A two-component theory of aerosol deposition in lung airways

Bulletin ol %lathematical Biology, Vol. 40, pp. 693 706 Pergamon Press Ltd. 1978. Printed in Great Britain ) Society f~r Malhe'natical Bk, l,~gy

A T W O - C O M P O N E N T THEORY OF AEROSOL DEPOSITION IN LUNG AIRWAYS

IIC. P. Yu Department of Engineering Science, Aerospace Engineering and Nuclear Engineering, State University of New York at Buffalo, Buffalo, New York 14214, U.S.A.

The deposition of aerosol particles in the human lung airways is due to two distinct mechanisms. One is by direct deposition resulting from diffusion, sedimentation and impaction as the aerosol moves in and out of the tung. The other is an indirect mechanism by which particles are transported mechanically from the tidal air to the residential air and eventually captured by the airways due to intrinsic particle motion. This last mechanism is not well understood at present. Using a trumpet airway model constructed from Weibel's data, a two-component theory is developed. In this theory, the particle concentrations in the airways and the alveoli at a given airway depth are considered to be quantitatively different. This difference in concentrations will cause a net mixing between the tidal and residential aerosol as the aerosol is breathed in and out. A distribution parameter is then introduced to account for the distribution of ventilation. The effect of intrinsic particle motion on the aerosol mixing is also included. From this theory, total and regional deposition in the lung at the steady mouth breathing without pause is calculated for several different respiratory cycles. The results agree reasonably well with the experimental data.

1. Introduction, When an aerosol is breathed into the lung, the loss of particles in the airways is due to two distinct mechanisms. One is by direct deposition such as diffusion, sedimentation and impaction, resulting from the intrinsic particle motion as the aerosol moves in and out of the lung. The other is an indirect mechanism by which particles are transported mechanically from the tidal air to the residential air and are eventually captured by the airways. The direct deposition is fairly well understood qualitatively although its exact amount relies upon the detailed information of the local lung geometry and the flow field which are often very complex. The indirect mechanism due to air exchange is determined principally by the dynamics of the lung, i.e. the 693

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interaction of lung elasticity and fluid mechanics of the air. Our understanding of this latter subject is still very limited at the present time. There have been a number of theoretical studies on particle deposition in the lung (Findeisen, 1935; Landahl, 1950, 1963; Beeckmans, 1965 a,b; Yu and Taulbee, 1973, 1975; Taulbee and Yu, 1975). From these theories, the dependence of the total deposition upon particle size was established and it was shown that the amount of deposition reaches a minimum at particle diameter of about 0.5/~m. However, the quantitative results of deposition depends upon the lung geometry adopted for the calculation and the manner that the indirect mechanism of deposition is accounted for. Since Findeisen's first study on particle deposition in 1935, considerable progress has been made on the anatomical model of the lung. The most recent morphometric measurements of the lung are due to Weibel (1963), Horsefield et al. (1971) and Raabe et al. (1976). Although the data of Horsefield et al. and Raabe et al. gave more details about the asymmetric characteristics of the airways, Weibel's symmetric, dichotomously branching airways (model A) has been widely used for theoretical study because of its simplicity. In Weibel's model, a completely bifurcating system consisting of 23 generations from the trachea to the alveolar sacs was assumed. Starting from the 17th generation, the airways become alveolated and the number of alveoli increases as the generation number increases. In the theories of Findeisen, Landahl and Beeckmans, the airways were considered to be a series of compartments in which the particles were wellmixed. This consideration overlooks the nonuniformity of particle concentration along and transverse to tli? airways within each compartment. Based on Weibel's data, Yu and Taulbee (1973) assumed a one-dimensional trumpet lung model with the cross sectional area of the trumpet varying with the airway depth. The particle concentration is then a function of the distance along the airways but its variation transverse to an airway cross section remains absent. The choice of the theoretical models described above has an important consequence in the treatment of air mixing process that accounts for the indirect deposition mechanism discussed previously. In the compartmental theory, air mixing was treated by introducing an empirical mixing function (Altshuler, 1959). In the trumpet model, this was accomplished by the use of an equivalent longitudinal diffusion coefficient. However, the determination of this diffusion coefficient is a very difficult task. Taulbee and Yu (1975) suggested that such a coefficient results principally from an asymmetrically branching system and derived an expression for the diffusion coefficient. Their idea implies that air mixing occurs only in an asymmetric lung. This is obviously too restrictive. Besides, a negative diffusion coefficient may exist during the exhalation phase if the air ventilation follows the pattern of first in and last out.

