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

How the kidney regulates magnesium: a modelling study

The kidneys are crucial for maintaining Mg2+ homeostasis. Along the proximal tubule and thick ascending limb, Mg2+ is reabsorbed paracellularly, while along the distal convoluted tubule (DCT), Mg2+ is reabsorbed transcellularly via transient receptor potential melastatin 6 (TRPM6). TRPM6 and other renal transporter expressions are regulated by sex hormones. To investigate renal Mg2 handling, we have developed sex-specific computational models of electrolyte transport along rat superficial nephron. Model simulations indicated that along the proximal tubule and thick ascending limb, Mg2+ and Na+ transport occur parallelly, but they are dissociated along the DCT. In addition, our models predicted higher paracellular Mg2+ permeability in females to attain similar cortical thick ascending limb fractional Mg2+ reabsorption in both sexes. Furthermore, DCT fractional Mg2+ reabsorption is higher in females than in males, allowing females to better fine-tune Mg2+ excretion. We validated our models by simulating the administration of three classes of diuretics. The model predicted significantly increased, marginally increased and significantly decreased Mg2+ excretions for loop, thiazide and K-sparing diuretics, respectively, aligning with experimental findings. The models can be used to conduct in silico studies on kidney adaptations to Mg2+ homeostasis alterations during conditions such as pregnancy, diabetes and chronic kidney disease.

Effects of the environment on the evolution of the vertebrate urinary tract

Evolution of the vertebrate urinary system occurs in response to numerous selective pressures, which have been incompletely characterized. Developing research into urinary evolution led to the occurrence of clinical applications and insights in paediatric urology, reproductive medicine, urolithiasis and other domains. Each nephron segment and urinary organ has functions that can be contextualized within an evolutionary framework. For example, the structure and function of the glomerulus and proximal tubule are highly conserved, enabling blood cells and proteins to be retained, and facilitating the elimination of oceanic Ca+ and Mg+. Urea emerged as an osmotic mediator during evolution, as cells of large organisms required increased precision in the internal regulation of salinity and solutes. As the first vertebrates moved from water to land, acid-base regulation was shifted from gills to skin and kidneys in amphibians. In reptiles and birds, solute regulation no longer occurred through the skin but through nasal salt glands and post-renally, within the cloaca and the rectum. In placental mammals, nasal salt glands are absent and the rectum and urinary tracts became separate, which limited post-renal urine concentration and led to the necessity of a kidney capable of high urine concentration. Considering the evolutionary and environmental selective pressures that have contributed to renal evolution can help to gain an increased understanding of renal physiology.

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.

Parameterization of Microsomal and Cytosolic Scaling Factors: Methodological and Biological Considerations for Scalar Derivation and Validation

Mathematical models that can predict the kinetics of compounds have been increasingly adopted for drug development and risk assessment. Data for these models may be generated from in vitro experimental systems containing enzymes contributing to metabolic clearance, such as subcellular tissue fractions including microsomes and cytosol. Extrapolation from these systems is facilitated by common scaling factors, known as microsomal protein per gram (MPPG) and cytosolic protein per gram (CPPG). Historically, parameterization of MPPG and CPPG has employed the use of recovery factors, commonly benchmarked to cytochromes P450 which work well in some contexts, but could be problematic for other enzymes. Here, we propose absolute quantification of protein content and supplementary assays to evaluate microsomal/cytosolic purity that should be employed. Examples include calculation of microsomal latency by mannose-6-phosphatase activity and immunoblotting of subcellular fractions with fraction-specific markers. Further considerations include tissue source, as disease states can affect enzyme expression and activity, and the methodology used for scalar parameterization. Regional- and organ-specific expression of enzymes, in addition to differences in organ physiology, is another important consideration. Because most efforts have focused on the liver that is, for the most part, homogeneous, derived scalars may not capture the heterogeneity of other major tissues contributing to xenobiotic metabolism including the kidneys and small intestine. Better understanding of these scalars, and how to appropriately derive them from extrahepatic tissues can provide support to the inferences made with physiologically based pharmacokinetic modeling, increase its accuracy in characterizing in vivo drug pharmacokinetics, and improve confidence in go-no-go decisions for clinical trials.

