INTRODUCTION

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TWO FEATURES OF THE DATA describing constricted lungs are pertinent to the work reported in this paper. First, airway resistance in constricted lungs is heterogeneous. Direct visualization using high-resolution computerized tomography (1, 5), alveolar capsule measurements (7), and indirect evidence from models fit to whole lung impedance data (4, 20) reveal nonuniform airway constriction. It has generally been assumed that this nonuniformity is a result of heterogeneous smooth muscle activation or tissue properties (3, 5, 7, 10, 20). Second, lung impedance decreases systematically with increasing tidal volume (25, 26). It has been hypothesized that the decrease in resistance that accompanies increased tidal volume is a result of smooth muscle dynamics (8, 9, 25-27). It is also well known that parenchymal tethering, which is an important component of airway mechanics (14), depends on lung volume. These ideas have not been developed to the point of predicting the quantitative dependence of lung resistance on tidal volume, and they provide no explanation for the corresponding dependance of elastance on tidal volume.

Bronchoconstriction primarily affects the lung periphery (16, 29), and our aim is to explain the properties of the constricted lung by modeling the mechanics of a constricted terminal airway. There are two crucial features of the model. First, the airway is surrounded by the parenchyma it serves so that flow through the airway affects its own peribronchial pressure. Second, the mechanical properties of the airway wall are determined by the dynamic constitutive properties of activated smooth muscle. Muscle length and airway radius are determined by peak transmural pressure.

The model equations are solved to obtain a pressure-tidal volume relation for the terminal airways. We find that, for some applied pressures, the resulting airway diameter and flow are not unique and that two stable solutions are obtained. Whole lung constriction is accordingly heterogeneous, with airways distributed between effectively open and nearly closed states. The number of airways in the open state increases as tidal volume increases. The total resistance and elastance of the model lung are computed as functions of tidal volume and positive end-expiratory pressure (PEEP), and the predictions agree with previously published experimental measurements. We conclude that heterogeneity is a result of the mechanics of bronchoconstriction and that the distribution of airways between two states determines lung impedance as a function of tidal volume and PEEP.

	MODELING

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A single terminal airway that feeds a respiratory acinus is represented by the model shown in Fig. 1. The pressure drop between the pressure at the airway entrance (Paw) and the pressure in the acinus (PA) drives flow through the airway with resistance (Raw) into the acinus with elastance. Smooth muscle is present in the airway wall, and the airway is embedded in the parenchyma it serves. The aim of the following analysis is to calculate the relationship between pressure and flow when the smooth muscle is fully activated.

View larger version (13K): [in this window] [in a new window]   Fig. 1.   Schematic of model airway and surrounding parenchyma. A: difference between airway entrance pressure (Paw) and alveolar pressure (PA) drives flow through a terminal airway with resistance (Raw). The region of parenchyma served by the airway has elastance (E). The airway is surrounded by the parenchyma it serves. B: muscle radius, nondimensionalized by the radius of an unconstricted airway (ro) at total lung capacity (TLC), is denoted by m. Inner radius, nondimensionalized by ro, is denoted by i. Lumen pressure (Plumen) acts on the inner surface of the airway, and PA and parenchymal tethering stress () contribute to peribronchial pressure.

The geometry of the airway cross section is shown in Fig. 1B. The radius of the layer of smooth muscle near the outer boundary of the unconstricted airway at total lung capacity (TLC) is denoted r_o. The ratio of the constricted muscle radius to r_o is taken as the basic measure of airway constriction and is denoted _m. The ratio of the inner radius to r_o is denoted _i. Because the volume of the submucosa remains constant, the cross-sectional area of the submucosa is a fixed fraction (f) of the unconstricted airway area. Thus _i, _m, and f are related by the following equation

(1)

Flow. A modified Poiseuille equation, taken from the low Reynolds number term in the empirical relation between flow and pressure reported by Reynolds and Lee (22, 23), is used to calculate Raw as a function of airway length (l), internal radius (r_oi), and gas viscosity (µ)

(2)

Paw oscillates around a mean pressure () with frequency  and amplitude Paw

(3)

where t means time. If Raw and elastance (E) remain nearly constant, PA is approximately sinusoidal with , , PA, and phase lag  

(4)

where

Airway mechanics. Raw is a function of _m, and _m is determined by a balance between the hoop stress due to airway smooth muscle tension (T), as described by the Law of Laplace, and transmural pressure (Ptm)

(5)

Equation 5 is the central relation used to determine _m, and the objective of the following development is to express both sides of this equation in terms of _m.

