A mathematical model of rat proximal tubule and loop of Henle

Regulation of glomerulotubular balance. IV. Implication of aquaporin 1 in flow-dependent proximal tubule transport and cell volume

The water channel aquaporin-1 (AQP1) is the principal water pathway for isotonic water reabsorption in the kidney proximal tubule (PT). We investigated flow-mediated fluid (Jv) and [Formula: see text] ([Formula: see text]) reabsorption in PTs of the mouse kidney by microperfusion in wild-type (WT) and AQP1 knockout (KO) mice. Experiments were simulated in an adaptation of a mathematical model of the rat PT. An increase in perfusion rate from 5 to 20 nL/min increased Jv and [Formula: see text] in PTs of WT mice. AQP1 KO mice significantly decreased Jv at low and high flow rates compared with control. In contrast, [Formula: see text] was not reduced at either low or high flow rates. Cell volume showed no significant difference between WT and AQP1 KO mice. Renal clearance experiments showed significantly higher urine flow in AQP1 KO mice, but there was no significant difference in either Na+ and K+ or [Formula: see text] excretion. Acid-base parameters of blood pH, Pco2, [Formula: see text], and urine pH were the same in both WT and KO mice. In model calculations, tubules whose tight junction (TJ) water permeability (Pf) was that assigned to the rat TJ, showed no difference in Jv between WT and KO, whereas TJ Pf set to 25% of the rat predicted Jv concordant with our observations from AQP1 KO. These results affirm the dominance of AQP1 in mediating isotonic water reabsorption by the mouse PT and demonstrate that flow-stimulated [Formula: see text] reabsorption is intact and independent of AQP1. With reference to the model, the findings also suggest that TJ water flux in the PT is less prominent in the mouse than in the rat kidney.NEW & NOTEWORTHY We found an absence of flow-dependent modulation of fluid absorption but no effect on either proximal tubule (PT) [Formula: see text] absorption or acid-base parameters in the aquaporin 1 (AQP1) knockout mouse. We affirmed the dominance of the water channel AQP1 in mediating isotonic water reabsorption by the mouse PT and demonstrated that flow-stimulated [Formula: see text] reabsorption is independent of AQP1. With reference to the model, the findings also suggest that tight junctional water flux in the PT is less prominent in the mouse than rat kidney.

A modular and reusable model of epithelial transport in the proximal convoluted tubule

We review a collection of published renal epithelial transport models, from which we build a consistent and reusable mathematical model able to reproduce many observations and predictions from the literature. The flexible modular model we present here can be adapted to specific configurations of epithelial transport, and in this work we focus on transport in the proximal convoluted tubule of the renal nephron. Our mathematical model of the epithelial proximal convoluted tubule describes the cellular and subcellular mechanisms of the transporters, intracellular buffering, solute fluxes, and other processes. We provide free and open access to the Python implementation to ensure our multiscale proximal tubule model is accessible; enabling the reader to explore the model through setting their own simulations, reproducibility tests, and sensitivity analyses.

A mathematical model of the rat kidney. IV. Whole kidney response to hyperkalemia

