Multistability in tubuloglomerular feedback and spectral complexity in spontaneously hypertensive rats

Optimizing SGLT inhibitor treatment for diabetes with chronic kidney diseases

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

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

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

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

Theoretical assessment of the Ca2+ oscillations in the afferent arteriole smooth muscle cell of the rat kidney

The afferent arteriole (AA) of rat kidney exhibits the myogenic response, in which the vessel constricts in response to an elevation in blood pressure and dilates in response to a pressure reduction. Additionally, the AA exhibits spontaneous oscillations in vascular tone at physiological luminal pressures. These time-periodic oscillations stem from the dynamic exchange of Ca[Formula: see text] between the cytosol and the sarcoplasmic reticulum, coupled to the stimulation of Ca[Formula: see text]-activated potassium and chloride channels, and to the modulation of voltage-gated L-type Ca[Formula: see text] channels. The effects of physiological factors, including blood pressure and vasoactive substances, on AA vasomotion remain to be well characterized. In this paper, we analyze a mathematical model of Ca[Formula: see text] signaling in an AA smooth muscle cell. The model represents detailed transmembrane ionic transport, intracellular Ca[Formula: see text] dynamics as well as kinetics of nitric oxide (NO) and superoxide (O[Formula: see text]) formation, diffusion and reaction. NO is an important factor in the maintenance of blood pressure and O[Formula: see text] has been shown to contribute significantly to the functional alternations of blood vessels in hypertension. We perform a bifurcation analysis of the model equations to assess the effect of luminal pressure, NO and O[Formula: see text] on the behaviors of limit cycle oscillations.

Biophysics and biofluid dynamics of primary cilia: evidence for and against the flow-sensing function

Primary cilia have been called “the forgotten organelle” for over 20 yr. As cilia now have their own journal and several books devoted to their study, perhaps it is time to reconsider the moniker “forgotten organelle.” In fact, during the drafting of this review, 12 relevant publications have been issued; we therefore apologize in advance for any relevant work we inadvertently omitted. What purpose is yet another ciliary review? The primary goal of this review is to specifically examine the evidence for and against the hypothesized flow-sensing function of primary cilia expressed by differentiated epithelia within a kidney tubule, bringing together differing disciplines and their respective conceptual and experimental approaches. We will show that understanding the biophysics/biomechanics of primary cilia provides essential information for understanding any potential role of ciliary function in disease. We will summarize experimental and mathematical models used to characterize renal fluid flow and incident force on primary cilia and to characterize the mechanical response of cilia to an externally applied force and discuss possible ciliary-mediated cell signaling pathways triggered by flow. Throughout, we stress the importance of separating the effects of fluid shear and stretch from the action of hydrodynamic drag.

Modeling the effects of positive and negative feedback in kidney blood flow control

Blood flow in the mammalian kidney is tightly autoregulated. One of the important autoregulation mechanisms is the myogenic response, which is activated by perturbations in blood pressure along the afferent arteriole. Another is the tubuloglomerular feedback, which is a negative feedback that responds to variations in tubular fluid [Cl(-)] at the macula densa.(1) When initiated, both the myogenic response and the tubuloglomerular feedback adjust the afferent arteriole muscle tone. A third mechanism is the connecting tubule glomerular feedback, which is a positive feedback mechanism located at the connecting tubule, downstream of the macula densa. The connecting tubule glomerular feedback is much less well studied. The goal of this study is to investigate the interactions among these feedback mechanisms and to better understand the effects of their interactions. To that end, we have developed a mathematical model of solute transport and blood flow control in the rat kidney. The model represents the myogenic response, tubuloglomerular feedback, and connecting tubule glomerular feedback. By conducting a bifurcation analysis, we studied the stability of the system under a range of physiologically-relevant parameters. The bifurcation results were confirmed by means of a comparison with numerical simulations. Additionally, we conducted numerical simulations to test the hypothesis that the interactions between the tubuloglomerular feedback and the connecting tubule glomerular feedback may give rise to a yet-to-be-explained low-frequency oscillation that has been observed in experimental records.

