A mathematical model of airway and pulmonary arteriole smooth muscle

Deciphering the underlying mechanisms of the pharyngeal pumping motions in Caenorhabditis elegans

The pharynx of the nematode Caenorhabditis elegans is a neuromuscular organ that exhibits typical pumping motions, which result in the intake of food particles from the environment. In-depth inspection reveals slightly different dynamics at the various pharyngeal areas, rather than synchronous pumping motions of the whole organ, which are important for its effective functioning. While the different pumping dynamics are well characterized, the underlying mechanisms that generate them are not known. In this study, the C. elegans pharynx was modeled in a bottom-up fashion, including all of the underlying biological processes that lead to, and including, its end function, food intake. The mathematical modeling of all processes allowed performing comprehensive, quantitative analyses of the system as a whole. Our analyses provided detailed explanations for the various pumping dynamics generated at the different pharyngeal areas; a fine-resolution description of muscle dynamics, both between and within different pharyngeal areas; a quantitative assessment of the values of many parameters of the system that are unavailable in the literature; and support for a functional role of the marginal cells, which are currently assumed to mainly have a structural role in the pharynx. In addition, our model predicted that in tiny organisms such as C. elegans, the generation of long-lasting action potentials must involve ions other than calcium. Our study exemplifies the power of mathematical models, which allow a more accurate, higher-resolution inspection of the studied system, and an easier and faster execution of in silico experiments than feasible in the lab.

Intrapulmonary arterial contraction assay reveals region-specific deregulation of vasoreactivity to lung injuries

Intrapulmonary arteries located in the proximal lung differ from those in the distal lung in size, cellular composition, and the surrounding microenvironment. However, whether these structural variations lead to region-specific regulation of vasoreactivity in homeostasis and following injury is unknown. Herein, we employ a two-step method of precision-cut lung slice (PCLS) preparation, which maintains almost intact intrapulmonary arteries, to assess contractile and relaxation responses of proximal preacinar arteries (PaAs) and distal intraacinar arteries (IaAs) in mice. We found that PaAs exhibited robust vasoconstriction in response to contractile agonists and significant nitric oxide (NO)-induced vasodilation. In comparison, IaAs were less contractile and displayed a greater relaxation response to NO. Furthermore, in a mouse model of pulmonary arterial hypertension (PAH) induced by chronic exposure to ovalbumin (OVA) allergen and hypoxia (OVA-HX), IaAs demonstrated a reduced vasocontraction despite vascular wall thickening with the emergence of new αSMA+ cells coexpressing markers of pericytes. In contrast, PaAs became hypercontractile and less responsive to NO. The reduction in relaxation of PaAs was associated with decreased expression of protein kinase G, a key component of the NO pathway, following chronic OVA-HX exposure. Taken together, the PCLS prepared using the modified preparation method enables functional evaluation of pulmonary arteries in different anatomical locations and reveals region-specific mechanisms underlying the pathophysiology of PAH in a mouse model.NEW & NOTEWORTHY Utilizing mouse precision-cut lung slices with preserved intrapulmonary vessels, we demonstrated a location-dependent structural and contractile regulation of pulmonary arteries in health and on noxious stimulations. For instance, chronic ovalbumin and hypoxic exposure increased pulmonary arterial pressure (PAH) by remodeling intraacinar arterioles to reduce vascular wall compliance while enhancing vasoconstriction in proximal preacinar arteries. These findings suggest region-specific mechanisms and therapeutic targets for pulmonary vascular diseases such as PAH.

A complete biomechanical model of Hydra contractile behaviors, from neural drive to muscle to movement

How does neural activity drive muscles to produce behavior? The recent development of genetic lines in Hydra that allow complete calcium imaging of both neuronal and muscle activity, as well as systematic machine learning quantification of behaviors, makes this small cnidarian an ideal model system to understand and model the complete transformation from neural firing to body movements. To achieve this, we have built a neuromechanical model of Hydra's fluid-filled hydrostatic skeleton, showing how drive by neuronal activity activates distinct patterns of muscle activity and body column biomechanics. Our model is based on experimental measurements of neuronal and muscle activity and assumes gap junctional coupling among muscle cells and calcium-dependent force generation by muscles. With these assumptions, we can robustly reproduce a basic set of Hydra's behaviors. We can further explain puzzling experimental observations, including the dual timescale kinetics observed in muscle activation and the engagement of ectodermal and endodermal muscles in different behaviors. This work delineates the spatiotemporal control space of Hydra movement and can serve as a template for future efforts to systematically decipher the transformations in the neural basis of behavior.

Modeling and experimental approaches for elucidating multi-scale uterine smooth muscle electro- and mechano-physiology: A review

The uterus provides protection and nourishment (via its blood supply) to a developing fetus, and contracts to deliver the baby at an appropriate time, thereby having a critical contribution to the life of every human. However, despite this vital role, it is an under-investigated organ, and gaps remain in our understanding of how contractions are initiated or coordinated. The uterus is a smooth muscle organ that undergoes variations in its contractile function in response to hormonal fluctuations, the extreme instance of this being during pregnancy and labor. Researchers typically use various approaches to studying this organ, such as experiments on uterine muscle cells, tissue samples, or the intact organ, or the employment of mathematical models to simulate the electrical, mechanical and ionic activity. The complexity exhibited in the coordinated contractions of the uterus remains a challenge to understand, requiring coordinated solutions from different research fields. This review investigates differences in the underlying physiology between human and common animal models utilized in experiments, and the experimental interventions and computational models used to assess uterine function. We look to a future of hybrid experimental interventions and modeling techniques that could be employed to improve the understanding of the mechanisms enabling the healthy function of the uterus.

