Parenchymal tethering, airway wall stiffness, and the dynamics of bronchoconstriction.

We do not yet have a good quantitative understanding of how the force-velocity properties of airway smooth muscle interact with the opposing loads of parenchymal tethering and airway wall stiffness to produce the dynamics of bronchoconstriction. We therefore developed a two-dimensional computational model of a dynamically narrowing airway embedded in uniformly elastic lung parenchyma and compared the predictions of the model to published measurements of airway resistance made in rats and rabbits during the development of bronchoconstriction following a bolus injection of methacholine. The model accurately reproduced the experimental time-courses of airway resistance as a function of both lung inflation pressure and tidal volume. The model also showed that the stiffness of the airway wall is similar in rats and rabbits, and significantly greater than that of the lung parenchyma. Our results indicate that the main features of the dynamical nature of bronchoconstriction in vivo can be understood in terms of the classic Hill force-velocity relationship operating against elastic loads provided by the surrounding lung parenchyma and an airway wall that is stiffer than the parenchyma.     

Contribution of tree structures in the lung to lung elastic recoil

The direct contribution of forces in tree structures in the lung to lung recoil pressure and changes in recoil pressure induced by alterations of the forces are analyzed. The analysis distinguishes the contributions of axial and circumferential tensions in the trees and indicates that only axial tensions directly contribute to static recoil. This contribution is derived from analysis of the axial forces transmitted across a random plane transecting the lung. The change in recoil pressure induced by changes in axial tension is similarly derived. Alterations of circumferential tensions in the trees indirectly change recoil by causing nonuniform deformations of the surrounding lung parenchyma, and a continuum elasticity solution for the stress induced by the deformations is derived. Sample calculations are presented for the airway tree based on available data on airway morphometric and mechanical properties. The increase in recoil pressure accompanying increases in axial and circumferential tensions with contraction of airway smooth muscle is also analyzed. The calculations indicate that axial stresses in the airway tree out to bronchioles directly contribute only a small fraction of the static recoil pressure. However, it is found that contraction of smooth muscle in these airways can increase recoil pressure appreciably (10-20%), mainly by the deformation of the parenchyma with increases in circumferential tension in smaller airways. The results indicate that the geometric and mechanical properties of the airway tree are such that only peripheral elements of the tree can substantially affect the elastic properties of the lung. The possible contributions of vascular trees for which data on mechanical and morphometric properties are more limited are also discussed.     

Effect of Parenchymal Stiffness on Canine Airway Size with Lung Inflation

Although airway patency is partially maintained by parenchymal tethering, this structural support is often ignored in many discussions of asthma. However, agonists that induce smooth muscle contraction also stiffen the parenchyma, so such parenchymal stiffening may serve as a defense mechanism to prevent airway narrowing or closure. To quantify this effect, specifically how changes in parenchymal stiffness alter airway size at different levels of lung inflation, in the present study, we devised a method to separate the effect of parenchymal stiffening from that of direct airway narrowing. Six anesthetized dogs were studied under four conditions: baseline, after whole lung aerosol histamine challenge, after local airway histamine challenge, and after complete relaxation of the airways. In each of these conditions, we used High resolution Computed Tomography to measure airway size and lung volume at five different airway pressures (0, 12, 25, 32, and 45 cm H₂O). Parenchymal stiffening had a protective effect on airway narrowing, a fact that may be important in the airway response to deep inspiration in asthma. When the parenchyma was stiffened by whole lung aerosol histamine challenge, at every lung volume above FRC, the airways were larger than when they were directly challenged with histamine to the same initial constriction. These results show for the first time that a stiff parenchyma per se minimizes the airway narrowing that occurs with histamine challenge at any lung volume. Thus in clinical asthma, it is not simply increased airway smooth muscle contraction, but perhaps a lack of homogeneous parenchymal stiffening that contributes to the symptomatic airway hyperresponsiveness.     

Effect of respiratory muscle tension on lung volume.

The chest wall is modeled as a linear system for which the displacements of points on the chest wall are proportional to the forces that act on the chest wall, namely, airway opening pressure and active tension in the respiratory muscles. A standard theorem of mechanics, the Maxwell reciprocity theorem, is invoked to show that the effect of active muscle tension on lung volume, or airway pressure if the airway is closed, is proportional to the change of muscle length in the relaxation maneuver. This relation was tested experimentally. The shortening of the cranial-caudal distance between a rib pair and the sternum was measured during a relaxation maneuver. These data were used to predict the respiratory effect of forces applied to the ribs and sternum. To test this prediction, a cranial force was applied to the rib pair and a caudal force was applied to the sternum, simulating the forces applied by active tension in the parasternal intercostal muscles. The change in airway pressure, with lung volume held constant, was measured. The measured change in airway pressure agreed well with the prediction. In some dogs, nonlinear deviations from the linear prediction occurred at higher loads. The model and the theorem offer the promise that existing data on the configuration of the chest wall during the relaxation maneuver can be used to compute the mechanical advantage of the respiratory muscles.     