The renal response to acute hyperkalemia is mediated by increased K⁺ secretion within the connecting tubule (CNT), flux that is modulated by tubular effects (e.g., aldosterone) in conjunction with increased luminal flow. There is ample evidence that peritubular K⁺ blunts Na⁺ reabsorption in the proximal tubule, thick ascending Henle limb, and distal convoluted tubule (DCT). Although any such reduction may augment CNT delivery, the relative contribution of each is uncertain. The kidney model of this laboratory was recently advanced with representation of the cortical labyrinth and medullary ray. Model tubules capture the impact of hyperkalemia to blunt Na⁺ reabsorption within each upstream segment. However, this forces the question of the extent to which increased Na⁺ delivery is transmitted past the macula densa and its tubuloglomerular feedback (TGF) signal. Beyond increasing macula densa Na⁺ delivery, peritubular K⁺ is predicted to raise cytosolic Cl^- and depolarize macula densa cells, which may also activate TGF. Thus, although the upstream reduction in Na⁺ transport may be larger, it appears that the DCT effect is critical to increasing CNT delivery. Beyond the flow effect, hyperkalemia reduces ammoniagenesis and reduced ammoniagenesis enhances K⁺ excretion. What this model provides is a possible mechanism. When cortical [Formula: see text] is taken up via peritubular Na⁺-K⁺([Formula: see text])-ATPase, it acidifies principal cells. Consequently, reduced ammoniagenesis increases principal cell pH, thereby increasing conductance of both the epithelial Na⁺ channel and renal outer medullary K⁺ channel, enhancing K⁺ excretion. In this model, the effect of aldosterone on principal cells, diminished DCT Na⁺ reabsorption, and reduced ammoniagenesis all provide relatively equal and additive contributions to renal K⁺ excretion.NEW & NOTEWORTHY Hyperkalemia blunts Na⁺ reabsorption along the nephron, and increased CNT Na⁺ delivery facilitates K⁺ secretion. The model suggests that tubuloglomerular feedback limits transmission of proximal effects past the macula densa, so that it is DCT transport that is critical. Hyperkalemia also reduces PCT ammoniagenesis, which enhances K⁺ excretion. The model suggests a mechanism, namely, that reduced cortical ammonia impacts CNT transport by raising cell pH and thus increasing both ENaC and ROMK conductance.

A mathematical model of the rat kidney. III. Ammonia transport

Ammonia generated within the kidney is partitioned into a urinary fraction (the key buffer for net acid excretion) and an aliquot delivered to the systemic circulation. The physiology of this partitioning has yet to be examined in a kidney model, and that was undertaken in this work. This involves explicit representation of the cortical labyrinth, so that cortical interstitial solute concentrations are computed rather than assigned. A detailed representation of cortical vasculature has been avoided by making the assumption that solute concentrations within the interstitium and peritubular capillaries are likely to be identical and that there is little to no modification of venous composition as blood flows to the renal vein. The model medullary ray has also been revised to include a segment of proximal straight tubule, which supplies ammonia to this region. The principal finding of this work is that cortical labyrinth interstitial ammonia concentration is likely to be several fold higher than systemic arterial ammonia. This elevation of interstitial ammonia enhances ammonia secretion in both the proximal convoluted tubule and distal convoluted tubule, with uptake by Na⁺-K⁺-ATPases of both segments. Model prediction of urinary ammonia excretion was concordant with measured values, but at the expense of greater ammoniagenesis, with high rates of renal venous ammonia flux. This derives from a limited capability of the model medulla to replicate the high interstitial ammonia concentrations that are required to drive collecting duct ammonia secretion. Thus, renal medullary ammonia trapping appears key to diverting ammonia from the renal vein to urine, but capturing the underlying physiology remains a challenge.NEW & NOTEWORTHY This is the first mathematical model to estimate solute concentrations within the kidney cortex. The model predicts cortical ammonia to be several fold greater than in the systemic circulation. This higher concentration drives ammonia secretion in proximal and distal tubules. The model reveals a gap in our understanding of how ammonia generated within the cortex is channeled efficiently into the final urine.

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.

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.

Claudins in Renal Physiology and Pathology

Claudins are integral proteins expressed at the tight junctions of epithelial and endothelial cells. In the mammalian kidney, every tubular segment express a specific set of claudins that give to that segment unique properties regarding permeability and selectivity of the paracellular pathway. So far, 3 claudins (10b, 16 and 19) have been causally traced to rare human syndromes: variants of CLDN10b cause HELIX syndrome and variants of CLDN16 or CLDN19 cause familial hypomagnesemia with hypercalciuria and nephrocalcinosis. The review summarizes our current knowledge on the physiology of mammalian tight junctions and paracellular ion transport, as well as on the role of the 3 above-mentioned claudins in health and disease. Claudin 14, although not having been causally linked to any rare renal disease, is also considered, because available evidence suggests that it may interact with claudin 16. Some single-nucleotide polymorphisms of CLDN14 are associated with urinary calcium excretion and/or kidney stones. For each claudin considered, the pattern of expression, the function and the human syndrome caused by pathogenic variants are described.

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.

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.

