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

Calcium Dynamics and Water Transport in Salivary Acinar Cells

Saliva is secreted from the acinar cells of the salivary glands, using mechanisms that are similar to other types of water-transporting epithelial cells. Using a combination of theoretical and experimental techniques, over the past 20 years we have continually developed and modified a quantitative model of saliva secretion, and how it is controlled by the dynamics of intracellular calcium. However, over approximately the past 5 years there have been significant developments both in our understanding of the underlying mechanisms and in the way these mechanisms should best be modelled. Here, we review the traditional understanding of how saliva is secreted, and describe how our work has suggested important modifications to this traditional view. We end with a brief description of the most recent data from living animals and discuss how this is now contributing to yet another iteration of model construction and experimental investigation.

Renal oxygenation: From data to insight

Computational models have made a major contribution to the field of physiology. As the complexity of our understanding of biological systems expands, the need for computational methods only increases. But collaboration between experimental physiologists and computational modellers (ie theoretical physiologists) is not easy. One of the major challenges is to break down the barriers created by differences in vocabulary and approach between the two disciplines. In this review, we have two major aims. Firstly, we wish to contribute to the effort to break down these barriers and so encourage more interdisciplinary collaboration. So, we begin with a "primer" on the ways in which computational models can help us understand physiology and pathophysiology. Second, we aim to provide an update of recent efforts in one specific area of physiology, renal oxygenation. This work is shedding new light on the causes and consequences of renal hypoxia. But as importantly, computational modelling is providing direction for experimental physiologists working in the field of renal oxygenation by: (a) generating new hypotheses that can be tested in experimental studies, (b) allowing experiments that are technically unfeasible to be simulated in silico, or variables that cannot be measured experimentally to be estimated, and (c) providing a means by which the quality of experimental data can be assessed. Critically, based on our experience, we strongly believe that experimental and theoretical physiology should not be seen as separate exercises. Rather, they should be integrated to permit an iterative process between modelling and experimentation.

A mathematical model of the rat kidney. II. Antidiuresis

Kidney water conservation requires a hypertonic medullary interstitium, NaCl in the outer medulla and NaCl and urea in the inner medulla, plus a vascular configuration that protects against washout. In this work, a multisolute model of the rat kidney is revisited to examine its capacity to simulate antidiuresis. The first step was to streamline model computation by parallelizing its Jacobian calculation, thus allowing finer medullary spatial resolution and more extensive examination of model parameters. It is found that outer medullary NaCl is modestly increased when transporter density in ascending Henle limbs from juxtamedullary nephrons is scaled to match the greater juxtamedullary solute flow. However, higher NaCl transport produces greater CO₂ generation and, by virtue of countercurrent vascular flows, establishment of high medullary Pco₂. This CO₂ gradient can be mitigated by assuming that a fraction of medullary transport is powered anaerobically. Reducing vascular flows or increasing vessel permeabilities does little to further increase outer medullary solute gradients. In contrast to medullary models of others, vessels in this model have solute reflection coefficients close to zero; increasing these coefficients provides little enhancement of solute profiles but does generate high interstitial pressures, which distort tubule architecture. Increasing medullary urea delivery via entering vasa recta increases inner medullary urea, although not nearly to levels found in rats. In summary, 1) medullary Na⁺ and urea gradients are not captured by the model and 2) the countercurrent architecture that provides antidiuresis also produces exaggerated Pco₂ profiles and is an unappreciated constraint on models of medullary function.

On the role of the epithelium in a model of sodium exchange in renal tubules

In this study we present a mathematical model describing the transport of sodium in a fluid circulating in a counter-current tubular architecture, which constitutes a simplified model of Henle's loop in a kidney nephron. The model explicitly takes into account the epithelial layer at the interface between the tubular lumen and the surrounding interstitium. In a specific range of parameters, we show that explicitly accounting for transport across the apical and basolateral membranes of epithelial cells, instead of assuming a single barrier, affects the axial concentration gradient, an essential determinant of the urinary concentrating capacity. We present the solution related to the stationary system, and we perform numerical simulations to understand the physiological behaviour of the system. We prove that when time grows large, our dynamic model converges towards the stationary system at an exponential rate. In order to prove rigorously this global asymptotic stability result, we study eigen-problems of an auxiliary linear operator and its dual.

Mathematical models for kidney function focusing on clinical interest

We give an overview of mathematical models of renal physiology and anatomy with the clinician in mind. Beyond the past focus on issues of local transport mechanisms along the nephron and the urine concentrating mechanism, recent models have brought insight into difficult problems such as renal ischemia (oxygen and CO₂ diffusion in the medulla) or calcium and potassium homeostasis. They have also provided revealing 3D reconstructions of the full trajectories of families of nephrons and collecting ducts through cortex and medulla. The recent appearance of sophisticated whole-kidney models representing nephrons and their associated renal vasculature promises more realistic simulation of renal pathologies and pharmacological treatments in the foreseeable future.

3D simulation of solutes concentration in urinary concentration mechanism in rat renal medulla

In this work, a mathematical model was developed to simulate the urinary concentration mechanism. A 3-D geometry was derived based on the detail physiological pictures of rat kidney. The approximate region of each tubule was obtained from the volume distribution of structures based on Walter Pfaller's monograph and Layton's region-based model. Mass and momentum balances were applied to solve for the change in solutes concentration and osmolality. The osmolality of short and long descending nephrons at the end of the outer medulla was obtained to be 530 mOsmol/kgH2O and 802 mOsmol/kgH2O, respectively, which were in acceptable agreement with experimental data. The fluid osmolality of the short and long ascending nephrons was also compatible with experimental data. The osmolality of CD fluid at the end of the inner medulla was determined to be 1198 mOsmol/kgH2O which was close the experimental data (1216 ± 118). Finally, the impact of the position of each tubule on the fluid osmolality and solutes concentration were obvious in the results; for example, short descending limb a1, which is the closest tubule to the collecting duct, had the highest urea concentration in all tubules. This reflects the important effect of the 3D modeling on the precise analysis of urinary concentration mechanism.