AEROSOL DEPOSITION IN HUMAN LUNG AIRWAYS

695

In this paper, a new direct way of treating the air mixing process is introduced. This idea is based upon the consideration that the particle concentration is nonuniform in the cross section of an alveolated airway. This approach avoids the difficulty of choosing an appropriate form for the diffusion coefficient. To make the analysis manageable, a two-component theory is proposed in which the particle concentrations in an airway and its attached alveoli at a given airway depth are uniform but different in quantity. In the nonalveolated airways, nonuniformity in the transverse direction is still neglected since most residential air stays in the alveolar region. As it will be seen the introduction of nonuniform particle concentration in the alveolated airways causes a net mixing between the tidal aerosol and the residential aerosol during breathing. This concept is compatible with the result of a recent model alveolar study by Cinkotai (1974) who showed that the interchange of the tidal and residential air takes place principally in the alveolar region and is caused by the nonuniform expansion and contraction of the walls. 2. Formuhttiml. The anatomical model of the lung adopted in the present study is the same as the trumpet model that we employed previously. Roughly speaking, the trumpet model is to approximate the many generations of the lung by a chamber model shaped like a trumpet (Figure 1). The cross-sectional

Summed Alveolar Cross Sectional Area A2(x)

\

Airwoy Length x

Mouth

C~SummedAirway Cross Sectional Area At(x)

Figure 1.

Trumpet model of lung airways

area of the chamber varies as a function of the generation number according to the data given by Weibel. In the last seven generations, additional volume which encircles the chamber is introduced to account for the alveolar volume. This alveolar volume is again a function of generation number from Weibel's data. The breathing process is then pictured as the movement of aerosol into

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and out of this chamber as the alveolar volume expands and contracts uniformly. The morphonletric measurements made by Weibel correspond to excised lungs at three-quarters maximum inflation. When breathing at the normal functional residual capacity, Weibel's data of airway dimensions should be scaled down. Since the detailed measurements of airway dimensions at different levels of lung inflation are not yet available, we assume that the airway diameters and lengths are proportional to the one-third power of the average lung volume but remain fixed during a breathing cycle. This cubic-root relationship between airway dimensions and lung volume was suggested by Hughes et al. (1972) from the measured data of dog lungs. The equations of mass balance for the particles along the airways can be established by considering a differential element along the airway depth. Let the particle concentration in the airways be Cl(x, t) while that in the alveolar region be C2(x, t), where x is the airway depth measured from the beginning of the trachea and t is the time. Then, during inhalation, we have for the airways 0 ~7(AIC~) =

~

(QC1)-qaCx-L

~- ( o

(1)

and for the alveoli 0 ~t (A2C2)=qaC1 - L 2 +4)

(2)

where An and A 2 a r e respectively the total cross sectional area (or volume per unit length) of the airways and alveoli at x, Q = A 1u is the air flowrate in which u is the convective velocity, qa is the flowrate per unit length from the airways to the alveolar region, L1 and L 2 a r e respectively the particle loss per unit length per unit time to the airway and alveolar surfaces due to deposition by impaction, diffusion and sedimentation, and ~b is the particle flux per unit length due to their intrinsic motion which we shall discuss later. Similarly, during exhalation, we have 0 & (A~C,)= -~xx(QC,)+fl%C2-L ,-~o

(3)

for the airways and 635 (A 2 C 2 ) = - flqaC2 -- L 2 + (9

for the alveoli.

(4)

AEROSOL DEPOSITION IN HUMAN LUNG AIRWAYS

697

The parameter fi indicates, on the average, alveolar particles with concentration tiC2 return to the airways during exhalation. The conservation of air mass requires C3 A 2

[?t -

(~Q ~x - %

(5)

Using this relation and the assumption that the airway dimensions remain fixed during breathing, we may rewrite, respectively, (1)-(4) in the following form: A1 ?C1

?t

-

~C 1

O~---L~

16)

-4,

cx

17)

A2 •C2?I - ~O~:x( C 2 - C 1 ) - L 2 +~ "~

A ~cC~ ~t _PQ (~x ( f l c-, - c ~ ) - Q A2 ('~C2 =~t

(1

3C 1

8x - L , - @

- fl ) e~,~ C ,_- L 2 + ~

(8)

