Search

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.

The ability of mammals to produce urine hyperosmotic to plasma requires the generation of a gradient of increasing osmolality along the medulla from the corticomedullary junction to the papilla tip. Countercurrent multiplication apparently establishes this gradient in the outer medulla, where there is substantial transepithelial reabsorption of NaCl from the water-impermeable thick ascending limbs of the loops of Henle. However, this process does not establish the much steeper osmotic gradient in the inner medulla, where there are no thick ascending limbs of the loops of Henle and the water-impermeable ascending thin limbs lack active transepithelial transport of NaCl or any other solute. The mechanism generating the osmotic gradient in the inner medulla remains an unsolved mystery, although it is generally considered to involve countercurrent flows in the tubules and vessels. A possible role for the three-dimensional interactions between these inner medullary tubules and vessels in the concentrating process is suggested by creation of physiologic models that depict the three-dimensional relationships of tubules and vessels and their solute and water permeabilities in rat kidneys and by creation of mathematical models based on biologic phenomena. The current mathematical model, which incorporates experimentally determined or estimated solute and water flows through clearly defined tubular and interstitial compartments, predicts a urine osmolality in good agreement with that observed in moderately antidiuretic rats. The current model provides substantially better predictions than previous models; however, the current model still fails to predict urine osmolalities of maximally concentrating rats.

The thick ascending limb (TAL) is a major NaCl reabsorbing site in the nephron. Efficient reabsorption along that segment is thought to be a consequence of the establishment of a strong transepithelial potential that drives paracellular Na+uptake. We used a multicell mathematical model of the TAL to estimate the efficiency of Na+transport along the TAL and to examine factors that determine transport efficiency, given the condition that TAL outflow must be adequately dilute. The TAL model consists of a series of epithelial cell models that represent all major solutes and transport pathways. Model equations describe luminal flows, based on mass conservation and electroneutrality constraints. Empirical descriptions of cell volume regulation (CVR) and pH control were implemented, together with the tubuloglomerular feedback (TGF) system. Transport efficiency was calculated as the ratio of total net Na+transport (i.e., paracellular and transcellular transport) to transcellular Na+transport. Model predictions suggest that 1) the transepithelial Na+concentration gradient is a major determinant of transport efficiency; 2) CVR in individual cells influences the distribution of net Na+transport along the TAL; 3) CVR responses in conjunction with TGF maintain luminal Na+concentration well above static head levels in the cortical TAL, thereby preventing large decreases in transport efficiency; and 4) under the condition that the distribution of Na+transport along the TAL is quasi-uniform, the tubular fluid axial Cl−concentration gradient near the macula densa is sufficiently steep to yield a TGF gain consistent with experimental data.

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.

The urine concentrating mechanism in the mammalian renal inner medulla (IM) is not understood, although it is generally considered to involve countercurrent flows in tubules and blood vessels. A possible role for the three-dimensional relationships of these tubules and vessels in the concentrating process is suggested by recent reconstructions from serial sections labelled with antibodies to tubular and vascular proteins and mathematical models based on these studies. The reconstructions revealed that the lower 60% of each descending thin limb (DTL) of Henle’s loops lacks water channels (aquaporin-1) and osmotic water permeability and ascending thin limbs (ATLs) begin with a prebend segment of constant length. In the outer zone of the IM (i) clusters of coalescing collecting ducts (CDs) form organizing motif for loops of Henle and vasa recta; (ii) DTLs and descending vasa recta (DVR) are arrayed outside CD clusters, whereas ATLs and ascending vasa recta (AVR) are uniformly distributed inside and outside clusters; (iii) within CD clusters, interstitial nodal spaces are formed by a CD on one side, AVR on two sides, and an ATL on the fourth side. These spaces may function as mixing chambers for urea from CDs and NaCl from ATLs. In the inner zone of the IM, cluster organization disappears and half of Henle’s loops have broad lateral bends wrapped around terminal CDs. Mathematical models based on these findings and involving solute mixing in the interstitial spaces can produce urine slightly more concentrated than that of a moderately antidiuretic rat but no higher.

