Analytical expressions for the steady-state concentrations of glucose, oxygen and gluconic acid in a composite membrane for closed-loop insulin delivery.

The mathematical model of Abdekhodaie and Wu (J Membr Sci 335:21-31, 2009) of glucose-responsive composite membranes for closed-loop insulin delivery is discussed. The glucose composite membrane contains nanoparticles of an anionic polymer, glucose oxidase and catalase embedded in a hydrophobic polymer. The model involves the system of nonlinear steady-state reaction-diffusion equations. Analytical expressions for the concentration of glucose, oxygen and gluconic acid are derived from these equations using the Adomian decomposition method. A comparison of the analytical approximation and numerical simulation is also presented. An agreement between analytical expressions and numerical results is observed.

High order accurate, one-sided finite-difference approximations to concentration gradients at the boundaries, for the simulation of electrochemical reaction-diffusion problems in one-dimensional space geometry.

Accurate calculation of concentration gradients at the boundaries is crucial in electrochemical kinetic simulations, owing to the frequent occurrence of gradient-dependent boundary conditions, and the importance of the gradient-dependent electric current. By using the information about higher spatial derivatives of the concentrations, contained in the time-dependent, kinetic reaction-diffusion partial differential equation(s) in one-dimensional space geometry, under appropriate assumptions it is possible to increase the accuracy orders of the conventional, one-sided n-point finite-difference formulae for the concentration gradients at the boundaries, without increasing n. In this way a new class of high order accurate gradient approximations is derived, and tested in simulations of potential-step chronoamperometric and current-step chronopotentiometric transients for the Reinert-Berg system. The new formulae possess advantages over the conventional gradient approximations. For example, they allow one to obtain a third order accuracy by using two space points only, or fourth order accuracy by using three points, and yet they yield smaller errors than the conventional four-point, or five-point formulae, respectively. Needing fewer points, for approximating the gradients with a given accuracy, simplifies also the solution of the linear algebraic equations arising from the application of implicit time integration schemes.

Use of the Numerov method to improve the accuracy of the spatial discretisation in finite-difference electrochemical kinetic simulations.

The fourth order accuracy of the spatial discretisation of time-dependent reaction-diffusion equations, in finite-difference electrochemical kinetic simulations in one space dimension, might well be achieved by means of the three-point Numerov method, instead of the 5(6)-point discretisation of second spatial derivatives, recently suggested in the literature. This is proven theoretically, and tested in simulations of potential-step chronoamperometric and current-step chronopotentiometric transients for the Reinert-Berg system, which is a classical example of electrochemical reaction-diffusion equations. Although less generally applicable than the 5(6)-point spatial scheme, the Numerov discretisation is easier to use, because it does not lead to increased linear equation matrix bandwidth, but results in quasi-block-tridiagonal matrices, similar to those for the conventional, second order accurate, three-point spatial discretisation. The simulations reveal that the Numerov method brings an improvement of accuracy and efficiency that is comparable with the one offered by the 5(6)-point spatial scheme.