Abstract
The scheme of Horsfield et al. for describing the pulmonary airway tree (J Appl Physiol 52: 21–26, 1982) catalogs each airway according to its order and the difference in order of its two daughters (denoted Δ). Although this scheme captures the natural asymmetry in the airway tree, it is still deterministic, because it assumes that all airways of a given order are the same; yet such variability is extremely important in determining the overall behavior of the lungs. We therefore analyzed complete lung lobes from three mature and two immature rabbits and determined the Horsfield order and Δ of every airway down to the terminal bronchioles. We also measured the diameter of each airway. This allowed us to determine the average structure of the rabbit airway tree, the variation about this average, and also how the structures of mature and immature airway trees compare. We found some variation in branching asymmetry and airway diameter at a given order between animals but no evidence of systematic differences in structure between mature and immature lungs. We found evidence of a difference in the branching structure of the peripheral vs. the central part of the airway tree (the break point being around order 20). We also determined the nature of the variation in Δ and diameter as a function of order, which should be valuable for the development of computer models seeking to encapsulate the naturally occurring regional variation in airway geometry in the normal rabbit lung.
 airway tree
 lung function
 heterogeneity
 selfsimilarity
 fractals
the branching pattern of the pulmonary airway tree is an important determinant of the distribution of ventilation to various lung regions. It also determines the overall resistance of the airways to airflow. Consequently, knowledge of the branching morphology of the airway tree is crucial for a complete understanding of the entire organ. Weibel's landmark idealization of the airway tree (12) assumes that each airway gives rise to a pair of identical daughters and captures the fact that increasingly distal airway generations are generally shorter and narrower than their parents. This structure has been utilized in models of the complete airway tree that are both fully deterministic (14) and stochastic (1). However, the Weibel model is perfectly symmetrical, because, at every bifurcation, the two downstream subtrees are identical. This is far from the case in reality. Horsfield et al. (5, 6) made a significant advance by modeling the airway tree in terms of an efficient recursive scheme that incorporates branching asymmetry. The Horsfield scheme catalogs airways according to their order, which starts at the terminal bronchioles (order 1) and increases as one moves proximally up the airway tree. In addition, each parent airway is considered to branch into two daughters, the respective orders of which are separated by an integer Δ. A number of computational studies have used the Horsfield scheme to model the airway tree to calculate its mechanical impedance (3, 4, 8,11) and to estimate regional ventilation distribution to the lung periphery (2).
Although the Horsfield scheme captures the natural asymmetry in the airway tree, it is still deterministic, because it assumes that all airways of a given order are the same. To try to account for the naturally variability of real lungs, some recent studies have utilized the Horsfield scheme with stochastic variability incorporated into the airway dimensions. Simulation results suggest (2, 3, 8,11) that such variability is extremely important in determining the overall behavior of the lungs, particularly with bronchoconstriction. It thus becomes a crucial issue to determine precisely the variability in airway dimensions and branching asymmetry at each order. This variability has only been guessed at in studies based on stochastic Horsfieldtype models, even though it is embodied in the original data of Raabe et al. (10), who obtained extensive morphometric data in humans, dogs, rats, and hamsters. However, because Raabe et al. studied only one or two of each species, limited information was obtained about interanimal variability. Furthermore, essentially nothing is known about how airway branching morphology changes with maturation.
In the present study we therefore analyzed every airway in one lobe from five rabbits, three mature and two immature, and determined the Horsfield order and Δ of every airway down to the terminal bronchioles. We also measured the diameter of each airway. This allowed us to determine the average structure of the rabbit airway tree, the variation about this average, and also how the structures of mature and immature airway trees compare.
METHODS
Freshly excised lungs from rabbits were degassed in a vacuum chamber, filled with 10% buffered formalin phosphate to an airway pressure of 20 cmH_{2}O, which was maintained with an infusion pump. After 7 days of fixation, 10% of the formalin volume was withdrawn from the trachea and an equal volume of a white silicone rubber material and catalyst (3110 RTV, Dow Corning, Midland, MI) was slowly infused into the airways. The silicone was allowed to cure for 2 days, and then the lung tissue was corroded in a 5.25% sodium hypochlorite solution, leaving the intact silicone cast of the airway tree. Under lowpower magnification, the acinar regions of the airway tree were pruned from the cast. Using a measuring magnifier (catalog no. 813435, Bausch and Lomb, Rochester, NY), we then measured and cataloged the remaining airways according to the following scheme. First, the trachea was denoted by the code 1. The two main stem bronchi were then denoted 11 and10, with 11 being the larger of the two. The nextgeneration airway was denoted by threedigit codes, the first two digits being the same as the parent bronchus and the third digit being 1 (for the larger daughter) or 0 (for the smaller daughter). This procedure was followed all the way to what we perceived to be terminal bronchioles, which were the last airway segments with no alveolar units. Thus, in general, an airway was denoted by the codexxx … xx, where the x is 0 or 1. The larger daughter of this airway was then denoted xxx… xx1 and the smaller daughter xxx …xx0.