Conduction of feedback-mediated signal in a computational model of coupled nephrons

The nephron in the kidney regulates its fluid flow by several autoregulatory mechanisms. Two primary mechanisms are the myogenic response and the tubuloglomerular feedback (TGF). The myogenic response is a property of the pre-glomerular vasculature in which a rise in intravascular pressure elicits vasoconstriction that generates a compensatory increase in vascular resistance. TGF is a negative feedback response that balances glomerular filtration with tubular reabsorptive capacity. While each nephron has its own autoregulatory response, the responses of the kidney's many nephrons do not act autonomously but are instead coupled through the pre-glomerular vasculature. To better understand the conduction of these signals along the pre-glomerular arterioles and the impacts of internephron coupling on nephron flow dynamics, we developed a mathematical model of renal haemodynamics of two neighbouring nephrons that are coupled in that their afferent arterioles arise from a common cortical radial artery. Simulations were conducted to estimate internephron coupling strength, determine its dependence on vascular properties and to investigate the effect of coupling on TGF-mediated flow oscillations. Simulation results suggest that reduced gap-junctional conductances may yield stronger internephron TGF coupling and highly irregular TGF-mediated oscillations in nephron dynamics, both of which experimentally have been associated with hypertensive rats.

Bifurcation study of blood flow control in the kidney

Renal blood flow is maintained within a narrow window by a set of intrinsic autoregulatory mechanisms. Here, a mathematical model of renal hemodynamics control in the rat kidney is used to understand the interactions between two major renal autoregulatory mechanisms: the myogenic response and tubuloglomerular feedback. A bifurcation analysis of the model equations is performed to assess the effects of the delay and sensitivity of the feedback system and the time constants governing the response of vessel diameter and smooth muscle tone. The results of the bifurcation analysis are verified using numerical simulations of the full nonlinear model. Both the analytical and numerical results predict the generation of limit cycle oscillations under certain physiologically relevant conditions, as observed in vivo.

Mathematical modeling of urea transport in the kidney

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

Mathematical modeling of renal hemodynamics in physiology and pathophysiology

In addition to the excretion of metabolic waste and toxin, the kidney plays an indispensable role in regulating the balance of water, electrolyte, acid-base, and blood pressure. For the kidney to maintain proper functions, hemodynamic control is crucial. In this review, we describe representative mathematical models that have been developed to better understand the kidney's autoregulatory processes. We consider mathematical models that simulate glomerular filtration, and renal blood flow regulation by means of the myogenic response and tubuloglomerular feedback. We discuss the extent to which these modeling efforts have expanded the understanding of renal functions in health and disease.

Mathematical modeling of kidney transport

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

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

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

Modeling Transport and Flow Regulatory Mechanisms of the Kidney

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

Mechanisms of pressure-diuresis and pressure-natriuresis in Dahl salt-resistant and Dahl salt-sensitive rats

Background

Data on blood flow regulation, renal filtration, and urine output in salt-sensitive Dahl S rats fed on high-salt (hypertensive) and low-salt (prehypertensive) diets and salt-resistant Dahl R rats fed on high-salt diets were analyzed using a mathematical model of renal blood flow regulation, glomerular filtration, and solute transport in a nephron.

Results

The mechanism of pressure-diuresis and pressure-natriuresis that emerges from simulation of the integrated systems is that relatively small increases in glomerular filtration that follow from increases in renal arterial pressure cause relatively large increases in urine and sodium output. Furthermore, analysis reveals the minimal differences between the experimental cases necessary to explain the observed data. It is determined that differences in renal afferent and efferent arterial resistances are able to explain all of the qualitative differences in observed flows, filtration rates, and glomerular pressure as well as the differences in the pressure-natriuresis and pressure-diuresis relationships in the three groups. The model is able to satisfactorily explain data from all three groups without varying parameters associated with glomerular filtration or solute transport in the nephron component of the model.

