Three-dimensional anatomy and renal concentrating mechanism. II. Sensitivity results.

A mathematical model has been developed to simulate hypertonic urine formation in the renal medulla. The model uses published values of membrane transport parameters, as have other models, but is unique in its representation of the three-dimensional anatomy of the medulla. The model successfully predicts measured fluid flows, osmolarities, and NaCl and urea concentrations. The model results are presented in the companion to this paper [A. S. Wexler, R. E. Kalaba, D. J. Marsh. Am. J. Physiol. 260 (Renal Fluid Electrolyte Physiol. 29): F368-F383, 1991.]. In this paper we provide tests of the sensitivity of model performance to variations in the description of the anatomy and in membrane transport parameters. From these studies we conclude that 1) strict counterflow arrangements are required in the outer stripe to prevent loss of NaCl to the systemic circulation, 2) the radial organization in the inner stripe materially improves performance of the inner medulla, 3) radial organization of the inner medulla is essential to hypertonic urine formation there, 4) the model is most sensitive to variation in collecting duct parameters, and 5) reabsorption of urea in the distal tubule improves system performance. The results support the claim that the three-dimensional structure, as captured in the model, provides a crucial framework for the production of hypertonic urine.

Three-dimensional anatomy and renal concentrating mechanism. I. Modeling results.

Simulations were performed to test the hypothesis that the three-dimensional organization of the renal medulla is essential for formation of hypertonic urine. As in previous models, representations of loops of Henle, distal tubules, collecting ducts, and vasa recta and recent estimates of tubule characteristics were included in a simulation of NaCl, urea, and fluid transport. In addition, this model specifies the relative positions of the medullary structures. By assuming that the structure of the minimum functional unit is a vascular bundle surrounded by tubules and ascending vessels, we have represented the three-dimensional organization of the medulla by a cylindrically symmetric two-dimensional model. The resulting set of equations gives rise to a nonlinear boundary value problem with linear boundary conditions, which was solved numerically via quasi linearization. Compared with previous simulations, the concentrations predicted by this model more accurately match measured quantities in two regards. First, papillary tip concentrations of NaCl and urea are significantly higher, and, second, a monotonic increase in osmolarity is observed in the inner medulla. The three-dimensional organization permitted development of local concentration gradients, which are essential to the final result.

Passive, one-dimensional countercurrent models do not simulate hypertonic urine formation.

Simulations were performed to test the ability of the countercurrent hypothesis to predict measured concentrations of NaCl and urea in the interstitium of the renal medulla. The simulations included one-dimensional representations of loops of Henle, distal tubules, collecting ducts, and vasa recta, and recent estimates of descending limb, thick ascending limb, and collecting duct transport parameters. The nonlinear two-point boundary value problem was solved numerically via quasi-linearization. The simulations failed to predict measured concentrations or concentration gradients of NaCl in the inner medulla. Including countertransport of urea and NaCl in thin ascending limbs added minimally to the performance of the system. The single most effective change in the model was the inclusion of a coefficient to permit preferential solute exchange among vasa recta. This result suggests that the three-dimensional ordering of blood vessels and tubules is an essential construct in the concentrating mechanism.

Automatic derivative evaluation in solving boundary value problems: the renal medulla.

Automatic evaluation of derivatives becomes essential when large systems of equations of many variables are to be solved. This paper presents a set of easy-to-use FORTRAN subroutines that perform automatic derivative evaluation. They were used in conjunction with the method of quasilinearization to solve a 13th-order boundary-value problem. This problem has been proposed as a test of numerical methods used to solve models of the renal concentrating mechanism. Quasilinearization gives the same result as has been reported by others with finite difference or multiple shooting methods. The approach described here offers the important potential advantage of being easier to apply to larger problems, which can be anticipated when attempts are made to simulate the three-dimensional structure of the renal medulla.