was promoted during expiration

360 Transport and Uptake of Inhaled Gases by the flow calf air that had horn strinn~H of SO2 during inspiration. This temporal countercurrent exchange process enhances the protective capability of the upper air- ways. That is, in addition to preventing pollutants from reaching lower airways, countercurrent exchange reduces pollutant loading in the upper airway mucosa. Up to this point, the discussion has fo- cused on physicochemical problems, namely, absorption rates and internal con- centration distributions of foreign gases. From a medical point of view, however, these physicochemical descriptions are use- ful only if they correspond to functional abnormalities. The connection between characterization of local dose on the one hand, and graded response of lung function on the other, has clearly been established for SO2. In particular, it is well known that acute exposure to SO2 leads to a reversible increase in airway flow resistance because of bronchoconstriction mediated by neural chemical sensors (Frank 1970~. Because these sensors are particularly abundant near the larynx and carina, it should be possible to correlate local dose at these sites with airway resistance. Amdur (1966) used pub- lished SO2 penetration values obtained by Strandberg (1964) on free-breathing rabbits in conjunction with her own airway flow- resistance values measured in guinea pigs and graphed the logarithm of airway resis- tance increase versus the logarithm of tra- cheal concentration. This local dose/re- sponse plot was linear with a slope of 0.6, corresponding to an approximate square root dependence of airway resistance in- crease on tracheal SO2 concentration. Going one step further, Kleinman (1984) hypothesized that if changes in airway re- sistance are directly related to the dose of SO2 reaching the postpharyngeal airways, then apparent differences in response that have been observed during rest, exercise, free breathing, and breathing through a mouthpiece (figure 24a) can be explained by the dependence of upper airway pene- tration on flow rate and on the point of air access. Kleinman presented a quantitative analysis in which equation 32 was used to convert the inhaled dose rate, Vpx, of the nine data points in figure 24a into their Is a) o Q x a) A 200 o c a)  C`7 LU oh CO UJ An: - CO As llJ of I 200 400 600 800 POSTPHARYNGEAL DOSE RATE, VpxO(~g/min) Figure 24. Dose/response data for increased airway resistance following exposure to SO2. A: Correlation with respect to inhaled dose. B: Improved correlation with respect to postpharyngeal dose. (Adapted with permission from Kleinman 1084, pp. 33, 35, and from the Air Pollution Control Association.) 400 300 100 400 300 200 100 A Mouth breathing (exercise) / Mouth breathing (rest) I_ g ~ ~ Natural breathing (exercise) I I I ~ I 1 1 1 1 1 1 1 200 400 600 800 1000 1200 INHALFn nORF RATE An. (,,~lmin] ~ ~Xj~-~ ~ B / / o corresponding postpharyngeal dose rate, Vpx. The flow fraction entering the nose, In, Divas evaluated as a function of airflow by assembling available human subject data; and the nasotracheal penetration, (PxO/Px)n' and orotracheal penetration, (PxO/Px'jm' were both modeled by a formula similar to equation 34, with separate values of KmS for the nose and for the mouth estimated on the basis of mixed data from dogs and man. When the change in airway resistance is replotted against the predicted values of postpharyngeal dose (figure 24b), the dose/ response correlation is considerably im- proved relative to the use of inhaled dose (figure 24a). Oulrey et al. (1983) performed a similar analysis of SO2 dose/response, but used a larger data set composed of 23 grouped measurements of airway resistance. They  concluded that the increase in specific air

lames S. Ultman 361 way resistance is best correlated with the square of postpharyngeal penetration. · Recommendation 5. Noninvasive pul- monary function tests such as the CO uptake method should be extended to the evaluation of pollutant transport in hu- mans. Summary Mathematical models can serve several pur- poses in the analysis of pollutant gas up- take. In order of increasing importance, a model can be used to correlate experimen- tal measurements made under differing en- vironmental or respiratory conditions; to extrapolate data obtained on laboratory an- imals to those values expected in humans; and to estimate sites and rates of uptake under conditions for which no data are available. There is no single model that best serves all these tasks. Rather, there may be a different model appropriate to each. For example, in data correlation and in extrap- olation, a compartment model can incor- porate important physiologic phenomena (for example, geometric asymmetry, air- flow nonuniformities, ventilation/perfu- sion inequalities) within a mathematical framework that requires little detailed in- formation for its development and only modest computational power for its imple- mentation. On the other hand, a distribut- ed-parameter model, which requires more elaborate input and more complex numer- ical algorithms to solve, is capable of pre- dicting outcomes from first principles. Whatever the nature of the model se- lected, it is assembled from four basic building blocks. First, an idealized geome- try accounting for the structure of the airways, tissue, and blood spaces must be decided upon. Then material balance equa- tions describing the time-dependent and possibly spatially distributed transport of pollutant are formulated. At the foundation of these material balances are the basic thermodynamic equilibrium, diffusional flux, and chemical reaction rate equations. Finally, using a specified set of pulmonary function parameters as forcing functions (for example, respiratory and pulmonary blood flows), the material balance equa- tions are solved to provide a numerical simulation of pollutant uptake. Experimental measurements are neces- sary to provide the geometric and physico- chemical data required as inputs to a model and also to validate predictions by the model. Ideally, this is accomplished by a combination of separate experiments. For example, basic thermodynamic and reac-  tion rate data can be obtained from in vitro systems such as isolated perfused lungs or excised tissue samples, whereas predicted uptake rates might be verified with nonin- vasive measurements in human subjects or invasive measurements in intact anlma s. Clearly, an adequate quantification of pollutant gas transport and uptake is an interdisciplinary problem. Its solution re- quires the modeling skills of engineers and physicists, as well as the biological exper- tise of biochemists, toxicologists, and physiologists. And if some scientists should choose to straddle two or more disciplines, then so much the better! Summary of Research Recommendations Recommendation 1 Objective. There is a critical need to quantify the chemical Basic Property Data interaction of specific pollutants with mucus, tissue, and blood. Besides determining solubility and Dyson coed~c~ents, it Is essential to determine the coefficients in reaction rate equations. The difficulty of this task is complicated by the fact that the associated thermodynamics are undoubtedly nonideal, and nonlin ear concentration effects are likely.

