Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Effect of alveolar wall shape on alveolar water stability To To the the Editor: Editor: In In the the paper paper in in which which he he described described the the low low surface surface tension tension of of bubbles bubbles formed formed of of fluid fluid taken taken from from the the lung, lung, Pattle Pattle (5) (5) mentioned mentioned that that a low low surface surface tension tension at at the the lung lung surface surface is is required required to to keep keep the the lung lung dry. dry. Clements Clements (1) modeled the alveolus as a spherical surface and made quantitative estimates of the effect of surface tension on the fluid balance of the alveolus. Reifenrath (6) pointed out that this equilibrium would be unstable if the liquid-air interface were spherical and surface tension were constant, and he suggested that the polyhedral shape of the alveolus has a regulating and stabilizing effect. He argued as follows. The lowest pressure in the liquid lining occurs above alveolar wall intersections where the radius of curvature of the wall R is least and the pressure difference across the liquid-air interface T/ This total water flux must be zero for equilibrium. If R due to surface tension T is greatest. Fluid that enters the wall permeability, interstitial pressure, and surface a polyhedral alveolus would collect to form pools in the tension are all uniform over the surface, equilibrium is corners. This would increase the radius of curvature of the surface over the pool, increase the pressure in the possible only if liquid lining at that point, and reduce the water flux. (PIS - PALV) c + 2n T = 0 (2) This description of the stabilizing effect of alveolar shape It is unlikely that this particular relation between the has been repeated in recent review articles (3, 4). There is a question about this logic that is difficult to parameters would be satisfied, and it is especially unlikely answer by qualitative reasoning. As water is added to a that this relation would be satisfied at all states of lung pool in a corner, the pressure in the pool rises, but the inflation and vascular pressure. If there is a net flux of water into the alveolus, water area covered by the pool also increases. To what extent does the increase in the area that is exposed to low would collect at the points of greatest wall curvature pressure near the corner offset the decrease in pressure where the pressure in the liquid lining is least. The gasliquid interface would not coincide with the alveolar wall at the corner? An analysis of a simple two-dimensional model in which wall permeability and interstitial pressure surface, and the contribution of the surface tension term to the integral of Eq. (I) would be slightly different. A are assumed to be uniform leads to the conclusion that detailed representation of a pool of water covering a the fluid balance is unstable for a wall, such as Reifenrath region around a point of maximum wall curvature is pictured, that is concave toward the alveolus. However, the fluid balance for a wall that is convex toward the shown in Fig. 2. For convenience, the coordinate s is alveolar lumen, such as that near a fold where fluid pools ALVEOLAR WALL CIRCUMFERENCE WITH C Ah Wa INTERSTITIAL PRESSURE Ts FIG. 1. An idealized Z-dimensional 2-dimensional representation of an alveolus. Net flux of water into the alveolus is zero only if (PIS - PALV)C + 2nT = 0. 222 FIG. 2. A pool of fluid covering a region around a point of maximum wall curvature. Flux of water into the alveolus through segment of the wall covered by the pool is greater than it would be if there were no pool and increases with increasing pool size. A concave alveolus is therefore unstable. It either dries out or fills with fluid. 0161-7567/81/oooO-oooO$O1.25 Copyright 0 1981 the American Physiological Society Downloaded from http://jap.physiology.org/ by 10.220.33.1 on June 17, 2017 are observed in electron micrograph, is stable. An idealized two dimensional representation of an alveolus is shown in Fig. 1. The driving force for the flow of water into the alveolus is the pressure difference between interstitial pressure PIS and the pressure in the liquid lining of the alveolus. The pressure in the liquid lining layer is alveolar gas pressure PALv minus surface tension T times the curvature of the gas-liquid interface. The curvature is equal to dO/ds where 0 is the inclination of the alveolar wall and s is a coordinate along the alveolar wall. The total flux of water into the alveolus Q is the integral over the circumference C of the driving force times the permeability of the wall K. c d6’ = K PIS-PALV+T~ ds Q (1) I0 ( > LETTERS TO THE 223 EDITOR measured from the point of maximum curvature, and for small values of s the inclination 8, relative to the inclination at the origin, can be described by the equation 8 = as - bs2 (3) where a and b are constants. The pressure within the pool must be nearly uniform since the resistance to flow between points within the pool is relatively small. Therefore, the curvature l/R of the gas-liquid interface covering the pool is constant. The gas-liquid interface and the wall are assumed to be tangent at the edges of the pool. The pressure in the liquid layer is reduced below alveolar pressure by T/R, but the water flux occurs across the wall surface element ds. In the interval from -sl to sl, the surface tension term in Eq. 1 should be replaced by (T/R) ds. ” T 2T ds E =P = 2T81 + 2T[(sl/R) - O,] (4 ALVEOLAR SPACE An idealized representation of the geometry near a fold is shown in Fig. 