scholarly journals A mathematical model of O2 transport in the rat outer medulla. II. Impact of outer medullary architecture

2009 ◽  
Vol 297 (2) ◽  
pp. F537-F548 ◽  
Author(s):  
Jing Chen ◽  
Aurélie Edwards ◽  
Anita T. Layton
Keyword(s):  
Mathematical Model ◽  
Anaerobic Metabolism ◽  
Critical Level ◽  
Vascular Bundles ◽  
Outer Medulla ◽  
O2 Transport ◽  
Vasa Recta ◽  
Parameter Variations ◽  
Complex Structural ◽  
The Impact

we extended the region-based mathematical model of the urine-concentrating mechanism in the rat outer medulla (OM) developed by Layton and Layton ( Am J Physiol Renal Physiol 289: F1346–F1366, 2005) to examine the impact of the complex structural organization of the OM on O2 transport and distribution. In the present study, we investigated the sensitivity of predicted Po2 profiles to several parameters that characterize the degree of OM regionalization, boundary conditions, structural dimensions, transmural transport properties, and relative positions and distributions of tubules and vessels. Our results suggest that the fraction of O2 supplied to descending vasa recta (DVR) that reaches the inner medulla, i.e., a measure of the axial Po2 gradient in the OM, is insensitive to parameter variations as a result of the sequestration of long DVR in the vascular bundles. In contrast, O2 distribution among the regions surrounding the vascular core strongly depends on the radial positions of medullary thick ascending limbs (mTALs) relative to the vascular core, the degree of regionalization, and the distribution of short DVR along the corticomedullary axis. Moreover, if it is assumed that the mTAL active Na+ transport rate decreases when mTAL Po2 falls below a critical level, O2 availability to mTALs has a significant impact on the concentrating capability of the model OM. The model also predicts that when the OM undergoes hypertrophy, its concentrating capability increases significantly only when anaerobic metabolism supports a substantial fraction of the mTAL active Na+ transport and is otherwise critically reduced by low interstitial and mTAL luminal Po2 in a hypertrophied OM.

scholarly journals A mathematical model of O2 transport in the rat outer medulla. I. Model formulation and baseline results

2009 ◽  
Vol 297 (2) ◽  
pp. F517-F536 ◽  
Author(s):  
Jing Chen ◽  
Anita T. Layton ◽  
Aurélie Edwards
Keyword(s):  
Mathematical Model ◽  
Structural Organization ◽  
Capillary Flow ◽  
Renal Medulla ◽  
Vascular Bundles ◽  
Thick Ascending Limb ◽  
Outer Medulla ◽  
Vasa Recta ◽  
Complex Structural ◽  
The Impact

The mammalian kidney is particularly vulnerable to hypoperfusion, because the O2 supply to the renal medulla barely exceeds its O2 requirements. In this study, we examined the impact of the complex structural organization of the rat outer medulla (OM) on O2 distribution. We extended the region-based mathematical model of the rat OM developed by Layton and Layton ( Am J Physiol Renal Physiol 289: F1346–F1366, 2005) to incorporate the transport of RBCs, Hb, and O2. We considered basal cellular O2 consumption and O2 consumption for active transport of NaCl across medullary thick ascending limb epithelia. Our model predicts that the structural organization of the OM results in significant Po2 gradients in the axial and radial directions. The segregation of descending vasa recta, the main supply of O2, at the center and immediate periphery of the vascular bundles gives rise to large radial differences in Po2 between regions, limits O2 reabsorption from long descending vasa recta, and helps preserve O2 delivery to the inner medulla. Under baseline conditions, significantly more O2 is transferred radially between regions by capillary flow, i.e., advection, than by diffusion. In agreement with experimental observations, our results suggest that 79% of the O2 supplied to the medulla is consumed in the OM and that medullary thick ascending limbs operate on the brink of hypoxia.

scholarly journals Nitric oxide and superoxide transport in a cross section of the rat outer medulla. I. Effects of low medullary oxygen tension

2010 ◽  
Vol 299 (3) ◽  
pp. F616-F633 ◽  
Author(s):  
Aurélie Edwards ◽  
Anita T. Layton
Keyword(s):  
Nitric Oxide ◽  
Cross Section ◽  
Vascular Bundle ◽  
Fluid Model ◽  
Vascular Bundles ◽  
Concentration Gradients ◽  
Outer Medulla ◽  
Vasa Recta ◽  
The Difference ◽  
The Impact