(9)

At any time t, the total alveolar volume is

VaT+ f'oQo dt V°r=°

(lO)

where Qo(t) is the air flowrate at the beginning of the trachea which is chosen to be located at x = 0 and V°r is the total alveolar volume at rest. Let F(x) be the fraction of VaT located between x = 0 and x. Then, from the assumption of uniform expansion and contraction of the alveolar volume during breathing, we have

(11)

Q(x,t)=[1-F(x)]Qo and A2 (x, t) = V,T d F = ( v ° r +

~, Qo dt~ dF

It is easy to show that these expressions for Q and

A2

(12)

satisfy the relation (5).

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Equations (6) (9), (11) and (12) are the basic equations which determine C1 and C,._ The function Al(x), F(x} and V,°r are given by Weibel's data. The expressions for [L d), L 1 and L2 will be specified and discussed in subsequent sections. Similar to our previous work, we introduce a stretched coordinate z based upon the generation number of the airways. Thus, z takes on values of 0 to 23 corresponding to locations at the end of trachea to the end of generation 23. Let 11 (z) be the airway length per generation. Then, Ii = d x / d z , d/dx =d/I 1 dz and all equations derived above can be written in terms of z. The initial and boundary conditions are as follows: At t = 0 when the lung is clear from aerosol, C1 = C2 =0. During inhalation, we have C1 = C10 at z = 0 . At z = 23, which is a hypothetical surface, C~ = 0. The boundary conditions for C2 are not required.

3. Particle Mixing. In the deadspace airways, the aerosol transport is dominated by longitudinal convection and mixing is negligible. This is confirmed experimentally by single-breath aerosol washout curves. In the alveolated airways, however, the mixing is considerable and it takes place in an airway tube principally in the transverse direction because of the large interface between the tidal air and the residential air in that direction. During inhalation, particles with concentration C1 move from the airways into the alveoli, mix with the particles already there, to reach a new concentration C2. Over the time of one breathing cycle, since the intrinsic mobility of particle is small, it is not possible for the particles inside an alveolus to attain the well-mixed state as required by the theory. The extent of mixing depends on the air velocity by which particles are transported. In reality, this velocity varies from alveolus to alveolus because of the nonuniform vertical distribution of ventilation in a real lung. Measurements (Robertson et al., 1969; Martin et al., 1972; Bake ez al., 1974) have shown that the vertical distribution of inspired gas depends upon the inspiratory flow rate and the distribution is more uniform at high flowrate. Since the alveolar size is much larger at the apex than the base of the lung, the effective alveolar particle concentration which returns to the airways during exhalation should be less at high flowrate. To take this into account, a distribution parameter fl is introduced so that fiC2 is the effective particle concentration which is transported from the alveoli to the airways during exhalation. To obtain an expression for fi, the dynamics of the lung must be considered. As an approximation, we assume fi = Q~d/Qo

for Q0 > Q*

(13)

fl=l

for Q 0 < Q ~

(14)

and

AEROSOL DEPOSITION IN HUMAN LUNG AIRWAYS

699

where Q* is a reference florwate. In our calculation, we choose Q* = 125 cm3/sec which is the flowrate at a quiet breathing condition. It should be noted that for fi :/: 1 the mixing is not exactly accounted for by the proposed two-component consideration and the two concentrations are only used for direct deposition calculations. When the particle size is extremely large or small, the transfer of particles between the airways and alveoli will be enhanced by intrinsic particle motion. For instance, at a flowrate of 250 cm3/sec, the air velocity to each alveolus is about 2 x 10 3cm/sec. This velocity is small compared with the settling velocity of a 3 lira unit-density particle which has a value of 3.2 x 10 2 cm/sec. Similar situation occurs for very small particles where the mean displacement of particle due to diffusion may exceed that due to convection. In the upper airways where the convective velocity is large, intrinsic particle motion is less important. Assuming that the distribution of the alveolar orientation is isotropic in space, the net transfer of the particles to the alveoli through their openings per unit airway length is then

1

dF

{15}

~s =~ uos,(C1 - C2 )N dx

where s~,=0.561 d~ is the area of each opening. The correction for the diffusion through the alveolar opening is more difficult to obtain. As an approximation, we assume that the diffusion flux ~b~ is proportional to the average concentration gradient between the airway and the attached alveoli. Then, dF