A new, region-based mathematical model of the urine concentrating mechanism of the rat renal inner medulla (IM) was used to investigate the significance of transport and structural properties revealed in recent studies that employed immunohistochemical methods combined with three-dimensional computerized reconstruction. The model simulates preferential interactions among tubules and vessels by representing two concentric regions. The inner region, which represents a collecting duct (CD) cluster, contains CDs, some ascending thin limbs (ATLs), and some ascending vasa recta; the outer region, which represents the intercluster region, contains descending thin limbs, descending vasa recta, remaining ATLs, and additional ascending vasa recta. In the upper portion of the IM, the model predicts that interstitial Na+and urea concentrations (and osmolality) in the CD clusters differ significantly from those in the intercluster regions: model calculations predict that those CD clusters have higher urea concentrations than the intercluster regions, a finding that is consistent with a concentrating mechanism that depends principally on the mixing of NaCl from ATLs and urea from CDs. In the lower IM, the model predicts that limited or nearly zero water permeability in descending thin limb segments will increase concentrating effectiveness by increasing the rate of solute-free water absorption. The model predicts that high urea permeabilities in the upper portions of ATLs and increased contact areas of longest loop bends with CDs both modestly increase concentrating capability. A surprising finding is that the concentrating capability of this region-based model falls short of the capability of a model IM that has radially homogeneous interstitial fluid at each level but is otherwise analogous to the region-based model.

In a mathematical model of the urine concentrating mechanism of the inner medulla of the rat kidney, a nonlinear optimization technique was used to estimate parameter sets that maximize the urine-to-plasma osmolality ratio (U/P) while maintaining the urine flow rate within a plausible physiologic range. The model, which used a central core formulation, represented loops of Henle turning at all levels of the inner medulla and a composite collecting duct (CD). The parameters varied were: water flow and urea concentration in tubular fluid entering the descending thin limbs and the composite CD at the outer-inner medullary boundary; scaling factors for the number of loops of Henle and CDs as a function of medullary depth; location and increase rate of the urea permeability profile along the CD; and a scaling factor for the maximum rate of NaCl transport from the CD. The optimization algorithm sought to maximize a quantity E that equaled U/P minus a penalty function for insufficient urine flow. Maxima of E were sought by changing parameter values in the direction in parameter space in which E increased. The algorithm attained a maximum E that increased urine osmolality and inner medullary concentrating capability by 37.5% and 80.2%, respectively, above base-case values; the corresponding urine flow rate and the concentrations of NaCl and urea were all within or near reported experimental ranges. Our results predict that urine osmolality is particularly sensitive to three parameters: the urea concentration in tubular fluid entering the CD at the outer-inner medullary boundary, the location and increase rate of the urea permeability profile along the CD, and the rate of decrease of the CD population (and thus of CD surface area) along the cortico-medullary axis.

The urine concentrating mechanism of the mammalian kidney, which can produce a urine that is substantially more concentrated than blood plasma during periods of water deprivation, is one of the enduring mysteries in traditional physiology. Owing to the complex lateral and axial relationships of tubules and vessels, in both the outer and inner medulla, the urine concentrating mechanism may only be fully understood in terms of the kidney’s three-dimensional functional architecture and its implications for preferential interactions among tubules and vessels.

The renal medulla produces concentrated urine through the generation of an osmotic gradient extending from the cortico-medullary boundary to the inner medullary tip. This gradient is generated in the outer medulla by the countercurrent multiplication of a comparatively small transepithelial difference in osmotic pressure. This small difference, called a single effect, arises from active NaCl reabsorption from thick ascending limbs, which dilutes ascending limb flow relative to flow in vessels and other tubules. In the inner medulla, the gradient may also be generated by the countercurrent multiplication of a single effect, but the single effect has not been definitively identified. There have been important recent advances in our understanding of key components of the urine concentrating mechanism. In particular, the identification and localization of key transport proteins for water, urea, and sodium, the elucidation of the role and regulation of osmoprotective osmolytes, better resolution of the anatomical relationships in the medulla, and improvements in mathematic modeling of the urine concentrating mechanism. Continued experimental investigation of transepithelial transport and its regulation, both in normal animals and in knock-out mice, and incorporation of the resulting information into mathematic simulations, may help to more fully elucidate the inner medullary urine concentrating mechanism.

