Dominant factors that govern pressure natriuresis in diuresis and antidiuresis: a mathematical model

Renal sympathetic activity: A key modulator of pressure natriuresis in hypertension

Hypertension is a complex disorder ensuing necessarily from alterations in the pressure-natriuresis relationship, the main determinant of long-term control of blood pressure. This mechanism sets natriuresis to the level of blood pressure, so that increasing pressure translates into higher osmotically driven diuresis to reduce volemia and control blood pressure. External factors affecting the renal handling of sodium regulate the pressure-natriuresis relationship so that more or less natriuresis is attained for each level of blood pressure. Hypertension can thus only develop following primary alterations in the pressure to natriuresis balance, or by abnormal activity of the regulation network. On the other hand, increased sympathetic tone is a very frequent finding in most forms of hypertension, long regarded as a key element in the pathophysiological scenario. In this article, we critically analyze the interplay of the renal component of the sympathetic nervous system and the pressure-natriuresis mechanism in the development of hypertension. A special focus is placed on discussing recent findings supporting a role of baroreceptors as a component, along with the afference of reno-renal reflex, of the input to the nucleus tractus solitarius, the central structure governing the long-term regulation of renal sympathetic efferent tone.

Renal blood flow and oxygenation

Our kidneys receive about one-fifth of the cardiac output at rest and have a low oxygen extraction ratio, but may sustain, under some conditions, hypoxic injuries that might lead to chronic kidney disease. This is due to large regional variations in renal blood flow and oxygenation, which are the prerequisite for some and the consequence of other kidney functions. The concurrent operation of these functions is reliant on a multitude of neuro-hormonal signaling cascades and feedback loops that also include the regulation of renal blood flow and tissue oxygenation. Starting with open questions on regulatory processes and disease mechanisms, we review herein the literature on renal blood flow and oxygenation. We assess the current understanding of renal blood flow regulation, reasons for disparities in oxygen delivery and consumption, and the consequences of disbalance between O2 delivery, consumption, and removal. We further consider methods for measuring and computing blood velocity, flow rate, oxygen partial pressure, and related parameters and point out how limitations of these methods constitute important hurdles in this area of research. We conclude that to obtain an integrated understanding of the relation between renal function and renal blood flow and oxygenation, combined experimental and computational modeling studies will be needed.

Numerical Modeling and Simulation of Blood Flow in a Rat Kidney: Coupling of the Myogenic Response and the Vascular Structure

A numerical simulation was carried out to investigate the blood flow behavior (i.e., flow rate and pressure) and coupling of a renal vascular network and the myogenic response to various conditions. A vascular segment and an entire kidney vascular network were modeled by assuming one single vessel as a straight pipe whose diameter was determined by Murray’s law. The myogenic response was tested on individual AA (afferent artery)–GC (glomerular capillaries)–EA (efferent artery) systems, thereby regulating blood flow throughout the vascular network. Blood flow in the vascular structure was calculated by network analysis based on Hagen–Poiseuille’s law to various boundary conditions. Simulation results demonstrated that, in the vascular segment, the inlet pressure Pinlet and the vascular structure act together on the myogenic response of each individual AA–GC–EA subsystem, such that the early-branching subsystems in the vascular network reached the well-regulated state first, with an interval of the inlet as Pinlet = 10.5–21.0 kPa, whereas the one that branched last exhibited a later interval with Pinlet = 13.0–24.0 kPa. In the entire vascular network, in contrast to the Pinlet interval (13.0–20.0 kPa) of the unified well-regulated state for all AA–GC–EA subsystems of the symmetric model, the asymmetric model exhibited the differences among subsystems with Pinlet ranging from 12.0–17.0 to 16.0–20.0 kPa, eventually achieving a well-regulated state of 13.0–18.5 kPa for the entire kidney. Furthermore, when Pinlet continued to rise (e.g., 21.0 kPa) beyond the vasoconstriction range of the myogenic response, high glomerular pressure was also related to vascular structure, where PGC of early-branching subsystems was 9.0 kPa and of late-branching one was 7.5 kPa. These findings demonstrate how the myogenic response regulates renal blood flow in vascular network system that comprises a large number of vessel elements.

