A Dynamic Numerical Method for Models of the Urine Concentrating Mechanism

Layton, Harold E.; Pitman, E. Bruce; Knepper, Mark A.

doi:10.1137/s0036139993252864

Cited by 26 publications

(19 citation statements)

References 31 publications

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“…Each distribution is constructed by formulating each dependent variable (e.g., concentration, flow rate, etc.) associated with a SDV or a SAV as a function of both the axial position and the medullary level at which the vas rectum terminates (a similar formulation has been used previously for loops of Henle) (42,49). Transmural vasa recta fluxes are scaled by the number of vasa recta, descending or ascending, per loop of Henle.…”

Section: Tubules and Vasa Rectamentioning

confidence: 99%

“…None of these models (including the one proposed in this study) has incorporated more than one fully realized spatial dimension, viz., the dimension corresponding to the corticomedullary axis. Indeed, in some model studies interstitial solute concentrations have been assumed to be uniform at each medullary level; i.e., tubules (and blood vessels, if represented) were assumed to interact with each other through a common surrounding medium in which solute concentrations varied only along the corticomedullary axis (47,49,60,72,73,85). However, this assumption appears to be inconsistent with anatomical studies.…”

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

Layton¹

2005

American Journal of Physiology-Renal Physiology

144

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Layton, Anita T., and Harold E. Layton. A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. I. Formulation and base-case results. Am J Physiol Renal Physiol 289: F1346 -F1366, 2005. First published May 24, 2005 doi:10.1152/ajprenal.00346.2003.-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. kidney; countercurrent multiplication; countercurrent exchange; NaCl transport; urea transport CONCENTRATED URINE, i.e., urine having an osmolality exceeding that of blood plasma, is produced by the absorption of water, in excess of solute, from the medullary collecting ducts (CDs). In the outer medulla (OM), water absorption from CDs is driven by vigorous active transport of NaCl from the thick ascending limbs (TALs) into the interstitium; at each medullary level, this transport results in an osmolality difference between the TALs and other medullary structures. This difference, frequently called the "single effect," is augmented (or "multiplied") by the countercurrent flow configuration of renal tubules and vasa recta, to generate a substantial osmolality gradient along the corticomedullary axis, a gradient that is believed to be common to all OM structures. In the inner medulla (IM), however, the means by which water is absorbed from CDs remains undetermined (20, 67).Substantial effort has been directed to constructing mathematical models...

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Section: Tubules and Vasa Rectamentioning

confidence: 99%

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confidence: 99%

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

Layton¹

2005

American Journal of Physiology-Renal Physiology

144

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“…These quantities are the collecting duct outflow variables, the central core concentration at the medullary tip (x = L), and the water and solute balance properties of the medulla [4,5,6,19,21,24]. The collecting duct outflow variables are of particular interest because the urine osmolality, urine solute concentration, and urine flow are key scientific results, and because the collecting duct fluid and solute outflow, being a small fraction of typical tubular flow, can be sensitive to changes in parameters or to the degree of spatial grid refinement.…”

Section: Convergence Studies and Comparison With The Explicit Modelmentioning

confidence: 99%

“…Numerical solutions to mathematical models of the urine concentrating mechanism have frequently been assessed on the basis of their mass conservation properties [6,8,21]. To assess solute and water conservation of the model, define F ss k (x) and F ss V (x) to be the total steady-state flow rates of solute and water through the renal medulla.…”

Section: Convergence Studies and Comparison With The Explicit Modelmentioning

confidence: 99%

“…1 This capability, which allows mammals to maintain a steady plasma osmolality during periods of water deprivation, is provided by the urine concentrating mechanism. Many model studies have sought to elucidate this mechanism, which is localized in the renal medulla and which depends on a countercurrent configuration of fluid flows in thousands of nearly parallel tubules [5,6,9,12,20,23].Models of the urine concentrating mechanism have usually been formulated as steady-state boundary-value problems involving differential equations expressing solute and water conservation. The solutions describe solute concentrations and fluid flow rates in interacting tubules having diameters of ∼20 μm and lengths of several millimeters.…”

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A Semi-Lagrangian Semi-Implicit Numerical Method for Models of the Urine Concentrating Mechanism

Layton¹

2002

SIAM J. Sci. Comput.

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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. AMS subject classifications. 65-04, 65M12, 65M25; 92-04, 92C35; 35-04, 35L45PII. S1064827500381781 1. Introduction. Mammals can produce urine that has a much higher osmolality than that of blood plasma. 1 This capability, which allows mammals to maintain a steady plasma osmolality during periods of water deprivation, is provided by the urine concentrating mechanism. Many model studies have sought to elucidate this mechanism, which is localized in the renal medulla and which depends on a countercurrent configuration of fluid flows in thousands of nearly parallel tubules [5,6,9,12,20,23].Models of the urine concentrating mechanism have usually been formulated as steady-state boundary-value problems involving differential equations expressing solute and water conservation. The solutions describe solute concentrations and fluid flow rates in interacting tubules having diameters of ∼20 μm and lengths of several millimeters. Models are typically formulated in one space dimension corresponding to the axis parallel to intratubular flow.The steady-state model problem consists of a nonlinear system of coupled, stiff ordinary differential equations (ODEs), which are usually solved by adaptations of Newton's method [16,18,20,23]. However, this formulation has to contend with the difficulty that intratubular flow may be transiently reversed relative to the assumed steady-state direction. This transient reversal of flow direction, which arises when

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Signal transduction in a compliant short loop of Henle

Pham

Ryu

2011

Numer Methods Biomed Eng

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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 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 thick ascending limb (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.

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A Dynamic Numerical Method for Models of the Urine Concentrating Mechanism

Cited by 26 publications

References 31 publications

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

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

A Semi-Lagrangian Semi-Implicit Numerical Method for Models of the Urine Concentrating Mechanism

Signal transduction in a compliant short loop of Henle

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