Renal tubular solute transport and oxygen consumption: insights from computational models

PURPOSE OF REVIEW

To maintain electrolyte homeostasis, the kidneys reabsorb more than 99% of the filtered Na under physiological conditions, resulting in less than 1% of the filtered Na excreted in urine. In contrast, due to distal tubular secretion, urinary K output may exceed filtered load. This review focuses on a relatively new methodology for investigating renal epithelial transport, computational modelling and highlights recent insights regarding renal Na and K transport and O2 consumption under pathophysiological conditions, with a focus on nephrectomy.

RECENT FINDINGS

Recent modelling studies investigated the extent to which the adaptive response to nephrectomy, which includes elevation in single-nephron glomerular filtration rate and tubular transport capacity, may achieve balance but increases O2 consumption per nephron. Simulation results pointed to potential mechanisms in a hemi-nephrectomized rat that may attenuate the natriuresis response under K load, or that may augment the natriuretic, diuretic and kaliuretic effects of sodium glucose cotransporter 2 inhibition.

SUMMARY

Computational models provide a systemic approach for investigating system perturbations, such as those induced by drug administration or genetic alterations. Thus, computational models can be a great asset in data interpretation concerning (but not limited to) renal tubular transport and metabolism.

A Glucose-Dependent Pharmacokinetic/ Pharmacodynamic Model of ACE Inhibition in Kidney Cells

Diabetic kidney disease (DKD) is a major cause of renal failure. Podocytes are terminally differentiated renal epithelial cells that are key targets of damage due to DKD. Podocytes express a glucose-stimulated local renin-angiotensin system (RAS) that produces angiotensin II (ANG II). Local RAS differs from systemic RAS, which has been studied widely. Hyperglycemia increases the production of ANG II by podocyte cells, leading to podocyte injury. Angiotensin-converting enzyme (ACE) is involved in the production of ANG II, and ACE inhibitors are drugs used to suppress elevated ANG II concentration. As systemic RAS differs from the local RAS in podocytes, ACE inhibitor drugs should act differently in local versus systemic contexts. Experimental and computational studies have considered the pharmacokinetics (PK) and pharmacodynamics (PD) of ACE inhibition of the systemic RAS. Here, a PK/PD model for ACE inhibition is developed for the local RAS in podocytes. The model takes constant or dynamic subject-specific glucose concentration input to predict the ANG II concentration and the corresponding effects of drug doses locally and systemically. The model is developed for normal and impaired renal function in combination with different glucose conditions, thus enabling the study of various pathophysiological conditions. Parameter uncertainty is also analyzed. Such a model can improve the study of the effects of drugs at the cellular level and can aid in development of therapeutic approaches to slow the progression of DKD.

Physiological relevance of epithelial geometry: New insights into the standing gradient model and the role of LI cadherin

We introduce a mathematical model of an absorbing leaky epithelium to reconsider the problem formulated by Diamond and Bossert in 1967: whether "… some distinctive physiological properties of epithelia might arise as geometrical consequences of epithelial ultrastructure". A standing gradient model of the intercellular cleft (IC) is presented that includes tight junctions (TJ) and ion channels uniformly distributed along the whole cleft. This nonlinear system has an intrinsic homogeneous concentration and the spatial scale necessary to establish it along the cleft. These parameters have not been elucidated so far. We further provide non-perturbative analytical approximations for a broad range of parameters. We found that narrowing of the IC increases ion concentration dramatically and can therefore prevent outflow through tight junctions (TJs) and the lateral membrane, as long as extremely high luminal osmolarities are not reached. Our model predicts that the system is to some extent self-regulating and thereby prevents fluxes into the lumen. Recent experimental evidence has shown that liver-intestine (LI) cadherin can control the up/down flux in intestines via regulation of the cleft width. This finding is in full agreement with predictions of our model. We suggest that LI-cadherin may increase water transport through epithelia via sequential narrowing of the cleft, starting from the highest concentration area at the beginning of the cleft and triggering a propagating squeezing motion.