A model of oxygen transport in the rat renal medulla

The renal medulla is prone to hypoxia. Medullary hypoxia is postulated to be a leading cause of acute kidney injury, so there is considerable interest in predicting the oxygen tension in the medulla. Therefore we have developed a computational model for blood and oxygen transport within a physiologically normal rat renal medulla, using a multilevel modeling approach. For the top-level model we use the theory of porous media and advection-dispersion transport through a realistic three-dimensional representation of the medulla’s gross anatomy to describe blood flow and oxygen transport throughout the renal medulla. For the lower-level models, we employ two-dimensional reaction-diffusion models describing the distribution of oxygen through tissue surrounding the vasculature. Steady-state model predictions at the two levels are satisfied simultaneously, through iteration between the levels. The computational model was validated by simulating eight sets of experimental data regarding renal oxygenation in rats (using 4 sets of control groups and 4 sets of treatment groups, described in 4 independent publications). Predicted medullary tissue oxygen tension or microvascular oxygen tension for control groups and for treatment groups that underwent moderate perturbation in hemodynamic and renal functions is within ±2 SE values observed experimentally. Diffusive shunting between descending and ascending vasa recta is predicted to be only 3% of the oxygen delivered. The validation tests confirm that the computational model is robust and capable of capturing the behavior of renal medullary oxygenation in both normal and early-stage pathological states in the rat.

Accounting for oxygen in the renal cortex: a computational study of factors that predispose the cortex to hypoxia

We develop a pseudo-three-dimensional model of oxygen transport for the renal cortex of the rat, incorporating both the axial and radial geometry of the preglomerular circulation and quantitative information regarding the surface areas and transport from the vasculature and renal corpuscles. The computational model was validated by simulating four sets of published experimental studies of renal oxygenation in rats. Under the control conditions, the predicted cortical tissue oxygen tension ([Formula: see text]) or microvascular oxygen tension (µPo2) were within ±1 SE of the mean value observed experimentally. The predicted [Formula: see text] or µPo2 in response to ischemia-reperfusion injury, acute hemodilution, blockade of nitric oxide synthase, or uncoupling mitochondrial respiration, were within ±2 SE observed experimentally. We performed a sensitivity analysis of the key model parameters to assess their individual or combined impact on the predicted [Formula: see text] and µPo2. The model parameters analyzed were as follows: 1) the major determinants of renal oxygen delivery ([Formula: see text]) (arterial blood Po2, hemoglobin concentration, and renal blood flow); 2) the major determinants of renal oxygen consumption (V̇o2) [glomerular filtration rate (GFR) and the efficiency of oxygen utilization for sodium reabsorption (β)]; and 3) peritubular capillary surface area (PCSA). Reductions in PCSA by 50% were found to profoundly increase the sensitivity of [Formula: see text] and µPo2 to the major the determinants of [Formula: see text] and V̇o2. The increasing likelihood of hypoxia with decreasing PCSA provides a potential explanation for the increased risk of acute kidney injury in some experimental animals and for patients with chronic kidney disease.

Modeling glucose metabolism and lactate production in the kidney

The metabolism of glucose provides most of the ATP required for energy-dependent transport processes. In the inner medulla of the mammalian kidney, limited blood flow and O2 supply yield low oxygen tension; therefore, a substantial fraction of the glucose metabolism in that region is anaerobic. Lactate is considered to be a waste product of anaerobic glycolysis, which yields two lactate molecules for each glucose molecule consumed, thereby likely leading to the production and accumulation of a significant amount of lactate in the inner medulla. To gain insights into the transport and metabolic processes in the kidney, we have developed a detailed mathematical model of the renal medulla of the rat kidney. The model represents the radial organization of the renal tubules and vessels, which centers around the vascular bundles in the outer medulla and around clusters of collecting ducts in the inner medulla. Model simulations yield significant radial gradients in interstitial fluid oxygen tension and glucose and lactate concentrations in the outer medulla and upper inner medulla. In the deep inner medulla, interstitial fluid concentrations become much more homogeneous, as the radial organization of tubules and vessels is not distinguishable. Using this model, we have identified parameters concerning glucose transport and basal metabolism, as well as lactate production via anaerobic glycolysis, that yield predicted blood glucose and lactate concentrations consistent with experimental measurements in the papillary tip. In addition, simulations indicate that the radial organization of the rat kidney may affect lactate buildup in the inner medulla.

A mathematical model of the rat kidney: K+-induced natriuresis

A model of the rat nephron (Weinstein. Am J Physiol Renal Physiol 308: F1098-F1118, 2015) has been extended with addition of medullary vasculature. Blood vessels contain solutes from the nephron model, plus additional species from the model of Atherton et al. (Am J Physiol Renal Fluid Electrolyte Physiol 247: F61-F72, 1984), representing hemoglobin buffering. In contrast to prior models of the urine-concentrating mechanism, reflection coefficients for DVR are near zero. Model unknowns are initial proximal tubule pressures and flows, connecting tubule pressure, and medullary interstitial pressures and concentrations. The model predicts outer medullary (OM) interstitial gradients for Na+, K+, CO2, and [Formula: see text], such that at OM-IM junction, the respective concentrations relative to plasma are 1.2, 3.0, 2.7, and 8.0; within IM, there is high urea and low [Formula: see text], with concentration ratios of 11 and 0.5 near the papillary tip. Quantitative similarities are noted between K+ and urea handling (medullary delivery and permeabilities). The model K+ gradient is physiologic, and the urea gradient is steeper due to restriction of urea permeability to distal collecting duct. Nevertheless, the predicted urea gradient is less than expected, suggesting reconsideration of proposals of an unrecognized reabsorptive urea flux. When plasma K+ is increased from 5.0 to 5.5 mM, Na+ and K+ excretion increase 2.3- and 1.3-fold, respectively. The natriuresis derives from a 3.3% decrease in proximal Na+ reabsorption and occurs despite delivery-driven increases in Na+ reabsorption in distal segments; kaliuresis derives from a 30% increase in connecting tubule Na+ delivery. Thus this model favors the importance of proximal over distal events in K+-induced diuresis.

