A reliable analysis of oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics

In this paper, we investigate the diffusion of oxygen in a spherical cell including nonlinear uptake kinetics. The Lane-Emden boundary value problem with Michaelis-Menten kinetics is used to model the dimensionless oxygen concentration in our analysis. We first convert the Lane-Emden equation to the equivalent Volterra integral form that incorporates the boundary condition at the cell's center, but which still leaves one unknown constant of integration, as an intermediate step. Next we evaluate the Volterra integral form of the concentration and its flux at the cell membrane and substitute them into the remaining boundary condition to determine the unknown constant of integration by appropriate algebraic manipulations. Upon substitution we have converted the equivalent Volterra integral form to the equivalent Fredholm-Volterra integral form, and use the Duan-Rach modified recursion scheme to effectively decompose the unknown constant of integration by formula. The Adomian decomposition method is then applied to solve the equivalent nonlinear Fredholm-Volterra integral representation of the Lane-Emden model for the concentration of oxygen within the spherical cell. Our approach shows enhancements over existing techniques.