Recent studies of three-dimensional architecture of rat renal inner medulla (IM) and expression of membrane proteins associated with fluid and solute transport in nephrons and vasculature have revealed structural and transport properties that likely impact the IM urine concentrating mechanism. These studies have shown that 1) IM descending thin limbs (DTLs) have at least two or three functionally distinct subsegments; 2) most ascending thin limbs (ATLs) and about half the ascending vasa recta (AVR) are arranged among clusters of collecting ducts (CDs), which form the organizing motif through the first 3–3.5 mm of the IM, whereas other ATLs and AVR, along with aquaporin-1-positive DTLs and urea transporter B-positive descending vasa recta (DVR), are external to the CD clusters; 3) ATLs, AVR, CDs, and interstitial cells delimit interstitial microdomains within the CD clusters; and 4) many of the longest loops of Henle form bends that include subsegments that run transversely along CDs that lie in the terminal 500 μm of the papilla tip. Based on a more comprehensive understanding of three-dimensional IM architecture, we distinguish two distinct countercurrent systems in the first 3–3.5 mm of the IM (an intra-CD cluster system and an inter-CD cluster system) and a third countercurrent system in the final 1.5–2 mm. Spatial arrangements of loop of Henle subsegments and multiple countercurrent systems throughout four distinct axial IM zones, as well as our initial mathematical model, are consistent with a solute-separation, solute-mixing mechanism for concentrating urine in the IM.

To better understand how the avian kidney's morphological and transepithelial transport properties affect the urine concentrating mechanism (UCM), an inverse problem was solved for a mathematical model of the quail UCM. In this model, a continuous, monotonically decreasing population distribution of tubes, as a function of medullary length, was used to represent the loops of Henle, which reach to varying levels along the avian medullary cones. A measure of concentrating mechanism efficiency - the ratio of the free-water absorption rate (FWA) to the total NaCl active transport rate (TAT) - was optimized by varying a set of parameters within bounds suggested by physiological experiments. Those parameters include transepithelial transport properties of renal tubules, length of the prebend enlargement of the descending limb (DL), DL and collecting duct (CD) inflows, plasma Na(+) concentration, length of the cortical thick ascending limbs, central core solute diffusivity, and population distribution of loops of Henle and of CDs along the medullary cone. By selecting parameter values that increase urine flow rate (while maintaining a sufficiently high urine-to-plasma osmolality ratio (U/P)) and that reduce TAT, the optimization algorithm identified a set of parameter values that increased efficiency by approximately 60% above base-case efficiency. Thus, higher efficiency can be achieved by increasing urine flow rather than increasing U/P. The algorithm also identified a set of parameters that reduced efficiency by approximately 70% via the production of a urine having near-plasma osmolality at near-base-case TAT. In separate studies, maximum efficiency was evaluated as selected parameters were varied over large ranges. Shorter cones were found to be more efficient than longer ones, and an optimal loop of Henle distribution was found that is consistent with experimental findings.

We have developed a highly detailed mathematical model for the urine concentrating mechanism (UCM) of the rat kidney outer medulla (OM). The model simulates preferential interactions among tubules and vessels by representing four concentric regions that are centered on a vascular bundle; tubules and vessels, or fractions thereof, are assigned to anatomically appropriate regions. Model parameters, which are based on the experimental literature, include transepithelial transport properties of short descending limbs inferred from immunohistochemical localization studies. The model equations, which are based on conservation of solutes and water and on standard expressions for transmural transport, were solved to steady state. Model simulations predict significantly differing interstitial NaCl and urea concentrations in adjoining regions. Active NaCl transport from thick ascending limbs (TALs), at rates inferred from the physiological literature, resulted in model osmolality profiles along the OM that are consistent with tissue slice experiments. TAL luminal NaCl concentrations at the corticomedullary boundary are consistent with tubuloglomerular feedback function. The model exhibited solute exchange, cycling, and sequestration patterns (in tubules, vessels, and regions) that are generally consistent with predictions in the physiological literature, including significant urea addition from long ascending vasa recta to inner-stripe short descending limbs. In a companion study (Layton AT and Layton HE. Am J Physiol Renal Physiol 289: F1367–F1381, 2005), the impact of model assumptions, medullary anatomy, and tubular segmentation on the UCM was investigated by means of extensive parameter studies.

