INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS THEORY RESULTS DISCUSSION REFERENCES

THE FACT THAT AIRWAY SMOOTH muscle (ASM) must develop tension to narrow an airway indicates that forces exist in and on the airway wall against which the muscle does work. The most obvious of these is the tension in the alveolar attachments that transfers pleural pressure to the airway wall (19). When the muscle shortens and the airway narrows, local deformation of the parenchyma causes an additional force that is usually referred to as the parenchymal interdependence force. For many years, much interest has centered on this force (5, 6, 8-12, 15-18). The interaction is usually described in terms of classical elasticity theory in which the parenchyma is treated as a continuum and the local distortion is governed by the local shear modulus (17).

It has also been proposed by three different groups that the folding of the epithelial membrane could provide an additional force that opposes the muscle (13, 14, 24, 31). Because it is not clear exactly which structures inside the muscle wall contribute to this force, we will use the term "folding membrane" from now on to denote the total structure that provides the force. In Lambert's analysis (13), elastic shell theory was used to show that an increased number of folds required an increased pressure difference across the folding membrane. A further development of the analysis showed that, as the airway narrowed, more folds developed (14). The key property of the folding membrane that determines the pressure difference required to produce a given level of folding was shown to be its flexural rigidity, D, which in turn depends on both the Young modulus, E, and the thickness of the tissue that folded. There are no data for these quantities for airway folding membrane, although there are data for the basement membrane of renal tubules, presumably a similar material (29, 30).

The data in our recent article (21) indicate that this model explains at least some of the observations of membrane folding in airways in which ASM has been activated. In particular, the walls of these airways remained circular as the lung was deflated, and the number of membrane folds (n) was shown to increase when the luminal area (A_i) was reduced. However, airways in which ASM was not activated collapsed in a different pattern when the lung was deflated; the entire airway wall underwent a buckling into an oval shape (the fundamental mode of collapse of an elastic tube) after the development of only a few folds in the folding membrane. After this buckling, membrane folds continued to increase as the airway flattened further and its long axis shortened.

The relationship of fold development in the membrane to the external load causing the folding is determined by D. Thus evaluation of D is essential to an assessment of the mechanical significance of folding. For instance, knowing the value of D for an airway will enable us to know whether membrane folding provides significant resistance to muscle shortening. In this study, we will use the observation of elastic buckling of the entire airway wall to obtain a value for the D of the folding membrane.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS THEORY RESULTS DISCUSSION REFERENCES

The data presented here are taken from those obtained in the previously reported study mentioned above (21). Briefly, we used New Zealand White rabbits divided into two groups: one was exposed to nebulized and infused carbachol (the Carb group), and the other, the control (Cont) group, to saline. After challenge (real or sham), lungs in the open-chest animals were brought to a lung recoil pressure (PL), either by deflation from PL = 4 cmH₂O to one of 4, 0, or 2 cmH₂O or by holding at a positive end-expiratory pressure of 4, 7, or 10 cmH₂O before infusion of Formalin and removal from the animal. After removal, the lungs were immersed in a bath of Formalin for 24 h before blocks were cut and sections prepared for morphometry. The 5-µm sections were stained with Masson's trichrome.

Using standard morphometric techniques, we measured A_i, area inside the basement membrane (A_bm), and the subdivisions of wall area as well as the perimeter of the epithelial basement membrane (P_bm). P_bm was our reference measure for airway size because it is independent of muscle shortening or lung inflation (7). The distribution of P_bm was the same in both groups of rabbits.

The data were pooled at each PL for each treatment to obtain overall trends with PL. They were also put into bins based on P_bm at each PL. After some preliminary analysis of the distribution of airway sizes, it was decided to use the bins shown in Table 1. The numbers of airways studied at each PL are given in Table 2. Results for the Carb airways were presented in the previous report (21). Mean values and SE of relevant measures of airway size and deformation were calculated for the pooled data and by bin. Where possible, data were normalized on an appropriate variable. This was often the calculated value of that variable for a fully dilated airway with a circular basement membrane under the assumption that the basement membrane does not stretch. This was done to obtain the greatest possible generalization of the results by removing the size of the airway from the variable being examined.

