INTRODUCTION

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IN THE CONSTRICTED LUNG, ventilation is nonuniform. Indeed, the increase in dynamic lung elastance (EL,dyn) that results from this heterogeneity is almost as large as the increase in airway resistance (Raw) (21, 22, 25, 26). In addition, heterogeneous ventilation impairs gas transport, and blood gases are adversely affected (30, 31). Impedance and gas mixing in constricted lungs have been measured by several groups.

A number of models of the constricted lung have been proposed (2, 8, 18, 24, 28). In most models, an ad hoc distribution of Raw is postulated, and the parameters that describe that distribution are determined by fit to experimental data. Recently, we described a model for the mechanics of a constricted terminal airway that results in a prediction for the distribution of Raw (1). The model predicts that the constricted airway is bistable. From this, we constructed a model of the whole lung in which the terminal airways are partitioned between the two stable states and the ventilation distribution is bimodal. Thus the distribution of Raw is obtained from the modeling, not from ad hoc assumptions. The modeling implies that pulmonary heterogeneity during bronchoconstriction is primarily the result of mechanical instability, rather than the heterogeneity of airway properties or level of activation.

We present data on impedance and gas mixing in constricted lungs, and we test an extended version of our model against these data. In five normal subjects, lung impedance and expired gas concentrations were measured during an He-washin maneuver in the control state and at several levels of lung constriction induced by inhalation of nebulized methacholine. Taken together, these data on impedance and gas mixing, obtained simultaneously, provide more information about the distribution of ventilation than does gas mixing or impedance data alone. The airway model was extended to describe the scaling with muscle activation, and a two-compartment model for the whole lung was constructed. Values of impedance and gas concentration during He washin were calculated from the lung model. The calculated values agree well with the experimental data.

	METHODS

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Three female and two male volunteers with previously normal clinical methacholine challenges were studied. All read and signed a consent form approved by the Mayo Institutional Review Board.

Data Acquisition

A Hans Rudolph nonrebreathing Y valve was connected to a low-dead-space switching valve, allowing a rapid change of inspired gas from room air to experimental gas (0.7% acetylene-9% He-21% O₂-69.3% N₂). To begin a washin, the switching valve was thrown during expiration so that the next inspiration began with test gas. The fractional concentration of He (C) in the inspired and expired gas was measured at the mouth by a mass spectrometer (Perkin-Elmer, Norwalk, CT) during the entire breath.

Protocol

Data Analysis

Modeling

Airway mechanics. Figure 1 shows the model for a terminal airway and the acinus it serves. Volumes are normalized by the volume of the acinus at TLC, and  denotes the difference between acinar volume and acinar residual volume (RV). Pleural pressure is taken as the reference pressure. Alveolar pressure (PA), as a function of time (t), is given by the product of acinar elastance (E) and  

(1)

The airway has length l, and the resistance of the airway segment from the acinus to a point x is denoted Raw(x). Equation 2 describes lumen pressure (P_lumen) as a function of x and t

(2)

Raw(x) is found from the viscous term of the empirical Raw formula reported by Reynolds (23), where µ and r denote the viscosity of air and the inner radius of the airway, respectively

(3)

The difficulty in evaluating Eq. 3 lies in determining r(x), and this requires descriptions of airway geometry and mechanics. In the airway wall, a layer of smooth muscle surrounds the airway mucosa. The unconstricted outer radius of the muscle layer at TLC is denoted r_o. Muscle radius in the constricted state, normalized by r_o, is denoted . The radius of the airway lumen [r(x)] depends on the outer radius of the airway and the thickness of the airway wall. The mucosal and smooth muscle layers are effectively incompressible and are assumed to occupy 16% of the airway volume in the control state (15). Thus , r_o, and r are related by geometry

(4)

Mechanical equilibrium is described by the law of Laplace: the hoop stress due to smooth muscle tension (T) must balance the transmural pressure difference (Ptm) acting across the airway wall

(5)

During breathing, the muscle in the airway wall is periodically stretched. The force-length curve for periodically stretched smooth muscle is much steeper than the isometric force-length curve (6, 9, 11, 27). As a result, for periodic force oscillations, muscle length is nearly constant. Furthermore, except for the recent report of Latourelle et al. (17), the published data describing periodically stretched muscle show that peak muscle tension matches the isometric tension (T_iso) at the same muscle length (6, 9, 27). These two features of the data are incorporated into the model as follows. First, fluctuations in muscle length during periodic force cycling are assumed to be negligible. That is, muscle length and airway radius are assumed to be constant over a cycle. Second, we assume that muscle length is set by the condition that length takes the value for which isometric muscle tension matches the peak tension during a cycle. Consequently, the muscle length-Ptm relation is given by the Laplace relation and the isometric length-tension curve. This relation is expressed in Eq. 6, where  denotes the angular frequency of breathing