(/)d = DBSa(C1 -- C2

)/{dl/2 + 9d2/14)N dx

(16}

where d~/2 + 9d2/14 is the distance from the axis of an airway to the center of an alveolus. Adding (15) and (16), we find the total net particle transfer from the airways to the alveoli per unit length due to both sedimentation and diffusion to be

[u 0 14D8 "] dF q~ =~bs+ {/)d=~4 --t 7(tx+9d2/] sa(C1-C2)N dx

4. Deposition Formulae.

(17)

The functions La and L2 in (6)--(9) represent respectively the direct deposition loss to the surfaces of the airways and alveoli per unit time and unit airway depth. The losses are due to the mechanisms of impaction, Brownian diffusion and sedimentation. The determination of L l

700

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and L 2 by the combined mechanisms under realistic airway geometry and flow field is practically impossible. Similar to the previous studies, we shall adopt the formulae which were derived for idealized situations. We assume that the combined deposition loss is the superposition of individual mechanisms, i.e., El = Lxi + L1a + LI~ L 2 =

L2a + L2s

(18) (19)

where the subscripts i, d and s refer respectively to the deposition by impaction, diffusion and sedimentation. The impaction loss in the alveoli is neglected because of the extremely small velocity there. Moreover, we assume that Lid, Lza, L~s and Lzs have the same expressions for inhalation and exhalation, and L~i = 0 during exhalation. For impaction during inhalation, we use the correlation OC,

Lli-~ 1.3 ~-.- I St ll

(20)

where St=pd~u/18 Itdl is the Stokes number in which p is the particle mass density, dp the particle diameter, # the air viscosity and u and di are measured in the daughter tube. Expression (20) falls within the scattering of the data from the bifurcating tube model measurements by Johnston et al. (1975) and also gives a reasonable fit to the data obtained by Schlesinger and Lippmann (1972) who took measurements in hollow Silastic casts of the upper airways of the human lung. However, because these measurements were all made in steady flow the applicability of (20) to the lung may have to be limited to slow breathing processes only. Gormley and Kennedy (1948) and Ingham (1975) obtained the result for the diffusion deposition in a circular tube with Poisseulle's flow. Their solution depends upon a nondimensional parameter A defined by DBll/ud2~ where DB is the Brownian diffusion coefficient, u is the average flow velocity, l 1 and dl are, respectively, the length and diameter of the tube. In the case of the lung, A ~ 1, an asymptotic expression was obtained for deposition. Using their result, one finds

L1 a _

QC1

(6.41 A 2/3 - 4 . 8 A - 1.123 A 4/3 ) (1 --0~)

(21)

11

where e is the fraction of the airways surface being alveolated. This fraction depends upon the generation number and is given by Weibel (1963).

AEROSOL DEPOSITION IN H U M A N L U N G AIRWAYS

701

The deposition due to settling was obtained by Pich (1972) for the Poisseulle flow in a horizontal tube. The more general case of inclined tubes with upflow and downflow was also studied by Wang (1975). Wang's results are more accurate for the application to the lung but the expressions are too cumbersome for our use. In Pich's result, the deposition depends upon the parameter e defined as 3Ugll/4Ud1, where u s is the settling velocity. Using his result and assuming that the airways orientation is uniformly distributed in space, we obtain

L,~ -2QC' •

{ 2 , g ' N ~ - - - g ' 2 / 3 - - g '1/3

N / i - - g '2/3 + s i n - ' ~,'~/3} (1 --cO

{22)

gl I

where e' = (~z/4)~:and the factor re/4 is introduced to account for the orientation effect of the airways approximately. To find an expression for deposition in the alveolar region, we assume that an alveolus can be represented by a spherical surface with a flat opening as suggested by Weibel (1963). The angle between the axis of symmetry and a line from the center of the sphere to the tip of the opening is 2.27 radians and is assumed to be constant during expansion and contraction. If statistically the distribution of the alveolar orientation is isotropic in space, it is not difficult to show that the deposition flux to the alveolar wall due to settling is 0.645 d22uoC2 where d2 is the alveolar diameter. The diffusion loss to the wall is calculated by assuming spherical symmetry. Using the series solution by Fuchs (1964) and taking the first term of the series in the manner of Harris and Timbrell (1975), one finds that the diffusion flux to be 16.975 d2DBC 2. The volume of each alveolus with the assumed geometry is found to be 0.478 d 3. During breathing, d 2 varies with time and Va T

0 + = Vat

Qo dt = 0.478 Nd32

(23)

o

from which d 2 c a n be determined. In (23), N is the total number of alveoli in the lung and has a value of 2.98 x l0 s according to Weibel (1963). The expressions for Lza and L2~ are then found to be Lza = N dF

dxx (16.975 N2DBC2)

dF L2~ = N ~ xx (0.645 dZuoC2) where N dF/dx is the number of alveoli per unit airway length.

