Quantitative analysis of mass and energy balance in non-ideal models of the renal counterflow system.

A modified Newton-Raphson method for solving finite difference equations for the renal counterflow system is described. The method has proved generally stable and efficient, and has given significant computational results for a variety of models: calculations on single solute models of the coupled vasa recta nephron counterflow system have shown that for large water and solute permeabilities of the exchanging membranes, behavior of the non-ideal system approaches that of the previously described ideal central core model. Concentration by salt and urea mixing in two solute models has been analyzed and previous conclusions from central core models have been found to remain valid in non-ideal systems. The numerical solutions have set some order of magnitude bounds on permeability requirements for concentration in different types of non-ideal systems. Finally, from the detailed concentration profiles it has been possible to relate the rate of free energy creation and dissipation from transmembrane transport of solutes and water to the net rate of free energy efflux from the counterflow system, and so to compute in a given model the fraction of power used for solute concentration.