Annals of Biomedical Engineering, Vol. 20, pp. 481-494, 1992 Printed in the USA. All rights reserved.
0090-6964/92 $5.00 + .00 Copyright 9 1992 Pergamon Press Ltd.
A Pore Transport Model for Pulmonary Alveolar Epithelium T. Chandra,* I.F. Miller,l and D.B. Yeatesi~: *Department of Chemical Engineering tDepartment of Medicine University of Illinois at Chicago ~& V.A. West Side, 820 South Damen Chicago, IL
(Received 1/10/90) Hydrodynamic heteropore f l o w models f o r transport o f solutes across alveolar epithelial tissue have been developed. A two-size cylindrical pore model and a similar parallel-plate model were formulated, tested and used to predict effective pore sizes f r o m literature data on transport in bullfrog, canine and rat lungs. The best f i t equivalent pore-size estimates were obtained using a modified, nonlinear least squares procedure, with alveolar surface area to volume ratio (S/V) and small-pore area fraction o f total pore area as parameters. Small-pore and large-pore width estimates o f 4 nm (84% o f total f l o w area) and 10 nm, respectively, with an average deviation o f 20~o f r o m experimentally derived permeabilities were obtained f r o m the bullfrog alveolar epithelium parallel-plate pore model (13 solutes, diameters 0.3 to 2.8 nm). The equivalent cylindrical pore model diameter estimates were 5 nm and 10 nm, with small-pore area fraction and percentage deviations similar to the parallel-plate model estimates. Eighty-eight percent o f the bulk water driven by a sucrose osmotic gradient was predicted to be transported through the small pores. The rat alveolus parallel-plate pore model (6 solutes) yielded small-pore and large-pore widths o f 0.4 nm and 50 nm, respectively. Clearance rate-constant data f o r dextran macromolecules (3,000 to 250, 000 Daltons), using a single parallel-plate pore model, resulted in a pore width estimate o f 98 nm f o r canine alveoli with an average deviation o f the predicted rate constants o f 18% from literature experimental values. In all cases tested, the parallel-plate pore model predicted lower small-pore size estimates than did the cylindrical pore model, and both models had appreciably smaller percentage deviations f r o m experimental data than previous models. Keywords--Hydrodynamic model, Pore flow, Membrane transport, Alveolus, Epithelium, Permeability. INTRODUCTION
T h e t r a n s p o r t o f n o n v o l a t i l e m a t e r i a l s a c r o s s t h e p u l m o n a r y e p i t h e l i u m is o f c o n s i d e r a b l e t o x i c o l o g i c a l , p h a r m a c o l o g i c a l a n d p h y s i o l o g i c a l i n t e r e s t . F i r s t o f all, t h e
Acknowledgment-This report is based on a thesis submitted by Tarun Chandra, in partial fulfillment of the requirements for the M.S. degree in Chemical Engineering at the University of Illinois at Chicago. Address correspondence to I.F. Miller, Section of Environmental and Occupational Medicine, Room 230, Benjamin Goldberg Research Center, 1940 West Taylor, University of Illinois at Chicago, Chicago, IL 60612. 481
482
T. Chandra, LF. Miller, and D.B. Yeates
response to inhaled irritants and toxins is critically dependent on the transport processes that carry them to target organs. Secondly, there is considerable current interest in the possible use of the lungs to deliver systemic drugs via aerosol. Finally, aerosolized compounds can be used as probes to elucidate the permeability characteristics of the pulmonary epithelium and, thus, can be used to explore structural models of tissue pathophysiology. The conditions under which aerosolized particles can be used as a probe of epithelial structure depend on such factors as the size, chemical composition and site of deposition of the particles. Submicron-sized particles deposit primarily in the distal airways, while larger particles deposit more proximally (21,24). In the bronchial region, mucociliary transport provides the main means whereby deposited particles are removed. In the alveolar region, there is no mucociliary transport, and macrophageal action, vesicular transport and diffusion provide the clearance mechanisms (7,21). Thus, the clearance of aerosolized particles as probes of epithelial structure can be used in the alveolar region, but with less success in the bronchial region. The transport of solutes across the alveolar epithelium depends in large part on their lipid solubilities. Lipophilic solutes can transport transcellularly and are rapidly removed from the lungs (5). On the other hand, lipophobic solutes are restricted to the aqueous intercellular spaces, and transport much more slowly. The clearance of lipophobic solutes, of differing molecular size, from the alveolar space can thus be used as a probe to model the structure of the aqueous intercellular spaces ("pores") in the alveolar epithelium. Attempts to produce such structural models of the alveolar epithelium have used either thermodynamic or hydrodynamic approaches. Thermodynamic approaches relate fluxes of volume and solute to gradients o f pressure and concentration and provide no direct structural information. Hydrodynamic approaches model the structure of the barrier by, for example, assuming identical cylindrical water-filled pores and using the Hagen-Poiseuille equation to estimate the apparent pore diameter from flux data (25). Improved hydrodynamic models use multiple pore populations. For example, Durbin, et al. (10), successfully modelled the frog gastric mucosa with two pore populations: 93~ of the available pore area was attributed to pores of 0.5 nm diameter, and 7~ to pores of 12 nm diameter, a model very similar to that developed by Kim and Crandall for the bullfrog alveolar epithelium (20). Altamirano and Martinoya (1) also arrived at almost the same model for canine gastric mucosa, as did Gatzy and Boucher (16) for excised canine tracheal and bronchial airways. Normand et al. (23) estimated a pore diameter of 1.48 nm for the fetal lamb alveolar epithelium, while Enna and Schanker (1 l) proposed three pore sizes to account for observed absorption rates of substances ranging from 60 to 75,000 Daltons in molecular weight. All such hydrodynamic models suffer from two major problems. First of all, there are insufficient data in the literature to adequately test all the models and to rank them in terms of reliability and predictability. Secondly, the epithelial barrier is a very complex structure that must be simplified in any attempt at modelling. For example, the small pores are probably associated with the tight junctions between the cells, which have been shown to be highly dynamic structures (18). Also, excised tracheal epithelial tissue has been found to exhibit an osmotic permeability from the apical side that is smaller than from the submucosal side (22). How such real tissue properties can be incorporated into a useful model is prob-
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483
lematic. Although Deen, et al. (9) considered bulk flow in modelling glomerular filtration, no model in the literature has accounted for bulk fluid transport in the alveolus. In fact, all the hydrodynamic models of pulmonary epithelium reported heretofore completely ignore both pore complexity and bulk water flow. They assume a homogeneous epithelial barrier crossed by fixed pores of circular cross-section through which only solutes can penetrate. With the presumptions that the "pores" can be identified with intercellular spaces and the tight junctions, and that the known secretory properties of the epithelium ought to impact solute transport across the tissue, the question arises as to the impact of pore irregularity and bulk fluid transport on transport models of the alveolar epithelial barrier. Breslau and Miller (3) developed a hydrodynamic model for electroosmotic transport across ion exchange membranes, in which they showed that a flat-plate pore model accounted much better for pore irregularities than did a cylindrical pore model. The model also accounted for bulk fluid flow across the membrane barrier better than previous models. We attempted to determine whether such an approach would allow the inclusion of pore irregularities and bulk fluid flow in models of the alveolar epithelial harrier, and whether the inclusion of these factors makes a significant impact on the predictions of such models. Herein, we describe a hydrodynamic model for solute transport across the alveolar-capillary barrier that is based on the widely accepted two pore models. Our model considers bulk fluid flow across the barrier, and evaluates both circular and parallel-plate pore models in an attempt to evaluate the effects of pore morphometry. Evaluation of the model using permeability data from normal bullfrog alveolar epithelium and retention data from healthy rat and canine lung gave very good data fits. MODEL
The alveolus was modelled as a sphere whose inner surface was covered by a uniform layer of surfactant. Aerosol deposited solute was assumed to dissolve instantaneously in the surfactant layer. The solute was then assumed to diffuse through the intercellular spaces (pores) in the epithelial layer. The epithelial layer was assumed to be a plane sheet of thickness 6 (300 nm) containing two different pore populations (Fig. 1.) Following Breslau and Miller (3), we modelled the epithelium as a barrier containing two populations of straight waterfilled parallel-plate or cylindrical pores. All transport processes, whether diffusively or osmotically driven, were assumed to occur at unsteady state in a direction perpendicular to the epithelial surface. As with all previously reported models, the interstitium and capillary endothelium were assumed to provide minimal resistance to transport. Since the blood volume is so large compared to the surfactant volume, the blood was treated as an infinite sink in the model, where the solute concentration is effectively zero. From this model, the rate of loss of solute from the alveolar spaces was determined by applying Fick's Second Law, with a volume flow term included, to each pore subpopulation present, and then summing over the subpopulations, Eq. 1:
Oc(x, t) at
-
=
D
02c(x, t) OEx
Oc(x, t) Ox
Jv -
,
(1)
484
T. Chandra, I.F. Miller, and D.B. Yeates
C
A L U E 0 L U S
A P I L L A R Y d X=6
X=O
FIGURE 1. Diagram of epithelial parallel plate pore model.
