Three-dimensional simulation of urine concentrating mechanism in a functional unit of rat outer medulla. I. Model structure 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. 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.

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A Novel Approach for Compensating the Significance of Tubule’s Architecture in Urine Concentrating Mechanism of Renal Medulla

Volume 3B: Biomedical and Biotechnology Engineering ◽

10.1115/imece2013-63747 ◽

2013 ◽

Author(s):

Salman Sohrabi ◽

Seyyed Mahdi Nemati Mehr ◽

Pedram Falsafi

Keyword(s):

Water Exchange ◽

Renal Medulla ◽

Spatial Dimension ◽

Model Structure ◽

Concentrating Mechanism ◽

Novel Approach ◽

Mathematical Simulations ◽

Spatial Dimensions ◽

Region Approach ◽

Urine Concentrating Mechanism

Many theories and mathematical simulations have been proposed concerning urine concentrating mechanism (UCM). The WKM and region approach are the two most valuable methods for compensating the effect of tubule’s architecture in renal medulla. They both have tried to simulate tubule’s confinement within a particular region mathematically in one spatial dimension. In this study, continuity, momentum and species transport equations along with standard expressions for transtubular solutes and water transports on tubule’s membrane were solved numerically in three spatial dimensions which practically is the main significance of our novel approach. Model structure has been chosen as simple as possible to minimize the effect of other factors in tubule’s solute and water exchange. It has been tried to simulate the preferential interaction between tubules by introducing different diffusion coefficients for solutes in the intermediate media in order that changing this physical parameter directly could influence tubule’s confinement with respect to each other. The results have been discussed in detail and then the effect of solute’s diffusivity on UCM has been investigated subsequently. In overall, it has been found out that this simulation can validate the integrity of our proposed approach for further investigation in this field.

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

AJP Renal Physiology ◽

10.1152/ajprenal.00203.2010 ◽

2011 ◽

Vol 300 (2) ◽

pp. F356-F371 ◽

Cited By ~ 44

Author(s):

Anita T. Layton

Keyword(s):

Mathematical Model ◽

Collecting Duct ◽

Renal Medulla ◽

Base Case ◽

Nacl Transport ◽

Rat Urine ◽

Concentrating Mechanism ◽

Steady State Model ◽

Model Equations ◽

Urine Concentrating Mechanism

A new, region-based mathematical model of the urine concentrating mechanism of the rat renal medulla was used to investigate the significance of transport and structural properties revealed in anatomic studies. The model simulates preferential interactions among tubules and vessels by representing concentric regions that are centered on a vascular bundle in the outer medulla (OM) and on a collecting duct cluster in the inner medulla (IM). Particularly noteworthy features of this model include highly urea-permeable and water-impermeable segments of the long descending limbs and highly urea-permeable ascending thin limbs. Indeed, this is the first detailed mathematical model of the rat urine concentrating mechanism that represents high long-loop urea permeabilities and that produces a substantial axial osmolality gradient in the IM. That axial osmolality gradient is attributable to the increasing urea concentration gradient. 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 that the interstitial NaCl and urea concentrations in adjoining regions differ substantially in the OM but not in the IM. In the OM, active NaCl transport from thick ascending limbs, at rates inferred from the physiological literature, resulted in a concentrating effect such that the intratubular fluid osmolality of the collecting duct increases ∼2.5 times along the OM. As a result of the separation of urea from NaCl and the subsequent mixing of that urea and NaCl in the interstitium and vasculature of the IM, collecting duct fluid osmolality further increases by a factor of ∼1.55 along the IM.

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A mathematical model of the urine concentrating mechanism in the rat renal medulla. II. Functional implications of three-dimensional architecture

AJP Renal Physiology ◽

10.1152/ajprenal.00204.2010 ◽

2011 ◽

Vol 300 (2) ◽

pp. F372-F384 ◽

Cited By ~ 27

Author(s):

Anita T. Layton

Keyword(s):

