Abstract
In 11 isolated dog lung lobes, we studied the size distribution of recruited alveolar volumes that become available for gas exchange during inflation from the collapsed state. Three catheters were wedged into 2mmdiameter airways at total lung capacity. Smallamplitude pseudorandom pressure oscillations between 1 and 47 Hz were led into the catheters, and the input impedances of the regions subtended by the catheters were continuously recorded using a wave tube technique during inflation from −5 cmH_{2}O transpulmonary pressure to total lung capacity. The impedance data were fit with a model to obtain regional tissue elastance (Eti) as a function of inflation. First, Eti was high and decreased in discrete jumps as more groups of alveoli were recruited. By assuming that the number of opened alveoli is inversely proportional to Eti, we calculated from the jumps in Eti the distribution of the discrete increments in the number of opened alveoli. This distribution was in good agreement with model simulations in which airways open in cascade or avalanches. Implications for mechanical ventilation may be found in these results.
 atelectasis
 lung elastance
 avalanches
 power law
 percolation
 gas exchange
the inspired air at low lung volumes is preferentially distributed to the upper regions of the lung as a result of the presence of airway closure (19). Airways start to close off when lung volume is lowered below the closing volume (CV) (16). In normal lungs, functional residual capacity (FRC) represents a higher lung volume than CV; hence, during normal breathing, endexpiratory lung volume does not reach CV, and closure does not take place. However, in the immature lung (33), with advancing age (16), in obesity (11), in emphysema (10), and possibly in other lung diseases such as asthma (28), closure may occur during normal breathing at end expiration. The transpulmonary pressure (Ptp, defined as airway pressure minus pleural pressure) at which the closed airways reopen during inspiration is always higher than the Ptp at which closure develops (22). Thus closure can easily lead to an inhomogeneous alveolar ventilation and, hence, an impaired gas exchange (4).
With regard to lung function in the presence of airway closure, the most important quantity is the amount of alveolar volume available for gas exchange. This alveolar volume is decreased at end expiration and is recruited during inspiration when airways reopen. Whereas the physical factors determining the actual process of closure and reopening in individual airways have been studied in great detail (8, 13, 23, 24), very few studies have addressed how airways reopen in situ (20, 22, 26, 34). The fact that airways constitute a tree structure may lead to interactions among reopening of airway segments that are otherwise spatially well separated. It is not clear how such a spatial interaction during the reopening process can influence the distribution of recruited alveolar volumes and, hence, gas exchange in the lung.
Recently, Peták et al. (26) and Otis et al. (21, 22) studied airway closure and opening by measuring the terminal airway resistance (R_{t}) during deflation and inflation and found that, during inflation, R_{t} decreased in a series of discrete jumps. A statistical interpretation of this process was provided by Suki et al. (32). According to this interpretation, airways open in cascades or avalanches triggered by overcoming a hierarchy of critical opening threshold pressures along the airway tree. More recently, Barabási et al. (3) developed an analytic statistical mechanical model of the first avalanches during an inflation by mapping the inflation problem to a percolation problem in a tree structure. This model has been further developed by Sujeer et al. (30) to include all avalanches during an inflation and to predict the distribution of the sizes of alveolar volumes that open via avalanches. Their simulations predicted that this distribution is wide following a power law and is independent of airway wall and alveolar tissue elasticity.
The purpose of this study is to experimentally test these predictions by indirectly measuring the sizes of terminal air spaces that open during inflation and by comparing their distribution with that predicted by previous model simulations (30). To achieve this goal, we used a technique developed by Hantos et al. (9) that is able to measure the input impedance of small subtrees of the tracheobronchial tree in isolated lungs. We measured these impedances during inflation and then fit the spectra with a model from which we can estimate the regional tissue elastance (Eti) of the subtrees as a function of inflation pressure. We found that, during inflation, Eti decreases in many discrete steps spanning a wide range of sizes. By assuming that Eti is inversely related to the size of the alveolar space that communicates with the trachea, we estimated the distribution of these steplike volume changes in terminal air spaces due to airway opening.
METHODS
Preparation of lobes.