Results from the previous study that are relevant here are as follows. Carb airways developed epithelial folds at high PL, the number of folds increased with decreasing PL to a maximum at PL = 0 cmH₂O, and then the airways underwent whole wall buckling and flattened with further reduction in PL. In contrast, although Cont airways developed a few epithelial folds at high PL, the number increased only slightly with reduction in PL before the entire airway buckled and flattened, after which epithelial fold number increased with further reduction in PL. Larger airways underwent whole wall buckling at higher values of PL than did the smaller airways.

	THEORY

TOP ABSTRACT INTRODUCTION METHODS THEORY RESULTS DISCUSSION REFERENCES

We are concerned with the onset of whole wall buckling in Cont airways. This occurs at a value of PL that is less than that at which epithelial folds first appear when PL is reduced from a high value. We will use this observation of whole wall buckling to evaluate the D of the folding membrane. To do this, we will use a simplified model of the airway wall that consists of a tube of folding membrane internal to, and separated from, a tube of smooth muscle. Any mechanical contribution of the adventitial layer is excluded.

A schematic diagram of a narrowed airway that contains a folded membrane but that has not yet undergone whole wall buckling is shown in Fig. 1. With pleural pressure as reference, the pressures indicated in Fig. 1 are related as follows

(1)

where P_fm is the pressure difference across the folding membrane, P_m is the pressure difference across the smooth muscle, P_g is the pressure in the highly deformable region between the ASM and the folding membrane (the "gap"), and Ppb is the peribronchial pressure (or, to put it another way, peribronchial radial stress). We are assuming that there is no change in Ppb across the adventitial tissue.

View larger version (39K): [in this window] [in a new window]   Fig. 1.   Schematic diagram of constricted airway showing pressures [peribronchial (Ppb), gap (Pg), lung recoil (PL)] discussed in text. Other symbols refer to morphometric parameters. WAg, cross-sectional wall area of gap; Pbm, perimeter of epithelial basement membrane; Abm, area inside basement membrane; Pi, luminal perimeter; Ai, luminal area.

Elastic buckling of a tube requires an inwardly directed pressure difference (that is, for an airway, Ppb > PL). Thus the question arises as to how such a pressure difference could be generated. In a fully inflated lung, the pressure difference across the wall of an airway is directed outwards (Ppb < PL), resisting the tendency of the airway to narrow under the influence of the elastic forces in its wall. Our data show that, as the lung is deflated from high volumes, the Cont airways narrow less than would a circular hole in the parenchyma (21). Thus in the vicinity of such an airway, the parenchyma is stretched less than it would be in the vicinity of the circular hole, and Ppb must be greater than pleural pressure. At sufficiently small lung volumes, Ppb could be greater than PL, thus providing the necessary condition for whole wall buckling. The buckling that occurred at PL = ~2 cmH₂O for the smaller airways indicates that, at about this pressure, Ppb exceeded PL and the airways were being compressed from outside. Thus at some point in the transition from the airway being held open to its being compressed, Ppb = PL (at which point the local tension in the parenchymal tissue is zero) and (from Eq. 1)

(2)

Another way of obtaining this result is to think of the tension in the airway wall when Ppb = PL. At this point, the airway wall must be in a state of zero net tension (by the Laplace law) because, if the net tension were greater than zero, the airway wall would remain circular and the airway would narrow as the lung deflated until the tension became zero. Thus at the PL at which net tension is zero, the tendency of the passive stress in the ASM to narrow the airway is exactly counterbalanced by the "springiness" of the folding membrane, which tends to keep the airway open. At this PL, the airway would still be circular.

P_m (Eq. 2) can be related to the wall tension and thus wall circumferential stress () and muscle thickness (T_m) through the Laplace equation (Eq. 3)

(3)

Assuming that the only significant source of tension in the airway wall is the ASM and because the ASM in these Cont airways has not been challenged,  is the passive stress in the ASM. Thus T_m is the circumferential tension/unit axial length in the airway wall; r_m is the radius of the muscle layer. T_m/r_m can be evaluated from our morphometric measurements, and  can be estimated from published data for rabbit trachealis (22). Thus by calculating P_m using Eq. 3, we can obtain P_fm from Eq. 2 for the circular airway on the verge of whole wall buckling.

