A mathematical model of calcium-induced fluid secretion in airway epithelium

Mathematical models of cystic fibrosis as a systemic disease

Cystic fibrosis (CF) is widely known as a disease of the lung, even though it is in truth a systemic disease, whose symptoms typically manifest in gastrointestinal dysfunction first. CF ultimately impairs not only the pancreas and intestine but also the lungs, gonads, liver, kidneys, bones, and the cardiovascular system. It is caused by one of several mutations in the gene of the epithelial ion channel protein CFTR. Intense research and improved antimicrobial treatments during the past eight decades have steadily increased the predicted life expectancy of a person with CF (pwCF) from a few weeks to over 50 years. Moreover, several drugs ameliorating the sequelae of the disease have become available in recent years, and notable treatments of the root cause of the disease have recently generated substantial improvements in health for some but not all pwCF. Yet, numerous fundamental questions remain unanswered. Complicating CF, for instance in the lung, is the fact that the associated insufficient chloride secretion typically perturbs the electrochemical balance across epithelia and, in the airways, leads to the accumulation of thick, viscous mucus and mucus plaques that cannot be cleared effectively and provide a rich breeding ground for a spectrum of bacterial and fungal communities. The subsequent infections often become chronic and respond poorly to antibiotic treatments, with outcomes sometimes only weakly correlated with the drug susceptibility of the target pathogen. Furthermore, in contrast to rapidly resolved acute infections with a single target pathogen, chronic infections commonly involve multi-species bacterial communities, called "infection microbiomes," that develop their own ecological and evolutionary dynamics. It is presently impossible to devise mathematical models of CF in its entirety, but it is feasible to design models for many of the distinct drivers of the disease. Building upon these growing yet isolated modeling efforts, we discuss in the following the feasibility of a multi-scale modeling framework, known as template-and-anchor modeling, that allows the gradual integration of refined sub-models with different granularity. The article first reviews the most important biomedical aspects of CF and subsequently describes mathematical modeling approaches that already exist or have the potential to deepen our understanding of the multitude aspects of the disease and their interrelationships. The conceptual ideas behind the approaches proposed here do not only pertain to CF but are translatable to other systemic diseases. This article is categorized under: Congenital Diseases > Computational Models.

Physiology and pathophysiology of human airway mucus

The mucus clearance system is the dominant mechanical host defense system of the human lung. Mucus is cleared from the lung by cilia and airflow, including both two-phase gas-liquid pumping and cough-dependent mechanisms, and mucus transport rates are heavily dependent on mucus concentration. Importantly, mucus transport rates are accurately predicted by the gel-on-brush model of the mucociliary apparatus from the relative osmotic moduli of the mucus and periciliary-glycocalyceal (PCL-G) layers. The fluid available to hydrate mucus is generated by transepithelial fluid transport. Feedback interactions between mucus concentrations and cilia beating, via purinergic signaling, coordinate Na+ absorptive vs Cl- secretory rates to maintain mucus hydration in health. In disease, mucus becomes hyperconcentrated (dehydrated). Multiple mechanisms derange the ion transport pathways that normally hydrate mucus in muco-obstructive lung diseases, e.g., cystic fibrosis (CF), chronic obstructive pulmonary disease (COPD), non-CF bronchiectasis (NCFB), and primary ciliary dyskinesia (PCD). A key step in muco-obstructive disease pathogenesis is the osmotic compression of the mucus layer onto the airway surface with the formation of adherent mucus plaques and plugs, particularly in distal airways. Mucus plaques create locally hypoxic conditions and produce airflow obstruction, inflammation, infection, and, ultimately, airway wall damage. Therapies to clear adherent mucus with hydrating and mucolytic agents are rational, and strategies to develop these agents are reviewed.

Calcium Dynamics and Water Transport in Salivary Acinar Cells

Saliva is secreted from the acinar cells of the salivary glands, using mechanisms that are similar to other types of water-transporting epithelial cells. Using a combination of theoretical and experimental techniques, over the past 20 years we have continually developed and modified a quantitative model of saliva secretion, and how it is controlled by the dynamics of intracellular calcium. However, over approximately the past 5 years there have been significant developments both in our understanding of the underlying mechanisms and in the way these mechanisms should best be modelled. Here, we review the traditional understanding of how saliva is secreted, and describe how our work has suggested important modifications to this traditional view. We end with a brief description of the most recent data from living animals and discuss how this is now contributing to yet another iteration of model construction and experimental investigation.

