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Published in final edited form as: Respir Physiol Neurobiol. 2012 Nov 2;185(2):211–216. doi: 10.1016/j.resp.2012.10.015

AIRWAY-PARENCHYMAL INTERDEPENDENCE IN THE LUNG SLICE

Baoshun Ma 1, Michael Sanderson 2, Jason HT Bates 1,*
PMCID: PMC3529993  NIHMSID: NIHMS419415  PMID: 23128069

Abstract

The explanted lung slice has become a popular in vitro system for studying how airways contract. Because the forces of airway-parenchymal interdependence are such important modulators of airway narrowing, it is of significant interest to understand how the parenchyma around a constricting airway in a lung slice behaves. We have previously shown that the predictions of the 2-dimensional distortion field around a constricting airway are substantially different depending on whether the parenchyma is modeled as an elastic continuum versus a network of hexagonally arranged springs, which raises the question as to which model best explains the lung slice. We treated lung slices with methacholine and then followed the movement of a set of parenchymal landmarks around the airway as it narrowed. The resulting parenchymal displacement field was compared to the displacement fields predicted by the continuum and hexagonal spring network models. The predictions of the continuum model were much closer to the measured data than were those of the hexagonal spring network model, suggesting that the parenchyma in the lung slice behaves like an elastic continuum rather than a network of discrete springs. This may be because the alveoli of the lung slice are filled with agarose in order to provide structural stability, causing the parenchyma in the slice to act like a true mechanical continuum. How the air-filled parenchyma in the intact lung behave in vivo remains an open question.

Keywords: continuum model, spring network model, finite element method, parenchymal strain field

1. INTRODUCTION

The forces of airway-parenchymal interdependence are important in determining the speed and degree of airway narrowing following activation of the airway smooth muscle. Consequently, a detailed knowledge of the stresses and strains that arise in the parenchyma surrounding a contracting airway is critical for elucidating the mechanical factors that can potentially give rise to, for example, the airway hyperresponsiveness of asthma. We recently performed a two-dimensional computational analysis of dynamic airway-parenchymal interdependence, and found that significantly different results were obtained depending on whether we modeled the parenchyma as a network of discrete hexagonally-arranged springs or as an elastic continuum (Ma and Bates, 2012). In particular, the spring network model predicted that the radial displacement and force fields created by a contracting airway would propagate substantially further away from the airway wall than when the parenchyma was modeled as a continuum. This has major implications for the extent to which one particular contracting airway might interfere with the narrowing of adjacent airways, which in turn would influence the degree of heterogeneity of narrowing that might be possible throughout the lungs.

It is therefore vital to compare these models to the distortion of the parenchyma in a real lung. Traditionally, the parenchyma has been modeled as a uniform continuum (Lai-Fook, 1979, 1981). On the other hand, the space-filling geometry of packed alveoli suggests the plausibility of representing the parenchyma in two dimensions as a hexagonal spring network. Settling the matter is not straightforward in the intact lung because of the difficulty of observing strain fields in the interior of the parenchyma, so the issue remains unresolved. However, the local distortions induced in the parenchyma around a contracting airway can readily be determined in explanted lung slices (Adler et al., 1998), which is an experimental system that has recently gained popularity as a means of studying airway mechanics in situ (Adler et al., 1998; Bai and Sanderson, 2006; Brook et al., 2010; Del Moral and Warburton, 2010; Khan et al., 2010; Lavoie et al., 2012; Seow, 2007). Understanding the nature of airway-parenchymal interdependence in the lung slice is therefore an interesting question in its own right, and would have clear relevance to the intact lung. Accordingly, the goal of the present study was to determine whether a slice of parenchyma around a contracting airway behaves as if it is an elastic continuum or a hexagonally arranged network of springs.

2. METHODS

2.1 Explanted lung slices: measurement and analysis

We studied a single lung slice from two mice. The slices were prepared following procedures in previous studies (Bai and Sanderson, 2006; Sanderson, 2011) and chosen to contain a well-defined and roughly circular airway cross section. We selected airways from a central region (generation 3 to 5) as these have been shown to be the most responsive in terms of contraction and Ca2+ signaling. Methacholine was added to each lung slice to elicit an immediate and rapid contraction of the airway that was recorded at 30 images/min with a phase-contrast microscope. One of the lung slices was challenged with a low methacholine concentration of 200 nM (the low-dose slice), while the other slice was challenged with a high concentration of 1000 nM (the high-dose slice). The high-dose slice had received prior treatments of 100 and 200 nM interspersed with a saline wash to remove the methacholine and allow the airway to relax. The initial configuration of the high-dose slice, its configuration post-contraction with 1000 nM methacholine, and its subsequent relaxed configuration following saline wash are presented in Fig. 1. These studies were approved by the Institutional Animal Care and Use Committee of the University of Massachusetts.