(6)

The length of the unconstricted muscle at TLC is assumed to be the optimal length for isometric muscle contraction, and isometric tension at optimal length is denoted T_o. For suboptimal muscle lengths, isometric tension is described by a linear function of muscle length or, equivalently, a linear function of  

(7)

Substituting Eq. 7 into Eq. 6 yields a relation between  and peak Ptm

(8)

Ptm has three components: P_lumen, PA, and peribronchial tethering stress (P_tether)

(9)

The first two terms on the right-hand side of Eq. 9 are described by Eqs. 1 and 2. In the absence of flow, P_lumen equals PA and Ptm is given by P_tether. P_tether depends on PA, the shear modulus of the lung, and the magnitude of parenchymal distortion. Parenchymal distortion is described by , the fractional difference between the airway radius and the radius of an unstressed hole in the parenchyma, and P_tether is given by the continuum model of Lai-Fook (14)

(10)

where

(11)

With these substitutions, Ptm is expressed in terms of Raw(x), (x), (t), and E

(12)

For a given volume oscillation, (t), Eqs. 3 and 8, together with subsidiary Eqs. 4, 11, and 12, form a complete set of equations for the variables (x) and Raw(x).

View larger version (11K): [in this window] [in a new window]   Fig. 1.   Model airway and the acinus it serves. Inner radius of airway [r(x)] varies as a function of axial position (x), and lumen pressure [Plumen(x,t)] is a function of x and time (t). Acinus has elastance (E) and is distended by a time-varying alveolar pressure [PA(t)].

View larger version (10K): [in this window] [in a new window]   Fig. 2.   Tidal volume vs. peak lumen pressure at airway entrance at 3 levels of smooth muscle activation. Tidal volume is scaled by volume of the acinus at total lung capacity. In the model, level of smooth muscle activation is described by hoop stress (To/ro, where To is isometric tension at optimal muscle length and ro is unconstricted outer radius of the muscle layer at total lung capacity) produced by isometric smooth muscle contraction at optimal length. Curves show results of model calculations for To/ro = 18, 24, and 30 cmH2O.

Scaling with methacholine dose. The hoop stress produced by isometric contraction at optimal muscle length is given by the value of T_o/r_o. With increasing doses of methacholine, peak muscle tension and T_o/r_o increase. T_o/r_o provides a natural pressure scale for airway mechanics. After scaling pressures and volumes by T_o/r_o and T_o/Er_o, respectively, Eqs. 2, 4, 8, and 12 are completely independent of T_o/r_o, and the scaled version of Eq. 11 has only a weak dependence on T_o/r_o through terms involving the cube root of . To scale the boundary condition, we assumed that the minimum Ptp (P_min) is equal to 0.1 (T_o/r_o). If the normalized model were completely independent of T_o/r_o, it would yield a "universal solution curve" that could be multiplied by the appropriate scaling factors to obtain the solution to the unscaled equations for any value of T_o/r_o. Although the model has some dependence on T_o/r_o, Fig. 3 shows that a universal curve provides a good approximation to the exact solution. After pressures and volumes are scaled, the model outputs for T_o/r_o = 18, 24, and 30 nearly coincide. The curve for T_o/r_o = 24 is taken as an approximate universal solution curve.

View larger version (14K): [in this window] [in a new window]   Fig. 3.   Normalized tidal volume vs. normalized peak airway entrance pressure. Curves are the same as those shown in Fig. 2 with pressure scaled by To/ro and volume scaled by To/(Ero). Range of normalized peak pressures is small, and curves for the 3 levels of muscle activation nearly coincide. The 2 stable states coexist for a small range of normalized peak pressures. For simplicity, one value was chosen. This value [P*/(To/ro), where P* is peak pressure at which open and closed units coexist] is marked by arrow.