where Jv = volume flux (m3/m 2 - s) c -- concentration of the diffusing solute (kmol/m3). Assuming that the hydrostatic pressure gradient is balanced by the colloidal osmotic pressure, the volume flow through the intercellular pores was calculated from Eq. 2: (2)
Jv = oR T L p A c ,
where a R T Lp
= = = =
reflection coefficient of the diffusing solute gas constant = 8314.3 N - m / k m o l - K absolute temperature -- 310 K hydraulic conductivity = 2.63 x 10 -12 mE/N-s [bullfrog alveolar epithelium (20)1 Ac = diffusing solute concentration difference across the m e m b r a n e (kmol/m3).
D, the solute diffusion coefficient within the pores, was calculated f r o m the hydrodynamic pore model, Eqs. 3a and 3b, developed by Kawabe et al. (19), and extended by Breslau and Miller (3) to cover the range where a/rp approaches 1: for a/rp < 0.4 D=Do
1-
a
1-
1 . 0 0 4 a +0.41825- - 0 " 1 6 9 ~ - - 0 . 2 4 5 7 rp r~ r~;
rp
(3a)
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485
and for a/rp > 0.95
D= ~Do(1-~)/((0.58461
+ 3-~5 [ ~ -
l])ln(2r~-~aa) )
(3b)
for the parallel-plate pore. In order to cover the entire range of values for a/r,, the Breslau and Miller (3) approximation was used. For the cylindrical pore model solute diffusion coefficient, Eq. 4 was used for a/rp < 0.4: =
-
rp / \
1 - 2.104 -- + 2.090 . ~ - 0.95
rp
rh
,
(4)
where Do = free diffusivity of the solute (m2/s) a = diameter of the diffusing solute (nm) rp = diameter of the cylindrical pore (nm). For a/rp > 0.4, the numerical solution of Wang (27) was used. For the parallel-plate pores, the solute reflection coefficient o is given by Eq. 5a formulated by Anderson and Adamski (2): a 2
o=20
a 3
-1.0
.
(5a)
For cylindrical pores e was estimated using Eq. 5b (20): a 2
a 3
a 4
a = 5.333 ~p2 - 6.667 ~p3 + 2.333 ~
.
(5b)
The initial condition was: no solute present in the tissue barrier initially, Eq. 6:
c(x,O) = 0 .
(6)
The boundary conditions were: 1. Zero solute concentration on the blood side, Eq. 7: c(0, t) = 0 .
(7)
2, Rate of loss of solute from the surfactant layer equals the rate crossing the surfactant-epithelial interface, Eq. 8:
D
Oc(~,t) = P [ c o - c(6, t)] , Ox
(8)
where P is the permeability of the tissue barrier to the diffusing species, (m/s) and co is the solute concentration in the surfactant layer (kmol/m3).
486
T. Chandra, LF. Miller, and D.B. Yeates
These equations can be solved by standard methods (6) if the epithelial surface concentration is constant. However, in the case in aerosol deposited solute clearance situations, the epithelial surface concentration is not constant. To resolve this conflict, the entire time period o f interest was divided into a series o f time increments within each of which the surface concentration was assumed constant, yielding Eq. 9: (~($, {) =
2 sinh(2R/2) e x p l R ( $ - 1)/21 (R/L + 2)sinh(R/2) + (R/L)cosh(R/2) (9)
o. -
L sin (~'/3n) exp{ - k ~ { ] , [ 1 / ( R + 2 L ) + 1]kn sin/3.
2 e x p ( R ( ~ - 1)/2) ~
where the parameters are defined by Eqs. 10, 11, 12, 13 and 14: C(~?, {) =
c(.~, {)
(10)
Co
J~6 R = -D .~ =
(11)
x -
(12)
Dt { = --
(13)
6P L = -- . D
(14)
/3~ is the nth root of Eq. 15: /3n cot /3, + (A + L) = 0
.
(15)
A and kn are given by Eq. 16 and Eq. 17, respectively: R
A = -2
(16)
)x. =/32 + A 2 .
(17)
F r o m the calculated concentration profile, the amount of solute loss per unit area f r o m the alveolar space, Mr, during the time increment At was determined from Eq. 18: Mt ~2
r2 O(
(18)
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487
If S is the surface area of the alveolar epithelial membrane available for diffusion and V is the volume of the surfactant compartment, the alveolar surface concentration after the m th time increment was then determined from Eq. 19: ~O(~.-m+l) :
Co(tm+l) _ 1 Co(tin)
(19)
S6 Mt V 6Co(tm)
Equations 18 and 19 yield Eq. 20:
C([m+~) = 1 - S_66[ LR A { - 2L e x p ( - R / 2 ) V I_ [ e R ( R + L ) - L}
~o cosec03n) i~+,] x Y] exp{-X,,{} 9 (20) 1 /3n{L(1 + L ) + L R / 2 + B 2} The first ten terms of the summation were used for calculations. Due to computational limitations the time interval was chosen to be 1 millisecond. If a two-pore system is assumed, the decrease in alveolar surface solute concentration can be determined from the contributions to transport of each subpopulation by Eq. 21:
S
Co(tm+l) =Co(tm)
1 -- ~ (FLMtL + {1 - F L } M t s )
] tm+l ,,,
(21)
where FL is the area fraction of the large pores and S / V is the alveolar surface to volume ratio. Assuming the solute molecules were well characterized, the only unknowns were the sizes of the small and large pores, and the fraction of the total pore area available to the small pores. These were determined by use of a nonlinear least squares regression procedure that predicted the solute permeabilities, which were then compared to literature values using a modified form of the Levenberg-Marquardt algorithm (4). The large pore width was first determined by fitting the model to transport data for solutes larger than erythritol (diameter = 0.7 nm). The small pore width and area fraction were then simultaneously determined by fitting the model to literature data for solutes smaller than erythritol, taking into account transport through the large pores. An optimal fit was obtained by minimizing the difference between predicted and measured permeabilities for all the solutes tested. For osmotically induced bulk water flow governed by Poiseuille's law through parallel-plate pores, the ratio of water flux through the large and small flat-plate pores is given by Eq. 22:
J'o
U,r?o, '
where N I / N s is the relative number of large pores to small pores.
(22)
488
T. Chandra, LF. Miller, and D.B. Yeates TABLE 1. Normal bullfrog alveolar epithelial permeability. Comparison between theoretical and literature values (8).
Solute
Solute Radius (nm)
Permeability (Theoretical) ( x l 0 -7 cm/s)
Permeability (Literature) ( x l O -7 cm/s)
Water Formamide Acetamide Urea Ethylene glycol Glycerol Erythritol 2-deoxy glucose Mannitol Sucrose Raffinose Cyanacobalamin Inulin
O. 15 0.21 0.24 0.24 0.25 0.30 0.35 0.36 0.43 0.52 0.61 0.72 1.40
375.00 195.40 158.70 18.40 99.80 24.30 8.01 4.02 3.69 2.54 2.10 0.41 0.66
369.9 190.8 154.2 17.9 96.2 23.1 7.52 3.74 3.40 2.38 1.77 0.32 0.25
Small pore width = 4 nm. Large pore width = 10 nm.
