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

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.     

Urine concentrating mechanism: impact of vascular and tubular architecture and a proposed descending limb urea-Na+ cotransporter

We extended a region-based mathematical model of the renal medulla of the rat kidney, previously developed by us, to represent new anatomic findings on the vascular architecture in the rat inner medulla (IM). In the outer medulla (OM), tubules and vessels are organized around tightly packed vascular bundles; in the IM, the organization is centered around collecting duct clusters. In particular, the model represents the separation of descending vasa recta from the descending limbs of loops of Henle, and the model represents a papillary segment of the descending thin limb that is water impermeable and highly urea permeable. Model results suggest that, despite the compartmentalization of IM blood flow, IM interstitial fluid composition is substantially more homogeneous compared with OM. We used the model to study medullary blood flow in antidiuresis and the effects of vascular countercurrent exchange. We also hypothesize that the terminal aquaporin-1 null segment of the long descending thin limbs may express a urea-Na(+) or urea-Cl(-) cotransporter. As urea diffuses from the urea-rich papillary interstitium into the descending thin limb luminal fluid, NaCl is secreted via the cotransporter against its concentration gradient. That NaCl is then reabsorbed near the loop bend, raising the interstitial fluid osmolality and promoting water reabsorption from the IM collecting ducts. Indeed, the model predicts that the presence of the urea-Na(+) or urea- Cl(-) cotransporter facilitates the cycling of NaCl within the IM and yields a loop-bend fluid composition consistent with experimental data.     

Role of UTB Urea Transporters in the Urine Concentrating Mechanism of the Rat Kidney

A mathematical model of the renal medulla of the rat kidney was used to investigate urine concentrating mechanism function in animals lacking the UTB urea transporter. The UTB transporter is believed to mediate countercurrent urea exchange between descending vasa recta (DVR) and ascending vasa recta (AVR) by facilitating urea transport across DVR endothelia. The model represents the outer medulla (OM) and inner medulla (IM), with the actions of the cortex incorporated via boundary conditions. Blood flow in the model vasculature is divided into plasma and red blood cell compartments. In the base-case model configuration tubular dimensions and transport parameters are based on, or estimated from, experimental measurements or immunohistochemical evidence in wild-type rats. The base-case model configuration generated an osmolality gradient along the cortico-medullary axis that is consistent with measurements from rats in a moderately antidiuretic state. When expression of UTB was eliminated in the model, model results indicated that, relative to wild-type, the OM cortico-medullary osmolality gradient and the net urea flow through the OM were little affected by absence of UTB transporter. However, because urea transfer from AVR to DVR was much reduced, urea trapping by countercurrent exchange was significantly compromised. Consequently, urine urea concentration and osmolality were decreased by 12% and 8.9% from base case, respectively, with most of the reduction attributable to the impaired IM concentrating mechanism. These results indicate that the in vivo urine concentrating defect in knockout mouse, reported by Yang et al. (J Biol Chem 277(12), 10633–10637, 2002), is not attributable to an OM concentrating mechanism defect, but that reduced urea trapping by long vasa recta plays a significant role in compromising the concentrating mechanism of the IM. Moreover, model results are in general agreement with the explanation of knockout renal function proposed by Yang et al.     

Maximum Urine Concentrating Capability in a Mathematical Model of the Inner Medulla of the Rat Kidney

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.     

Concentrating Engines and the Kidney: II. Multisolute Central Core Systems

The analysis of the central core model of the renal medulla is extended to multisolute systems. It is shown that total solute concentration obeys the same differential equations for core and ascending limb as in a single solute system. Equations are derived for the concentration of individual solutes. Application of these equations to a two solute system shows that a central core system can concentrate with all transport being down a concentration gradient. This analysis applied to the renal medulla shows that mixing of urea from the collecting duct (CD) and salt from the loop of Henle in the central core of the inner medulla contributes to the concentration of urine during antidiuresis. It also sets criteria for completely passive function of the loop in the inner medulla, but whether these are satisfied cannot be determined from present experimental data.     