Whole lung mechanics. Given this model of airway mechanics, a two-compartment lung model immediately follows. In the lung, the peripheral airways are subjected to nearly identical driving pressures. However, the airways are bistable, and some airways will be in the open state, while others are nearly closed. The collection of open airways and the collection of closed airways form two distinct functional compartments. The fraction of terminal airways in the open compartment is denoted , and the value of , together with the values of _o and _c, determine the mechanical properties of the whole lung. For any , the resistance of the constricted peripheral lung (R_periph) and EL,dyn are given by Eqs. 13 and 14, where EL denotes the static elastance of the entire network (25 cmH₂O/TLC)

(13)

(14)

The magnitude of peripheral airway impedance (Z_periph) is given by

(15)

The number of closed and open airways and, hence,  is determined by the requirement that the pressure drop over the peripheral airways equals the product of Z_periph and whole lung VT

(16)

P* and the left-hand side of Eq. 16 increase with increasing T_o/r_o, and the right-hand side of Eq. 16 must increase in kind. Consequently, if VT is constant, Eq. 16 requires that  and the number of open airways decrease with increasing muscle activation. The values of T_o/r_o used to simulate the three effective doses of methacholine were 14, 19, and 32 cmH₂O, and the corresponding values of  were 0.83, 0.62, and 0.34.

Gas mixing. The same two-compartment model is used to describe gas mixing in the constricted lung. The two compartments are assumed to be well mixed and to be connected to a serial, or "stacked," dead space with no axial mixing. When gas is flowing into both compartments, the gas most recently expired into the dead space is the first to be reinspired. When gas is flowing out of both compartments, the concentration of test gas delivered to the dead space is a flow-weighted average of the gas concentration in each compartment. Because the flow delivered to the closed compartment lags that delivered to the open compartment, there are two intervals in each breath during which one compartment expires as the other inspires. The open and closed airways are assumed to be interspersed, and the scale of the heterogeneity is assumed to be small, so that during these intervals the gas expired by one compartment is directly inspired by the other.

	RESULTS

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Anthropometric data and baseline pulmonary function data for the five subjects are shown in Table 1. For each subject, EL,dyn at baseline (E₀) was used to scale the changes in lung impedance that resulted from methacholine constriction. Figures 4 and 5 describe these changes. The ratio of EL,dyn to E₀ at each of the three effective methacholine doses is shown in Fig. 4, and the increase in normalized whole lung viscance, (RL  R₀)/E₀, where R₀ is resistance at baseline, is shown in Fig. 5. Although the response to a particular methacholine dose varied considerably among subjects, EL,dyn and RL/E₀ increased systematically with increasing methacholine dose for each subject. At the highest dose of methacholine, EL,dyn averaged 2.6 E₀, and the increase in lung viscance averaged 2.7 E₀.

View larger version (6K): [in this window] [in a new window]   Fig. 4.   Effect of methacholine dose on dynamic lung elastance (EL,dyn): data and model predictions. Ratio of EL,dyn to control elastance (E0) is shown for 3 methacholine doses. Individual data (small circles) and means ± SE (large circles) are shown with model predictions (connected by solid lines) for each dose.

View larger version (5K): [in this window] [in a new window]   Fig. 5.   Effect of methacholine dose on whole lung resistance (RL): data and model predictions. Increase in whole lung viscance (RL  R0, where  is angular frequency and R0 is baseline resistance) normalized by baseline EL,dyn (EO) is shown for each methacholine dose. Individual data (small circles) and means ± SE (large circles) are shown with model predictions (connected by solid lines) for each dose.

The data describing gas mixing in the unconstricted and constricted states are shown in Figs. 6-8. Figure 6 shows the average value of ln(1  C_EE) as a function of n. As the dose increases, the plots of ln(1  C_EE) vs. n become more curved and indicate increased heterogeneity. The curvature of the washin plots at each dose is shown in Fig. 7. Similar to the impedance data, these data vary among subjects, but every subject followed the same trend: the magnitudes of initial slope and curvature increased with each successive dose. Figure 8 shows the average value of S_N, as a function of n, for each methacholine dose. S_N increased with n, and the rate of increase increased with methacholine dose.

View larger version (15K): [in this window] [in a new window]   Fig. 6.   Effect of methacholine dose on end-expiratory concentration of He (CEE) during multibreath washin. Value of ln(1  CEE) averaged among subjects is plotted vs. breath number (n) for control state and each methacholine dose. Values are means ± SE.

View larger version (8K): [in this window] [in a new window]   Fig. 7.   Washin curvature vs. methacholine dose: data and model predictions. Curvatures of washin plots for control state and for each dose of methacholine are shown. Individual data (small circles) and means ± SE (large circles) are shown with model predictions (connected by solid lines).

