Annals ofBiomedicalEngineering, Vol. 18, pp. 123-133, 1990 Printed in the USA. All rights reserved.

0090-6964/90 $3.00 + .00 Copyright 9 1990 Pergamon Press plc

Fractal Character of Pulmonary Microvascular Permeability J a m e s E. M c N a m e e Department of Physiology University of South Carolina School of Medicine Columbia, SC 29208 The pulmonary microvasculature offers a heterogeneous barrier to the motion o f large solutes as they pass between blood and lymph. While this barrier has been approximated by a f e w discrete pathways or by statistical ensembles o f many pathways, these descriptions only partly capture the structural and functional properties o f the pulmonary microcirculation. The concept that this barrier may be a fractal object is explored. Endothelial cleft geometry displays scaling in junctional path length and self-similarity in its spatial organization. It is shown that a fractal cleft produces heterogeneous spaces capable o f transporting water and macromolecules. Cleft location, size, and depth are characterized, in part, by a fractal dimension o f approximately O.8. The consequences for transport through a fractal barrier are then determined. Predicted sieving o f macromolecules by a fractal barrier is found to be consistent with lung microvascular transport data. Nonlinear transport phenomena are one consequence o f a barrier having a fractional dimension. Keywords-Mathematical model, Macromolecules, Dimension, Sieving.

INTRODUCTION A primary function of the human lung is to bring air and blood into intimate contact. The lung achieves this by leading both fluids through an intertwined network of bifurcating cylinders. At each branching generation a pair of smaller diameter cylinders divides the flow. After repeated division a 70 cm 3 volume o f blood, a small 3-dimensional object, is almost completely transformed into a thin film of blood, a large 2-dimensional object, which coats 70-100 m 2 of alveolar surface. In its passage through the pulmonary circulation, not only is blood spread into an unusual shape but this shape also possesses new properties. One of these properties is size selectivity. Airways allow the passage of gas molecules but remove most inhaled particles long before they reach the end of the respiratory tract. Similarly, the p u l m o n a r y microcirculation allows molecules and only the smallest formed elements in blood to reach the alveolar capillary. Successively dividing things into smaller pieces can produce objects having unusual forms and new properties. Repeatedly fracturing a 3-dimensional geometric shape in

Address correspondenceto James E. McNamee, Ph.D., Department of Physiology, Universityof South Carolina School of Medicine, Columbia, SC 29208. 123

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FIGURE 1. A branching respiratory " t r e e " made by recursive construction. The tree is composed of a trunk and two thinner perpendicular branches. At each stage of generation the trunks are left untouched and every branch become a trunk for the next generation. The figure has a dimension slightly less than 2,

a deterministic manner can create an object whose form approaches that of a 2-dimensional plane. Mandelbrot (6) has called objects which exhibit intermediate dimensions fractals. Figure 1, adapted from Mandelbrot, is a 2-dimensional projection of a 3-dimensional branching network. It illustrates how a cylinder can be fashioned into a plane-filling network of ever finer tubules. The similarity between this fractal curve and a photograph of the conducting airways or pulmonary arterial tree is crude but evocative. It suggests nature has found fractal geometry to be a useful way to selectively improve mass transfer. Not only does the lung take advantage of fractal geometry to improve gas transport, it may use a similar scheme to enhance or impede the motion of other solutes between blood and pulmonary tissue. The lung's microvascular barrier is permeable to many substances in addition to the respiratory gases. Perhaps this is necessary in order to provide a rich supply of substrates for lung cell metabolism or in order to maintain a vigorous defense against infection from airborne assaults. Regardless of its purpose, the unique geometry and unusual function of the pulmonary circulation have made description of its microvascular barrier difficult. This paper describes how the lung's blood-lymph barrier can be viewed as a fractal object and it will demonstrate some properties the pulmonary microcirculation may possess. Much of what follows may also apply to the microvascular barrier in other organs. LUNG MICROVASCULAR BARRIER DESCRIPTIONS Experimental Data