TESTS OF T H E M O D E L The model was tested with data of Crandall and Kim (8) and Gatzy (15) for the permeability of the bullfrog alveolar epithelium. A best fit was obtained from the flatplate pore model with a small-pore width of 4 nm, a large-pore width of 10 nm, and a large-pore area fraction of 0.16. F r o m Eq. 22 the small pores accounted for 88% of the bulk water flow. The measured and predicted permeabilities for 13 solutes are compared in Table 1 and in Fig. 2. The cylindrical-pore model yielded a small-pore diameter of 5 nm and a large-pore diameter of 10 nm for the same solutes and the same large-pore area fraction as the flat-plate pore model. Both models yielded an average deviation of 20% from the literature permeabilities with the only m a j o r deviations occurring for raffinose, cyanacobalamin, and inulin. Data on first order rate constants for solute loss (6 solutes) from rat lung (14) were then fit with the parallel-plate pore model. The rate constant, K was defined as K = P S / V , where P is the permeability o f the tissue barrier, and S and V are defined as in Eq. 19. O p t i m u m pore widths were 0.4 nm (small pore) and 50 n m (large pore), and a large-pore area fraction of 0.02. The predicted and measured rate constants are presented in Table 2 and Fig. 3. D a t a for first order rate constants for the loss of Dextrans o f molecular weights ranging f r o m 3,000 to 500,000 Daltons (12,13,17) from the canine alveolus yielded a parallel-plate best fit large-pore width of 98 nm with an average deviation of 18% between predicted and measured rate constants (Table 3 and Fig. 4). A first order rate constant for Dextran of 60,000-90,000 Daltons (26), reported to be three times larger than expected from the other data reported, was not used. DISCUSSION As can be seen f r o m the figures, the model predicted transport rates for a wide variety of solutes in three different animal models extremely well. In the case of the
Pore Transport Model
489
~o 1#
i,o, lo" 10"
10' 10' 10' LITERATURE PERMEABILITY X 107 cm/s
FIGURE 2. Comparison between predicted and literature permeability values for bullfrog alveolar epithelium. Small-pore width = 4 nm; large-pore width = 10 nm.
bullfrog epithelium, parallel-plate pore width estimates of 4 nm and 10 nm were somewhat different from the diameters determined by Kim and Crandall (1 nm and 10 nm) (20), using a cylindrical-pore model. The only substantial difference between experimental and predicted values for permeability were for raffinose, cyanacobalamin and
TABLE 2. Normal rat lung retention rate constants for lipophobic solutes, Comparison between theoretical and literature values.
Solute
Solute Radius (nm)
Rate Constant (Theoretical) ( x 10-5/s)
Rate Constant (Literature) ( x 1O-S/s)
Urea Erythritol Mannitol Sucrose Cyanacobalamin Inulin
0.24 0.35 0.43 O. 52 0.72 1.40
288.70 35.80 18.23 14.70 6.38 4.84
288.30 35.00 17.83 14.50 6.42 5.13
Small pore width = 0.4 nm. Large pore width = 50.0 nm.
490
T. Chandra, LF. Miller, and D.B. Yeates
|
ld. X IZ
| |
r
Z
o o
|
ld
|
MJ
1r
0.0
1).2
0.4
().6
1).8
1'.0
;.2
;.4
1.6
SOLUTE RADIUS nm FIGURE 3. Comparison between calculated and predicted rate constant values for rat lung. Smallpore width = 0 . 4 nm; large-pore width = B0 nm. * = literature values, o = predicted values.
inulin, three of the largest solutes. Since the largest solutes have the smallest permeabilities, which presumably are the most difficult to measure accurately, it is likely that the observed predictive error is a result of scatter in the experimental data. There appears to be no experimental criterion for choosing between the parallel-plate and cylindrical models, since both fit experimental data equally well, and there is no independent experimental approach to distinguish between them. However, following Breslau and Miller (3), a flat-plate model is likely to fit an irregular pore better than does a circular pore model because o f the nature of the functional relationship between pore conductance and the ratio of solute size to pore size, as represented by Eqs. 3 and 4. The functional relationship is highly nonlinear and
TABLE 3. Canine lung retention rate constants for dextrans, Comparison between theoretical and literature values (23). Dextran Molecular Wt (Daltons)
Solute Radius (nm)
3000 10400 20000 150000-170000 250000
1.0 2.0 3.2 10.0 14.5
Large pore width = 98 nm.
Ref.
Rate Constant (Theoretical) (x lO-e/s)
Rate Constant (Literature) ( x l O 6/s)
12 12 13 17 12
3.36 3.20 2.82 0.80 0.22
3.40 3.28 2.97 1.10 0.48
Pore Transport Model
491
4.0-
J 3.0-
x Z
2.0
o o ~
1.0-
r
0.0
2.0
4.0
.0
8.0
10.0
1 .0
14.0
16.0
SOLUTE RADIUS nm FIGURE 4. Comparison between predicted and literature rate constants for canine lung. Large-pore width = 9 8 n m . * -- literature values. 9 = predicted values.
the pore conductance approaches zero very quickly as solute size approaches the pore width. Thus, drag forces on the solute are concentrated at points of closest approach. This nonlinear relationship between pore conductance and solute size has several consequences. First of all, the diffusing solute particle will tend to find the pore central axis very quickly, since unbalanced forces created by different points of closest approach will drive the particle to the axis. Secondly, since a circular pore would cause drag forces to be evenly distributed around the particle, while a parallel-plate pore would have drag forces concentrated at the two ends of a diameter perpendicular to the walls, the morphologically irregular pore would behave much more like a parallel-plate pore than like a circular pore, insofar as pore conductance is concerned. Another consequence is that solute transport would be expected to transport water through the pores via a solute drag mechanism. The amount of water transported would depend on the particular configuration of the pore and so could, in principle, be predicted from a particular pore model. Unfortunately, there are insufficient data available to predict the extent of water flow due to solute drag in the work reported herein. In the case of Dextran loss from the canine alveolus, the model predicted a pore width of 98 nm, substantially larger than any of the pore widths reported in the literature (1,10,11,15,20), although none of the literature values are for canine alveolus. Since the Dextran molecules used were much larger than the sizes of solutes tested in other studies, pore widths less than 2 nm could not be detected. The predicted apparent large pore size results from the fact that even the largest Dextran (250,000 Daltons) had a measurable rate of loss (rate constant = 4.8 x 10-7/s) from the alveolus. If other mechanisms than pore flow contributed to the measured losses, the actual pore dimensions could be much smaller than those predicted by the model. The model fit experimental data well in spite of such simplifying assumptions as
492
T. Chandra, LF. Miller, and D.B. Yeates
spherical alveoli, uniform surfactant layers, straight-through flat-plate or cylindrical pores, etc. Thus it may be concluded that either the assumptions are reasonable pictures of reality, or that the model is insensitive to changes in these particular properties. This question may be resolved when significantly more aerosol clearance data on alveolar permeability of lipophobic solutes, together with the effects of agents that alter the pore dimensions in a predictable manner, are available to test and refine the model. Since the reliability of a regression model is dependent on the number of observations, the availability of more data from animal models utilized will also improve the reliability of the results. The sensitivity of the flat-plate pore model was estimated from the sum of the squared errors for permeability, 0p, defined as in Eq. 23:
(~P= s=l ~ [ PexpS_p~x~s-Ptheors] 2
(23)
where n = the total number of probe solutes Pexps = the experimentally measured permeability Ptheors = the permeability obtained f r o m the model. The sensitivity analysis was performed for (a) small pore, (b) large pore, and (c) pore area fractions. For the bullfrog alveolar epithelium model, 0p was found to be very sensitive to initial small pore estimates in the proximity of the size of the smaller solutes tested. 0p was not greatly influenced by the initial large-pore estimates but passed through a minimum for a large pore width of 10 nm. The estimation of osmotically induced water fluxes (Jr) required knowledge of the concentration gradient across the pulmonary barrier. These concentration gradients were obtained from the literature (20) for sucrose and raffinose and, because of a lack of data for other solutes, it was assumed to hold for all the solutes tested in the model. In the absence of an osmotic water flux, the parallel-plate pore width estimates for the bullfrog alveolar epithelium were 2.4 nm and 13 nm. This is consistent with the observation that a back flux of water towards the alveolar lumen is likely to reduce the solute flux towards the capillary endothelium and hence a larger small-pore size (4 nm) would be required to account for the observed permeability. To estimate the osmotic water flux, a solute concentration of 0.6 k m o l / m 3, based on the osmotic gradients for sucrose and raffinose used by Kim and Crandall (20), was used for all the solutes tested in the model. This resulted in water velocities of up to 4 • 10 -8 m / s for the bullfrog alveolar epithelium. Although pore flow models similar to that presented herein are highly simplified pictures of reality, they have potential practical utility. For example, they can be used to predict the clearance of solutes from the alveolar spaces, and the effects of permeability-altering agents on pore dimensions. The close data fits to literature values suggest that the model has the potential to be a powerful investigative tool. Further refinements like accounting for ion transport, better evaluation of water transport both by osmosis and by solvent drag, and determination of the effects of permeability altering agents could add to the accuracy o f the model. The choice between the flat-plate and the cylindrical pore model can be made more definitive, using independent data that can distinguish between them.