View larger version (17K): [in this window] [in a new window]   Fig. 8.   Effect of methacholine dose on normalized slope of phase III (SN): data and model predictions. Value of SN (mean ± SE) is plotted against n. Experimental values obtained in control state and after each dose of methacholine are connected by the 4 interrupted curves. Model calculations for the same 4 lung states are connected by solid lines.

Model predictions for lung impedance and gas mixing are plotted with the corresponding experimental data.

	DISCUSSION

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Data

The data describing lung impedance vs. methacholine dose shown in Figs. 4 and 5 are similar, in magnitude and variability among subjects, to the data reported by Pellegrino and colleagues (21, 22). In both studies, EL,dyn and RL increased systematically with increasing dose. At the highest dose, we measured an average EL,dyn of 2.6 E₀ and an average increase in RL of 2.7 E₀. At their highest dose, Officer et al. (21) report an EL,dyn that is 2.3 times the control value. For an assumed frequency of 15 breaths/min, their data yield an increase in dimensionless lung viscance of 2.6.

Likewise, our data describing gas mixing are similar to the results reported by Verbanck et al. (29). Although Verbanck et al. use a different measure of washout curvature, their results are qualitatively similar to ours. The values of S_N they report, similar to the data shown in Fig. 8, progressively increase with breath number and methacholine dose. At the 16th breath of the washin and the highest dose, the value of S_N shown in Fig. 8 is 0.65 liter¹. For our subjects, 16 breaths corresponds to an average lung turnover number of 3.7, and at that turnover number, Verbanck et al. report an S_N near 0.7 liter¹.

We also describe a model that provides a unified and quantitative description of impedance and gas mixing in the constricted lung. The model is based on an analysis of terminal airway mechanics. The basic treatment of airway mechanics is the same as that proposed by Gunst et al. (12) and subsequently used by others (16, 19) to describe the static equilibrium of the constricted airway. Previously, we extended this airway model by including dynamic muscle properties and analyzing the response of the airway to an oscillatory driving pressure (1). Here, the model is extended further. First, the requirements of mechanical equilibrium are imposed at each point along the length of the airway. Second, a scaling with degree of smooth muscle activation is obtained.

Airway Model

Fredberg and colleagues (5, 6) and Latourelle et al. (17) emphasized the significance of the dynamic properties of airway smooth muscle, and they have suggested that, when coupled with tidal stretching, these properties result in a vicious cycle of airway closure or, alternatively, a virtuous cycle of airway opening. In drawing this conclusion, they focused on the fact that stiffness decreases and length increases as the amplitude of applied force is increased. The dynamic properties of airway smooth muscle are also central to the quantitative model of airway instability presented here. However, we have neglected the change in muscle length that occurs over the course of a single breath and, hence, any changes in dynamic airway stiffness; in our model, muscle length depends only on peak applied force. We expect that, in reality, an airway in the open state is more compliant than an airway in the closed state, but this is not an essential feature of the instability that we describe. In our model, the feedback between Raw and airway tethering is essential to the instability. Without including the resistive pressure drop, the airway model shows no instability.

The level of muscle activation is described by T_o, the isometric tension at optimum muscle length. The corresponding hoop stress is T_o/r_o, and T_o/r_o provides a natural pressure scale for constricted airway mechanics. Figure 3 shows that the peak pressure at which the two stable states coexist is, to a good approximation, proportional to T_o/r_o. Pressures and volumes scale with T_o/r_o, but the caliber and resistance of the two stable airway states are independent of the level of muscle activation. Thus P* ~ 0.4 T_o/r_o, and the viscances of well-open and nearly closed units, described in terms of their viscance-to-elastance ratios, are approximately _o = 0.8 and _c = 16, independent of the level of muscle activation.

Whole Lung Model

The relation of  to muscle activation explains the relation of lung impedance to methacholine dose shown in Figs. 4 and 5. As T_o/r_o increases, airways close, tidal flow is forced into a smaller volume fraction of the lung, and EL,dyn increases. We have assumed that the added resistance of the constricted lung is primarily due to the increased resistance of the peripheral airways, and the two-compartment model represents this additional resistance. Peripheral airways flip from the well-open to the nearly closed state as T_o/r_o increases, and R_periph, RL, and the viscance of the constricted lung increase.