Sex differences in solute transport along the nephrons: effects of Na+ transport inhibition

Each day, ~1.7 kg of NaCl and 180 liters of water are reabsorbed by nephron segments in humans, with urinary excretion fine tuned to meet homeostatic requirements. These tasks are coordinated by a spectrum of renal Na⁺ transporters and channels. The goal of the present study was to investigate the extent to which inhibitors of transepithelial Na⁺ transport (T_Na) along the nephron alter urinary solute excretion and how those effects may vary between male and female subjects. To accomplish that goal, we developed sex-specific multinephron models that represent detailed transcellular and paracellular transport processes along the nephrons of male and female rat kidneys. We simulated inhibition of Na⁺/H⁺ exchanger 3 (NHE3), bumetanide-sensitive Na⁺-K⁺-2Cl^- cotransporter (NKCC2), Na⁺-Cl^- cotransporter (NCC), and amiloride-sensitive epithelial Na⁺ channel (ENaC). NHE3 inhibition simulations predicted a substantially reduced proximal tubule T_Na, and NKCC2 inhibition substantially reduced thick ascending limb T_Na. Both gave rise to diuresis, natriuresis, and kaliuresis, with those effects stronger in female rats. While NCC inhibition was predicted to have only minor impact on renal T_Na, it nonetheless had a notable effect of enhancing excretion of Na⁺, K⁺, and Cl^-, particularly in female rats. Inhibition of ENaC was predicted to have opposite effects on the excretion of Na⁺ (increased) and K⁺ (decreased) and to have only a minor impact on whole kidney T_Na. Unlike inhibition of other transporters, ENaC inhibition induced stronger natriuresis and diuresis in male rats than female rats. Overall, model predictions agreed well with measured changes in Na⁺ and K⁺ excretion in response to diuretics and Na⁺ transporter mutations.

Solute and water transport along an inner medullary collecting duct undergoing peristaltic contractions

The mechanism by which solutes accumulate in the inner medulla of the mammalian kidney has remained incompletely understood. That persistent mystery has led to hypotheses based on the peristaltic contractions of the pelvic wall smooth muscles. It has been demonstrated the peristaltic contractions propel fluid down the collecting duct in boluses. In antidiuresis, boluses are sufficiently short that collecting ducts may be collapsed most of the time. In this study, we investigated the mechanism by which about half of the bolus volume is reabsorbed into the collecting duct cells despite the short contact time. To accomplish this, we developed a dynamic mathematical model of solute and water transport along a collecting duct of a rat papilla undergoing peristaltic contractions. The model predicts that, given preexisting axial concentration gradients along the loops of Henle, ∼40% of the bolus volume is reabsorbed as the bolus flows down the inner medullary collecting duct. Additionally, simulation results suggest that while the contraction-induced luminal hydrostatic pressure facilitates water extraction from the bolus, that pressure is not necessary to concentrate the bolus. Also, neither the negative interstitial pressure generated during the relaxation phase nor the concentrating effect of hyaluronic acid has a significant effect on bolus concentration. Taken together, these findings indicate that the high collecting duct apical water permeability allows a substantial amount of water to be extracted from the bolus, despite its short transit time. However, the potential role of the peristaltic waves in the urine-concentrating mechanism remains to be revealed.

Optimizing SGLT inhibitor treatment for diabetes with chronic kidney diseases

Diabetes induces glomerular hyperfiltration, affects kidney function, and may lead to chronic kidney diseases. A novel therapeutic treatment for diabetic patients targets the sodium-glucose cotransporter isoform 2 (SGLT2) in the kidney. SGLT2 inhibitors enhance urinary glucose, [Formula: see text] and fluid excretion and lower hyperglycemia in diabetes by inhibiting [Formula: see text] and glucose reabsorption along the proximal convoluted tubule. A goal of this study is to predict the effects of SGLT2 inhibitors in diabetic patients with and without chronic kidney diseases. To that end, we applied computational rat kidney models to assess how SGLT2 inhibition affects renal solute transport and metabolism when nephron population are normal or reduced (the latter simulates chronic kidney disease). The model predicts that SGLT2 inhibition induces glucosuria and natriuresis, with those effects enhanced in a remnant kidney. The model also predicts that the [Formula: see text] transport load and thus oxygen consumption of the S3 segment are increased under SGLT2 inhibition, a consequence that may increase the risk of hypoxia for that segment. To protect the vulnerable S3 segment, we explore dual SGLT2/SGLT1 inhibition and seek to determine the optimal combination that would yield sufficient urinary glucose excretion while limiting the metabolic load on the S3 segment. The model predicts that the optimal combination of SGLT2/SGLT1 inhibition lowers the oxygen requirements of key tubular segments, but decreases urine flow and [Formula: see text] excretion; the latter effect may limit the cardiovascular protection of the treatment.