Finally, the theory of membrane folding that we are using (13, 14) relates P_fm to the D of the folding membrane (Eq. 4)

(4)

where P_th is the theoretical, nondimensional value of the transmembrane pressure difference used in the published analysis of folding (14). Increases in P_th lead to deeper folds and, when the folds encounter the geometric constraint of the muscle wall, to increases in fold number. As can be seen from Eq. 4, P_fm depends inversely on the cube of the radius of the membrane tube (r_fm). This value is calculated when the membrane is circular in shape. Its value can be estimated from the morphometric data

(5)

Combining Eqs. 2, 4, and 5 yields Eq. 6 from which D will be evaluated

(6)

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS THEORY RESULTS DISCUSSION REFERENCES

The aim in presenting the following sequence of results is to use Eq. 6 to calculate a value for D. To do this, the value of PL at which the wall first buckles must be identified. The histological evidence showed that whole wall buckling occurred between 4 and 0 cmH₂O (21). This corresponds to the steeply sloping part of the normalized A_i vs. PL curve (Fig. 2, in which A represents luminal area in a fully dilated airway). We will assume that the small airways and the 1-mm airways commence buckling at PL = ~2 cmH₂O, because there is a large decrease in A_i between PL = 2 and 0 cmH₂O. By using the same criterion, the 1.5-mm airways appear to buckle at ~4 cmH₂O. However, the size of the error bar shows that there is substantial heterogeneity in A_i values at this value of PL, indicating that many of these airways are already buckled. We chose not to use the data at PL = 7 cmH₂O because the values for T_m/r_m at this PL and at PL = 10 cmH₂O appear to be anomalous (Fig. 3). We have no explanation for these anomalous values. This consideration also rules out using data for the large airways. Thus we will evaluate D from Eq. 6 only for the small and 1-mm airways. The steps in the calculation are as follows.

View larger version (25K): [in this window] [in a new window]   Fig. 2.   Normalized luminal area (Ai/A, where Ai is luminal area and A is luminal area in fully dilated airway) vs. PL for control rabbit membranous bronchioles. The airway size categories are given in Table 1. Values are means ± SE. There is insufficient data from small airways at PL = 7 and 10 cmH2O to calculate meaningful averages.

View larger version (45K): [in this window] [in a new window]   Fig. 3.   Muscle thickness (Tm)-to-muscle radius (rm) ratio vs. PL. Values are means ± SE. Binned data, control airways, and airway size categories are given in Table 1.

P_bm was obtained morphometrically for every airway.

We evaluated P_th using the published theoretical results and our morphometric data. WA_g represents the cross-sectional area of the "gap". This area, normalized on A (the A_bm in a fully dilated airway) was denoted A_sub in the theoretical paper (14). Figure 9A of that paper is a graph of A_bm/A against P_th for three values of WA_g/A: 0.05, 0.1, and 0.15. A value of 0.05 lies within the observed range for rabbits (21). Thus from this graph, choosing WA_g/A to be 0.05 and using the mean morphometric values for A_bm/A, we obtained the values for P_th given in Table 3. These values are nondimensional and thus have no units.

P_m can be calculated from Eq. 3 if  and T_m/r_m are known. We calculated T_m/r_m from the measured area of smooth muscle and the smooth muscle perimeter at the PL of interest. The results for all airways are shown in Fig. 3 for all PL values. We used the values for small and 1-mm airways at PL = 2 cmH₂O. The rest of the data set is included to provide a context for the values that we used.

To obtain a value for the muscle  from the published data, we need to know muscle length (L). Because the muscle is the outer boundary of the combined areas of the gap and the A_bm, L normalized on P_bm is given by Eq. 7

(7)

L values are usually reported normalized on the length at which the muscle develops maximal isometric stress, L_max. We believe it is reasonable to expect that, in vivo, L_max occurs at a reasonably high value of PL. We chose to use the length at PL = 7 cmH₂O as L_max. The exact value of PL is not very important because L changes very little at high values of PL. Thus L/L_max can be estimated by taking the value of L/P_bm at the PL of interest and dividing by the value at PL = 7 cmH₂O (Table 3).

In a study conducted in our laboratory, the stress generated by rabbit tracheal smooth muscle as a function of L was measured for the trachealis muscle (22). We assumed that those results were applicable to the ASM in the rabbit airways in the present study. We fitted a cubic curve to the passive stress data to have a simple representation for the purposes of interpolation (Fig. 4). We were then able to use Eq. 3 to estimate  in the ASM and, using T_m/r_m, the value of P_m (Table 3).