Figure 1.

Figure 1

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Images of the high-dose lung slice before airway constriction (left), in the fully constricted state (middle), and after the methacholine (1000 nM) was washed away to allow the airway to relax (right). The hatched structures in the periphery of the image are nylon mesh that anchored the borders of the slice in place.

The displacement generated within the parenchyma of a lung slice was determined by analyzing the sequence of images of the slice obtained during airway contraction, with the assistance of custom software developed in Matlab (ver. 7, MathWorks, Inc.). First, a visually obvious anatomical landmark was identified in the pre-contraction image. The changing positions of the landmark were then followed through the sequence of images until contraction was complete, and a displacement vector defined between the start and end positions of the landmark (Adler et al., 1998; Brook et al., 2010). This process was repeated for approximately 150 different landmarks distributed around the image field.

To determine the size and shape of the airway lumen in the lung slice, 20 points were placed around the inner border of the airway wall. The perimeter (P) of the lumen was approximated by that of the closed polygon formed by these 20 points. These 20 points were also used as seed points to create a triangular mesh inside the polygon. The area of the airway lumen (A) was computed as the sum of the surface areas of all the triangles in the mesh. The hydraulic diameter, D, of the airway was defined as

D=4A∕P (1)

The initial value of D is designated D1, while the value at the peak of airway narrowing is designated D2. The airway lumen remained elliptical during contraction, so its major and minor diameters could be defined as follows. Euclidean distances between all pairs of the 20 points on the lumen boundary were computed, and the major diameter was defined as the distance between the two most widely separated points. The minor diameter was defined as the distance between the two boundary tangents that were parallel to and on opposite sides of the major diameter. The aspect ratio was defined as the ratio of major to minor diameters.

2.2 Hexagonal spring network model

We constructed a model of the lung slice in which the parenchyma was represented as a hexagonally arranged network of springs, as we have described in detail previously (Ma and Bates, 2012), where the hexagons represent cuts through individual alveoli. The network was square, and an airway was created at the center by eliminating springs within a closed loop of edges that defined the border of the lumen, as illustrated in Fig. 2A. This defined the pre-contraction geometry of the model. The airway lumen size and shape of the model during the subsequent contraction were analyzed using the same methods as for the lung slice above. Each spring in the network was assigned the same linear force-length relationship, and the intersections of the springs were represented as pin joints. The nodes on the boundary of the network were fixed in place. During airway contraction, quasi-static force equilibrium was assumed, and the configuration of the model for which the forces at each node were in balance at each time step was determined using custom software based on the finite element method as described previously (Ma and Bates, 2012).

Figure 2.

Figure 2

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A) The hexagonal spring network model of the parenchyma with an airway at its center. Each linear element in the model represents a spring with a specified stiffness and elastic equilibrium (zero stress) length. B) The continuum model showing the mesh used to calculate parenchymal properties. The mesh becomes finer nearer the airway border in order to accurately model the inward movement of the parenchyma near the contracting airway.

The dynamics of shortening of the airway wall was assumed to follow the hyperbolic Hill velocity-force relation for smooth muscle, as we have shown previously that this relationship can be used to mimic the dynamic changes in airway resistance in intact animals and in explants (Bates et al., 2009; Bates and Lauzon, 2005, 2007; Khan et al., 2010). The relationship between force and velocity that we used is (Bates and Lauzon, 2007)

v={bF0−FF+a,whenF<F0bF0−FF0+a,whenF≥F0} (2)

where v is the contraction velocity of the airway perimeter, F is the increased hoop stress generated in the loop of springs defining the lumen border as contraction proceeded, F0 corresponds to isometric (maximum) muscle force, and a and b are constants, where we assume a = F0/4 (Blanc et al., 2003).

The model airway was contracted in a step-wise fashion by shortening the airway circumference by a length equal to the product of v (Eq. 2) and the time step size, the latter being chosen small enough to provide a smooth and accurate solution. This was achieved by progressively decreasing the elastic zero-load lengths of the loop of springs that defined the lumen border and determining the resulting elastic equilibrium configuration of the model until the airway had shortened by the desired amount. At each time step, the shortened circumference of the loop of airway springs pulled increasingly inward on the attached parenchymal springs, thereby progressively increasing the hoop stress in the loop with a commensurate reduction in shortening velocity as defined by Eq. 2.