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493
P o s s i b l y , r e f l e c t i o n c o e f f i c i e n t d a t a f o r a s e r i e s o f w e l l - c h a r a c t e r i z e d s o l u t e s will allow such a selection to be made.
REFERENCES 1. Altamirano, M.; Martinoya, C. The permeability of the gastric mucosa of the dog. J. Physiol. (London) 184:771-790; 1966. 2. Anderson, J.L.; Adamski, R.P. Solute concentration effects on membrane reflection coefficients. Am. Inst. Chem. Eng. (AIChE Syrup. Set. 227.) 79:84-92; 1983. 3. Breslau, B.R.; Miller, I.F.I.E.C. fund. A hydrodynamic model for electroosmosis. 10:554-565; 1971. 4. Brown, K.M.; Dennis, J.E. Derivative free analogues of the L-M and game algorithm for non-linear least squares approximation. Numerische Mathematik 18:289-297; 1972. 5. Brown, R.A.; Schanker, L.S. Absorption of aerosolized drugs from the rat lung. Drug metabolism and disposition 11:355-360; 1983. 6. Carslaw, H.S.; Jaeger, J.C. Conduction of heat in solids. Oxford: Clarendon Press; 1959: p. 391. 7. Chopra, S.K.; Taplin, G.V.; Tashkin, D.T.; Elam, D. Lung clearance of soluble radio-aerosols of different molecular weights in systemic sclerosis. Thorax 34:63-67; 1979. 8. Crandall, E.D.; Kim, K.J. Transport of water and solutes across the excised bullfrog alveolar epithelium. J. Appl. Physiol. 50:1263-1271; 1981. 9. Deen, W.M.; Bridges, C.R.; Brenner, B.M.; Myers, B.D. Heteroporous model of glomerular size selectivity: Application to normal and nephrotic humans. Am. J. Physiol. 249:F374-F389; 1985. 10. Durbin, R.P.; Frank, H.; Solomon, A.K. Water flow through frog gastric mucosa. J. Gen. Physiol. 39:535-551; 1956. 11. Enna, S.J.; Schanker, L.S. Absorption of drugs from the rat lung. Am. J. Physiol. 223:1277-1231; 1972. 12. Fischer, P.; Glauser, F.L.; Miller, J.E.; Lewis, J.; Egan, P. The effects of ethchlorvynol on pulmonary alveolar membrane permeability. Am. Rev. Respir. Dis. 116:901-906; 1977. 13. Fischer, P.; Miller, J.E.; Glauser, F.L. The pulmonary alveolar capillary membrane (ACM) during hemorrhageic hypotension in dogs. Surg. Gynecol. Obstet. 146:383-386; 1978. 14. Gardiner, T.H.; Schanker, L.S. Effect of papain induced emphysema on permeability of rat lungs to drugs. Proc. Soc. Exp. Biol. Med. 149:972-977; 1975. 15. Gatzy, J.T. Paths of hydrophilic solute flow across excised bullfrog lung. Expt. Lung Res. 3:147-161; 1982. 16. Gatzy, J.T.; Boucher, R.C. Paths of hydrophilic solute flow across excised canine airways (abstract). Fed. Proc. 41:1244; 1982. 17. Glauser, F.L.; Miller, J.E.; Falls, R. Effects of acid aspiration on pulmonary alveolar epithelial membrane permeability. Chest. 76:201-205; 1975. 18. Gumbiner, B. Structure, biochemistry and assembly of epithelial tight junctions. Am. J. Physiol. 253 (Cell Physiol. 22): C749-C758; 1987. 19. Kawabe, H.; Jacobson, H.; Miller, I.F.; Gregor, H.P. Functional properties of cation exchange membranes as related to their structures. J. Coll. Int. Sci. 21:79; 1966. 20. Kim, K.J.; Crandall, E.D. Heteropore population of bullfrog alveolar epithelium. J. Appl. Physiol. 54:140-146; 1983. 21. Lippmann, M.; Yeates, D.B.; Albert, R.E. Deposition, retention, and clearance of inhaled aerosol particles. Br. J. Ind. Med. 37:337-362; 1980. 22. Man, S.F.P.; Hulbert, W.; Park, D.S.K.; Thomson, A.B.R.; Hogg, J.C. Asymmetry of canine tracheal epithelium: Osmotically induced changes. J. Appl. Physiol. 57:1338-1346; 1984. 23. Normand, I.C.S.; Olver, R.E.; Reynolds, E.O.R.; Strang, L.B. Permeability of lung capillaries and alveoli to non-electrolytes in the fetal lamb. J. Physiol. (London) 219:303-330; 1971. 24. Oberdorster, G.; Utell, R.; Morrow, P.E.; Hyde, R.W.; Weber, D.A. Bronchial and alveolar absorption of inhaled 99rnTc-DTPA. Am. Rev. Respir. Dis. 134:944-950; 1986. 25. Taylor, A.E.; Gaar, K.A. Estimation of equivalent pore radii of pulmonary capillary and alveolar membranes. Am. J. Physiol. 218:1133-1140; 1970. 26. Theodore, J.; Robin, E.D.; Gaudio, R; Acevedo, J. Transalveolar transport of large polar solutes (sucrose, inulin and dextran). Am. J. Physiol. 229:989-996; 1975. 27. Wang, H. Viscous flow in a cylindrical tube containing a line of spherical particles. Ph.D. Dissertation, Columbia University, New York, NY; 1967.
494
T. Chandra, L F . Miller, and D.B. Yeates
NOMENCLATURE C Co D
Do j~
Lp g~ P
Pexps Ptheors
R rp S
T X
V 6 AC
diameter of the diffusing solute (nm) concentration of the diffusing solute (kmol/m 3) = solute concentration in the surfactant layer (kmol/m 3) = solute diffusion coefficient within the pore (mZ/s) = free diffusivity of the solute (mZ/s) = area fraction of the large pores = volume flux (m3/mZ-s) = hydraulic conductivity (mZ/N-s) = amount of solute loss per unit area (kmol/m 2) = permeability of the tissue barrier to diffusing species (m/s) experimentally measured permeability (m/s) theoretical permeability (m/s) -- gas constant (N-m/kmol-K) = width of parallel-plate pore (nm) or diameter of cylindrical pore (nm) = cross-sectional area of the alveolar epithelial membrane available for diffusion (m 2) absolute temperature (K) = time after mtn time increment (s) = distance measured into epithelial membrane (m) = volume of the alveolar compartment (m 3) = thickness of epithelial layer (m) = concentration gradient across the membrane (kmol/m 3) sum of the squared errors for permeability -~- reflection coefficient of the diffusing solute =
=
O"
Subscripts:
1, L s,S
= large pore = small pore
Superscripts."