Mathematical Model ◽

Collecting Duct ◽

Rat Kidney ◽

Renal Medulla ◽

Base Case ◽

Unexpected Finding ◽

Thick Ascending Limb ◽

Model Calculations ◽

Concentrating Mechanism ◽

Urine Concentrating Mechanism

In a companion study [Layton AT. A mathematical model of the urine concentrating mechanism in the rat renal medulla. I. Formulation and base-case results. Am J Physiol Renal Physiol. (First published November 10, 2010). 10.1152/ajprenal.00203.2010] a region-based mathematical model was formulated for the urine concentrating mechanism in the renal medulla of the rat kidney. In the present study, we investigated model sensitivity to some of the fundamental structural assumptions. An unexpected finding is that the concentrating capability of this region-based model falls short of the capability of models that have radially homogeneous interstitial fluid at each level of only the inner medulla (IM) or of both the outer medulla and IM, but are otherwise analogous to the region-based model. Nonetheless, model results reveal the functional significance of several aspects of tubular segmentation and heterogeneity: 1) the exclusion of ascending thin limbs that reach into the deep IM from the collecting duct clusters in the upper IM promotes urea cycling within the IM; 2) the high urea permeability of the lower IM thin limb segments allows their tubular fluid urea content to equilibrate with the surrounding interstitium; 3) the aquaporin-1-null terminal descending limb segments prevent water entry and maintain the transepithelial NaCl concentration gradient; 4) a higher thick ascending limb Na+ active transport rate in the inner stripe augments concentrating capability without a corresponding increase in energy expenditure for transport; 5) active Na+ reabsorption from the collecting duct elevates its tubular fluid urea concentration. Model calculations predict that these aspects of tubular segmentation and heterogeneity promote effective urine concentrating functions.

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The Mammalian Urine Concentrating Mechanism: Hypotheses and Uncertainties

Physiology ◽

10.1152/physiol.00013.2009 ◽

2009 ◽

Vol 24 (4) ◽

pp. 250-256 ◽

Cited By ~ 32

Author(s):

Anita T. Layton ◽

Harold E. Layton ◽

William H. Dantzler ◽

Thomas L. Pannabecker

Keyword(s):

Blood Plasma ◽

Water Deprivation ◽

Three Dimensional ◽

Functional Architecture ◽

Mammalian Kidney ◽

Concentrating Mechanism ◽

Preferential Interactions ◽

Urine Concentrating Mechanism

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.

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

AJP Renal Physiology ◽

10.1152/ajprenal.00347.2003 ◽

2005 ◽

Vol 289 (6) ◽

pp. F1367-F1381 ◽

Cited By ~ 40

Author(s):

Anita T. Layton ◽

Harold E. Layton

Keyword(s):

Mathematical Model ◽

Boundary Conditions ◽

Rat Kidney ◽

Vascular Bundles ◽

Thick Ascending Limb ◽

Outer Medulla ◽

Model Calculations ◽

Concentrating Mechanism ◽

Vasa Recta ◽

Urine Concentrating Mechanism

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.

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Numerical methods for three-dimensional models of the urine concentrating mechanism

Applied Mathematics and Computation ◽

10.1016/0096-3003(91)90082-x ◽

1991 ◽

Vol 45 (2) ◽

pp. 219-240 ◽

Cited By ~ 8

Author(s):

Anthony S. Wexler ◽

Donald J. Marsh

Keyword(s):

Numerical Methods ◽

Three Dimensional ◽

Concentrating Mechanism ◽

Dimensional Models ◽

Three Dimensional Models ◽

Urine Concentrating Mechanism

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Urine concentrating mechanism in the inner medulla of the mammalian kidney: role of three-dimensional architecture

Acta Physiologica ◽

10.1111/j.1748-1716.2010.02214.x ◽

2010 ◽

Vol 202 (3) ◽

pp. 361-378 ◽

Cited By ~ 24

Author(s):

W. H. Dantzler ◽

T. L. Pannabecker ◽

A. T. Layton ◽

H. E. Layton

Keyword(s):

Three Dimensional ◽

Mammalian Kidney ◽

Concentrating Mechanism ◽

Urine Concentrating Mechanism

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

AJP Renal Physiology ◽

10.1152/ajprenal.90252.2008 ◽

2008 ◽

Vol 295 (5) ◽

pp. F1271-F1285 ◽

Cited By ~ 54

Author(s):

Thomas L. Pannabecker ◽

William H. Dantzler ◽

Harold E. Layton ◽

Anita T. Layton

Keyword(s):

Three Dimensional ◽

Cluster System ◽

Urea Transporter ◽

Comprehensive Understanding ◽

Concentrating Mechanism ◽

Vasa Recta ◽

Collecting Ducts ◽

Renal Inner Medulla ◽

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.

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