Next, the airway codes were analyzed to assign each airway a Horsfield order (H). Inasmuch as H counts from the lung periphery, H for any particular airway is equal to one more than the number of airway bifurcations between it and the terminal bronchiole that is reached by proceeding along the principal daughter airway at each bifurcation. In other words, if the airway in question has the code xxx, …, xx, then the airways with codes xxx … xx1, xxx …xx11, xxx … xx111, etc., are searched for until there is no airway found having the code given by that of the previous airway with 1 appended (meaning that the previous airway is a terminal bronchiole). The number of 1's appended to the original airway code xxx … xx is then 1 less than H for the airway in question. Having found Hfor each airway, the Δ for each airway was simply taken as the difference in H of its two daughters (the Δ of a terminal bronchiole is by definition equal to 0).
In five rabbit lungs, three mature and two immature, we cataloged every single airway in the right upper lobe. In addition, this procedure was followed for all the airways of the major pathway and all the minor daughter airways coming off the main pathway in six mature and six immature rabbit lungs.
RESULTS
We analyzed our data in a variety of ways to try to capture the essential structural features of the rabbit airway tree. First, we calculated, in the five complete lobes studied, the number of terminal airways subtended by each individual airway. This gives a measure of how much of the lung parenchyma is served by each airway and is shown in Fig. 1. There was considerable overlap between the two groups (although we cannot test this statistically because of the small number of animals involved), so we conclude that there is no difference in structure between mature and immature rabbit lungs as far as number of subtended terminal airways is concerned.
A related reflection of airway tree structure is provided by Δ, which is a measure, at any airway bifurcation, of the degree of asymmetry in the tree. Figure 2 shows Δ vs. order, again for all airways in the five lobes studied, together with linear regression lines through the individual data sets. These regression lines do not appear to pass through the middle of the data points, because there are between 2,000 and 3,000 airways in each lobe, so many of the symbols shown in Fig. 2 represent the superposition of numerous individual data points. The great majority of points were clustered around 0 and thus drove the regression. Only 1–4% of the airways had negative Δ. Although this should not, strictly speaking, ever happen, because Δ is supposed to be the difference in pathway order between major and minor daughters, very occasionally the airway identified at a particular bifurcation as being the minor daughter actually had the longer pathway. This may have occurred because of the natural random variation in airway radii. In other words, there is a nonzero probability that the true major daughter at a bifurcation will have an anomalously small radius while its corresponding minor daughter has a large one, causing them to be wrongly identified. Alternatively, some airways with larger diameters may not have the longest pathway.
Figure 3 shows the natural logarithm of airway diameter along the main branch of six mature lungs and six immature lungs vs. order. The immature lungs are shifted downward in a fashion virtually parallel to that of the mature lungs. This indicates that, being a semilogarithmic plot, the airways along the main pathway of the immature lungs were simply scaled versions of the corresponding mature airways.
Figure 4 shows airway diameter vs. order for the airways of the main pathway for the whole lung and the main pathway in the right lower lobe for three mature and three immature rabbits. The smaller airways (those of order 1 through aboutorder 20) seem to have similar structures whether they come from the main pathway of the entire lung or from the main pathway in a minor lobe. Beyond order 20, the airways of a given order in the lobe are smaller than the airways of the same order in the main pathway of the entire lung.
Figure 5 shows the ratio of the diameters of the minor daughter to those of the major daughter at each bifurcation along the main pathway of six mature and six immature lungs. The diameter of the minor daughter is almost always smaller than that of the major daughter, as expected. The diameter ratio decreases with increasing order, meaning that the two daughters are more similar in size in the distal airways.
Table 1 lists the mean and SD of Δ and diameter vs. order for all three mature lobes. Mean Δ increases quite linearly with order and so does its SD up to about order 28(after which there were probably too few airways to obtain a good estimate of SD). Table 2 shows the corresponding information for the two immature lobes studied. Inasmuch as immature lungs are smaller than mature lungs, we cannot pool all the five lobes together but must consider them in two groups. Airway diameter increases roughly linearly with order, as does its SD up to about order 20. Again, for higher orders we probably do not have enough airways to obtain good statistics on airway diameter. Tables 3, 4and 5 present airway diameter vs. generation number from the central to the peripheral airways for lobes and the main axial pathway of the lung. This format enables investigators to use the rabbit airway data with a Weibel model.