1 s
= large pore = small pore
0090-6964/92 $5.00 + .00 Copyright 9 1992 Pergamon Press Ltd.
A Pore Transport Model for Pulmonary Alveolar Epithelium T. Chandra,* I.F. Miller,l and D.B. Yeatesi~: *Department of Chemical Engineering tDepartment of Medicine University of Illinois at Chicago ~& V.A. West Side, 820 South Damen Chicago, IL
(Received 1/10/90) Hydrodynamic heteropore f l o w models f o r transport o f solutes across alveolar epithelial tissue have been developed. A two-size cylindrical pore model and a similar parallel-plate model were formulated, tested and used to predict effective pore sizes f r o m literature data on transport in bullfrog, canine and rat lungs. The best f i t equivalent pore-size estimates were obtained using a modified, nonlinear least squares procedure, with alveolar surface area to volume ratio (S/V) and small-pore area fraction o f total pore area as parameters. Small-pore and large-pore width estimates o f 4 nm (84% o f total f l o w area) and 10 nm, respectively, with an average deviation o f 20~o f r o m experimentally derived permeabilities were obtained f r o m the bullfrog alveolar epithelium parallel-plate pore model (13 solutes, diameters 0.3 to 2.8 nm). The equivalent cylindrical pore model diameter estimates were 5 nm and 10 nm, with small-pore area fraction and percentage deviations similar to the parallel-plate model estimates. Eighty-eight percent o f the bulk water driven by a sucrose osmotic gradient was predicted to be transported through the small pores. The rat alveolus parallel-plate pore model (6 solutes) yielded small-pore and large-pore widths o f 0.4 nm and 50 nm, respectively. Clearance rate-constant data f o r dextran macromolecules (3,000 to 250, 000 Daltons), using a single parallel-plate pore model, resulted in a pore width estimate o f 98 nm f o r canine alveoli with an average deviation o f the predicted rate constants o f 18% from literature experimental values. In all cases tested, the parallel-plate pore model predicted lower small-pore size estimates than did the cylindrical pore model, and both models had appreciably smaller percentage deviations f r o m experimental data than previous models. Keywords--Hydrodynamic model, Pore flow, Membrane transport, Alveolus, Epithelium, Permeability. INTRODUCTION
T h e t r a n s p o r t o f n o n v o l a t i l e m a t e r i a l s a c r o s s t h e p u l m o n a r y e p i t h e l i u m is o f c o n s i d e r a b l e t o x i c o l o g i c a l , p h a r m a c o l o g i c a l a n d p h y s i o l o g i c a l i n t e r e s t . F i r s t o f all, t h e
Acknowledgment-This report is based on a thesis submitted by Tarun Chandra, in partial fulfillment of the requirements for the M.S. degree in Chemical Engineering at the University of Illinois at Chicago. Address correspondence to I.F. Miller, Section of Environmental and Occupational Medicine, Room 230, Benjamin Goldberg Research Center, 1940 West Taylor, University of Illinois at Chicago, Chicago, IL 60612. 481
482
T. Chandra, LF. Miller, and D.B. Yeates
response to inhaled irritants and toxins is critically dependent on the transport processes that carry them to target organs. Secondly, there is considerable current interest in the possible use of the lungs to deliver systemic drugs via aerosol. Finally, aerosolized compounds can be used as probes to elucidate the permeability characteristics of the pulmonary epithelium and, thus, can be used to explore structural models of tissue pathophysiology. The conditions under which aerosolized particles can be used as a probe of epithelial structure depend on such factors as the size, chemical composition and site of deposition of the particles. Submicron-sized particles deposit primarily in the distal airways, while larger particles deposit more proximally (21,24). In the bronchial region, mucociliary transport provides the main means whereby deposited particles are removed. In the alveolar region, there is no mucociliary transport, and macrophageal action, vesicular transport and diffusion provide the clearance mechanisms (7,21). Thus, the clearance of aerosolized particles as probes of epithelial structure can be used in the alveolar region, but with less success in the bronchial region. The transport of solutes across the alveolar epithelium depends in large part on their lipid solubilities. Lipophilic solutes can transport transcellularly and are rapidly removed from the lungs (5). On the other hand, lipophobic solutes are restricted to the aqueous intercellular spaces, and transport much more slowly. The clearance of lipophobic solutes, of differing molecular size, from the alveolar space can thus be used as a probe to model the structure of the aqueous intercellular spaces ("pores") in the alveolar epithelium. Attempts to produce such structural models of the alveolar epithelium have used either thermodynamic or hydrodynamic approaches. Thermodynamic approaches relate fluxes of volume and solute to gradients o f pressure and concentration and provide no direct structural information. Hydrodynamic approaches model the structure of the barrier by, for example, assuming identical cylindrical water-filled pores and using the Hagen-Poiseuille equation to estimate the apparent pore diameter from flux data (25). Improved hydrodynamic models use multiple pore populations. For example, Durbin, et al. (10), successfully modelled the frog gastric mucosa with two pore populations: 93~ of the available pore area was attributed to pores of 0.5 nm diameter, and 7~ to pores of 12 nm diameter, a model very similar to that developed by Kim and Crandall for the bullfrog alveolar epithelium (20). Altamirano and Martinoya (1) also arrived at almost the same model for canine gastric mucosa, as did Gatzy and Boucher (16) for excised canine tracheal and bronchial airways. Normand et al. (23) estimated a pore diameter of 1.48 nm for the fetal lamb alveolar epithelium, while Enna and Schanker (1 l) proposed three pore sizes to account for observed absorption rates of substances ranging from 60 to 75,000 Daltons in molecular weight. All such hydrodynamic models suffer from two major problems. First of all, there are insufficient data in the literature to adequately test all the models and to rank them in terms of reliability and predictability. Secondly, the epithelial barrier is a very complex structure that must be simplified in any attempt at modelling. For example, the small pores are probably associated with the tight junctions between the cells, which have been shown to be highly dynamic structures (18). Also, excised tracheal epithelial tissue has been found to exhibit an osmotic permeability from the apical side that is smaller than from the submucosal side (22). How such real tissue properties can be incorporated into a useful model is prob-
Pore Transport Model
483
lematic. Although Deen, et al. (9) considered bulk flow in modelling glomerular filtration, no model in the literature has accounted for bulk fluid transport in the alveolus. In fact, all the hydrodynamic models of pulmonary epithelium reported heretofore completely ignore both pore complexity and bulk water flow. They assume a homogeneous epithelial barrier crossed by fixed pores of circular cross-section through which only solutes can penetrate. With the presumptions that the "pores" can be identified with intercellular spaces and the tight junctions, and that the known secretory properties of the epithelium ought to impact solute transport across the tissue, the question arises as to the impact of pore irregularity and bulk fluid transport on transport models of the alveolar epithelial barrier. Breslau and Miller (3) developed a hydrodynamic model for electroosmotic transport across ion exchange membranes, in which they showed that a flat-plate pore model accounted much better for pore irregularities than did a cylindrical pore model. The model also accounted for bulk fluid flow across the membrane barrier better than previous models. We attempted to determine whether such an approach would allow the inclusion of pore irregularities and bulk fluid flow in models of the alveolar epithelial harrier, and whether the inclusion of these factors makes a significant impact on the predictions of such models. Herein, we describe a hydrodynamic model for solute transport across the alveolar-capillary barrier that is based on the widely accepted two pore models. Our model considers bulk fluid flow across the barrier, and evaluates both circular and parallel-plate pore models in an attempt to evaluate the effects of pore morphometry. Evaluation of the model using permeability data from normal bullfrog alveolar epithelium and retention data from healthy rat and canine lung gave very good data fits. MODEL
The alveolus was modelled as a sphere whose inner surface was covered by a uniform layer of surfactant. Aerosol deposited solute was assumed to dissolve instantaneously in the surfactant layer. The solute was then assumed to diffuse through the intercellular spaces (pores) in the epithelial layer. The epithelial layer was assumed to be a plane sheet of thickness 6 (300 nm) containing two different pore populations (Fig. 1.) Following Breslau and Miller (3), we modelled the epithelium as a barrier containing two populations of straight waterfilled parallel-plate or cylindrical pores. All transport processes, whether diffusively or osmotically driven, were assumed to occur at unsteady state in a direction perpendicular to the epithelial surface. As with all previously reported models, the interstitium and capillary endothelium were assumed to provide minimal resistance to transport. Since the blood volume is so large compared to the surfactant volume, the blood was treated as an infinite sink in the model, where the solute concentration is effectively zero. From this model, the rate of loss of solute from the alveolar spaces was determined by applying Fick's Second Law, with a volume flow term included, to each pore subpopulation present, and then summing over the subpopulations, Eq. 1:
Oc(x, t) at
-
=
D
02c(x, t) OEx
Oc(x, t) Ox
Jv -
,
(1)
484
T. Chandra, I.F. Miller, and D.B. Yeates
C
A L U E 0 L U S
A P I L L A R Y d X=6
X=O
FIGURE 1. Diagram of epithelial parallel plate pore model.
