To evaluate the effect of increasing smooth muscle activation on the distribution of ventilation, lung impedance and expired gas concentrations were measured during a 16-breath He-washin maneuver in five nonasthmatic subjects at baseline and after each of three doses of aerosolized methacholine. Values of dynamic lung elastance (El,dyn), the curvature of washin plots, and the normalized slope of phase III (S N) were obtained. At the highest dose, El,dyn was 2.6 times the control value andS N for the 16th breath was 0.65 liter−1. A previously described model of a constricted terminal airway was extended to include variable muscle activation, and the extended model was tested against these data. The model predicts that the constricted airway has two stable states. The impedances of the two stable states are independent of smooth muscle activation, but driving pressure and the number of airways in the high-resistance state increase with increasing muscle activation. Model predictions and experimental data agree well. We conclude that, as a result of the bistability of the terminal airways, the ventilation distribution in the constricted lung is bimodal.
- mathematical model
- helium washin
- phase III
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.
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.
The procedures used for acquiring lung impedance (3) and He-washin (4) data have been described previously and are briefly reviewed here. Subjects wore a noseclip and breathed through a mouthpiece into a pneumotachograph equipped with a differential pressure transducer. The transducer signal was digitized and linearized using the technique of Yeh et al. (34). The flow signal was integrated to obtain volume. The pneumotachograph was calibrated before each study using a 3-liter syringe, and integrated volumes were required to be within ±3% of syringe volume. Calibrations were performed separately using room air and test gas so that lung impedance measurements could be obtained with either gas. Transpulmonary pressure (Ptp) was taken as the difference between esophageal balloon pressure and mouth pressure.
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.
For each subject, target tidal volume (Vt) was fixed at 10 ml/kg body wt. Subjects were shown the target volume and their volume trace on a video screen and were asked to time their breathing to a metronome paced at 0.33 Hz. After a practice period, the experiment began with a control 16- to 19-breath He washin and a forced vital capacity (FVC) maneuver. Measurements were suspended for a few minutes to allow the test gas to wash out, and the washin was repeated. Using moderate-sized breaths at a rate of 10 breaths/min, the subjects then inhaled a nebulized mist for 1 min. A washin and an FVC maneuver were performed 3 min after nebulizer treatment and repeated 6 min later. The dosing schedule was saline followed by methacholine at 0.1, 0.4, 1.6, 6.4, and 25 mg/ml. At 10 min after each nebulizer treatment, the next dose of methacholine was administered. The washin and FVC studies were repeated for each dose. A reduction of forced expiratory volume in 1 s below 80% of control was the preestablished guideline to end the experiment. None of the subjects experienced a drop of this magnitude before the 25 mg/ml dose. All subjects were given two puffs of an albuterol (Ventolin, Glaxo-Wellcome) rescue inhaler after completion of the experiment. At the highest three doses, data showed notable changes from baseline, and only data for the control state and these three doses are reported
Pressure-volume and flow-pressure loops were displayed for each run and each breath. Aberrant breaths were excluded from the analysis. El,dyn was determined as the ratio of the difference in Ptp to the difference in lung volume at points of zero flow. Resistance (Rl) was obtained by application of the method of Mead and Whittenberger (20) to flow data in the range of ±1 l/s. For each breath, Vt and end-expiratory He concentration (CEE) were taken directly from the data stream. Dead space volume (Vd) was calculated for each breath in the control state using a modification of Fowler's method (4, 7). Function residual capacity (FRC) was found using the gas dilution data for the control state and the computational model described by Johnson et al. (13). Total lung capacity (TLC) was computed as the sum of inspiratory capacity, obtained at the end of the baseline washin maneuver, and FRC. The trace of He concentration vs. expired volume for each breath was analyzed to obtain the slope of the best-fit line to phase III. The slope was divided by CEE to obtain the normalized slope of phase III (S N).