We obtained 11 lung lobes from 8 mongrel dogs weighing 18–24 kg. The animals were anesthetized with pentobarbital sodium (30 mg/kg), treated with heparin (5,000 units), and exsanguinated via a femoral artery. The lungs were removed, and selected lobes were cannulated in the main bronchus with an 8 to 12mmID metal tube. First, the lobe was inflated to a positive airway pressure of 30 cmH_{2}O and checked for leaks. The bronchial cannula was attached to a short tube mounted in the lid of an airtight glass box (15 liters). A schematic drawing of the setup is shown in Fig. 1. The cannula was open to atmosphere so that Ptp, measured in the box with respect to atmosphere with a transducer (model MP45, Validyne; 50 cmH_{2}O), could be conveniently adjusted by pumping or sucking air into or out of the box with the use of a dualmembrane pump (model MP 03 Ez, Otto Huber). The lobe was suspended in the closed box and reinflated to 30 cmH_{2}O Ptp by creating a −30cmH_{2}O pressure in the box. A slightly curved 20 to 30cmlong (L _{2}) polyethylene catheter (1.53 mm ID, 2 mm OD) with a bellshaped metal ending was led through the lid into the main bronchus until it wedged in a peripheral airway. After a deflation to 5 cmH_{2}O Ptp, the catheter was gently pulled back to ensure that the rim of the metal ending was fixed in the bronchial wall. This procedure was repeated with two more catheters that were guided to different peripheral regions. The bottom of the box was covered with wet gauze to keep the lobe surface moist.
Impedance measurements and signal processing.
The three wedged catheters were connected to a loudspeakerinbox system through identical tubes of the same polyethylene material (42 cm long, L
_{1}; Fig. 1). These sections served as wave guides and were equipped with side taps and miniature transducers (model 33NA002D, ICS) to measure the lateral pressures at their distal ends (P_{2}) and the common entrance pressure in the loudspeaker chamber (P_{1}). The loudspeaker was driven by a computergenerated pseudorandom signal having a period of 1 s and containing 17 discrete frequency components between 1 and 47 Hz. The spectrum of the signal was flat, and the phase angles were chosen to minimize the peaktopeak value of the signal. The power amplifier of the loudspeaker was adjusted so that the peaktopeak value of P_{1} was <1 cmH_{2}O. Forced oscillations were continuously delivered, and P_{1} and P_{2} were measured while the lobe was slowly inflated from −5 to 30 cmH_{2}O Ptp in ∼160 s. The signals P_{1} and P_{2} were lowpass filtered (5thorder Butterworth, 50Hz corner frequency) and digitized at a sampling rate of 256 Hz. The inflation recordings were split into 160 short recordings, each containing 256 time points. The pressure transfer functions P_{1}/P_{2} were computed using fast Fourier transformation for each recording of 256 points, providing a 1Hz frequency resolution. From the P_{1}/P_{2} spectra, the input impedance of the subtrees (Z_{p}) subtended by the catheters was derived as the load impedance seen at the distal end of the wedged catheter, as described in detail previously (9)
The ability of the entire system to determine impedance over a wide range of magnitudes was tested in two different ways. First, small glass bottles of different sizes were measured with elastance values similar to those of the airway subtrees. The length of the catheters and the input amplitude were varied to find their optimal values. Second, the catheter end where P_{1} is measured was blocked, and the pressuretopressure ratio (P_{1}/P_{2}) in the closed tube was determined over the 1 to 47Hz frequency range. P_{1}/P_{2} can then be predicted using wave propagation theory in rigid tubes, which provides a validation of the technique as described previously (15). Both tests were satisfactory, providing evidence that the resolution (ratio of the smallest to the largest elastance) of the wave tube technique was ≤0.001.
Parameter estimation and statistical analysis.
The Z_{p} spectra were evaluated on the basis of several simple models of the airway tree and the alveoli. A simplistic view of two neighboring subtrees in the lung and two collateral airways connecting them is shown in Fig.2 A. An equivalent electrical model of this structure is shown in Fig. 2 B. R_{c,1} and R_{c,2} are the resistances of the collateral airways. The electrical model is a simplified version of the model introduced recently by Hantos et al. (9). The model includes two airway resistances (R_{1} and R_{2}) in series, representing the regular airways between the catheter end and the alveoli. Another resistance (R_{c}) is placed as a shunt pathway between R_{1} and R_{2} to account for the resistance of the collateral airways connecting the two subtrees (Fig.2 A). Thus R_{1} can be interpreted as an equivalent resistance of all the airways between the end of the catheter and the collateral airway, and R_{2} models all the airways that are peripheral to the collateral airway. The parenchymal tissues are modeled by an ideal elastic component, Eti, connected in series with R_{2}. Thus R_{c} is in parallel with R_{2}and Eti, which means that it is connected to ground in the electrical model in Fig. 2 B, since the collateral airway will shunt part of the input flow to the atmosphere through the neighboring subtree shown in Fig. 2 A.