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.

Mathematical model of hemodynamic mechanisms and consequences of glomerular hypertension in diabetic mice

Many preclinically promising therapies for diabetic kidney disease fail to provide efficacy in humans, reflecting limited quantitative translational understanding between rodent models and human disease. To quantitatively bridge interspecies differences, we adapted a mathematical model of renal function from human to mice, and incorporated adaptive and pathological mechanisms of diabetes and nephrectomy to describe experimentally observed changes in glomerular filtration rate (GFR) and proteinuria in db/db and db/db UNX (uninephrectomy) mouse models. Changing a small number of parameters, the model reproduced interspecies differences in renal function. Accounting for glucose and Na⁺ reabsorption through sodium glucose cotransporter 2 (SGLT2), increasing blood glucose and Na⁺ intake from normal to db/db levels mathematically reproduced glomerular hyperfiltration observed experimentally in db/db mice. This resulted from increased proximal tubule sodium reabsorption, which elevated glomerular capillary hydrostatic pressure (P_gc) in order to restore sodium balance through increased GFR. Incorporating adaptive and injurious effects of elevated P_gc, we showed that preglomerular arteriole hypertrophy allowed more direct transmission of pressure to the glomerulus with a smaller mean arterial pressure rise; Glomerular hypertrophy allowed a higher GFR for a given P_gc; and P_gc-driven glomerulosclerosis and nephron loss reduced GFR over time, while further increasing P_gc and causing moderate proteinuria, in agreement with experimental data. UNX imposed on diabetes increased P_gc further, causing faster GFR decline and extensive proteinuria, also in agreement with experimental data. The model provides a mechanistic explanation for hyperfiltration and proteinuria progression that will facilitate translation of efficacy for novel therapies from mouse models to human.

Many drugs for diabetic kidney disease appear to work in rodents, but fail in humans, reflecting incomplete understanding of disease processes. A team led by Melissa Hallow at the University of Georgia has developed a mathematical model that explains how elevated blood glucose in diabetes causes kidney injury in mice. They first showed that normal human, rat, or mouse kidney physiology could be reproduced with the same model by changing a small number of parameters. They then showed that diabetes-induced increases in sodium reabsorption cause unintuitive changes in kidney function that increase pressure on glomerular capillaries, causing protein leakage and nephron loss. The model reproduced faster disease progression observed in diabetic mice who have had one kidney removed. This mathematical understanding of diabetic kidney injury may improve translation of novel therapies from mice to human.

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.

Renal medullary and urinary oxygen tension during cardiopulmonary bypass in the rat

Renal hypoxia could result from a mismatch in renal oxygen supply and demand, particularly in the renal medulla. Medullary hypoxic damage is believed to give rise to acute kidney injury, which is a prevalent complication of cardiac surgery performed on cardiopulmonary bypass (CPB). To determine the mechanisms that could lead to medullary hypoxia during CPB in the rat kidney, we developed a mathematical model which incorporates (i) autoregulation of renal blood flow and glomerular filtration rate, (ii) detailed oxygen transport and utilization in the renal medulla and (iii) oxygen transport along the ureter. Within the outer medulla, the lowest interstitial tissue P$_{\rm O2}$, which is an indicator of renal hypoxia, is predicted near the thick ascending limbs. Interstitial tissue P$_{\rm O2}$ exhibits a general decrease along the inner medullary axis, but urine P$_{\rm O2}$ increases significantly along the ureter. Thus, bladder urinary P$_{\rm O2}$ is predicted to be substantially higher than medullary P$_{\rm O2}$. The model is used to identify the phase of cardiac surgery performed on CPB that is associated with the highest risk for hypoxic kidney injury. Simulation results indicate that the outer medulla's vulnerability to hypoxic injury depends, in part, on the extent to which medullary blood flow is autoregulated. With imperfect medullary blood flow autoregulation, the model predicts that the rewarming phase of CPB, in which medullary blood flow is low but medullary oxygen consumption remains high, is the phase in which the kidney is most likely to suffer hypoxic injury.