DISCUSSION
The present stateoftheart in the characterization of airway tree morphology is the Horsfield asymmetric branching scheme, which has been described in a variety of species including humans (5, 6, 9,10, 15) and has been used in a number of computer models of the lung (26, 8, 11). Some of these models are stochastic, allowing for random variations in airway properties (2, 3, 11). These models have shown that the bronchoconstrictive response of the lungs is determined not only by mean airway properties (e.g., dimensions and amounts of smooth muscle) but also by how these properties are distributed about the mean. Most modeling studies of the airway tree have merely guessed at what this variability might be. A notable exception is that of Kitaoka and Suki (7), who modeled a stochastic airway tree by assuming that airway radius has a powerlaw relationship to mean flow along the airway. They were able to reproduce the statistical features of the data of Raabe et al. (10). However, their model was not specified in terms of the Horsfield orderΔ scheme, and they did not use it to calculate lung function characteristics. The purpose of our study was to characterize the airway tree of rabbit lungs in terms of mean structural properties and their variation about the mean and to characterize this variation in a way that can be readily incorporated into an orderΔtype model. To do this, we determined the branching pattern and diameter of every airway in a complete lobe in mature and immature lungs. This enabled us to also address the further issue of whether there are important differences between the airway structures of mature and immature lungs.
If the airway tree structure is established at birth, one would expect the total number of terminal airways in the lung to be about the same for mature and immature animals. In particular, one would expect the total number of terminal airways to be the same between mature and immature animals. In support of this is our observation that, in the five rabbits we studied, no major differences were seen between the total number of terminal airways and the way in which the number of subtended airways varied with airway order (Fig. 1). There was some degree of interanimal variation, however. For example, the total number of terminal airways varied between animals by a factor of ∼2 from ∼800 to ∼1,600 (Fig. 1). This presumably could reflect natural variation. Alternatively, this observation may reflect the fact that we were not entirely consistent in our identification of terminal airways. When the lung casts were prepared, we had to judge where the conducting airways ended and the parenchyma began, so there may have been some variation in precisely which structures were identified as terminal airways. However, we can estimate how such variation would lead to variation in the total number of terminal airways identified. In a dichotomously branching airway tree, each additional generation in the tree doubles the number of airways. Therefore, uncertainty by one generation in the identification of a terminal airway in such a tree could lead to variation in the total number of airways by a factor of 2. Of course, the airway tree is not dichotomously branching in the rabbit, so uncertainty by one generation produces a less than twofold variation in the number of terminal airways. However, the value of Δ near the terminal end of the tree is close to 1 (corresponding to dichotomously branching), so it would seem that the identification of terminal airways in these rabbit lobes was done consistently between lobes to within about one airway generation.
If the structures of the five lobes were identical but the terminal airways in some of the lobes were one generation different from those in the others, then the curves in Fig. 1 would be superimposed by horizontally shifting them by one order. This is clearly not possible; the higherorder airways subtending a given number of terminal airways differ in their orders by up to 5 or more. We can thus conclude that, with respect to branching pattern, the airway structures in these lobes were indeed fundamentally different and that the variation between lobes evident in Fig. 1 is not merely due to variations in the identification of terminal airways. Specifically, the dependence of Δ on order must have been different among the different animals. This is further supported by Fig. 2, which shows subtle but significant differences in the regression slopes of Δ on order.
We thus conclude that different animals have different mean airway branching structures. The next question is whether these structural differences are larger between mature and immature animals than between animals of the same level of maturity. Figures 1 and 2 suggest not. Additionally, as Fig. 3 shows with respect to airway diameters along the main pathway, the immature airway tree is essentially a small version of the mature tree. In other words, airway diameter at a given order in a mature lung is essentially a constant factor larger than its immature equivalent. Thus the branching structure of the airways appears to be formed at birth and to be preserved throughout life, implying that maturation occurs via a selfsimilar growth process.