Conclusions

Thus the differences between the experimental groups are explained solely in terms of difference in blood flow regulation. This finding is consistent with the hypothesis that, if a shift in the pressure-natriuresis relationship is the primary cause of elevated arterial pressure in the Dahl S rat, then alternation in how renal afferent and efferent arterial resistances are regulated represents the primary cause of chronic hypertension in the Dahl S rat.

Signal transduction in a compliant short loop of Henle

To study the transformation of fluctuations in filtration rate into tubular fluid chloride concentration oscillations alongside the macula densa, we have developed a mathematical model for tubuloglomerular feedback (TGF) signal transduction along the pars recta, the descending limb, and the thick ascending limb (TAL) of a short-looped nephron. The model tubules are assumed to have compliant walls and, thus, a tubular radius that depends on the transmural pressure difference. Previously, it has been predicted that TGF transduction by the TAL is a generator of nonlinearities: if a sinusoidal oscillation is added to a constant TAL flow rate, then the time required for a fluid element to traverse the TAL is oscillatory in time but nonsinusoidal. The results from the new model simulations presented here predict that TGF transduction by the loop of Henle is also, in the same sense, a generator of nonlinearities. Thus, this model predicts that oscillations in tubular fluid alongside the macula densa will be nonsinusoidal and will exhibit harmonics of sinusoidal perturbations of pars recta flow. Model results also indicate that the loop acts as a low-pass filter in the transduction of the TGF signal.

Modeling TGF-mediated flow dynamics in a system of three coupled nephrons

This paper focuses on a mathematical model of a system of three closely coupled nephrons and accompanying analytical and computational analysis. In our previous modeling efforts, we have shown how coupling magnifies the tendency of many coupled identical nephrons to oscillate owing to tubuloglomerular feedback (TGF) mechanism. However, in this study, our focus is on the coupled nonidentical nephrons and their dynamics due to the TGF system. Our detailed analytical and computational results suggest that systems of three nonidentical nephrons coupled to their nearest neighbors are prone to be found in an oscillatory state, relative to a single-nephron case with the same properties; however, their steady-state regions are not necessarily as small as it was predicted from the system of many coupled identical nephrons cases.

Signal transduction in a compliant thick ascending limb

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

Feedback-mediated dynamics in a model of coupled nephrons with compliant thick ascending limbs

The tubuloglomerular feedback (TGF) system in the kidney, a key regulator of glomerular filtration rate, has been shown in physiologic experiments in rats to mediate oscillations in thick ascending limb (TAL) tubular fluid pressure, flow, and NaCl concentration. In spontaneously hypertensive rats, TGF-mediated flow oscillations may be highly irregular. We conducted a bifurcation analysis of a mathematical model of nephrons that are coupled through their TGF systems; the TALs of these nephrons are assumed to have compliant tubular walls. A characteristic equation was derived for a model of two coupled nephrons. Analysis of that characteristic equation has revealed a number of parameter regions having the potential for differing stable dynamic states. Numerical solutions of the full equations for two model nephrons exhibit a variety of behaviors in these regions. Also, model results suggest that the stability of the TGF system is reduced by the compliance of TAL walls and by internephron coupling; as a result, the likelihood of the emergence of sustained oscillations in tubular fluid pressure and flow is increased. Based on information provided by the characteristic equation, we identified parameters with which the model predicts irregular tubular flow oscillations that exhibit a degree of complexity that may help explain the emergence of irregular oscillations in spontaneously hypertensive rats.

