Abstract
Butler, J. P., E. H. Oldmixon, and F. G. Hoppin, Jr.Dihedral angles of septal “bend” structures in lung parenchyma. J. Appl. Physiol. 81: 1800–1806, 1996.—Alveolar parenchyma comprises two interacting tensile systems: the cable system (a network of linear condensations of connective tissue) and the membrane system (a network of quasiplanar alveolar septa). Inferences can be drawn about the mechanics of this structure from its configuration. We reported earlier (E. H. Oldmixon, J. P. Butler, and F. G. Hoppin, Jr. J. Appl. Physiol. 64: 299–307, 1988) that the angles between alveolar septa at the common threeway junctions (J) are nearly uniform, indicating that septal tensions are also nearly uniform. We now report on the interseptal angles at the next most common class of septal junction (B), a structure where two septa meet along a segment of the cable system. We find, first, that the distributions of interseptal angles at B junctions have means >120°, are narrow, and have few, if any, angles <120°. The findings of uniform 120° angles at J junctions and a cutoff below 120° at B junctions are also characteristic of soap films supported on a frame, which follows the physical principle of surface area minimization. We suggest that this principle may be operative in parenchymal development and remodeling.
 stereology
 microanatomy
 connective tissue
 tension
 micromechanics
tensions in septa and connective tissue are essential to the arrangement and stability of the gasexchanging surfaces of alveolar parenchyma, the distribution of gas and ventilation within the lung, and the mechanical behavior of the intrapulmonary vasculature and airways. The roles of the various tensionbearing components and the patterns of tensions within the parenchyma, however, are not as yet fully understood.
The configuration of parenchyma at the level of alveoli and alveolar septa may provide a window into some of these issues. Like a parachute, parenchyma maintains its configuration and function by tensions in its parts. Its structure can be considered as comprising tensed cables and membranes. The membranes are the approximately planar alveolar septa. Each septum bears tension in its two airliquid interfaces, in its framework of fine elastin and collagen, and probably in its basement membranes. The cables are linear curved bundles of elastin and collagen, with varying amounts of smooth muscle, interconnecting to form a network comprising alveolar entrance rings (which bound the alveolar duct) and branches (which circle out to the outer margins of the alveolar ductal unit). It is important to recognize that we are dividing the stressbearing features of the parenchyma along the functional lines of belonging to either the membrane or cable class. Thus connective tissue is certainly present in both the septa and the cables; that portion of connective tissue in the septa is lumped together with surface tension forces at the airliquid interface in determining the net septal tension, whereas the condensed portion of the connective tissue tenting pairs of septa is defined to be strictly within the cable system.
The interaction between these membrane and cable systems in canine lungs has been described partially in terms of the relative total lengths of classes of linear features along which their components meet (Fig. 1) (8). The “junction” ( J) class has the greatest (45%) total length of linear features in the parenchyma; a J feature is a linear junction along which three septa join. Next in total length (25%) is the “bend” (B) class; a B feature is a linear junction of two septa, meeting at a distinguishable angle. The “end” (E) class has the third greatest total length (19%); an E feature is a linear border of a single septum, typically the free end of a septum at the alveolar entrance. The J features do not incorporate cables (9) and thus lie entirely within the membrane system. By contrast, the B and E features always incorporate a cable system segment, and cables are seen almost exclusively in these features. Being curved and tense, these cables provide stress equilibrium to the supported septa. Thus the Bs and Es are the exclusive sites of mechanical interaction between the cable and membrane systems.
In a previous publication (6), we noted that the dihedral angles between the three septa that meet at the J features are remarkably uniform around 120°, which implies that tensions in the adjoining septa are similarly uniform. In this paper, we report on the dihedral angles between the two septa that meet at the B features. The inferred distributions of dihedral angles are again narrow but here they are clustered >120°, with few, if any, angles <120°.
Together, these findings support a mechanism of minimal surface area and locally uniform septal tension. The configurations are strikingly similar to what is seen in a relatively simple, wellunderstood physical system; namely, soap films supported on a framework, e.g., of wire or string. We suggest that such physical mechanisms in parenchyma may operate during the development and growth of the parenchyma and, conceivably, assist in maintaining parenchymal configuration in the adult lung.
MATERIALS AND METHODS
Animal protocol, tissue preparation, and angle measurements.