The model parameters were estimated by means of a global optimization procedure (5) minimizing the rootmeansquare error between measured and model impedances. R_{1} and R_{2}, however, were not fit simultaneously, because the features in the Z_{p} spectra did not allow simultaneous and unique estimation of R_{1} and R_{2}. For each data set corresponding to a single inflation and a single catheter (160 impedance spectra), first R_{1} was fixed to zero (model A in Fig. 2 C) and the parameters including R_{2} were determined. Next, R_{2} was fixed to zero (model B in Fig. 2 D), and all parameters including R_{1} were determined for the same data set. These two models differ in the way they represent the major location of the collateral airway resistance. Model A incorporates the notion that the collateral airway is closer to the end of the catheter. Thus R_{c,2} is large (or infinite) and negligible compared with R_{c,1}; hence, R_{1} is neglected. Model B places the collateral airway closer to the alveoli. Thus R_{c,1} is large (or infinite) and negligible compared with R_{c,2}; hence, R_{2} is neglected. For a given data set corresponding to a single inflation, the final model parameters were selected on the basis of which model produced smaller errors. However, for a given data set corresponding to one inflation and one region, only one of the models, model A or model B, was used for all 160 Z_{p} spectra. The corresponding resistance from model A or model B was simply denoted by R. Time series were then formed from the model parameters as a function of inflation time. The statistical properties of these time series were evaluated by calculating their probability density distribution function.
Simulation studies.
We used the model developed by Sujeer et al. (30) to interpret our measured regional airway resistance and Eti time series. Briefly, the periphery of the airway tree was modeled as a symmetrical binary tree with airway segments that can be closed or opened. Attime 0, all airways are assumed to be closed. Lung inflation is simulated by applying an external pressure (P_{E}) at the top of the tree and gradually increasing P_{E} at a slow rate. Airways are labeled (i,j) with a generation numberi (i = 0, … ,M), whereM = 12 is the order of the tree (i = 0 denotes the root of the tree), and a column number j(j = 0, … ,2^{i} ^{ − 1}). A critical opening threshold pressure P_{i,j} is assigned to each airway (i,j), which pops open instantaneously whenever P_{i,j} is smaller than or equal to the pressure in its parent. All pressures are normalized so that, during inflation, P_{E} increases from 0 to 1, which corresponds to Ptp decreasing from 0 to −30 cmH_{2}O at total lung capacity (TLC). The values of P_{i,j} were thus between 0 and 1 and were taken from a uniform distribution (3, 30, 32). The alveoli are represented by the lastgeneration segments in the model. Because of the lack of data in the literature, we assume that these segments behave the same way as the small airways; that is, they are assigned a threshold pressure that is uniformly distributed between 0 and 1.
The inflation process is simulated in the lung model by increasing P_{E} in small increments. P_{E} is initially assigned the value P_{0,0}, the critical opening threshold pressure of the root or airway (0,0). Since an airway opens when the pressure in its parent equals or exceeds its critical opening threshold pressure, the airway (0,0) now opens, and its pressure is set equal to P_{E}. Next, the two airways (1,0) and (1,1) are tested to see if they can be opened by this value of P_{E} (the present pressure in their parent airway), that is, whether P_{E} > P_{1,0} and/or P_{E} > P_{1,1}. If one or both conditions are met, then the airways (1,0) and/or (1,1) are also opened. This opening is then continued sequentially down the tree until no airway is found with its P_{i,j} < P_{E}. Of particular interest is the fact that a small increase in P_{E} can lead to an “avalanche” in which many airways open simultaneously (32). When the first avalanche stops, the critical opening threshold pressures of those airways that are still closed but with parents that are now open are examined. P_{E} is then incremented to the smallest of these threshold pressures, and the pressure in all open airways is updated to this new value. This process is iterated until all airways open. A sequence of avalanches filling a small fivegeneration tree is demonstrated in Fig.3.