(6)

(7)

(8)

(9)

Muscle. To evaluate the left side of Eq. 5, a description of the constitutive properties of activated smooth muscle is required. In the past decade, several researchers have investigated the response of activated smooth muscle to periodic stretch (9, 11-13, 27). Plots representative of the data on cyclically driven smooth muscle are shown in Fig. 2, along with an approximate isometric force-length curve. These data suggest a rich, nonlinear relation between force and length, but two simple observations seem to describe muscle behavior to a first approximation.

View larger version (11K): [in this window] [in a new window]   Fig. 2.   Representative tension-length behavior of activated smooth muscle. The tension-length relationship for isometrically contracted muscle is shown by the thin line. Maximum isometric tension (To) occurs at muscle length Lo. Tension-length loops (thick loops), obtained from periodic stretch, are much stiffer than the isometric line. Only small changes in muscle length result from large variations in applied tension. Peak tension during periodic stretch is approximately equal to isometric force at the same length.

(10)

Final equation. T_max occurs at the time during the cycle when Ptm is maximum. At that time, T is given by Eq. 10 and Ptm is equal to its maximum value (P). Therefore, at the time when T and Ptm are at maximum, Eq. 5 has the form

(11)

The value of _m was obtained from this equation using the numerical methods described below.

Parameter values. The parameters that describe the geometry and physical properties of the airway and parenchyma are chosen as follows. The fraction of an unconstricted airway cross section occupied by the submucosa is set at 16% (18). By direct application of Eq. 1, the airway is thus closed when _m  0.4. r_o = 0.027 cm is calculated from the inner radius of terminal airways of the dog lung given by Horsfield et al. (15). Airway length is fixed by requiring that the model airway have the same effective resistance as the combination of Horsfield generations 5 and 6, the terminal and preterminal airways (l = 0.3 cm). Elastance (= 25 kPa/ml) of a single acinar unit is determined by partitioning the elastance of a whole dog lung as measured by Salerno et al. (25) (2.5 kPa/l) into the 10⁴ terminal units of the Horsfield model. V_TLC is the volume of a parenchymal unit at 20 cmH₂O, assuming that V_RV is 20% V_TLC. Finally, the value of T_o is obtained from the ratio of T_o/r_o (40 cmH₂O) reported by Gunst and Stropp (12) for small airways.

Numerical methods. For given values of _m and P, Ptm was calculated as a function of time from Eqs. 1-4 and 6-9, and its peak value, P, was identified. P and _m were varied in increments of 0.25 cmH₂O and 0.001, respectively, and a table of values of P was obtained. This table describes the right side of Eq. 11, and _m was obtained by finding the intersection of these values with the function of _m given by the left side of Eq. 11. Once the values of _m and Raw corresponding to a given applied pressure were computed, the tidal volume carried by the airway was determined by integrating the equations of flow.

	RESULTS

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The equations describing both sides of Eq. 11 are plotted in Fig. 3. The family of thin curves represents P and thus the right-hand side of the equation. Each curve shows the P exerted on the airway by its surroundings for fixed P and varying _m. The single dark curve represents the left side of Eq. 11, the peak hoop stress developed by smooth muscle as a function of _m. The value of _m that satisfies the equations of airway mechanics for a given applied P is determined by the intersection of the dark and light curves. For P < 13.8 cmH₂O, there are no points of intersection, hoop stress is greater than Ptm, and the airway lumen is closed. For intermediate values of P (13.8 cmH₂O < P <15.5 cmH₂O), there are two or three intersections, and the value of _m is not uniquely specified. For P > 15.5 cmH₂O, there is only one intersection, and the airway is well open. The solution curve for Eq. 11 is plotted in Fig. 4, which shows the relationship between _m and P. The tidal volume carried by a single airway, normalized by V_TLC, is plotted as a function of P in Fig. 5.

View larger version (17K): [in this window] [in a new window]   Fig. 3.   Peak transmural pressure (P) vs. m. Each of the thin curves shows P as a function of m for a fixed-peak pressure at the airway entrance. The single thick line shows the peak hoop stress produced by airway smooth muscle.

View larger version (12K): [in this window] [in a new window]   Fig. 4.   m vs. peak airway entrance pressure.