View larger version (13K): [in this window] [in a new window]   Fig. 4.   Trachealis stress vs. normalized muscle length (L/Lmax, where L is muscle length and Lmax is length at which muscle develops maximal isometric stress) for passively generated stress (22). Continuous line is fitted cubic equation.

D can be calculated (Eq. 6) from the values thus obtained. The results are shown in Table 3. Although D is the basic property of the membrane that determines its buckling and folding behavior, it is not a familiar quantity, and its significance is, therefore, difficult to assess. A better sense of its significance can be obtained from its use in evaluating the P_fm (Eq. 4). This equation relates theoretical, nondimensional pressure to dimensioned pressure through the conversion factor D/r. Values of this factor for the two airway sizes are given in Table 3. Thus a value of P_th of 99 (the critical pressure for developing 10 folds) corresponds to ~0.5 cmH₂O for the 1-mm airways.

Knowledge of D/r enables the calculation of dimensioned A_i-transmural pressure (Ptm) curves for a model airway that stays circular with reductions in Ptm difference (Fig. 5). To obtain this curve, we used our simple model of a 1-mm airway consisting of a muscle layer separated from a folding membrane by the gap. There is neither parenchyma nor adventitia in this model. The pressure difference across this airway wall (Ptm) was calculated as follows
Positive values of Ptm indicate that luminal pressure is greater than Ppb. P_fm was calculated by taking the previously published (14) theoretically derived A_i-Ptm curve for an airway with WA_g/A = 0.05 and applying the scale factor of 0.5 cmH₂O to P_th (Eq. 4 and Table 3). P_m was calculated from the passive muscle data as described above. The curve is given in Fig. 5. We have tried to indicate how whole wall buckling might appear on this plot by including a cartoon curve (labeled with a question mark) starting at slightly negative Ptm. A negative value is required to initiate buckling. We have not tried to calculate the curve once whole wall buckling has started. The stress field around the buckled airway is nonuniform because the airway is being compressed along its long diameter and stretched along its short diameter. An analysis of this deformation is beyond the scope of this paper.

View larger version (14K): [in this window] [in a new window]   Fig. 5.   Normalized luminal area vs. transmural pressure (Ptm) difference for model airway with unchallenged muscle and Pbm = 1 mm. Solid curve is for an airway that remains circular. Dashed curve is a cartoon of what might happen once the airway undergoes whole wall buckling. Vertical dashed line indicates the approximate Ptm at which buckling may occur.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS THEORY RESULTS DISCUSSION REFERENCES

This is the first time that a value for D of airway folding membrane has been reported. Our estimated values for D and D/r show that physiologically significant pressure differences are required to cause folding of the airway folding membrane in peripheral airways. Thus membrane folding provides significant opposition to ASM shortening in small airways. These data and analyses prompt us to suggest that airway membrane folding acts to prevent excessive narrowing and that the airway wall thickening which occurs in disease may, by increasing D, be a protective mechanism.

We are not aware of reports of D for any other biological membrane with which we could make comparisons. However, D can be calculated from the formula for a tube (27) if the E and folding membrane thickness (T_fm) are known (Eq. 8)

(8)

where , the third physical quantity, is the Poisson ratio of the material. Its value is not known for collagenous membranes but is usually assumed to be 0.5, the value for incompressible materials. Values of E and T_fm (~5 MPa and 100 nm, respectively) have been published for renal tubules (29, 30). These yield a value for D of 6 × 10¹⁶ Pa · m³, a value much less than our value for airway folding membrane. However, bigger tubes (airways) need stiffer walls to resist the same Ptm values as smaller tubes (renal tubules). Thus a fairer comparison is between values of D/r, which is ~0.6 Pa for renal tubules and thus lies between our two airway estimates.

Our estimates of the value of D depend crucially on the trachealis passive stress curve that we have used. As far as we know, these are the only published data on passive stress in rabbit trachealis. When we used the dog data of Gunst and Stropp (4), our values for D were approximately the same as reported in Table 3. However, use of the dog data of Okazawa and colleagues (20) resulted in values that were a factor of 10 smaller.

