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.

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

The goal of this study is to investigate the functional implications of the sexual dimorphism in transporter patterns along the proximal tubule. To do so, we have developed sex-specific computational models of solute and water transport in the proximal convoluted tubule of the rat kidney. The models account for the sex differences in expression levels of the apical and basolateral transporters, in single-nephron glomerular filtration rate, and in tubular dimensions.

Advances in understanding the urine-concentrating mechanism.

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.

Urine-concentrating mechanism in the inner medulla: function of the thin limbs of the loops of Henle.

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.

Fluid dilution and efficiency of Na(+) transport in a mathematical model of a thick ascending limb cell.

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 O(2) availability 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.

Transport efficiency and workload distribution in a mathematical model of the thick ascending limb.

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.

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.

Countercurrent multiplication may not explain the axial osmolality gradient in the outer medulla of the rat kidney.

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.

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 mathematical model of the myogenic response to systolic pressure in the afferent arteriole.

Elevations in systolic blood pressure are believed to be closely linked to the pathogenesis and progression of renal diseases. It has been hypothesized that the afferent arteriole (AA) protects the glomerulus from the damaging effects of hypertension by sensing increases in systolic blood pressure and responding with a compensatory vasoconstriction (Loutzenhiser R, Bidani A, Chilton L. Circ Res 90: 1316-1324, 2002).

Functional implications of the three-dimensional architecture of the rat renal inner medulla.

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.

Hyperfiltration and inner stripe hypertrophy may explain findings by Gamble and coworkers.

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.

Maximum urine concentrating capability in a mathematical model of the inner medulla of the rat kidney.

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 mammalian urine concentrating mechanism: hypotheses and uncertainties.

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 physiology of urinary concentration: an update.

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.

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.

Role of three-dimensional architecture in the urine concentrating mechanism of the rat renal inner medulla.

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 microm of the papilla tip.

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.

An optimization algorithm for a distributed-loop model of an avian urine concentrating mechanism.

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.

Multistability in tubuloglomerular feedback and spectral complexity in spontaneously hypertensive rats.

Single-nephron proximal tubule pressure in spontaneously hypertensive rats (SHR) can exhibit highly irregular oscillations similar to deterministic chaos. We used a mathematical model of tubuloglomerular feedback (TGF) to investigate potential sources of the irregular oscillations and the corresponding complex power spectra in SHR. A bifurcation analysis of the TGF model equations, for nonzero thick ascending limb (TAL) NaCl permeability, was performed by finding roots of the characteristic equation, and numerical simulations of model solutions were conducted to assist in the interpretation of the analysis.

A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. I. Formulation and base-case results.

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.

A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. II. Parameter sensitivity and tubular inhomogeneity.

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).

Feedback-mediated dynamics in two coupled nephrons.

Previously, we developed a dynamic model for the tubuloglomerular feedback (TGF) system in a single, short-looped nephron of the mammalian kidney. In that model, a semi-linear hyperbolic partial differential equation was used to represent two fundamental processes of solute transport in the nephron's thick ascending limb (TAL): chloride advection by fluid flow along the TAL lumen and transepithelial chloride transport from the lumen to the interstitium. An empirical function and a time delay were used to relate glomerular filtration rate to the chloride concentration at the macula densa of the TAL.

Two modes for concentrating urine in rat inner medulla.

We used a mathematical model of the urine concentrating mechanism of rat inner medulla (IM) to investigate the implications of experimental studies in which immunohistochemical methods were combined with three-dimensional computerized reconstruction of renal tubules. The mathematical model represents a distribution of loops of Henle with loop bends at all levels of the IM, and the vasculature is represented by means of the central core assumption. Based on immunohistochemical evidence, descending limb portions that reach into the papilla are assumed to be only moderately water permeable or to be water impermeable, and only prebend segments and ascending thin limbs are assumed to be NaCl permeable.

A region-based model framework for the rat urine concentrating mechanism.

The highly structured organization of tubules and blood vessels in the outer medulla of the mammalian kidney is believed to result in preferential interactions among tubules and vessels; such interactions may promote solute cycling and enhance urine concentrating capability. In this study, we formulate a new model framework for the urine concentrating mechanism in the outer medulla of the rat kidney. The model simulates preferential interactions among tubules and vessels by representing two concentric regions and by specifying the fractions of tubules and vessels assigned to each of the regions.