A multiscale sliding filament model of lymphatic muscle pumping

The lymphatics maintain fluid balance by returning interstitial fluid to veins via contraction/compression of vessel segments with check valves. Disruption of lymphatic pumping can result in a condition called lymphedema with interstitial fluid accumulation. Lymphedema treatments are often ineffective, which is partially attributable to insufficient understanding of specialized lymphatic muscle lining the vessels. This muscle exhibits cardiac-like phasic contractions and smooth muscle-like tonic contractions to generate and regulate flow. To understand the relationship between this sub-cellular contractile machinery and organ-level pumping, we have developed a multiscale computational model of phasic and tonic contractions in lymphatic muscle and coupled it to a lymphangion pumping model. Our model uses the sliding filament model (Huxley in Prog Biophys Biophys Chem 7:255-318, 1957) and its adaptation for smooth muscle (Mijailovich in Biophys J 79(5):2667-2681, 2000). Multiple structural arrangements of contractile components and viscoelastic elements were trialed but only one provided physiologic results. We then coupled this model with our previous lumped parameter model of the lymphangion to relate results to experiments. We show that the model produces similar pressure, diameter, and flow tracings to experiments on rat mesenteric lymphatics. This model provides the first estimates of lymphatic muscle contraction energetics and the ability to assess the potential effects of sub-cellular level phenomena such as calcium oscillations on lymphangion outflow. The maximum efficiency value predicted (40%) is at the upper end of estimates for other muscle types. Spontaneous calcium oscillations during diastole were found to increase outflow up to approximately 50% in the range of frequencies and amplitudes tested.

Reduced biomechanical models for precision-cut lung-slice stretching experiments

Precision-cut lung-slices (PCLS), in which viable airways embedded within lung parenchyma are stretched or induced to contract, are a widely used ex vivo assay to investigate bronchoconstriction and, more recently, mechanical activation of pro-remodelling cytokines in asthmatic airways. We develop a nonlinear fibre-reinforced biomechanical model accounting for smooth muscle contraction and extracellular matrix strain-stiffening. Through numerical simulation, we describe the stresses and contractile responses of an airway within a PCLS of finite thickness, exposing the importance of smooth muscle contraction on the local stress state within the airway. We then consider two simplifying limits of the model (a membrane representation and an asymptotic reduction in the thin-PCLS-limit), that permit analytical progress. Comparison against numerical solution of the full problem shows that the asymptotic reduction successfully captures the key elements of the full model behaviour. The more tractable reduced model that we develop is suitable to be employed in investigations to elucidate the time-dependent feedback mechanisms linking airway mechanics and cytokine activation in asthma.

Research progress in the mechanism of calcium ion on contraction and relaxation of airway smooth muscle cells

lntracellular calcium ion is the key secondary messenger system of the cellular processes in airway smooth muscle cells(ASMc). The treatment and regulation of Ca²⁺ in airway smooth muscle (ASM) is, in part, to associated with many airway diseases such as asthma, COPD and pulmonary fibrosis. The mechanism of contraction and relaxation of ASM is a concerned aspect in airway diseases. This review emphasizes established and recent discoveries whice show the research progress of Ca²⁺ on cell contraction and relaxation in ASM in recent years, to provide theoretical support and new targets for clinical prevention and treatment of perioperative bronchospasm and variousrespiratory related diseases.

Model discrimination for Ca2+ -dependent regulation of myosin light chain kinase in smooth muscle contraction

Excitation-contraction coupling in smooth muscle is mediated by the Ca²⁺ - and calmodulin-dependent regulation of myosin light chain kinase. The precise mechanism of this regulation remains controversial, and several mathematical models have been proposed for the interaction of the three species. These models have previously been analyzed at steady state primarily by numerical simulation of differential equations, for which parameter values must be estimated from data. Here, we use the linear framework for timescale separation to demonstrate that models of this general kind can be solved analytically for an equilibrium steady state, without having to determine parameter values. This analysis leads to parameter-independent methods for discriminating between the models, for which we propose experiments that could be performed with existing methods.

A theoretical model of inflammation- and mechanotransduction-driven asthmatic airway remodelling

Inflammation, airway hyper-responsiveness and airway remodelling are well-established hallmarks of asthma, but their inter-relationships remain elusive. In order to obtain a better understanding of their inter-dependence, we develop a mechanochemical morphoelastic model of the airway wall accounting for local volume changes in airway smooth muscle (ASM) and extracellular matrix in response to transient inflammatory or contractile agonist challenges. We use constrained mixture theory, together with a multiplicative decomposition of growth from the elastic deformation, to model the airway wall as a nonlinear fibre-reinforced elastic cylinder. Local contractile agonist drives ASM cell contraction, generating mechanical stresses in the tissue that drive further release of mitogenic mediators and contractile agonists via underlying mechanotransductive signalling pathways. Our model predictions are consistent with previously described inflammation-induced remodelling within an axisymmetric airway geometry. Additionally, our simulations reveal novel mechanotransductive feedback by which hyper-responsive airways exhibit increased remodelling, for example, via stress-induced release of pro-mitogenic and pro-contractile cytokines. Simulation results also reveal emergence of a persistent contractile tone observed in asthmatics, via either a pathological mechanotransductive feedback loop, a failure to clear agonists from the tissue, or a combination of both. Furthermore, we identify various parameter combinations that may contribute to the existence of different asthma phenotypes, and we illustrate a combination of factors which may predispose severe asthmatics to fatal bronchospasms.

