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 80s 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
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 80s experimental period. Lung resistance also steadily increased throughout the 80s 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 constantphase 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 selfsimilarity 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 selfsimilarity 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.
Airway tree.
In Horsfield’s model of the canine lung (15), the airways are grouped into 47 orders, with all airways within each order having the same dimensions. The trachea is of order 47, the terminal bronchioles are oforder 5, and the alveoli are oforder 1, giving a total of 300,153 individual airways leading to 150,077 alveolar (tissue) units. The equations governing the airway dimensions in the Horsfield model are itemized in Table 1. Each airway in the Horsfield model divides into two branch airways that, in general, are of different sizes. This asymmetrical branching pattern is determined by a connectivity parameter (Δ) such that, for an airway of thek ^{th} order, one branch is of order k − 1, and the other oforder k − 1 − Δk. The values of Δk for each order of airway are specified in Table 1.
The measurements comprising Horsfield’s model were made on lungs inflated to total lung capacity (TLC) at a pressure (Ptlc) equal to 25 cmH_{2}O. The dimensions at any lower inflation pressure are, of course, smaller because the airway walls are compliant. The volume of a simple compartment with linearly elastic walls is proportional to the transmural pressure, and therefore the linear dimensions change in proportion to the cube root of transmural pressure. Airway walls, however, especially those of the central airways, contain considerable amounts of relatively noncompliant cartilage. By making the simplifying assumption that changes in transmural pressure stretch only the compliant portion of the wall material, the fractional change in overall linear dimension (radius or length) caused by a small change in transmural pressure is given by
Impedance.
Because our model was only required to simulate the impedance of the lung to small amplitude lowfrequency (≤6Hz) oscillations applied at the trachea at fixed mean lung volumes, we ignored volumedependent nonlinearities and turbulent flow effects. The series flow impedance (Zf; in cmH_{2}O ⋅ l^{−1} ⋅ s^{−1}) of a single airway is, therefore, given by the linear flow equation
The impedance of each tissue unit (Zti) is defined by the structural damping model, in which the real and imaginary parts of the impedance are related through a constant hysteresivity factor (η) (10). Zti is therefore
The respiratory system impedance observed at the trachea is composed of contributions from all the airways and tissue units that comprise the lung model. Because of the branching structure of the airway tree, wherein every airway connects to two smaller branch airways (until the most peripheral airways, which connect to a tissue unit instead of further branches), it is possible to implement a recursive algorithm that, starting from the trachea, traverses each branch in the tree. The input impedance (Zin) looking into any airway segment comprises the series Zf and Zs, together with a parallel combination of the two branch input impedances, Zb_{1} and Zb_{2}. By making the simplification that Zs acts at a single point halfway along the airway length (an assumption that is valid at the low frequencies that we consider because even at 6 Hz the wavelength is much greater than the airway length), Zin becomes
To compute Zin for any particular airway, it is necessary to first compute the branch input impedances Zb_{1} and Zb_{2}. These can be obtained by recursively evaluating Eq. 7 for each branch in turn. These calculations also require knowledge of their respective branch Zin, and thus the recursion process must be continued until a tissue unit is reached that contains no further branches and for which Zo = Zti. The recursion then unwinds itself and the Zin of each branch up the tree can be computed, finally providing the overall Zin at the trachea. Note that it is not possible to employ any of the computational shortcuts that can be invoked for a selfconsistent tree (7, 17) because the stochastic heterogeneity that we introduce destroys the selfsimilarity in Horsfield’s branching pattern.
Bronchoconstriction.
In a previous experimental study (3), a bolus of histamine was administered to the vena cava and so reached the lung periphery with little dispersion. This elicited a rapid response of contractile elements in the lung periphery. After some transit delay to reach the bronchial circulation, the histamine then presented a more diffuse profile to the central ASM, eliciting a more gradual increase in airway resistance. The magnitude of the parenchymal response was similar to that seen in tissue studies (6, 33). In our simulation model, we attempted to reproduce these features by using two time courses, Dc(t) and Dp(t) for the reactions in the central airways and peripheral tissues, respectively. Each time course was modeled as a twoexponential curve
For airways close to the periphery (orders 5 and less), a combination of the peripheral Dp(t) and central Dc(t) time courses was used to represent some degree of overlap in the regions affected by the pulmonary and bronchial circulations (20). We arbitrarily quantified this gradual overlap in the simplest way by means of a linear transition zone; thus
We modeled bronchoconstriction in the periphery tissue by increasing Eti and hysteresivity parameters with the Dp(t) according to
Following Moreno et al. (30), we computed the constricted radius (r
_{iC}) of an airway after ASM shortening according to
The fraction c decreases from the trachea to the peripheral airways according to the empirical piecewise function (simplified from that in Ref. 12)
The parameter ASM_{max} defines the proportional shortening of ASM in the airway wall when the bronchoconstriction timecourse function D(t) equals unity. It is a simplifying parameter embodying all factors contributing to the ASM responsiveness. For the simulations reported here, we set ASM_{max} = 0.3, which is below the critical value identified by Wiggs et al. (38), where the smallest airways close completely under maximal bronchoconstriction [D(t) = 1]. Note, however, that, when we introduce heterogeneity by stochastically varying ASM_{max}, this condition no longer holds and some airways may close completely.