Understanding sex differences in long-term blood pressure regulation: insights from experimental studies and computational modeling

Sex differences in blood pressure and the prevalence of hypertension are found in humans and animal models. Moreover, there has been a recent explosion of data concerning sex differences in nitric oxide, the renin-angiotensin-aldosterone system, inflammation, and kidney function. These data have the potential to reveal the mechanisms underlying male-female differences in blood pressure control. To elucidate the interactions among the multitude of physiological processes involved, one may apply computational models. In this review, we describe published computational models that represent key players in blood pressure regulation, and highlight sex-specific models and their findings.

A computational model of epithelial solute and water transport along a human nephron

We have developed the first computational model of solute and water transport from Bowman space to the papillary tip of the nephron of a human kidney. The nephron is represented as a tubule lined by a layer of epithelial cells, with apical and basolateral transporters that vary according to cell type. The model is formulated for steady state, and consists of a large system of coupled ordinary differential equations and algebraic equations. Model solution describes luminal fluid flow, hydrostatic pressure, luminal fluid solute concentrations, cytosolic solute concentrations, epithelial membrane potential, and transcellular and paracellular fluxes. We found that if we assume that the transporter density and permeabilities are taken to be the same between the human and rat nephrons (with the exception of a glucose transporter along the proximal tubule and the H⁺-pump along the collecting duct), the model yields segmental deliveries and urinary excretion of volume and key solutes that are consistent with human data. The model predicted that the human nephron exhibits glomerulotubular balance, such that proximal tubular Na⁺ reabsorption varies proportionally to the single-nephron glomerular filtration rate. To simulate the action of a novel diabetic treatment, we inhibited the Na⁺-glucose cotransporter 2 (SGLT2) along the proximal convoluted tubule. Simulation results predicted that the segment’s Na⁺ reabsorption decreased significantly, resulting in natriuresis and osmotic diuresis.

In addition to its well-known function of waste removal from the body, the kidney is also responsible for the critical regulation of the body’s salt, potassium, acid content, and blood pressure. The kidneys perform these life-sustaining task by filtering and returning to blood stream about 200 quarts of blood every 24 hours. What isn’t returned to blood stream is excreted as urine. The production of urine involves highly complex steps of secretion and reabsorption. To study these processes without employing invasive experimental procedures, we developed the first computational model of the human nephron (which is the functional unit of a kidney). The model contains detailed representation of the transport processes that take place in the epithelial cells that form the walls of the nephron. Using that model, we conducted simulations to predict how much filtered solutes and and water is transported along each individual and functionally distinct nephron segment. We conducted these simulations under normal physiological conditions, and under pharmacological conditions. The nephron model can be used as an essential component in an integrated model of kidney function in humans.

Mammalian urine concentration: a review of renal medullary architecture and membrane transporters

Mammalian kidneys play an essential role in balancing internal water and salt concentrations. When water needs to be conserved, the renal medulla produces concentrated urine. Central to this process of urine concentration is an osmotic gradient that increases from the corticomedullary boundary to the inner medullary tip. How this gradient is generated and maintained has been the subject of study since the 1940s. While it is generally accepted that the outer medulla contributes to the gradient by means of an active process involving countercurrent multiplication, the source of the gradient in the inner medulla is unclear. The last two decades have witnessed advances in our understanding of the urine-concentrating mechanism. Details of medullary architecture and permeability properties of the tubules and vessels suggest that the functional and anatomic relationships of these structures may contribute to the osmotic gradient necessary to concentrate urine. Additionally, we are learning more about the membrane transporters involved and their regulatory mechanisms. The role of medullary architecture and membrane transporters in the mammalian urine-concentrating mechanism are the focus of this review.