Knowledge about pulmonary microvascular barrier function comes largely from measurements o f the clearance of tracer molecules from the pulmonary circulation and from the steady state concentration of endogenous and tracer solutes in lung blood and lymph. Several investigators have measured transport through the lung's microvascular barrier for a variety of solutes (2,3,7,9,12,16). Despite varied experimental preparations and methodologies investigators from different laboratories agree

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on certain observations. Small hydrophilic solutes the size o f sucrose diffuse within seconds from blood to lung tissue. Large macromolecules traveling primarily by convection cross the lung's microvascular barrier with a half-time of hours. Molecules of intermediate size move by a combination of convection and diffusion and leave the circulation over the course of several minutes. As molecules leave blood they are hindered in proportion to their size and their charge (11). Agreement ends at describing where and how hinderance to transport occurs. Theoretical Models

Molecules differing in size by less than I nm move from blood to lung tissue as though they traversed a barrier of uniform porosity. But when a collection of molecules whose sizes span several nanometers cross the blood-tissue barrier, they move as though hindered by a heterogeneous structure (3). No homogeneous pathway can be found to explain the transport of plasma solutes in the lung (8,16). These same investigators developed mathematical models of transport o f plasma solutes through the lung's microvascular barrier (2,3,7,9,12,16). Despite similar data, investigators differ in the numbers, kinds and descriptions of pathways offered to explain solute hinderance. Experimental error may account for some of the diversity between characterizations of transport but the differences cannot be attributed to measurement error alone. At least two points o f agreement are found among their models. No less than two distinct pathways are needed to explain macromolecular transport (2,3,9) and at least one other is required to explain the motion o f smaller solutes (16). One reason for the divergence among descriptions may be that inappropriate models were forced to fit the data. Rather than being composed of a few uniform pathways acting in parallel, as these models assume, it could be that the pulmonary microvascular barrier is an ensemble of diverse paths. The possibility of structural heterogeneity and the failure of discrete pathway models to converge towards a consistent set of parameter values led to the hypothesis of a distribution of pathways in the endothelial junctions (8,16). A heterogeneous distribution o f transport pathways was postulated to arise from the complicated geometry of the intercellular cleft. The pulmonary vasculature is lined by a layer of endothelial cells. Water and hydrophilic solutes move from blood to surrounding tissues by traversing the cleft between endothelial cells. Clefts are intermittently joined by points of contact between apposing endothelial membranes. These punctate junctions are believed to restrict the free movement of fluid and solutes. A few hundred punctate junctions scattered irregularly along each micrometer o f cleft length are sufficient to break the space between endothelial cells into diverse pathways capable of sieving plasma macromolecules. Repeatedly breaking a space produces a set o f many smaller spaces. Fracturing spaces randomly produces a distribution of spaces whose sizes are described by a lognormal probability density function (1). The resulting lognormal distribution of pathway sizes is sufficient to account for the sieving of plasma macromolecules through the lung microvascular barrier (8,16). The consequences implied by this model for transport have not been explored.