Feedback-mediated dynamics in a model of a compliant thick ascending limb

The tubuloglomerular feedback (TGF) system in the kidney, which is a key regulator of filtration rate, has been shown in physiologic experiments in rats to mediate oscillations in tubular fluid pressure and flow, and in NaCl concentration in the tubular fluid of the thick ascending limb (TAL). In this study, we developed a mathematical model of the TGF system that represents NaCl transport along a TAL with compliant walls. The model was used to investigate the dynamic behaviors of the TGF system. A bifurcation analysis of the TGF model equations was performed by deriving and finding roots of the characteristic equation, which arises from a linearization of the model equations. Numerical simulations of the full model equations were conducted to assist in the interpretation of the bifurcation analysis. These techniques revealed a complex parameter region that allows a variety of qualitatively different model solutions: a regime having one stable, time-independent steady-state solution; regimes having one stable oscillatory solution only; and regimes having multiple possible stable oscillatory solutions. Model results suggest that the compliance of the TAL walls increases the tendency of the model TGF system to oscillate.

Coupling-induced complexity in nephron models of renal blood flow regulation

Tubular pressure and nephron blood flow time series display two interacting oscillations in rats with normal blood pressure. Tubuloglomerular feedback (TGF) senses NaCl concentration in tubular fluid at the macula densa, adjusts vascular resistance of the nephron's afferent arteriole, and generates the slower, larger-amplitude oscillations (0.02-0.04 Hz). The faster smaller oscillations (0.1-0.2 Hz) result from spontaneous contractions of vascular smooth muscle triggered by cyclic variations in membrane electrical potential. The two mechanisms interact in each nephron and combine to act as a high-pass filter, adjusting diameter of the afferent arteriole to limit changes of glomerular pressure caused by fluctuations of blood pressure. The oscillations become irregular in animals with chronic high blood pressure. TGF feedback gain is increased in hypertensive rats, leading to a stronger interaction between the two mechanisms. With a mathematical model that simulates tubular and arteriolar dynamics, we tested whether an increase in the interaction between TGF and the myogenic mechanism can cause the transition from periodic to irregular dynamics. A one-dimensional bifurcation analysis, using the coefficient that couples TGF and the myogenic mechanism as a bifurcation parameter, shows some regions with chaotic dynamics. With two nephrons coupled electrotonically, the chaotic regions become larger. The results support the hypothesis that increased oscillator interactions contribute to the transition to irregular fluctuations, especially when neighboring nephrons are coupled, which is the case in vivo.

Modeling transport in the kidney: investigating function and dysfunction

Mathematical models of water and solute transport in the kidney have significantly expanded our understanding of renal function in both health and disease. This review describes recent theoretical developments and emphasizes the relevance of model findings to major unresolved questions and controversies. These include the fundamental processes by which urine is concentrated in the inner medulla, the ultrastructural basis of proteinuria, irregular flow oscillation patterns in spontaneously hypertensive rats, and the mechanisms underlying the hypotensive effects of thiazides. Macroscopic models of water, NaCl, and urea transport in populations of nephrons have served to test, confirm, or refute a number of hypotheses related to the urine concentrating mechanism. Other macroscopic models focus on the mechanisms, role, and irregularities of renal hemodynamic control and on the regulation of renal oxygenation. At the mesoscale, models of glomerular filtration have yielded significant insight into the ultrastructural basis underlying a number of disorders. At the cellular scale, models of epithelial solute transport and pericyte Ca2+ signaling are being used to elucidate transport pathways and the effects of hormones and drugs. Areas where further theoretical progress is conditional on experimental advances are also identified.

