A mathematical model of the urine concentrating mechanism in the rat renal medulla. II. Functional implications of three-dimensional architecture

Abstract

In a companion study [Layton AT. A mathematical model of the urine concentrating mechanism in the rat renal medulla. I. Formulation and base-case results. Am J Physiol Renal Physiol. (First published November 10, 2010). 10.1152/ajprenal.00203.2010] a region-based mathematical model was formulated for the urine concentrating mechanism in the renal medulla of the rat kidney. In the present study, we investigated model sensitivity to some of the fundamental structural assumptions. An unexpected finding is that the concentrating capability of this region-based model falls short of the capability of models that have radially homogeneous interstitial fluid at each level of only the inner medulla (IM) or of both the outer medulla and IM, but are otherwise analogous to the region-based model. Nonetheless, model results reveal the functional significance of several aspects of tubular segmentation and heterogeneity: 1) the exclusion of ascending thin limbs that reach into the deep IM from the collecting duct clusters in the upper IM promotes urea cycling within the IM; 2) the high urea permeability of the lower IM thin limb segments allows their tubular fluid urea content to equilibrate with the surrounding interstitium; 3) the aquaporin-1-null terminal descending limb segments prevent water entry and maintain the transepithelial NaCl concentration gradient; 4) a higher thick ascending limb Na(+) active transport rate in the inner stripe augments concentrating capability without a corresponding increase in energy expenditure for transport; 5) active Na(+) reabsorption from the collecting duct elevates its tubular fluid urea concentration. Model calculations predict that these aspects of tubular segmentation and heterogeneity promote effective urine concentrating functions.

Fig. 1.

Region and collecting duct (CD) osmolality profiles, as a function of medullary depth, for varying degrees of regionalization, obtained by increasing region boundary Na⁺ and urea permeabilities in selected portions of the medulla. A: base case. B: homogeneous inner medulla (IM) case. C: homogeneous medulla case. D: CD osmolality profiles for 3 cases.

Fig. 2.

IM region urea concentration and CD osmolality obtained by assigning an intracluster position to the long ascending limbs (LALs) associated with loops that reach into the deep IM (denoted LAL_L). Base-case profiles are included for comparison. The “intracluster LAL_L” case yielded lower interstitial urea concentrations and an impaired concentrating effect.

Fig. 3.

Parameter studies for water permeability of LDL3. Shown is CD fluid osmolality as a function of medullary depth. Results indicate that higher LDL3 water permeability reduces concentrating effect.

Fig. 4.

Parameter studies for urea permeabilities of LDL3 and thin ascending limb. Shown are loop urea concentration profiles, obtained for the 5 alternative cases and for the base case, as a function of IM depth. Solid lines, descending limb; dashed lines, ascending limb. Model's urine concentrating capability was substantially impaired in all alternative cases (see Table 1).

Fig. 5.

Osmolalities and concentration profiles of concentric regions, tubules, and vasa recta, obtained for the “pipe mode,” which assumes low loop urea permeabilities. 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. Compared with the base-case model (Fig. 2 in Ref. 19), the pipe mode predicts a low fluid osmolality along the LDL3 segment, but a substantially higher urine osmolality.

Fig. 6.

Water flows in tubules and vasa recta, given per individual tubule or vessel. Results were obtained for the pipe mode. Notation is analogous to that used in Fig. 5. 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. 7.

CD tubular fluid osmolalities for differing thick ascending limb (TAL) Na⁺ active transport rates. “High,” “Average,” and “Low” cases correspond to uniform TAL Na⁺ transport rates of 25.9, 19.95, and 10.5 nmol·cm⁻²·s⁻¹, respectively. High transport rate generates the highest CD fluid osmolality, but at a high energy cost. Both average and low transport rates decrease CD fluid osmolality substantially.

Fig. 8.

Parameter studies for differing CD Na⁺ active transport rates. “High” and “Low” cases correspond to IMCD Na⁺ transport rates 1.5 and 0.5 times of base-case value; “Average” case corresponds to uniform IMCD Na⁺ transport rate of 5.525 nmol·cm⁻²·s⁻¹; “No OM” case corresponds to zero OMCD Na⁺ transport rate. High transport rate generates the highest CD fluid osmolality, but at a low urine flow rate. Both average and low transport rates decrease CD fluid osmolality substantially.

Fig. 9.

Parameter studies for differing CD urea permeability profiles. Case 1, higher urea permeability (1 × 10⁻⁵ cm/s) in OMCD and initial IMCD; case 2, IMCD urea permeability increases abruptly at x = L_OM + 0.7L_IM; case 3, same as case 2 but with initial IMCD urea permeability increased by 10-fold from base case. A: CD tubular fluid osmolality. B and C: CD Na⁺ and urea concentrations. D: CD water flow rate. The alternative CD urea permeability profiles yielded lower urine osmolalities.

Fig. 10.

CD tubular fluid osmolality obtained for simulations of Clc-k1-null and urea transporter UT-A1/A3-null rats. Both cases show substantial impairment in urine concentrating capability.