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ONE-DIMENSIONAL F R A C T A L M O D E L OF L U N G MICROVASCULATURE Consider the luminal end of the endothelial cleft as a 1-dimensional line segment. Each time a point of contact forms between neighboring endothelial membranes it interrupts the line to form two slightly smaller segments. Repeated formation of cell-tocell punctate junctions along the length of the endothelial cleft transforms the single line into a set of many smaller segments. Isolated points of contact distributed randomly along the cleft cause the spaces between endothelial cells to assume a lognormal size distribution which is consistent with the sieving behavior of the pulmonary microcirculation. However, the modest number of submicroscopic punctate junctions needed to confer macromolecular selectivity to the endothelial barrier leaves more than 99~ of cleft length available for transport. In contrast, morphologic and physiologic measurements of lung and skeletal muscle microvessels suggest that as little as 8~ of cleft length is available for transport of even the smallest solutes (14). Furthermore, Wissig has shown by freeze fracture examination of the endothelial cleft that there is a clear pattern of junction arrangement paralleling the luminal cleft border (18). Individual junctions are grouped into serpentine curves resembling strands of beads. Strand junctions are of variable length and may be continuous or interrupted. The irregularly placed punctate junction model of the pulmonary endothelial cleft overestimates the length of cleft available for transport and it is incapable of accounting for the organization of punctate junctions found within the barrier. A more suitable description should accommodate the 1-dimensional organization of the cleft and allow access of molecules to only a limited portion of cleft length. One such model is presented in the following example. Let the length of a cleft between two endothelial cell be represented by a pair of parallel line segments 25/zm long. Space between the lines represents the region through which solute and solvent exchange can occur. Allow a tightly spaced string of punctate junctions to form a strand which joins a portion, say 1/6, of the cleft. The strand restricts the movement of water and solutes to the remainder of the cleft. This is represented diagrammatically by the black bar in the topmost part of Fig. 2. The two smaller spaces created by the strand become sites for formation of second generation strands. Each new generation fractures the remaining spaces. Repeating this process indefinitely yields a collection of discrete points that form a Cantor set. Although individual points have a Euclidean dimension of 0 the set is a fractal object that has a dimension of log(2)/ log(12/5). In practice, it is impossible to endlessly divide the endothelial cleft. The biologically realizable collection of cleft spaces represents a finite portion of a Cantor set. It is unlikely that strands are created or placed with precise regularity. When the location of a new strand becomes a matter of chance, the size of each space varies randomly obeying a lognormal probability density function. The logmean and log standard deviation of the probability function depend on how frequently spaces are divided and how uniformly the division is made. The fraction of cleft length occupied by spaces is proportional to the fractal dimension of the set of spaces. It is possible to determine the properties of the endothelial cleft described by this example. Let division of the cleft proceed until the size of a space to be occupied by a new strand becomes smaller than the size of a punctate junction. A +25~ varia-

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[

glll FIGURE 2. Successive generations in joining apposing endothelial membranes within a cleft. Onesixth of the remaining open space (white) in the previous generation is occupied by a strand of punctate junctions (black) in the current generation. The number of spaces and strands doubles and their size is reduced by a factor of 5/12 with each generation.

tion in the location of successive strands within a space produces a distribution of interendothelial pathways having a logmean of - 0 . 1 and a log standard deviation o f 1.06. These values correspond closely to those describing the sieving characteristics of the pulmonary microvascular barrier, - 0 . 2 and 1.08, respectively (8). The fractal dimension of the set of spaces in a 1-dimensional cleft model can be estimated by tiling the luminal opening of the cleft with N line segments each of length L so that all spaces between the strands are covered. The tiling process is repeated using successively shorter line segments and new values of N are determined. The relation between N and L is given by: N = L -D

(1)

where D is the H a u s d o r f f Besicovitch or fractal dimension o f the set o f spaces (6). A cleft constructed using the rules described above has a fractal dimension of approximately 0.79. Spaces through which transport of water and small solutes could occur account for approximately 10~ of cleft length. A 1-dimensional fractal model of the endothelial cleft is consistent with the structure of the pulmonary microvascular barrier. The fractal model is also consistent with the diversity of transport pathways implied by macromolecular sieving data and with the limited availability o f cleft length for the transport of smaller solutes. A MULTIDIMENSIONAL FRACTAL DESCRIPTION Morphologic measurements suggest that not only are intercellular pathways fractal in one dimension (i.e., in terms of their luminal openings) but that their fractal qualities may extend into other dimensions. The depth of the endothelial cleft has a fractal dimension of 1.11 (8). This observation implies that microvascular transport pathways should be viewed as fractal objects o f higher dimension. Strands of punctate junc-