Thickness of the airway surface liquid layer in the lung is affected in cystic fibrosis by compromised synergistic regulation of the ENaC ion channel

The lung epithelium is lined with a layer of airway surface liquid (ASL) that is crucial for healthy lung function. ASL thickness is controlled by two ion channels: epithelium sodium channel (ENaC) and cystic fibrosis (CF) transmembrane conductance regulator (CFTR). Here, we present a minimal mathematical model of ENaC, CFTR and ASL regulation that sheds light on the control of ENaC by the short palate lung and nasal epithelial clone 1 (SPLUNC1) protein and by phosphatidylinositol 4,5-biphosphate (PI(4,5)P₂). The model, despite its simplicity, yields a good fit to experimental observations and is an effective tool for exploring the interplay between ENaC, CFTR and ASL. Steady-state data and dynamic information constrain the model's parameters without ambiguities. Testing the hypothesis that PI(4,5)P₂ protects ENaC from ubiquitination suggests that this protection does not improve the model results and that the control of the ENaC opening probability by PI(4,5)P₂ is sufficient to explain all available data. The model analysis further demonstrates that ASL at the steady state is sensitive to small changes in PI(4,5)P₂ abundance, particularly in CF conditions, which suggests that manipulation of phosphoinositide metabolism may promote therapeutic benefits for CF patients.

The in vitro effect of nebulised hypertonic saline on human bronchial epithelium

Inhaled hypertonic saline (HS) is an effective therapy for muco-obstructive lung diseases. However, the mechanism of action and principles pertinent to HS administration remain unclear.An in vitro system aerosolised HS to epithelial cells at rates comparable to in vivo conditions. Airway surface liquid (ASL) volume and cell height responses were measured by confocal microscopy under normal and hyperconcentrated mucus states.Aerosolised HS produced a rapid increase in ASL height and decrease in cell height. Added ASL volume was quickly reabsorbed following termination of nebulisation, although cell height did not recover within the same time frame. ASL volume responses to repeated HS administrations were blunted, but could be restored by a hypotonic saline bolus interposed between HS administrations. HS-induced ASL hydration was prolonged with hyperconcentrated mucus on the airway surface, with more modest reductions in cell volume.Aerosolised HS produced osmotically induced increases in ASL height that were limited by active sodium absorption and cell volume-induced reductions in cell water permeability. Mucus on airway surfaces prolonged the effect of HS via mucus-dependent osmotic forces, suggesting that the duration of action of HS is increased in patients with hyperconcentrated mucus.

An integrated mathematical epithelial cell model for airway surface liquid regulation by mechanical forces

A robust method based on reverse engineering was utilized to construct the ion-channel conductance functions for airway epithelial sodium channels (ENaC), the cystic fibrosis transmembrane conductance regulator (CFTR), and calcium-activated chloride channels (CaCC). The ion-channel conductance models for both normal (NL) and cystic fibrosis (CF) airway epithelia were developed and then coupled to an adenosine triphosphate (ATP) metabolism model and a fluid transport model (collectively called the integrated cell model) to investigate airway surface liquid (ASL) volume regulation and hence mucus concentration, by mechanical forces in NL and CF human airways. The epithelial cell models for NL and CF required differences in Cl^- secretion (decreased in CF) and Na⁺ absorption (raised in CF) to reproduce behaviors similar to in vitro epithelial cells exposed to mechanical forces (cyclic shear stress, cyclic compressive pressure and cilial strain) and selected modulators of ion channels and ATP release. The epithelial cell models were then used to investigate the effects of mechanical forces and evaporative flux on ASL and mucus homeostasis in both NL and CF airway epithelia. Because of reduced CF ASL volumes, CF mucus concentrations increased and produced a greater dependence of ASL volume regulation on cilia-mucus-ATP release interactions in CF than NL epithelial nodules. Similarly, the CF model was less tolerant to evaporation induced ASL volume reduction at all ATP release rates than the NL model. Consequently, this reverse engineered model appears to provide a robust tool for investigating CF pathophysiology and novel therapies.

