Journal of Applied Physiology
Vol. 82, No. 5, pp. 1616-1625, May 1997
GAS EXCHANGE, MECHANICS, AND AIRWAYS

Effect of stochastic heterogeneity on lung impedance during acute bronchoconstriction: a model analysis

C. William Thorpe and Jason H. T. Bates

Meakins-Christie Laboratories and Department of Biomedical Engineering, McGill University, Montréal, Québec, Canada H2X 2P2

ABSTRACT

INTRODUCTION

METHODS

RESULTS

DISCUSSION

ACKNOWLEDGEMENTS

FOOTNOTES

REFERENCES

ABSTRACT

Thorpe, C. William, and Jason H. T. Bates. Effect of stochastic heterogeneity on lung impedance during acute bronchoconstriction: a model analysis. J. Appl. Physiol. 82(5): 1616-1625, 1997.In a previous study (J. H. T. Bates, A. M. Lauzon, G. S. Dechman, G. N. Maksym, and T. F. Schuessler. J. Appl. Physiol. 76: 616-626, 1994), we investigated the acute changes in isovolume lung mechanics immediately after a bolus injection of histamine. We found that dynamic resistance and elastance increased progressively in the 80-s period after injection, whereas the estimated tissue hysteresivity reached a stable plateau after ~25 s. In the present study, we developed a computer model of the lung to investigate the mechanisms responsible for these observations. The model conforms to Horsfield's morphometry, with the addition of compliant airways and structural damping tissue units. Using this model, we simulated the time course of acute bronchoconstriction after intravenous administration of a bolus of bronchial agonist. Heterogeneity was induced by randomly varying the values of the maximal airway smooth muscle contraction and the tissue response to the agonist. Our results demonstrate that much of the increase in lung impedance observed in our previous study can be produced purely by the effects of airway heterogeneity. However, we were only able to reproduce the plateauing of hysteresivity by assigning a minimum radius to each airway, beyond which it would immediately snap completely shut. We propose that airway closure played an important role in our experimental observations.

pulmonary mechanics; airway resistance; tissue resistance; ventilation inhomogeneity; airway closure; hysteresivity

INTRODUCTION

THE APPEARANCE OF REGIONAL mechanical heterogeneity throughout the lung after bronchoconstriction has been well established in studies using the alveolar capsule (8, 23) and the alveolar capsule oscillator (1, 5, 28) techniques. Widespread heterogeneity in the responses of individual airways has also been demonstrated by in vitro studies on lung explants (4). In principle, such inhomogeneities could have an influence on the overall mechanical properties of the lung in the manner first elucidated by Otis et al. (31). In a recent study, the changes in canine lung mechanics occurring during the 80 s after a bolus injection of histamine at constant lung volume were investigated (3). The static elastance rose to a plateau within 20 s, but at high histamine dose the dynamic elastance continued to increase throughout the 80-s experimental period. Lung resistance also steadily increased throughout the 80-s period, but the hysteresivity followed a similar time course to that of static elastance. We postulated that these results could be explained by the progressive development of mechanical heterogeneity in the airways that increased the dynamic impedance but not the static recoil of the lung. However, the precise nature of this heterogeneity is difficult to ascertain in an intact lung. We therefore decided to simulate the processes of bronchoconstriction in a computer model that was as anatomically accurate as present knowledge permits but, in addition, allowed us to introduce controlled levels of heterogeneity. The model was based on the Horsfield characterization of the canine airway system (15), adapted to allow random variations in regional properties. In this study we use our model to investigate what kinds of heterogeneity in the lung can reproduce the observations of our previous experimental study (3).

METHODS

We represented the lung as a branching structure of compliant airways on the basis of Horsfield's morphometry, terminating each branch with a constant-phase tissue unit. Because we wanted to examine the effect of distributed inhomogeneities, every individual airway and tissue unit in the tree were explicitly represented in the computational model. This is in contrast to some other implementors, who have taken advantage of the self-similarity of the tree by storing only the main branch together with the appropriate connecting information (7, 12, 18, 38). We could not do this, however, because the self-similarity was destroyed by the heterogeneities that we applied. We used a singly linked tree structure in which each node, representing a single airway, was linked to the two immediately peripheral airways. Each branch terminated in a tissue unit, which itself contained no links. To reduce the memory requirements, the nominal parameter values for each airway order were stored in a separate parameter array so that the nodes in the tree structure only needed to contain values representing the deviation of that airway from its nominal characteristics.

Table 1. Dimensions of Horsfield's branching model of canine lung Order k   Diameter, cm Length, cm 47 2 2.1 20.0 46 2 2.1  0.75 45 2 2.03  1.8 26 . . 44 10 e0.0685k  2.87 e0.0538k  2.35 14 . . 25 4 e0.0685k  2.87 e0.0538k  2.35  7 . . 13 4 e0.115k  3.47 e0.0538k  2.35  2 . . 6 0 e0.115k  3.47 e0.0538k  2.35 k, Order of airways; , connectivity parameter. Measurements were performed on a lung inflated to 25 cmH2O before fixing (see text for details).