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 ν 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 ν Equation 1The 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 (Plumen) as a function of x and t Equation 2Raw(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 Equation 3The 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 byr o, is denoted ρ. 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 ρ, r o , and r are related by geometry Equation 4Mechanical 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 Equation 5During 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 (Tiso) 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 ω denotes the angular frequency of breathing Equation 6The 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 To. For suboptimal muscle lengths, isometric tension is described by a linear function of muscle length or, equivalently, a linear function of ρ Equation 7Substituting Eq. 7 into Eq. 6 yields a relation between ρ and peak Ptm Equation 8Ptm has three components: Plumen, pa, and peribronchial tethering stress (Ptether) Equation 9The first two terms on the right-hand side of Eq. 9 are described by Eqs. 1 and 2 . In the absence of flow, Plumen equals Pa and Ptm is given by Ptether. Ptether depends on Pa, the shear modulus of the lung, and the magnitude of parenchymal distortion. Parenchymal distortion is described by δ, the fractional difference between the airway radius and the radius of an unstressed hole in the parenchyma, and Ptether is given by the continuum model of Lai-Fook (14) Equation 10where Equation 11With these substitutions, Ptm is expressed in terms of Raw(x), ρ(x), ν(t), and E Equation 12For a given volume oscillation, ν(t), Eqs.3 and 8 , together with subsidiary Eqs. 4,11 , and 12 , form a complete set of equations for the variables ρ(x) and Raw(x).
The parameters describing airway geometry were taken from the literature. The inner radius of a 12th-generation airway reported by Weibel (32), together with Eq. 4 , determinesr o. Airway length, l, is chosen so that the unconstricted model airway has the same resistance as a composite terminal airway consisting of Weibel generations 12, 13, and 14. TLC is 6 liters, and acinar volume at TLC is determined by dividing TLC among 212 terminal units. Acinar elastance (E) and RV are taken as 25 cmH2O/acinar TLC and 0.3 acinar TLC, respectively.
A sinusoidal volume oscillation with amplitude νtidal/2, mean value ν̄, and ω = 2π/3 rad/s was imposed. For each value of νtidal, the model system was solved numerically using Mathematica, and ν̄ was adjusted so that the minimum Ptp was equal to a fixed value. By substituting the expression for Ptm(x,t) given by Eq. 12 into the right-hand side of Eq. 8 , ρ(x) was determined as a function of Raw(x). With this function and the relation between ρ(x) and r(x) given byEq. 4 , Raw(x) was determined by numerical evaluation of the integral in Eq. 3 . Plumen(x,t) was obtained from Eq. 2 , and Paw, the peak lumen pressure at the airway entrance (x = l), was determined. The process was repeated for To /r o = 18, 24, and 30 cmH2O, and for each value of To /r o,νtidal was plotted against Paw. The resulting curves are shown in Fig. 2.
Scaling with methacholine dose.
The hoop stress produced by isometric contraction at optimal muscle length is given by the value of To/r o. With increasing doses of methacholine, peak muscle tension and To /r o increase. To /r o provides a natural pressure scale for airway mechanics. After scaling pressures and volumes by To /r o and To /Er o, respectively,Eqs. 2, 4, 8 , and 12 are completely independent of To /r o, and the scaled version ofEq. 11 has only a weak dependence on To /r o through terms involving the cube root of ν. To scale the boundary condition, we assumed that the minimum Ptp (Pmin) is equal to 0.1 (To /r o). If the normalized model were completely independent of To /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 To /r o. Although the model has some dependence on To/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 To /r o = 18, 24, and 30 nearly coincide. The curve for To /r o = 24 is taken as an approximate universal solution curve.
As we explained previously (1), the sigmoidal Vt vs. peak pressure curve indicates that constricted terminal airways are bistable with well-open and nearly closed stable states. The two stable states coexist for a small range of pressures, but, for simplicity, we chose one value in this range. Because pressures and Vt approximately scale with To /r o, their ratio and the resistances of the two stable states are independent of muscle activation. The ratios of ωRaw to parenchymal elastance (E) for the open and closed units are denoted ηo and ηc, respectively. Values of ηo and ηc and the peak pressure at which they coexist (P*) are taken from the universal solution curve. These values are ηo = 0.8, ηc = 16, and P* = 0.43 To /r o.