Airway and Parenchymal Strains during Bronchoconstriction in the Precision Cut Lung Slice

The precision-cut lung slice (PCLS) is a powerful tool for studying airway reactivity, but biomechanical measurements to date have largely focused on changes in airway caliber. Here we describe an image processing tool that reveals the associated spatio-temporal changes in airway and parenchymal strains. Displacements of sub-regions within the PCLS are tracked in phase-contrast movies acquired after addition of contractile and relaxing drugs. From displacement maps, strains are determined across the entire PCLS or along user-specified directions. In a representative mouse PCLS challenged with 10(-4)M methacholine, as lumen area decreased, compressive circumferential strains were highest in the 50 μm closest to the airway lumen while expansive radial strains were highest in the region 50-100 μm from the lumen. However, at any given distance from the airway the strain distribution varied substantially in the vicinity of neighboring small airways and blood vessels. Upon challenge with the relaxant agonist chloroquine, although most strains disappeared, residual positive strains remained a long time after addition of chloroquine, predominantly in the radial direction. Taken together, these findings establish strain mapping as a new tool to elucidate local dynamic mechanical events within the constricting airway and its supporting parenchyma.

Systems-level airway models of bronchoconstriction

Understanding lung and airway behavior presents a number of challenges, both experimental and theoretical, but the potential rewards are great in terms of both potential treatments for disease and interesting biophysical phenomena. This presents an opportunity for modeling to contribute to greater understanding, and here, we focus on modeling efforts that work toward understanding the behavior of airways in vivo, with an emphasis on asthma. We look particularly at those models that address not just isolated airways but many of the important ways in which airways are coupled both with each other and with other structures. This includes both interesting phenomena involving the airways and the layer of airway smooth muscle that surrounds them, and also the emergence of spatial ventilation patterns via dynamic airway interaction. WIREs Syst Biol Med 2016, 8:459-467. doi: 10.1002/wsbm.1349 For further resources related to this article, please visit the WIREs website.

Nonlinear compliance modulates dynamic bronchoconstriction in a multiscale airway model

The role of breathing and deep inspirations (DI) in modulating airway hyperresponsiveness remains poorly understood. In particular, DIs are potent bronchodilators of constricted airways in nonasthmatic subjects but not in asthmatic subjects. Additionally, length fluctuations (mimicking DIs) have been shown to reduce mean contractile force when applied to airway smooth muscle (ASM) cells and tissue strips. However, these observations are not recapitulated on application of transmural pressure (PTM) oscillations (that mimic tidal breathing and DIs) in isolated intact airways. To shed light on this paradox, we have developed a biomechanical model of the intact airway, accounting for strain-stiffening due to collagen recruitment (a large component of the extracellular matrix (ECM)), and dynamic actomyosin-driven force generation by ASM cells. In agreement with intact airway studies, our model shows that PTM fluctuations at particular mean transmural pressures can lead to only limited bronchodilation. However, our model predicts that moving the airway to a more compliant point on the static pressure-radius relationship (which may involve reducing mean PTM), before applying pressure fluctuations, can generate greater bronchodilation. This difference arises from competition between passive strain-stiffening of ECM and force generation by ASM yielding a highly nonlinear relationship between effective airway stiffness and PTM, which is modified by the presence of contractile agonist. Effectively, the airway at its most compliant may allow for greater strain to be transmitted to subcellular contractile machinery. The model predictions lead us to hypothesize that the maximum possible bronchodilation of an airway depends on its static compliance at the PTM about which the fluctuations are applied. We suggest the design of additional experimental protocols to test this hypothesis.

Static and dynamic stress heterogeneity in a multiscale model of the asthmatic airway wall

Airway hyperresponsiveness (AHR) is a key characteristic of asthma that remains poorly understood. Tidal breathing and deep inspiration ordinarily cause rapid relaxation of airway smooth muscle (ASM) (as demonstrated via application of length fluctuations to tissue strips) and are therefore implicated in modulation of AHR, but in some cases (such as application of transmural pressure oscillations to isolated intact airways) this mechanism fails. Here we use a multiscale biomechanical model for intact airways that incorporates strain stiffening due to collagen recruitment and dynamic force generation by ASM cells to show that the geometry of the airway, together with interplay between dynamic active and passive forces, gives rise to large stress and compliance heterogeneities across the airway wall that are absent in tissue strips. We show further that these stress heterogeneities result in auxotonic loading conditions that are currently not replicated in tissue-strip experiments; stresses in the strip are similar to hoop stress only at the outer airway wall and are under- or overestimates of stresses at the lumen. Taken together these results suggest that a previously underappreciated factor, stress heterogeneities within the airway wall and consequent ASM cellular response to this micromechanical environment, could contribute to AHR and should be explored further both theoretically and experimentally.

A biomechanical model for fluidization of cells under dynamic strain

Recent experiments have investigated the response of smooth muscle cells to transient stretch-compress (SC) and compress-stretch (CS) maneuvers. The results indicate that the transient SC maneuver causes a sudden fluidization of the cell while the CS maneuver does not. To understand this asymmetric behavior, we have built a biomechanical model to probe the response of stress fibers to the two maneuvers. The model couples the cross-bridge cycle of myosin motors with a viscoelastic Kelvin-Voigt element that represents the stress fiber. Simulation results point to the sensitivity of the myosin detachment rate to tension as the cause for the asymmetric response of the stress fiber to the CS and SC maneuvers. For the SC maneuver, the initial stretch increases the tension in the stress fiber and suppresses myosin detachment. The subsequent compression then causes a large proportion of the myosin population to disengage rapidly from actin filaments. This leads to the disassembly of the stress fibers and the observed fluidization. In contrast, the CS maneuver only produces a mild loss of myosin motors and no fluidization.