Bronchoconstriction changes the airway Cw by increasing the stiffness of the ASM (29). We modeled this increasing stiffness by means of
Stochastic inhomogeneity.
There are several parameters in our structural lung model that can be expected to vary from unit to unit within the lung. Variability in airway radius and wall thickness may be due to the presence of secretions as well as to intrinsic variability in the airway structure. The response of individual bronchi to bronchoconstricting agents may also vary, due either to intrinsic ASM factors or to circulation (i.e., drug delivery) differences. Indeed, recent studies have shown a wide degree of heterogeneity in the sensitivity of ASM to bronchoconstricting agents (4). Similarly, the responses of the tissue units can be expected to have some stochastic distribution. In this study, we are simply interested in generating heterogeneity, without regard to any detailed mechanism by which it is achieved. Therefore, we modeled the stochastic variability in the lung by applying a normally distributed random perturbation to the response parameter associated with each airway and tissue unit in the entire lung tree (i.e., the ASM_{max} and ΔEmax, respectively). The SD of the perturbation ranged from 0 (deterministic) to 40% of each parameter’s nominal value.
Simulation protocol.
We implemented the model in the Oberon2 (ETH, Zurich) programming language. Simulations were performed on an IBM RS6000/390 computer. On this machine, ∼7 s of computation are required to calculate the impedance of the lung model at each simulation instant. Approximately 20 MB of memory were required to contain the complete airway tree structure.
After the protocol of the experiments reported in Ref. 3, we computed the time course of changes to the overall lung impedance for a period of 80 s after administration of a bronchoconstricting bolus. Results were calculated at 5s time steps during the simulation period. The simulation protocol proceeded as follows.
First, the lung model was inflated to the desired operating pressure of 5 cmH_{2}O by adjusting all airway dimensions according to Eq. 1 . Next, randomization factors were generated for each airway and tissue element in the lung according to the desired degree of heterogeneity. Third, for each time step of the simulation, the airways and tissue units were bronchoconstricted according to the value of the simulated bronchoconstriction time course at that instant. Finally, the lung impedance at each time step was computed by recursively following each path in the model and combining the impedances of each branch as described above.
In accordance with our earlier experimental protocol (3), the lung impedance was computed at both 1 and 6 Hz. The impedance at 1 Hz was fit to a model comprising both elastance and resistance, whereas at 6 Hz only the real part was retained. In the results that follow, we denote the lung’s dynamic elastance at 1 Hz by El
_{1}, the overall lung resistance at 1 Hz by Rl
_{1}, and the resistance at 6 Hz by Rl
_{6}. We take the difference Rl
_{1} − Rl
_{6} to be an approximation to the tissue resistance at 1 Hz. From these values we define the measured hysteresivity η′ to be
To examine the variability in response obtained from different realizations of the heterogeneous lung model, we performed 10 simulations at each configuration with different stochastic randomizations, then computed the mean and SD of each set of 10 simulated signals.
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 Rl _{6}, El _{1}, Rl _{1} − Rl _{6}, and η′ 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 Rl _{6}, El _{1}, and Rl _{1} − Rl _{6} shown in Fig. 3 all continue to increase progressively throughout the 80s 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 η′ in Fig. 3 also continue to increase progressively. This is in contrast to our experimental findings, in which η′ reached a plateau at ∼25 s.
To shed light on this discrepancy, we investigated the behavior of a simple twobranch 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 Rl _{6} and Rl _{1} − Rl _{6}, which produces a corresponding peak in η′. Once closure occurs, however, the model reverts to homogeneity again and the η′ anomalies disappear. Thus there is a critical range of heterogeneous airway narrowing that elevates η′. In the complete stochastic lung model, we can expect progressively more airways to enter this range as bronchoconstriction proceeds, which would explain why the η′ 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 η′ 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 0.1mm 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 El _{1} and Rl _{1} − Rl _{6} yet has attenuated the increase in η′.
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 airwaybranching 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 highfrequency 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 (El
_{1}), the resistance at 6 and 1 Hz (Rl
_{6} and Rl
_{1} − Rl
_{6}), and the η′] closely follows the applied bronchoconstriction time course, as shown by the lower curve in each panel of Fig. 3. (Note that the value of η′ calculated by means ofEq. 18
is approximately
This result matches those of a previous experimental study (3), in which the dynamic elastance El _{1} and resistance Rl _{1} − Rl _{6}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 El _{1} 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 El _{1} and Rl _{1} − Rl _{6} 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 El _{1} and Rl _{1} − Rl _{6} 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 El _{1} 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 El _{1} and the decrease in Rl _{6} 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 η′ 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 El _{1} and Rl _{1}− Rl _{6} 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 η′ 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 η′.