Mathematical model reveals role of nucleotide signaling in airway surface liquid homeostasis and its dysregulation in cystic fibrosis

Mucociliary clearance is composed of three components (i.e., mucin secretion, airway surface hydration, and ciliary-activity) which function coordinately to clear inhaled microbes and other foreign particles from airway surfaces. Airway surface hydration is maintained by water fluxes driven predominantly by active chloride and sodium ion transport. The ion channels that mediate electrogenic ion transport are regulated by extracellular purinergic signals that signal through G protein-coupled receptors. These purinoreceptors and the signaling pathways they activate have been identified as possible therapeutic targets for treating lung disease. A systems-level description of airway surface liquid (ASL) homeostasis could accelerate development of such therapies. Accordingly, we developed a mathematical model to describe the dynamic coupling of ion and water transport to extracellular purinergic signaling. We trained our model from steady-state and time-dependent experimental measurements made using normal and cystic fibrosis (CF) cultured human airway epithelium. To reproduce CF conditions, reduced chloride secretion, increased potassium secretion, and increased sodium absorption were required. The model accurately predicted ASL height under basal normal and CF conditions and the collapse of surface hydration due to the accelerated nucleotide metabolism associated with CF exacerbations. Finally, the model predicted a therapeutic strategy to deliver nucleotide receptor agonists to effectively rehydrate the ASL of CF airways.

A spatial model of fluid recycling in the airways of the lung

The genetic disease cystic fibrosis (CF) is a mutation in the cystic fibrosis transmembrane conductance regulator (CFTR) gene, and results in viscous mucus and impaired mucociliary clearance leading to chronic recurring pulmonary infections. Although extensive experimental research has been conducted over the last few decades, CF lung pathophysiology remains controversial. There are two competing explanations for the observed depletion of periciliary liquid (PCL) in CF lungs. The low volume hypothesis assumes fluid hyperabsorption through surface epithelia due to an over-active epithelial Na(+) channel (ENaC), and the low secretion hypothesis assumes inspissated mucins secreted from glands due to lack of serous fluid secreted from gland acini. We present a spatial mathematical model that reflects in vivo fluid recycling via submucosal gland (SMG) secretion, and absorption through surface epithelia. We then test the model in CF conditions by increasing ENaC open probability and decreasing SMG flux while simultaneously reducing CFTR open probability. Increasing ENaC activity only results in increased fluid absorption across surface epithelia, as seen in in vitro experiments. However, combining potential CF mechanisms results in markedly less fluid absorbed while providing the largest reduction in PCL volume, suggesting that a compromise in gland fluid secretion dominates over increased ENaC activity to decrease the amount of fluid transported transcellularly in CF lungs in vivo. Model results also indicate that a spatial model is necessary for an accurate calculation of total fluid transport, as the effects of spatial gradients can be severe, particularly in close proximity to the SMGs.

Biophysical model of ion transport across human respiratory epithelia allows quantification of ion permeabilities

Lung health and normal mucus clearance depend on adequate hydration of airway surfaces. Because transepithelial osmotic gradients drive water flows, sufficient hydration of the airway surface liquid depends on a balance between ion secretion and absorption by respiratory epithelia. In vitro experiments using cultures of primary human nasal epithelia and human bronchial epithelia have established many of the biophysical processes involved in airway surface liquid homeostasis. Most experimental studies, however, have focused on the apical membrane, despite the fact that ion transport across respiratory epithelia involves both cellular and paracellular pathways. In fact, the ion permeabilities of the basolateral membrane and paracellular pathway remain largely unknown. Here we use a biophysical model for water and ion transport to quantify ion permeabilities of all pathways (apical, basolateral, paracellular) in human nasal epithelia cultures using experimental (Ussing Chamber and microelectrode) data reported in the literature. We derive analytical formulas for the steady-state short-circuit current and membrane potential, which are for polarized epithelia the equivalent of the Goldman-Hodgkin-Katz equation for single isolated cells. These relations allow parameter estimation to be performed efficiently. By providing a method to quantify all the ion permeabilities of respiratory epithelia, the model may aid us in understanding the physiology that regulates normal airway surface hydration.