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 φ, and the value of φ, together with the values of ηo and ηc, determine the mechanical properties of the whole lung. For any φ, the resistance of the constricted peripheral lung (Rperiph) and El,dyn are given by Eqs.13 and 14 , where El denotes the static elastance of the entire network (25 cmH2O/TLC) Equation 13 Equation 14The magnitude of peripheral airway impedance (Zperiph) is given by Equation 15The number of closed and open airways and, hence, φ is determined by the requirement that the pressure drop over the peripheral airways equals the product of Zperiph and whole lung Vt Equation 16P* and the left-hand side of Eq. 16 increase with increasing To /r o, and the right-hand side of Eq. 16 must increase in kind. Consequently, if Vt is constant, Eq. 16 requires that φ and the number of open airways decrease with increasing muscle activation. The values of To /r o used to simulate the three effective doses of methacholine were 14, 19, and 32 cmH2O, and the corresponding values of φ were 0.83, 0.62, and 0.34.
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.
The fraction of the lung in each compartment and the phase and amplitude of the Vt each receives are obtained from the mechanical model decribed above. The values of FRC, Vd, and Vt, as fractions of TLC, are taken as the average experimental values. Under these assumptions, the standard compartment modeling equations were solved using Mathematica. C at the mouth was calculated as a function of expired volume. Values of CEE and the slope of phase III were obtained from the concentration vs. expired volume curves at each breath number (n) and at each methacholine dose. For each dose, ln(1 − CEE) was plotted vs. n, and a quadratic polynomial in n was fit to the resulting curve. The initial slope of the logarithmic curve was taken to be the coefficient of the linear term, and the curvature was taken to be twice the coefficient of the quadratic term. As a result of the phase difference between the open and closed compartments, C changes during the course of expiration. A line was fit to the second half of the C vs. expired volume curve, and for each breath, S N was obtained by dividing the slope of that line by CEE.
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 and5 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, (ωRl − ωR0)/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 ωRl/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.
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 − CEE) as a function of n. As the dose increases, the plots of ln(1 − CEE) 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 ofS N, as a function of n, for each methacholine dose. S N increased withn, and the rate of increase increased with methacholine dose.
Model predictions for lung impedance and gas mixing are plotted with the corresponding experimental data.
Measurements of impedance and measurements of gas mixing have long but, for the most part, distinct histories in the study of the constricted lung. Lung resistance and elastance measurements of Officer et al. (21) and of Pellegrino and colleagues (22) during methacholine challenge and measurement of nitrogen washout by Verbanck et al. (29) during histamine provocation are two recent examples. Compliance and gas mixing efficiency are smaller in the constricted than in the normal lung. The decreases in both are the result of an increased heterogeneity of ventilation, and compliance and gas mixing data provide information about that heterogeneity. However, compliance and gas mixing depend on different features of the ventilation distribution and, therefore, provide complementary information about the state of the constricted lung. Here we report data on lung impedance and gas mixing measured simultaneously at different degrees of constriction. Thus these data provide complementary information about the ventilation distribution in the same lung and at the same degree of constriction.
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 ωRl increased systematically with increasing dose. At the highest dose, we measured an average El,dyn of 2.6 E0 and an average increase in ωRl 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 Nthey 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 ofS N shown in Fig. 8 is 0.65 liter−1. For our subjects, 16 breaths corresponds to an average lung turnover number of 3.7, and at that turnover number, Verbanck et al. report anS N near 0.7 liter−1.
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.
