# A computational study of the physicochemical influence of surfactant under bulk-equilibrium conditions on pulmonary airway reopening

In this investigation, we have developed a model of airway reopening that includes the physicochemical influence of surfactant. Our goal is to emulate pulmonary airway reopening phenomena and elucidate the role of surfactant during the reopening process. The airway is modeled as a flexible-walled channel, where the channel walls are membranes supported with a transverse elasticity E and longitudinal tension T$\sb0.$ Lining fluid of viscosity $\mu$ and film thickness H hold the walls in apposition. Airway reopening occurs when an inviscid, semi-infinite bubble of air progresses steadily at velocity U and separate the walls. Surfactant molecules exist in the lining fluid (C*) and the air-fluid interface $(\Gamma$*). Bulk equilibrium is assumed (C* = C$\sb0)$ and the kinetic transfer of surfactant with the interface occurs with a rate k. Surfactant also diffuses along the interface with diffusivity $\rm D\sb{int}.$ The equilibrium condition between the bulk and surface surfactant concentrations is based upon Henry's isotherm $\rm(\Gamma\sb{eq} = KC\sb0).$ Surfactant affects the surface tension equation of state, a relationship between the surface tension $(\gamma$*) and surface surfactant concentration. This relationship is modeled with a linear expansion of the equilibrium surface tension $\rm\gamma\sb{eq}.$ A set of dimensionless governing equations that represent the hydrodynamics and surfactant transport of the system are formulated. Several dimensionless parameters are created showing the relative importance of different fundamental properties of the system: the Capillary number $\rm Ca = \mu U/\gamma\sb{eq},$ the surface Elasticity number $\rm E1 = {-}(d\gamma$*/d$\Gamma$*)$\rm(\Gamma\sb{eq}/\gamma\sb{eq}),$ the wall Elasticity number $\rm\beta = EH\sp2/\gamma\sb{eq},$ the wall tension ratio $\rm\eta = T\sb0/\gamma\sb{eq},$ the modified Stanton number St$\sb\lambda = {{k/K}\over{U/H}},$ and the surface Peclet number $\rm Pe\sb{int} = UH/D\sb{int}.$ The analytical equations are solved by using a composite, computational technique of Boundary Element Method for the hydrodynamics and Finite-Difference Method for the transport characteristics. By investigating a range of values for the dimensionless parameters, the influence of the fundamental properties on the airway reopening characteristics was determined. The results compare favorably with prior studies of airway reopening and other bulk equilibrium models of free-surface flows. Physiologically, the computational results help explain the conditions of infant Respiratory Distress Syndrome, cystic fibrosis, and emphysema. This model provides a foundation for future investigations of the physicochemical influence of surfactant on pulmonary airway reopening