The influence of ligament modelling strategies on the predictive capability of finite element models of the human knee joint

In finite element (FE) models knee ligaments can represented either by a group of one-dimensional springs, or by three-dimensional continuum elements based on segmentations. Continuum models closer approximate the anatomy, and facilitate ligament wrapping, while spring models are computationally less expensive. The mechanical properties of ligaments can be based on literature, or adjusted specifically for the subject. In the current study we investigated the effect of ligament modelling strategy on the predictive capability of FE models of the human knee joint. The effect of literature-based versus specimen-specific optimized material parameters was evaluated. Experiments were performed on three human cadaver knees, which were modelled in FE models with ligaments represented either using springs, or using continuum representations. In spring representation collateral ligaments were each modelled with three and cruciate ligaments with two single-element bundles. Stiffness parameters and pre-strains were optimized based on laxity tests for both approaches. Validation experiments were conducted to evaluate the outcomes of the FE models.Models (both spring and continuum) with subject-specific properties improved the predicted kinematics and contact outcome parameters. Models incorporating literature-based parameters, and particularly the spring models (with the representations implemented in this study), led to relatively high errors in kinematics and contact pressures. Using a continuum modelling approach resulted in more accurate contact outcome variables than the spring representation with two (cruciate ligaments) and three (collateral ligaments) single-element-bundle representations. However, when the prediction of joint kinematics is of main interest, spring ligament models provide a faster option with acceptable outcome.     

An elastic network model based on the structure of the red blood cell membrane skeleton.

A finite element network model has been developed to predict the macroscopic elastic shear modulus and the area expansion modulus of the red blood cell (RBC) membrane skeleton on the basis of its microstructure. The topological organization of connections between spectrin molecules is represented by the edges of a random Delaunay triangulation, and the elasticity of an individual spectrin molecule is represented by the spring constant, K, for a linear spring element. The model network is subjected to deformations by prescribing nodal displacements on the boundary. The positions of internal nodes are computed by the finite element program. The average response of the network is used to compute the shear modulus (mu) and area expansion modulus (kappa) for the corresponding effective continuum. For networks with a moderate degree of randomness, this model predicts mu/K = 0.45 and kappa/K = 0.90 in small deformations. These results are consistent with previous computational models and experimental estimates of the ratio mu/kappa. This model also predicts that the elastic moduli vary by 20% or more in networks with varying degrees of randomness. In large deformations, mu increases as a cubic function of the extension ratio lambda 1, with mu/K = 0.62 when lambda 1 = 1.5.     

Airway closure with high PEEP in vivo.

When airway smooth muscle is contracted in vitro, the airway lumen continues to narrow with increasing concentrations of agonist until complete airway closure occurs. Although there remains some controversy regarding whether airways can close in vivo, recent work has clearly demonstrated that, if the airway is sufficiently stimulated with contractile agonists, complete closure of even large cartilaginous conducting airways can readily occur with the lung at functional residual capacity (Brown RH and Mitzner W. J Appl Physiol 85: 2012-2017, 1998). This result suggests that the tethering of airways in situ by parenchymal attachments is small at functional residual capacity. However, at lung volumes above functional residual capacity, the outward tethering of airways should increase, because both the parenchymal shear modulus and tethering forces increase in proportion to the transpulmonary pressure. In the present study, we tested whether we could prevent airway closure in vivo by increasing lung volume with positive end-expiratory pressure (PEEP). Airway smooth muscle was stimulated with increasing methacholine doses delivered directly to airway smooth muscle at three levels of PEEP (0, 6, and 10 cmH(2)O). Our results show that increased lung volume shifted the airway methacholine dose-response curve to the right, but, in many airways in most animals, airway closure still occurred even at the highest levels of PEEP.     

Comparison of elastic properties and contractile responses of isolated airway segments from mature and immature rabbits.