2.3 Continuum model

The model with the parenchyma represented as a continuum was implemented in a similar fashion to the hexagonal spring model above, and as we have described in detail previously (Ma and Bates, 2012), except that here the parenchymal domain was triangulated as illustrated in Fig. 2B. Each triangle in the mesh was regarded as an elastic constant-strain element and the equilibrium configuration of the model was determined using a finite element procedure. We assumed a plane stress condition and linear elasticity for the material of the parenchyma. The bulk modulus of the parenchyma was assigned a value of 1, while the shear modulus was set to be 5% of the bulk modulus. Airway contraction was achieved by applying an inward force (i.e. toward the centroid of the airway lumen) at each of the nodes lying around the boundary of the airway lumen. The applied force was selected by trial and error to produce a degree of airway constriction that matched experimental observation.

3. RESULTS

3.1 Time history of airway contraction

The results we obtained from the low-dose slice and high-dose slice were similar. Airway contraction was complete in 10-15 s following application of methacholine, with most of the contraction occurring in the first 4 s. The contracted airway lumen area shown in Fig. 1 is 31% of that before contraction (Fig. 1A), and it returned to about 88% of its original value after methacholine removal (Fig. 1C). During contraction, D (Eq. 1) decreased from 0.24 mm to 0.13 mm, while the aspect ratio increased from 1.48 to 1.57, so the lumen was somewhat more elliptical when contracted. Because the airway in the hexagonal spring network model could be contracted in accord with the Hill force-velocity relationship (Eq. 2), with appropriate choice of its parameters values the model was able to accurately mimic the experimentally observed time-history of the lumen area of the slice, as shown in Fig. 3.

Figure 3.

Figure 3

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Comparison of the time-history of airway lumen area during airway contraction between experiments and spring network model.

3.2 Parenchymal displacement fields

The experimentally determined parenchymal displacement field shown in Fig. 4 is not perfectly axi-symmetric, in contrast to the displacement fields predicted by the models. Nevertheless, the magnitude of the experimental displacement field decreased with distance away from the lumen boundary, as it did in the models. For ease of comparison, the magnitudes of the experimental and simulated displacement fields were normalized with respect to the mean displacement at the lumen boundary, equal to D1 – D2. The normalized displacement fields were then expressed as functions of radial distance from the airway centroid in the fully contracted state normalized to D1. Figure 5 shows that the experimental displacement field fell off hyperbolically with radial distance so that by about 3D1 to 4D1 away from the airway center the parenchymal displacement was only about 10% of its maximum value at the lumen border.

Figure 4.

Figure 4

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Parenchymal displacement field showing the movements of each landmark that occurred over the course of airway narrowing in the high-dose slice. Arrows indicate direction and distance of movement.

Figure 5.

Figure 5

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Comparison of normalized displacement fields between the low-dose slice and high-dose slice experimental data sets and the continuum model when the ratio of parenchyma to airway size in the model is comparable to that of the lung slice.

The normalized displacement field predicted by the continuum model matched the experimental displacement field closely (Fig. 5). By contrast, the displacement field predicted by the spring network model declined significantly more slowly with radial distance (Fig. 6). These results would seem to suggest that the hexagonal spring network model should be discarded in favor of the continuum model as a description of the parenchyma in a lung slice. Interestingly, however, we found that the spring network model could be made to give a good prediction of the experimental data if we increased both the distance to the fixed boundary of the parenchyma in the model by a factor of 3, and normalized the distance axis in the plot by 3D1 instead of D1 (Fig. 6). We explored this procedure for a variety of different model geometries and found that the predictions of the continuum and spring network models matched well if this procedure was followed for the spring network model.

Figure 6.

Figure 6

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Comparison of normalized displacement field between the low-dose slice and high-dose slice experimental data sets and the hexagonal spring network model when the ratio of network to airway size in the model is comparable to that of the lung slice. Also shown is the prediction of the spring network model when the dimensions of the parenchymal network are increased 3 fold and the results are plotted with the normalized radial distance reduced by 3 fold.