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FIGURE 3. Left. Schematic diagram of the luminal surface of the pulmonary microcirculation showing adjacent cells and an endothelial cleft (shaded area), Right. A perpendicular section through the cleft revealing strands of punctate junctions (heavy lines) between adjacent cells.

tions are layered within the cleft bending freely, approaching or intersecting other strands. Electron microscopists observe that nearly all endothelial clefts in continuous capillaries appear to be fused at one or more points throughout their depth. Straight routes of passage through the depth of the endothelial junction are rare. In an effort to determine the 3-dimensional structure of cleft pathways, Bundgaard (4) prepared ultrathin serial sections through capillary endothelial junctions of rat cardiac muscle. By tracking the location of points of membrane apposition through successive sections he was able to trace the course of strands and spaces along the length and depth of a cleft. Figure 3 schematically depicts how one portion o f a endothelial junction might look. The panel on the left shows a face view of microvascular wall as seen from the center of a capillary. The cleft is represented by the shaded region between endothelial cells. On the right, the cleft has been cut by a plane running parallel to the lateral margin of the endothelial cells and perpendicular to the plane of the page. This section is magnified to show strands which join adjacent endothelial cell membranes in the interior of the cleft. Bundgaard demonstrated that strands form staggered lines of contact between endothelial cell surfaces. They obstruct the radial movement of macromolecules from plasma to interstitium. It is unlikely that a transverse section will avoid intersection with strands in all layers simultaneously. For this reason intercellular junctions appear to be tightly sealed at one or more places in randomly sampled electronmicrographs (14). While strands may divert diffusion and convection that need not halt solute and water exchange through the cleft. Although the work of Wissig and Bundgaard pertain to skeletal and cardiac muscle capillaries, their results may also be applicable to lung microvessels. Perry (14) showed that the structural features of the cleft region including the length, width, depth and appearance of tight junctions for pulmonary capillaries are similar to those in skeletal muscle. CONSEQUENCES OF F R A C T A L S T R U C T U R E Considered in one dimension the endothelial cleft can be characterized as a set of point-like spaces whose fractal dimension is approximately 0.79. In view of the junc-

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tional organization known to exist within the intercellular region a higher dimensional model of the cleft consists of a plane whose length and depth coincide with the size and shape o f the apposing endothelial cell surfaces. This 2-dimensional plane is traversed by fractal strands of the type described above. To simplify analysis, let those strands lie in the plane, parallel to the luminal end of the cleft and let each one be a replica, on a smaller scale, of its luminal neighbor. This construction creates a barrier composed of ever smaller spaces. The barrier and each of its layers is self-similar. Transport through such a barrier is governed by the local Kedem-Katchalsky fluid, Jv, and solute flux, Js, equations:

(2)

Js = - w R T [ ~ x ]

+(1-o)CJv.

(3)

Both fluid volume and solute mass fluxes are conserved through each layer. Therefore, fluxes must be equal from one side of the cleft to the other and from one barrier to the next. Patlak et al. (13) solved these equations for the case of two dissimilar barriers bounded by unstirred regions of fluid. The solution is, in general, nonlinear and exhibits the property flux rectification. In the present study a slightly different set of boundary conditions were chosen to approximate the environment of the pulmonary microvascular barrier under conditions o f a sieving experiment: luminal or upstream solute concentration, Co, and abluminal or downstream hydrostatic pressure were assumed to remain constant while upstream hydrostatic pressure was set at various constant values. Writing and integrating these local equations for a barrier composed of a single layer of strands give expressions for volume flux and the steady state concentration o f solute leaving the downstream layer of the barrier (Cl) as a function o f upstream solute concentration, Co, layer transport coefficients (~o and of) and driving pressure: CI = otCo + 3

(4)

where 3 = ( J s / P ) ( 1 - a ) / P e , o~ = c pe, Pe = ( 1 - a f ) J v / P , P = o~RT/~ and 6 is the thickness of the layer. For a set of N layers (0, 1, 2 . . . . N - 1) separated by N - 1 well-mixed spaces the solute concentration downstream from the ith layer, Ci+l, becomes:

Ci+l = aiCi + 3i

(5)

where Ci is the upstream solute concentration and, as before, O~i and ~i are functions of ith layer's transport parameters and overall solute and volume fluxes. An interative quality begins to appear in the form o f the transport equations. Solute concentration is repeatedly scaled (ct) and translated (/3) as it traverses each layer o f the barrier.