Model limitations.
Our model represents the structure of the lung with as much accuracy as currently available data and our computational resources permit. However, several simplifying assumptions were made that may have had an effect on the accuracy of our simulation results. Although we believe that our overall conclusions, which follow from the introduction of stochastic heterogeneity in the model, are robust, it is worth pointing out the assumptions to which our model is sensitive.
The airway wall thickness has a strong influence on the amount of airway closure induced by a particular level of bronchoconstriction. It also affects the airway Cw. Our values of Cw are, in general, similar to those predicted by the formulas used by Suki et al. (36), which explicitly represent the wall thickness and the elastic properties of the wall tissues. Indeed, at TLC the total Cw equals 3.7 ml/cmH_{2}O for both our formulations and those of Suki et al., but at a positive endexpiratory pressure of 5 cmH_{2}O the total Cw under our scheme increases to 6.3 ml/cmH_{2}O. This increase in Cw as the lung volume decreases accounts for the increase in slope of the airway pressurevolume relationship at low volumes.
Determining the airway wall viscance is problematic because it requires assumptions about the wall thickness, composition, and the effect of the interstitial tissues. Although we did experiment, including estimates of the wall viscance [by using the formulation of Suki et al. (36)], it made little difference to the simulation results at the frequencies that we considered for this study. We therefore decided not to include it in our simulation.
The accurate simulation of the bronchoconstriction time course requires numerous assumptions about processes such as the dynamics of drug dispersal in the circulatory system and the dose responsiveness of the ASM and peripheral tissues. Because the purpose of this paper was to examine how heterogeneity affected the dynamic impedance of the lung, we decided to subsume all these processes into a simple time course that, in itself, matched the experimental static response of the lung to bronchoconstriction.
The form of the heterogeneity itself was chosen arbitrarily to conform to a Gaussian probability distribution function. There are almost no data in the literature to support or refute this choice. One might argue that mechanical interdependence should induce some degree of spatial coherence in regional mechanical properties. On the other hand, secretions within an airway can affect its resistance drastically in a way that would seem to be random and uncorrelated with respect to neighboring airways. It is also possible that a skewed distribution of airway properties may be more appropriate than the symmetric Gaussian distribution employed here. The identification of these distributions from morphometric studies would therefore appear to be an important area of research for the future (see Ref. 4).
The effect of lung volume on the response to bronchoconstriction is dependent on several factors, including the relationship between pressure and airway dimension, the way in which forces of interdependence and parenchymal tethering change with inflation, and how the ASM constriction characteristics are affected by changes in airway dimension. Our model contained no interdependence mechanism, partly because we wished to demonstrate that inhomogeneous and progressive airway closure alone could account for the increase in effective dynamic impedance seen during acute bronchoconstriction. As for parenchymal tethering, several recent studies have shown how the tethering forces can modulate the ASMconstrictive forces such that the effective ASM shortening is increased at low lung volume (21, 25). Because we subsumed all the factors relating to maximal constriction into the factor ASM_{max}, we did not include any separate tethering mechanism in our model.
The functional relationship between pressure and airway dimension (Eq. 1 ) used here was chosen because of its simplicity. The use of a different formulation, such as that Lambert proposed for Weibel’s model (22), would have had little bearing on the primary result of this study, which was to show how the progressive introduction of airway heterogeneity during bronchoconstriction could affect the overall lung impedance. However, a different dimensionpressure relationship could have changed the precise values of resistance and elastance that we obtained. As implemented, our simulation results were very similar to previous experimental results (3), confirming the validity of the model. Only Rl _{6} was smaller (by a factor of 4 at baseline) from all of the experimental data. We could have tried to optimize the Horsfield dimension to reduce this difference, but because the individual dogs themselves showed a wide range of responses, we decided to let the model stand on its own.
In conclusion, we have developed a computational model of the lung that embodies much of the physical structure of the lung and is able to reproduce the key features of data obtained in a previous experimental study of acute induced bronchoconstriction in dogs. Simulations with our model strongly support the notion that much of the apparent increase in Eti and resistance observed during bronchoconstriction can be attributed to the development of mechanical heterogeneity throughout the lung. However, our results also indicate that such inhomogeneities must develop to the extent that a significant proportion of the lung periphery is isolated from the central airways by peripheral airway closure before much effect on overall lung impedance is seen. We also found that, by making the airways snap shut once a critical degree of narrowing had been achieved, the time course of η′ more closely resembled the experimental.
Acknowledgments
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 ChercheurBoursier of the Fonds de la Recherche en Santé du Québec.
Footnotes

Address for reprint requests: J. H. T. Bates, MeakinsChristie Laboratories, 3626 St. Urbain St., Montreal, Quebec, Canada, H2X 2P2 (Email: Jason{at}Meakins.LAN.McGill.ca).
 Copyright © 1997 the American Physiological Society