In a companion study (Layton AT and Layton HE. Am J Physiol Renal Physiol 289: F1346–F1366, 2005), a region-based mathematical model was formulated for the urine concentrating mechanism (UCM) in the outer medulla (OM) of the rat kidney. In the present study, we quantified the sensitivity of that model to several structural assumptions, including the degree of regionalization and the degree of inclusion of short descending limbs (SDLs) in the vascular bundles of the inner stripe (IS). Also, we quantified model sensitivity to several parameters that have not been well characterized in the experimental literature, including boundary conditions, short vasa recta distribution, and ascending vasa recta (AVR) solute permeabilities. These studies indicate that regionalization elevates the osmolality of the fluid delivered into the inner medulla via the collecting ducts; that model predictions are not significantly sensitive to boundary conditions; and that short vasa recta distribution and AVR permeabilities significantly impact concentrating capability. Moreover, we investigated, in the context of the UCM, the functional significance of several aspects of tubular segmentation and heterogeneity: SDL segments in the IS that are likely to be impermeable to water but highly permeable to urea; a prebend segment of SDLs that may be functionally like thick ascending limb (TAL); differing IS and outer stripe Na+ active transport rates in TAL; and potential active urea secretion into the proximal straight tubules. Model calculations predict that these aspects of tubular of segmentation and heterogeneity generally enhance solute cycling or promote effective UCM function.

A mathematical model previously formulated by us predicts that limit-cycle oscillations (LCO) in nephron flow are mediated by tubuloglomerular feedback (TGF) and that the LCO arise from a bifurcation that depends heavily on the feedback gain magnitude, γ, and on its relationship to a theoretically determined critical value of gain, γc. In this study, we used that model to show how sustained perturbations in proximal tubule flow, a common experimental maneuver, can initiate or terminate LCO by changing the values of γ and γc, thus changing the sign of γ - γc. This result may help explain experiments in which intratubular pressure oscillations were initiated by the sustained introduction or removal of fluid from the proximal tubule (Leyssac PP and Baumbach L. Acta Physiol Scand 117: 415–419, 1983). In addition, our model predicts that, for a range of TGF sensitivities, sustained perturbations that initiate or terminate LCO can yield substantial and abrupt changes in both distal NaCl delivery and NaCl delivery compensation, changes that may play an important role in the response to physiological challenge.

The urine concentrating mechanism of mammals and birds depends on a counterflow configuration of thousands of nearly parallel tubules in the medulla of the kidney. Along the course of a renal tubule, cell type may change abruptly, resulting in abrupt changes in the physical characteristics and transmural transport properties of the tubule. A mathematical model that faithfully represents these abrupt changes will have jump discontinuities in model parameters. Without proper treatment, such discontinuities may cause unrealistic transmural fluxes and introduce suboptimal spatial convergence in the numerical solution to the model equations. In this study, we show how to treat discontinuous parameters in the context of a previously developed numerical method that is based on the semi-Lagrangian semi-implicit method and Newton's method. The numerical solutions have physically plausible fluxes at the discontinuities and the solutions converge at second order, as is appropriate for the method.

Mathematical models of the urine concentrating mechanism consist of large systems of coupled differential equations. The numerical methods that have usually been used to solve the steady-state formulation of these equations involve implicit Newton-type solvers that are limited by numerical instability attributed to transient flow reversal. Dynamic numerical methods, which solve the dynamic formulation of the equations by means of a direction-sensitive time integration until a steady state is reached, are stable in the presence of transient flow reversal. However, when an explicit, Eulerian-based dynamic method is used, prohibitively small time steps may be required owing to the CFL condition and the stiffness of the problem. In this report, we describe a semi-Lagrangian semi-implicit (SLSI) method for solving the system of hyperbolic partial differential equations that arises in the dynamic formulation. The semi-Lagrangian scheme advances the solution in time by integrating backward along flow trajectories, thus allowing large time steps while maintaining stability. The semi-implicit approach controls stiffness by averaging transtubular transport terms in time along flow trajectories. For sufficiently refined spatial grids, the SLSI method computes stable and accurate solutions with substantially reduced computation costs.

The renal medulla produces concentrated urine through the generation of an osmotic gradient that progressively increases from the cortico-medullary boundary to the inner medullary tip. In the outer medulla, the osmolality gradient arises principally from vigorous active transport of NaCl, without accompanying water, from the thick ascending limbs of short- and long-looped nephrons. In the inner medulla, the source of the osmotic gradient has not been identified. Recently, there have been important advances in our understanding of key components of the urine-concentrating mechanism, including (a) better understanding of the regulation of water, urea, and sodium transport proteins; (b) better resolution of the anatomical relationships in the medulla; and (c) improvements in mathematical modeling of the urine-concentrating mechanism. Continued experimental investigation of signaling pathways regulating transepithelial transport, both in normal animals and in knockout mice, and incorporation of the resulting information into mathematical simulations may help to more fully elucidate the mechanism for concentrating urine in the inner medulla.

