A novel method for the solution of convection-diffusion problems with applications to nitric oxide production and transport in vitro

Nitric Oxide (NO) has been widely studied as the endothelial derived factor responsible for acute vasomotor responses to flow and vessel remodeling in response to chronic changes in flow. In this study we developed a steady state analytical solution to the problem of mass transport under forced convection and a dynamic/steady numerical model to investigate the production and transport of NO generated by a monolayer of cultured endothelial cells exposed to flow in a parallel plate flow chamber. The analytical solution was obtained by generalizing the method of substitution of variables in order to include any type of boundary condition. At first, an approximated velocity profile was employed to reduce the governing Partial Differential Equation (PDE) to an Ordinary Differential Equation (ODE). Subsequently, the governing PDE was reduced to a set of recursive ODEs for a generic fully developed velocity profile. The approximated ODE was reduced to a known quantum mechanical ODE and solved by hypergeometric functions. The analytical solution indicates that it may not be possible, at steady-state, to determine a priori the existence of a diffusion boundary layer. A critical ratio between length and height of the conduit for which the analytical solution applies has been developed. Numerical results are in excellent agreement with the approximated solution of the governing PDE. Numerical modeling was also used to suggest a quantitative relationship between shear stress and NO production rate. NO production was described as a combination of a basal production rate term and a shear-dependent term, which is shown to influence the nature of mass transport in proximity of the boundary. The steady state NO concentration near the endothelial surface exhibits a biphasic dependence on shear stress, in which at low flow, NO concentration decreases owing to the enhanced removal by convective transport while only at higher shear stresses does the increased production cause an increase in NO concentration. The unsteady response to step changes in flow exhibits transient fluctuations in NO that can be explained by time-dependent changes in the diffusive and convective mass transport as the concentration profile evolves.