A mathematical model of the urine concentrating mechanism in the rat renal medulla. I. Formulation and base-case results

Abstract

A new, region-based mathematical model of the urine concentrating mechanism of the rat renal medulla was used to investigate the significance of transport and structural properties revealed in anatomic studies. The model simulates preferential interactions among tubules and vessels by representing concentric regions that are centered on a vascular bundle in the outer medulla (OM) and on a collecting duct cluster in the inner medulla (IM). Particularly noteworthy features of this model include highly urea-permeable and water-impermeable segments of the long descending limbs and highly urea-permeable ascending thin limbs. Indeed, this is the first detailed mathematical model of the rat urine concentrating mechanism that represents high long-loop urea permeabilities and that produces a substantial axial osmolality gradient in the IM. That axial osmolality gradient is attributable to the increasing urea concentration gradient. The model equations, which are based on conservation of solutes and water and on standard expressions for transmural transport, were solved to steady state. Model simulations predict that the interstitial NaCl and urea concentrations in adjoining regions differ substantially in the OM but not in the IM. In the OM, active NaCl transport from thick ascending limbs, at rates inferred from the physiological literature, resulted in a concentrating effect such that the intratubular fluid osmolality of the collecting duct increases ~2.5 times along the OM. As a result of the separation of urea from NaCl and the subsequent mixing of that urea and NaCl in the interstitium and vasculature of the IM, collecting duct fluid osmolality further increases by a factor of ~1.55 along the IM.

Fig. 1.

Schematic diagram of a cross section through the outer stripe, inner stripe, upper inner medulla (IM), mid-IM, and deep IM, showing concentric regions and relative positions of tubules and vessels. Decimal numbers indicate relative interaction weightings with regions. R1, R2, R3, and R4, concentric regions in the OM; R5 and R6, concentric regions in the IM; SDL, descending limbs of short loops of Henle; SAL, ascending limbs of long loops of Henle; LDL, descending limb of long loop of Henle; LAL, ascending limb of long loop of Henle. Subscripts “S,” “M,” and “L” associated with a LDL or LAL denote limbs that turn with the first millimeter of the IM (S), within the mid-IM (M), or reach into the deep IM (L). CD, collecting duct; SDV, short descending vasa recta; SAV3 and SAV4, 2 populations of short ascending vasa recta; LDV, long descending vas rectum; LAV1, LAV2, LAV5, and LAV6: populations of long ascending vasa recta.

Fig. 2.

Osmolalities and concentration profiles of concentric regions, tubules, and vasa recta. The ordinate is identified at the top of each column: column A, osmolality; column B, Na⁺ or NR concentration; column C, urea concentration. The topmost row, indicated by 0, contains profiles for the interstitia of the regions. Row 1 contains profiles for the vasa recta; row 2, loops of Henle; and row 3, CD. Note variation, among panels, in ordinate scalings.

Fig. 3.

Water and solute flows in regions, tubules, and vasa recta, given per individual tubule or vessel. Notation is analogous to that used in Fig. 2; however, the ordinates, given above each column, are water flow (column A); Na⁺ flow (column B); and urea flow (column C). Negative flows in ascending limbs and ascending vasa recta are directed toward the cortex; flows in descending structures are directed toward the papillary tip. No axial flow is assumed in the regions.

Fig. 4.

Water and solute flows in IM vasa recta, given per nephron. Notation is analogous to that used in Fig. 3.

Fig. 5.

Model sensitivity results in which CD boundary water flow rate was varied by ±10% and ±20%. A: CD tubular fluid osmolality. B: CD water flow rate. C: CD urea flow rate. Results suggest that a higher CD boundary water flow rate imposes a larger load on the concentrating mechanism of the model renal medulla and reduces its concentrating effectiveness.

Fig. 6.

Model sensitivity results in which fractional urea reabsorption from the distal tubules was varied from 0.1 to 0.3 (base-case value, 0.2). Model results indicate that the effectiveness of the IM concentrating mechanism depends on the delivery of sufficient urea to the IM.