Abstract
The effect of bronchoconstriction on airway resistance is known to be spatially heterogeneous and dependent on tidal volume. We present a model of a single terminal airway that explains these features. The model describes a feedback between flow and airway resistance mediated by parenchymal interdependence and the mechanics of activated smooth muscle. The pressuretidal volume relationship for a constricted terminal airway is computed and shown to be sigmoidal. Constricted terminal airways are predicted to have two stable states: one effectively open and one nearly closed. We argue that the heterogeneity of whole lung constriction is a consequence of this behavior. Airways are partitioned between the two states to accommodate total flow, and changes in tidal volume and endexpiratory pressure affect the number of airways in each state. Quantitative predictions for whole lung resistance and elastance agree with data from previously published studies on lung impedance.
 airway mechanics
 lung resistance
 lung elastance
 mathematical model
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 highresolution 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, 2527). 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 pressuretidal 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 endexpiratory 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
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.
The geometry of the airway cross section is shown in Fig.1
B. 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 ρ_{m}. The ratio of the inner radius to r
_{o} is denoted ρ_{i}. Because the volume of the submucosa remains constant, the crosssectional area of the submucosa is a fixed fraction (f) of the unconstricted airway area. Thus ρ_{i}, ρ_{m}, and f are related by the following equation
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
_{o}ρ_{i}), and gas viscosity (μ)
Airway mechanics.
Raw is a function of ρ_{m}, and ρ_{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)
Ptm is the difference between lumen pressure (P_{lumen}) and peribronchial pressure acting on the outer airway surface. Both Pa and parenchymal tethering stress (τ) contribute to the latter
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, 1113,27). Plots representative of the data on cyclically driven smooth muscle are shown in Fig. 2, along with an approximate isometric forcelength 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.
First, smooth muscle is much stiffer in response to a periodic stimulus than to a quasistatic one. With a stimulus near breathing frequency, more than an 80% reduction in applied force is required to reduce muscle length by 10% (9, 11, 27). This fact justifies the earlier assumption that airway radius and muscle length are nearly constant over the course of a single breath. Second, the peak applied force and peak muscle length are anchored along the isometric forcelength curve (9, 11, 27). In other words, the maximum force imposed during a cycle approximately equals the force obtained during isometric contraction at the peak muscle length.
An approximate constitutive law for cyclically driven smooth muscle can be formulated from these rules. The isometric forcelength curve is described by the equation T/T_{o} = 1.25λ − 0.25, where T_{o} denotes the isometric tension at optimal length (L
_{o}) and λ denotes muscle length nondimensionalized by L
_{o}. During a force oscillation, muscle length is nearly constant, and maximum tension (T_{max}) is the isometric force at that length. Thus the relation between T_{max} and λ during a force oscillation is the same as the relation between T and λ during an isometric contraction. Furthermore, it is assumed that muscle length isL
_{o} when airway radius isr
_{o}. Thus λ = ρ_{m} and T_{max} is related to ρ_{m} by the equation
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
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 ρ_{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 10^{4} terminal units of the Horsfield model. V_{TLC} is the volume of a parenchymal unit at 20 cmH_{2}O, 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 cmH_{2}O) reported by Gunst and Stropp (12) for small airways.
The imposed stimulus is chosen to simulate the experimental protocol of Salerno et al. (25). In this experiment, resistance and elastance were measured in methacholineconstricted dog lungs at two tidal volumes, and a PEEP of 5 cmH_{2}O was imposed at both tidal volumes. To simulate this experiment, PEEP is held at 5 cmH_{2}O by varying P̄ as ∥Paw∥ is varied. The frequency of breathing is 0.33 Hz in both the computational and experimental protocols.
Numerical methods.
For given values of ρ_{m} and P
RESULTS
The equations describing both sides of Eq. 11
are plotted in Fig. 3. The family of thin curves represents P
DISCUSSION
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, passivespecific elastance (∼4 cmH_{2}O) (13) is small compared with the specific elastance of activated muscle (∼100 cmH_{2}O). 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 pressuretidal 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. 35 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
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 ρ_{m} at the intersecting points of Fig. 3 are plotted in Fig. 4. For the range of P
Alternatively, the curve shown in Fig. 4 can be understood directly. As ρ_{m} grows, peak muscle hoop stress increases, and a greater P
The plot of tidal volume vs. P
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 ofregion 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 4and 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
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 cmH_{2}O, 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.
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 methacholineconstricted rabbit. Salerno et al. reported data on both lung resistance and elastance in the methacholineconstricted 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
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 ventilationtoperfusion (V˙/Q˙) ratio in human asthmatic subjects and in bronchoconstricted dogs (24) and observed sharply bimodalV˙/Q˙ distributions. The peaks occurred at V˙/Q˙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 V˙/Q˙ values at the peaks of theV˙/Q˙ 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 pressureflow 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.
Acknowledgments
This work was funded by a Whitaker Foundation Graduate Fellowship.
Footnotes

Address for reprint requests and other correspondence: T. A. Wilson, 107 Akerman Hall, 110 Union St. SE, Minneapolis, MN 55455 (Email: wilson{at}aem.umn.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “advertisement” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
 Copyright © 2001 the American Physiological Society