where Jv = volume flux (m3/m 2 - s) c -- concentration of the diffusing solute (kmol/m3). Assuming that the hydrostatic pressure gradient is balanced by the colloidal osmotic pressure, the volume flow through the intercellular pores was calculated from Eq. 2: (2)
Jv = oR T L p A c ,
where a R T Lp
= = = =
reflection coefficient of the diffusing solute gas constant = 8314.3 N - m / k m o l - K absolute temperature -- 310 K hydraulic conductivity = 2.63 x 10 -12 mE/N-s [bullfrog alveolar epithelium (20)1 Ac = diffusing solute concentration difference across the m e m b r a n e (kmol/m3).
D, the solute diffusion coefficient within the pores, was calculated f r o m the hydrodynamic pore model, Eqs. 3a and 3b, developed by Kawabe et al. (19), and extended by Breslau and Miller (3) to cover the range where a/rp approaches 1: for a/rp < 0.4 D=Do
1-
a
1-
1 . 0 0 4 a +0.41825- - 0 " 1 6 9 ~ - - 0 . 2 4 5 7 rp r~ r~;
rp
(3a)
Pore Transport Model
485
and for a/rp > 0.95
D= ~Do(1-~)/((0.58461
+ 3-~5 [ ~ -
l])ln(2r~-~aa) )
(3b)
for the parallel-plate pore. In order to cover the entire range of values for a/r,, the Breslau and Miller (3) approximation was used. For the cylindrical pore model solute diffusion coefficient, Eq. 4 was used for a/rp < 0.4: =
-
rp / \
1 - 2.104 -- + 2.090 . ~ - 0.95
rp
rh
,
(4)
where Do = free diffusivity of the solute (m2/s) a = diameter of the diffusing solute (nm) rp = diameter of the cylindrical pore (nm). For a/rp > 0.4, the numerical solution of Wang (27) was used. For the parallel-plate pores, the solute reflection coefficient o is given by Eq. 5a formulated by Anderson and Adamski (2): a 2
o=20
a 3
-1.0
.
(5a)
For cylindrical pores e was estimated using Eq. 5b (20): a 2
a 3
a 4
a = 5.333 ~p2 - 6.667 ~p3 + 2.333 ~
.
(5b)
The initial condition was: no solute present in the tissue barrier initially, Eq. 6:
c(x,O) = 0 .
(6)
The boundary conditions were: 1. Zero solute concentration on the blood side, Eq. 7: c(0, t) = 0 .
(7)
2, Rate of loss of solute from the surfactant layer equals the rate crossing the surfactant-epithelial interface, Eq. 8:
D
Oc(~,t) = P [ c o - c(6, t)] , Ox
(8)
where P is the permeability of the tissue barrier to the diffusing species, (m/s) and co is the solute concentration in the surfactant layer (kmol/m3).
486
T. Chandra, LF. Miller, and D.B. Yeates
These equations can be solved by standard methods (6) if the epithelial surface concentration is constant. However, in the case in aerosol deposited solute clearance situations, the epithelial surface concentration is not constant. To resolve this conflict, the entire time period o f interest was divided into a series o f time increments within each of which the surface concentration was assumed constant, yielding Eq. 9: (~($, {) =
2 sinh(2R/2) e x p l R ( $ - 1)/21 (R/L + 2)sinh(R/2) + (R/L)cosh(R/2) (9)
o. -
L sin (~'/3n) exp{ - k ~ { ] , [ 1 / ( R + 2 L ) + 1]kn sin/3.
2 e x p ( R ( ~ - 1)/2) ~
where the parameters are defined by Eqs. 10, 11, 12, 13 and 14: C(~?, {) =
c(.~, {)
(10)
Co
J~6 R = -D .~ =
(11)
x -
(12)
Dt { = --
(13)
6P L = -- . D
(14)
/3~ is the nth root of Eq. 15: /3n cot /3, + (A + L) = 0
.
(15)
A and kn are given by Eq. 16 and Eq. 17, respectively: R
A = -2
(16)
)x. =/32 + A 2 .
(17)
F r o m the calculated concentration profile, the amount of solute loss per unit area f r o m the alveolar space, Mr, during the time increment At was determined from Eq. 18: Mt ~2
r2 O(
(18)
Pore Transport Model
487
If S is the surface area of the alveolar epithelial membrane available for diffusion and V is the volume of the surfactant compartment, the alveolar surface concentration after the m th time increment was then determined from Eq. 19: ~O(~.-m+l) :
Co(tm+l) _ 1 Co(tin)
(19)
S6 Mt V 6Co(tm)
Equations 18 and 19 yield Eq. 20:
C([m+~) = 1 - S_66[ LR A { - 2L e x p ( - R / 2 ) V I_ [ e R ( R + L ) - L}
~o cosec03n) i~+,] x Y] exp{-X,,{} 9 (20) 1 /3n{L(1 + L ) + L R / 2 + B 2} The first ten terms of the summation were used for calculations. Due to computational limitations the time interval was chosen to be 1 millisecond. If a two-pore system is assumed, the decrease in alveolar surface solute concentration can be determined from the contributions to transport of each subpopulation by Eq. 21:
S
Co(tm+l) =Co(tm)
1 -- ~ (FLMtL + {1 - F L } M t s )
] tm+l ,,,
(21)
where FL is the area fraction of the large pores and S / V is the alveolar surface to volume ratio. Assuming the solute molecules were well characterized, the only unknowns were the sizes of the small and large pores, and the fraction of the total pore area available to the small pores. These were determined by use of a nonlinear least squares regression procedure that predicted the solute permeabilities, which were then compared to literature values using a modified form of the Levenberg-Marquardt algorithm (4). The large pore width was first determined by fitting the model to transport data for solutes larger than erythritol (diameter = 0.7 nm). The small pore width and area fraction were then simultaneously determined by fitting the model to literature data for solutes smaller than erythritol, taking into account transport through the large pores. An optimal fit was obtained by minimizing the difference between predicted and measured permeabilities for all the solutes tested. For osmotically induced bulk water flow governed by Poiseuille's law through parallel-plate pores, the ratio of water flux through the large and small flat-plate pores is given by Eq. 22:
J'o
U,r?o, '
where N I / N s is the relative number of large pores to small pores.
(22)
488
T. Chandra, LF. Miller, and D.B. Yeates TABLE 1. Normal bullfrog alveolar epithelial permeability. Comparison between theoretical and literature values (8).
Solute
Solute Radius (nm)
Permeability (Theoretical) ( x l 0 -7 cm/s)
Permeability (Literature) ( x l O -7 cm/s)
Water Formamide Acetamide Urea Ethylene glycol Glycerol Erythritol 2-deoxy glucose Mannitol Sucrose Raffinose Cyanacobalamin Inulin
O. 15 0.21 0.24 0.24 0.25 0.30 0.35 0.36 0.43 0.52 0.61 0.72 1.40
375.00 195.40 158.70 18.40 99.80 24.30 8.01 4.02 3.69 2.54 2.10 0.41 0.66
369.9 190.8 154.2 17.9 96.2 23.1 7.52 3.74 3.40 2.38 1.77 0.32 0.25
Small pore width = 4 nm. Large pore width = 10 nm.