The muscle strip experiments that provided the data that we used for passive muscle stress were "static" in that the muscle was stretched and allowed to come to a steady value of stress. It is now well established that cycling greatly reduces the active stress that the muscle can generate (2, 3, 23, 25, 26). There appear to be no data on the effect of cycling on passive stress. In the experiments described here, the rabbits were tidally ventilated at 40 breaths/min until death. Thus the airway L and stress were also cycled. However, the cycling was stopped before fixation was started. If some of what we are referring to as passive stress arises from spontaneous muscle tone, then it would be reasonable to expect that this would have been reduced by respiratory cycling, thus leading to an overestimate in our values for D if the lower level of  persisted until fixation prevented any further changes in geometry.

A curve for challenged airways similar to that shown in Fig. 5 can be calculated. However, its accuracy depends crucially on the value for the maximal stress of the ASM. Because the L and force were being cycled by tidal breathing, we cannot use the published value of maximal stress and have no reasonable way of estimating the reduction in this value caused by the cycling. Therefore, we chose not to present such a curve.

We cannot reliably evaluate E for airway folding membrane from our results because we could not obtain a value for T_fm. However, if we assume that the membrane has the same value of E as renal tubules, then, using Eqs. 4 and 8, we estimate the value of T_fm/r_fm to be ~0.01 in the small and 1-mm airways. This is about the same value reported for renal basement membrane (30). Thus it appears possible that the folding membrane has a similar value of E as that of basement membrane in renal tubules.

A value for E on the order of 5 MPa is in poor agreement with data obtained from mucosal strips taken from tracheas of both sheep (1) and rabbits (28), which were in the range of 3-24 kPa. A value for E of 24 kPa would require T_fm/r_fm to be ~0.06, which is twice the maximum possible value based on the observed values of WA_g/A in these airways. The difficulty with the mucosal strip data appears to lie in identifying the load-bearing tissue and assessing its cross-sectional area to convert the measured tension to stress. It would appear either that both mucosal strip experiments greatly overestimated the amount of tissue that is significant in membrane folding or that the properties of tracheal mucosal tissue are very different from those of folding membrane in membranous bronchioles. Our estimates for the value of D depend on knowing the critical buckling pressure of the airway wall. Because we did not design the experiment to evaluate this quantity, we can make only an educated guess as to its value. However, most of the parameters that contribute to the calculation of D are insensitive to the exact value of PL at which they are evaluated. We estimated P_th using WA_g/A and A_bm/A because there is a published relationship between these quantities. We could also have used the number of folds in conjunction with either of these normalized areas. Doing so yields values of D that are up to 10 times larger than those reported here. One of the reasons is that the mean values of these morphometric quantities taken together do not yield a point on the surface predicted by theory [Fig. 3A in the theoretical paper (14)]. It would be unreasonable to expect that the simple averages that we took should yield such a point because the relationship between the quantities is nonlinear and it is not known how to average the data in an appropriate fashion. In addition, the theory is highly idealized in its representation of the material in the gap as being incapable of sustaining shear forces. The effect of fixation shrinkage on the results should be minimal because an attempt was made to correct for this before data analysis took place.

The folding membrane analysis ignored the epithelium. Whereas the presence of epithelium is not expected to have any significant effect on the initiation of elastic collapse, it could have a significant effect on area reduction at large transmembrane pressure differences because it would prevent the close juxtaposition of opposite walls of folding membrane. We believe that the theoretical results are valid at the transmembrane pressures and small fold numbers that existed under the conditions analyzed here.

In summary, we have used the observation of elastic buckling of the entire airway wall to open a window on the stiffness of one of the components of that wall: the folding (epithelial) membrane. We have deduced that the D of the folding membrane in small membranous bronchioles from rabbits is of the order of 10¹² Pa · m³. This value leads to the conclusion that folding of the epithelial membrane provides a significant load for ASM in small membranous bronchioles, especially at small values of PL. Unloaded ASM is capable of shortening to 20% of its starting length, a magnitude of shortening that would result in complete airway closure. These data lead us to suggest that at least one of the reasons that airways do not easily close when the smooth muscle is stimulated is the load provided by the folding of the mucosal membrane. To the extent that the thickening of this membrane, which occurs in diseases such as asthma and COPD, is associated with a preservation of the mechanical properties of the materials that make up the wall, thickening will increase the D and serve as a protective mechanism against excessive airway narrowing.