View larger version (10K): [in this window] [in a new window]   Fig. 5.   Acinar tidal volume vs. peak airway entrance pressure. Acinar tidal volume is nondimensionalized by the volume of an acinus at TLC (VTLC). The curve can be divided into 3 regions according to the sign of its slope. In the regions with positive slope, regions I and III, the airway is nearly closed and well open, respectively. In the transition region (region II), the slope is negative, and the solution to the flow equations is unstable.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODELING RESULTS DISCUSSION REFERENCES

Our model of the geometry and mechanics of the constricted airway is the standard model. Gunst et al. (14) described this model and used it to estimate the value of transpulmonary pressure (Ptp) at which interdependence forces are large enough to open a maximally constricted airway. Subsequently, Macklem (21) and Lambert et al. (19), among others, have used similar models. Macklem calculated the equilibrium value of muscle tension as a function of airway radius and Ptp. He noted that the plots of tension vs. radius at fixed Ptp have maxima and that the airway would snap shut if muscle force exceeded the maximum. Lambert et al. used the model to evaluate the effects of muscle hypertrophy and airway inflammation on airway resistance. In these applications, static equilibrium was studied, whereas here we study the dynamic behavior of the model.

Several simplifications have been made in this application of the model. First, the passive stiffness of the airway wall has been neglected. The stiffness of a relaxed airway depends on diameter and is large near maximum airway diameter (12, 13). However, the model concerns constricted airways with small diameters, and, for small diameters, passive-specific elastance (~4 cmH₂O) (13) is small compared with the specific elastance of activated muscle (~100 cmH₂O). Second, the model ignores tissue resistance. Although tissue resistance is significant in the normal lung, the resistance of the constricted lung is thought to be primarily due to the increased resistance of peripheral airways (16, 29). Baseline tissue and central airway resistance are added to peripheral airway resistance in comparing model results with whole lung data. Finally, the model ignores the effect of interregional tethering. Local parenchymal distortion is assumed to depend only on local alveolar volume, and the influence of adjacent units is assumed to be small. Although the simplifications listed above may affect the quantitative predictions of the model, we believe that their effects are small and would not change the qualitative features of the results.

Two key assumptions underlie the model and are responsible for the main features of the predicted pressure-tidal volume relationship. Taken together, these assumptions lead to the feedback between flow and resistance. First, it is assumed that the airway is embedded in the parenchyma it serves. Therefore, PA in the subserved parenchyma is the same as PA in the parenchyma that surrounds the airway. As a result, flow through an airway affects peribronchial pressure and feeds back on airway mechanics. Second, it is assumed that airway radius is determined by peak Ptm and remains constant over a breath. This assumption is founded on experimental observations of smooth muscle strips and excised airways (9, 11, 13, 27). Recently, Fredberg (6) drew attention to the dynamic properties of smooth muscle and emphasized the idea that, in the ventilated lung, dynamic equilibrium is the crucial determinant of the state of the airway. Fredberg et al. (8) described two mechanical consequences of periodic stretch: the anchoring of peak length to peak tension as described by Eq. 10 and a decrease in stiffness with increasing stretch amplitude. Although muscle stiffness decreases with increasing tidal stretch amplitude, it remains much stiffer than the passive airway wall, and Fredberg et al. (8) estimated that tidal breathing would induce strains of only 4%. Here, we incorporate these ideas into the model for the dynamic equilibrium of the airway.

The qualitative features of the results presented in Figs. 3-5 can be explained as follows. We begin by examining the family of light curves shown in Fig. 3. These curves show the relation between P and _m for fixed peak entrance pressures. Equations 6, 7, and 9 show that Ptm depends on _m through the contributions of PA and . Changing airway radius affects both the fraction of airway opening pressure transmitted to the acinus and the distortion of the parenchyma. For _m near 0.4 (near closure), Raw is high and PA remains near and insensitive to _m. However, as _m increases, x and  fall and P decreases. For intermediate values of _m (0.42 < _m < 0.48), flow and peak PA grow rapidly as _m increases. As a result, P increases with _m in this region. At higher values of _m (_m > 0.48), Raw is small, and peak PA is approximately equal to P. Here, as in the region near _m = 0.4, PA is insensitive to _m and P decreases with increasing _m because  decreases with increasing _m. The equilibrium values of _m are determined by the intersections of these curves with the bold line describing peak muscle hoop stress as a function of _m. For some curves, multiple intersections occur because P changes rapidly with _m as the airway opens, whereas peak muscle hoop stress increases slowly with _m.

The shape of the curve shown in Fig. 4 is a consequence of the shapes of the curves in Fig. 3. That is, the values of _m at the intersecting points of Fig. 3 are plotted in Fig. 4. For the range of P for which multiple intersections exist, _m has multiple values at a single value of P.

