Mathematical description of geometric and kinematic aspects of smooth muscle plasticity and some related morphometrics.

Despite considerable investigation, the mechanisms underlying the functional properties of smooth muscle are poorly understood. This can be attributed, at least in part, to a lack of knowledge about the structure and organization of the contractile apparatus inside the muscle cell. Recent observations of the plasticity of smooth muscle and of morphometry of the cell have provided enough information for us to propose a quantitative, although highly simplified, model for the geometric arrangement of contractile units and their collective kinematic functions in smooth muscle, particularly airway smooth muscle. We propose that, to a considerable extent, contractile machinery restructures upon activation of the muscle and adapts to cell geometry at the time of activation. We assume that, under steady-state conditions, the geometric arrangement of contractile units and the filaments within these units determines the kinematic characteristics of the muscle. The model successfully predicts the results of experiments on airway smooth muscle plasticity relating to maximal force generation, maximal velocity of shortening, and the variation of compliance with adapted length. The model is also concordant with morphometric observations that show an increase in myosin filament density when muscle is adapted to a longer length. The model provides a framework for design of experiments to quantitatively test various aspects of smooth muscle plasticity in terms of geometric arrangement of contractile units and the muscle's mechanical properties.

Computational model of airway narrowing: mature vs. immature rabbit.

Immature rabbits have greater maximal airway narrowing and greater maximal fold increases in airway resistance during bronchoconstriction than mature animals. We have previously demonstrated that excised immature rabbit lungs have more distensible airways, a lower shear modulus, and structural differences in the relative composition and thickness of anatomically similar airways. In the present study, we incorporated anatomic and physiological data for mature and immature rabbits into a computational model of airway narrowing. We then investigated the relative importance of maturational differences in these factors as determinants of the greater airway narrowing that occurs in the immature animal. The immature model demonstrated greater sensitivity to agonist, as well as a greater maximal fold increase in airway resistance. Exchanging values for airway compliance between the mature and immature models resulted in the mature model exhibiting a greater maximal airway response than the immature model. In contrast, exchanging the shear moduli or the composition of the airway wall relative to the airway size produced relatively small changes in airway reactivity. Our results strongly suggest that the mechanical properties of the airway, i.e., greater compliance of the immature airway, can be an important factor contributing to the greater airway narrowing of the immature animal.

Analysis of tracheal mechanics and applications.

We have developed a mathematical model for a tracheal ring that consists of a "horseshoe" of cartilage with its tips joined by a membrane. The ring is subjected to a uniform transmural pressure (Ptm) difference. The model was used to calculate the cross-sectional area (A) of the trachea. Whereas the mechanics of the deformation of the cartilage were analyzed using elastica theory, the posterior membrane was treated as a simple membrane that is inextensible under changes in Ptm. The membrane can be specified to be of any length less than baseline and thus can represent a posterior membrane under tension. The cartilage can have specifiable nonuniform unstressed curvature as well as nonuniform bending stiffness. We have investigated the effect on the tracheal A-Ptm curve of posterior membrane length and tensile force in the membrane, cartilage shape and elasticity, and localized weakening of the cartilage. The model predictions are in good agreement with magnetic resonance imaging data from rabbit tracheas and show that the shape of the horseshoe as well as the posterior membrane force are important determinants of tracheal compliance.

Stiffness of peripheral airway folding membrane in rabbits.

We have observed that small, membranous bronchioles from rabbits, in which the smooth muscle is not activated, experience a critical elastic buckling involving the whole airway wall during deflation of the lung. This implies that, at some point during the deflation, the airway wall goes from being in a state of tension to a state of compression. At the transition, there is neither net tension nor net compression in the wall, and the transmural pressure difference must, therefore, be zero. Thus at this point, the pressure difference across the muscle that results from the passive stress in the muscle is just balanced by the pressure difference across the folded mucosal membrane. We estimated the muscle stress, and hence the pressure across the muscle, from published data on rabbit trachealis (Opazo-Saez A and Paré PD, J Appl Physiol 77: 1638-1643, 1994) and equated this to the pressure across the folded membrane. By using a theoretical prediction of this pressure (Lambert RK, Codd SL, Alley MR, and Pack RJ, J Appl Physiol 77: 1206-1216, 1994), together with the results of our morphometric measurements on these airways, we estimated that the flexural rigidity of the folding membrane in peripheral rabbit airways is of the order of 10(-12) Pa x m3. This value implies that, in these airways, membrane folding provides significant resistance to airway smooth muscle shortening.

