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

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.

Modeling the Steady-State Effects of Mean Arterial Pressure on the Kidneys

Goal: We describe the relationship between mean arterial pressure (MAP) and glomerular filtration rate (GFR) since therapies affecting MAP can have large effects on kidney function. Methods: We developed a closed-loop, steady-state mechanistic model of the human kidney with a reduced parameter set estimated from measurements. Results: The model was first validated against literature models. Further, GFR was validated against intensive care patient data (root mean squared error (RMSE) 13.5 mL/min) and against hypertensive patients receiving sodium nitroprusside (SNP) (RMSE less than 5 mL/min). A sensitivity analysis of the model reinforced the fact that vascular resistance is inversely related to GFR and showed that changes to either vascular resistance or renal autoregulation cause a significant change in sodium concentration in the descending limb of Henle. Conclusions: This model can be used to determine the impact of MAP on GFR and overall kidney health. The modeling framework lends itself to personalization of the model to a specific human.

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.

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

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

Mathematical models for kidney function focusing on clinical interest

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

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.

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

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

Implementation of a model of bodily fluids regulation

The classic model of blood pressure regulation by Guyton et al. (Annu Rev Physiol 34:13–46, 1972a; Ann Biomed Eng 1:254–281, 1972b) set a new standard for quantitative exploration of physiological function and led to important new insights, some of which still remain the focus of debate, such as whether the kidney plays the primary role in the genesis of hypertension (Montani et al. in Exp Physiol 24:41–54, 2009a; Exp Physiol 94:382–388, 2009b; Osborn et al. in Exp Physiol 94:389–396, 2009a; Exp Physiol 94:388–389, 2009b). Key to the success of this model was the fact that the authors made the computer code (in FORTRAN) freely available and eventually provided a convivial user interface for exploration of model behavior on early microcomputers (Montani et al. in Int J Bio-med Comput 24:41–54, 1989). Ikeda et al. (Ann Biomed Eng 7:135–166, 1979) developed an offshoot of the Guyton model targeting especially the regulation of body fluids and acid–base balance; their model provides extended renal and respiratory functions and would be a good basis for further extensions. In the interest of providing a simple, useable version of Ikeda et al.’s model and to facilitate further such extensions, we present a practical implementation of the model of Ikeda et al. (Ann Biomed Eng 7:135–166, 1979), using the ODE solver Berkeley Madonna.

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.

Regulation of calcium reabsorption along the rat nephron: a modeling study

We expanded a mathematical model of transepithelial transport along the rat nephron to include the transport of Ca2+ and probe the impact of calcium-sensing mechanisms on Ca2+ reabsorption. The model nephron extends from the medullary thick ascending limb (mTAL) to the inner medullary collecting duct (IMCD). Our model reproduces several experimental findings, such as measurements of luminal Ca2+ concentrations in cortical tubules, and the effects of furosemide or deletion of the transient receptor potential channel vanilloid subtype 5 (TRPV5) on urinary Ca2+ excretion. In vitro microperfusion of rat TAL has demonstrated that activation of the calcium-sensing receptor CaSR lowers the TAL permeability to Ca2+, PCaTAL (Loupy A, Ramakrishnan SK, Wootla B, Chambrey R, de la Faille R, Bourgeois S, Bruneval P, Mandet C, Christensen EI, Faure H, Cheval L, Laghmani K, Collet C, Eladari D, Dodd RH, Ruat M, Houillier P. J Clin Invest 122: 3355, 2012). Our results suggest that this regulatory mechanism significantly impacts renal Ca2+ handling: when plasma Ca2+ concentration ([Ca2+]) is raised by 10%, the CaSR-mediated reduction in PCaTAL per se is predicted to enhance urinary Ca2+ excretion by ∼30%. If high [Ca2+] also induces renal outer medullary potassium (ROMK) inhibition, urinary Ca2+ excretion is further raised. In vitro, increases in luminal [Ca2+] have been shown to activate H+-ATPase pumps in the outer medullary CD and to lower the water permeability of IMCD. Our model suggests that if these responses exhibit the sigmoidal dependence on luminal [Ca2+] that is characteristic of CaSR, then the impact of elevated Ca2+ levels in the CD on urinary volume and pH remains limited. Finally, our model suggests that CaSR inhibitors could significantly reduce urinary Ca2+ excretion in hypoparathyroidism, thereby reducing the risk of calcium stone formation.

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.

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.

A matrix-induced ion suppression method to normalize concentration in urinary metabolomics studies using flow injection analysis electrospray ionization mass spectrometry

Normalizing the total urine concentration is important for minimizing bias in urinary metabolomics analysis comparisons. In this study, we report a matrix-induced ion suppression (MIIS)-based method to normalize concentration using flow injection analysis coupled with electrospray ionization mass spectrometry (FIA-ESI-MS). An ion suppression indicator (ISI) was spiked into urine samples, and the intensity of the extracted ion chromatogram (EIC) for ISI in a urine matrix was subtracted by the EIC for a blank solution and used to calculate the extent to which the signal was reduced by the urine matrix. A series dilution of pooled urine samples was used to correlate the urine concentration and level of ion suppression for ISI. A regression equation was used to estimate the relative concentration of unknown urine samples. The MIIS method was validated for linearity, precision and accuracy. We obtained a good correlation using a quadratic regression model for 1- to 32-fold urine dilutions (R(2)=0.998). The reproducibility (n=4) and intermediate precision (n=3) were below 5% RSD, and the accuracy ranged from 97.15% to 102.10%. The established method was used to estimate the relative concentrations of 16 urine samples, and the results were compared with commonly used normalization methods. Pearson's correlation test was used to demonstrate that the MIIS method correlated highly with the creatinine and osmolarity methods; the correlation coefficients were 0.93 and 0.99, respectively. We successfully applied this method to a urinary metabolomics study on breast cancer. This study demonstrated the MIIS method is simple, accurate and can contribute to data integrity in urinary metabolomics studies.

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.

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

We have developed a whole kidney model of the urine concentrating mechanism and renal autoregulation. The model represents the tubuloglomerular feedback (TGF) and myogenic mechanisms, which together affect the resistance of the afferent arteriole and thus glomerular filtration rate. TGF is activated by fluctuations in macula densa [Cl(-)] and the myogefnic mechanism by changes in hydrostatic pressure. The model was used to investigate the relative contributions of medullary blood flow autoregulation and inhibition of transport in the proximal convoluted tubule to pressure natriuresis in both diuresis and antidiuresis. The model predicts that medullary blood flow autoregulation, which only affects the interstitial solute composition in the model, has negligible influence on the rate of NaCl excretion. However, it exerts a significant effect on urine flow, particularly in the antidiuretic kidney. This suggests that interstitial washout has significant implications for the maintenance of hydration status but little direct bearing on salt excretion, and that medullary blood flow may only play a signaling role for stimulating a pressure-natriuresis response. Inhibited reabsorption in the model proximal convoluted tubule is capable of driving pressure natriuresis when the known actions of vasopressin on the collecting duct epithelium are taken into account.