Immature rabbits have greater maximal airway narrowing with bronchoconstriction in vivo compared with mature animals. As isolated immature lungs have a lower shear modulus, it is unclear whether the greater airway narrowing in the immature lung is secondary to less tethering between the airways and the lung parenchyma or to differences in the mechanical properties of the mature and immature airways. In the present study, we compared the mechanical properties of fluid-filled, isolated, intraparenchymal airway segments of the same generation from mature and immature rabbits. Stimulation with ACh resulted in greater airway narrowing in immature than mature bronchi. The immature bronchi were more compliant, had a lower resting airway volume, and were more collapsible compared with the mature bronchi. When the airways were contracted with ACh under isovolume conditions, the immature bronchi generated greater active pressure, and they were more sensitive to ACh than were mature bronchi. Our results suggest that maturational differences in the structure and function of the airways in the absence of the lung parenchyma can account for the greater maximal narrowing of immature than mature airways in vivo.     

Hopping frequency in humans: a test of how springs set stride frequency in bouncing gaits.

The storage and recovery of elastic energy in muscle-tendon springs is important in running, hopping, trotting, and galloping. We hypothesized that animals select the stride frequency at which they behave most like simple spring-mass systems. If higher or lower frequencies are used, they will not behave like simple spring-mass systems, and the storage and recovery of elastic energy will be reduced. We tested the hypothesis by having humans hop forward on a treadmill over a range of speeds and hop in place over a range of frequencies. The body was modeled as a simple spring-mass system, and the properties of the spring were measured by use of a force platform. Our subjects used nearly the same frequency (the "preferred frequency," 2.2 hops/s) when they hopped forward on a treadmill and when they hopped in place. At this frequency, the body behaved like a simple spring-mass system. Contrary to our predictions, it also behaved like a simple spring-mass system when the subjects hopped at higher frequencies, up to the maximum they could achieve. However, at the higher frequencies, the time available to apply force to the ground (the ground contact time) was shorter, perhaps resulting in a higher cost of generating muscular force. At frequencies below the preferred frequency, as predicted by the hypothesis, the body did not behave in a springlike manner, and it appeared likely that the storage and recovery of elastic energy was reduced. The combination of springlike behavior and a long ground contact time at the preferred frequency should minimize the cost of generating muscular force.     

Longitudinal elastic wave propagation in pulmonary parenchyma

Consideration of the lung as an elastic continuum led us to investigate the possible propagation of elastic waves. Here the relevant stiffness and density are given by the Lamé constants and density of the parenchyma. To test this hypothesis, we measured propagation velocities (c) in dog lobes by recording transit times of a velocity impulse on one side of the lobe and the subsequent arrival on the other side. We compared our measured values of c with elastic longitudinal wave velocities (c long) predicted by values of elastic moduli given by Lai-Fook et al. (J. Appl. Physiol. 40: 508-513, 1976) as a function of translobar pressure (PL) and our measured densities. Good agreement was found between c and c long. Typical values of c ranged from 250-1,500 cm/s as PL ranged from 2-20 cmH2O. No systematic difference in the c-c long relation was found between inflation and deflation, suggesting that the elastic moduli of lungs are essentially a function of pressure. No significant effect was observed by changing the physical properties of the gas within the lobe [air vs. He vs. sulfur hexafluoride (SF6)], suggesting that indeed we were observing waves associated with the coupling of parenchymal density to parenchymal stiffness.     

Coupling between rib cage and abdominal compartments of the relaxed chest wall

The volume displacements of the rib cage and abdomen of relaxed seated subjects were measured as functions of pleural pressure with the chest wall expanded by airway pressure and with the chest wall distorted by an external force applied to the rib cage. From the measured displacements for the two independent loads, the three compliances that describe the mechanical properties of the relaxed chest wall modeled as a linear elastic system with two degrees of freedom were obtained. The cross compliance that describes the coupling between the rib cage and abdomen was found to be small and positive, 0.01-0.02 1/cmH2O. The displacement of the rib cage by the external force was consistent with the displacement predicted by use of standard methods for calculating the mechanical advantage of the force.     

A cellular model of lung elasticity

The mechanics of the lung parenchyma is studied using models comprised of line members interconnected to form 3-D cellular structures. The mechanical properties are represented as elastic constants of a continuum. These are determined by perturbing each individual cell from a reference state by an increment in stress which is superimposed upon the uniform stretching forces initially present in the members due to the transpulmonary pressure. A force balance on the distorted structure, together with a force-deformation law for the members, leads to a calculation of the strain increments of the members. Predictions based on the analysis of the 3-D isotropic dodecahedron are in good agreement with experimental values for the Young's, shear, and bulk moduli reported in the literature. The model provides an explanation for the dependence of the elastic moduli on transpulmonary pressure, the geometrical details of the structure, and the stress-strain law of the tissue.     

