No abstract

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.It has been proposed in previous papers (1-3) that the behavior of the intricately coupled nephrovascular counterflow system of the renal medulla (4) approaches as a limiting case that of a four-tube model: the vascular counterflow exchanger is represented by a single tube-the central core, closed at the papillary end and open at the corticomedullary junction-which exchanges with three other tubes corresponding respectively to ascending Henle's limb (AHL), descending Henle's limb (DHL), and collecting duct (CD). Under the assumption that total solute concentrations in core, DHL, and CD are nearly the same at each level of the medulla, it has been possible to develop an approximate analytic theory of the ideal central core concentrating engine and so of the medullary counterflow system. This assumption implies very high solute and water permeability of the vasa recta and very high osmotic water permeability and (or) solute permeability of DHL and CD. The behavior of non-ideal models with finite permeabilities will deviate from that of the ideal central core model. In general, the differential equations describing non-ideal models must be solved numerically. For certain single-solute models this has been done by converting the two-point boundary value problem to an initial value problem (5), but this method tends to be unstable, requiring very good initial estimates to converge, and does not extend readily to two-solute models.In this paper we outline a modified Newton-Raphson method for solving globally finite difference equations approximating the differential equations. The method has proved generally applicable to a variety of models of the renal counterflow system. In this paper we summarize some significant preliminary computational results. Detailed descriptions of both the method and the results are in preparation. [2) [3] [4] ...