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In the limit as Pe becomes very large in all layers, the concentration of solute in the well mixed space downstream from the ith layer approaches:

(1 - ao)

Ci=Co (1-ai)'

0
1

(6)

and CN = Co(l - o0).

(7)

The solid curve in Fig. 4 illustrates the results of solving these equations numerically for the sieving of a 1-mM albumin solution by a two-layer barrier. The reflection coefficient for albumin of the upstream layer is 0.55 and the downstream layer's reflection coefficient is 0.92. The value of Lp used for each layer was 33 • 10 -5 ml. s - ~ . m m H g -1 and the values of ~0RT for the upstream and downstream layers were 100 • 10 -5 and 8 x 10 -5 ml.s -1, respectively. Albumin concentration as it leaves the downstream layer is plotted as a function of transmembrane hydrostatic driving pressure. For reference, the dashed curves indicate the albumin sieving that would be produced by a barrier composed of only each single layer. Three phases of sieving behavior can be demonstrated by this multi-layer barrier. At very low sieving pressures, it acts like a single homogeneous membrane having the selectivity of the more restrictive layer. Solute concentration between layers gradually increases as driving pressure is increased. Diffusion is the principal driving force for transport at low pressure. This is illustrated in Fig. 4 for transmembrane pressures less than 5 mmHg.

e= 0.8

=

~

o

o

do

Transmembrane

5'o

8'o

Pressure

1 .o (mmHg)

FIGURE 4. Sieving as a function of hydrostatic driving pressure for a 1 mM albumin solution through a two-layer barrier (solid curve). The reflection coefficients of the upstream and downstream layers are 0 . 5 5 and 0.92, respectively. Dashed curves depict sieving of each layer in isolation.

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As filtration pressure increases sieving departs from the highly selective behavior of the downstream layer. Albumin concentration leaving the barrier reaches a minimum at a transmembrane pressure near the colloid osmotic pressure of the upstream solution. In this phase, solute concentration between layers exceeds the upstream concentration. Convection becomes dominant for the upstream layer while an enhanced solute concentration gradient at the downstream layer augments diffusive transport across that portion of the barrier. At transmembrane pressures above 30 m m H g the concentration of albumin leaving the barrier begins to increase. Once pressures exceed 125 mmHg albumin concentration plateaus at a value determined by the sieving coefficient of the upstream layer. In the limit when Jv becomes large and convective transport predominates in both layers, the concentration of solute leaving the barrier approaches the sieving limit of the upstream or first layer, as given by Eq. 7, not the sieving limit of the downstream or more selective layer. As long as plasma oncotic pressures are 20-25 mmHg, increasing lung microvascular hydrostatic pressure to as high as 40 m m H g is likely to elicit only the first two phases of multilayer barrier behavior. Interestingly, some investigators may have elicited third phase response from the pulmonary microvasculature in an indirect manner. Kramer et aL (5), after decreasing plasma protein concentration in sheep by means of plasmapheresis, measured increased fluxes of macromolecules and calculated a reversible increase in hydraulic conductivity through the pulmonary microvascular barrier. They interpreted their findings as evidence o f "pore stretching" and reversible lung injury. Their findings may alternatively be explained by noting that lowering upstream concentration of macromolecules to a multi-layer barrier moves the curves of Fig. 4 leftward. Downstream solute concentrations, if previously at a minimum, will increase. It also results in a larger Jv in response to any net filtration force. These results give the impression that the pulmonary microvasculature has been disrupted or reversibly injured by a fall in plasma protein concentration or oncotic pressure even though the structure o f the microvascular barrier remains unaltered. EVIDENCE OF F R A C T A L F U N C T I O N IN T H E PERIPHERAL MICROCIRCULATION The pulmonary microcirculation lacks the support of solid tissue structures found in other vascular beds. Consequently, lung microvessels may suffer irreversible damage when perfusion pressures exceed 65 mmHg (15). Muscle microcirculation is able to withstand higher pressures without hemorrhage. It should be possible to elicit the effects of multi-layer barrier geometry more easily from peripheral microvascular beds. Wolf et al. (19) measured the sieving of albumin from plasma to tissue in the isolated, perfused cat hind limb as a function of venous pressure. Using a mass balance technique they calculated the osmotic reflection coefficient, af, and hydraulic conductivity, Lp, for the microvasculature of cat skeletal muscle. Their data for af and Lp are plotted as open circles in Fig. 5. At venous pressures up to 40 m m H g af remained constant at approximately 0.8. As venous pressure was increased further albumin moved more readily from plasma to tissue and calculated af fell. Others have observed a decreased selectivity for macromolecules in response to increased venous pressure and attributed it to "pore stretching" or damage of the microvascular barrier (17). When venous pressure was