Functional implications of sexual dimorphism of transporter patterns along the rat proximal tubule: modeling and analysis

The goal of this study is to investigate the functional implications of the sexual dimorphism in transporter patterns along the proximal tubule. To do so, we have developed sex-specific computational models of solute and water transport in the proximal convoluted tubule of the rat kidney. The models account for the sex differences in expression levels of the apical and basolateral transporters, in single-nephron glomerular filtration rate, and in tubular dimensions. Model simulations predict that 70.6 and 38.7% of the filtered volume is reabsorbed by the proximal tubule of the male and female rat kidneys, respectively. The lower fractional volume reabsorption in females can be attributed to their smaller transport area and lower aquaporin-1 expression level. The latter also results in a larger contribution of the paracellular pathway to water transport. Correspondingly similar fractions (70.9 and 39.2%) of the filtered Na+ are reabsorbed by the male and female proximal tubule models, respectively. The lower fractional Na+ reabsorption in females is due primarily to their smaller transport area and lower Na+/H+ exchanger isoform 3 and claudin-2 expression levels. Notably, unlike most Na+ transporters, whose expression levels are lower in females, Na+-glucose cotransporter 2 (SGLT2) expression levels are 2.5-fold higher in females. Model simulations suggest that the higher SGLT2 expression in females may compensate for their lower tubular transport area to achieve a hyperglycemic tolerance similar to that of males.

Regulation of renal Na transporters in response to dietary K

Changes in the expression of Na transport proteins were measured in the kidneys of mice with increased dietary K intake for 1 wk. The epithelial Na channel (ENaC) was upregulated, with enhanced expression of full-length and cleaved forms of α-ENaC and cleaved γ-ENaC. At the same time, the amount of the NaCl cotransporter NCC and its phosphorylated form decreased by ~50% and ~80%, respectively. The expression of the phosphorylated form of the Na-K-2Cl cotransporter NKCC2 also decreased, despite an increase in overall protein content. The effect was stronger in males (80%) than in females (40%). This implies that less Na⁺ is reabsorbed in the thick ascending limb of Henle's loop and distal convoluted tubule along with Cl^-, whereas more is reabsorbed in the aldosterone-sensitive distal nephron in exchange for secreted K⁺. The abundance of the proximal tubule Na/H exchanger NHE3 decreased by ~40%, with similar effects in males and females. Time-course studies indicated that NCC and NHE3 proteins decreased progressively over 7 days on a high-K diet. Expression of mRNA encoding these proteins increased, implying that the decreased protein levels resulted from decreased rates of synthesis or increased rates of degradation. The potential importance of changes in NHE3, NKCC2, and NCC in promoting K⁺ excretion was assessed with a mathematical model. Simulations indicated that decreased NHE3 produced the largest effect. Regulation of proximal tubule Na⁺ transport may play a significant role in achieving K homeostasis.

A model of calcium transport and regulation in the proximal tubule

The objective of this study was to examine theoretically how Ca²⁺ reabsorption in the proximal tubule (PT) is modulated by Na⁺ and water fluxes, parathyroid hormone (PTH), Na⁺-glucose cotransporter (SGLT2) inhibitors, and acetazolamide. We expanded a previously published mathematical model of water and solute transport in the rat PT (Layton AT, Vallon V, Edwards A. Am J Physiol Renal Physiol 308: F1343–F1357, 2015) that did not include Ca²⁺. Our results indicate that Ca²⁺ reabsorption in the PT is primarily driven by the transepithelial Ca²⁺ concentration gradient that stems from water reabsorption, which is itself coupled to Na⁺ reabsorption. Simulated variations in permeability or transporter activity elicit opposite changes in paracellular and transcellular Ca²⁺ fluxes, whereas a simulated decrease in filtration rate lowers both fluxes. The model predicts that PTH-mediated inhibition of the apical Na⁺/H⁺ exchanger NHE3 reduces Na⁺ and Ca²⁺ transport to a similar extent. It also suggests that acetazolamide- and SGLT2 inhibitor-induced calciuria at least partly stems from reduced Ca²⁺ reabsorption in the PT. In addition, backleak of phosphate (PO₄) across tight junctions is predicted to reduce net PO₄ reabsorption by ~20% under normal conditions. When transcellular PO₄ transport is substantially reduced by PTH, paracellular PO₄ flux is reversed and contributes significantly to PO₄ reabsorption. Furthermore, PTH is predicted to exert an indirect impact on PO₄ reabsorption via its inhibitory action on NHE3. This model thus provides greater insight into the mechanisms that modulate Ca²⁺ and PO₄ reabsorption in the PT.