4. DISCUSSION

The forces of airway-parenchymal interdependence are very important in determining how the airways respond to challenge with a smooth muscle agonist, such as during a clinical methacholine challenge test or following endogenous histamine release from degranulating mast cells. This is clearly demonstrated by the high sensitivity of airways responsiveness to transpulmonary pressure, modest reductions of which can transform the behavior of a normal lung into that resembling a lung with severe asthma (Bates and Lauzon, 2007; Cojocaru et al., 2008; Ding et al., 1987). Accordingly, it is important to understand how the parenchyma deforms around a contracting airway and how this deformation leads to increased parenchymal stresses. Such stresses can not only severely limit the ability of the airway to narrow, but can also affect the abilities of other nearby airways to contract.

Conventional wisdom about the nature of airway-parenchymal interdependence is based on the classic work of Lai-Fook (Lai-Fook, 1979, 1981; Lai-Fook et al., 1977) who used continuum theory to predict how the forces opposing airway narrowing would build up in the parenchyma surrounding it, and how these forces would propagate radially from the airway wall. This theory has been used successfully in a number of studies to account for the time-course of airway narrowing in intact lungs (Bates et al., 2007; Bates and Lauzon, 2007; Cojocaru et al., 2008; Gunst et al., 1988). However, the lung parenchyma is more like an open-cell foam than a continuum, and in thin cross-section resembles a hexagonally arranged network of alveolar walls interspersed with larger alveolar ducts and small airways and vessels. Nevertheless, the Lai-Fook continuum theory has been widely accepted to represent parenchymal distortion mechanics, especially since Wilson (Wilson, 1972) showed a degree of theoretical agreement between the behavior of a hexagonal spring network and continuum theory. On the other hand, it should be noted that Wilson's analysis only applies to small deformations, and recent work by Breen et al. (Breen et al., 2012) indicates that the Lai-Fook theory may begin to break down when the radius of a contracting airway decreases by more than about 50%.

In any case, it was surprising that we found (Ma and Bates, 2012) that modeling the parenchyma as a network of hexagonally arranged springs around a contracting airway does not give the same results as when the parenchyma is modeled as an elastic continuum, either in terms of the parenchymal displacements or the radial propagation of stresses. These results raise the question as to how real lung parenchyma behaves. This is not easily addressed experimentally in the intact lung because of the practical difficulties associated with visualizing the parenchymal displacements and strains taking place within the center of the organ. The compromise made by Lai-Fook et al. (Lai-Fook et al., 1977) was to use a cylindrical core of parenchymal tissue removed from a lung lobe under tension. With this preparation, they found that the predictions of linear continuum theory matched experimental observation of the outward displacement of a cylindrical hole in the parenchyma, supporting the continuum model of parenchymal elasticity. However, what is needed to address the concern of the present study is an experimental system in which the parenchymal boundaries move inward, for example as can be seen around a contracting airway in a lung slice.

The lung slice would thus appear to be an excellent alternative experimental system for addressing parenchymal influences on the airway because both the displacements of the airway wall and the surrounding parenchyma are clearly visible (Fig. 4). Our results clearly indicate that the parenchyma in a lung slice behaves much more like a continuum (Fig. 5) than a hexagonal network of springs (Fig. 6), at least in terms of the regional displacements of the parenchyma induced by airway contraction (Fig. 4). We could not determine how local parenchymal stresses compare between experiment and model because we do not have a way of measuring stress experimentally, so we can only assume it falls off rapidly with distance from the airway lumen as indicated by Lai-Fook's studies (Lai-Fook, 1979; Lai-Fook and Hyatt, 1979; Lai-Fook et al., 1978). Nevertheless, these results cannot be directly extrapolated to the intact lung in vivo because of the agarose used to prepare the lung slices. Although the agarose was washed out from the lumen of the airway in the slices, it was retained within the alveoli in order to preserve their geometry, and could thus have rendered the parenchyma in the slice a true mechanical continuum. By contrast, in vivo alveoli are filled with air and have an air-liquid interface manifesting an important degree of surface tension, and so may have quite different mechanical properties.