Increased apical Na+ permeability in cystic fibrosis is supported by a quantitative model of epithelial ion transport

Cystic fibrosis (CF) is caused by mutations in the cystic fibrosis transmembrane conductance regulator (CFTR) gene, which encodes an anion channel. In the human lung CFTR loss causes abnormal ion transport across airway epithelial cells. As a result CF individuals produce thick mucus, suffer persistent bacterial infections and have a much reduced life expectancy. Trans-epithelial potential difference (V_t) measurements are routinely carried out on nasal epithelia of CF patients in the clinic. CF epithelia exhibit a hyperpolarised basal V_t and a larger V_t change in response to amiloride (a blocker of the epithelial Na⁺ channel, ENaC). Are these altered bioelectric properties solely a result of electrical coupling between the ENaC and CFTR currents, or are they due to an increased ENaC permeability associated with CFTR loss? To examine these issues we have developed a quantitative mathematical model of human nasal epithelial ion transport. We find that while the loss of CFTR permeability hyperpolarises V_t and also increases amiloride-sensitive V_t, these effects are too small to account for the magnitude of change observed in CF epithelia. Instead, a parallel increase in ENaC permeability is required to adequately fit observed experimental data. Our study provides quantitative predictions for the complex relationships between ionic permeabilities and nasal V_t, giving insights into the physiology of CF disease that have important implications for CF therapy.

A mechanochemical model for auto-regulation of lung airway surface layer volume

We develop a proof-of-principle model for auto-regulation of water volume in the lung airway surface layer (ASL) by coupling biochemical kinetics, transient ASL volume, and homeostatic mechanical stresses. The model is based on the hypothesis that ASL volume is sensed through soluble mediators and phasic stresses generated by beating cilia and air drag forces. Model parameters are fit based on the available data on human bronchial epithelial cell cultures. Simulations then demonstrate that homeostatic volume regulation is a natural consequence of the processes described. The model maintains ASL volume within a physiological range that modulates with phasic stress frequency and amplitude. Next, we show that the model successfully reproduces the responses of cell cultures to significant isotonic and hypotonic challenges, and to hypertonic saline, an effective therapy for mucus hydration in cystic fibrosis patients. These results compel an advanced airway hydration model with therapeutic value that will necessitate detailed kinetics of multiple molecular pathways, feedback to ASL viscoelasticity properties, and stress signaling from the ASL to the cilia and epithelial cells.

Image-based modeling of lung structure and function

The current state-of-the-art in image-based modeling allows derivation of patient-specific models of the lung, lobes, airways, and pulmonary vascular trees. The application of traditional engineering analyses of fluid and structural mechanics to image-based subject-specific models has the potential to provide new insight into structure-function relationships in the individual via functional interpretation that complements imaging and experimental studies. Three major issues that are encountered in studies of airflow through the bronchial airways are the representation of airway geometry, the imposition of physiological boundary conditions, and the treatment of turbulence. Here we review some efforts to resolve each of these issues, with particular focus on image-based models that have been developed to simulate airflow from the mouth to the terminal bronchiole, and subjected to physiologically meaningful boundary conditions via image registration and soft-tissue mechanics models.

Regulatory volume decrease and P receptor signaling in fish cells: mechanisms, physiology, and modeling approaches

For animal cell plasma membranes, the permeability of water is much higher than that of ions and other solutes, and exposure to hyposmotic conditions almost invariably causes rapid water influx and cell swelling. In this situation, cells deploy regulatory mechanisms to preserve membrane integrity and avoid lysis. The phenomenon of regulatory volume decrease, the partial or full restoration of cell volume following cell swelling, is well-studied in mammals, with uncountable investigations yielding details on the signaling network and the effector mechanisms involved in the process. In comparison, cells from other vertebrates and from invertebrates received little attention, despite of the fact that e.g. fish cells could present rewarding model systems given the diversity in ecology and lifestyle of this animal group that may be reflected by an equal diversity of physiological adaptive mechanisms, including those related to cell volume regulation. In this review, we therefore present an overview on the most relevant aspects known on hypotonic volume regulation presently known in fish, summarizing transporters and signaling pathways described so far, and then focus on an aspect we have particularly studied over the past years using fish cell models, i.e. the role of extracellular nucleotides in mediating cell volume recovery of swollen cells. We, furthermore, present diverse modeling approaches developed on the basis of data derived from studies with fish and other models and discuss their potential use for gaining insight into the theoretical framework of volume regulation.