Our model for airway mechanics is the standard model, modified by an approximate description of smooth muscle dynamics and an assumption about the relation of the flow through an airway to the tethering forces that act on that airway. We made the approximation that, for oscillatory flows at breathing frequencies, the fluctuations in muscle length and airway radius that occur over a cycle are negligible and that muscle length and airway radius are set at the values for which isometric force equals the peak force applied during a cycle. Thus the curves shown in Figs. 2 and 3 describe the locus of periodic solutions for oscillatory Vt with different amplitudes. The assumption that describes the relation between the flow through an airway and the tethering force that acts on that airway is the key assumption of the model. We assume that the Pa that is determined by the volume that passes through the airway is the same as the Pa that determines the magnitude of the tethering force. That is, we assume that the airway is embedded in the parenchyma it serves. This introduces a feedback between flow and resistance and, together with the properties of periodically stretched muscle, leads to the prediction that the curve of νtidal vs. peak Plumen at the airway entrance is sigmoidal (Fig. 2). A certain minimum value of airway entrance pressure is required to open the airway. For very low values of Vt, peak pressure increases because the resistive pressure drop increases with increasing flow. As Vt increases further, Pa increases and Ptether increases so that airway radius increases, the resistive drop decreases, and peak pressure decreases. At very high Vt, resistive pressure losses are negligible and peak pressure is proportional to Vt and acinar elastance. As a result of the sigmoidal Vt vs. peak pressure curve, the airway has two stable states for the same driving pressure. In one state, the airway is nearly closed, flow and, hence, Vt are small, peak Pa and parenchymal tethering stress are small, and the airway is in stable equilibrium. In the second, the airway is well open, flow and Vt are large, and peak Paand tethering stress are large enough to hold the airway open. In the intervening region of instability, airway viscance and parenchymal elastance are on the same order, and the derivative of Pawith respect to Raw is maximal. For a given pressure oscillation applied at the airway entrance, a transient increase in airway radius and the corresponding decrease in Raw would result in an increase in Pa and a corresponding increase in tethering stress that is larger than that required to maintain the new radius. Consequently, further airway distension would occur. Notably, a mechanical balance between airway viscance and parenchymal elastance characterizes the region of instability, and this balance is independent of the degree of muscle activation.
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 To, the isometric tension at optimum muscle length. The corresponding hoop stress is To /r o, and To /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 To /r o. Pressures and volumes scale with To /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 To /r o, and the viscances of well-open and nearly closed units, described in terms of their viscance-to-elastance ratios, are approximately ηo = 0.8 and ηc = 16, independent of the level of muscle activation.
Whole Lung Model
In the lung, the terminal airways are exposed to a common driving pressure, but the airways are bistable, so that some airways are well open and some are nearly closed. Although the open and closed airways may be anatomically interspersed, the collection of open airways and the collection of closed airways form two functionally distinct compartments. As To /r o increases, P* and the Vt delivered to each open airway increase. To maintain a fixed Vt, the fraction of airways in the open state, φ, must decrease and the fraction of airways in the closed state, (1 − φ), must increase.
The relation of φ to muscle activation explains the relation of lung impedance to methacholine dose shown in Figs. 4 and 5. As To /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 To /r oincreases, and Rperiph, 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, φ is 1, and the model lung is a single open compartment. As a result, CEE approaches 1 exponentially, and the plot of ln(1 − CEE) vs. n has zero curvature. However, for the constricted states, φ falls below 1, the model has two compartments, and CEE 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 To /r o, φ 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 − CEE) 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 Ncalculated 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 ωRperiph/E, Eqs. 13 and 14 impose only two constraints on the three variables φ, ηc, and ηo. As a result, for every value of ηo there is a corresponding φ and ηc that yield the desired El,dyn/E and ωRperiph/E. For a range of ηo, Fig. 9shows the value of ηc 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 ηo and ηc. The model matches the impedance and curvature data for only one pair of values of ηo and ηc. The values of ηo and ηc 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.
The values of the parameters used in the model, with the exception of To, were taken directly from the physiology literature, and the values of To that were used are well within the range described by Gunst and Stropp (10) for ex vivo airway contraction. In previous work (1), we showed that this model fits data describing the relation between lung impedance and Vt in animals (25, 26). Here we demonstrate that the model fits data describing lung impedance and gas mixing at different levels of smooth muscle activation in humans.
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 (V˙/Q˙) distribution, with a ∼10-fold difference in V˙/Q˙ 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 bimodalV˙/Q˙ distribution, with the two peaks separated by a 12-fold difference in V˙/Q˙.
The authors thank Jeffrey Fredberg for help with the revision of the manuscript and Kathy O'Malley for technical assistance.
This work was supported in part by National Institutes of Health General Clinical Research Center Grant MO1-RR-00585, the Mayo Clinic, National Heart, Lung, and Blood Institute Grant HL-52230, and a Whitaker Foundation Graduate Fellowship.
Address for reprint requests and other correspondence: T. A. Wilson, 107 Akerman Hall, 110 Union St. SE, Minneapolis, MN 55455 (E-mail:).
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- Copyright © 2003 the American Physiological Society