Airway and alveolar wall tissue elasticity is introduced by requiring that the diameters (and hence the volumes) of the open airways and alveoli depend on P_{E}. The diameter values are updated with each increase in P_{E} according to the following exponential pressurevolume relationship taken from the literature (27)
RESULTS
Two examples of the input impedance of a subtree separated by 2 s during an inflation are shown in Fig.4. The real parts are decreasing hyperbolically from a large value of ∼8,000 cmH_{2}O · l^{−1} · s at 1 Hz to a constant of ∼1,500 cmH_{2}O · l^{−1} · s at 40 Hz. The imaginary parts are negative and first decrease, showing a local minimum at ∼6 Hz, then increase similarly to the imaginary part of an ideal capacitor. During a slow inflation, one would expect that the magnitude of regional impedance increases with time, since with increasing lung volume, the airways and alveoli become stiffer as a result of stretching their walls. However, our data show that the magnitude of the impedance decreases with increasing time. This can only happen if there was an abrupt opening between the two recordings whereby a larger alveolar region popped open, which resulted in a decrease in impedance magnitude. Figure 4 also shows that the model fits the impedance data reasonably well, although some systematic errors can also be seen. The two Eti values obtained from the fits are 1.3 × 10^{5} and 1.1 × 10^{5}cmH_{2}O/l, corresponding to the solid and dashed lines, respectively.
The model parameters Eti, R, and R_{c} are shown as a function of inflation in Fig. 5 for one of the regions. As inflation progresses, all parameters decrease along hyperboliclike curves. The maximum and minimum values of Eti are 781,200 and 43,980 cmH_{2}O/l, covering a range of 1.5 orders of magnitude. However, the continuous decrease is interrupted by sudden changes or jump downs. Smaller jumps can also be seen as magnified in the inset for Eti. In the middle of inflation, Eti sometimes shows small increases, and, toward the end of inflation, Eti starts continuously increasing. This phenomenon is due to stiffening of the parenchyma. Interestingly, a large jump in R_{c} occurs simultaneously with Eti at ∼45 s, which is not seen in the series resistance R. These patterns changed from region to region and varied between two consecutive inflations even in the same region. Figure6 demonstrates that our numerical model simulation using a ninegeneration tree provides an Eti graph as a function of inflation time similar to that shown in Fig. 5. Because of the elasticity of the alveolar wall tissue in the model (Eq.3 ), the simulated Eti as a function of time can even mimic the small increases that follow a jump as well as the gradual increase toward the end of the inflation. Additionally, Fig. 6,inset, shows the jumps on a much smaller scale, similar to the experimental data in Fig. 5.
Examining all experimentally obtained Eti graphs (i.e., Eti in Fig.5 A), we were able to manually record 1,021 drops in Eti. From the jumps in Eti, dE (defined as in Eq. 5 ), a time series was formed and normalized with the maximum value of dE (Fig.7 A). For comparison, a similar time series of dE containing 1,021 elements obtained from the inflation simulations (Fig. 6 A) is also shown in Fig. 7 B. In the computer simulation, the numbers of terminal segments before (N ^{1}) and after an avalanche (N ^{2}) were recorded, and the dE was estimated according to Eq. 5 (where because of elastic walls Ea depends on inflation pressure P_{E}) and normalized with the largest dE value. Despite the fact that the modeling does not involve any curve fitting or use of measured model parameters, the simulated time series of dE is qualitatively similar to the experimental data both displaying many small jumps with intermittent large jumps. A quantitative comparison can be obtained by examining the statistical features such as the probability density distribution of the time series. The distributions of the experimentally obtained and the simulated dE time series were calculated by binning the dE values using equal size bins in the logarithmic domain. This results in a smoother estimation of the distribution especially for high values of dE, which do not occur frequently. There is a good agreement between the experimental (Fig.8 A) and the numerical (Fig.8 B) distributions of dE using a ninegeneration tree. Both distributions show a region of linear decrease on a loglog graph extending over about two decades of dE values. We also show the distribution of dE using a 12generation tree that exhibits a linear decrease of ∼7 decades on the loglog graph. This means that, over the region where the distributions decrease linearly on the loglog graph, the distributions must follow a power law: p(dE) ∼ dE^{−k}. The negative slope of the linear decrease is the exponent, k, in the power law, which can be estimated by a straightline fit to the distribution data. The value ofk was 1.71 for the measured and 1.5 for the simulated distribution, independently of the size of the tree.