Intraglomerular crosstalk elaborately regulates podocyte injury and repair in diabetic patients: insights from a 3D multiscale modeling study

Podocytes are mainly involved in the regulation of glomerular filtration rate (GFR) under physiological condition. Podocyte depletion is a crucial pathological alteration in diabetic nephropathy (DN) and results in a broad spectrum of clinical syndromes such as protein urine and renal insufficiency. Recent studies indicate that depleted podocytes can be regenerated via differentiation of the parietal epithelial cells (PECs), which serve as the local progenitors of podocytes. However, the podocyte regeneration process is regulated by a complicated mechanism of cell-cell interactions and cytokine stimulations, which has been studied in a piecemeal manner rather than systematically. To address this gap, we developed a high-resolution multi-scale multi-agent mathematical model in 3D, mimicking the in situ glomerulus anatomical structure and micro-environment, to simulate the podocyte regeneration process under various cytokine perturbations in healthy and diabetic conditions. Our model showed that, treatment with pigment epithelium derived factor (PEDF) or insulin-like growth factor-1 (IGF-1) alone merely ameliorated the glomerulus injury, while co-treatment with both cytokines replenished the damaged podocyte population gradually. In addition, our model suggested that continuous administration of PEDF instead of a bolus injection sustained the regeneration process of podocytes. Part of the results has been validated by our in vivo experiments. These results indicated that amelioration of the glomerular stress by PEDF and promotion of PEC differentiation by IGF-1 are equivalently critical for podocyte regeneration. Our 3D multi-scale model represents a powerful tool for understanding the signaling regulation and guiding the design of cytokine therapies in promoting podocyte regeneration.

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 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.

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.

Ammonia Transporters and Their Role in Acid-Base Balance

Acid-base homeostasis is critical to maintenance of normal health. Renal ammonia excretion is the quantitatively predominant component of renal net acid excretion, both under basal conditions and in response to acid-base disturbances. Although titratable acid excretion also contributes to renal net acid excretion, the quantitative contribution of titratable acid excretion is less than that of ammonia under basal conditions and is only a minor component of the adaptive response to acid-base disturbances. In contrast to other urinary solutes, ammonia is produced in the kidney and then is selectively transported either into the urine or the renal vein. The proportion of ammonia that the kidney produces that is excreted in the urine varies dramatically in response to physiological stimuli, and only urinary ammonia excretion contributes to acid-base homeostasis. As a result, selective and regulated renal ammonia transport by renal epithelial cells is central to acid-base homeostasis. Both molecular forms of ammonia, NH₃ and NH₄⁺, are transported by specific proteins, and regulation of these transport processes determines the eventual fate of the ammonia produced. In this review, we discuss these issues, and then discuss in detail the specific proteins involved in renal epithelial cell ammonia transport.

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.

Novel minimal physiologically-based model for the prediction of passive tubular reabsorption and renal excretion clearance

PURPOSE

Develop a minimal mechanistic model based on in vitro-in vivo extrapolation (IVIVE) principles to predict extent of passive tubular reabsorption. Assess the ability of the model developed to predict extent of passive tubular reabsorption (Freab) and renal excretion clearance (CLR) from in vitro permeability data and tubular physiological parameters.

METHODS

Model system parameters were informed by physiological data collated following extensive literature analysis. A database of clinical CLR was collated for 157 drugs. A subset of 45 drugs was selected for model validation; for those, Caco-2 permeability (Papp) data were measured under pH6.5-7.4 gradient conditions and used to predict Freab and subsequently CLR. An empirical calibration approach was proposed to account for the effect of inter-assay/laboratory variation in Papp on the IVIVE of Freab.

RESULTS

The 5-compartmental model accounted for regional differences in tubular surface area and flow rates and successfully predicted the extent of tubular reabsorption of 45 drugs for which filtration and reabsorption were contributing to renal excretion. Subsequently, predicted CLR was within 3-fold of the observed values for 87% of drugs in this dataset, with an overall gmfe of 1.96. Consideration of the empirical calibration method improved overall prediction of CLR (gmfe=1.73 for 34 drugs in the internal validation dataset), in particular for basic drugs and drugs with low extent of tubular reabsorption.

CONCLUSIONS

The novel 5-compartment model represents an important addition to the IVIVE toolbox for physiologically-based prediction of renal tubular reabsorption and CLR. Physiological basis of the model proposed allows its application in future mechanistic kidney models in preclinical species and human.