Compliance of peripheral airways deduced from morphometry.

Insights into airway mechanics were sought by applying morphometric techniques to rabbit lungs fixed at several lung recoil pressures. Rabbits were treated with either nebulized carbachol followed by iv administration of carbachol or with saline solution (sham). The lungs were held at one of six values of positive end-expiratory pressure (PEEP; 10, 7, 4, 2, 0, and -4 cmH(2)O) while the animal was killed and formalin was circulated through the lungs. The lungs were removed and left in a bath of formalin for 24 h. Standard airway morphometric measurements were made on membranous bronchiole slices taken from representative blocks of tissue. Reductions in PEEP produced the expected reductions in lumen area in the carbachol-treated airways but not in the sham-treated airways for PEEP > 2 cmH(2)O. Sham-treated airways remained more open than expected until they collapsed into an oval shape at PEEPs between 4 and 2 cmH(2)O. The carbachol-treated airways exhibited this behavior at PEEP = -4 cmH(2)O. The smallest airways, which had relatively thicker walls, collapsed less than larger airways. We postulate that this behavior implies that peribronchial stress is greater than lumen pressure on collapse into the oval shape. Resistance to buckling increases with the thickness-to-radius ratio of the airway wall, which explains why the smallest airways are the most open. The development of epithelial folds appeared to follow the theoretical prediction of a previous study (Lambert RK, Codd SL, Alley MR, and Pack RJ. J Appl Physiol 77: 1206-1216, 1994).

Cartilaginous airway wall dimensions and airway resistance in cystic fibrosis lungs.

It is not clear how airway pathology relates to the severity of airflow obstruction and increased bronchial responsiveness in cystic fibrosis (CF) patients. The aim of this study was to measure the airway dimensions of CF patients and to estimate the importance of these dimensions to airway resistance using a computational model. Airway dimensions were measured in lungs obtained from CF patients who had undergone lung transplantation (n=12), lobectomy (n=1), or autopsy (n=4). These dimensions were compared to those of airways from lobectomy specimens from 72 patients with various degrees of chronic obstructive pulmonary disease (COPD). The airway dimensions of the CF and COPD patients were introduced into a computational model to study their effect on airway resistance. The inner wall and smooth muscle areas of peripheral CF airways were increased 3.3- and 4.3-fold respectively compared to those of COPD airways. The epithelium was 53% greater in height in peripheral CF airways. The sensitivity and maximal plateau resistance of the computed dose/response curves were substantially increased in the CF patients compared to COPD patients. The changes in airway dimensions of cystic fibrosis patients probably contribute to the severe airflow obstruction, and to increased bronchial responsiveness, in these patients.

Lung parenchymal shear modulus, airway wall remodeling, and bronchial hyperresponsiveness.

When airways narrow, either through the action of smooth muscle shortening or during forced expiration, the lung parenchyma is locally distorted and provides an increased peribronchial stress that resists the narrowing. Although this interdependence has been well studied, the quantitative significance of airway remodeling to interdependence has not been elucidated. We have used an improved computational model of the bronchial response to smooth muscle agonists to investigate the relationships between airway narrowing (as indicated by airway resistance), parenchymal shear modulus, adventitial thickening, and inner wall thickening at lung recoil pressures of 4, 5, and 8 cmH2O. We have found that, at low recoil pressures, decreases in parenchymal shear modulus have a significant effect that is comparable to that of moderate thickening of the airway wall. At higher lung recoil pressures, the effect is negligible.

Effects of surface tension and intraluminal fluid on mechanics of small airways.