DISCUSSION
The purpose of this work was to experimentally determine the distribution of terminal air spaces that become sequentially open during inflation from the collapsed state of isolated dog lung lobes. For this purpose, we used the technique of Hantos et al. (9) to measure the input impedance of small subtrees of the lobes using 2mmOD catheters wedged into the peripheral airways. This measurement system could detect changes in the mechanical parameters of 12–20 generational subtrees according to the airway tree model of Horsfield et al. (12). In particular, this technique allowed us to detect small changes in regional Eti as a function of inflation pressure that could not be detected from pressureflow measurements at the trachea or from measurement of Ptp.
The primary findings of this study are that airflow resistance and Eti of the subtrees decrease in discrete jumps as a result of discrete openings during inflation. The magnitudes and patterns of these jumps are highly variable, demonstrating that airway reopening observed at the level of these subtrees is a stochastic process reminiscent of the jumps observed in the terminal airway resistances (21, 22,26). Thus the present data support the notion that airways open in cascades or avalanches (32). Additionally, the distribution of the jumps in Eti is in quantitative agreement with that predicted by a computational model based on the assumption that airways open in avalanches (30).
The most important limitation of the technique is that, to identify Eti, frequencies as low as possible must be included in the input signal. In the original study that introduced this catheter impedance technique, Hantos et al. (9) applied a frequency range of 0.1–48 Hz. The corresponding time resolution was 10 s. Such a poor time resolution would not have allowed us to detect the jumps in Eti seen in Fig. 5. Most of these jumps would have occurred within the time window of the Fourier transform, deteriorating the quality of the impedance spectrum and masking the discrete nature of the openings.
The lowest frequency in our study was chosen to be 1 Hz, which resulted in a 1s time resolution. The 1 Hz lowest frequency still allowed us to fit a simplified model to the impedance spectra. However, the general quality of the fits did not reach that obtained by Hantos et al. (9), where Z_{p} was ensemble averaged from several long steadystate recordings including many time windows. The reasons are most likely due to the facts that we had only a single time window for estimating the spectra and we did not include frequencies <1 Hz. The former can lead to less reliable impedance data, whereas the latter can result in reduced reliability of the parameter estimates. As a result, the fluctuations in the parameters in Fig. 5may, in fact, reflect the presence of numerous discrete opening events occurring within the time window of the Fourier transform (i.e., 1 s). The primary assumption behind the Fourier analysis is stationarity: when an opening occurs within a time window, the corresponding impedance estimate is deteriorated. Although the magnitudes of the series resistance R and the collateral resistance R_{c} were similar to those found by Hantos et al. (9), the number of jumps that could reliably be identified from the data were not sufficient to carry out a reliable statistical analysis. Additionally, these resistances do not have a clear relationship to the recruited alveolar space, and, hence, we only investigated Eti in this study. In general, for comparable Ptp, the frequency spectra and the values of Eti in this study were similar to those found by Hantos et al. (9). Since during inflation the incremental dynamic elastance of tissue units is expected to rise, the discrete drops in Eti provide evidence that airway opening occurs discontinuously, leading to opening of terminal air spaces of highly varying sizes.
The minimum value of Eti in Fig. 5 is 43,980 cmH_{2}O/l. This value corresponds to an almost completely open alveolar region inflated to a lung volume close to TLC. This minimum value is much larger than the elastance of the lung; however, it is quite reasonable when we compare it with the lung region supplied by the catheters. The outer diameter of the catheter was 2 mm, and it was fit into an airway at TLC. Thus the airway diameter into which the catheter was fixed must have been ∼2 mm. This corresponds to a 17 to 19generation tree in the airway model of Horsfield et al. (12). The number of terminal segments (alveolar ducts) supplied by such an airway is 331–574. The total number of terminal segments in the airway model of Horsfield et al. is ∼150,000. Since we always used the largest lobes from a lung, we estimate the total number of terminal segments in a lobe to be 40,000. If we assume that the catheter supplied 400 segments, the volume of such a region would scale with the ratio 400:40,000 = 0.01. The volume of a dog lobe at TLC is ∼300 ml; hence, the volume of the region is estimated to be 300 ml ∗ 0.01 = 3 ml. This is in excellent agreement with the estimates of the supplied volumes we obtained from casts of the peripheral airways as described by Hantos et al. (9). These casts were created by infusing the cast material through catheters similar to those used in the present study. The measured volumes of four casts were 2.8, 3.2, 3.3, and 4.7 ml. Thus the catheter sees ∼1% of the total volume of a lobe. The incremental elastance of a dog lung is 10–20 cmH_{2}O/l at FRC (25), and it would be ≥80 cmH_{2}O/l at TLC. Thus the lobe elastance close to TLC can be estimated to be ∼320 cmH_{2}O/l depending on the size of the lobe. If the elastance is inversely proportional to regional lung volume, then, on average, the elastance seen by the catheter would be 320 ÷ 0.01 = 32,000 cmH_{2}O/l, which is in the range of the minimum Eti of 44,000 cmH_{2}O/l shown in Fig. 5.