A quantitative systems physiology model of renal function and blood pressure regulation: Model description

Renal function plays a central role in cardiovascular, kidney, and multiple other diseases, and many existing and novel therapies act through renal mechanisms. Even with decades of accumulated knowledge of renal physiology, pathophysiology, and pharmacology, the dynamics of renal function remain difficult to understand and predict, often resulting in unexpected or counterintuitive therapy responses. Quantitative systems pharmacology modeling of renal function integrates this accumulated knowledge into a quantitative framework, allowing evaluation of competing hypotheses, identification of knowledge gaps, and generation of new experimentally testable hypotheses. Here we present a model of renal physiology and control mechanisms involved in maintaining sodium and water homeostasis. This model represents the core renal physiological processes involved in many research questions in drug development. The model runs in R and the code is made available. In a companion article, we present a case study using the model to explore mechanisms and pharmacology of salt‐sensitive hypertension.

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.

Primary proximal tubule hyperreabsorption and impaired tubular transport counterregulation determine glomerular hyperfiltration in diabetes: a modeling analysis

Glomerular hypertension and hyperfiltration in early diabetes are associated with development and progression of diabetic kidney disease. The tubular hypothesis of diabetic hyperfiltration proposes that it is initiated by a primary increase in sodium (Na) reabsorption in the proximal tubule (PT) and the resulting tubuloglomerular feedback (TGF) response and lowering of Bowman space pressure (P_Bow). Here we utilized a mathematical model of the human kidney to investigate over acute and chronic timescales the mechanisms responsible for the magnitude of the hyperfiltration response. The model implicates that the primary hyperreabsorption of Na in the PT produces a Na imbalance that is only partially restored by the hyperfiltration induced by TGF and changes in P_Bow Thus secondary adaptations are needed to restore Na balance. This may include neurohumoral transport regulation and/or pressure-natriuresis (i.e., the decrease in Na reabsorption in response to increased renal perfusion pressure). We explored the role of each tubular segment in contributing to this compensation and the consequences of impairment in tubular compensation. The simulations indicate that impaired secondary downregulation of transport potentiated the rise in glomerular hypertension and hyperfiltration needed to restore Na balance at a given level of primary PT hyperreabsorption. Therefore, we propose for the first time that both the extent of primary PT hyperreabsorption and the degree of impairment of the distal tubular responsiveness to regulatory signals determine the level of glomerular hypertension and hyperfiltration in the diabetic kidney, thereby extending the tubule-centric concept of diabetic hyperfiltration and potential therapeutic approaches beyond the proximal tubule.

Generation and phenotypic analysis of mice lacking all urea transporters

Urea transporters (UT) are a family of transmembrane urea-selective channel proteins expressed in multiple tissues and play an important role in the urine concentrating mechanism of the mammalian kidney. UT inhibitors have diuretic activity and could be developed as novel diuretics. To determine if functional deficiency of all UTs in all tissues causes physiological abnormality, we established a novel mouse model in which all UTs were knocked out by deleting an 87 kb of DNA fragment containing most parts of Slc14a1 and Slc14a2 genes. Western blot analysis and immunofluorescence confirmed that there is no expression of urea transporter in these all-UT-knockout mice. Daily urine output was nearly 3.5-fold higher, with significantly lower urine osmolality in all-UT-knockout mice than that in wild-type mice. All-UT-knockout mice were not able to increase urinary urea concentration and osmolality after water deprivation, acute urea loading, or high protein intake. A computational model that simulated UT-knockout mouse models identified the individual contribution of each UT in urine concentrating mechanism. Knocking out all UTs also decreased the blood pressure and promoted the maturation of the male reproductive system. Thus, functional deficiency of all UTs caused a urea-selective urine-concentrating defect with little physiological abnormality in extrarenal organs.