Tubuloglomerular feedback signal transduction in a short loop of henle

In previous studies, we used a mathematical model of the thick ascending limb (TAL) to investigate nonlinearities in the tubuloglomerular feedback (TGF) loop. That model does not represent other segments of the nephron, the water, and NaCl transport along which may impact fluid flow rate and NaCl transport along the TAL. To investigate the extent to which those transport processes affect TGF mediation, we have developed a mathematical model for TGF signal transduction in a short loop nephron. The model combines a simple representation of the renal cortex with a highly-detailed representation of the outer medulla (OM). The OM portion of the model is based on an OM urine concentrating mechanism model previously developed by Layton and Layton (Am. J. Renal 289:F1346-F1366, 2005a). When perturbations are applied to intratubular fluid flow at the proximal straight tubule entrance, the present model predicts oscillations in fluid flow and solute concentrations in the cortical TAL and interstitium, and in all tubules, vessels, and interstitium in the OM. Model results suggest that TGF signal transduction by the TAL is a generator of nonlinearities: if a sinusoidal oscillation is added to constant intratubular fluid flow, the time required for an element of tubular fluid to traverse the TAL is oscillatory, but nonsinusoidal; those results are consistent with our previous studies. As a consequence, oscillations in NaCl concentration in tubular fluid alongside the macula densa (MD) will be nonsinusoidal and contain harmonics of the original sinusoidal frequency. Also, the model predicts that the oscillations in NaCl concentration at the loop-bend fluid are smaller in amplitude than those at the MD, a result that further highlights the crucial role of TAL in the nonlinear transduction of TGF signal from SNGFR to MD NaCl concentration.

Multistable dynamics mediated by tubuloglomerular feedback in a model of coupled nephrons

To help elucidate the causes of irregular tubular flow oscillations found in the nephrons of spontaneously hypertensive rats (SHR), we have conducted a bifurcation analysis of a mathematical model of two nephrons that are coupled through their tubuloglomerular feedback (TGF) systems. This analysis was motivated by a previous modeling study which predicts that NaCl backleak from a nephron's thick ascending limb permits multiple stable oscillatory states that are mediated by TGF (Layton et al. in Am. J. Physiol. Renal Physiol. 291:F79-F97, 2006); that prediction served as the basis for a comprehensive, multifaceted hypothesis for the emergence of irregular flow oscillations in SHR. However, in that study, we used a characteristic equation obtained via linearization from a single-nephron model, in conjunction with numerical solutions of the full, nonlinear model equations for two and three coupled nephrons. In the present study, we have derived a characteristic equation for a model of any finite number of mutually coupled nephrons having NaCl backleak. Analysis of that characteristic equation for the case of two coupled nephrons has revealed a number of parameter regions having the potential for differing stable dynamic states. Numerical solutions of the full equations for two model nephrons exhibit a variety of behaviors in these regions. Some behaviors exhibit a degree of complexity that is consistent with our hypothesis for the emergence of irregular oscillations in SHR.

Theoretical models for regulation of blood flow

Blood-flow rate in the normal microcirculation is regulated to meet the metabolic demands of the tissues, which vary widely with position and with time, but is relatively unaffected by changes of arterial pressure over a considerable range. The regulation of blood flow is achieved by the combined effects of multiple interacting mechanisms, including sensitivity to pressure, flow rate, metabolite levels, and neural signals. The main effectors of flow regulation, the arterioles and small arteries, are located at a distance from the regions of tissue that they supply. Flow regulation requires the sensing of metabolic and hemodynamic conditions and the transfer of information about tissue metabolic status to upstream vessels. Theoretical approaches can contribute to the understanding of flow regulation by providing quantitative descriptions of the mechanisms involved, by showing how these mechanisms interact in networks of interconnected microvessels supplying metabolically active tissues, and by establishing relationships between regulatory processes occurring at the microvascular level and variations of metabolic activity and perfusion in whole tissues. Here, a review is presented of previous and current theoretical approaches for investigating the regulation of blood flow in the microcirculation.

Analysis of nonstationarity in renal autoregulation mechanisms using time-varying transfer and coherence functions

