Mucus plugging in the small airways is a central feature of cystic fibrosis (CF). This thesis presents theoretical modelling of surface-tension-driven flows, including flows that can lead to mucus plug formation, and of inhaled drug deposition in lungs affected by obstructions in the small airways. First, we investigate the Rayleigh-Plateau instability of a viscoplastic liquid film coating the interior of a tube, a model for the flow of mucus in an airway that accounts for the mucus yield stress. We use long-wave theory to derive an evolution equation for the liquid layer thickness, and derive a simpler evolution equation in the thin film limit. Using numerical solutions and asymptotic analyses, we determine how raising the capillary Bingham number, which measures yield stress relative to capillary stresses, increases the minimum size of free-surface perturbation required to trigger instability, decreases the final peak height of thin layers, and can increase the critical layer thickness required for plug formation to occur. We discuss how these results indicate a mechanism for how mucolytic drugs, which lower mucus yield stress, could trigger plugging in some airways. Second, we extend our first model to study the impact of surfactant on the Rayleigh-Plateau instability. We quantify how increasing the surfactant strength can reduce the deformation caused to a thin film and slow the dynamics. When Marangoni effects are strong, the dynamics are the same as for a layer without surfactant, but with time slowed by a factor of four and the capillary Bingham number doubled. Numerical solutions of the long-wave equations demonstrate that the critical thickness required for plugging can be raised by increasing the surfactant strength when the liquid has a yield stress. This suggests a new mechanism for how surfactant deficiency in CF may promote plugging. Third, we investigate the dynamics of a viscoplastic layer coating a vertical tube. We use thin-film theory to derive an evolution equation, which we solve numerically and analyse using matched asymptotics in the limit of small Bond number. We quantify the impact of viscoplasticity on the formation and motion of liquid collars. We show that for viscoplastic collars, unlike for Newtonian collars, there is a range of film thicknesses ahead of the collar for which a stable, steadily-translating state is reached. Finally, we present a whole-lung model of inhaled particle deposition. The model does not assume uniform ventilation of the lungs. Instead, it calculates air flow rates through each airway, with the resistance of each airway depending on its diameter, so simulated ventilation patterns respond to airway constriction. We apply constrictions to the distal airways, representative of CF disease, to demonstrate the capability of the model to predict the resulting impacts on particle deposition throughout the lungs. Applying constrictions can increase deposition rates in central airways away from sites of constriction due to enhanced inertial impaction. Our model marks a significant advancement on many established particle-deposition models, and could be used as a framework for future modelling of drug delivery in patients with airway disease.
Theoretical models of mucus plugging and inhaled drug delivery in cystic fibrosis
Shemilt, J. (Author). 1 Aug 2025
Student thesis: Phd