Dynamic contrast-enhanced quantitative susceptibility mapping with ultrashort echo time MRI for evaluating renal function

Dynamic contrast-enhanced (DCE) MRI can provide key insight into renal function. DCE MRI is typically achieved through an injection of a gadolinium (Gd)-based contrast agent, which has desirable T1 quenching and tracer kinetics. However, significant T2* blooming effects and signal voids can arise when Gd becomes very concentrated, especially in the renal medulla and pelvis. One MRI sequence designed to alleviate T2* effects is the ultrashort echo time (UTE) sequence. In the present study, we observed T2* blooming in the inner medulla of the mouse kidney, despite using UTE at an echo time of 20 microseconds and a low dose of 0.03 mmol/kg Gd. We applied quantitative susceptibility mapping (QSM) and resolved the signal void into a positive susceptibility signal. The susceptibility values [in parts per million (ppm)] were converted into molar concentrations of Gd using a calibration curve. We determined the concentrating mechanism (referred to as the concentrating index) as a ratio of maximum Gd concentration in the inner medulla to the renal artery. The concentrating index was assessed longitudinally over a 17-wk course (3, 5, 7, 9, 13, 17 wk of age). We conclude that the UTE-based DCE method is limited in resolving extreme T2* content caused by the kidney's strong concentrating mechanism. QSM was able to resolve and confirm the source of the blooming effect to be the large positive susceptibility of concentrated Gd. UTE with QSM can complement traditional magnitude UTE and offer a powerful tool to study renal pathophysiology.

Recent advances in renal hemodynamics: insights from bench experiments and computer simulations

It has been long known that the kidney plays an essential role in the control of body fluids and blood pressure and that impairment of renal function may lead to the development of diseases such as hypertension (Guyton AC, Coleman TG, Granger Annu Rev Physiol 34: 13-46, 1972). In this review, we highlight recent advances in our understanding of renal hemodynamics, obtained from experimental and theoretical studies. Some of these studies were published in response to a recent Call for Papers of this journal: Renal Hemodynamics: Integrating with the Nephron and Beyond.

A mathematical model of rat proximal tubule and loop of Henle

Proximal tubule and loop of Henle function are coupled, with proximal transport determining loop fluid composition, and loop transport modulating glomerular filtration via tubuloglomerular feedback (TGF). To examine this interaction, we begin with published models of the superficial rat proximal convoluted tubule (PCT; including flow-dependent transport in a compliant tubule), and the rat thick ascending Henle limb (AHL). Transport parameters for this PCT are scaled down to represent the proximal straight tubule (PST), which is connected to the thick AHL via a short descending limb. Transport parameters for superficial PCT and PST are scaled up for a juxtamedullary nephron, and connected to AHL via outer and inner medullary descending limbs, and inner medullary thin AHL. Medullary interstitial solute concentrations are specified. End-AHL hydrostatic pressure is determined by distal nephron flow resistance, and the TGF signal is represented as a linear function of end-AHL cytosolic Cl concentration. These two distal conditions required iterative solution of the model. Model calculations capture inner medullary countercurrent flux of urea, and also suggest the presence of an outer medullary countercurrent flux of ammonia, with reabsorption in AHL and secretion in PST. For a realistically strong TGF signal, there is the expected homeostatic impact on distal flows, and in addition, a homeostatic effect on proximal tubule pressure. The model glycosuria threshold is compatible with rat data, and predicted glucose excretion with selective 1Na(+):1glucose cotransporter (SGLT2) inhibition comports with observations in the mouse. Model calculations suggest that enhanced proximal tubule Na(+) reabsorption during hyperglycemia is sufficient to activate TGF and contribute to diabetic hyperfiltration.

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.