SGLT2 inhibition in a kidney with reduced nephron number: modeling and analysis of solute transport and metabolism

Sodium-glucose cotransporter 2 (SGLT2) inhibitors enhance urinary glucose, Na+ and fluid excretion, and lower hyperglycemia in diabetes by targeting Na+ and glucose reabsorption along the proximal convoluted tubule. A goal of this study was to predict the effects of SGLT2 inhibitors in diabetic and nondiabetic patients with chronic kidney disease. To that end, we employed computational rat kidney models to explore how SGLT2 inhibition affects renal solute transport and metabolism when nephron populations are normal or reduced. Model simulations suggested that in a nondiabetic rat, acute and chronic SGLT2 inhibition induces glucosuria, diuresis, natriuresis, and kaliuresis. Those effects were stronger with chronic SGLT2 inhibition (due to SGLT1 downregulation) and tempered by nephron loss. In a diabetic rat with normal nephron number, acute SGLT2 inhibition similarly elevated urine fluid, Na+, and K+ excretion, whereas the urinary excretory effects of chronic SGLT2 inhibition were attenuated in proportion to its plasma glucose level lowering effect. Nephron loss in a diabetic kidney was predicted to lower the glucosuric and blood glucose-reducing effect of chronic SGLT2 inhibition, but due to the high luminal glucose delivery in the remaining hyperfiltering nephrons, nephron loss enhanced proximal tubular paracellular Na+ secretion, thereby augmenting the natriuretic, diuretic, and kaliuretic effects. A proposed shift in oxygen-consuming active transport to the outer medulla, which may simulate systemic hypoxia and enhance erythropoiesis, was also preserved with nephron loss. These effects may contribute to the protective effects of SGLT2 inhibitors on blood pressure and heart failure observed in diabetic patients with chronic kidney diseases.

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.

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.

Semi-mechanistic kidney model incorporating physiologically-relevant fluid reabsorption and transporter-mediated renal reabsorption: pharmacokinetics of γ-hydroxybutyric acid and L-lactate in rats

This study developed a semi-mechanistic kidney model incorporating physiologically-relevant fluid reabsorption and transporter-mediated active reabsorption. The model was applied to data for the drug of abuse γ-hydroxybutyric acid (GHB), which exhibits monocarboxylate transporter (MCT1/SMCT1)-mediated renal reabsorption. The kidney model consists of various nephron segments--proximal tubules, Loop-of-Henle, distal tubules, and collecting ducts--where the segmental fluid flow rates, volumes, and sequential reabsorption were incorporated as functions of the glomerular filtration rate. The active renal reabsorption was modeled as vectorial transport across proximal tubule cells. In addition, the model included physiological blood, liver, and remainder compartments. The population pharmacokinetic modeling was performed using ADAPT5 for GHB blood concentration-time data and cumulative amount excreted unchanged into urine data (200-1000 mg/kg IV bolus doses) from rats [Felmlee et al (PMID: 20461486)]. Simulations assessed the effects of inhibition (R = [I]/KI = 0-100) of renal reabsorption on systemic exposure (AUC) and renal clearance of GHB. Visual predictive checks and other model diagnostic plots indicated that the model reasonably captured GHB concentrations. Simulations demonstrated that the inhibition of renal reabsorption significantly increased GHB renal clearance and decreased AUC. Model validation was performed using a separate dataset. Furthermore, our model successfully evaluated the pharmacokinetics of L-lactate using data obtained from Morse et al (PMID: 24854892). In conclusion, we developed a semi-mechanistic kidney model that can be used to evaluate transporter-mediated active renal reabsorption of drugs by the kidney.

Aquaporins in desert rodent physiology

Desert rodents face a sizeable challenge in maintaining salt and water homeostasis due to their life in an arid environment. A number of their organ systems exhibit functional characteristics that limit water loss above that which occurs in non-desert species under similar conditions. These systems include renal, pulmonary, gastrointestinal, nasal, and skin epithelia. The desert rodent kidney preserves body water by producing a highly concentrated urine that reaches a maximum osmolality nearly three times that of the common laboratory rat. The precise mechanism by which urine is concentrated in any mammal is unknown. Insights into the process may be more apparent in species that produce highly concentrated urine. Aquaporin water channels play a fundamental role in water transport in several desert rodent organ systems. The role of aquaporins in facilitating highly effective water preservation in desert rodents is only beginning to be explored. The organ systems of desert rodents and their associated AQPs are described.

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.