A mathematical model was used to investigate how concentrated urine is produced within the medullary cones of the quail kidney. Model simulations were consistent with a concentrating mechanism based on single-solute countercurrent multiplication and on NaCl cycling from ascending to descending limbs of loops of Henle. The model predicted a urine-to-plasma (U/P) osmolality ratio of ∼2.26, a value consistent with maximum avian U/P osmolality ratios. Active NaCl transport from descending limb prebend thick segments contributed 70% of concentrating capability. NaCl entry and water extraction provided 80 and 20%, respectively, of the concentrating effect in descending limb flow. Parameter studies indicated that urine osmolality is sensitive to the rate of fluid entry into descending limbs and collecting ducts at the cone base. Parameter studies also indicated that the energetic cost of concentrating urine is sensitive to loop of Henle population as a function of medullary depth: as the fraction of loops reaching the cone tip increased above anatomic values, urine osmolality increased only marginally, and, ultimately, urine osmolality decreased.

Flow in arterioles is determined by a number of interacting factors, including perfusion pressure, neural stimulation, vasoactive substances, the intrinsic contractility of arteriolar walls, and wall shear stress. We have developed a two-dimensional model of arteriolar fluid flow and mass transport. The model includes a phenomenological representation of the myogenic response of the arteriolar wall, in which an increase in perfusion pressure stimulates vasoconstriction. The model also includes the release, advection, diffusion, degradation, and dilatory action of nitric oxide (NO), a potent, but short-lived, vasodilatory agent. Parameters for the model were taken primarily from the experimental literature of the rat renal afferent arteriole. Solutions to the incompressible Navier-Stokes equations were approximated by means of a splitting that used upwind differencing for the inertial term and a spectral method for the viscous term and incompressibility condition. The immersed boundary method was used to include the forces arising from the arteriolar walls. The advection of NO was computed by means of a high-order flux-corrected transport scheme; the diffusion of NO was computed by a spectral solver. Simulations demonstrated the efficacy of the numerical methods employed, and grid refinement studies confirmed anticipated first-order temporal convergence and demonstrated second-order spatial convergence in key quantities. By providing information about the effective width of the immersed boundary and sheer stress magnitude near that boundary, the grid refinement studies indicate the degree of spatial refinement required for quantitatively reliable simulations. Owing to the dominating effect of NO advection, relative to degradation and diffusion, simulations indicate that NO has the capacity to produce dilation along the entire length of the arteriole.

A simple mathematical model was used to investigate the spectral properties of the tubuloglomerular feedback (TGF) system. A perturbation, consisting of small-amplitude broad-band forcing, was applied to simulated thick ascending limb (TAL) flow, and the resulting spectral response of the TGF pathway was assessed by computing a power spectrum from resulting TGF-regulated TAL flow. Power spectra were computed for both open- and closed-feedback-loop cases. Open-feedback-loop power spectra are consistent with a mathematical analysis that predicts a nodal pattern in TAL frequency response, with nodes corresponding to frequencies where oscillatory flow has a TAL transit time that equals the steady-state fluid transit time. Closed-feedback-loop spectra are dominated by the open-loop spectral response, provided that γ, the magnitude of feedback gain, is less than the critical value γc required for emergence of a sustained TGF-mediated oscillation. For γ exceeding γc, closed-loop spectra have peaks corresponding to the fundamental frequency of the TGF-mediated oscillation and its harmonics. The harmonics, expressed in a nonsinusoidal waveform for tubular flow, are introduced by nonlinear elements of the TGF pathway, notably TAL transit time and the TGF response curve. The effect of transit time on the flow waveform leads to crests that are broader than troughs and to an asymmetry in the magnitudes of increasing and decreasing slopes. For feedback gain magnitude that is sufficiently large, the TGF response curve tends to give a square waveshape to the waveform. Published waveforms and power spectra of in vivo TGF oscillations have features consistent with the predictions of this analysis.