The time series of Eti decreases via smaller and intermittent larger jumps. However, Eti also shows some smaller occasional increases and later a continuous increase toward the end of inflation (Fig. 5). The deterministic increase toward the end of inflation is due to stiffening of the alveolar and airway walls with increasing mean distension. Since in the numerical simulation model we also included alveolar wall elasticity, the Eti predicted by the model will also increase with inflation (Fig. 6). However, we are interested in the rate of decrease in Eti, which was qualitatively similar in the simulations (Fig. 6) and in the measured Eti (Fig. 5). Our simulations in Fig. 6 show that the increases in Eti after a large drop can also be due to stiffening of alveolar tissue. We cannot exclude the possibility, however, that some of the small increases in Eti are due to measurement noise or systematic errors in the fitting of the impedance data. The electrical models (Fig. 2, C and D) we fit to the impedance data are gross simplifications of the airway structure. During inflation, new collateral channels can open, and the model chosen for that particular inflation may not be the optimal representation of the structure. For example, if first R_{c,2} was open, the tree could be modeled as the network in Fig. 2 D. However, when R_{c,1} also opens during the same inflation, the tree should be modeled by one of the configurations shown in Fig. 2, Cand B. These model errors (systematic differences between model and data), may occasionally result in an increase in Eti. Unfortunately, these errors are not uniformly distributed. The reason is that the magnitude of impedance decreases more than an order of magnitude from the beginning of inflation to the end, and the absolute error in fitting is a function of the magnitude of the impedance. To see the effects of these fitting errors on our experimental distribution function, we estimated from Fig. 5 A a maximum value of 0.2 × 10^{5} cmH_{2}O/l for this deterministic error caused by the fitting procedure. This value corresponds to 0.0027 on the normalized elastance jump scale shown in Fig. 7 A. Rejecting all values of the normalized elastance jumps <0.0027 from the calculation of the distribution in Fig.8 A results in omitting the first three points from the distribution. These first three points, however, were not used in fitting a straight line to the tail of the distribution on the loglog graph. Thus we conclude that possible systematic model errors have no effect on the power law tail of the distribution and, hence, the numerical value of its exponent.
The time series of dE shows many small jumps as well as large jumps (Fig. 7 A). The simulation results (Fig. 7 B) are similar to the experimentally derived time series. The probability density distributions of dE estimated from the experimental and simulated data are also similar, showing a linear decrease on the loglog graph over two decades of dE values (Fig. 8). In the numerical simulations, every jump can be detected, including the opening of a single terminal unit. Thus, with the use of the 12generation tree, the corresponding distribution is much wider, following a power law for very small dE values. The fact that the distributions become flat for small dE (dE < 10^{−7} for a 12generation tree and dE < 10^{−2} for a 9generation tree) indicates a finite size effect: the smallest jumps are those that correspond to the opening of a single alveolus. Since the walls of the alveoli are nonlinearly elastic, there is a range of dE values (approximately between 5 × 10^{−8} and 10^{−7} for the 12generation tree) corresponding to the opening of a single alveolus where the distribution is more similar to a Gaussian distribution. In contrast, the small jumps are not easily identified from the experimental time series. Toward the end of inflation, adding a small volume (due to opening) to a large volume (already open) will cause Eti to decrease by such a small amount that it can be easily within the experimental noise level. Also, our technique, unfortunately, cannot differentiate among openings occurring independently at different locations but at the same inflation pressure. For example, if two separate openings were triggered within 1 s and in the same time window, we would detect it as a single event with a larger change in Eti. The result is that, instead of two small jumps, we would detect one larger jump. All these effects will reduce the number of small dE values and can lead to a saturation of the dE distribution at much larger values of dE (∼10^{−3}) than in the computer simulations (<10^{−7}).