Airway constriction is accompanied by folding of the mucosa to form ridges that run axially along the inner surface of the airways. The mucosa has been modeled (R. K. Lambert. J. Appl. Physiol. 71:666-673, 1991) as a thin elastic layer with a finite bending stiffness, and the contribution of its bending stiffness to airway elastance has been computed. In this study, we extend that work by including surface tension and intraluminal fluid in the model. With surface tension, the pressure on the inner surface of the elastic mucosa is modified by the pressure difference across the air-liquid interface. As folds form in the mucosa, intraluminal fluid collects in pools in the depressions formed by the folds, and the curvature of the air-liquid interface becomes nonuniform. If the amount of intraluminal fluid is small, < 2% of luminal volume, the pools of intraluminal fluid are small, the air-liquid interface nearly coincides with the surface of the mucosa, and the area of the air-liquid interface remains constant as airway cross-sectional area decreases. In that case, surface energy is independent of airway area, and surface tension has no effect on airway mechanics. If the amount of intraluminal fluid is > 2%, the area of the air-liquid interface decreases as airway cross-sectional area decreases. and surface tension contributes to airway compression. The model predicts that surface tension plus intraluminal fluid can cause an instability in the area-pressure curve of small airways. This instability provides a mechanism for abrupt airway closure and abrupt reopening at a higher opening pressure.

Determinants of airway smooth muscle shortening in excised canine lobes.

There is marked heterogeneity of airway narrowing in intraparenchymal airways in response to bronchoconstricting stimuli. We hypothesized that this heterogeneity results from variations in the structure of the airway wall. Freshly excised dog lung lobes were inflated to transpulmonary pressures (PL) of between 5 and 15 cmH2O, and an aerosol containing a high concentration of carbachol was administered. The lobes were fixed and processed for light-microscopic examination and morphometric analysis of membranous airway dimensions. The relationships of smooth muscle shortening to PL and airway dimensions were analyzed using multiple linear regression. The results show that airway smooth muscle shortening was greater at lower PL and in airways with larger internal perimeter and a greater number of folds per internal perimeter and that it was less in airways with greater inner wall area. We conclude that the magnitude and variability of airway smooth muscle shortening and airway narrowing in response to maximal constricting stimuli are influenced by mechanical factors related to airway wall geometry.

Physical determinants of bronchial mucosal folding.

It has recently been proposed, on the basis of a theoretical analysis, that the folding of the mucosa provides a significant component of airway stiffness. The model predicted that the stiffness of an airway was directly related to the number of epithelial folds that developed. In this study we examine the possibility that the folding pattern is determined by the physical requirements that the folding membrane must stay within the boundary of the smooth muscle wall, that the submucosal mass is constant, and that the strain energy of the folding membrane is the minimum possible within the geometric constraints. Model predictions are compared with morphometric data from the noncartilaginous airways of 17 sheep lungs. The data are in agreement with our predictions, which are based on the assumption that the folding membrane thickness is proportional to the submucosal thickness (in a fully dilated airway). The outcome of this analysis is that the increase in intrinsic stiffness of the folding membrane resulting from the increased thickness outweighs the decrease in stiffness conferred by the fewer folds required by the thicker submucosa. It is suggested that the increase in folding membrane thickness observed in asthma could be viewed as a protective mechanism that tends to reduce hyperresponsiveness.

Proposed nomenclature for quantifying subdivisions of the bronchial wall.

There is increasing interest in the structural components of the airway wall because of the airway remodeling that is observed in conditions such as asthma and chronic obstructive pulmonary disease and because of their contribution to changes in airway mechanics. This interest has stimulated several groups to make morphometric measurements on airway cross sections, and their results have been reported using a variety of nomenclature. We propose the adoption of a standard system of nomenclature that is based on accepted terms for subdivisions of the airway wall and has been agreed to by several groups working in this field.

Tensile stiffness of ovine tracheal wall.