362 Transport and Uptake of Inhaled Gases Motivation. The descriptions of pollutant chemistry that are con- tained in this chapter were intended to be illustrative of quantitative methods, but were not completely accurate. In particular, physical properties in biological media were represented by aqueous values, and in the absence of chemical rate data, it was necessary to assume either instantaneous reaction or first-order kinetics. To develop reliable mathematical models and provide sound interpretation of absorption data, basic property data remain to be established, even for the most common pollutants. For example, the explanation of concentration effects in O3 and SO2 uptake data is still somewhat speculative, largely because of our ignorance of the underlying chemistry. Approaches. The use of isolated tissue preparations, such as tissue cultures or excised organ segments, could provide more direct measurements of biochemical properties than are possible in the entire organ. For example, by incorporating such a preparation into a flow-through or batch reactor, it is possible to determine a reaction rate expression using the well-established engineering principles of reactor design. Recommendation 2 Objective. A serious effort is needed to analyze mass transport Individual Mass through individual diffusion barriers, particularly the mucous Transfer Coefficients layer, the bronchial wall, and the alveolar capillary network. Undoubtedly, the sparsely perfused bronchial wall will require different mass transfer theory than the richly perfused plate-and post structure of the alveolar walls. And the analysis of diffusion through the mucous blanket, because it may have a discontinuous dynamically changing conformation, poses unique challenges. Motivation. At the core of any mathematical model of pollutant uptake are the individual mass transfer coefficients for the diffusion barriers. The values of mass transfer coefficients presented in this chapter were merely estimates. Considerable refinement is neces sary. Approaches. Although some physical modeling may be appro priate, it is also possible to perform computations based on existing geometric and hydrodynamic data. DuBois and Rogers (1968) have illustrated the application of diffusion theory to the bronchial wall; mechanical engineers have reported methods for analyzing trans port in interrupted flows similar to those in the pulmonary circulation (Wieting 1975~; and chemical engineers have developed a surface renewal theory to deal with dynamically changing inter faces such as the gas/mucus boundary (Astarita 1967~. Recommendation 3 Objective. To analyze total uptake of pollutants and to predict Ventilation and dose distribution, mathematical models that account for ventilation Perfusion Effects and perfusion limitations, including their regional distribution, should be developed and validated. Motivation. It is clear that there is a nonuniform distribution of ventilation and perfusion, even in a normal lung. And the possi bility of diffusion and perfusion limitations exists for all pollutant gases. These interrelated phenomena have not been systematically investigated for pollutant gases, and yet it seems likely that they will have an impact on uptake distribution.

.Iames S. Ultman 363 Approaches. A combination of measurement and mathematical modeling is necessary. The use of isolated perfused lung prepara- tions could allow the measurement of total uptake under conditions where the overall ventilation/perfusion ratio is controlled. More- over, intravascular tracer methods previously developed for deter- mining ventilation/perfusion ratios in humans (Wagner 1981) might also be applied to an isolated lung. In analyzing data, distributed-param eter models may be unnecessarily complicated. Multicompartment lumped-parameter models are probably more appropriate. Recommendation 4 Objective. Using a consistent experimental protocol, total up Extrapolation take of select pollutants should be measured in different animal Modeling species and then used to develop basic rules of extrapolation. More specifically, these uptake data could be correlated using known interspecies differences in lung volume, surface area, breathing rate, and rate-limiting mass transfer coefficients within the framework of an appropriate mathematical model. Motivation. There is virtually no information in the literature that allows prediction of uptake by the human lung from data obtained in smaller laboratory animals. This represents a critical problem in setting air quality standards in cases where measure ments on humans do not exist or cannot be taken. Approaches. Yokoyama (1984) described an enclosure, similar to a closed-circuit metabolic chamber, in which the total O3 uptake by a free-breathing rat could be monitored, without the need to anesthetize the animal. Using such a device, or possibly several chambers of different sizes, it would be possible to amass total uptake data on a series of different animal species. This data base could be analyzed with a simple compartment model that treats the animal and the chamber as two separate subsystems. Recommendation 5 Objective. Noninvasive pulmonary function tests such as the CO Noninvasive uptake method should be extended to the evaluation of pollutant gas Methods transport in humans. The data from such experiments, particularly when several indicator gases are used simultaneously, can be analyzed with an appropriate mathematical model to extract considerable information about regional inhomogeneities in uptake rate and dose. Motivation. To date, most pollutant uptake data have been obtained in animals using protocols that required heavy sedation, and in some cases, extreme surgical procedures. Moreover, these measurements were made in the absence of complementary tests that characterize other important functional features such as the distribution of ventilation. Approaches. It would be useful to extend the methodology of Saidel et al. (1973) to soluble and reactive pollutants. Naturally, the multicompartment model used by these investigators to simulate CO uptake must be generalized to include absorption into the upper airway and conducting airway compartments. Also, because of limitations in gas analyzer response and the potential health hazard during continual exposure, it may not be practical to use the steady-state uptake technique; the single-breath, breath-holding method may be a better choice.

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