In the unconstricted lung, the curvature of a multibreath washin results from regional variations in lung properties (33). These regional variations are ignored in the model, and the model cannot explain the control value of washin curvature. That is, in the control state,  is 1, and the model lung is a single open compartment. As a result, C_EE approaches 1 exponentially, and the plot of ln(1  C_EE) vs. n has zero curvature. However, for the constricted states,  falls below 1, the model has two compartments, and C_EE is the sum of two exponentials. The closed compartment receives a smaller specific ventilation than the open compartment and has a longer time constant. At higher values of T_o/r_o,  decreases further, the closed compartment receives a greater fraction of the tidal ventilation, the influence of the second exponential is increased, and the plot of ln(1  C_EE) vs. n becomes more curved. Increasing levels of muscle activation result in more heterogeneous gas mixing as ventilation is more evenly divided between the open and closed compartments.

The initial slope of the gas washin is dominated by the rate constant of the open compartment. At higher levels of muscle activation, the number of open units decreases, but each remaining open unit accommodates a larger tidal flow and equilibrates with the test gas at a faster rate. Thus the initial slope increases with increasing methacholine dose.

Anatomic and temporal heterogeneity of ventilation are required for the nonzero values of S_N shown in Fig. 8. In the control state, the model lung is uniform and cannot explain the small, nonzero value of S_N. However, the constricted lung model has anatomic and temporal heterogeneity. First, the open units receive a larger VT than the closed units, and as a result, the concentration of He is larger in the open compartment. Second, the closed units have higher impedance, and the flow from the closed compartment lags behind the flow from the open compartment. In combination, these two effects cause the concentration of He in the expired gas to decrease as expired volume increases and the closed compartment contributes a greater fraction of the expired gas. At the highest methacholine dose, the values of S_N calculated from the model are larger than those obtained experimentally. Secondary mixing mechanisms, such as axial diffusion and turbulent mixing, which have been neglected, would be expected to reduce the values of S_N and washin curvature.

One might suspect that any two-compartment model that matches the impedance data would, by necessity, also match the gas mixing data. However, the two data sets contain different information and provide separate tests of the model. For given values of EL,dyn/E and R_periph/E, Eqs. 13 and 14 impose only two constraints on the three variables , _c, and _o. As a result, for every value of _o there is a corresponding  and _c that yield the desired EL,dyn/E and R_periph/E. For a range of _o, Fig. 9 shows the value of _c required for a model lung to have the same impedance as the measured values at the highest methacholine dose. Figure 9 also shows the value of washin curvature predicted for each of these pairs of _o and _c. The model matches the impedance and curvature data for only one pair of values of _o and _c. The values of _o and _c that provide the best fit to gas mixing and lung impedance data are near the values of 0.8 and 16 predicted by the airway model.

View larger version (11K): [in this window] [in a new window]   Fig. 9.   Effect of airway parameters on gas mixing. Hysteresivities of low- and high-resistance states are denoted o and c, respectively. For each value of o, dashed curve shows value of c that is required to match experimental impedance data at the highest dose of methacholine. Solid curve shows value of washin curvature calculated from a 2-compartment model using value of o shown on abscissa and required value of c. At highest dose of methacholine, experimental value of washin curvatures was 0.0064, and parameter values that provide best simultaneous fit to mechanics and gas mixing data are approximately o = 0.8 and c = 16, obtained from airway model.

Conclusions

Although the parameters used in the model are obtained from anatomic data, they are the average values of parameters that vary throughout the lung. The model of this average terminal unit was further simplified by ignoring secondary effects such as the effects of tissue resistance, interregional tethering, passive airway stiffness, the variation of airway radius throughout a breath, and secondary gas mixing mechanisms. Despite these simplifications, model predictions and experimental data are in good quantitative agreement, and the bistability of the constricted airway and the consequent behavior of the constricted lung model are robust to changes in parameter values. As a result, we believe the model accurately represents the fundamental features of the constricted lung. Most strikingly, the bistable airway model explains the long-standing observation that constricted lungs have a bimodal ventilation-perfusion (/) distribution, with a ~10-fold difference in / separating the two peaks. This observation was first reported by Wagner et al. (31) using the multiple inert gas elimination technique and was recently confirmed by Vidal Melo and colleagues (30) with functional positron emission tomography imaging. In our model, the impedances of the open and closed airways differ by a factor of , and the model predicts a bimodal / distribution, with the two peaks separated by a 12-fold difference in /.