(24) (25)

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5. Results amt Discussion. Equations (6) (9) with the appropriate initial and boundary conditions were solved numerically using finite differences. To test the theory, calculations were first made for those cases where measurements were made by Heyder et al. (1975) at steady mouth breathing using di-2-ethylhexyl sebacate aerosol carrying no electric charge (specific gravity 0.917). These measurements were carefully controlled and taken under the condition that the subjects breathed with constant respiratory flowrates and no pauses. Thus, the data from these measurements are typically lower than others. In our calculation, the number of breathing cycles needed to reach the steady state varies with particle size and breathing condition. Typically, 10 cycles are sufficient. For better accuracy, all the results given here were obtained using 25 cycles. Figures 2 and 3 show the total deposition data at steady mouth breathing by Heyder et al. for five human subjects with an average FRC of 3310cm 3. The solid curves in these figures are the predicted total deposition. For simplicity, all calculations were made for a sawtooth-like time function of lung volume with equal inhalation and exhalation time period. This is very close to the experimental recordings. Except for the case of 2sec breathing period and 1000cm 3 tidal volume, Figures 2 and 3 show that agreement between the theory and the experiment is good. The higher predicted deposition for 2 sec breathing period and 1000 cm 3 tidal volume appears to be caused by the improper use of the steady flow impaction deposition formula at extremely large flowrate. It should be also 1.0 Flowrote= 250crn3s-~ O9-e 2000crn3_16s o t500cm3-t2s 018 l l ~OOOcm~- 8S

0.7-0 500cm3-4s

y J

.,~

:-,-E_-0.6 o.5

t

~_. 0.4

}

0.3

0.2 0.1

o6 o'., o'.8 ,'2

2'.o 2'4 2'.8 3'.2

dp/zm

Figure 2. Total deposition versus particle diameter at 250cm3/sec flowrate and various breathing periods. Comparison between theoretical prediction (solid linest and experimental data of five subjects by Heyder et al. (1975). The bars of the experimental points indicate variation of individual deposition.

AEROSOL DEPOSITION IN HUMAN LUNG AIRWAYS

703

t.0 0.9 0.8 .-~ 0.7

Tidal Volume= tO00cm 3 125cm3s-~_ t6s o 250cm3$-t-8s • 500cm3s- t - 4s o lO00cm3s-t-2s . / •

~, 0.5 ~_ 0.4 03

o.,

.ZJY/

0.t

;

I

I

I

0.4 0.8 t.2

I

I

1.6 20

I

[

I

2.4 2.8 3.2

dp/.Lm

Figure 3. Total deposition versus particle diameter at 1000 cm 3 tidal volume and various breathing periods. Comparison between theoretical prediction (solid lines) and experimental data of five subjects by Heyder et al. (1975). The bars of the experimental points indicate variation of individual deposition

pointed out that the introduction of fi in (3) and (4) improve deposition results considerably. Without fl, the calculated results do not depend on the flowrate as much as they should do. Figure 4 shows the comparison between the theory and the experimental data recently compiled by Lippmann and Altshuler (1976) for the experiments with uncontrolled normal breathing. These data show a greater amount of scattering than that obtained by Heyder et al. Because of the differences in breathing period and tidal volume for each experiment, several theoretical curves are plotted there for comparison. The agreement is again reasonable although the predicted deposition results are slightly lower. This discrepancy may be caused by the absence of pause in the theoretical breathing cycle adopted for the calculation. Figure 4 also reveals that a good portion of the scattering in the experimental data results from the differences in breathing condition used in different experiments. Reliable experimental data on regional deposition are relatively scarce. Using a tubular continuous filter bed as a theoretical analogy for the respiratory tract, Altshuler et al. (1966) estimated regional deposition from mouth breathing experiments using monodisperse triphenyl phosphate aerosol on three subjects. Figure 5 shows their data for the alveolar deposition along with the theoretical prediction. It is seen that the theoretical values are lower than the experimental ones particularly for small particle sizes. The disagreement may again be attributed to the absence of pause in theoretical calculation.