Airway smooth muscle in asthma: linking contraction and mechanotransduction to disease pathogenesis and remodelling

Asthma is an obstructive airway disease, with a heterogeneous and multifactorial pathogenesis. Although generally considered to be a disease principally driven by chronic inflammation, it is becoming increasingly recognised that the immune component of the pathology poorly correlates with the clinical symptoms of asthma, thus highlighting a potentially central role for non-immune cells. In this context airway smooth muscle (ASM) may be a key player, as it comprises a significant proportion of the airway wall and is the ultimate effector of acute airway narrowing. Historically, the contribution of ASM to asthma pathogenesis has been contentious, yet emerging evidence suggests that ASM contractile activation imparts chronic effects that extend well beyond the temporary effects of bronchoconstriction. In this review article we describe the effects that ASM contraction, in combination with cellular mechanotransduction and novel contraction-inflammation synergies, contribute to asthma pathogenesis. Specific emphasis will be placed on the effects that ASM contraction exerts on the mechanical properties of the airway wall, as well as novel mechanisms by which ASM contraction may contribute to more established features of asthma such as airway wall remodelling.

Ca2+ handling and sensitivity in airway smooth muscle: emerging concepts for mechanistic understanding and therapeutic targeting

Free calcium ions within the cytosol serve as a key secondary messenger system for a diverse range of cellular processes. Dysregulation of cytosolic Ca(2+) handling in airway smooth muscle (ASM) has been implicated in asthma, and it has been hypothesised that this leads, at least in part, to associated changes in both the architecture and function of the lung. Significant research is therefore directed towards furthering our understanding of the mechanisms which control ASM cytosolic calcium, in addition to those regulating the sensitivity of its downstream effector targets to calcium. Key aspects of the recent developments in this field were discussed at the 8th Young Investigators' Symposium on Smooth Muscle (2013, Groningen, The Netherlands), and are outlined in this review.

Phenotype, endotype and patient-specific computational modelling for optimal treatment design in asthma

Understanding and treatment of asthma is significantly complicated by the heterogeneous spectrum of phenotypes associated with the disease. Recent advances in phenotype classification promise more targeted therapies, but these categories are based on constellations of largely external measurements and are not necessarily indicative of underlying pathophysiology. We propose that computational modelling is a valuable tool that allows the disease spectrum to be decomposed not into phenotypes but rather into groups organized by underlying dysfunction, referred to by some authors as endotypes. By breaking down the asthmatic spectrum in this way, therapies can be targeted more directly to the underlying defects. This would be not only an important improvement in its own right, but also an important step toward the ultimate goal of patient-specific modelling.

A constitutive model for cytoskeletal contractility of smooth muscle cells

The constitutive model presented in this article aims to describe the main bio-chemo-mechanical features involved in the contractile response of smooth muscle cells, in which the biochemical response is modelled by extending the four-state Hai–Murphy model to isotonic contraction of the cells and the mechanical response is mainly modelled based on the phosphorylation-dependent hyperbolic relation between isotonic shortening strain rate and tension. The one-dimensional version of the model is used to simulate shortening-induced deactivation with good agreement with selected experimental measurements. The results suggest that the Hai–Murphy biochemical model neglects the strain rate effect on the kinetics of cross-bridge interactions with actin filaments in the isotonic contractions. The two-dimensional version and three-dimensional versions of the model are developed using the homogenization method under finite strain continuum mechanics framework. The two-dimensional constitutive model is used to simulate swine carotid media strips under electrical field stimulation, experimentally investigated by Singer and Murphy, and contraction of a hollow airway and a hollow arteriole buried in a soft matrix subjected to multiple calcium ion stimulations. It is found that the transverse deformation may have significant influence on the response of the swine carotid medium. In both cases, the orientation of the maximal value of attached myosin is aligned with the orientation of maximum principal stress.

Modeling the intracellular organization of calcium signaling

Calcium (Ca²⁺) is a key signaling ion that plays a fundamental role in many cellular processes in most types of tissues and organisms. The versatility of this signaling pathway is remarkable. Depending on the cell type and the stimulus, intracellular Ca²⁺ increases can last over different periods, as short spikes or more sustained signals. From a spatial point of view, they can be localized or invade the whole cell. Such a richness of behaviors is possible thanks to numerous exchange processes with the external medium or internal Ca²⁺ pools, mainly the endoplasmic or sarcoplasmic reticulum and mitochondria. These fluxes are also highly regulated. In order to get an accurate description of the spatiotemporal organization of Ca²⁺ signaling, it is useful to resort to modeling. Thus, each flux can be described by an appropriate kinetic expression. Ca²⁺ dynamics in a given cell type can then be simulated by a modular approach, consisting of the assembly of computational descriptions of the appropriate fluxes and regulations. Modeling can also be used to get insight into the mechanisms of decoding of the Ca²⁺ signals responsible for cellular responses. Cells can use frequency or amplitude coding, as well as take profit of Ca²⁺ oscillations to increase their sensitivity to small average Ca²⁺ increases.

Activation of store-operated calcium entry in airway smooth muscle cells: insight from a mathematical model

Intracellular dynamics of airway smooth muscle cells (ASMC) mediate ASMC contraction and proliferation, and thus play a key role in airway hyper-responsiveness (AHR) and remodelling in asthma. We evaluate the importance of store-operated entry (SOCE) in these dynamics by constructing a mathematical model of ASMC signaling based on experimental data from lung slices. The model confirms that SOCE is elicited upon sufficient depletion of the sarcoplasmic reticulum (SR), while receptor-operated entry (ROCE) is inhibited in such conditions. It also shows that SOCE can sustain agonist-induced oscillations in the absence of other influx. SOCE up-regulation may thus contribute to AHR by increasing the oscillation frequency that in turn regulates ASMC contraction. The model also provides an explanation for the failure of the SERCA pump blocker CPA to clamp the cytosolic of ASMC in lung slices, by showing that CPA is unable to maintain the SR empty of . This prediction is confirmed by experimental data from mouse lung slices, and strongly suggests that CPA only partially inhibits SERCA in ASMC.