Alternatively, the curve shown in Fig. 4 can be understood directly. As _m grows, peak muscle hoop stress increases, and a greater P is required to maintain mechanical equilibrium. For _m near 0.4, PA changes slowly with _m, and P can only increase by increasing P. In the middle range of _m, flow and maximum PA increase rapidly with _m, and P increases despite the decrease in P. Finally, with the airway effectively open, peak PA equals P, and an increase in applied pressure is required to increase _m further.

The plot of tidal volume vs. P shown in Fig. 5 is similar to the plot of _m vs. P in Fig. 4. For values of _m near 0.4 (region I), the airway is nearly closed and tidal volume increases slowly with P. At intermediate values of _m (region II), flow increases rapidly with increasing _m, and tidal volume increases despite the decrease in P. Finally, for larger values of _m (region III), Raw is negligible and tidal volume is proportional to Paw/E.

These results for a single terminal airway have important implications for whole lung mechanics. The existence of multiple equilibrium solutions at a given applied pressure implies that heterogeneous constriction does not require heterogeneous muscle activation or nonuniform tissue properties. In fact, the negative slope of region II ensures that heterogeneous constriction will occur. For a wide range of tidal volumes, equal partitioning of ventilation among the units would require that each airway carry a tidal volume in region II. This solution is unstable to perturbation. Two airways, connected in parallel, can satisfy the equilibrium flow equations by carrying equal tidal volumes and operating in region II. However, disturbances will cause one airway to be larger than the other and carry a larger flow. Figures 4 and 5 show that increasing the radius and flow lowers the pressure required to maintain equilibrium, whereas decreasing the radius and flow increases the required pressure. Because parallel airways are subject to the same pressure drop, airway entrance pressure expands the slightly larger airway while it is unable to resist the further constriction of the smaller airway. The radii of the two airways diverge until they reach the regions of Fig. 4 with a positive slope. Total tidal volume is maintained, but it is preferentially distributed toward the larger airway. One airway is nearly closed, the other open. For tidal volumes in the range of 4-30% TLC, terminal airways will partition themselves into two groups: airways that are nearly closed (region I) and airways that are effectively open (region III). Changes in tidal volume are accommodated by redistributing the number of airways in each group.

For sufficiently small tidal volumes (<4% TLC) and applied pressures (P < 14 cmH₂O), there is only one equilibrium solution to the airway equations, and ventilation is uniformly distributed. For this range, as tidal volume increases, each unit carries a greater flow and the applied pressure increases as described in region I of Fig. 5. When P reaches ~15.5 cmH₂O, the model predicts that regional tidal volume no longer increases smoothly with applied pressure. At that pressure, some airways jump to the open solution described by region III. Further increases in tidal volume cause a larger fraction of units to open. For tidal volumes that are sufficient to force all airways open (>40% TLC), ventilation is uniform and pressure increases as described by region III.

A simple relationship between whole lung impedance and tidal volume immediately results from this model. As tidal volume increases and airways pop open, the pressure drop over the terminal airways remains constant. Thus the impedance of the terminal airways is inversely proportional to tidal volume. Increasing the tidal volume results in an increased number of open airways and decreased impedance. Experimental measurements of whole lung impedance exhibit this behavior (25, 26).

The two components of lung impedance, resistance and elastance, can also be calculated. Peak airway entrance pressure is assumed fixed at 15.5 cmH₂O, and the two stable solutions for airway radius are obtained from Fig. 4. The number of units in each region is determined by requiring that the total tidal volume carried by the terminal airways be equal to the imposed tidal volume, and the complex impedance of the terminal airways is computed accordingly. Elastance and resistance are determined by separating the real and imaginary parts of total impedance. The baseline resistance of the lung reported by Salerno et al. (25) is taken to represent central airway and tissue resistance and is added to peripheral resistance.

The quantitative dependence of whole lung resistance and elastance on tidal volume predicted by the model is shown in Fig. 6. The ratio of elastance and resistance for the constricted lung to baseline values is shown, plotted vs. tidal volume, as a fraction of TLC. At very small tidal volumes, all of the airways are constricted, and resistance is high. Because the distribution of resistance is uniform, elastance equals its baseline value. This low value of elastance seems unrealistic. Intrinsic variability of smooth muscle mass, activation, and variable airway geometry have not been included in the model, and these would be expected to be particularly important when the airway is nearly closed. As tidal volume increases, the first airways open, and elastance rises sharply because a large fraction of total flow is forced into the small region of parenchyma served by the open airways. For further increases in tidal volume, both elastance and resistance fall as more airways open. When all of the airways have opened, resistance is again uniform and elastance returns to its baseline value.