Oxygen transport in a cross section of the rat inner medulla: impact of heterogeneous distribution of nephrons and vessels

We have developed a highly detailed mathematical model of oxygen transport in a cross section of the upper inner medulla of the rat kidney. The model is used to study the impact of the structured organization of nephrons and vessels revealed in anatomic studies, in which descending vasa recta are found to lie distant from clusters of collecting ducts. Specifically, we formulated a two-dimensional oxygen transport model, in which the positions and physical dimensions of renal tubules and vessels are based on an image obtained by immunochemical techniques (T. Pannabecker and W. Dantzler, Three-dimensional architecture of inner medullary vasa recta, Am. J. Physiol. Renal Physiol. 290 (2006) F1355-F1366). The model represents oxygen diffusion through interstitium and other renal structures, oxygen consumption by the Na(+)/K(+)-ATPase activities of the collecting ducts, and basal metabolic consumption. Model simulations yield marked variations in interstitial PO2, which can be attributed, in large part, to the heterogeneities in the position and physical dimensions of the collecting ducts. Further, results of a sensitivity study suggest that medullary oxygenation is highly sensitive to medullary blood flow, and that, at high active consumption rates, localized patches of tissue may be vulnerable to hypoxic injury.

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.

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.

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.

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)

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.

New insights into urea and glucose handling by the kidney, and the urine concentrating mechanism

The mechanism by which urine is concentrated in the mammalian kidney remains incompletely understood. Urea is the dominant urinary osmole in most mammals and may be concentrated a 100-fold above its plasma level in humans and even more in rodents. Several facilitated urea transporters have been cloned. The phenotypes of mice with deletion of the transporters expressed in the kidney have challenged two previously well-accepted paradigms regarding urea and sodium handling in the renal medulla but have provided no alternative explanation for the accumulation of solutes that occurs in the inner medulla. In this review, we present evidence supporting the existence of an active urea secretion in the pars recta of the proximal tubule and explain how it changes our views regarding intrarenal urea handling and UT-A2 function. The transporter responsible for this secretion could be SGLT1, a sodium-glucose cotransporter that also transports urea. Glucagon may have a role in the regulation of this secretion. Further, we describe a possible transfer of osmotic energy from the outer to the inner medulla via an intrarenal Cori cycle converting glucose to lactate and back. Finally, we propose that an active urea transporter, expressed in the urothelium, may continuously reclaim urea that diffuses out of the ureter and bladder. These hypotheses are all based on published findings. They may not all be confirmed later on, but we hope they will stimulate further research in new directions.

Architecture of kangaroo rat inner medulla: segmentation of descending thin limb of Henle's loop

We hypothesize that the inner medulla of the kangaroo rat Dipodomys merriami, a desert rodent that concentrates its urine to more than 6,000 mosmol/kgH(2)O water, provides unique examples of architectural features necessary for production of highly concentrated urine. To investigate this architecture, inner medullary nephron segments in the initial 3,000 μm below the outer medulla were assessed with digital reconstructions from physical tissue sections. Descending thin limbs of Henle (DTLs), ascending thin limbs of Henle (ATLs), and collecting ducts (CDs) were identified by immunofluorescence using antibodies that label segment-specific proteins associated with transepithelial water flux (aquaporin 1 and 2, AQP1 and AQP2) and chloride flux (the chloride channel ClC-K1); all tubules and vessels were labeled with wheat germ agglutinin. In the outer 3,000 μm of the inner medulla, AQP1-positive DTLs lie at the periphery of groups of CDs. ATLs lie inside and outside the groups of CDs. Immunohistochemistry and reconstructions of loops that form their bends in the outer 3,000 μm of the inner medulla show that, relative to loop length, the AQP1-positive segment of the kangaroo rat is significantly longer than that of the Munich-Wistar rat. The length of ClC-K1 expression in the prebend region at the terminal end of the descending side of the loop in kangaroo rat is about 50% shorter than that of the Munich-Wistar rat. Tubular fluid of the kangaroo rat DTL may approach osmotic equilibrium with interstitial fluid by water reabsorption along a relatively longer tubule length, compared with Munich-Wistar rat. A relatively shorter-length prebend segment may promote a steeper reabsorptive driving force at the loop bend. These structural features predict functionality that is potentially significant in the production of a high urine osmolality in the kangaroo rat.

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.

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.