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

2009 ◽  
Author(s):  
Raymond Mejia
Keyword(s):  
Mathematical Model ◽  
Rat Kidney

scholarly journals A mathematical model of the myogenic response to systolic pressure in the afferent arteriole

2011 ◽  
Vol 300 (3) ◽  
pp. F669-F681 ◽  
Author(s):  
Jing Chen ◽  
Ioannis Sgouralis ◽  
Leon C. Moore ◽  
Harold E. Layton ◽  
Anita T. Layton
Keyword(s):  
Blood Pressure ◽  
Mathematical Model ◽  
Systolic Blood Pressure ◽  
Rat Kidney ◽  
Systolic Pressure ◽  
Vessel Diameter ◽  
Renal Diseases ◽  
Myogenic Response ◽  
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). To investigate this hypothesis, we developed a mathematical model of the myogenic response of an AA wall, based on an arteriole model (Gonzalez-Fernandez JM, Ermentrout B. Math Biosci 119: 127–167, 1994). The model incorporates ionic transport, cell membrane potential, contraction of the AA smooth muscle cell, and the mechanics of a thick-walled cylinder. The model represents a myogenic response based on a pressure-induced shift in the voltage dependence of calcium channel openings: with increasing transmural pressure, model vessel diameter decreases; and with decreasing pressure, vessel diameter increases. Furthermore, the model myogenic mechanism includes a rate-sensitive component that yields constriction and dilation kinetics similar to behaviors observed in vitro. A parameter set is identified based on physical dimensions of an AA in a rat kidney. Model results suggest that the interaction of Ca2+ and K+ fluxes mediated by voltage-gated and voltage-calcium-gated channels, respectively, gives rise to periodicity in the transport of the two ions. This results in a time-periodic cytoplasmic calcium concentration, myosin light chain phosphorylation, and cross-bridge formation with the attending muscle stress. Furthermore, the model predicts myogenic responses that agree with experimental observations, most notably those which demonstrate that the renal AA constricts in response to increases in both steady and systolic blood pressures. The myogenic model captures these essential functions of the renal AA, and it may prove useful as a fundamental component in a multiscale model of the renal microvasculature suitable for investigations of the pathogenesis of hypertensive renal diseases.

scholarly journals A mathematical model of the rat kidney: K+-induced natriuresis

2017 ◽  
Vol 312 (6) ◽  
pp. F925-F950 ◽  
Author(s):  
Alan M. Weinstein
Keyword(s):  
Mathematical Model ◽  
Proximal Tubule ◽  
Blood Vessels ◽  
Collecting Duct ◽  
Rat Kidney ◽  
Reflection Coefficients ◽  
Additional Species ◽  
Connecting Tubule ◽  
Prior Models ◽  
Urine Concentrating Mechanism

A model of the rat nephron (Weinstein. Am J Physiol Renal Physiol 308: F1098–F1118, 2015) has been extended with addition of medullary vasculature. Blood vessels contain solutes from the nephron model, plus additional species from the model of Atherton et al. ( Am J Physiol Renal Fluid Electrolyte Physiol 247: F61–F72, 1984), representing hemoglobin buffering. In contrast to prior models of the urine-concentrating mechanism, reflection coefficients for DVR are near zero. Model unknowns are initial proximal tubule pressures and flows, connecting tubule pressure, and medullary interstitial pressures and concentrations. The model predicts outer medullary (OM) interstitial gradients for Na+, K+, CO2, and [Formula: see text], such that at OM-IM junction, the respective concentrations relative to plasma are 1.2, 3.0, 2.7, and 8.0; within IM, there is high urea and low [Formula: see text], with concentration ratios of 11 and 0.5 near the papillary tip. Quantitative similarities are noted between K+and urea handling (medullary delivery and permeabilities). The model K+gradient is physiologic, and the urea gradient is steeper due to restriction of urea permeability to distal collecting duct. Nevertheless, the predicted urea gradient is less than expected, suggesting reconsideration of proposals of an unrecognized reabsorptive urea flux. When plasma K+is increased from 5.0 to 5.5 mM, Na+and K+excretion increase 2.3- and 1.3-fold, respectively. The natriuresis derives from a 3.3% decrease in proximal Na+reabsorption and occurs despite delivery-driven increases in Na+reabsorption in distal segments; kaliuresis derives from a 30% increase in connecting tubule Na+delivery. Thus this model favors the importance of proximal over distal events in K+-induced diuresis.

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 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.

scholarly journals Dynamic myogenic autoregulation in the rat kidney: a whole-organ model

2008 ◽  
Vol 294 (6) ◽  
pp. F1453-F1464 ◽  
Author(s):  
N. Kleinstreuer ◽  
T. David ◽  
M. J. Plank ◽  
Z. Endre
Keyword(s):  
Mathematical Model ◽  
Rat Kidney ◽  
Morphological Data ◽  
Myogenic Response ◽  
Wall Tension ◽  
Renal Vasculature ◽  
Renal Autoregulation ◽  
Organ Model ◽  
Endothelial Nitric Oxide

A transient 1D mathematical model of whole-organ renal autoregulation in the rat is presented, examining the myogenic response on multiple levels of the renal vasculature. Morphological data derived from micro-CT imaging were employed to divide the vasculature via a Strahler ordering scheme. A previously published model of the myogenic response based on wall tension is expanded and adapted to fit the response of each level, corresponding to a distally dominant resistance distribution with the highest contributions localized to the afferent arterioles and interlobular arteries. The mathematical model was further developed to include the effects of in vivo viscosity variation and flow-induced dilation via endothelial nitric oxide production. Computer simulations of the autoregulatory response to pressure perturbations were examined and compared with experimental data. The model supports the hypothesis that change in circumferential wall tension is the catalyst for the myogenic response. The model provides a basis for examining the steady state and transient characteristics of the whole-organ renal myogenic response in the rat, as well as the modulatory influences of metabolic and hemodynamic factors.

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