Sex-specific computational models of the spontaneously hypertensive rat kidneys: factors affecting nitric oxide bioavailability

The goals of this study were to 1) develop a computational model of solute transport and oxygenation in the kidney of the female spontaneously hypertensive rat (SHR), and 2) apply that model to investigate sex differences in nitric oxide (NO) levels in SHR and their effects on medullary oxygenation and oxidative stress. To accomplish these goals, we first measured NO synthase (NOS) 1 and NOS3 protein expression levels in total renal microvessels of male and female SHR. We found that the expression of both NOS1 and NOS3 is higher in the renal vasculature of females compared with males. To predict the implications of that finding on medullary oxygenation and oxidative stress levels, we developed a detailed computational model of the female SHR kidney. The model was based on a published male kidney model and represents solute transport and the biochemical reactions among O₂, NO, and superoxide ([Formula: see text]) in the renal medulla. Model simulations conducted using both male and female SHR kidney models predicted significant radial gradients in interstitial fluid oxygen tension (Po₂) and NO and [Formula: see text] concentration in the outer medulla and upper inner medulla. The models also predicted that increases in endothelial NO-generating capacity, even when limited to specific vascular segments, may substantially raise medullary NO and Po₂ levels. Other potential sex differences in SHR, including [Formula: see text] production rate, are predicted to significantly impact oxidative stress levels, but effects on NO concentration and Po₂ are limited.

Modeling Glucose Metabolism in the Kidney

The mammalian kidney consumes a large amount of energy to support the reabsorptive work it needs to excrete metabolic wastes and to maintain homeostasis. Part of that energy is supplied via the metabolism of glucose. To gain insights into the transport and metabolic processes in the kidney, we have developed a detailed model of the renal medulla of the rat kidney. The model represents water and solute flows, transmural fluxes, and biochemical reactions in the luminal fluid of the nephrons and vessels. In particular, the model simulates the metabolism of oxygen and glucose. Using that model, we have identified parameters concerning glucose transport and basal metabolism that yield predicted blood glucose concentrations that are consistent with experimental measurements. The model predicts substantial axial gradients in blood glucose levels along various medullary structures. Furthermore, the model predicts that in the inner medulla, owing to the relatively limited blood flow and low tissue oxygen tension, anaerobic metabolism of glucose dominates.

Impact of nitric-oxide-mediated vasodilation and oxidative stress on renal medullary oxygenation: a modeling study

The goal of this study was to investigate the effects of nitric oxide (NO)-mediated vasodilation in preventing medullary hypoxia, as well as the likely pathways by which superoxide (O2(-)) conversely enhances medullary hypoxia. To do so, we expanded a previously developed mathematical model of solute transport in the renal medulla that accounts for the reciprocal interactions among oxygen (O2), NO, and O2(-) to include the vasoactive effects of NO on medullary descending vasa recta. The model represents the radial organization of the vessels and tubules, centered around vascular bundles in the outer medulla and collecting ducts in the inner medulla. Model simulations suggest that NO helps to prevent medullary hypoxia both by inducing vasodilation of the descending vasa recta (thus increasing O2 supply) and by reducing the active sodium transport rate (thus reducing O2 consumption). That is, the vasodilative properties of NO significantly contribute to maintaining sufficient medullary oxygenation. The model further predicts that a reduction in tubular transport efficiency (i.e., the ratio of active sodium transport per O2 consumption) is the main factor by which increased O2(-) levels lead to hypoxia, whereas hyperfiltration is not a likely pathway to medullary hypoxia due to oxidative stress. Finally, our results suggest that further increasing the radial separation between vessels and tubules would reduce the diffusion of NO towards descending vasa recta in the inner medulla, thereby diminishing its vasoactive effects therein and reducing O2 delivery to the papillary tip.

Mathematical modeling of urea transport in the kidney

Mathematical modeling techniques have been useful in providing insights into biological systems, including the kidney. This article considers some of the mathematical models that concern urea transport in the kidney. Modeling simulations have been conducted to investigate, in the context of urea cycling and urine concentration, the effects of hypothetical active urea secretion into pars recta. Simulation results suggest that active urea secretion induces a "urea-selective" improvement in urine concentrating ability. Mathematical models have also been built to study the implications of the highly structured organization of tubules and vessels in the renal medulla on urea sequestration and cycling. The goal of this article is to show how physiological problems can be formulated and studied mathematically, and how such models may provide insights into renal functions.

A mathematical model of the rat nephron: glucose transport

Mathematical models of the proximal tubule (PT), loop of Henle (LOH), and distal nephron have been combined to simulate transport by rat renal tubules. The ensemble is composed of 24,000 superficial (SF) nephrons and 12,000 juxtamedullary (JM) nephrons in 5 classes (according to LOH length); all coalesce into 7,200 connecting tubules (CNT). Medullary interstitial solute concentrations are specified. The model equations require that each nephron glomerular filtration rate (GFR) satisfies a tubuloglomerular feedback (TGF) relationship, and each initial hydrostatic pressure yields a common CNT pressure; that common CNT pressure is determined from an overall distal hydraulic resistance to flow. By virtue of the greater GFR for JM nephrons, fluid delivery to SF and JM tubules is comparable. Glucose reabsorption is restricted to the PT, cotransported with one Na in the convoluted tubule (SGLT2), and two Na in the straight tubule (SGLT1). Increasing ambient glucose from 5 to 10 mM increases proximal Na reabsorption and decreases distal delivery. This is mitigated by a TGF-mediated increase in GFR, and may thus be an etiology for TGF-mediated glomerular hyperfiltration. With SGLT2 inhibition by 95%, the model predicts that under normoglycemic conditions about 60% of filtered glucose will still be reabsorbed, so that profound glycosuria is not to be expected. Compared with glucose-driven osmotic diuresis, SGLT2 inhibition provokes greater natriuresis. When hyperglycemia is superimposed on SGLT2 inhibition, the model suggests that natriuresis may be severe, reflecting synergy of a proximal diuretic and osmotic diuresis. In sum, the model captures TGF-mediated diabetic hyperfiltration and predicts glomerular protection with SGLT2 inhibition.

Impacts of nitric oxide and superoxide on renal medullary oxygen transport and urine concentration

The goal of this study was to investigate the reciprocal interactions among oxygen (O2), nitric oxide (NO), and superoxide (O2 (-)) and their effects on medullary oxygenation and urinary output. To accomplish that goal, we developed a detailed mathematical model of solute transport in the renal medulla of the rat kidney. The model represents the radial organization of the renal tubules and vessels, which centers around the vascular bundles in the outer medulla and around clusters of collecting ducts in the inner medulla. Model simulations yield significant radial gradients in interstitial fluid oxygen tension (Po2) and NO and O2 (-) concentration in the OM and upper IM. In the deep inner medulla, interstitial fluid concentrations become much more homogeneous, as the radial organization of tubules and vessels is not distinguishable. The model further predicts that due to the nonlinear interactions among O2, NO, and O2 (-), the effects of NO and O2 (-) on sodium transport, osmolality, and medullary oxygenation cannot be gleaned by considering each solute's effect in isolation. An additional simulation suggests that a sufficiently large reduction in tubular transport efficiency may be the key contributing factor, more so than oxidative stress alone, to hypertension-induced medullary hypoxia. Moreover, model predictions suggest that urine Po2 could serve as a biomarker for medullary hypoxia and a predictor of the risk for hospital-acquired acute kidney injury.