Polyol determination along the rat nephron

The polyols sorbitol and inositol were determined in single freshly microdissected tubule segments of rat kidney. Twenty different structures were separated from six different kidney zones reaching from cortex to papillary tip. Picomol amounts of sorbitol and inositol were quantitated by use of an enzymatic bioluminescence procedure. Experimental conditions (700 mosmol/kg, 4 degrees C) were chosen to assure constant polyol concentrations over 3 h dissection period. Sorbitol exhibited a concentration gradient in the collecting duct system from the outer/inner medullary border (3.9 +/- 0.5 pmol/mm) to the papillary tip (78.8 +/- 6.9 pmol/mm). In the same region descending and ascending limbs of Henle's loop contained 1.5 +/- 0.5 to 5.3 +/- 1.6 pmol/mm and 2.5 +/- 0.8 to 8.35 +/- 1.5 pmol/mm, respectively. In contrast, all outer medullary and cortical structures had lower sorbitol concentrations. Inositol amounts increased continuously in the collecting duct from cortex (5.3 +/- 0.5 pmol/mm) to inner medulla (30.7 +/- 3.8 pmol/mm). This polyol was also found in thick ascending limb of Henle's loop (6.2 +/- 1.1 pmol/mm in cortex to 11.2 +/- 1.4 pmol/mm in outer medulla) and in proximal tubules (5.6 +/- 1.2 pmol/mm in S1 and 4.5 +/- 1.5 pmol/mm in S3). When related to cellular volume measured by planimetry, intracellular sorbitol concentration was calculated to be 51 mmol/l in papillary collecting duct and inositol 28 mmol/l in outer medullary thick ascending limb cells. These data confirm the role of sorbitol in the renal concentrating process in papilla. Inositol seems to have additional function in thick ascending limb of Henle's loop and the proximal tubule.     

Production of urea from arginine in pars recta and collecting duct of the rat kidney

Urea production from arginine was studied in vitro in the kidney of normal rats in tubule suspensions of the four different renal zones (cortex, outer and inner stripe of outer medulla, and inner medulla), and in individual microdissected nephron segments. Tissue was incubated with L-[guanido-14C]-arginine to measure cellular arginase activity. Addition of urease to the incubate freed 14CO2 from the 14C-urea formed by arginase and released from the cells. CO2 was trapped in KOH and counted. These experiments revealed that significant amounts of urea are produced in the outer stripe and in the inner medulla. This intrarenal urea generation takes place mainly in the proximal straight tubule and in the collecting duct, with increasing activity in these two structures from superficial to deep regions of the kidney. Urea is known to play a critical role in the urinary concentrating process. The fact that some urea can be produced in the mammalian kidney, and that the two structures showing this capacity are straight portions of the renal tubular system descending along the corticopapillary axis suggest that this urea production might play a role in the formation and/or maintenance of the medullary urea concentration gradient.     

Modulation of outer medullary NaCl transport and oxygenation by nitric oxide and superoxide

We expanded our region-based model of water and solute exchanges in the rat outer medulla to incorporate the transport of nitric oxide (NO) and superoxide (O(2)(-)) and to examine the impact of NO-O(2)(-) interactions on medullary thick ascending limb (mTAL) NaCl reabsorption and oxygen (O(2)) consumption, under both physiological and pathological conditions. Our results suggest that NaCl transport and the concentrating capacity of the outer medulla are substantially modulated by basal levels of NO and O(2)(-). Moreover, the effect of each solute on NaCl reabsorption cannot be considered in isolation, given the feedback loops resulting from three-way interactions between O(2), NO, and O(2)(-). Notwithstanding vasoactive effects, our model predicts that in the absence of O(2)(-)-mediated stimulation of NaCl active transport, the outer medullary concentrating capacity (evaluated as the collecting duct fluid osmolality at the outer-inner medullary junction) would be ～40% lower. Conversely, without NO-induced inhibition of NaCl active transport, the outer medullary concentrating capacity would increase by ～70%, but only if that anaerobic metabolism can provide up to half the maximal energy requirements of the outer medulla. The model suggests that in addition to scavenging NO, O(2)(-) modulates NO levels indirectly via its stimulation of mTAL metabolism, leading to reduction of O(2) as a substrate for NO. When O(2)(-) levels are raised 10-fold, as in hypertensive animals, mTAL NaCl reabsorption is significantly enhanced, even as the inefficient use of O(2) exacerbates hypoxia in the outer medulla. Conversely, an increase in tubular and vascular flows is predicted to substantially reduce mTAL NaCl reabsorption. In conclusion, our model suggests that the complex interactions between NO, O(2)(-), and O(2) significantly impact the O(2) balance and NaCl reabsorption in the outer medulla.     