The epithelial folding that occurs during bronchoconstriction requires that the pressure on the muscle side of the folding membrane be greater than that on the lumen side. The pressure required for a given level of folding depends on the elastic properties of the tissue and on the geometry of the folding. To quantify the elastic properties, uniaxial tensile stiffness of the tracheal inner wall of nine sheep was measured in two directions: parallel to the tracheal axis and circumferentially. The tissue showed anisotropic behavior, being approximately three times stiffer longitudinally than circumferentially. Histological examination showed that collagen in the lamina propria was randomly arranged, whereas there were straight elastin fibers aligned with the tracheal axis. This observation could explain the observed elastic anisotropy. Mechanical removal of the epithelium had no effect on tensile stiffness. It was also found that the tissue was under tension in situ. When a strip was excised, its length decreased by > or = 30%. After allowing for the systematic errors inherent in this experiment, the in situ circumferential tensile stiffness is estimated to be > or = 20 kPa. If the equivalent tissue in the bronchioles has the same tensile stiffness as that in the trachea, the forces required to fold the membrane are significant at small transbronchial pressure differences and increase in the presence of membrane thickening such as that seen in asthma.

Functional significance of increased airway smooth muscle in asthma and COPD.

Using a computational model, we investigated the effect of the morphologically determined increased airway smooth muscle mass, adventitial mass, and submucosal mass observed in patients with asthma and chronic obstructive pulmonary disease (COPD) on the increase in airway resistance in response to a bronchoconstricting stimulus. The computational model of Wiggs et al. (J. Appl. Physiol. 69: 849-860, 1990) was modified in such a way that smooth muscle shortening was limited by the maximal stress that the muscle could develop at the constricted length. Increased adventitial thickness was found to increase constriction by reducing parenchymal interdependence. Increased submucosal thickness led to greater luminal occlusion for any degree of smooth muscle shortening. Increased muscle thickness allowed greater smooth muscle shortening against the elastic loads provided by parenchymal interdependence and airway wall stiffness. We found that for constant airway mechanics, as reflected by the passive area-pressure curves of the airways, the increased muscle mass is likely to be the most important abnormality responsible for the increased resistance observed in response to bronchoconstricting stimuli in asthma and COPD. For a given maximal muscle stress, greater muscle thickness allows the development of greater tension and thus more constriction of the lumen.

Role of bronchial basement membrane in airway collapse.

Bronchial basement membrane is an elastic structure that has the potential to be load bearing and thus to contribute to the mechanical stiffness of the bronchus. To investigate this possible role, the membrane was modeled as a thin-walled linearly elastic tube surrounded by a uniform liquid on the outside and by air on the inside. When the external pressure on such a tube exceeds the internal pressure by a critical amount, the tube buckles reversibly into two or more folds. The critical buckling pressure varies as the square of the number of folds. The analysis was used to investigate the collapse behavior of the model tube into patterns ranging from 2 to 24 folds. This showed that the resistance to collapse increases rapidly as the number of folds increases. Data in the literature lead to the conclusion that the pressures involved in collapsing the tubes are probably in the physiological range. It is suggested, on the basis of the model results reported here, that bronchial hyperresponsiveness could be related to the number of folds into which the basement membrane buckles when the bronchial muscle contracts. A reduced number of folds would yield an increased response.

A method for estimating the Young's modulus of complete tracheal cartilage rings.

Cartilage is primarily responsible for maintaining the stability of the large airways; yet very little is known about the mechanical properties of airway cartilage. This work establishes a technique whereby average values for the equilibrium modulus of excised tracheal cartilage rings can be obtained. An apparatus was designed to apply preset deformations to a tracheal segment and to monitor the deforming force. Segments of four human tracheae obtained postmortem and containing three rings were mounted in the apparatus after being stripped of posterior membrane. The load-deformation behavior was analyzed with a model on the basis of thin curved beam theory. Agreement between predicted deformed shapes and those observed was good in three of the four cases and in the case of a short length of longitudinally split rubber tube. The technique is suitable for comparing mechanical properties of cartilage before and after an intervention.

Simulation of the effects of mechanical nonhomogeneities on expiratory flow from human lungs.