Concentrated urine is produced through the generation of an osmotic gradient in the renal medulla. The gradient in the outer medulla is widely believed to be generated by means of a mechanism in which an osmotic pressure difference is amplified by countercurrent multiplication. This difference arises from active NaCl reabsorption from thick ascending limbs, which dilutes ascending limb flow relative to flow in vessels and other tubules. In the inner medulla, the mechanisms responsible for generating the osmotic gradient have not been definitively identified. Important advances in understanding the urine concentrating mechanism include: identification and localization of key transport proteins for water, sodium, and urea; better resolution of the anatomical relationships in the medulla; and improvements in mathematical modeling of the urine concentrating mechanism. Continued experimental investigation of transepithelial transport and its regulation, both in normal animals and in knock-out mice, and incorporation of the resulting information into mathematical simulations, may help to more fully elucidate the inner medullary urine concentrating mechanism.

The urine concentrating effect of the mammalian renal inner medulla has been attributed to countercurrent multiplication of a transepithelial osmotic difference arising from passive absorption of NaCl from thin ascending limbs of long loops of Henle. This study assesses, both mathematically and experimentally, whether the permeability criteria for effective function of this passive hypothesis are consistent with transport properties measured in long loops of Henle of chinchilla. Mathematical simulations incorporating loop of Henle transepithelial permeabilities idealized for the passive hypothesis generated a steep inner medullary osmotic gradient, confirming the fundamental feasibility of the passive hypothesis. However, when permeabilities measured in chinchilla were used, no inner medullary gradient was generated. A key parameter in the apparent failure of the passive hypothesis is the long-loop descending limb (LDL) urea permeability, which must be small to prevent significant transepithelial urea flux into inner medullary LDL. Consequently, experiments in isolated perfused thin LDL were conducted to determine whether the urea permeability may be lower under conditions more nearly resembling those in the inner medulla. LDL segments were dissected from 30–70% of the distance along the inner medullary axis of the chinchilla kidney. The factors tested were NaCl concentration (125–400 mM in perfusate and bath), urea concentration (5–500 mM in perfusate and bath), calcium concentration (2–8 mM in perfusate and bath), and protamine concentration (300 micrograms/ml in perfusate). None of these factors significantly altered the measured urea permeability, which exceeded 20 x 10(-5) cm/s for all conditions. Simulation results show that this moderately high urea permeability in LDL is an order of magnitude too high for effective operation of the passive countercurrent multiplier.

Dynamic models of the urine concentrating mechanism consist of large systems of hyperbolic partial differential equations, with stiff source terms, coupled with fluid conservation relations. Efforts to solve these equations numerically with explicit methods have been frustrated by numerical instability and by long computation times. As a consequence, most models have been reformulated as steady-state boundary value problems, which have usually been solved by an adaptation of Newton’s method. Nonetheless, difficulties arise in finding conditions that lead to stable convergence, especially when the very large membrane permeabilities measured in experiments are used. In this report, an explicit method, previously introduced to solve the model equations of a single renal tubule, is extended to solve a large-scale model of the urine concentrating mechanism. This explicit method tracks concentration profiles in the upwind direction and thereby avoids instability arising from flow reversal. To attain second-orde...

Thick ascending limb (TAL) cells are capable of reducing tubular fluid Na+concentration to as low as ∼25 mM, and yet they are thought to transport Na+efficiently owing to passive paracellular Na+absorption. Transport efficiency in the TAL is of particular importance in the outer medulla where O2availability is limited by low blood flow. We used a mathematical model of a TAL cell to estimate the efficiency of Na+transport and to examine how tubular dilution and cell volume regulation influence transport efficiency. The TAL cell model represents 13 major solutes and the associated transporters and channels; model equations are based on mass conservation and electroneutrality constraints. We analyzed TAL transport in cells with conditions relevant to the inner stripe of the outer medulla, the cortico-medullary junction, and the distal cortical TAL. At each location Na+transport efficiency was computed as functions of changes in luminal NaCl concentration ([NaCl]), [K+], [NH4+], junctional Na+permeability, and apical K+permeability. Na+transport efficiency was calculated as the ratio of total net Na+transport to transcellular Na+transport. Transport efficiency is predicted to be highest at the cortico-medullary boundary where the transepithelial Na+gradient is the smallest. Transport efficiency is lowest in the cortex where luminal [NaCl] approaches static head.

