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

2005 ◽  
Vol 289 (6) ◽  
pp. F1346-F1366 ◽  
Author(s):  
Anita T. Layton ◽  
Harold E. Layton
Keyword(s):  
Mathematical Model ◽  
Rat Kidney ◽  
Tubuloglomerular Feedback ◽  
Transepithelial Transport ◽  
Base Case ◽  
Model Parameters ◽  
Outer Medulla ◽  
Concentrating Mechanism ◽  
The Impact ◽  
Urine Concentrating Mechanism

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.

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

2011 ◽  
Vol 300 (2) ◽  
pp. F372-F384 ◽  
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.

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

2005 ◽  
Vol 289 (6) ◽  
pp. F1367-F1381 ◽  
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.

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

2011 ◽  
Vol 300 (2) ◽  
pp. F356-F371 ◽  
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.

scholarly journals A mathematical model of the rat kidney. II. Antidiuresis

2020 ◽  
Vol 318 (4) ◽  
pp. F936-F955
Author(s):  
Alan M. Weinstein
Keyword(s):  
Mathematical Model ◽  
Water Conservation ◽  
Rat Kidney ◽  
Model Parameters ◽  
Reflection Coefficients ◽  
Model Computation ◽  
Outer Medulla ◽  
Nacl Transport ◽  
Solute Flow ◽  
Vasa Recta

Kidney water conservation requires a hypertonic medullary interstitium, NaCl in the outer medulla and NaCl and urea in the inner medulla, plus a vascular configuration that protects against washout. In this work, a multisolute model of the rat kidney is revisited to examine its capacity to simulate antidiuresis. The first step was to streamline model computation by parallelizing its Jacobian calculation, thus allowing finer medullary spatial resolution and more extensive examination of model parameters. It is found that outer medullary NaCl is modestly increased when transporter density in ascending Henle limbs from juxtamedullary nephrons is scaled to match the greater juxtamedullary solute flow. However, higher NaCl transport produces greater CO2 generation and, by virtue of countercurrent vascular flows, establishment of high medullary Pco2. This CO2 gradient can be mitigated by assuming that a fraction of medullary transport is powered anaerobically. Reducing vascular flows or increasing vessel permeabilities does little to further increase outer medullary solute gradients. In contrast to medullary models of others, vessels in this model have solute reflection coefficients close to zero; increasing these coefficients provides little enhancement of solute profiles but does generate high interstitial pressures, which distort tubule architecture. Increasing medullary urea delivery via entering vasa recta increases inner medullary urea, although not nearly to levels found in rats. In summary, 1) medullary Na+ and urea gradients are not captured by the model and 2) the countercurrent architecture that provides antidiuresis also produces exaggerated Pco2 profiles and is an unappreciated constraint on models of medullary function.

A mathematical model of the rat kidney IV: whole-kidney response to hyperkalemia

2022 ◽  
Author(s):  
Alan Mark Weinstein
Keyword(s):  
Rat Kidney ◽  
Macula Densa ◽  
Tubuloglomerular Feedback ◽  
Na Transport ◽  
Ample Evidence ◽  
Principal Cells ◽  
Relative Contribution ◽  
K Secretion ◽  
Luminal Flow ◽  
The Impact

The renal response to acute hyperkalemia is mediated by increased K secretion within connecting tubule (CNT), flux that is modulated by tubular effects (e.g. aldosterone) in conjunction with increased luminal flow. There is ample evidence that peritubular K blunts Na reabsorption in proximal tubule, thick ascending Henle limb, and distal convoluted tubule (DCT). While any such reduction may augment CNT delivery, the relative contribution of each is uncertain. The kidney model of this lab was recently advanced with representation of cortical labyrinth and medullary ray. Model tubules capture the impact of hyperkalemia to blunt Na reabsorption within each upstream segment. However, this forces the question of the extent to which increased Na delivery is transmitted past macula densa and its tubuloglomerular feedback (TGF) signal. Beyond increasing macula densa Na delivery, peritubular K is predicted to raise cytosolic Cl and depolarize macula densa cells, which may also activate TGF. Thus, although upstream reduction in Na transport may be larger, it appears that the DCT effect is critical to increasing CNT delivery. Beyond the flow effect, hyperkalemia reduces ammoniagenesis and reduced ammoniagenesis enhances K excretion. What this model provides is a possible mechanism. When cortical NH4 is taken up via peritubular Na,K(NH4)-ATPase, it acidifies principal cells. Consequently, reduced ammoniagenesis increases principal cell pH, thereby increasing conductance of both ENaC and ROMK, enhancing K excretion. In this model, aldosterone's effect on principal cells, diminished DCT Na reabsorption, and reduced ammoniagenesis, all provide relatively equal and additive contributions to renal K excretion.

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