This revised proposal to develop a mathematical model for the stability of pulmonary airways is a basic inquiry into the mechanisms that lead to airway closure of the lung in health, disease, and its modifications by therapies. Closure results from a surface tension instability that tends to pull the airway to a smaller diameter while potentially driving the liquid lining into the formation of a plug, given the right conditions. A plug blocks the lumen, stopping gas exchange and delivery of aerosol medications. It can be the terminal event in an asthmatic attack, for example. From purely the fluid mechanical perspective, the plug may or may not form depending on the liquid physical properties. Significant non-Newtonian fluid properties in disease (asthma, emphysema, cystic fibrosis) include viscoelasticity, shear-thinning viscosity, and a yield stress, all of which can alter the fluid movement. Surfactants at the air-liquid interface also influence the instability by reducing surface tension and creating surface tension gradients. The lung volume at which closure occurs is the closing volume, and it is a common pulmonary function test that is often difficult to interpret, due largely to inadequate models. This surface-tension/fluid system couples to the airway and lung elastic properties. New to this revision is the inclusion of non-linear airway elasticity, airway smooth muscle dynamics, and effects of a surrounding parenchymal tethering. This novel combination of effects will allow the development of a theory which can incorporate the simultaneous issues presented by diseases such as asthma and emphysema where bronchoconstriction, parenchymal tissue elasticity changes, and fluid property modifications (non-Newtonian fluids) acting together present a complex mechanical situation; The proposed model will provide a platform for predicting the impact of these interactions on closing volumes, as well as predict the effectiveness of therapies which may include those aimed at surfactants, or liquid material properties, or bronchoconstriction or the parenchyma. This proposal outlines a set of mathematical models, building from the simplest to the more complex, that examine the stability of a liquid-lined airway tube as it depends on (1) Newtonian vs non-Newtonian fluids, (2) rigid vs flexible tubes with parenchyma and smooth muscle tone, (3) axisymmetric vs non-axisymmetric deformations, (4) effects of surfactants, (5) effects of a forced, oscillatory core flow. ? ? ?
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