Modelling airway smooth muscle passive length adaptation via thick filament length distributions

We present a new model of airway smooth muscle (ASM), which surrounds and constricts every airway in the lung and thus plays a central role in the airway constriction associated with asthma. This new model of ASM is based on an extension of sliding filament/crossbridge theory, which explicitly incorporates the length distribution of thick sliding filaments to account for a phenomenon known as dynamic passive length adaptation; the model exhibits good agreement with experimental data for ASM force-length behaviour across multiple scales. Principally these are (nonlinear) force-length loops at short timescales (seconds), parabolic force-length curves at medium timescales (minutes) and length adaptation at longer timescales. This represents a significant improvement on the widely-used crossbridge models which work so well in or near the isometric regime, and may have significant implications for studies which rely on crossbridge or other dynamic airway smooth muscle models, and thus both airway and lung dynamics.

The importance of synergy between deep inspirations and fluidization in reversing airway closure

Deep inspirations (DIs) and airway smooth muscle fluidization are two widely studied phenomena in asthma research, particularly for their ability (or inability) to counteract severe airway constriction. For example, DIs have been shown effectively to reverse airway constriction in normal subjects, but this is impaired in asthmatics. Fluidization is a connected phenomenon, wherein the ability of airway smooth muscle (ASM, which surrounds and constricts the airways) to exert force is decreased by applied strain. A maneuver which sufficiently strains the ASM, then, such as a DI, is thought to reduce the force generating capacity of the muscle via fluidization and hence reverse or prevent airway constriction. Understanding these two phenomena is considered key to understanding the pathophysiology of asthma and airway hyper-responsiveness, and while both have been extensively studied, the mechanism by which DIs fail in asthmatics remains elusive. Here we show for the first time the synergistic interaction between DIs and fluidization which allows the combination to provide near complete reversal of airway closure where neither is effective alone. This relies not just on the traditional model of airway bistability between open and closed states, but also the critical addition of previously-unknown oscillatory and chaotic dynamics. It also allows us to explore the types of subtle change which can cause this interaction to fail, and thus could provide the missing link to explain DI failure in asthmatics.

Quantifying parenchymal tethering in a finite element simulation of a human lung slice under bronchoconstriction

Airway hyper-responsiveness (AHR), a hallmark of asthma, is a highly complex phenomenon characterised by multiple processes manifesting over a large range of length and time scales. Multiscale computational models have been derived to embody the experimental understanding of AHR. While current models differ in their derivation, a common assumption is that the increase in parenchymal tethering pressure P(teth) during airway constriction can be described using the model proposed by Lai-Fook (1979), which is based on intact lung experimental data for elastic moduli over a range of inflation pressures. Here we reexamine this relationship for consistency with a nonlinear elastic material law that has been parameterised to the pressure-volume behaviour of the intact lung. We show that the nonlinear law and Lai-Fook's relationship are consistent for small constrictions, but diverge when the constriction becomes large.

A multi-scale approach to airway hyperresponsiveness: from molecule to organ

Airway hyperresponsiveness (AHR), a characteristic of asthma that involves an excessive reduction in airway caliber, is a complex mechanism reflecting multiple processes that manifest over a large range of length and time scales. At one extreme, molecular interactions determine the force generated by airway smooth muscle (ASM). At the other, the spatially distributed constriction of the branching airways leads to breathing difficulties. Similarly, asthma therapies act at the molecular scale while clinical outcomes are determined by lung function. These extremes are linked by events operating over intermediate scales of length and time. Thus, AHR is an emergent phenomenon that limits our understanding of asthma and confounds the interpretation of studies that address physiological mechanisms over a limited range of scales. A solution is a modular computational model that integrates experimental and mathematical data from multiple scales. This includes, at the molecular scale, kinetics, and force production of actin-myosin contractile proteins during cross-bridge and latch-state cycling; at the cellular scale, Ca²⁺ signaling mechanisms that regulate ASM force production; at the tissue scale, forces acting between contracting ASM and opposing viscoelastic tissue that determine airway narrowing; at the organ scale, the topographic distribution of ASM contraction dynamics that determine mechanical impedance of the lung. At each scale, models are constructed with iterations between theory and experimentation to identify the parameters that link adjacent scales. This modular model establishes algorithms for modeling over a wide range of scales and provides a framework for the inclusion of other responses such as inflammation or therapeutic regimes. The goal is to develop this lung model so that it can make predictions about bronchoconstriction and identify the pathophysiologic mechanisms having the greatest impact on AHR and its therapy.

A three-dimensional chemo-mechanical continuum model for smooth muscle contraction

Based on two fields, namely the placement and the calcium concentration, a chemo-mechanically coupled three-dimensional model, describing the contractile behaviour of smooth muscles, is presented by means of a strain energy function. The strain energy function (Schmitz and Böl, 2011) is additively decomposed into a passive part, relating to elastin and collagen, and an active calcium-driven part related to the chemical contraction of the smooth muscle cells. For the description of the calcium phase the four state cross-bridge model of Hai and Murphy (Hai and Murphy, 1988) has been implemented into the finite element method. Beside three-dimensional illustrative boundary-value problems demonstrating the features of the presented modelling concept, simulations on an idealised artery document the applicability of the model to more realistic geometries.