Another structural issue in the airway tree is whether it obeys a fixed set of scaling rules at every level. In other words, is the branching pattern, on average, purely a function of order? Figure 4 suggests that this is the case for the distal lung, up until about order 20, because the relationship of diameter to order is the same for the main pathway of the lung as it is for the main pathway of the right upper lobe (which does not include the airways of the main pathway of the lung). However, for airways larger than about order 20(i.e., the proximal part of the tree), the relationship for the main pathway diverges somewhat from that of the lobe, both in mature (Fig.4 A) and immature (Fig. 4 B) lungs. That is, there appears to be a subtle break point at about order 20, supporting the notion that distal airways follow a scaling rule different from that of proximal airways. In particular, the proximal lung is more asymmetric than the distal lung, in terms of Δ and in terms of daughter diameter ratios. The suggestion of slight differences between proximal and distal lung regions is also seen in Fig. 5, which shows how minor daughter airways scale in terms of radius compared with their respective major counterparts. There is a noticeable change in the orderindependent scaling in the airway structure around order 20. Thus, toward the periphery (beloworder 20), the structure appears selfsimilar, inasmuch as the subtrees along the main pathway and in the lower lobe are statistically similar. For the more central airways (beyondorder 20), the asymmetry of the structure changes. Although the functional significance of this transition is not clear, one may speculate that the role of the more central airways is to direct flow appropriately to different macroscopic regions of the lung according to their respective volumes. In contrast, within each region, the peripheral airways assume a uniform spacefilling role.
The most important new information provided by our study concerns the variability of airway properties throughout the tree, between animals, and as a function of maturation. Such information is crucial for a complete understanding of overall lung function. Modeling studies (13, 8, 11) have demonstrated that consideration of the distribution of airway properties, in addition to their mean values, is crucial for a complete understanding of lung function. Our data can now be used to construct stochastic models of the airway tree based on the Horsfield asymmetric branching morphology while also allowing for random variation within this structure. Tables 1 and 2show how the SDs of Δ and diameter scale with order and indicate that, to a reasonable degree of approximation, the coefficients of variation (CV) of these quantities can be taken as approximately constant below order 20. Over this range, Δ has a mean CV of ∼0.3, while diameter has a mean CV of ∼0.25. Thus there seems to be a certain inherent heterogeneity to the rabbit lung that is independent of maturation and scale. Part of this heterogeneity may have been preprogrammed into the airway structure from the beginning of its development, and part may have arisen through chance interactions with the environment as development progressed. In any case, the independence of variability with order conforms to the theory advanced by West (13) that fractal structures are relatively insensitive to structural errors occurring at any particular level (compared with structures following a fixed scaling rule with generation), supporting the notion that the peripheral airways are selfsimilar and fractal.
One concern we had about our data was the rather large total number of airway orders identified in our rabbit lungs (∼40). This seemed somewhat high given the size of the rabbit compared with airway tree analyses reported in larger species (6, 10). We wondered whether we might have identified minor branching structures in the airway trees of our rabbits that were functionally insignificant and that might have been ignored in previous studies of the airway tree by workers using other morphometric methods. Including such minor branches would lead to a higher total number of airway generations than would otherwise be the case. We reason that an insignificant airway branch would be characterized by a large difference between the diameters of the two daughter airways at the bifurcation leading to the branch; the minor daughter leading to the insignificant branch should have a much smaller diameter than the major daughter leading to the remainder of the lobe. To test whether we had included negligible branches in our analysis, we calculated the minortomajor daughter diameter ratio at every bifurcation in each complete lobe studied and defined insignificant airways as those subtended by a daughter airway having a diameter ratio less than a specified threshold. We found almost no insignificant airways when this threshold value was <0.2, and then the number of insignificant airways climbed steadily to include all the airways in the lobe when the threshold reached 1. The number of small daughter airways defined in this way behaved similarly with the value of the threshold. Therefore, we were not able to identify any clear group of airways in our lobes that could be defined as negligible compared with the others in terms of the amount of lung they subtended.
In summary, we analyzed the complete airway tree in three mature and two immature rabbit lung lobes. We found some variation in branching asymmetry and airway diameter at a given order between animals but no evidence of systematic differences in structure between mature and immature lungs. We found some evidence of a difference in the branching structure of the peripheral vs. the central parts of the airway tree (the break point being around order 20). We also determined the nature of the variation in Δ and diameter as a function of order, which should be valuable for the development of computer models seeking to encapsulate the naturally occurring regional variation in airway geometry in the normal rabbit lung.
Acknowledgments
This work was supported by National Heart, Lung, and Blood Institute Grant HL48522.
Footnotes

Address for reprint requests and other correspondence: J. H. T. Bates, Colchester Research Facility, 208 South Park Dr., Suite 2, Colchester, VT 05446 (Email:jhtbates{at}zoo.uvm.edu).

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 Copyright © 2001 the American Physiological Society