Water is the most abundant constituent in the body. Vasopressin secretion, water ingestion, and the renal concentrating mechanism collaborate to maintain human body fluid osmolality nearly constant. Abnormalities in these processes cause hyponatremia, hypernatremia, and polyuria. The primary hormonal control of renal water excretion is by vasopressin (also named antidiuretic hormone). Thirst and vasopressin release from the posterior pituitary are under the control of osmoreceptive neurons in the central nervous system. The kidney maintains blood plasma osmolality and sodium concentration nearly constant by means of mechanisms that independently regulate water and sodium excretion. The renal medulla produces concentrated urine through the generation of an osmotic gradient extending from the cortico-medullary boundary to the inner medullary tip. This gradient is generated in the outer medulla by the countercurrent multiplication of a comparatively small transepithelial difference in osmotic pressure. This small difference, called a single effect, arises from active NaCl reabsorption from thick ascending limbs, which dilutes ascending limb flow relative to flow in vessels and other tubules. In the inner medulla, the gradient may also be generated by the countercurrent multiplication of a single effect, but the single effect has not been definitively identified. Continued experimental investigation and incorporation of the resulting information into mathematic simulations may help to more fully elucidate the inner medullary urine concentrating mechanism.

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.

Method-dependent mechanisms that may affect dynamic numerical solutions of a hyperbolic partial differential equation that models concentration profiles in renal tubules are described. Some numerical methods that have been applied to the equation are summarized, and ways by which the methods may misrepresent true solutions are analysed. Comparison of these methods demonstrates the need for thoughtful application of computational mathematics when simulating complicated time-dependent phenomena.

We show that an explicit method for solving hyperbolic partial differential equations can be applied to a model of a renal tubule to obtain both dynamic and steady-state solutions. Appropriate implementation of this method eliminates numerical instability arising from reversal of intratubular flow direction. To obtain second-order convergence in space and time, we employ the recently developed ENO (Essentially Non-Oscillatory) methodology. We present examples of computed flows and concentration profiles in representative model contexts. Finally, we indicate briefly how model tubules may be coupled to construct large-scale simulations of the renal counterflow system.

It has become widely accepted that the osmolality gradient along the corticomedullary axis of the mammalian outer medulla is generated and sustained by a process of countercurrent multiplication: active NaCl absorption from thick ascending limbs is coupled with the counterflow configuration of the descending and ascending limbs of the loops of Henle to generate an axial osmolality gradient along the outer medulla. However, aspects of anatomic structure (e.g., the physical separation of the descending limbs of short loops of Henle from contiguous ascending limbs), recent physiologic experiments (e.g., those that suggest that the thin descending limbs of short loops of Henle have a low osmotic water permeability), and mathematical modeling studies (e.g., those that predict that water-permeable descending limbs of short loops are not required for the generation of an axial osmolality gradient) suggest that countercurrent multiplication may be an incomplete, or perhaps even erroneous, explanation. We propose an alternative explanation for the axial osmolality gradient: we regard the thick limbs as NaCl sources for the surrounding interstitium, and we hypothesize that the increasing axial osmolality gradient along the outer medulla is primarily sustained by an increasing ratio, as a function of increasing medullary depth, of NaCl absorption (from thick limbs) to water absorption (from thin descending limbs of long loops of Henle and, in antidiuresis, from collecting ducts). We further hypothesize that ascending vasa recta that are external to vascular bundles will carry, toward the cortex, an absorbate that at each medullary level is hyperosmotic relative to the adjacent interstitium.

We have developed a mathematical model of a rat afferent arteriolar smooth muscle cell. The model is based on a model for small vessels by Gonzalez-Fernandez & Ermentrout (Math Biosci 119:127, 1994...