Urine-concentrating mechanism in the inner medulla: function of the thin limbs of the loops of Henle

The ability of mammals to produce urine hyperosmotic to plasma requires the generation of a gradient of increasing osmolality along the medulla from the corticomedullary junction to the papilla tip. Countercurrent multiplication apparently establishes this gradient in the outer medulla, where there is substantial transepithelial reabsorption of NaCl from the water-impermeable thick ascending limbs of the loops of Henle. However, this process does not establish the much steeper osmotic gradient in the inner medulla, where there are no thick ascending limbs of the loops of Henle and the water-impermeable ascending thin limbs lack active transepithelial transport of NaCl or any other solute. The mechanism generating the osmotic gradient in the inner medulla remains an unsolved mystery, although it is generally considered to involve countercurrent flows in the tubules and vessels. A possible role for the three-dimensional interactions between these inner medullary tubules and vessels in the concentrating process is suggested by creation of physiologic models that depict the three-dimensional relationships of tubules and vessels and their solute and water permeabilities in rat kidneys and by creation of mathematical models based on biologic phenomena. The current mathematical model, which incorporates experimentally determined or estimated solute and water flows through clearly defined tubular and interstitial compartments, predicts a urine osmolality in good agreement with that observed in moderately antidiuretic rats. The current model provides substantially better predictions than previous models; however, the current model still fails to predict urine osmolalities of maximally concentrating rats.

Three-dimensional simulation of urine concentrating mechanism in a functional unit of rat outer medulla. I. Model structure and base case results

The urine formation and excretion system have long been of interest for mathematicians and physiologists to elucidate the obscurities within the process happens in renal tissue. In this study, a novel three-dimensional approach is utilized for modeling the urine concentrating mechanism in rat renal outer medulla which is essentially focused on demonstrating the significance of tubule's architecture revealed in anatomic studies and physiological literature. Since nephrons and vasculatures work interdependently through a highly structured arrangement in outer medulla which is dominated by vascular bundles, a detailed functional unit is proposed based on this specific configuration. Furthermore, due to relatively lesser influence of vasa recta on interstitial medullary osmolality and osmotic gradients as well as model structure simplicity, central core assumption is employed. The model equations are based on three spatial dimensional mass, momentum and species transport equations as well as standard expressions for solutes and water transmural transport. Our model can simulate preferential interactions between different tubules and it is shown that such interactions promote solute cycling and subsequently, enhance urine-concentrating capability. The numerical results are well consistent with tissue slice experiments and moreover, our model predicts more corticomedullary osmolality gradient in outer medulla than previous influential 1-D simulations.

Targeted delivery of solutes and oxygen in the renal medulla: role of microvessel architecture

Renal medullary function is characterized by corticopapillary concentration gradients of various molecules. One example is the generally decreasing axial gradient in oxygen tension (Po2). Another example, found in animals in the antidiuretic state, is a generally increasing axial solute gradient, consisting mostly of NaCl and urea. This osmolality gradient, which plays a principal role in the urine concentrating mechanism, is generally considered to involve countercurrent multiplication and countercurrent exchange, although the underlying mechanism is not fully understood. Radial oxygen and solute gradients in the transverse dimension of the medullary parenchyma have been hypothesized to occur, although strong experimental evidence in support of these gradients remains lacking. This review considers anatomic features of the renal medulla that may impact the formation and maintenance of oxygen and solute gradients. A better understanding of medullary architecture is essential for more clearly defining the compartment-to-compartment flows taken by fluid and molecules that are important in producing axial and radial gradients. Preferential interactions between nephron and vascular segments provide clues as to how tubular and interstitial oxygen flows contribute to safeguarding active transport pathways in renal function in health and disease.

Impact of renal medullary three-dimensional architecture on oxygen transport

We have developed a highly detailed mathematical model of solute transport in the renal medulla of the rat kidney to study the impact of the structured organization of nephrons and vessels revealed in anatomic studies. The model represents the arrangement of tubules around a vascular bundle in the outer medulla and around a collecting duct cluster in the upper inner medulla. Model simulations yield marked gradients in intrabundle and interbundle interstitial fluid oxygen tension (PO2), NaCl concentration, and osmolality in the outer medulla, owing to the vigorous active reabsorption of NaCl by the thick ascending limbs. In the inner medulla, where the thin ascending limbs do not mediate significant active NaCl transport, interstitial fluid composition becomes much more homogeneous with respect to NaCl, urea, and osmolality. Nonetheless, a substantial PO2 gradient remains, owing to the relatively high oxygen demand of the inner medullary collecting ducts. Perhaps more importantly, the model predicts that in the absence of the three-dimensional medullary architecture, oxygen delivery to the inner medulla would drastically decrease, with the terminal inner medulla nearly completely deprived of oxygen. Thus model results suggest that the functional role of the three-dimensional medullary architecture may be to preserve oxygen delivery to the papilla. Additionally, a simulation that represents low medullary blood flow suggests that the separation of thick limbs from the vascular bundles substantially increases the risk of the segments to hypoxic injury. When nephrons and vessels are more homogeneously distributed, luminal PO2 in the thick ascending limb of superficial nephrons increases by 66% in the inner stripe. Furthermore, simulations predict that owing to the Bohr effect, the presumed greater acidity of blood in the interbundle regions, where thick ascending limbs are located, relative to that in the vascular bundles, facilitates the delivery of O2 to support the high metabolic requirements of the thick limbs and raises NaCl reabsorption.

Advances in Understanding the Urine-Concentrating Mechanism

The renal medulla produces concentrated urine through the generation of an osmotic gradient that progressively increases from the cortico-medullary boundary to the inner medullary tip. In the outer medulla, the osmolality gradient arises principally from vigorous active transport of NaCl, without accompanying water, from the thick ascending limbs of short- and long-looped nephrons. In the inner medulla, the source of the osmotic gradient has not been identified. Recently, there have been important advances in our understanding of key components of the urine-concentrating mechanism, including (a) better understanding of the regulation of water, urea, and sodium transport proteins; (b) better resolution of the anatomical relationships in the medulla; and (c) improvements in mathematical modeling of the urine-concentrating mechanism. Continued experimental investigation of signaling pathways regulating transepithelial transport, both in normal animals and in knockout mice, and incorporation of the resulting information into mathematical simulations may help to more fully elucidate the mechanism for concentrating urine in the inner medulla.