TESTS OF T H E M O D E L The model was tested with data of Crandall and Kim (8) and Gatzy (15) for the permeability of the bullfrog alveolar epithelium. A best fit was obtained from the flatplate pore model with a small-pore width of 4 nm, a large-pore width of 10 nm, and a large-pore area fraction of 0.16. F r o m Eq. 22 the small pores accounted for 88% of the bulk water flow. The measured and predicted permeabilities for 13 solutes are compared in Table 1 and in Fig. 2. The cylindrical-pore model yielded a small-pore diameter of 5 nm and a large-pore diameter of 10 nm for the same solutes and the same large-pore area fraction as the flat-plate pore model. Both models yielded an average deviation of 20% from the literature permeabilities with the only m a j o r deviations occurring for raffinose, cyanacobalamin, and inulin. Data on first order rate constants for solute loss (6 solutes) from rat lung (14) were then fit with the parallel-plate pore model. The rate constant, K was defined as K = P S / V , where P is the permeability o f the tissue barrier, and S and V are defined as in Eq. 19. O p t i m u m pore widths were 0.4 nm (small pore) and 50 n m (large pore), and a large-pore area fraction of 0.02. The predicted and measured rate constants are presented in Table 2 and Fig. 3. D a t a for first order rate constants for the loss of Dextrans o f molecular weights ranging f r o m 3,000 to 500,000 Daltons (12,13,17) from the canine alveolus yielded a parallel-plate best fit large-pore width of 98 nm with an average deviation of 18% between predicted and measured rate constants (Table 3 and Fig. 4). A first order rate constant for Dextran of 60,000-90,000 Daltons (26), reported to be three times larger than expected from the other data reported, was not used. DISCUSSION As can be seen f r o m the figures, the model predicted transport rates for a wide variety of solutes in three different animal models extremely well. In the case of the
Pore Transport Model
489
~o 1#
i,o, lo" 10"
10' 10' 10' LITERATURE PERMEABILITY X 107 cm/s
FIGURE 2. Comparison between predicted and literature permeability values for bullfrog alveolar epithelium. Small-pore width = 4 nm; large-pore width = 10 nm.
bullfrog epithelium, parallel-plate pore width estimates of 4 nm and 10 nm were somewhat different from the diameters determined by Kim and Crandall (1 nm and 10 nm) (20), using a cylindrical-pore model. The only substantial difference between experimental and predicted values for permeability were for raffinose, cyanacobalamin and
TABLE 2. Normal rat lung retention rate constants for lipophobic solutes, Comparison between theoretical and literature values.
Solute
Solute Radius (nm)
Rate Constant (Theoretical) ( x 10-5/s)
Rate Constant (Literature) ( x 1O-S/s)
Urea Erythritol Mannitol Sucrose Cyanacobalamin Inulin
0.24 0.35 0.43 O. 52 0.72 1.40
288.70 35.80 18.23 14.70 6.38 4.84
288.30 35.00 17.83 14.50 6.42 5.13
Small pore width = 0.4 nm. Large pore width = 50.0 nm.
490
T. Chandra, LF. Miller, and D.B. Yeates
|
ld. X IZ
| |
r
Z
o o
|
ld
|
MJ
1r
0.0
1).2
0.4
().6
1).8
1'.0
;.2
;.4
1.6
SOLUTE RADIUS nm FIGURE 3. Comparison between calculated and predicted rate constant values for rat lung. Smallpore width = 0 . 4 nm; large-pore width = B0 nm. * = literature values, o = predicted values.
inulin, three of the largest solutes. Since the largest solutes have the smallest permeabilities, which presumably are the most difficult to measure accurately, it is likely that the observed predictive error is a result of scatter in the experimental data. There appears to be no experimental criterion for choosing between the parallel-plate and cylindrical models, since both fit experimental data equally well, and there is no independent experimental approach to distinguish between them. However, following Breslau and Miller (3), a flat-plate model is likely to fit an irregular pore better than does a circular pore model because o f the nature of the functional relationship between pore conductance and the ratio of solute size to pore size, as represented by Eqs. 3 and 4. The functional relationship is highly nonlinear and
TABLE 3. Canine lung retention rate constants for dextrans, Comparison between theoretical and literature values (23). Dextran Molecular Wt (Daltons)
Solute Radius (nm)
3000 10400 20000 150000-170000 250000
1.0 2.0 3.2 10.0 14.5
Large pore width = 98 nm.
Ref.
Rate Constant (Theoretical) (x lO-e/s)
Rate Constant (Literature) ( x l O 6/s)
12 12 13 17 12
3.36 3.20 2.82 0.80 0.22
3.40 3.28 2.97 1.10 0.48
Pore Transport Model
491
4.0-
J 3.0-
x Z
2.0
o o ~
1.0-
r
0.0
2.0
4.0
.0
8.0
10.0
1 .0
14.0
16.0
SOLUTE RADIUS nm FIGURE 4. Comparison between predicted and literature rate constants for canine lung. Large-pore width = 9 8 n m . * -- literature values. 9 = predicted values.
the pore conductance approaches zero very quickly as solute size approaches the pore width. Thus, drag forces on the solute are concentrated at points of closest approach. This nonlinear relationship between pore conductance and solute size has several consequences. First of all, the diffusing solute particle will tend to find the pore central axis very quickly, since unbalanced forces created by different points of closest approach will drive the particle to the axis. Secondly, since a circular pore would cause drag forces to be evenly distributed around the particle, while a parallel-plate pore would have drag forces concentrated at the two ends of a diameter perpendicular to the walls, the morphologically irregular pore would behave much more like a parallel-plate pore than like a circular pore, insofar as pore conductance is concerned. Another consequence is that solute transport would be expected to transport water through the pores via a solute drag mechanism. The amount of water transported would depend on the particular configuration of the pore and so could, in principle, be predicted from a particular pore model. Unfortunately, there are insufficient data available to predict the extent of water flow due to solute drag in the work reported herein. In the case of Dextran loss from the canine alveolus, the model predicted a pore width of 98 nm, substantially larger than any of the pore widths reported in the literature (1,10,11,15,20), although none of the literature values are for canine alveolus. Since the Dextran molecules used were much larger than the sizes of solutes tested in other studies, pore widths less than 2 nm could not be detected. The predicted apparent large pore size results from the fact that even the largest Dextran (250,000 Daltons) had a measurable rate of loss (rate constant = 4.8 x 10-7/s) from the alveolus. If other mechanisms than pore flow contributed to the measured losses, the actual pore dimensions could be much smaller than those predicted by the model. The model fit experimental data well in spite of such simplifying assumptions as
492
T. Chandra, LF. Miller, and D.B. Yeates
spherical alveoli, uniform surfactant layers, straight-through flat-plate or cylindrical pores, etc. Thus it may be concluded that either the assumptions are reasonable pictures of reality, or that the model is insensitive to changes in these particular properties. This question may be resolved when significantly more aerosol clearance data on alveolar permeability of lipophobic solutes, together with the effects of agents that alter the pore dimensions in a predictable manner, are available to test and refine the model. Since the reliability of a regression model is dependent on the number of observations, the availability of more data from animal models utilized will also improve the reliability of the results. The sensitivity of the flat-plate pore model was estimated from the sum of the squared errors for permeability, 0p, defined as in Eq. 23:
(~P= s=l ~ [ PexpS_p~x~s-Ptheors] 2
(23)
where n = the total number of probe solutes Pexps = the experimentally measured permeability Ptheors = the permeability obtained f r o m the model. The sensitivity analysis was performed for (a) small pore, (b) large pore, and (c) pore area fractions. For the bullfrog alveolar epithelium model, 0p was found to be very sensitive to initial small pore estimates in the proximity of the size of the smaller solutes tested. 0p was not greatly influenced by the initial large-pore estimates but passed through a minimum for a large pore width of 10 nm. The estimation of osmotically induced water fluxes (Jr) required knowledge of the concentration gradient across the pulmonary barrier. These concentration gradients were obtained from the literature (20) for sucrose and raffinose and, because of a lack of data for other solutes, it was assumed to hold for all the solutes tested in the model. In the absence of an osmotic water flux, the parallel-plate pore width estimates for the bullfrog alveolar epithelium were 2.4 nm and 13 nm. This is consistent with the observation that a back flux of water towards the alveolar lumen is likely to reduce the solute flux towards the capillary endothelium and hence a larger small-pore size (4 nm) would be required to account for the observed permeability. To estimate the osmotic water flux, a solute concentration of 0.6 k m o l / m 3, based on the osmotic gradients for sucrose and raffinose used by Kim and Crandall (20), was used for all the solutes tested in the model. This resulted in water velocities of up to 4 • 10 -8 m / s for the bullfrog alveolar epithelium. Although pore flow models similar to that presented herein are highly simplified pictures of reality, they have potential practical utility. For example, they can be used to predict the clearance of solutes from the alveolar spaces, and the effects of permeability-altering agents on pore dimensions. The close data fits to literature values suggest that the model has the potential to be a powerful investigative tool. Further refinements like accounting for ion transport, better evaluation of water transport both by osmosis and by solvent drag, and determination of the effects of permeability altering agents could add to the accuracy o f the model. The choice between the flat-plate and the cylindrical pore model can be made more definitive, using independent data that can distinguish between them.