Simulations conducted in a mathematical model were used to exemplify the hypothesis that elevated solute concentrations and tubular flows at the boundary of the renal outer and inner medullas of rats may contribute to increased urine osmolalities and urine flow rates. Such elevated quantities at that boundary may arise from hyperfiltration and from inner stripe hypertrophy, which are correlated with increased concentrating activity (Bankir L, Kriz W. Kidney Int. 47: 7–24, 1995). The simulations used the region-based model for the rat inner medulla that was presented in the companion study (Layton AT, Pannabecker TL, Dantzler WH, Layton HE. Am J Physiol Renal Physiol 298: F000–F000, 2010). The simulations were suggested by experiments which were conducted in rat by Gamble et al. (Gamble JL, McKhann CF, Butler AM, Tuthill E. Am J Physiol 109: 139–154, 1934) in which the ratio of NaCl to urea in the diet was systematically varied in eight successive 5-day intervals. The simulations predict that changes in boundary conditions at the boundary of the outer and inner medulla, accompanied by plausible modifications in transport properties of the collecting duct system, can significantly increase urine osmolality and flow rate. This hyperfiltration-hypertrophy hypothesis may explain the finding by Gamble et al. that the maximum urine osmolality attained from supplemental feeding of urea and NaCl in the eight intervals depends on NaCl being the initial predominant solute and on urea being the final predominant solute, because urea in sufficient quantity appears to stimulate concentrating activity. More generally, the hypothesis suggests that high osmolalities and urine flow rates may depend, in large part, on adaptive modifications of cortical hemodynamics and on outer medullary structure and not entirely on an extraordinary concentrating capability that is intrinsic to the inner medulla.

Recent micropuncture studies in rats have demonstrated the existence of oscillatory states in nephron filtration mediated by tubuloglomerular feedback (TGF). We develop a minimal mathematical model of the TGF system, consisting of a first-order hyperbolic partial differential equation describing thick ascending limb (TAL) NaCl reabsorption and an empirical feedback relation. An analytic bifurcation analysis of this model provides fundamental insight into how oscillatory states depend on the physiological parameters of the model. In the special case of no solute backleak in the TAL, the emergence of oscillations explicitly depends on two nondimensional parameters. The first corresponds to the delay time of the TGF response across the juxtaglomerular apparatus, and the second corresponds to the product of the slope of the TGF response curve at the steady-state operating point and the space derivative of the steady-state NaCl concentration profile in the TAL at the macula densa. Numerical calculations for the case without TAL backleak are consistent with this result. Numerical simulation of the more general case with TAL backleak shows that the bifurcation analysis still provides useful predictions concerning nephron dynamics. With typical parameter values, the analysis predicts that the TGF system will be in oscillatory state. However, the system is near enough to the boundary of the nonoscillatory region so that small changes in parameter values could result in nonoscillatory behavior.

Continuously distributed loops of Henle were used in a central core model of the rat kidney's urine-concentrating mechanism to investigate the importance of overlapping loops for three different modes of urea transport in the long loops of Henle: 1) urea-impermeable loops, 2) urea-permeable loops (as indicated by perfused tubule experiments), and 3) loops with urea-permeable descending limbs and active urea transport out of thin ascending limbs. Mode 1 produces high papillary tip osmolality in accordance with tissue slice experiments, but the relative contribution of urea to the osmolality of the central core and the long descending limbs is below experimental measurements. Mode 2 generates no significant osmolality increase in the inner medulla, in agreement with other model studies. Mode 3 produces high papillary tip osmolality with a substantial contribution of urea to the osmolality of the core and the descending limbs, which is more in accordance with experiments. The results suggest that 1) overlapping loops may produce a cascade effect that contributes to the inner medullary concentrating mechanism and that 2) new experiments are needed to more certainly ascertain the urea transport characteristics of the thin ascending limbs.

In the mammalian kidney the number of loops of Henle decreases as a function of medullary depth. The role of this decreasing loop population was studied in a steady-state, central core model of the renal inner medulla under simple assumptions: there is no axial diffusion in the central core; the osmolalities in the central core, the descending limbs, and the collecting ducts are equal at each medullary level; and the concentration gradient is generated through the reabsorption of solute from the water-impermeable ascending limbs. A continuous approximation to the loop distribution in rats was based on experimental data. When solute is transported from the ascending limbs with a spatially uniform transport rate, similar in magnitude to the transport rate from the thick ascending limbs of the outer medulla, a moderate gradient is generated in the inner medulla. A steeper gradient, however, is generated by a transport rate that is largest near the turns in the loops, but which is scaled so that the total solute transport is unchanged. When loop distributions that decrease more slowly than those found in rats are used in the model, concentrating capability is decreased for both transport-rate assumptions. These results indicate that the conclusions reached in an earlier study under less accurate physiological assumptions also hold in a central core model.