The experimentally determined exponent is ∼13% larger than the numerical one (1.7 vs. 1.5). Several factors could contribute to this discrepancy. First, the tail of the power law distribution is determined by the large values of dE. We point out that one needs many large values of dE to reliably estimate the tail. The number of experimental dE values was only 1,021; hence, the number of large dE values was much less than in the simulations. Second, after insertion of the catheters, the lobes could not be completely degassed. Thus a certain amount of trapped air must have remained in the region supplied by the catheters. To study the effect of trapped air on the distributions of the elastance jumps and the recruited volumes, we repeated some of the simulations using a tree model in which we allowed for trapped air. This was achieved by setting the threshold pressure of a given percentage of the alveoli (or end tips of the tree) to zero and connecting them to the root of the tree. We then inflated the model 10,000 times (see methods) and calculated the dE and the recruited volume distribution as a function of the percentage of trapped air. The nature of the distributions was invariant; that is, the elastance jumps and the volume increments followed a power law distribution. However, the exponent k of the elastance distribution was sensitive to the amount of trapped air, increasing from 1.5 with no trapped air to 2 with ∼20% trapped air (Fig.9). The experimental value ofk = 1.71 corresponds to 1–3% trapped air in the region subtended by the catheter. Because of this dependence ofk on trapped air, one should not use Eq. 5 to transform the experimental dE distribution to estimate the distribution of the volume increments. Instead, we can use the full statistics of the simulation (including 100,000 jumps) to estimate the distribution of alveolar volumes that become open during inflation. This distribution is a power law (Fig. 10) with an exponent of 2, in agreement with the predictions of Sujeer et al. (30). Additionally, this distribution was completely independent of the amount of trapped air in the model.
The significance of a power law distribution is that the tail of the distribution is very long compared with, for example, a normal distribution. The tail of a distribution is representative of the relative frequency of occurrence of rare events. Since the tail of a power law can be orders of magnitude larger than the tail of a Gaussian model, the probability of a rare event is also orders of magnitude higher in the power law than in the Gaussian model. Therefore, the process or phenomenon described by a power law distribution does not have a characteristic scale or size that would be largely preferred over other sizes; hence, the power law distribution is said to be “scale free” (29). In our case, a rare event represents a large alveolar region suddenly popping open. If, for example, the alveoli would tend to open in groups of 10–15, then the likelihood of finding a rare event (e.g., a large atelectatic region simultaneously opening) would be small, and the recruited volumes would follow a normal distribution with a mean corresponding to the air volume of ∼13 alveoli. The fact, however, that the volume distribution is a power law implies that the probability of having a large alveolar region opening simultaneously is quite high and can be orders of magnitude higher than that for a normal distribution. As a consequence, the measured volumes do not represent the average size of any known physiological unit or structure (e.g., the acinus). Instead, the volume distribution represents a process that can generate a scalefree power law distribution. We argue that the only process that can lead to a power lawrecruited volume distribution is airway opening via avalanches. The model developed by Sujeer et al. (30) also predicts that the volume distribution is a power law. The power law volume distribution in that model was obtained by assuming that there is an interaction between reopening of airways and the number of alveoli, because the critical opening pressures are distributed over a tree structure. In particular, airways open in cascades or avalanches, which results in a widely varying number of terminal airways (Fig. 3) and, hence, of recruited alveoli during inflation, the distribution of which follows a power law functional form.