Molecular Mechanisms of Urea Transport

Physiologic data provided evidence for specific urea transporter proteins in red blood cells and kidney inner medulla. During the past decade, molecular approaches resulted in the cloning of several urea transporter cDNA isoforms derived from two gene families: UT-A and UT-B. Polyclonal antibodies were generated to the cloned urea transporter proteins, and their use in integrative animal studies resulted in several novel findings, including: (1) UT-B is the Kidd blood group antigen; (2) UT-B is also expressed in many non-renal tissues and endothelial cells; (3) vasopressin increases UT-A1 phosphorylation in rat inner medullary collecting duct; (4) the surprising finding that UT-A1 protein abundance and urea transport are increased in the inner medulla during conditions in which urine concentrating ability is reduced; and (5) UT-A protein abundance is increased in uremia in both liver and heart. This review will summarize the knowledge gained from studying molecular mechanisms of urea transport and from integrative studies into urea transporter protein regulation.     

Basolateral glucose transport in distal segments of the dog nephron

The transport of glucose by canine thick ascending limbs (TAL) and inner medullary collecting ducts (IMCD) was studied using tubule suspensions and membrane vesicles. The uptake of D-[14C(U)]glucose by a suspension of intact TAL tubules was reduced largely by phloretin (Pt), moderately by phlorizin (Pz), and completely suppressed by a combination of both agents. A selective effect of Pz on the transport of [14C]alpha-methyl-D-glucoside, but not on 2-[3H]deoxyglucose, was also observed in TAL tubules. In contrast, glucose transport was unaffected by Pz but entirely suppressed by Pt alone in IMCD tubules. The metabolism of glucose was largely suppressed by Pt but unaffected by Pz in both types of tubules. Membrane vesicles were prepared from the red medulla and the white papilla or from TAL and IMCD tubules isolated from these tissues. Vesicle preparations from both tissues demonstrated a predominant carrier-mediated, sodium-independent, Pt- and cytochalasin B-sensitive glucose transport. Following purification of basolateral membrane on a Percoll gradient, the sodium-insensitive D-[14C(U)]glucose transport activity copurified with the activity of the basolateral marker Na(+)-K+ ATPase in both tissues. However, a small sodium-dependent and Pz-sensitive component of glucose transport was found in membrane vesicles prepared from the red medulla or from thick ascending limb tubules but not from the papilla nor collecting duct tubules. The kinetic analysis of the major sodium-independent processes showed that the affinity of the transporter for glucose was greater in collecting ducts (Km = 2.3 mM) than in thick ascending limbs (Km = 4.9 mM). We conclude that glucose gains access into the cells largely through a basolateral facilitated diffusion process in both segments. However a small sodium-glucose cotransport is also detected in membranes of TAL tubules. The transport of glucose presents an axial differentiation in the affinity of glucose transporters in the renal medulla, ensuring an adequate supply of glucose to the glycolytic inner medullary structures.     