(1)

(2)

1

1

(3)

1

1

3

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

RESULTS

The curves displayed in Fig. 3 show the behavior of the model with and without stochastic variability in the individual responsiveness factors. Five sets of curves are shown, corresponding to the time courses of the simulated RL₆, EL₁, RL₁  RL₆, and and the percentage of tissue units that become isolated from the trachea by airway closure. In each panel, the lower solid curve corresponds to the case of zero applied heterogeneity (i.e., the deterministic Horsfield model). The upper two solid curves in each panel show the results obtained with different levels of stochastic heterogeneity, with SD equal to 20 and 40% of the nominal values, respectively. The dashed lines shown in three of the panels indicate the true tissue responses, which were obtained by combining all the Zti in parallel at each time step during the simulation. The ten different realizations of the stochastic lung model gave very similar results, with the maximum SD from the mean responses shown being only 1%.

With the larger degree of stochastic heterogeneity, the curves of RL₆, EL₁, and RL₁  RL₆ shown in Fig. 3 all continue to increase progressively throughout the 80-s simulation period. This is the key feature observed in a previous experimental study (3) that led us to postulate that developing heterogeneity plays an important role in the acute bronchoconstriction time course. However, our simulated curves of in Fig. 3 also continue to increase progressively. This is in contrast to our experimental findings, in which reached a plateau at ~25 s.

To shed light on this discrepancy, we investigated the behavior of a simple two-branch airway model as closure is approached, first in one airway and then in the other. The results are shown in Fig. 4, which contains the same five panels as Fig. 3. For this simulation, we kept Zti constant. The points at which the two airways close are indicated in each panel. The key finding is that airway closure is immediately preceded by a sharp peak in both RL₆ and RL₁  RL₆, which produces a corresponding peak in . Once closure occurs, however, the model reverts to homogeneity again and the anomalies disappear. Thus there is a critical range of heterogeneous airway narrowing that elevates . In the complete stochastic lung model, we can expect progressively more airways to enter this range as bronchoconstriction proceeds, which would explain why the curves shown in Fig. 3 do not reach a plateau. Note that, after both airways have closed in the simulation shown in Fig. 4, the parameter values reflect only the impedance of the central airway.

The above result suggests that our simulated curves will reach a stable plateau if we prevent the degree of heterogeneity from ever reaching the critical range. We did this by assigning a minimum radius to each airway so that any further narrowing resulted in immediate complete closure. This is reminiscent of the sudden formation of liquid bridges in airways shown by the analysis of Halpern and Grotberg (13). Figure 5 shows the results of applying a 0.1-mm closure threshold to each airway in the complete model together with the uppermost curves from Fig. 3 for comparison. The closure threshold mechanism produces more total airway closure, and more pronounced increases in EL₁ and RL₁  RL₆ yet has attenuated the increase in .

Figure 6 shows the average radius of the airways in the model after the maximum bronchoconstriction during the simulation has been attained. As expected, smaller airways constrict proportionally more than larger ones, with the same average reduction in size for both the deterministic and stochastic implementations. Also shown in Fig. 6 is the proportion of airways that become closed for each order of the airway tree. Closure only occurs in airways smaller than order 20 (2 mm diameter). The closure threshold affects airways smaller than 1 mm in diameter.

DISCUSSION

Although experimental data obtained from a single port such as the trachea are insufficient to uniquely ascertain the distributed structure of the lung, it is possible to construct a distributed model of the lung on the basis of physical measurements of its structural components such as the airway dimensions described by Weibel (37) and Horsfield et al. (15). A model of lung impedance based on these descriptions, therefore, contains many compartments that together make up the overall response. Weibel's airway-branching scheme is symmetrical, however, so only series heterogeneity (implicit in the 23 airway generations) is represented. Horsfield introduced asymmetric branching, which provides a distribution of path lengths and, consequently, some degree of both series and parallel heterogeneity. Several authors have developed models of lung impedance on the basis of one or other of these structures. For example, Pedley et al. (32) analyzed the pressures and flows that can be expected at each generation of Weibel's (37) airway tree when high-frequency oscillations are applied at the mouth. Wiggs et al. (38) similarly based their model of airway narrowing on Weibel's morphometric data. Fredberg et al. (7, 9) and Jackson et al. (17) used Horsfield's asymmetrical branching model in their simulations of the frequency dependence of airway impedance. These studies simplified the tree structure by having all airways in each equivalent generation of the model identical, thereby making the implicit assumption that any natural variability somehow averages out when the overall response is considered. However, as pointed out by Bates (2), the combined impedance of parallel branches depends not only on the mean impedance of those branches but also on their distribution about the mean.

Jackson et al. (18) utilized Horsfield's model and were able to compute a measure of ventilation inhomogeneity within the lung. They found it to be surprisingly low, probably because Zti (which were all equal) dominated the differences in path impedances. Their overall impedance showed a good correspondence with experimental data in the frequency range 5-60 Hz. Lutchen et al. (24) introduced further heterogeneity in the model by reducing the diameters of specific airway orders encompassing up to 80% of the periphery. They found that extreme levels of diameter reduction were required to change the frequency dependence of lung impedance. Hantos et al. (14) examined peripheral inhomogeneity by appending 900 parallel peripheral units to a parametric model with a single lumped element for the central airway resistance. Although they assigned widely spread random values to the peripheral units, they found that the overall effect of their inhomogeneity was negligible.