4. The alveolar wall is represented as two circular arcs of radius c that meet to form a cusp at the origin. The liquid-air interface is again assumed to be an arc of constant curvature that is tangent to the wall at the edge of the pool. In this case, the integral over the region covered by the pool is again 2T81 + 2T[(sJR) - 811.The additional term 2T[(sJR) - &] is again positive. In terms of the parameters of the wall 7T 2T[(sl/R l micrograph of a rabbit point to fluid pools in h/4 - > 1 - COS(Sl/C) 5 (5) This additional term is a decreasing function of R and s1 so that, as the pool size increases, the tendency for fluid to enter the alveolus decreases, and a stable equilibrium is possible. If there are n folds around the circumference C, the equation for equilibrium is (PIS - PALV)C (6) h/a) 77 1 - cos(sl/a) - Z =0 The radius of curvature of the surface of the pool can lie between zero and infinity. As the radius of curvature and sl approach zero, the pool shrinks into the cusp of the fold. As R becomes large and s1 approaches 42, the pool fills the space between adjacent points of maximum wall height. A value of R and the corresponding sl/c that satisfies Eq. 6 can therefore be found for any value of PIS which satisfies the inequality PIS < PALV FIG. 3. Drawing of a transmission electron lung at 40% of total lung capacity (2). Arrows cusp-shaped crevices in alveolar surface. - 811= 2T T C - - The alveolar space can never dry out. A small pool of fluid exists in the cusp; no matter how negative interstitial pressure is, the pools cover the region of convex wall curvature, the stabilizing effect of convex curvature is lost, and the alveolus floods. The surface of the pools shown in Fig. 3 have a radius of about 0.5 ,um. The radius c of the wall near the fold is approximately 5 pm. From the relation, c/R = cos(sl/c)], sl/c is found to be 0.34. The cash/d / [l - Downloaded from http://jap.physiology.org/ by 10.220.33.1 on June 17, 2017 The first term 2T& is the value the integral would have if the liquid-air interface coincided with the wall. The additional term 2T[ (sl/R) - 011 is positive, since the length of wall covered . bY the pool is greater than the liquid-air interface on the surface of the pool. Furthermore, this term, which is approximately equal to Ta2bs14/6,increases with the size of the pool. The presence of water in the alveolus increases the tendency for water to enter the alveolus. If surface tension is constant, a stable equilibrium for water in the alveolar space is not possible for a concave alveolus. If (PIS - PALV)C + 2nT < 0, the alveolus dries out. If (PIS - PALV)C + 277T > 0, water enters the alveolus and forms pools that augment the water flux. If (PIS PALV)C + 27~T = 0, a slight perturbation that forms a pool of water leads to alveolar flooding. Transmission electron micrographs show folds and crevices in the alveolar walls that are partially filled with fluid. A drawing, patterned after an electron micrograph presented by Gil et al. (2) is shown in Fig. 3. Arrows point to fluid pools in the folds where the walls are convex inward. FIG. 4. A model for the geometry of the wall near a fold. Wall is represented as 2 circular arcs of radius c. Pool surface has a radius of curvature R and is tangent to the wall at the edge. Force for water flow into the alveolus decreases with increasing pool size. This geometry provides a stable fluid equilibrium for the alveolus. 224 folds occur on each side of a capillary and hence C/n is approximately 10 pm. If C is taken to be 200 r_crnand T is assumed to be 1 dyn/cm, PIS, as computed from Eq. 6, is 9 cmH20 less than PALL The value of Prs for alveolar flooding would be 0.4 cmHz0 below PALV. The contribution of the regions near the folds is significant. Osmotic pressure has not been included in the equations. If the concentrations of osmotically effective solutes are different on the two sides of the alveolar wall, the osmotic pressure difference should appear in Eq. 6. Interstitial pressure has been assumed to be uniform. Since the stresses are probably not uniform in the alveolar wall, fluid pressures may be nonuniform also and this would affect the analysis. Only the effect of alveolar wall geometry on fluid equilibrium has been included in the model. The objective of the letter is to point out that the regions of convex wall curvature provide a geometry that can produce a stable alveolar fluid balance. LETTERS This work was supported Institutes of Health. by Grant HL-21584 TO from THE EDITOR the National REFERENCES J. A. Pulmonary edema and permeability of alveolar 1. CLEMENTS, membranes. AMA Arch. Environ. HeaZth 2: 280-283, 1961. 2. GIL, J., H. BACHOFEN, P. GEHR, AND E. R. WEIBEL, Alveolar volume-surface area relation in air- and saline-filled lungs fixed by vascular perfusion. J. Appl. Physiol.: Respirat. Environ. Exercise Physiol. 47: 990-1001, 1979. 3. HOPPIN, F.B., JR., AND J. HILDEBRANDT. Mechanicalpropertiesof the lung. In: Bioengineering Aspects of the lung, edited by J. B. West. New York: Dekker, 1977, vol. 3, chapt. 2, p. 83-162. 4. MACKLEM, P. T. Respiratory mechanics. Am. Rev. Physiol. 40: 157184, 1978. 5. PATTLE, R. E. Properties, function, and origin of the alveolar lining layer. Proc. Ry. Sot. London 148: 217-240, 1958. R. The significance of alveolar geometry and surface 6. REIFENRATH, tension in the respiratory mechanics of the lung. Respir. Physiol. 24: 115-137, 1975. Downloaded from http://jap.physiology.org/ by 10.220.33.1 on June 17, 2017 Theodore A Wilson Department of Aerospace, Engineering, and Mechanics University of Minnesota Minneapolis, Minnesota 55455