Structure and function of the thin limbs of the loop of Henle

The thin limbs of the loop of Henle, which comprise the intermediate segment, connect the proximal tubule to the distal tubule and lie entirely within the renal medulla. The descending thin limb consists of at least two or three morphologically and functionally distinct subsegments and participates in transepithelial transport of NaCl, urea, and water. Only one functionally distinct segment is recognized for the ascending thin limb, which carries out transepithelial transport of NaCl and urea in the reabsorptive and/or secretory directions. Membrane transporters involved with passive transcellular Cl, urea, and water fluxes have been characterized for thin limbs; however, these pathways do not account for all transepithelial fluid and solute fluxes that have been measured in vivo. The paracellular pathway has been proposed to play an important role in transepithelial Na and urea fluxes in defined thin-limb subsegments. As the transport pathways become clearer, the overall function of the thin limbs is becoming better understood. Primary and secondary signaling pathways and protein-protein interactions are increasingly recognized as important modulators of thin-limb cell function and cell metabolism. These functions must be investigated under diverse extracellular conditions, particularly for those cells of the deep inner medulla that function in an environment of wide variation in hyperosmolality. Transgenic mouse models of several key water and solute transport proteins have provided significant insights into thin-limb function. An understanding of the overall architecture of the medulla, including juxtapositions of thin limbs with collecting ducts, thick ascending limbs, and vasa recta, is essential for understanding the role of the kidney in maintaining Na and water homeostasis, and for understanding the urine concentrating mechanism.

Hormonal regulation of salt and water excretion: a mathematical model of whole kidney function and pressure natriuresis

We present a lumped-nephron model that explicitly represents the main features of the underlying physiology, incorporating the major hormonal regulatory effects on both tubular and vascular function, and that accurately simulates hormonal regulation of renal salt and water excretion. This is the first model to explicitly couple glomerulovascular and medullary dynamics, and it is much more detailed in structure than existing whole organ models and renal portions of multiorgan models. In contrast to previous medullary models, which have only considered the antidiuretic state, our model is able to regulate water and sodium excretion over a variety of experimental conditions in good agreement with data from experimental studies of the rat. Since the properties of the vasculature and epithelia are explicitly represented, they can be altered to simulate pathophysiological conditions and pharmacological interventions. The model serves as an appropriate starting point for simulations of physiological, pathophysiological, and pharmacological renal conditions and for exploring the relationship between the extrarenal environment and renal excretory function in physiological and pathophysiological contexts.

Impacts of Active Urea Secretion into Pars Recta on Urine Concentration and Urea Excretion Rate

It has been observed experimentally that early distal tubular urea flow exceeds urea delivery by the proximal convoluted tubule to the pars recta and loop of Henle. Moreover, the fractional excretion of urea in the urine may exceed values compatible with the reabsorption known to occur in the proximal convoluted tubule in the cortex. A likely explanation for these observations is that urea may be actively secreted into the pars recta, as proposed in a few studies. However, this hypothesis has yet to be demonstrated experimentally. In this study, we used a mathematical model of the renal medulla of the rat kidney to investigate the impacts of active urea secretion in the intrarenal handling of urea and in the urine concentrating ability. The model represents only the outer and inner medullary zones, with the actions taking place in the cortex incorporated via boundary conditions. Blood flow in the model vasculature is divided into plasma and red blood cell compartments. We compared urea flow rates and other related model variables without and with the hypothetical active urea secretion in the pars recta. The simulation suggests that active urea secretion induces a “urea-selective” improvement in urine concentrating ability by enhancing the efficiency of urea excretion without requiring a higher urine flow rate, and with only modest changes in the excretion of other solutes. These results should encourage experimental studies in order to assess the existence of an active urea secretion in the rodent kidney.

Mathematical modeling of kidney transport

In addition to metabolic waste and toxin excretion, the kidney also plays an indispensable role in regulating the balance of water, electrolytes, nitrogen, and acid-base. In this review, we describe representative mathematical models that have been developed to better understand kidney physiology and pathophysiology, including the regulation of glomerular filtration, the regulation of renal blood flow by means of the tubuloglomerular feedback mechanisms and of the myogenic mechanism, the urine concentrating mechanism, epithelial transport, and regulation of renal oxygen transport. We discuss the extent to which these modeling efforts have expanded our understanding of renal function in both health and disease.

Comparative physiology and architecture associated with the mammalian urine concentrating mechanism: role of inner medullary water and urea transport pathways in the rodent medulla

Comparative studies of renal structure and function have potential to provide insights into the urine-concentrating mechanism of the mammalian kidney. This review focuses on the tubular transport pathways for water and urea that play key roles in fluid and solute movements between various compartments of the rodent renal inner medulla. Information on aquaporin water channel and urea transporter expression has increased our understanding of functional segmentation of medullary thin limbs of Henle's loops, collecting ducts, and vasa recta. A more complete understanding of membrane transporters and medullary architecture has identified new and potentially significant interactions between these structures and the interstitium. These interactions are now being introduced into our concept of how the inner medullary urine-concentrating mechanism works. A variety of regulatory pathways lead directly or indirectly to variable patterns of fluid and solute movements among the interstitial and tissue compartments. Animals with the ability to produce highly concentrated urine, such as desert species, are considered to exemplify tubular structure and function that optimize urine concentration. These species may provide unique insights into the urine-concentrating process.(1)

Tubular fluid flow and distal NaCl delivery mediated by tubuloglomerular feedback in the rat kidney

The glomerular filtration rate in the kidney is controlled, in part, by the tubuloglomerular feedback (TGF) system, which is a negative feedback loop that mediates oscillations in tubular fluid flow and in fluid NaCl concentration of the loop of Henle. In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a short loop of Henle with compliant walls. The proximal tubule and the outer-stripe segment of the descending limb are assumed to be highly water permeable; the thick ascending limb (TAL) is assumed to be water impermeable and have active NaCl transport. A bifurcation analysis of the TGF model equations was performed by computing parameter boundaries, as functions of TGF gain and delay, that separate differing model behaviors. The analysis revealed a complex parameter region that allows a variety of qualitatively different model equations: a regime having one stable, time-independent steady-state solution and regimes having stable oscillatory solutions of different frequencies. A comparison with a previous model, which represents only the TAL explicitly and other segments using phenomenological relations, indicates that explicit representation of the proximal tubule and descending limb of the loop of Henle lowers the stability of the TGF system. Model simulations also suggest that the onset of limit-cycle oscillations results in increases in the time-averaged distal NaCl delivery, whereas distal fluid delivery is not much affected.