The relevance of the hexagonal spring network model for the intact lung in vivo thus remains uncertain despite its apparent structural similarity to the 2-dimensional geometry of closely packed alveoli. That is, in the absence of definitive experimental data, we cannot say whether the living air-filled lung behaves more like a hexagonal spring network or a continuum. We concluded in our previous study (Ma and Bates, 2012) that the different behavior of the hexagonal spring network relative to the continuum is related to the relative stabilities of the two models under shear, the continuum model being much more stable than the network model. In any case, the hexagonal spring network model remains useful because its predictions can be made to match experimental observations of radial parenchymal displacement in the agarose-filled lung slice if the dimensions of the network are made 3 times larger than the experimental system and the normalized displacement is plotted against a horizontal distance axis that is scaled to be three times the actual initial radius (i.e. 3D1 instead of D1). We confirmed with additional simulations (data not shown) that this scaling effect applies to the overall dimensions of the model and not to the size of the individual hexagonal cells in the network relative to the size of the airway. Therefore, provided this scaling is taken into account, the hexagonal spring network model can be exploited for its ready use in, for example, simulating the time-course of airway narrowing (Fig. 3). Also, it is straightforward to assign arbitrary constitutive properties to individual springs in the hexagonal network model in order to represent arbitrary degrees of regional mechanical heterogeneity. The functional significance of such heterogeneity remains unclear, yet it is an important feature of real lung slices as evidenced by lack of radial symmetry of the strain field about the airway center (Fig. 4 and (Adler et al., 1998)), so spring network models offer an approach to investigating this issue.

Nevertheless, even though we can scale the hexagonal spring network model so that it predicts a realistic displacement field, there remain important differences between its predictions and those of the continuum model in terms of the radial force field. The force fields are proportional to the gradients of the strain fields, and it is possible for imperceptible differences in the latter to give rise to significant differences in the former. This appears to be the case for the hexagonal spring network model versus the continuum model because the former predicts force to asymptote to a finite plateau value that continues virtually to the tissue border, while force in the continuum model falls off hyperbolically with radial distance (Ma and Bates, 2012).

The results of this study are contingent upon its limitations, one of which might appear to be the fact that we studied only two different lung slices. This might seem to raise the question as to the generality of our experimental findings, particular in terms of the effects of structural heterogeneities between different slices. We note however, that the parenchymal displacement data we obtained consist of a great many individual points take from all angles around each airway. The degree of local structural heterogeneity in a given lung slice varied substantially, as exemplified by Fig. 4. Despite this, however, and with the fact that the methacholine challenge doses were very different for the two slices, the displacement data all followed the same relationship with radial distance as shown in Fig. 5. Our conclusions related to the experimental measurements thus have considerable statistical support.

The main limitations of our study are thus manifest in the various simplifying assumptions we made in constructing the computational models. For example, the models are strictly 2-dimensional, whereas the explanted lung slice is actually 3 dimensional in that it has a finite thickness. This means that the alveolar walls are not anatomically equivalent to one-dimensional spring elements but rather correspond to 2-dimensional elastic sheets. This may have contributed to the apparent continuum nature of the explant over and above the fact that its alveolar spaces are filled with agarose. We also neglected the finite thickness of the airway wall. In reality, the inner rim of the parenchyma is separated from the lumen border by the airway wall thickness. On the other hand, Brook et al. (Brook et al., 2010) took this into account in their simulations of the static parenchymal deformation field based on continuum theory and found results similar to those of Lai-Fook (Lai-Fook, 1979; Lai-Fook et al., 1978; Lai-Fook et al., 1977). Finally, our models also assumed the parenchyma to be homogeneous, but even visual inspection of the lung slices (Fig. 1) show this was not actually the case. For example, one can see a structure just above the airway in Figs. 1 and 4 that may be a small artery. Lai-Fook (Lai-Fook and Kallok, 1982) showed that the close juxtaposition of vessels and airways can lead to interaction effects, and indeed this may have been responsible for some of the circumferential asymmetries evident in the experimental strain field shown in Fig. 4. Nevertheless, we assume that these effects average out to give radial displacement behavior that is accounted for in its main features by homogeneous computational models of the parenchyma.

In summary, we compared experimentally determined displacement fields around a contracting airway in an explanted lung slice to predictions made by computational models in which the parenchyma is represented as a hexagonal spring network and as an elastic continuum. The experimental observations were more closely predicted by the continuum model, validating the classic theory of Lai-Fook for this system. What pertains in the intact lung in vivo remains an open question.

Highlights.

! ! We measured the parenchymal distortion around a contracting airway in a lung slice

! ! The parenchyma behaved more like an elastic continuum than a spring network

! ! The spring network model predictions were accurate when appropriately normalized

! ! How the parenchyma in an intact lung behaves in vivo remains an open question

ACKNOWLEDGEMENTS

This study was supported by NIH grant HL-103405.

Footnotes

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