132

,I.E. McNamee 1.0 .

m

o.g.

e ro

0.8.

ro

0.7.

o

0.6.

0

0

0.5

0

0.02

1

0

u

u

u

~

0 On u

0.00 _I I 40

I

I 80

I

I 80

Venous Pressure (mmHfK) FIGURE 5. Osmotic reflection coefficient and hydraulic conductivity in the cat hindlimb as functions o f v e n o u s pressure. Open circ|es are t h e d a t a o f W o l f e t a l . (19). Solid c u r v e s represent t h e values predicted by a two-layer barrier.

returned to normal a/also returned to its control value. Had the microvascular barrier been damaged or irreversibly stretched, ~/should have remained below its control value. Furthermore, despite a substantial fall in of, Lp remained nearly constant while venous pressure rose to 80 mmHg. A "stretched" pathway should have produced a measurable increase in Lp (10,15) in conjunction with the falling reflection coefficient. Their observations are not consistent with the concept of a stretched pathway. The solid curve at the top of Fig. 5 is the predicted osmotic reflection coefficient produced by a two-layer barrier whose luminal reflection coefficient for albumin is 0.40 and whose abluminal reflection coefficient is 0.88. The solid curve at the bottom of Fig. 5 is the empirical Lp of this barrier, calculated as the ratio of Jv to net filtration pressure. Lp remains almost constant over this range of pressures. The behavior of a multi-layer fractal barrier is consistent with the sieving of albumin observed by Wolf et al. (19) for intravascular hydrostatic pressures up to 80 mmHg. SUMMARY The pulmonary microcirculation appears to exhibit both fractal structure and fractal function. Endothelial cleft geometry displays scaling in junctional path length and self-similarity in its spatial organization. A fractal structure can produce heterogeneous spaces capable of transporting water and solutes within the endothelial cleft. Cleft spaces vary in number, size and depth and are determined, in part, by the fractal

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d i m e n s i o n o f t h o s e spaces a n d t h e r e g u l a r i t y w i t h w h i c h s t r a n d j u n c t i o n s a r e f o r m e d b e t w e e n e n d o t h e l i a l ceils. S i e v i n g o f m a c r o m o l e c u l e s b y t h e p u l m o n a r y m i c r o v a s c u lar b a r r i e r is c o n s i s t e n t w i t h a s p e c t r u m o f t r a n s p o r t p a t h w a y sizes o r i g i n a t i n g f r o m repeated strand junction formation. Analyzing microvascular morphometry and transport characteristics using fractals allows n e w i n t e r p r e t a t i o n o f existing d a t a a n d r e c o n c i l i a t i o n o f p r e v i o u s l y u n e x p l a i n e d experimental observations. A fractal approach can suggest new experiments which may reveal important insights about the function of the pulmonary microvascular b a r r i e r in h e a l t h a n d disease.

