Spatially discrete models of counter-current mass transport for application to the kidney

Within the kidney the transport of many solute species and water is accomplished by the parallel operation of several counter-current transport systems. The simulation of these processes requires the solution of a non-linear, two-point boundary value problem with many simultaneous equations. Standard formulations of the problem involve the solution of a set of one-dimensional, plug-transport equations. Rather than seek an accurate solution of this spatially continuous model, which is only an approximation to the real system, we investigated spatially discrete analogs. The motivation for use of spatially discrete models is to simulate with much more efficiency the detailed simultaneous transport processes of many solutes and water. The membrane flux equations of these models describe solvent drag as well as passive and active transport of solutes. In this study, we investigated a perfectly mixed compartment model and an improved Euler (semi-discretized) model. The solutions of these models were compared to the numerical solution of a standard spatially continuous model. For efficient numerical solution of the continuous model, the equations were arranged to yield a band matrix structure and a triangular decomposition procedure was applied. The model solutions were compared for convergence, efficiency, and accuracy for a variety of parameter values. With membrane parameter values typical of tubule segments of the rat kidney, a spatial step size as large as 1 mm (roughly one-seventh the length of the renal medulla of the rat) was found to be suitable for both discrete models.