704

C.P.

YU No of Tidal Resp. Subj's Vol. Rate

Source 0 , 9 " I. o

2 []

kippmonn

22

Landahletcl,195t

0.7

o

2 05

15

15

t5

4 (~) Altshuleretol,1966 3 0 5

15

o __( /,

3.-(~- Landuhletal,t952 0.~

o°

1.1-*.3 14

5 •

GiocomelliMclt0ni etal,1972

6 •

Martens and Jocobi, 1973

2

t5

tO

i2

t

tO

14

o o

o

/o[~

°

°°

o

T

06

o

o°

o

P .

o 0

g 0.5

0.4 B4

~- 0.3 0.2

0.1,0

o.~

\3 0'.2

I

o'.5

i

~.o

~

~

lo

20

Aerodynomic Diometer (fl.rn) Figure 4. Total deposition versus aerodynamic diameter. Comparison between theoretical prediction (solid lines) and experimental data compiled by Lippmann and Altshuler (1976I from various sources indicated in the figure. FRC for theoretical calculation was 2400 cm 3 No of Tidal Rasp. $ubils VoL Rate

0.7

Source

0 . 6 - 1. * Lippmonn 22 11-'3 t4 • 2 0 AltshuleretoL,|966 3 0.5 t5 • • •

0.5

g :~ 0.4

g_ o

~

0.2 0.1

%1

o'.2

0:5

1.o

2

5

to

20

Aerodynomic Diorneter (fl.m) Figure 5. Alveolar deposition versus aerodynamic diameter. Comparison between theoretical prediction )solid lines) and experimental data compiled by Lippmann and Altshuler (1976) from various sources indicated in the figure. FRC for theoretical calculation was 2400 cm 3

AEROSOL DEPOSITION IN HUMAN LUNG AIRWAYS

705

It should be pointed out that the total deposition results obtained from the present theory does not differ significantly from the values that we obtained from our previous diffusion model (see Figures 4 and 5, Yu and Taulbee, 1975) although somewhat different deposition formulae were used in the two cases. However, there is a distinct difference between the two models when singlebreath washout curves for 0.5/am particle are computed, and the present model agrees far better with the experimental recordings than the diffusion model (Taulbee, Yu and Heyder, 1978). Since 0.5/am particle is a good tracer of the air flow due to its low intrinsic mobility, the air mixing process proposed in this study appear to be an appropriate description of the real physical process that occurs in the lung airways. The author would like to express his sincere thanks to Dr. C. N. Davies who first introduced the lung deposition problem to him. He is also indebted to Dr. D. B. Taulbee for many contributions and to Dr. C. S. Liu for reading the manuscript. K. C. Fan and S. Rajaram obtained all the numerical results and their assistance is deeply appreciated. This work was supported jointly by the National Institute of Environmental Health Sciences and the Environmental Protection Agency under Grant No. R01 ES1239.

LITERATURE Altshuler, B. 1959. "'Calculation of Regional Deposition of Aerosol in the Respiratory Tract." Bull. Math. Biophys. 21,257 270. Bake, B., Wood, L., Murphy, B., Macklem, P. T. and Milic-Emili, J. 1974. "Effect of lnspiratory Flow Rate on Regional Distribution of Inspired Gas." J. ,4ppl. Physiol., 37, 8 17. Beeckmans, J. M. 1965a. "'The Deposition of Aerosols in the Respiratory Tract: 1. Mathematical Analysis and Comparison with Experimental Data." Can. J. Physiol. Pharm., 43, 157-172. 1965. '~Correction Factor for Size-Selective Sampling Results Based on a New Computed Alveolar Deposition Curve." Ann. Occup. Hyg., 8, 221 231. Cinkotai, F. F. 1974. "Fluid Flow in a Model Alveolar Sac." J. Appl. Physiol., 37, 249 251. Findeisen, W. 1935. "Uber das Absetzen Kleiner, in der Lull Suspendierten Teilchen in dcr menschlichen Lunge bei der Atmung." Pfiueger Arch. Ges. Physiol., 236, 367 379. Fuchs, N. A. 1964• The Mechanics of Aerosols. New York: Pergamon Press. Gormley, P. G. and Kennedy, M. 1949. "Diffusion from a Stream Flowing through a Cylindrical Tube." Proc. R. Irish. Acad., 52, 163 169. Harris, R. L., Jr. and Timbrell, V. 1975. "~The Influence of Fiber Shape in Lung Deposition Mathematical Estimates." British Occupational Hygiene Society, 4th International Symposium on Inhaled Particles, Edinburgh, September 22 26. Heyder, J., Armbruster, L., Gebhart, J., Grein, E. and Stahlhofen, W. 1975. "Total Deposition of Aerosol Particles in the Human Respiratory Tract at Nose and Mouth Breathing." J. Aerosol Sci.,6,311 328. Horsefield, K., Dart, G., Olson, D. E., Filey, G. F. and Cumming, G. 1971. "'Models of the Hu man Bronchial Tree." J. Appl. Physiol., 3!, 207 217. Hughes, J. M. B., Hoppin, F. G., Jr. and Mead, J. 1972• "Effect of Lung Inflation on Bronchial Length and Diameter in Exercised Lungs." J. Appl. Physiol., 32, 25-35.