The exponent of a power law distribution fully characterizes the distribution, because the knowledge of the exponent allows us to predict the likelihood of one event compared with another event. The actual numerical value of the exponent has the following significance. First, the smaller the exponent, the slower is the decrease of the tail of the distribution and, hence, the higher the probability of finding rare events. Second, if the process or phenomenon can be mapped onto an existing class of models, a theoretical value for the exponent may be possible. For example, if we associate the normalized reopening pressures with probabilities of airways becoming open, we can map the airway reopening problem onto a sequence of randomly occupying segments in an abstract tree with a given probability p(3), a process called percolation (29). Then increasing the inflation pressure P_{E} in the lung corresponds to increasing p in percolation. As P_{E} increases, more and more airways become open, which then corresponds to more clusters of connected segments becoming occupied in the equivalent percolation process. A key quantity in percolation is the distribution of cluster sizes, which is known to follow a power law for certain critical values of p (at the percolation threshold when a large cluster spanning the entire system appears). However, the cluster size distribution is similar to the distribution of recruited volumes. Thus one might expect that known concepts and exponents from percolation theory (29) may be applied to airway reopening. Indeed, the exponent 2 for the volume distribution can be predicted from percolation (30). Another attractive feature of this mapping is that, for example, in percolation, the cluster size distribution is independent of the details of the system (29). In the model of Sujeer et al. (30), the volume distribution is only marginally affected by airway wall and alveolar wall elasticity or asymmetry in the airway tree: the power law is maintained with the same exponent, but the region over which the power law holds is slightly extended by elasticity and reduced by asymmetry. The primary reason for this robust behavior of the volume distribution is that it has a strong relationship to percolation. Since percolation is a purely geometrical problem, the distribution of recruited volumes is also a geometrical problem and should indeed be invariant of other details of the system. Indeed, power law distributions in the intensity of crackle sound detected during lung inflation similarly arise from the geometric tree structure of the lung (2). In contrast, the distribution of the elastance jumps is not a purely geometrical problem. The reason is that adding a given volume to the open region (adding a constant toN ^{1} and N ^{2} in Eq.5 ) is a nontrivial transformation of the random variable dE inEq. 5, which alters the exponent k of the distribution. Thus we conclude that the power law volume distribution arises from avalanches of airway reopenings triggered by overcoming a hierarchy of critical opening threshold pressures that are distributed widely and relatively independently of generation within the last 10–14 generations of the airway tree in the collapsed lung (32). Our experimental and simulation data in Figs.59 support this interpretation.
Other factors that may influence airway reopening include the nature of the distribution of threshold pressures (30). If, for example, the distribution of threshold pressures is very narrow with a strong generation dependence, such that the threshold pressures do not overlap in consecutive generations, airways do not open in cascade and the size distribution of alveolar volumes is not a power law (30). Additional important factors include the physical properties of the surface film (8, 23) and parenchymal tethering (24, 34). The surface tension, the viscosity, and the nonNewtonian viscoelastic properties (13) of the airway lining fluid certainly have an effect on the dynamics of the local process of opening an airway segment. Parenchymal tethering may also introduce spatial correlations among the reopening of airway segments. However, as long as the distribution of airway opening threshold pressures remains relatively wide and independent of generation, the volume distribution and, hence, the global effect of airway closure on gas exchange are not likely to be influenced much by these factors (30). On the other hand, the opening threshold pressures may change as a result of increased surface tension in certain diseases such as respiratory distress syndrome (1). The effects on reopening of increased film surface tension as in respiratory distress syndrome, increased film viscosity as in cystic fibrosis, or reduced parenchymal tethering as in emphysema were investigated by Perun and Gaver (23, 24). They showed that these factors can prolong the reopening time or increase the critical opening threshold pressures affecting pulmonary function. Sufficiently long reopening times may result in a slow opening process that occurs sequentially, rather than in avalanches. Additionally, if alterations in these physical factors lead to a significant change in the distribution of threshold pressures, the alveolar volume distribution will change, which in turn will also result in significant changes in the pressurevolume curve of the lung, further hindering gas exchange (30).
The implications of knowing the volume distribution are that after a longterm mechanical ventilation “the magnitude and timing of pressure excursions applied at the airway entrance during artificial ventilation may be critical in triggering the avalanche process of alveolar recruitment” (32) and, hence, can have a significant influence on the average number of open airways in a breathing cycle. Indeed, recently, Lefevre et al. (17) found that, in a porcine model of lung injury, introducing “biological variability” in mechanical ventilation by choosing the frequency and tidal volume of ventilation from a normal distribution significantly increases lung compliance and improves gas exchange in the lung. Since airway reopening is a stochastic process (32), variability in tidal volume of mechanical ventilation may help opening of closed airways along the highly nonlinear pressurevolume curve of the atelectatic regions compared with fixedfrequency and volume ventilation (31). Thus our data support the findings of Lefevre et al. and the predictions of Suki et al. (31), which may therefore have applications in the optimization of ventilation strategies for individuals suffering from lung diseases with significant airway closure and alveolar collapse.
Acknowledgments
This study was supported by National Science Foundation Grant BES9813599 and Hungarian Scientific Research Fund Grant OTKA T 016308.
Footnotes

Address for reprint requests and other correspondence: B. Suki, Dept. of Biomedical Engineering, Boston University, 44 Cummington St., Boston, MA 02215 (Email:bsuki{at}bu.edu).

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 Copyright © 2000 the American Physiological Society