Renal pericytes: regulators of medullary blood flow

Regulation of medullary blood flow (MBF) is essential in maintaining normal kidney function. Blood flow to the medulla is supplied by the descending vasa recta (DVR), which arise from the efferent arterioles of juxtamedullary glomeruli. DVR are composed of a continuous endothelium, intercalated with smooth muscle-like cells called pericytes. Pericytes have been shown to alter the diameter of isolated and in situ DVR in response to vasoactive stimuli that are transmitted via a network of autocrine and paracrine signalling pathways. Vasoactive stimuli can be released by neighbouring tubular epithelial, endothelial, red blood cells and neuronal cells in response to changes in NaCl transport and oxygen tension. The experimentally described sensitivity of pericytes to these stimuli strongly suggests their leading role in the phenomenon of MBF autoregulation. Because the debate on autoregulation of MBF fervently continues, we discuss the evidence favouring a physiological role for pericytes in the regulation of MBF and describe their potential role in tubulo-vascular cross-talk in this region of the kidney. Our review also considers current methods used to explore pericyte activity and function in the renal medulla.

Axial compartmentation of descending and ascending thin limbs of Henle's loops

In the inner medulla, radial organization of nephrons and blood vessels around collecting duct (CD) clusters leads to two lateral interstitial regions and preferential intersegmental fluid and solute flows. As the descending (DTLs) and ascending thin limbs (ATLs) pass through these regions, their transepithelial fluid and solute flows are influenced by variable transepithelial solute gradients and structure-to-structure interactions. The goal of this study was to quantify structure-to-structure interactions, so as to better understand compartmentation and flows of transepithelial water, NaCl, and urea and generation of the axial osmotic gradient. To accomplish this, we determined lateral distances of AQP1-positive and AQP1-negative DTLs and ATLs from their nearest CDs, so as to gauge interactions with intercluster and intracluster lateral regions and interactions with interstitial nodal spaces (INSs). DTLs express reduced AQP1 and low transepithelial water permeability along their deepest segments. Deep AQP1-null segments, prebend segments, and ATLs lie equally near to CDs. Prebend segments and ATLs abut CDs and INSs throughout much of their descent and ascent, respectively; however, the distal 30% of ATLs of the longest loops lie distant from CDs as they approach the outer medullary boundary and have minimal interaction with INSs. These relationships occur regardless of loop length. Finally, we show that ascending vasa recta separate intercluster AQP1-positive DTLs from descending vasa recta, thereby minimizing dilution of gradients that drive solute secretion. We hypothesize that DTLs and ATLs enter and exit CD clusters in an orchestrated fashion that is important for generation of the corticopapillary solute gradient by minimizing NaCl and urea loss.

Architecture of vasa recta in the renal inner medulla of the desert rodent Dipodomys merriami: potential impact on the urine concentrating mechanism

We hypothesize that the inner medulla of the kangaroo rat Dipodomys merriami, a desert rodent that concentrates its urine to over 6,000 mosmol/kg H(2)O, provides unique examples of architectural features necessary for production of highly concentrated urine. To investigate this architecture, inner medullary vascular segments in the outer inner medulla were assessed with immunofluorescence and digital reconstructions from tissue sections. Descending vasa recta (DVR) expressing the urea transporter UT-B and the water channel aquaporin 1 lie at the periphery of groups of collecting ducts (CDs) that coalesce in their descent through the inner medulla. Ascending vasa recta (AVR) lie inside and outside groups of CDs. DVR peel away from vascular bundles at a uniform rate as they descend the inner medulla, and feed into networks of AVR that are associated with organized clusters of CDs. These AVR form interstitial nodal spaces, with each space composed of a single CD, two AVR, and one or more ascending thin limbs or prebend segments, an architecture that may lead to solute compartmentation and fluid fluxes essential to the urine concentrating mechanism. Although we have identified several apparent differences, the tubulovascular architecture of the kangaroo rat inner medulla is remarkably similar to that of the Munich Wistar rat at the level of our analyses. More detailed studies are required for identifying interspecies functional differences.

Modeling Transport and Flow Regulatory Mechanisms of the Kidney

The kidney plays an indispensable role in the regulation of whole-organism water balance, electrolyte balance, and acid-base balance, and in the excretion of metabolic wastes and toxins. In this paper, we review representative mathematical models that have been developed to better understand kidney physiology and pathophysiology, including the regulation of glomerular filtration, the regulation of renal blood flow by means of the tubuloglomerular feedback mechanisms and of the myogenic mechanism, the urine concentrating mechanism, and regulation of renal oxygen transport. We discuss how such modeling efforts have significantly expanded our understanding of renal function in both health and disease.

Impact of nitric oxide-mediated vasodilation on outer medullary NaCl transport and oxygenation

The present study aimed to elucidate the reciprocal interactions between oxygen (O(2)), nitric oxide (NO), and superoxide (O(2)(-)) and their effects on vascular and tubular function in the outer medulla. We expanded our region-based model of transport in the rat outer medulla (Edwards A, Layton AT. Am J Physiol Renal Physiol 301: F979-F996, 2011) to incorporate the effects of NO on descending vasa recta (DVR) diameter and blood flow. Our model predicts that the segregation of long DVR in the center of vascular bundles, away from tubular segments, gives rise to large radial NO concentration gradients that in turn result in differential regulation of vasoactivity in short and long DVR. The relative isolation of long DVR shields them from changes in the rate of NaCl reabsorption, and hence from changes in O(2) requirements, by medullary thick ascending limbs (mTALs), thereby preserving O(2) delivery to the inner medulla. The model also predicts that O(2)(-) can sufficiently decrease the bioavailability of NO in the interbundle region to affect the diameter of short DVR, suggesting that the experimentally observed effects of O(2)(-) on medullary blood flow may be at least partly mediated by NO. In addition, our results indicate that the tubulovascular cross talk of NO, that is, the diffusion of NO produced by mTAL epithelia toward adjacent DVR, helps to maintain blood flow and O(2) supply to the interbundle region even under basal conditions. NO also acts to preserve local O(2) availability by inhibiting the rate of active Na(+) transport, thereby reducing the O(2) requirements of mTALs. The dual regulation by NO of oxygen supply and demand is predicted to significantly attenuate the hypoxic effects of angiotensin II.