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Ingham, D. B. 1975. "Diffusion of Aerosols from a Stream Flowing through a Cylindrical Tube." J. Aerosol Sci., 6, 125 132. Johnston, J. R., Isles, K. D. and Muir, D. C. F. 1975. "'Inertia Deposition of Particles in Human Branching Airways." British Occupational Hygiene Society, 4th lntermltio,al Symposi,m on Inhaled Particles, Edinburgh, September 22 26. Landahl, H. D. 1950. "On the Removal of Air-borne Droplets by the Human Respiratory Tract: I. The Lung." Bull. Math. Biophys., 12, 43 56. .1963. "Particle Removal by the Respiratory System. Note on the Removal of Airborne Particulates by the Human Respiratory Tract with Particular Reference to the Role of Diffusion." lbid, 25, 29 39. Lippmann, M. and Altshuler, B. 1976. "'Regional Deposition of Aerosols in Air Pollution and the Lung." Aharonson, E. F., Ben-David, A. aud Klingberg M. A. (Eds.). New York: John Wiley and Sons. Martin, R. R., Anthonisen, N. R. and Zutter, M. 1972. "Flow Dependence of the lntrapulmonary Distribution of Inspired Boluses of 133 Xe in Smokers and Nonsmokers." Clin. Sci., 43,319 329. Pich, J. 1972. "'Theory of Gravitational Deposition of Particles from Laminar Flows in Channels." J. Aerosol Sci., 3, 351 361. Raabe, O. G., Yeh, H. C., Schum, G. M. and Phalen, R. F. 1976. "Tracheobronchial Geometry: Human, Dog, Rat, Hamster." Report LF-53, Lovelace Foundation for Medical Education and Research, AIburquerque, NM. Robertson, P. C., Anthonisen, N. R. and Ross, D. 1969. "Effect of Inspiratory Flow Rate on Regional Distribution of Inspired Gas." J. Appl. Physiol., 26, 438 443. Schlesinger, R. B. and l, ippman, M. 1972. "Particle Deposition in Casts of the Human Upper Tracheobronchial Tree." Am. Ind. Hyg. Assoc. J., 33, 237 ~261. Taulbee, D. B. and Yu, C. P. 1975. "A Theory of Aerosol Deposition in the Human Respiratory Tract." J. Appl. Physiol., 38, 77 85. Taulbee, D. B., Yu, C. P. and Heyder, J. 1978. "'Aerosol Transport in Human Lung from Analysis of Single Breaths." J. Appl. Physiol., in press. Wang, C.-S. 1975. "'Gravitational Deposition of Particles from Laminar Flows in Inclined Channels.'" J. Aerosol Sci., 6, 191 204. Weibel, E. R. 1963. Morphometry o[the Human Lung. Berlin: Springer-Verlag. Yu, C. P. and Taulbee, D. B. 1973. "'A Model for Predicting Particle Deposition in Human Respiratory Tract." Bull. Am. Phys. Soc., 18, 1486. Yu, C. P. and Taulbee, D. B. 1975. "A Theory of Predicting Respiratory Tract Deposition of Inhaled Particles in Man.'" British Occupational Hygiene Society, 4th Internatiomd Symposium on Inhaled Particles, Edinburgh, September 22 26. RECEIVED 2 - 1 5 - 7 7 REVISED 7-15-77