Maximum urine concentrating capability in a mathematical model of the inner medulla of the rat kidney

Abstract

In a mathematical model of the urine concentrating mechanism of the inner medulla of the rat kidney, a nonlinear optimization technique was used to estimate parameter sets that maximize the urine-to-plasma osmolality ratio (U/P) while maintaining the urine flow rate within a plausible physiologic range. The model, which used a central core formulation, represented loops of Henle turning at all levels of the inner medulla and a composite collecting duct (CD). The parameters varied were: water flow and urea concentration in tubular fluid entering the descending thin limbs and the composite CD at the outer-inner medullary boundary; scaling factors for the number of loops of Henle and CDs as a function of medullary depth; location and increase rate of the urea permeability profile along the CD; and a scaling factor for the maximum rate of NaCl transport from the CD. The optimization algorithm sought to maximize a quantity E that equaled U/P minus a penalty function for insufficient urine flow. Maxima of E were sought by changing parameter values in the direction in parameter space in which E increased. The algorithm attained a maximum E that increased urine osmolality and inner medullary concentrating capability by 37.5% and 80.2%, respectively, above base-case values; the corresponding urine flow rate and the concentrations of NaCl and urea were all within or near reported experimental ranges. Our results predict that urine osmolality is particularly sensitive to three parameters: the urea concentration in tubular fluid entering the CD at the outer-inner medullary boundary, the location and increase rate of the urea permeability profile along the CD, and the rate of decrease of the CD population (and thus of CD surface area) along the cortico-medullary axis.

Figure 1

Schematic diagram of the central core (CC) model formulation showing six representative loops of Henle and a composite collecting duct (CD); in the numerical formulation of the model, 300 loops of Henle were represented to approximate a continuously decreasing number of loops as a function of medullary depth x. LDL2_S, a descending thin limb (DTL) of a loop of Henle that turns within the first millimeter of the inner medulla (IM); LDL2_S does not contain significant aquaporin-1 (Pannabecker et al., 2004) and is therefore assumed to be water-impermeable. LDL2, a segment that makes up the first ~40% of a DTL that reaches beyond the first millimeter of the IM; this segment contains plentiful aquaporin-1. LDL3, a segment that makes up most of the remaining ~60% of the DTL, which does not contain significant aquaporin-1, but may be moderately permeable to water (Chou and Knepper, 1992). The prebend segment (PBE) is considered to have the same transport properties as the ATL (ascending thin limb), which has no significant aquaporin-1 and is assumed to be water-impermeable. The collecting duct (CD) is permeable to water and has varying degrees of urea permeability and NaCl active transport, as a function of x.

Figure 2

CD transepithelial transport properties. A, CD urea permeability as a function of IM depth for three values of λ. B, CD maximum Na⁺ active transport as a function of normalized IM depth (i.e., x/L) for three values of β.

Figure 3

Profiles of tubular fluid osmolality as a function of IM depth. Osmolality profiles for longest loop of Henle (DTL and ATL), collecting duct (CD), and central core (CC): A1, base-case parameters; B1, parameters that maximize E. Osmolality profiles for loops of Henle reaching to 20%, 40%, 70%, and 100% of the IM depth: A2, base-case parameters; B2, parameters that maximize E. Segments with flow direction from x/L = 0 to x/L = 1 (DTL and CD) are indicated with black solid curves whereas segments with flow direction from x/L = 1 to x/L = 0 (ATL and CC) are indicated with dashed curves. Because of overlapping, the total fluid carried by all the ATLs to the OM is dilute relative to the flow in other tubules at the OM-IM boundary.

Figure 4

Profiles of tubular fluid concentrations in longest DTL, longest ATL, CD, and CC as a function of normalized IM depth. Sodium and CD nonreabsorbable solute concentration profiles: A1, base-case parameters; B1, parameters that maximize E. Dashed curve, CD nonreabsorbable solute (NR) concentration (NR concentration is 0 in loops of Henle and in CC). Urea concentration profiles: A2, base-case parameters; B2, parameters that maximize E.

Figure 5

Profiles of tubular flow rate as a function of normalized IM depth. Aggregate fluid flow rate (per nephron, based on 38,000 nephrons/kidney) in DTLs and ATLs, and fluid flow rate (per nephron) in CD and CC: A1, base-case parameters; B1, parameters that maximize E. CD solute flow rates (per nephron): A2, base-case parameters; B2, parameters that maximize E.