View larger version (11K): [in this window] [in a new window]   Fig. 6.   Effect of tidal volume on constricted lung impedance, model predictions, and experimental data. A: solid curve shows the model prediction for the ratio of constricted lung resistance to baseline resistance as a function of tidal volume.  and , Data of Shen et al. (26) and Salerno et al. (25), respectively. B: solid curve shows the predicted ratio of constricted lung elastance to baseline elastance vs. tidal volume. , Data of Salerno et al. (25).

Also shown in Fig. 6 are the experimental data of Shen et al. (26) and Salerno et al. (25). Shen et al. measured the effect of tidal volume on lung resistance in the methacholine-constricted rabbit. Salerno et al. reported data on both lung resistance and elastance in the methacholine-constricted dog. The predictions of the model agree with the reported data.

The data of Balassy et al. (2) provide another example of data that describe the dependence of lung impedance on a ventilation parameter. In contrast to the protocols of Shen et al. (26) and Salerno et al. (25), Balassy et al. measured lung impedance for a fixed tidal volume and a range of values of PEEP. To simulate this experiment, we repeated the calculations described above for a tidal volume of 350 ml and different values of PEEP. The parameter values were the same except for the value of elastance, which was adjusted to match the data of Balassy et al. for lung elastance as a function of PEEP in the control state. The results are shown, together with the data of Balassy et al., in Fig. 7. These results can be explained as follows. For different values of PEEP, the curves of airway radius and tidal volume vs. P are similar to the curves shown in Figs. 4 and 5. P decreases slightly with increasing PEEP. Because the amplitude of the pressure oscillation decreases, the tidal volume for the regions served by open airways decreases, and more of the lung must open as PEEP increases. The lung is fully opened at a PEEP of ~7 cmH₂O, and, for further increases in PEEP, peak pressure increases, the airways open further, and resistance decreases sharply. For lower values of PEEP, the model values of resistance and elastance are lower and higher, respectively, than the experimental values. Both the model and data show a sharp decrease in resistance at a PEEP of ~7 cmH₂O.

View larger version (13K): [in this window] [in a new window]   Fig. 7.   Effect of positive end-expiratory pressure (PEEP) on constricted lung impedance. Ratio of constricted lung resistance and elastance to baseline values calculated from the model (solid and dashed lines, respectively) and data reported by Balassy et al. (2) are shown ( and , respectively).

These two examples show the dependence of lung impedance on two of the parameters that describe ventilation: tidal volume and PEEP. For a wide range of these two parameters, the model gives values of resistance and elastance that agree reasonably well with the data. We would like to comment that the model for lung impedance contains no adjustable parameters. It is entirely based on the model for the mechanics of an airway, and the values of the parameters of the airway model are all obtained from data in the literature.

Some observations that are consistent with the predictions of the model can be cited. Brown et al. (5) compared different mechanisms for the delivery of histamine and concluded that the heterogeneity of constriction was a result of local mechanisms rather than nonuniform agonist delivery. Using the alveolar capsule technique, Fredberg et al. (7) observed distinctly different airway responses to constriction: relatively slight response in some airways and closure or near closure in others. Furthermore, they noted a redistribution of flow toward the less affected units. Finally, Wagner et al. (28) measured the ventilation-to-perfusion (/) ratio in human asthmatic subjects and in bronchoconstricted dogs (24) and observed sharply bimodal / distributions. The peaks occurred at / values near 1 and 0.07. The ratio of the ventilations for the regions served by the open and nearly closed airways in our model is about the same as the ratio of the / values at the peaks of the / distribution.

Other groups have modeled whole airway networks (3, 10, 20) and have included inertial forces and airway compliance to model constricted lung behavior at higher frequencies. The scope of our work is more limited. We have modeled the mechanics of a single terminal airway to explain the heterogeneity of constriction that has been assumed in previous work. The novel feature of this model is the feedback between regional flow and peribronchial pressure. This feedback results in a sigmoidal pressure-flow relationship and two stable solutions for the flow in a terminal airway. The model yields quantitative predictions for the dependence of constricted whole lung impedance on tidal volume and PEEP that match experimental data. We conclude that the heterogeneity of whole lung constriction is a result of terminal airway mechanics. Although variations in muscle activation and inherent tissue properties may add to this nonuniformity, the distribution of airway constriction is inherently bimodal and dependent on tidal volume.