REFERENCES 1. Aitchison, J.; Brown, J.A.C. The lognormal distribution. Cambridge, MA: Cambridge University Press; 1957. 2. Blake, L.; Staub, N.C. Pulmonary vascular transport in sheep: a mathematical model. Microvasc. Res. 12:197-220; 1976. 3. Boyd, R.D.H.; Humphries, J.R.; Normand, I.C.S.; Reynolds, E.O.R.; Strang, L.B. Permeability of lung capillaries to macromolecules in foetal and newborn lambs and sheep. J. Physiol. (London). 201:567-588; 1969. 4. Bundgaard, M. The three-dimensional organization of tight junctions in a capillary endothelium revealed by serial-section electron microscopy. J. Ultrastruct. Res. 88:1-17; 1984. 5. Kramer, G.C.; Harms, B.A.; Bodai, B.I.; Renkin, E.M.; Demling, R.H. Effects of hypoproteinemia and increased vascular pressure on lung fluid balance in sheep. J. Appl. Physiol.: Respirat. Environ., Exercise Physiol. 55:1514-1522; 1983. 6. Mandelbrot, B.B. The fractal geometry of nature. San Francisco, CA: W.H. Freeman and Co.; 1983. 7. McNamee, J.E. Restricted dextran transport in the sheep lung lymph preparation. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 54:914-918; 1983. 8. McNamee, J.E. Distribution of transvascular pathway sizes through the pulmonary microvascular barrier. Ann. Biomed. Engg. 15:139-155; 1987. 9. McNamee, J.E.; Staub, N.C. Pore models of the sheep lung microvascular barrier using new data on protein tracers. Microvasc. Res. 18:229-244; 1979. 10. Nicolaysen, G.; Waaler, B.A.; Aarseth, P. On the existence of stretchable pores in the exchange vessels of the isolated rabbit lung preparation. Lymphology 12:210-207; 1979. 11. Parker, J.C. Transvascular clearance and distribution of charged macromolecules in ANTU lung injury. J. Appl. Physiol. 60:1221-1229; 1986. 12. Parker, J.C.; Parker, R.E.; Granger, D.N.' Taylor, A.E. Vascular permeability and transvascular fluid and protein transport in the dog lung. Circ. Res. 48:549-561; 1981. 13. Patlak, C.S., Goldstein, D.A.; Hoffman, J.F. The flow of solute and solvent across a two-membrane system. J. Theor. Biol. 5:426-442; 1963. 14. Perry, M. Capillary filtration and permeability coefficients calculated from measurements of interendothelial cell junctions in the rabbit lung and skeletal muscle. Microvasc. Res. 19:142-157; 1980. 15. Rippe, B.; Townsley, M.; Thigpen, J.' Parker, J.C.; Korthuis, R.J.; Taylor, A.E. Effects of vascular pressure on the pulmonary microvasculature in isolated dog lungs. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 57:233-239, 1984. 16. Roselli, R.J.; Parker, R.E.; Harris, T.R. Comparison between pore model predictions and sheep lung fluid and protein transport. Microvasc. Res. 29:320-339, 1985. 17. Shirley, H.H., Jr.; Wolfram, C.G.; Wasserman, K.; Mayerson, H.S. Capillary permeability to macromolecules: stretched pore phenomenon. Am. J. Physiol. 190:189-193; 1957. 18. Wissig, S. Identification of the small pore in muscle capillaries. Acta Physiol. Scand. (suppl.) 463:3344; 1979. 19. Wolf, M.B.; Porter, L.P.; Watson, P.D. High venous pressure increases water and protein permeability in the cat hindlimb. Am. J. Physiol. 26:H2025-H2032; 1989.