Computational modeling of airway and pulmonary vascular structure and function: development of a "lung physiome"

Computational models of lung structure and function necessarily span multiple spatial and temporal scales, i.e., dynamic molecular interactions give rise to whole organ function, and the link between these scales cannot be fully understood if only molecular or organ-level function is considered. Here, we review progress in constructing multiscale finite element models of lung structure and function that are aimed at providing a computational framework for bridging the spatial scales from molecular to whole organ. These include structural models of the intact lung, embedded models of the pulmonary airways that couple to model lung tissue, and models of the pulmonary vasculature that account for distinct structural differences at the extra- and intra-acinar levels. Biophysically based functional models for tissue deformation, pulmonary blood flow, and airway bronchoconstriction are also described. The development of these advanced multiscale models has led to a better understanding of complex physiological mechanisms that govern regional lung perfusion and emergent heterogeneity during bronchoconstriction.

Mathematical modeling of calcium signaling during sperm hyperactivation

Mammalian sperm must hyperactivate in order to fertilize oocytes. Hyperactivation is characterized by highly asymmetrical flagellar bending. It serves to move sperm out of the oviductal reservoir and to penetrate viscoelastic fluids, such as the cumulus matrix. It is absolutely required for sperm penetration of the oocyte zona pellucida. In order for sperm to hyperactivate, cytoplasmic Ca(2+) levels in the flagellum must increase. The major mechanism for providing Ca(2+) to the flagellum, at least in mice, are CatSper channels in the plasma membrane of the principal piece of the flagellum, because sperm from CatSper null males are unable to hyperactivate. There is some evidence for the existence of other types of Ca(2+) channels in sperm, but their roles in hyperactivation have not been clearly established. Another Ca(2+) source for hyperactivation is the store in the redundant nuclear envelope of sperm. To stabilize levels of cytoplasmic Ca(2+), sperm contain Ca(2+) ATPase and exchangers. The interactions between channels, Ca(2+) ATPases, and exchangers are poorly understood; however, mathematical modeling can help to elucidate how they work together to produce the patterns of changes in Ca(2+) levels that have been observed in sperm. Mathematical models can reveal interesting and unexpected relationships, suggesting experiments to be performed in the laboratory. Mathematical analysis of Ca(2+) dynamics has been used to develop a model for Ca(2+) clearance and for CatSper-mediated Ca(2+) dynamics. Models may also be used to understand how Ca(2+) patterns produce flagellar bending patterns of sperm in fluids of low and high viscosity and elasticity.

Exploring lung physiology in health and disease with lung slices

The development of therapeutic approaches to treat lung disease requires an understanding of both the normal and disease physiology of the lung. Although traditional experimental approaches only address either organ or cellular physiology, the use of lung slice preparations provides a unique approach to investigate integrated physiology that links the cellular and organ responses. Living lung slices are robust and can be prepared from a variety of species, including humans, and they retain many aspects of the cellular and structural organization of the lung. Functional portions of intrapulmonary airways, arterioles and veins are present within the alveoli parenchyma. The dynamics of macroscopic changes of contraction and relaxation associated with the airways and vessels are readily observed with conventional low-magnification microscopy. The microscopic changes associated with cellular events, that determine the macroscopic responses, can be observed with confocal or two-photon microscopy. To investigate disease processes, lung slices can either be prepared from animal models of disease or animals exposed to disease invoking conditions. Alternatively, the lung slices themselves can be experimentally manipulated. Because of the ability to observe changes in cell physiology and how these responses manifest themselves at the level of the organ, lung slices have become a standard tool for the investigation of lung disease.

Calcium oscillations

Calcium signaling results from a complex interplay between activation and inactivation of intracellular and extracellular calcium permeable channels. This complexity is obvious from the pattern of calcium signals observed with modest, physiological concentrations of calcium-mobilizing agonists, which typically present as sequential regenerative discharges of stored calcium, a process referred to as calcium oscillations. In this review, we discuss recent advances in understanding the underlying mechanism of calcium oscillations through the power of mathematical modeling. We also summarize recent findings on the role of calcium entry through store-operated channels in sustaining calcium oscillations and in the mechanism by which calcium oscillations couple to downstream effectors.

Possible role of differential growth in airway wall remodeling in asthma

Airway remodeling in patients with chronic asthma is characterized by a thickening of the airway walls. It has been demonstrated in previous theoretical models that this change in thickness can have an important mechanical effect on the properties of the wall, in particular on the phenomenon of mucosal folding induced by smooth muscle contraction. In this paper, we present a model for mucosal folding of the airway in the context of growth. The airway is modeled as a bilayered cylindrical tube, with both geometric and material nonlinearities accounted for via the theory of finite elasticity. Growth is incorporated into the model through the theory of morphoelasticity. We explore a range of growth possibilities, allowing for anisotropic growth as well as different growth rates in each layer. Such nonuniform growth, referred to as differential growth, can change the properties of the material beyond geometrical changes through the generation of residual stresses. We demonstrate that differential growth can have a dramatic impact on mucosal folding, in particular on the critical pressure needed to induce folding, the buckling pattern, as well as airway narrowing. We conclude that growth may be an important component in airway remodeling.

Multiscale mathematical models of airway constriction and disease

Loss of lung function in airway disease frequently involves many complex phenomena and interconnected underlying causes. In many conditions, such as asthmatic airway hyper-responsiveness, hypothesised underlying causes span multiple spatial scales. In cases like this, it is insufficient to take a reductionist approach, wherein each subsystem (at a given spatial scale) is considered in isolation and then the whole is taken to be merely the sum of the parts; this is because there can be significant and important interactions and synergies between spatial scales. Experimentally this can manifest as, for example, significant differences between behaviour in isolated tissue and that seen in vivo, while from a modelling perspective, it necessitates multiscale modelling approaches. Because it is precisely in these complex environs that models have the greatest potential to improve understanding of underlying behaviours, these multiscale models are of particular importance. This paper reviews several examples of multiscale models from the most important models in the literature, with a particular emphasis on those concerned with airway hyper-responsiveness and airway constriction.