The theory of urine formation in water diuresis with implications for antidiuresis

We investigate a model of the renal medulla in which active NaCl transport is restricted to the thick ascending limb of Henle's loop. The model contains a vas rectum, a loop of Henle, salt, and water. The model generates interstitial osmolality curves consonant with the known functioning of the kidney in water diuresis. Using data from the white rat and the curves generated by the model, one can predict the permeability of the thin limb of Henle's loop to NaCl and the percentage of total renal blood flow entering the inner medulla. In this model interstitial osmolality at the papilla can be about twice plasma osmolality, so that NaCl transport restricted to the outer medulla can contribute significantly to the work required in producing a hypertonic urine. However, the interstitial osmolality monotonically decreases proceeding from the junction of the outer and inner medulla to the papilla, and the maximum interstitial osmolality in the outer medulla is greater than the maximum interstitial osmolality in the inner medulla. Thus we infer that a source of active transport located in the inner medulla is needed to explain the high osmolalities observed in hydropenia. A sketch of an alternative model, a “lineal multiplication mechanism”, for the renal concentrating process is presented in which active transport in the inner medulla is restricted to active salt transport by the collecting duct. The lineal multiplication mechanism makes no use of counter-current multipliers in the inner medulla. The research of this author was supported in part by NIH Grant AM06864-03 and a Career Scientist Award from the Health Research Council of New York City, Contr. No. 1391. The research of this author was supported in part by the Office of Naval Research, U.S. Navy under Contr. N(onr) 595(17). The research of this author was supported in part by Grant NSF GP-2067 from the National Science Foundation and was performed at the University of Maryland.     

Renal countercurrent system: Role of collecting duct convergence and pelvic urea predicted from a mathematical model

A differential equation model of the renal countercurrent system has been developed and physiological data from nephron segments were incorporated together with recently suggested urea recycling from renal pelvis to inner medulla and, particularly, an exponential reduction in the number of collecting tubules towards the renal papilla. The role of these features for the countercurrent concentrating mechanism has been studied by simulation runs. The computations, using the multiple shooting method, provide predictions about concentration profiles for salt and urea in tubes (nephron segments) and in the central core along the entire medullary countercurrent system. The results indicate that this model, without active salt or urea transport in the inner medulla, yields concentration gradients along the medullary axis compatible with those measured in the tissue.     

An inverse algorithm for a mathematical model of an avian urine concentrating mechanism

A nonlinear optimization technique, in conjunction with a single-nephron, single-solute mathematical model of the quail urine concentrating mechanism, was used to estimate parameter sets that optimize a measure of concentrating mechanism efficiency, viz., the ratio of the free-water absorption rate to the total NaCl active transport rate. The optimization algorithm, which is independent of the numerical method used to solve the model equations, runs in a few minutes on a 1000 MHz desktop computer. The parameters varied were: tubular permeabilities to water and solute; maximum active solute transport rates of the ascending limb of Henle and the collecting duct (CD); length of the prebend enlargement (PBE) of the descending limb; fractional solute delivery to the CD; solute concentration of tubular fluid entering the CD at the cortico-medullary boundary; and rate of exponential CD population decrease along the medullary cone. Using a base-case parameter set and parameter bounds suggested by physiologic experiments, the optimization algorithm identified a maximum-efficiency set of parameter values that increased efficiency by 40% above base-case efficiency; a minimum-efficiency set reduced efficiency by about 41%. When maximum-efficiency parameter values were computed as medullary length varied over the physiologic range, the PBE was found to make up 88% of a short medullary cone but only 8% of a long medullary cone.     

Monitoring urea transport in rat kidney in vivo using hyperpolarized ¹³C magnetic resonance imaging

Urea functions as a key osmolyte in the urinary concentrating mechanism of the inner medulla. The urea transporter UT-A1 is upregulated by antidiuretic hormone, facilitating faster equilibration of urea between the lumen and interstitium of the inner medullary collecting duct, resulting in the formation of more highly concentrated urine. New methods in dynamic nuclear polarization, providing ～50,000-fold enhancement of nuclear magnetic resonance signals in the liquid state, offer a novel means to monitor this process in vivo using magnetic resonance imaging. In this study, we detected significant signal differences in the rat kidney between acute diuretic and antidiuretic states, using dynamic (13)C magnetic resonance imaging following a bolus infusion of hyperpolarized [(13)C]urea. More rapid medullary enhancement was observed under antidiuresis, consistent with known upregulation of UT-A1.     