The extent to which renal blood flow dynamics vary in time and whether such variation contributes substantively to dynamic complexity have emerged as important questions. Data from Sprague-Dawley rats (SDR) and spontaneously hypertensive rats (SHR) were analyzed by time-varying transfer functions (TVTF) and time-varying coherence functions (TVCF). Both TVTF and TVCF allow quantification of nonstationarity in the frequency ranges associated with the autoregulatory mechanisms. TVTF analysis shows that autoregulatory gain in SDR and SHR varies in time and that SHR exhibit significantly more nonstationarity than SDR. TVTF gain in the frequency range associated with the myogenic mechanism was significantly higher in SDR than in SHR, but no statistical difference was found with tubuloglomerular (TGF) gain. Furthermore, TVCF analysis revealed that the coherence in both strains is significantly nonstationary and that low-frequency coherence was negatively correlated with autoregulatory gain. TVCF in the frequency range from 0.1 to 0.3 Hz was significantly higher in SDR (7 out of 7, >0.5) than in SHR (5 out of 6, <0.5), and consistent for all time points. For TGF frequency range (0.03-0.05 Hz), coherence exhibited substantial nonstationarity in both strains. Five of six SHR had mean coherence (<0.5), while four of seven SDR exhibited coherence (<0.5). Together, these results demonstrate substantial nonstationarity in autoregulatory dynamics in both SHR and SDR. Furthermore, they indicate that the nonstationarity accounts for most of the dynamic complexity in SDR, but that it accounts for only a part of the dynamic complexity in SHR.

Synchronization among mechanisms of renal autoregulation is reduced in hypertensive rats

We searched for synchronization among autoregulation mechanisms using wavelet transforms applied to tubular pressure recordings in nephron pairs from the surface of rat kidneys. Nephrons have two oscillatory modes in the regulation of their pressures and flows: a faster (100–200 mHz) myogenic mode, and a slower (20–40 mHz) oscillation in tubuloglomerular feedback (TGF). These mechanisms interact; the TGF mode modulates both the amplitude and the frequency of the myogenic mode. Nephrons also communicate with each other using vascular signals triggered by membrane events in arteriolar smooth muscle cells. In addition, the TGF oscillation changes in hypertension to an irregular fluctuation with characteristics of deterministic chaos. The analysis shows that, within single nephrons of normotensive rats, the myogenic mode and TGF are synchronized at discrete frequency ratios, with 5:1 most common. There is no distinct synchronization ratio in spontaneously hypertensive rats (SHR). In normotensive rats, full synchronization of both TGF and myogenic modes is the most probable state for pairs of nephrons originating in a common cortical radial artery. For SHR, full synchronization is less probable; most common in SHR is a state of partial synchronization with entrainment between neighboring nephrons for only one of the modes. Modulation of the myogenic mode by the TGF mode is much stronger in hypertensive than in normotensive rats. Synchronization among nephrons forms the basis for an integrated reaction to blood pressure fluctuations. Reduced synchronization in SHR suggests that the effectiveness of the coordinated response is impaired in hypertension.

Effect of tubular inhomogeneities on filter properties of thick ascending limb of Henle's loop

We used a simple mathematical model of rat thick ascending limb (TAL) of the loop of Henle to predict the impact of spatially inhomogeneous NaCl permeability, spatially inhomogeneous NaCl active transport, and spatially inhomogeneous tubular radius on luminal NaCl concentration when sustained, sinusoidal perturbations were superimposed on steady-state TAL flow. A mathematical model previously devised by us that used homogeneous TAL transport and fixed TAL radius predicted that such perturbations result in TAL luminal fluid NaCl concentration profiles that are standing waves. That study also predicted that nodes in NaCl concentration occur at the end of the TAL when the tubular fluid transit time equals the period of a periodic perturbation, and that, for non-nodal periods, sinusoidal perturbations generate non-sinusoidal oscillations (and thus a series of harmonics) in NaCl concentration at the TAL end. In the present study we find that the inhomogeneities transform the standing waves and their associated nodes into approximate standing waves and approximate nodes. The impact of inhomogeneous NaCl permeability is small. However, for inhomogeneous active transport or inhomogeneous radius, the oscillations for non-nodal periods tend to be less sinusoidal and more distorted than in the homogeneous case and to thus have stronger harmonics. Both the homogeneous and non-homogeneous cases predict that the TAL, in its transduction of flow oscillations into concentration oscillations, acts as a low-pass filter, but the inhomogeneities result in a less effective filter that has accentuated non-linearities.