Stereocilia membrane deformation: implications for the gating spring and mechanotransduction channel

In hair cells, although mechanotransduction channels have been localized to tips of shorter stereocilia of the mechanically sensitive hair bundle, little is known about how force is transmitted to the channel. Here, we use a biophysical model of the membrane-channel complex to analyze the nature of the gating spring compliance and channel arrangement. We use a triangulated surface model and Monte Carlo simulation to compute the deformation of the membrane under the action of tip link force. We show that depending on the gating spring stiffness, the compliant component of the gating spring arises from either the membrane alone or a combination of the membrane and a tether that connects the channel to the actin cytoskeleton. If a bundle is characterized by relatively soft gating springs, such as those of the bullfrog sacculus, the need for membrane reinforcement by channel tethering then depends on membrane parameters. With stiffer gating springs, such as those from rat outer hair cells, the channel must be tethered for all biophysically realistic parameters of the membrane. We compute the membrane forces (resultants), which depend on membrane tension, bending modulus, and curvature, and show that they can determine the fate of the channel.     

New perspectives on the mechanical basis for airway hyperreactivity and airway hypersensitivity in asthma.

We revisit the airway wall model of Lambert et. al. (Lambert RK, Wiggs BR, Kuwano K, Hogg JC, and Pare PD. J Appl Physiol 74: 2771-2781, 1993). We examine in detail the notion of a general airway bistability such that the airway lumen can suddenly decrease from a relatively open to a relatively closed condition without needing additional increase in active airway smooth muscle (ASM) tension during the stimulation. The onset of this bistability is an emergent consequence of the balance of forces associated with airway wall properties, parenchymal tissue properties, maximum lung elastic recoil, and the maximum stress that the ASM can generate. In healthy lungs, we find that all these properties reside in conditions that largely prevent the emergence of the bistability even during maximum ASM stimulation. In asthmatic airways, however, the airway wall and ASM remodeling conditions can tip the balance so as to promote the onset of the bistability at a lower dose of ASM stimulation (enhanced sensitivity) and then work to amplify the maximum constriction reached by each airway (enhanced reactivity). Hence, a larger fraction of asthmatic airways can display overall airway hyperreactivity. Simulations studies examine the role of increasing ASM maximum tension, airway wall stiffening, reduced lung volume, and decreased parenchymal tethering. Results predict that the single most important factor causing this airway hyperreactivity is amplified maximum ASM tension and not a thickening of the airway wall per se.     

Gating mechanisms of mechanosensitive channels of large conductance, I: a continuum mechanics-based hierarchical framework

A hierarchical simulation framework that integrates information from molecular dynamics (MD) simulations into a continuum model is established to study the mechanical response of mechanosensitive channel of large-conductance (MscL) using the finite element method (FEM). The proposed MD-decorated FEM (MDeFEM) approach is used to explore the detailed gating mechanisms of the MscL in Escherichia coli embedded in a palmitoyloleoylphosphatidylethanolamine lipid bilayer. In Part I of this study, the framework of MDeFEM is established. The transmembrane and cytoplasmic helices are taken to be elastic rods, the loops are modeled as springs, and the lipid bilayer is approximated by a three-layer sheet. The mechanical properties of the continuum components, as well as their interactions, are derived from molecular simulations based on atomic force fields. In addition, analytical closed-form continuum model and elastic network model are established to complement the MDeFEM approach and to capture the most essential features of gating. In Part II of this study, the detailed gating mechanisms of E. coli-MscL under various types of loading are presented and compared with experiments, structural model, and all-atom simulations, as well as the analytical models established in Part I. It is envisioned that such a hierarchical multiscale framework will find great value in the study of a variety of biological processes involving complex mechanical deformations such as muscle contraction and mechanotransduction.     

Bronchodilatory effect of deep inspiration on the dynamics of bronchoconstriction in mice.