In the model presented here, we started with Horsfield's airway structure but then added heterogeneity by stochastically varying the response magnitude of each airway to a simulated bronchoconstriction time course. Without such heterogeneity (i.e., the deterministic Horsfield model), the computed response [characterized by the elastance at 1 Hz (EL₁), the resistance at 6 and 1 Hz (RL₆ and RL₁  RL₆), and the ] closely follows the applied bronchoconstriction time course, as shown by the lower curve in each panel of Fig. 3. (Note that the value of calculated by means of Eq. 18 is approximately of the true value for the deterministic case because we compute it from RL₁ and RL₆.) In other words, the overall response is simply a scaled version of the response of each individual element. However, when stochastic variability is applied to the airway responses, the overall dynamic response no longer follows the (average) response of the individual elements of the lung. In particular, the apparent Zti (EL₁ and RL₁  RL₆) rises dramatically as airways progressively close. Thus the differences among the three curves in Fig. 3 are due solely to the introduction of heterogeneity in the airway structure as bronchoconstriction proceeds. The average response across all airways in any order remains the same in each case.

This result matches those of a previous experimental study (3), in which the dynamic elastance EL₁ and resistance RL₁  RL₆ continued to rise out to 80 s (at the largest dose of histamine) even though the static elastic recoil pressure of the lung reached a plateau after ~25 s. The increase of EL₁ with bronchoconstriction is well known in the literature (27, 29, 34), and explanations of this phenomenon often involve a presumed stiffening of the lung tissue through interdependence with the airway tree (29, 35). The explanation previously postulated (3) and supported by the results shown in Figs. 3, 4, 5 is that the increases in EL₁ and RL₁  RL₆ are due largely to the development of severe inhomogeneity in the airway tree that progressively isolates parts of the peripheral tissue from the central airways. Indeed, as illustrated in Fig. 4, the increases in EL₁ and RL₁  RL₆ appear to be directly related to the proportion of terminal units that are isolated from the central airways by peripheral airway closure. This process requires no interdependence mechanism to stiffen the lung tissue, such as might be mediated by parenchymal tethering. This phenomenon is supported by the experimental data and stochastic model of Hubmayr et al. (16), who also concluded that increased dynamic impedance was caused by heterogeneous airway closure at the level of the terminal bronchioles.

The phenomenon of airway closure increases overall EL₁ by effectively removing part of the lung tissue. This can be demonstrated with a simple model containing only two compartments connected either in series or parallel; eliminate one of the compartments and the total elastance is raised to equal that of the remaining compartment (Fig. 4). However, changes in elastance tend to be unphysiologically abrupt with such simple models. Even the more complicated airway tree models such as that used by Wiggs et al. (38) demonstrate the same rapid transition in overall impedance as the point of total closure in any one airway generation is approached. In contrast, when the airway diameters are stochastically distributed across the lung, the rise in EL₁ and the decrease in RL₆ are much more reminiscent of experimental findings (3) because the shutting down of lung regions is now more gradual as bronchoconstriction develops.

Perhaps the most interesting insight given by our simulation results is that imposing a minimum diameter that airways can narrow to, beyond which they snap completely shut, results in a time course for that resembles previous experimental observations (3). Indeed, such a closing process in vivo is to be expected in view of the liquid bridge formation that is known to rapidly occur in very narrow airways (13). This process effectively limits the degree of heterogeneity that can develop in the lungs because once the critical diameter has been reached an airway closes off completely, thereby eliminating the downstream segment completely while leaving the remainder of the lung to act as a more homogeneous whole. Lutchen et al. (24) also found that there was a progressive change from homogeneous through inhomogeneous and back to homogeneous behavior in their model as an airway was progressively constricted. In our simulation we found that a limiting diameter of 0.1 mm gave quite realistic results. Increasing the limiting diameter beyond 0.1 mm resulted in too much airway closure such that EL₁ and RL₁  RL₆ rose to unrealistic values. This implicates the smaller airways as being the major site of inhomogeneity in the airway tree. Interestingly, in a previous experimental study (3) it was found that rose transiently to a large value in the two most reactive dogs when studied at a low lung volume. In view of the current modeling results, we might interpret these observations as being due to some of the larger airways being able to constrict in an extremely heterogeneous manner, while still remaining patent. By the mechanism elucidated in Fig. 4, this would produce a large increase in .

ACKNOWLEDGEMENTS

This work was supported by the Medical Research Council of Canada and the J. T. Costello Memorial Research Fund. J. H. T. Bates is a Chercheur-Boursier of the Fonds de la Recherche en Santé du Québec.

FOOTNOTES

Address for reprint requests: J. H. T. Bates, Meakins-Christie Laboratories, 3626 St. Urbain St., Montreal, Quebec, Canada, H2X 2P2 (E-mail: Jason{at}Meakins.LAN.McGill.ca).

Received 11 June 1996; accepted in final form 18 December 1996.

REFERENCES