Interactions contributing to kidney blood flow autoregulation

PURPOSE OF REVIEW

Autoregulation of renal blood flow has traditionally been considered to stabilize glomerular filtration, and thus tubular load, in the face of blood pressure fluctuations. This view arose because of the contribution of tubuloglomerular feedback, which senses distal tubular fluid composition, to regulation and autoregulation of renal blood flow. Studies have indicated a more important role for the myogenic mechanism. It has been proposed that the 'purpose' of autoregulation is to defend glomerular structure. Both these views may be incomplete because neither takes into consideration the complex interactions between tubuloglomerular feedback and the myogenic mechanism and among nephrons whose afferent arterioles derived from a common interlobular artery.

RECENT FINDINGS

Recent findings indicate that it is now indisputable that effective autoregulation is necessary for defense of glomerular structure. Extensive modulation of the myogenic mechanism by tubuloglomerular feedback has been shown using a variety of experimental designs that have illuminated one pathway (neuronal nitric oxide synthase at the macula densa) by which this occurs.

SUMMARY

These findings indicate that the myogenic mechanism can no longer be considered as a purely vascular mechanism in the kidney and instead receives information via tubuloglomerular feedback about the status of renal function.

Parameter estimation of feedback gain in a stochastic model of renal hemodynamics: differences between spontaneously hypertensive and Sprague-Dawley rats

Proximal tubular pressure shows periodic self-sustained oscillations in normotensive rats but highly irregular fluctuations in spontaneously hypertensive rats (SHR). Although we have suggested that the irregular fluctuations in SHR represent low-dimensional deterministic chaos in tubuloglomerular feedback (TGF), they could also arise from other mechanisms, such as intrinsic instabilities in preglomerular vessels or inputs from neighboring, coupled nephrons. To test this possibility, we applied a parameter estimation procedure to a model of TGF, where a stochastic process was added to represent mechanisms not included explicitly in the model. In its deterministic version, the model can have chaotic dynamics arising from TGF. The model introduces random fluctuations into a parameter that determines the gain of TGF. The model shows a rich variety of dynamics ranging from low-dimensional deterministic oscillations and chaos to high-dimensional random fluctuations. To fit the data from normotensive rats, the model must introduce only a small variation in the feedback gain, and its estimates of that gain agree well with experimental values. These results support the use of the deterministic model of nephron dynamics in normotensive rats. In contrast, the irregular tubular pressure fluctuations in SHR were best described by a model dominated by random parameter fluctuations. The results point to the failure of simple mathematical models of nephron dynamics adequately to describe processes that are important for the irregular tubular pressure fluctuations and the need to consider other factors, such as differences in vascular function or nephron-nephron interactions, in further work on this problem.

Assessment of renal autoregulation

The kidney displays highly efficient autoregulation so that under steady-state conditions renal blood flow (RBF) is independent of blood pressure over a wide range of pressure. Autoregulation occurs in the preglomerular microcirculation and is mediated by two, perhaps three, mechanisms. The faster myogenic mechanism and the slower tubuloglomerular feedback contribute both directly and interactively to autoregulation of RBF and of glomerular capillary pressure. Multiple experiments have been used to study autoregulation and can be considered as variants of two basic designs. The first measures RBF after multiple stepwise changes in renal perfusion pressure to assess how a biological condition or experimental maneuver affects the overall pressure-flow relationship. The second uses time-series analysis to better understand the operation of multiple controllers operating in parallel on the same vascular smooth muscle. There are conceptual and experimental limitations to all current experimental designs so that no one design adequately describes autoregulation. In particular, it is clear that the efficiency of autoregulation varies with time and that most current techniques do not adequately address this issue. Also, the time-varying and nonadditive interaction between the myogenic mechanism and tubuloglomerular feedback underscores the difficulty of dissecting their contributions to autoregulation. We consider the modulation of autoregulation by nitric oxide and use it to illustrate the necessity for multiple experimental designs, often applied iteratively.