We recently developed a computational model of an airway embedded in elastic parenchyma (Bates JH, Lauzon AM. J Appl Physiol 102: 1912-1920, 2007) that accurately mimics the time dependence of airway resistance on tidal volume and positive end-expiratory pressure (PEEP) following methacholine injection in normal animals. In the present study, we compared the model predictions of bronchodilation induced by a deep inflation (DI) of the lung following administration of the bronchial agonist methacholine to corresponding experimental measurements made in mice. We found that a DI in mice caused an immediate reduction in airway resistance when it was administered soon after intravenous injection of methacholine, while the airway smooth muscle was in the process of contracting. However, the magnitude of the reduction in resistance was greater and its subsequent rate of increase less than that predicted by the model. We found that this effect was most pronounced when the DI was given within approximately 3 s following methacholine injection, again in contrast to the predictions of the model. The reduction of airway resistance was virtually independent of the rate of lung inflation during the DI, however, which agrees with model predictions. We conclude that while the model accounts for a substantial fraction of the post-DI reduction in airway resistance seen experimentally, there remain important differences between prediction and experiment that suggest that the effects of a DI are not simply due to eccentric contraction of the airway smooth muscle.     

A validated three-dimensional computational model of a human knee joint

This paper presents a three-dimensional finite element tibio-femoral joint model of a human knee that was validated using experimental data. The geometry of the joint model was obtained from magnetic resonance (MR) images of a cadaveric knee specimen. The same specimen was biomechanically tested using a robotic/universal force-moment sensor (UFS) system and knee kinematic data under anterior-posterior tibial loads (up to 100 N) were obtained. In the finite element model (FEM), cartilage was modeled as an elastic material, ligaments were represented as nonlinear elastic springs, and menisci were simulated by equivalent-resistance springs. Reference lengths (zero-load lengths) of the ligaments and stiffness of the meniscus springs were estimated using an optimization procedure that involved the minimization of the differences between the kinematics predicted by the model and those obtained experimentally. The joint kinematics and in-situ forces in the ligaments in response to axial tibial moments of up to 10 Nm were calculated using the model and were compared with published experimental data on knee specimens. It was also demonstrated that the equivalent-resistance springs representing the menisci are important for accurate calculation of knee kinematics. Thus, the methodology developed in this study can be a valuable tool for further analysis of knee joint function and could serve as a step toward the development of more advanced computational knee models.     

A computational model of ameboid deformation and locomotion

Traditional continuum models of ameboid deformation and locomotion are limited by the computational difficulties intrinsic in free boundary conditions. A new model using the immersed boundary method overcomes these difficulties by representing the cell as a force field immersed in fluid domain. The forces can be derived from a direct mechanical interpretation of such cell components as the cell membrane, the actin cortex, and the transmembrane adhesions between the cytoskeleton and the substratum. The numerical cytoskeleton, modeled as a dynamic network of immersed springs, is able to qualitatively model the passive mechanical behavior of a shear-thinning viscoelastic fluid (Bottino 1997). The same network is used to generate active protrusive and contractile forces. When coordinated with the attachment-detachment cycle of the cell's adhesions to the substratum, these forces produce directed locomotion of the model ameba. With this model it is possible to study the effects of altering the numerical parameters upon the motility of the model cell in a manner suggestive of genetic deletion experiments. In the context of this ameboid cell model and its numerical implementation, simulations involving multicellular interaction, detailed internal signaling, and complex substrate geometries are tractable. Received: 5 January 1998 / Revised version: 23 March 1998 / Accepted: 26 March 1998     

Airway remodeling in asthma amplifies heterogeneities in smooth muscle shortening causing hyperresponsiveness.

Although airway remodeling and inflammation in asthma can amplify the constriction response of a single airway, their influence on the structural changes in the whole airway network is unknown. We present a morphometric model of the human lung that incorporates cross-sectional wall areas corresponding to the adventitia, airway smooth muscle (ASM), and mucosa for healthy and mildly and severely asthmatic airways and the influence of parenchymal tethering. A heterogeneous ASM percent shortening stimulus is imposed, causing distinct constriction patterns for healthy and asthmatic airways. We calculate lung resistance and elastance from 0.1 to 5 Hz. We show that, for a given ASM stimulus, the distribution of wall area in asthmatic subjects will amplify not only the mean but the heterogeneity of constriction in the lung periphery. Moreover, heterogeneous ASM shortening that would produce only mild changes in the healthy lung can cause hyperresponsive changes in lung resistance and elastance at typical breathing rates in the asthmatic lung, even with relatively small increases in airway resistance. This condition arises when airway closures occur randomly in the lung periphery. We suggest that heterogeneity is a crucial determinant of hyperresponsiveness in asthma and that acute asthma is more a consequence of extensive airway wall inflammation and remodeling, predisposing the lung to produce an acute pattern of heterogeneous constriction.