A functional model of the rat kidney

Summary A mathematical model of the nephron was developed by writing a set of material balance equations for the flow of urea, salt and water along the length of the nephron. The geometric proportions have been elaborated in a foregoing study and are taken here as a basis, in particular the model configuration of the collecting duct system. The medullary interstitial solute concentration profiles are taken to increase linearly in outer and inner zone. The several transepithelial fluxes are driven by diffusion, osmosis, solvent drag and active transport. The development of osmotic gradient in the inner medulla is taken here to be caused by active secretion of salt into the descending limb of Henle's loop. The parameters in the flux equations for all parts of the nephron and the concentration values at the end of each tubular section are determined by collecting and averaging the values given in literature and by extrapolating the measurement data.The simulation of the model equations with these averaged parameters resulted in concentration profiles which at the ends of the several tubular sections were consistent with the values observed in experimental investigations.This work was supported by the Deutsche Forschungsgemeinschaft.     

Isolated interstitial nodal spaces may facilitate preferential solute and fluid mixing in the rat renal inner medulla

Recent anatomic findings indicate that in the upper inner medulla of the rodent kidney, tubules, and vessels are organized around clusters of collecting ducts (CDs). Within CD clusters, CDs and some of the ascending vasa recta (AVR) and ascending thin limbs (ATLs), when viewed in transverse sections, form interstitial nodal spaces, which are arrayed at structured intervals throughout the inner medulla. These spaces, or microdomains, are bordered on one side by a single CD, on the opposite side by one or more ATLs, and on the other two sides by AVR. To study the interactions among these CDs, ATLs, and AVR, we have developed a mathematical compartment model, which simulates steady-state solute exchange through the microdomain at a given inner medullary level. Fluid in all compartments contains Na(+), Cl(-), urea and, in the microdomain, negative fixed charges that represent macromolecules (e.g., hyaluronan) balanced by Na(+). Fluid entry into AVR is assumed to be driven by hydraulic and oncotic pressures. Model results suggest that the isolated microdomains facilitate solute and fluid mixing among the CDs, ATLs, and AVR, promote water withdrawal from CDs, and consequently may play an important role in generating the inner medullary osmotic gradient.     

Effect of mineralocorticoid deficiency on ion and urea transporters and aquaporin water channels in the rat

Mineralocorticoid deficiency is associated with impaired urinary concentration and dilution. The present investigation was undertaken to determine the effects of selective mineralocorticoid deficiency on renal sodium and urea transporters and aquaporin water channels and whether these perturbations can be reversed by maintenance of extracellular fluid volume. Mineralocorticoid deficiency was induced by bilateral adrenalectomies with glucocorticoid replacement. Mineralocorticoid deficient rats receiving plain drinking water (MDW) were compared with mineralocorticoid deficient rats receiving saline-drinking water (MDS) in order to maintain extracellular fluid volume, and with controls (CTL). In MDW rats, there was a significant decrease in renal outer medulla Na-K-2Cl co-transporter and outer medulla Na-K-ATPase as well as an increase in inner medulla aquaporins 2 and 3. There were no significant changes in aquaporin-1, aquaporin-4, or urea transporters. These alterations were reversed with maintenance of extracellular fluid volume in MDS rats. Our findings indicate that mineralocorticoid deficiency in the rat is associated with alterations in factors involved in the countercurrent concentrating mechanism (Na-K-2Cl, Na-K-ATPase) and osmotic water equilibration in the collecting duct (AQP2, AQP3). Maintenance of sodium balance and extracellular fluid volume is associated with normalization of these perturbations.