Emergence of airway smooth muscle functions related to structural malleability

The function of a complex system such as a smooth muscle cell is the result of the active interaction among molecules and molecular aggregates. Emergent macroscopic manifestations of these molecular interactions, such as the length-force relationship and its associated length adaptation, are well documented, but the molecular constituents and organization that give rise to these emergent muscle behaviors remain largely unknown. In this minireview, we describe emergent properties of airway smooth muscle that seem to have originated from inherent fragility of the cellular structures, which has been increasingly recognized as a unique and important smooth muscle attribute. We also describe molecular interactions (based on direct and indirect evidence) that may confer malleability on fragile structural elements that in turn may allow the muscle to adapt to large and frequent changes in cell dimensions. Understanding how smooth muscle works may hinge on how well we can relate molecular events to its emergent macroscopic functions.

A mathematical analysis of agonist- and KCl-induced Ca(2+) oscillations in mouse airway smooth muscle cells

Airway hyperresponsiveness is a major characteristic of asthma and is generally ascribed to excessive airway narrowing associated with the contraction of airway smooth muscle cells (ASMCs). ASMC contraction is initiated by a rise in intracellular calcium concentration ([Ca(2+)](i)), observed as oscillatory Ca(2+) waves that can be induced by either agonist or high extracellular K(+) (KCl). In this work, we present a model of oscillatory Ca(2+) waves based on experimental data that incorporate both the inositol trisphosphate receptor and the ryanodine receptor. We then combined this Ca(2+) model and our modified actin-myosin cross-bridge model to investigate the role and contribution of oscillatory Ca(2+) waves to contractile force generation in mouse ASMCs. The model predicts that: 1), the difference in behavior of agonist- and KCl-induced Ca(2+) waves results principally from the fact that the sarcoplasmic reticulum is depleted during agonist-induced oscillations, but is overfilled during KCl-induced oscillations; 2), regardless of the order in which agonist and KCl are added into the cell, the resulting [Ca(2+)](i) oscillations will always be the short-period, agonist-induced-like oscillations; and 3), both the inositol trisphosphate receptor and the ryanodine receptor densities are higher toward one end of the cell. In addition, our results indicate that oscillatory Ca(2+) waves generate less contraction than whole-cell Ca(2+) oscillations induced by the same agonist concentration. This is due to the spatial inhomogeneity of the receptor distributions.

The effect of intracellular calcium oscillations on fluid secretion in airway epithelium

Airway epithelium has been shown to elicit fluid secretion after a rise in intracellular calcium. This rise in intracellular calcium has been shown to display complex oscillations in many species after the binding of particular agonists to extracellular receptors. Fluid secreted by the airway epithelium is used to maintain the depth of the periciliary liquid (PCL) above the apical membrane of the epithelial cells lining the bronchial airways. Previous mathematical models have been published which separately consider the electrophysiology involved in regulating periciliary liquid depth, and the transmission of intracellular calcium waves in airway epithelial tissue. In this paper we present a mathematical model that combines these previous models and allows the effect of oscillations in intracellular calcium on fluid secretion by airway epithelial cells to be investigated. We show that an oscillatory calcium response produces different fluid secretion properties to that elicited by a tonic rise in intracellular calcium. These differences are shown to be due to saturation of the Ca(2+) activated ion channels.

A multiscale, spatially distributed model of asthmatic airway hyper-responsiveness

We present a multiscale, spatially distributed model of lung and airway behaviour with the goal of furthering the understanding of airway hyper-responsiveness and asthma. The model provides an initial computational framework for linking events at the cellular and molecular levels, such as Ca(2+) and crossbridge dynamics, to events at the level of the entire organ. At the organ level, parenchymal tissue is modelled using a continuum approach as a compressible, hyperelastic material in three dimensions, with expansion and recoil of lung tissue due to tidal breathing. The governing equations of finite elasticity deformation are solved using a finite element method. The airway tree is embedded in this tissue, where each airway is modelled with its own airway wall, smooth muscle and surrounding parenchyma. The tissue model is then linked to models of the crossbridge mechanics and their control by Ca(2+) dynamics, thus providing a link to molecular and cellular mechanisms in airway smooth muscle cells. By incorporating and coupling the models at these scales, we obtain a detailed, computational multiscale model incorporating important physiological phenomena associated with asthma.

Nitric oxide induces airway smooth muscle cell relaxation by decreasing the frequency of agonist-induced Ca2+ oscillations

Nitric oxide (NO) induces airway smooth muscle cell (SMC) relaxation, but the underlying mechanism is not well understood. Consequently, we investigated the effects of NO on airway SMC contraction, Ca²⁺ signaling, and Ca²⁺ sensitivity in mouse lung slices with phase-contrast and confocal microscopy. Airways that were contracted in response to the agonist 5-hydroxytryptamine (5-HT) transiently relaxed in response to the NO donor, NOC-5. This NO-induced relaxation was enhanced by zaprinast or vardenafil, two selective inhibitors of cGMP-specific phosphodiesterase-5, but blocked by ODQ, an inhibitor of soluble guanylyl cyclase, and by Rp-8-pCPT-cGMPS, an inhibitor of protein kinase G (PKG). Simultaneous measurements of airway caliber and SMC [Ca²⁺]_i revealed that airway contraction induced by 5-HT correlated with the occurrence of Ca²⁺ oscillations in the airway SMCs. Airway relaxation induced by NOC-5 was accompanied by a decrease in the frequency of these Ca²⁺ oscillations. The cGMP analogues and selective PKG activators 8Br-cGMP and 8pCPT-cGMP also induced airway relaxation and decreased the frequency of the Ca²⁺ oscillations. NOC-5 inhibited the increase of [Ca²⁺]_i and contraction induced by the photolytic release of inositol 1,4,5-trisphosphate (IP₃) in airway SMCs. The effect of NO on the Ca²⁺ sensitivity of the airway SMCs was examined in lung slices permeabilized to Ca²⁺ by treatment with caffeine and ryanodine. Neither NOC-5 nor 8pCPT-cGMP induced relaxation in agonist-contracted Ca²⁺-permeabilized airways. Consequently, we conclude that NO, acting via the cGMP–PKG pathway, induced airway SMC relaxation by predominately inhibiting the release of Ca²⁺ via the IP₃ receptor to decrease the frequency of agonist-induced Ca²⁺ oscillations.