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 biophysical model for integration of electrical, osmotic, and pH regulation in the human bronchial epithelium

A dynamical biophysical model for the functioning of an epithelium is presented. This model integrates the electrical and osmotic behaviors of the epithelium, taking into account intracellular conditions. The specific tissue modeled is the human bronchial epithelium, which is of particular interest, as it is the location of the most common lethal symptoms of cystic fibrosis. The model is implemented in a modular form to facilitate future application of the code to other epithelial tissue by inputting different transporters, channels, and geometric parameters. The model includes pH regulation as an integral component of overall regulation of epithelial function, through the interdependence of pH, bicarbonate concentration, and current. The procedures for specification, the validation of the model, and parametric studies are presented using available experimental data of cultured human bronchial epithelium. Parametric studies are performed to elucidate a), the contribution of basolateral chloride channels to the short-circuit current functional form, and b), the role that regulation of basolateral potassium conductance plays in epithelial function.

The role of airway epithelium in replenishment of evaporated airway surface liquid from the human conducting airways

This article presents a multi-scale computational model describing the transport of water vapor and heat within the human conducting airways and its interaction with cellular fluid transport kinetics. This tight coupling between the cell and the evaporative flux allows the periciliary liquid (PCL) depth to be investigated within the context of a geometric framework of the human conducting airways with spatial and temporal variations. Within the in vivo airway, the epithelium is not the only source of fluid available for hydration of the PCL, and fluid may also be supplied from submucosal glands (SMGs) or via axial transport of the PCL. The model predicts that without fluid supplied by either SMGs or via PCL transport, significant dehydration would occur under normal breathing conditions. Previous studies have suggested that PCL transport from the periphery to the trachea would require absorption of the fluid by the epithelium; here we show that this can theoretically be sustained by the evaporative load under normal breathing conditions. SMGs could also provide a significant supply of fluid for airway hydration, a hypothesis which is corroborated by comparing the distribution of SMGs as a function of airway generation with the distribution of airway evaporative flux.

A dynamic model of saliva secretion

We construct a mathematical model of the parotid acinar cell with the aim of investigating how the distribution of K(+) and Cl(-) channels affects saliva production. Secretion of fluid is initiated by Ca(2+) signals acting on Ca(2+) dependent K(+) and Cl(-) channels. The opening of these channels facilitates the movement of Cl(-) ions into the lumen which water follows by osmosis. We use recent results into both the release of Ca(2+) from internal stores via the inositol (1,4,5)-trisphosphate receptor (IP(3)R) and IP(3) dynamics to create a physiologically realistic Ca(2+) model which is able to recreate important experimentally observed behaviours seen in parotid acinar cells. We formulate an equivalent electrical circuit diagram for the movement of ions responsible for water flow which enables us to calculate and include distinct apical and basal membrane potentials to the model. We show that maximum saliva production occurs when a small amount of K(+) conductance is located at the apical membrane, with the majority in the basal membrane. The maximum fluid output is found to coincide with a minimum in the apical membrane potential. The traditional model whereby all Cl(-) channels are located in the apical membrane is shown to be the most efficient Cl(-) channel distribution.

Mathematical modelling of calcium wave propagation in mammalian airway epithelium: evidence for regenerative ATP release

Airway epithelium has been shown to exhibit intracellular calcium waves after mechanical stimulation. Two classes of mechanism have been proposed to explain calcium wave propagation: diffusion through gap junctions of the intracellular messenger inositol 1,4,5-trisphosphate (IP3), and diffusion of paracrine extracellular messengers such as ATP. We have used single cell recordings of airway epithelium to parameterize a model of an airway epithelial cell. This was then incorporated into a spatial model of a cell culture where both mechanisms for calcium wave propagation are possible. It is shown that a decreasing return on the radius of Ca2+ wave propagation is achieved as the amount of ATP released from the stimulated cell increases. It is therefore shown that for a Ca2+ wave to propagate large distances, a significant fraction of the intracellular ATP pool would be required to be released. Further to this, the radial distribution of maximal calcium response from the stimulated cell does not produce the same flat profile of maximal calcium response seen in experiential studies. This suggests that an additional mechanism is important in Ca2+ wave propagation, such as regenerative release of ATP from cells downstream of the stimulated cell.