Signal transduction in a compliant thick ascending limb

In several previous studies, we used a mathematical model of the thick ascending limb (TAL) to investigate nonlinearities in the tubuloglomerular feedback (TGF) loop. That model, which represents the TAL as a rigid tube, predicts that TGF signal transduction by the TAL is a generator of nonlinearities: if a sinusoidal oscillation is added to constant intratubular fluid flow, the time interval required for an element of tubular fluid to traverse the TAL, as a function of time, is oscillatory and periodic but not sinusoidal. As a consequence, NaCl concentration in tubular fluid alongside the macula densa will be nonsinusoidal and thus contain harmonics of the original sinusoidal frequency. We hypothesized that the complexity found in power spectra based on in vivo time series of key TGF variables arises in part from those harmonics and that nonlinearities in TGF-mediated oscillations may result in increased NaCl delivery to the distal nephron. To investigate the possibility that a more realistic model of the TAL would damp the harmonics, we have conducted new studies in a model TAL that has compliant walls and thus a tubular radius that depends on transmural pressure. These studies predict that compliant TAL walls do not damp, but instead intensify, the harmonics. In addition, our results predict that mean TAL flow strongly influences the shape of the NaCl concentration waveform at the macula densa. This is a consequence of the inverse relationship between flow speed and transit time, which produces asymmetry between up- and downslopes of the oscillation, and the nonlinearity of TAL NaCl absorption at low flow rates, which broadens the trough of the oscillation relative to the peak. The dependence of waveform shape on mean TAL flow may be the source of the variable degree of distortion, relative to a sine wave, seen in experimental recordings of TGF-mediated oscillations.

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.

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.

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.

Countercurrent multiplication may not explain the axial osmolality gradient in the outer medulla of the rat kidney

It has become widely accepted that the osmolality gradient along the corticomedullary axis of the mammalian outer medulla is generated and sustained by a process of countercurrent multiplication: active NaCl absorption from thick ascending limbs is coupled with the counterflow configuration of the descending and ascending limbs of the loops of Henle to generate an axial osmolality gradient along the outer medulla. However, aspects of anatomic structure (e.g., the physical separation of the descending limbs of short loops of Henle from contiguous ascending limbs), recent physiologic experiments (e.g., those that suggest that the thin descending limbs of short loops of Henle have a low osmotic water permeability), and mathematical modeling studies (e.g., those that predict that water-permeable descending limbs of short loops are not required for the generation of an axial osmolality gradient) suggest that countercurrent multiplication may be an incomplete, or perhaps even erroneous, explanation. We propose an alternative explanation for the axial osmolality gradient: we regard the thick limbs as NaCl sources for the surrounding interstitium, and we hypothesize that the increasing axial osmolality gradient along the outer medulla is primarily sustained by an increasing ratio, as a function of increasing medullary depth, of NaCl absorption (from thick limbs) to water absorption (from thin descending limbs of long loops of Henle and, in antidiuresis, from collecting ducts). We further hypothesize that ascending vasa recta that are external to vascular bundles will carry, toward the cortex, an absorbate that at each medullary level is hyperosmotic relative to the adjacent interstitium.

Urine concentrating mechanism in the inner medulla of the mammalian kidney: role of three-dimensional architecture

The urine concentrating mechanism in the mammalian renal inner medulla (IM) is not understood, although it is generally considered to involve countercurrent flows in tubules and blood vessels. A possible role for the three-dimensional relationships of these tubules and vessels in the concentrating process is suggested by recent reconstructions from serial sections labelled with antibodies to tubular and vascular proteins and mathematical models based on these studies. The reconstructions revealed that the lower 60% of each descending thin limb (DTL) of Henle's loops lacks water channels (aquaporin-1) and osmotic water permeability and ascending thin limbs (ATLs) begin with a prebend segment of constant length. In the outer zone of the IM (i) clusters of coalescing collecting ducts (CDs) form organizing motif for loops of Henle and vasa recta; (ii) DTLs and descending vasa recta (DVR) are arrayed outside CD clusters, whereas ATLs and ascending vasa recta (AVR) are uniformly distributed inside and outside clusters; (iii) within CD clusters, interstitial nodal spaces are formed by a CD on one side, AVR on two sides, and an ATL on the fourth side. These spaces may function as mixing chambers for urea from CDs and NaCl from ATLs. In the inner zone of the IM, cluster organization disappears and half of Henle's loops have broad lateral bends wrapped around terminal CDs. Mathematical models based on these findings and involving solute mixing in the interstitial spaces can produce urine slightly more concentrated than that of a moderately antidiuretic rat but no higher.

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

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.

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.

Nitric oxide and superoxide transport in a cross section of the rat outer medulla. I. Effects of low medullary oxygen tension

To examine the impact of the complex radial organization of the rat outer medulla (OM) on the distribution of nitric oxide (NO), superoxide (O(2)(-)) and total peroxynitrite (ONOO), we developed a mathematical model that simulates the transport of those species in a cross section of the rat OM. To simulate the preferential interactions among tubules and vessels that arise from their relative radial positions in the OM, we adopted the region-based approach developed by Layton and Layton (Am J Physiol Renal Physiol 289: F1346-F1366, 2005). In that approach, the structural organization of the OM is represented by means of four concentric regions centered on a vascular bundle. The model predicts the concentrations of NO, O(2)(-), and ONOO in the tubular and vascular lumen, epithelial and endothelial cells, red blood cells (RBCs), and interstitial fluid. Model results suggest that the large gradients in Po(2) from the core of the vascular bundle toward its periphery, which stem from the segregation of O(2)-supplying descending vasa recta (DVR) within the vascular bundles, in turn generate steep radial NO and O(2)(-) concentration gradients, since the synthesis of both solutes is O(2) dependent. Without the rate-limiting effects of O(2), NO concentration would be lowest in the vascular bundle core, that is, the region with the highest density of RBCs, which act as a sink for NO. Our results also suggest that, under basal conditions, the difference in NO concentrations between DVR that reach into the inner medulla and those that turn within the OM should lead to differences in vasodilation and preferentially increase blood flow to the inner medulla.