A computational model for expiration from lungs with mechanical nonhomogeneities was used to investigate the effect of such nonhomogeneities on the distribution of expiratory flow and the development of alveolar pressure differences between regions. The nonhomogeneities used were a modest constriction of the peripheral airways and a 50% difference in compliance between regions. The model contains only two mechanically different regions but allows these to be as grossly distributed as left lung-right lung or to be distributed as a set of identical pairs of parallel nonhomogeneous regions with flows from each merging in a specified bronchial generation. The site of flow merging had no effect on the flow-volume curve but had a significant effect on the development of alveolar pressure differences (delta PA). With the peripheral constriction, greater values of delta PA developed when flows were merged peripherally rather than centrally. The opposite was true in the case of a compliance nonhomogeneity. The delta PA values were smaller at submaximal flows. Plots of delta PA vs. lung volume were similar to those obtained experimentally. These results were interpreted in terms of the expression used for the fluid mechanics of the merging flows. delta PA was greater when the viscosity of the expired gas was increased or when its density was reduced. Partial forced expirations were shown to indicate the presence of mechanical nonhomogeneity.

A new computational model for expiratory flow from nonhomogeneous human lungs.

A model has been developed for expiration from human lungs in which the mechanical properties of the airways and parenchyma can be varied between regions. The model is based on an existing homogeneous model. The fluid mechanical problem of the merging of dissimilar flows from adjacent regions is underspecified by the conservation laws of mass and energy. An existing, empirically derived result, provides the required extra equation. Model simulation of a nonhomogeneously distributed mild constriction of the peripheral airways gives results for maximal flows and alveolar pressure differences which are in good agreement with recent experimental findings.

In vitro tracheal mechanics by nuclear magnetic resonance imaging.

Images of rabbit tracheal cross sections were obtained at a series of transmural pressures ranging from 22 to -95 cmH2O by use of a nuclear magnetic resonance imaging microscope. The excised, washed tracheas were immersed in a solution of phosphate-buffered saline made up in deuterium oxide (D2O, pH 7.3). The images are maps of proton density in the image slice (2.5 mm thick). All but one series of images showed a collapse process in which the trachealis muscle invaginated asymmetrically, i.e., the muscle appeared to favor one side of the cartilage ring system more than the other. The connecting tissue between the cartilage rings appeared to be more compliant than the rings themselves, thus suggesting that the tracheal lumen became corrugated at negative pressures. In the plane of a cartilage ring, the lumen appeared to remain patent at pressures as low as -95 cmH2O. However, between rings, where the tracheal wall was more compliant, the lumen appeared to be totally occluded at -53 cmH2O. Lumen areas in both the plane of the cartilage rings and in a plane between rings were measured from each series of printed images for six tracheas. These measurements, when normalized, averaged, and plotted against transmural pressure gave asymptotic logarithmic compliances (n1 in the model of Lambert et al., J. Appl. Physiol. 52: 44-56, 1982) of 1.2 +/- 0.4 and 20 +/- 7 for the interring and ring regions, respectively. These values are greater than the critical value of 0.5 (J. Appl. Physiol. 62: 2426-2435, 1987) and are thus consistent with wave speed flow limitation being possible anywhere in the trachea during forced expiration.

Bronchial mechanical properties and maximal expiratory flows.

Flow limitation in a collapsible elastic tube is dependent on the area (A) vs. pressure (P) relationship (the "tube law") for the tube. In this paper, a tube law in which A varies as (1-P)-n1 at negative pressures is assumed. It is shown that wave-speed limitation is possible at negative pressures only if n1 is greater than 0.5. Dissipative limitation is also investigated. Viscous limitation can occur if n1 is greater than 0.5, and turbulent limitation can occur if n1 is not less than 0.4. For values of n1 less than 0.4, flow cannot be limited at negative pressures. Model simulations are used to show that a combination of a value of n1 less than 0.3 together with an area minimum in the bronchial tree produce a minimum (a "hook") in the flow-volume curve. In the vicinity of such hooks, density dependence exceeds the usually accepted theoretical maximum value. Simulations also show that, when n1 is sufficiently small, apparently supramaximal flows appear to be possible.