A computational model for simulating solute transport and oxygen consumption along the nephrons

The goal of this study was to investigate water and solute transport, with a focus on sodium transport (T_Na) and metabolism along individual nephron segments under differing physiological and pathophysiological conditions. To accomplish this goal, we developed a computational model of solute transport and oxygen consumption (Q_O2 ) along different nephron populations of a rat kidney. The model represents detailed epithelial and paracellular transport processes along both the superficial and juxtamedullary nephrons, with the loop of Henle of each model nephron extending to differing depths of the inner medulla. We used the model to assess how changes in T_Na may alter Q_O2 in different nephron segments and how shifting the T_Na sites alters overall kidney Q_O2 Under baseline conditions, the model predicted a whole kidney T_Na/Q_O2 , which denotes the number of moles of Na⁺ reabsorbed per moles of O₂ consumed, of ∼15, with T_Na efficiency predicted to be significantly greater in cortical nephron segments than in medullary segments. The T_Na/Q_O2 ratio was generally similar among the superficial and juxtamedullary nephron segments, except for the proximal tubule, where T_Na/Q_O2 was ∼20% higher in superficial nephrons, due to the larger luminal flow along the juxtamedullary proximal tubules and the resulting higher, flow-induced transcellular transport. Moreover, the model predicted that an increase in single-nephron glomerular filtration rate does not significantly affect T_Na/Q_O2 in the proximal tubules but generally increases T_Na/Q_O2 along downstream segments. The latter result can be attributed to the generally higher luminal [Na⁺], which raises paracellular T_Na Consequently, vulnerable medullary segments, such as the S3 segment and medullary thick ascending limb, may be relatively protected from flow-induced increases in Q_O2 under pathophysiological conditions.

Key to Opening Kidney for In Vitro-In Vivo Extrapolation Entrance in Health and Disease: Part I: In Vitro Systems and Physiological Data

The programme for the 2015 AAPS Annual Meeting and Exhibition (Orlando, FL; 25-29 October 2015) included a sunrise session presenting an overview of the state-of-the-art tools for in vitro-in vivo extrapolation (IVIVE) and mechanistic prediction of renal drug disposition. These concepts are based on approaches developed for prediction of hepatic clearance, with consideration of scaling factors physiologically relevant to kidney and the unique and complex structural organisation of this organ. Physiologically relevant kidney models require a number of parameters for mechanistic description of processes, supported by quantitative information on renal physiology (system parameters) and in vitro/in silico drug-related data. This review expands upon the themes raised during the session and highlights the importance of high quality in vitro drug data generated in appropriate experimental setup and robust system-related information for successful IVIVE of renal drug disposition. The different in vitro systems available for studying renal drug metabolism and transport are summarised and recent developments involving state-of-the-art technologies highlighted. Current gaps and uncertainties associated with system parameters related to human kidney for the development of physiologically based pharmacokinetic (PBPK) model and quantitative prediction of renal drug disposition, excretion, and/or metabolism are identified.

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.

Mathematical Model of Ammonia Handling in the Rat Renal Medulla

The kidney is one of the main organs that produces ammonia and release it into the circulation. Under normal conditions, between 30 and 50% of the ammonia produced in the kidney is excreted in the urine, the rest being absorbed into the systemic circulation via the renal vein. In acidosis and in some pathological conditions, the proportion of urinary excretion can increase to 70% of the ammonia produced in the kidney. Mechanisms regulating the balance between urinary excretion and renal vein release are not fully understood. We developed a mathematical model that reflects current thinking about renal ammonia handling in order to investigate the role of each tubular segment and identify some of the components which might control this balance. The model treats the movements of water, sodium chloride, urea, NH₃ and NH4+, and non-reabsorbable solute in an idealized renal medulla of the rat at steady state. A parameter study was performed to identify the transport parameters and microenvironmental conditions that most affect the rate of urinary ammonia excretion. Our results suggest that urinary ammonia excretion is mainly determined by those parameters that affect ammonia recycling in the loops of Henle. In particular, our results suggest a critical role for interstitial pH in the outer medulla and for luminal pH along the inner medullary collecting ducts.

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.