Influence of airway wall stiffness and parenchymal tethering on the dynamics of bronchoconstriction

Understanding how tissue remodeling affects airway responsiveness is of key importance, but experimental data bearing on this issue remain scant. We used lung explants to investigate the effects of enzymatic digestion on the rate and magnitude of airway narrowing induced by acetylcholine. To link the observed changes in narrowing dynamics to the degree of alteration in tissue mechanics, we compared our experimental results with predictions made by a computational model of a dynamically contracting elastic airway embedded in elastic parenchyma. We found that treatment of explanted airways with two different proteases (elastase and collagenase) resulted in differential effects on the dynamics of airway narrowing following application of ACh. Histological corroboration of these different effects is manifest in different patterns of elimination of collagen and elastin from within the airway wall and the surrounding parenchyma. Simulations with a computational model of a dynamically contracting airway embedded in elastic parenchyma suggest that elastase exerts its functional effects predominately through a reduction in parenchymal tethering, while the effects of collagenase are more related to a reduction in airway wall stiffness. We conclude that airway and parenchymal remodeling as a result of protease activity can have varied effects on the loads opposing ASM shortening, with corresponding consequences for airway responsiveness.

A biomechanical model of agonist-initiated contraction in the asthmatic airway

This paper presents a modelling framework in which the local stress environment of airway smooth muscle (ASM) cells may be predicted and cellular responses to local stress may be investigated. We consider an elastic axisymmetric model of a layer of connective tissue and circumferential ASM fibres embedded in parenchymal tissue and model the active contractile force generated by ASM via a stress acting along the fibres. A constitutive law is proposed that accounts for active and passive material properties as well as the proportion of muscle to connective tissue. The model predicts significantly different contractile responses depending on the proportion of muscle to connective tissue in the remodelled airway. We find that radial and hoop-stress distributions in remodelled muscle layers are highly heterogenous with distinct regions of compression and tension. Such patterns of stress are likely to have important implications, from a mechano-transduction perspective, on contractility, short-term cytoskeletal adaptation and long-term airway remodelling in asthma.

Human airway contraction and formoterol-induced relaxation is determined by Ca2+ oscillations and Ca2+ sensitivity

The etiology of airway hyperresponsiveness associated with asthma requires an understanding of the regulatory mechanisms mediating human airway smooth muscle cell (SMC) contraction. The objective of this study was to determine how human airway SMC contraction (induced by histamine) and relaxation (induced by formoterol) are regulated by Ca(2+) oscillations and Ca(2+) sensitivity. The responses of human small airways and their associated SMCs were studied in human lung slices cut from agarose-inflated lungs. Airway contraction was measured with phase-contrast video microscopy. Ca(2+) signaling and Ca(2+) sensitivity of airway SMCs were measured with two-photon fluorescence microscopy and Ca(2+)-permeabilized lung slices. The agonist histamine induced contraction of human small airways by stimulating both an increase in intracellular Ca(2+) concentration in the SMCs in the form of oscillatory Ca(2+) waves and an increase in Ca(2+) sensitivity. The frequency of the Ca(2+) oscillations increased with histamine concentration, and correlated with increased contraction. Formoterol induced airway relaxation at low concentrations by initially decreasing SMC Ca(2+) sensitivity. At higher concentrations, formoterol additionally slowed or inhibited the Ca(2+) oscillations of the SMCs to relax the airways. The action of formoterol was only slowly reversed. Human lung slices provide a powerful experimental assay for the investigation of small airway physiology and pharmacology. Histamine induces contraction by simultaneously increasing SMC Ca(2+) signaling and Ca(2+) sensitivity. Formoterol induces long-lasting relaxation by initially reducing the Ca(2+) sensitivity and, subsequently, the frequency of the Ca(2+) oscillations of the airway SMCs.

Airway mechanics and methods used to visualize smooth muscle dynamics in vitro

Contraction of airway smooth muscle (ASM) is regulated by the physiological, structural and mechanical environment in the lung. We review two in vitro techniques, lung slices and airway segment preparations, that enable in situ ASM contraction and airway narrowing to be visualized. Lung slices and airway segment approaches bridge a gap between cell culture and isolated ASM, and whole animal studies. Imaging techniques enable key upstream events involved in airway narrowing, such as ASM cell signalling and structural and mechanical events impinging on ASM, to be investigated.

Ion channel regulation of intracellular calcium and airway smooth muscle function

Airway hyper-responsiveness associated with asthma is mediated by airway smooth muscle cells (SMCs) and has a complicated etiology involving increases in cell contraction and proliferation and the secretion of inflammatory mediators. Although these pathological changes are diverse, a common feature associated with their regulation is a change in intracellular Ca(2+) concentration ([Ca(2+)](i)). Because the [Ca(2+)](i) itself is a function of the activity and expression of a variety of ion channels, in both the plasma membrane and sarcoplasmic reticulum of the SMC, the modification of this ion channel activity may predispose airway SMCs to hyper-responsiveness. Our objective is to review how ion channels determine the [Ca(2+)](i) and influence the function of airway SMCs and emphasize the potential of ion channels as sites for therapeutic approaches to asthma.