The data to be presented were obtained from the same samples previously reported in the study of J features (6). In brief, four dogs were anesthetized with pentobarbital sodium, intubated, and ventilated on ambient air. Heparin was administered intravenously. The chest was opened with the lungs inflated. Each lower lobe was cannulated for closedcircuit intravascular perfusion via the pulmonary artery and left atrium, dissected free, weighed, and displaced for determination of volume. The lungs were inflated three times from airway pressure of 2 to 30 cmH_{2}O and deflated as necessary to the target volume. The lobar gas volume at 30 cmH_{2}O was defined as Vl _{30}. Each lobe was held at 1.0 Vl _{30}, 0.8 Vl _{30}, 0.6 Vl _{30}, or 0.4 Vl _{30}while it was perfused, in a procedure designed to preserve alveolar configuration (5), with lactated Ringer solution supplemented with 3% dextran, glutaraldehyde, osmium tetroxide, uranyl acetate, and a dehydrating gradient of ethanol. Lactated Ringer rinses were used between fixatives and a 0.9% saline rinse before the ethanol gradient. Samples were cut at increasing depths beneath the pleura, then held in absolute ethanol, transferred to propylene oxide, and embedded in Spurr’s epoxy. Sections 1.6 μm thick were cut, stained for transmission light microscopy with phloxine B and methylene blueAzur II, and viewed with 454nm yellowgreen light. This emphasizes connective tissue cables and counterstains the remaining tissue (7). Photomicrographs were taken at ×512 magnification on finegrained monochrome negative film, starting at an arbitrary location on a section and skipping alternate fields in directions of height and width. The frames were backprojected onto a frosted screen for a ×580 final magnification.
Locations where two relatively straight septa came together at a welldefined and measurable angle were identified as traces of B features. All B feature traces seen in a systematic scan of each field were identified. If one septum meeting a B extended beyond the screen edge, that B was passed over. Angles were measured from the onscreen projections with a goniometer (6). Roughly 1,250–1,300 angles in five or more samples were measured from each lobe.
We recognize that the identification of B features is potentially ambiguous. This may become a problem when such features are randomly sectioned, because the traces that are seen for any single dihedral range from very large angles to very small ones. The former can appear as additional waviness on section and are easily missed. The latter can masquerade as E features. Both of these cases are addressed in the methodology of accounting for “censored” data, described below, wherein we make use of known distributions of angles seen on random sections and estimate quantitatively the effect of missing both high and low angles. As shown below and in Table 2 the actual magnitude of the fraction of missing angles is <5% in all cases studied. An additional ambiguity may arise at the transition of a cable from an E configuration to a B configuration (e.g., in the EBJ node illustrated in Fig. 4). Generally, the neighborhood of nodes where linear features join is small compared with the length of the features themselves and, therefore, this ambiguity has little impact on the conclusions. That the node neighborhoods are small is verified independently by results for other types of nodes (8). In that work, it was shown that a JJJJ node (which class supplies 26% of parenchymal nodes) may be sectioned to produce a septal X trace but that X border traces are only 0.9% of all border traces, or ∼1.8% of J border traces. Another node, BBJJ, supplies 12% of parenchymal nodes and may show a T border trace when sectioned, but Ts provide only 0.9% of border traces. Thus it is reasonable that the transitional zone into a presumably comparably small neighborhood of an EBJ node should be small as well.
Stereology.
The stereologic theory that permits us to draw inferences about the underlying dihedral septal angles (α) from the random angles (A ) of their traces seen on random section is developed at length in earlier papers (1, 6, 10). We summarize it briefly here, with particular attention to its application to the B structures. ( contains errata for Ref. 6.)
Each observed A may be equal to, greater than, or less than the true α for that dihedral, depending on the orientation of the section to the septa; moreover, the αs are themselves a distributed quantity. Recovering the full distribution of α is very difficult, but its mean and variance can be estimated by application of the following principles. First, the mean of A is equal to α; more generally, for a distribution of αs, the mean A is equal to the mean of α weighted by the lengths of the B features (1). Second, the variance can be estimated from the observed “smearing” (6) of the cumulative distribution function (cdf ) for a single α (10). Just as in the cdf values for the trihedrals at J features (6), comparison of the observed cdf with the cdf for a single α (where α is the mean A ) shows only modest differences, suggesting that the true distribution of α is narrow. In particular, the frequency density of A for a single α is logarithmically infinite at the point where A = α and, therefore, the slope of the cdf is infinite at that point. The slope of the observed cdf thus can be used to infer the net variance (Var_{net}), which in this case is given by the sum of the variance of α (Var_{α}) and the instrument and operator variance [Var(A )_{o}]. The desired quantity Var_{α} is thus given by
Dealing with the censor, and estimation of
α ¯
and Var_{net}.
