Analysing Airway Dynamics Using 2D Iterated Maps and Systems of ODEs

Abstract

The airway, providing a means for respiration, is important to human health. An understanding of the behaviour of the airways and their major structural components, such as airway smooth muscle, may give important insights into human health and disease processes. As such, the dynamics of both the airways and airway smooth muscle have been popular areas of research. Airway and airway smooth muscle dynamics have previously been studied separately from one another. However, it is known that there are important connections between the behaviour of these two systems. In this work, an extension of an existing 1D iterated map for modelling airway dynamics to a 2D iterated map, which includes a non-constant force-length relationship for airway smooth muscle, is examined. Adding a force-length relationship and length adaptation to the original 1D map is shown to not affect the stability of stationary points or the period-two oscillatory dynamics in the region of interest. However, a period-three oscillation was discovered which was not seen in the 1D map, implying the oscillatory dynamics have been changed by adding the force-length relationship and length adaptation. The work has extended this 2D iterated map into a 2D system of ODEs and examined the behaviour of this new, continuous model, through analytical and numerical approaches. The continuous model has been found to lack the oscillatory behaviour associated with the discrete maps. This approach for producing the continuous model and the subsequent analysis is expected to have important implications in the future for coupling airway dynamics with full PDE-based airway smooth muscle dynamics. The behavioural similarities and differences between the 1D, 2D discrete and ODE models have been examined, showing that much of the behaviour of this family of models is highly similar, however, differences, particularly in respect to the appearance of oscillatory behaviour, have been shown to exist.