Pore Transport Model
493
P o s s i b l y , r e f l e c t i o n c o e f f i c i e n t d a t a f o r a s e r i e s o f w e l l - c h a r a c t e r i z e d s o l u t e s will allow such a selection to be made.
REFERENCES 1. Altamirano, M.; Martinoya, C. The permeability of the gastric mucosa of the dog. J. Physiol. (London) 184:771-790; 1966. 2. Anderson, J.L.; Adamski, R.P. Solute concentration effects on membrane reflection coefficients. Am. Inst. Chem. Eng. (AIChE Syrup. Set. 227.) 79:84-92; 1983. 3. Breslau, B.R.; Miller, I.F.I.E.C. fund. A hydrodynamic model for electroosmosis. 10:554-565; 1971. 4. Brown, K.M.; Dennis, J.E. Derivative free analogues of the L-M and game algorithm for non-linear least squares approximation. Numerische Mathematik 18:289-297; 1972. 5. Brown, R.A.; Schanker, L.S. Absorption of aerosolized drugs from the rat lung. Drug metabolism and disposition 11:355-360; 1983. 6. Carslaw, H.S.; Jaeger, J.C. Conduction of heat in solids. Oxford: Clarendon Press; 1959: p. 391. 7. Chopra, S.K.; Taplin, G.V.; Tashkin, D.T.; Elam, D. Lung clearance of soluble radio-aerosols of different molecular weights in systemic sclerosis. Thorax 34:63-67; 1979. 8. Crandall, E.D.; Kim, K.J. Transport of water and solutes across the excised bullfrog alveolar epithelium. J. Appl. Physiol. 50:1263-1271; 1981. 9. Deen, W.M.; Bridges, C.R.; Brenner, B.M.; Myers, B.D. Heteroporous model of glomerular size selectivity: Application to normal and nephrotic humans. Am. J. Physiol. 249:F374-F389; 1985. 10. Durbin, R.P.; Frank, H.; Solomon, A.K. Water flow through frog gastric mucosa. J. Gen. Physiol. 39:535-551; 1956. 11. Enna, S.J.; Schanker, L.S. Absorption of drugs from the rat lung. Am. J. Physiol. 223:1277-1231; 1972. 12. Fischer, P.; Glauser, F.L.; Miller, J.E.; Lewis, J.; Egan, P. The effects of ethchlorvynol on pulmonary alveolar membrane permeability. Am. Rev. Respir. Dis. 116:901-906; 1977. 13. Fischer, P.; Miller, J.E.; Glauser, F.L. The pulmonary alveolar capillary membrane (ACM) during hemorrhageic hypotension in dogs. Surg. Gynecol. Obstet. 146:383-386; 1978. 14. Gardiner, T.H.; Schanker, L.S. Effect of papain induced emphysema on permeability of rat lungs to drugs. Proc. Soc. Exp. Biol. Med. 149:972-977; 1975. 15. Gatzy, J.T. Paths of hydrophilic solute flow across excised bullfrog lung. Expt. Lung Res. 3:147-161; 1982. 16. Gatzy, J.T.; Boucher, R.C. Paths of hydrophilic solute flow across excised canine airways (abstract). Fed. Proc. 41:1244; 1982. 17. Glauser, F.L.; Miller, J.E.; Falls, R. Effects of acid aspiration on pulmonary alveolar epithelial membrane permeability. Chest. 76:201-205; 1975. 18. Gumbiner, B. Structure, biochemistry and assembly of epithelial tight junctions. Am. J. Physiol. 253 (Cell Physiol. 22): C749-C758; 1987. 19. Kawabe, H.; Jacobson, H.; Miller, I.F.; Gregor, H.P. Functional properties of cation exchange membranes as related to their structures. J. Coll. Int. Sci. 21:79; 1966. 20. Kim, K.J.; Crandall, E.D. Heteropore population of bullfrog alveolar epithelium. J. Appl. Physiol. 54:140-146; 1983. 21. Lippmann, M.; Yeates, D.B.; Albert, R.E. Deposition, retention, and clearance of inhaled aerosol particles. Br. J. Ind. Med. 37:337-362; 1980. 22. Man, S.F.P.; Hulbert, W.; Park, D.S.K.; Thomson, A.B.R.; Hogg, J.C. Asymmetry of canine tracheal epithelium: Osmotically induced changes. J. Appl. Physiol. 57:1338-1346; 1984. 23. Normand, I.C.S.; Olver, R.E.; Reynolds, E.O.R.; Strang, L.B. Permeability of lung capillaries and alveoli to non-electrolytes in the fetal lamb. J. Physiol. (London) 219:303-330; 1971. 24. Oberdorster, G.; Utell, R.; Morrow, P.E.; Hyde, R.W.; Weber, D.A. Bronchial and alveolar absorption of inhaled 99rnTc-DTPA. Am. Rev. Respir. Dis. 134:944-950; 1986. 25. Taylor, A.E.; Gaar, K.A. Estimation of equivalent pore radii of pulmonary capillary and alveolar membranes. Am. J. Physiol. 218:1133-1140; 1970. 26. Theodore, J.; Robin, E.D.; Gaudio, R; Acevedo, J. Transalveolar transport of large polar solutes (sucrose, inulin and dextran). Am. J. Physiol. 229:989-996; 1975. 27. Wang, H. Viscous flow in a cylindrical tube containing a line of spherical particles. Ph.D. Dissertation, Columbia University, New York, NY; 1967.
494
T. Chandra, L F . Miller, and D.B. Yeates
NOMENCLATURE C Co D
Do j~
Lp g~ P
Pexps Ptheors
R rp S
T X
V 6 AC
diameter of the diffusing solute (nm) concentration of the diffusing solute (kmol/m 3) = solute concentration in the surfactant layer (kmol/m 3) = solute diffusion coefficient within the pore (mZ/s) = free diffusivity of the solute (mZ/s) = area fraction of the large pores = volume flux (m3/mZ-s) = hydraulic conductivity (mZ/N-s) = amount of solute loss per unit area (kmol/m 2) = permeability of the tissue barrier to diffusing species (m/s) experimentally measured permeability (m/s) theoretical permeability (m/s) -- gas constant (N-m/kmol-K) = width of parallel-plate pore (nm) or diameter of cylindrical pore (nm) = cross-sectional area of the alveolar epithelial membrane available for diffusion (m 2) absolute temperature (K) = time after mtn time increment (s) = distance measured into epithelial membrane (m) = volume of the alveolar compartment (m 3) = thickness of epithelial layer (m) = concentration gradient across the membrane (kmol/m 3) sum of the squared errors for permeability -~- reflection coefficient of the diffusing solute =
=
O"
Subscripts:
1, L s,S
= large pore = small pore
Superscripts."
1 s
= large pore = small pore