Of particular concern is the possibility of systematic error introduced by failing to record angles that are either too small or too large to be identified on section. Such a censor must operate, since both high and low angles are underrepresented in our data; the cdf of the observed angles (A ) lies below the limiting curve of a single α at low angles and above it at high angles. The methodology by which we estimate the mean α
Briefly, the method is to estimate the censored fraction (i.e., the fraction of angles missing from the data) by direct comparison of the observed cdf measured [F
_{ob}(A )], with the cdf Φ(A, α) that would obtain from a single true underlying dihedral angle given by α. The censored fractions are separately estimated for low and high angles by the locally maximal difference of the two cdf curves, since in the absence of a censorF
_{ob}(A ) cannot be less than Φ(A, α) at low angles nor can it be greater than Φ(A, α) at high angles. This fraction is then used to correct the observed mean angle
Specifically, let the total censored fraction m be written asm
_{lo} + m
_{hi}, wherem
_{lo} is the fraction of angles censored at low angles and m
_{hi} is the fraction of angles censored at high angles. Further, let
The first equation shows the relationship between the slope sof the observed cdf and the SD of the underlying distribution of α; this follows from the logarithmic singularity in the density function for A
The second equation describes how the observed slope of the cdf departs from the actual slope due to the censor. This follows from the normalization requirement that cdf values range from 0 to 1. Thus
The third and fourth equations are the means by which we estimate the fraction of angles censored. This is the same method as used in Ref. 6but computed separately into low and high regions. These equations also implicitly define A
_{hi} andA
_{lo}, the “typical” size of angles characterizing where the censorship occurs
The fifth and final relationship is the effect of the censor on the estimate of
RESULTS
Table 1 displays the raw statistical data for all studies. The wide SD values are mostly associated with the random sectioning rather than with an underlying distribution of α. The computation procedure outlined above results in estimates of the fraction of angles censored (m = m
_{lo} + m
_{hi}), the corrected mean observed angle that estimates the mean underlying dihedral angle
The mean dihedral angles
The SDs of the distributions of angles (ς_{α}), as estimated by Eq. 1, with Var_{net} given by the above procedure, are shown in Fig. 3. The distributions are narrow, particularly at high lung volumes. ς_{α} ranges from ∼6° at 0.4 Vl
_{30} to ∼3° at 1.0 Vl
_{30}. As described in detail below, an angle of 120° plays a very special role in the minimization of surface energy when such energy is monotonically associated with surface area. Note that our results for
DISCUSSION
The findings of this study and the earlier report (6) are strikingly similar to what is seen in an opencell array of soap films suspended on a frame of, e.g., wire or string. First, the topology is similar. At their borders, three soap films may meet to form a J feature, two films may meet along an element of the supporting frame to form a B feature, or one single film may adjoin an element of the supporting frame to form an E feature. These, as noted above, are the dominant features of alveolar parenchyma.
Second, the dihedral angles at the J and B features are similar in the soap film and parenchymal structures. In soap films, the trihedral J features have dihedral angles of 120° (see the work of Plateau regarding 120° angles and tetrahedral angles among edges, accessibly and engagingly recounted in Ref. 11). The B features, where two films meet, supported by the frame, may have dihedral angles >120°, but angles <120° are not seen. These configurations reflect the principle that the films assume the local least energy configuration and two properties of soap films, namely, 1) that the tension in the films is uniform (i.e., the same everywhere) and 2) that the surface area is not conserved but can be created or destroyed. It follows from property 1 that minimization of the total energy associated with the surface is equivalent to minimization of surface area, and this is achieved locally at a trihedral junction (J) where three films meet with common angles of 120°. A B feature with dihedral angles >120° also has locally minimal surface area. On the other hand, a B feature in which the two films meet at <120° is unstable. Its configuration is not of minimum area and, because ofproperty 2, the junction of the two films can drop away from the supporting frame, with the development of a new film. This new configuration has a lower surface area and thus lower surface energy than the original. Illustrating this evolution, both in supported soap films and parenchyma, one may find the configuration shown in Fig.4, where a J structure with three dihedrals of 120° is continuous with a B structure with a single dihedral >120°. Note that the cable in the E feature associated with the original J continues as the curved cable in the B necessary to support the septal tensions. The location where the E, B, and J features meet has been called an EBJ node, and its frequency in parenchyma has been estimated (8).
Our earlier data were consistent with a narrow range of angles of ∼120° at the J features (6). Our present data for the dihedral angles at B features are consistent with the presence of angles >120° and an effective cutoff of angles <120°. The mean dihedral angle
where
Because P(X < −t ),t > 0 satisfies the same probability law, we may rephrase our results in the more general context of unimodal distributions by saying that fewer than ∼10% of B features may violate the critical angle of 120°. This bound is much less at high lung volumes. Figure5 shows these upper bounds for the separate data sets.
