Mathematical Modelling of Airway Smooth Muscle

Abstract

The bronchial tree consists of thousands of airways, from the trachea down to the small terminal bronchioles. Surrounding each airway is a layer of Airway Smooth Muscle (ASM) tissue. It is thought that abnormalities in the properties of the muscle play a key role in the pathogenesis of asthma and airway hyper-responsiveness. Specifically, stiffening of the ASM tissue or a weakened sensitivity in its response to stretch can lead to sustained airway narrowing, one of the major characteristics of airway diseases. A long-term goal is to learn more about the effects of increased muscle mass or changes in the heterogeneity of the lung response on diseases such as asthma. This entails the formulation of a mathematical representation of the entire bronchial tree. The first step towards this goal is to represent a single airway. By coupling a model of the airway radius with a model of its ASM lining, a description of the response of a single airway under various chemical and mechanical stimuli can be obtained. The force generated by the muscle will determine the extent to which the airway will narrow. A sophisticated ASM model developed previously is capable of reproducing many of the observed properties from tissue strip experiments but is difficult to analyse and computationally expensive to solve. Employing such a complex model to describe the ASM generated force would make a representation of the entire tree almost impossible. As such, our goal is to determine whether this model could be replaced with a vastly simplified model. We demonstrate that, under some circumstances, complex ASM properties at the tissue level can have a profound impact on the airway response. Naturally, we want to further our understanding of these properties, particularly passive mechanisms such as cytoskeletal remodelling. Our interest in these processes stems from the roles they are likely to play in increasing the cell's resistance, stiffening the muscle to prevent airway relaxation and modulating the sensitivity of the cell to stretch. We are particularly concerned with how this behaviour is regulated, its implications in airway disorders and how such passive dynamics interact with their active counterparts. It is found that a simplified model can be employed to describe the ASM generated force for large airways. However, for a small range of parameter values, the small airway response to imposed pressure oscillations varies significantly depending upon the complexity of the model used to describe the ASM force.In addition, we find that both crosslinkers and latchbridges are needed to provide a complete representation of ASM contraction. Specifically, latchbridges are required to explain the decrease in phosphorylation following muscle activation while crosslinkers are required to explain the independence of passive stiffness on myosin light chain kinase activity as well as the passive length-tension curves observed by Naghshin et. al. [93]. A constitutive formulation, typical of that found in the complex fluids literature, can be employed as a method of combining the active and passive sides of ASM contraction. In addition, this formulation improves upon the predicted force-length relationship observed experimentally by several groups.