Nitric oxide and superoxide transport in a cross section of the rat outer medulla. II. Reciprocal interactions and tubulovascular cross talk

In a companion study (Edwards A and Layton AT. Am J Physiol Renal Physiol. doi:10.1152/ajprenal.00680.2009), we developed a mathematical model of nitric oxide (NO), superoxide (O(2)(-)), and total peroxynitrite (ONOO) transport in mid-outer stripe and mid-inner stripe cross sections of the rat outer medulla (OM). We examined how the three-dimensional architecture of the rat OM, together with low medullary oxygen tension (Po(2)), affects the distribution of NO, O(2)(-), and ONOO in the rat OM. In the current study, we sought to determine generation rate and permeability values that are compatible with measurements of medullary NO concentrations and to assess the importance of tubulovascular cross talk and NO-O(2)(-) interactions under physiological conditions. Our results suggest that the main determinants of NO concentrations in the rat OM are the rate of vascular and tubular NO synthesis under hypoxic conditions, and the red blood cell (RBC) permeability to NO (P(NO)(RBC)). The lower the P(NO)(RBC), the lower the amount of NO that is scavenged by hemoglobin species, and the higher the extra-erythrocyte NO concentrations. In addition, our results indicate that basal endothelial NO production acts to significantly limit NaCl reabsorption across medullary thick ascending limbs and to sustain medullary perfusion, whereas basal epithelial NO production has a smaller impact on NaCl transport and a negligible effect on vascular tone. Our model also predicts that O(2)(-) consumption by NO significantly reduces medullary O(2)(-) concentrations, but that O(2)(-) , when present at subnanomolar concentrations, has a small impact on medullary NO bioavailability.

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.

Architecture of inner medullary descending and ascending vasa recta: pathways for countercurrent exchange

Pathways and densities of descending vasa recta (DVR) and ascending vasa recta (AVR) in the outer zone of the inner medulla (IM) were evaluated to better understand medullary countercurrent exchange. Nearly all urea transporter B (UT-B)-positive DVR, those vessels exhibiting a continuous endothelium, descend with little or no branching exclusively through the intercluster region. All DVR have a terminal fenestrated (PV-1-positive) segment that partially overlaps with the UT-B-positive segment. This fenestrated segment descends a distance equal to approximately 15% of the length of the connecting UT-B-positive segment before formation of the first branch. The onset of branching is indicative of vessel entry into the intracluster region. The number density of UT-B-positive DVR at 3,000 mum below the OM-IM boundary is approximately 60% lower than the density at 400 mum below the OM-IM boundary, a result of DVR joining to fenestrated interconnecting vessels and an overall decline in UT-B expression. AVR that lie in the intercluster region (designated AVR(2)) lie distant from CDs and ascend to the OM-IM boundary with little or no branching. AVR(2a) represent a subcategory of AVR(2) that abut DVR. The mean DVR length (combined UT-B- and PV-1-positive segments) nearly equals the mean AVR(2a) length, implying a degree of overall equivalence in fluid and solute countercurrent exchange may exist. The AVR(2)/DVR ratio is approximately 2:1, and the AVR(2a)/DVR ratio is approximately 1:1; however, the AVR/DVR ratio determined for the full complement of fenestrated vessels is approximately 4:1. The excess fenestrated vessels include vessels of the intracluster region (designated AVR(1)). Countercurrent exchange between vasa recta occurs predominantly in the intercluster region. This architecture supports previous functional estimates of capillary fluid uptake in the renal IM.

Effects of pH and medullary blood flow on oxygen transport and sodium reabsorption in the rat outer medulla

We used a mathematical model of O(2) transport and the urine concentrating mechanism of the outer medulla of the rat kidney to study the effects of blood pH and medullary blood flow on O(2) availability and Na(+) reabsorption. The model predicts that in vivo paracellular Na(+) fluxes across medullary thick ascending limbs (mTALs) are small relative to transcellular Na(+) fluxes and that paracellular fluxes favor Na(+) reabsorption from the lumen along most of the mTAL segments. In addition, model results suggest that blood pH has a significant impact on O(2) transport and Na(+) reabsorption owing to the Bohr effect, according to which a lower pH reduces the binding affinity of hemoglobin for O(2). Thus our model predicts that the presumed greater acidity of blood in the interbundle regions, where mTALs are located, relative to that in the vascular bundles, facilitates the delivery of O(2) to support the high metabolic requirements of the mTALs and raises the concentrating capability of the outer medulla. Model results also suggest that increases in vascular and tubular flow rates result in disproportional, smaller increases in active O(2) consumption and mTAL active Na(+) transport, despite the higher delivery of O(2) and Na(+). That is, at a sufficiently high medullary O(2) supply, O(2) demand in the outer medulla does not adjust precisely to changes in O(2) delivery.

Functional implications of the three-dimensional architecture of the rat renal inner medulla

A new, region-based mathematical model of the urine concentrating mechanism of the rat renal inner medulla (IM) was used to investigate the significance of transport and structural properties revealed in recent studies that employed immunohistochemical methods combined with three-dimensional computerized reconstruction. The model simulates preferential interactions among tubules and vessels by representing two concentric regions. The inner region, which represents a collecting duct (CD) cluster, contains CDs, some ascending thin limbs (ATLs), and some ascending vasa recta; the outer region, which represents the intercluster region, contains descending thin limbs, descending vasa recta, remaining ATLs, and additional ascending vasa recta. In the upper portion of the IM, the model predicts that interstitial Na(+) and urea concentrations (and osmolality) in the CD clusters differ significantly from those in the intercluster regions: model calculations predict that those CD clusters have higher urea concentrations than the intercluster regions, a finding that is consistent with a concentrating mechanism that depends principally on the mixing of NaCl from ATLs and urea from CDs. In the lower IM, the model predicts that limited or nearly zero water permeability in descending thin limb segments will increase concentrating effectiveness by increasing the rate of solute-free water absorption. The model predicts that high urea permeabilities in the upper portions of ATLs and increased contact areas of longest loop bends with CDs both modestly increase concentrating capability. A surprising finding is that the concentrating capability of this region-based model falls short of the capability of a model IM that has radially homogeneous interstitial fluid at each level but is otherwise analogous to the region-based model.