Observe that the measure of the departure of
Is it reasonable to think that alveolar parenchyma has the two requisite properties to behave as a minimalarea system 1) local uniformity of septal tension and 2) the ability to create or destroy surface area? The septum has three major tensionbearing structural components; namely, the two airliquid interfaces, the epithelial and endothelial basement membranes, and the finely woven connective tissue in the interstitial space. Net septal tension is the sum of these contributions and has been observed to be remarkably uniform among adjacent septa over the entire vital capacity (6). Consider the contributors separately. The tension in the airliquid interface likely has the requisite properties. Tension is likely to be uniform locally because of the spreading of surfactant and, over time, to be independent of the creation or destruction of surface due to the biological regulation of the surface concentration of surfactant. The solid tissue components of the septum are likely also to bear significant tensions over the range of lung volumes studied (9); might these tensions be uniform? One possibility is that the airliquid interface, by virtue of the uniformity of its tensions, would continually bias the membrane system toward that of uniform total tension and that tissue components would grow during postnatal development or remodel to conform progressively to that configuration. Another possibility is that the solid tissue components develop with uniform tensions from the start. The nature of basement membranes is generally to cover surfaces smoothly. We speculate that their development is likely to be by planar growth in response to traction, resulting in uniform tension. Similar behavior by the fine connective tissue seems conceivable. Uniformity of tensions could arise prenatally in the solid tissues of the septa, before the airliquid interface and its associated surface tension are established. In this context, it is important to note that the two requirements considered here do not involve a particular quantitative level of stresses or tensions. The absence of an airliquid interface in the fetal lung certainly implies a much lower state of septal tension than in postnatal lung, but even very small stresses, if spatially uniform and under conditions of being able to modulate tissue area, suffice to ensure at least an approximate satisfaction of the minimumarea principle and, in consequence, a structural configuration consistent with pulmonary parenchyma.
Departures of geometry from selfsimilarity.
We find that, over the physiological range of lung volumes,
Given the heterogeneity of the structural components of the alveolar parenchyma, it is remarkable that there is so little change in configuration over the volume range. In the context of a tensed cablemembrane structure (2, 3, 12), the septal membranes are in part mechanically in series with the alveolar ductal cables. It follows that if their mechanical properties are perfectly matched, expansion will be geometrically similar. The contrapositive of this argument is that any observed departure from selfsimilarity implies differences of properties. In particular, the earlier observations of relative widening of the alveolar duct during inflation suggest that the septal membrane component is somewhat stiffer in extension than are the cables. This latter mechanism is also supported by the present result, which showed the angles at the B structures widening with inflation. Estimating the net effect of septa or of cables requires consideration of tensionbearing elements and geometry. Specifically, mechanical equilibrium demands that net septal tension in the plane bisecting the two septa, given by the product of twice septal tension and the cosine of the half dihedral angle, is equal to the ratio of the cable force to its radius of curvature. In our case,
In summary, our data on the dihedral angles of alveolar septal bends in pulmonary parenchyma suggest that 1) the architectural configuration of alveolar parenchyma and the surface and septal tissue tensions adapt to minimize surface area of the cable and membrane network; and 2) there is a modest increase with lung volume in the contribution of net septal tension over the resultant balance in the cables, which slightly alters the resulting configuration at high lung volumes.
Acknowledgments
The authors thank Judy Blake for painstaking data collection and tabulation and Dr. J. Reeds for calling our attention to Mallows’ work.
Appendix
There are substantive errata to the paper by Oldmixon et al. (6), which, since the analytic methodology of this work largely parallels that paper, deserve correction.
Equation 12
should read
The sequence following Eq. 15 should read “These were solved by successive iteration of the sequence ς_{i+1} = ς{s[m(ς_{i})]}, beginning with ς_{1} = ς(s _{ob}).”
The end of the next paragraph should read “If a pairi, j appears, we take i ≠ j. Let the tension in septum S _{i} beT _{i}.”
Equation 17
should read
The beginning of the next paragraph should read “Let E be the expectation operator over the triples (a _{1}, a _{2}, a _{3}). The fact that a _{1} + a _{2} +a _{3} = 2π implies that e _{1} +e _{2} + e _{3} = 0.”
The sentence following Eq. 19 should read “These same remarks apply in an identical fashion to the properties oft
_{i}; i.e., Et
_{i}= 0,
Equation 21
should read
Finally, the column headed Var_{net}, degrees^{2}, in Table 2, should read “
Footnotes

Address for reprint requests: J. P. Butler, Physiology Program, Harvard School of Public Health, 665 Huntington Ave., Boston, MA 02115.

This work was supported by the National Heart, Lung, and Blood Institute Grant HL26863.
 Copyright © 1996 the American Physiological Society