To examine the impact of the complex radial organization of the rat outer medulla (OM) on the distribution of nitric oxide (NO), superoxide (O2−) and total peroxynitrite (ONOO), we developed a mathematical model that simulates the transport of those species in a cross section of the rat OM. To simulate the preferential interactions among tubules and vessels that arise from their relative radial positions in the OM, we adopted the region-based approach developed by Layton and Layton ( Am J Physiol Renal Physiol 289: F1346–F1366, 2005). In that approach, the structural organization of the OM is represented by means of four concentric regions centered on a vascular bundle. The model predicts the concentrations of NO, O2−, and ONOO in the tubular and vascular lumen, epithelial and endothelial cells, red blood cells (RBCs), and interstitial fluid. Model results suggest that the large gradients in Po2 from the core of the vascular bundle toward its periphery, which stem from the segregation of O2-supplying descending vasa recta (DVR) within the vascular bundles, in turn generate steep radial NO and O2− concentration gradients, since the synthesis of both solutes is O2 dependent. Without the rate-limiting effects of O2, NO concentration would be lowest in the vascular bundle core, that is, the region with the highest density of RBCs, which act as a sink for NO. Our results also suggest that, under basal conditions, the difference in NO concentrations between DVR that reach into the inner medulla and those that turn within the OM should lead to differences in vasodilation and preferentially increase blood flow to the inner medulla.

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 Impact of nitric oxide-mediated vasodilation on outer medullary NaCl transport and oxygenation

2012 ◽  
Vol 303 (7) ◽  
pp. F907-F917 ◽  
Author(s):  
Aurélie Edwards ◽  
Anita T. Layton
Keyword(s):  
Nitric Oxide ◽  
Blood Flow ◽  
Supply And Demand ◽  
Vascular Bundles ◽  
Concentration Gradients ◽  
Outer Medulla ◽  
Nacl Transport ◽  
Reciprocal Interactions ◽  
Medullary Blood Flow ◽  
Vasa Recta

The present study aimed to elucidate the reciprocal interactions between oxygen (O2), nitric oxide (NO), and superoxide (O2−) and their effects on vascular and tubular function in the outer medulla. We expanded our region-based model of transport in the rat outer medulla (Edwards A, Layton AT. Am J Physiol Renal Physiol 301: F979–F996, 2011) to incorporate the effects of NO on descending vasa recta (DVR) diameter and blood flow. Our model predicts that the segregation of long DVR in the center of vascular bundles, away from tubular segments, gives rise to large radial NO concentration gradients that in turn result in differential regulation of vasoactivity in short and long DVR. The relative isolation of long DVR shields them from changes in the rate of NaCl reabsorption, and hence from changes in O2 requirements, by medullary thick ascending limbs (mTALs), thereby preserving O2 delivery to the inner medulla. The model also predicts that O2− can sufficiently decrease the bioavailability of NO in the interbundle region to affect the diameter of short DVR, suggesting that the experimentally observed effects of O2− on medullary blood flow may be at least partly mediated by NO. In addition, our results indicate that the tubulovascular cross talk of NO, that is, the diffusion of NO produced by mTAL epithelia toward adjacent DVR, helps to maintain blood flow and O2 supply to the interbundle region even under basal conditions. NO also acts to preserve local O2 availability by inhibiting the rate of active Na+ transport, thereby reducing the O2 requirements of mTALs. The dual regulation by NO of oxygen supply and demand is predicted to significantly attenuate the hypoxic effects of angiotensin II.

scholarly journals Impact of renal medullary three-dimensional architecture on oxygen transport

2014 ◽  
Vol 307 (3) ◽  
pp. F263-F272 ◽  
Author(s):  
Brendan C. Fry ◽  
Aurélie Edwards ◽  
Ioannis Sgouralis ◽  
Anita T. Layton
Keyword(s):  
Oxygen Delivery ◽  
Interstitial Fluid ◽  
Collecting Duct ◽  
Rat Kidney ◽  
Three Dimensional ◽  
Fluid Composition ◽  
Vascular Bundles ◽  
Thick Ascending Limb ◽  
Outer Medulla ◽  
The Impact

We have developed a highly detailed mathematical model of solute transport in the renal medulla of the rat kidney to study the impact of the structured organization of nephrons and vessels revealed in anatomic studies. The model represents the arrangement of tubules around a vascular bundle in the outer medulla and around a collecting duct cluster in the upper inner medulla. Model simulations yield marked gradients in intrabundle and interbundle interstitial fluid oxygen tension (Po2), NaCl concentration, and osmolality in the outer medulla, owing to the vigorous active reabsorption of NaCl by the thick ascending limbs. In the inner medulla, where the thin ascending limbs do not mediate significant active NaCl transport, interstitial fluid composition becomes much more homogeneous with respect to NaCl, urea, and osmolality. Nonetheless, a substantial Po2 gradient remains, owing to the relatively high oxygen demand of the inner medullary collecting ducts. Perhaps more importantly, the model predicts that in the absence of the three-dimensional medullary architecture, oxygen delivery to the inner medulla would drastically decrease, with the terminal inner medulla nearly completely deprived of oxygen. Thus model results suggest that the functional role of the three-dimensional medullary architecture may be to preserve oxygen delivery to the papilla. Additionally, a simulation that represents low medullary blood flow suggests that the separation of thick limbs from the vascular bundles substantially increases the risk of the segments to hypoxic injury. When nephrons and vessels are more homogeneously distributed, luminal Po2 in the thick ascending limb of superficial nephrons increases by 66% in the inner stripe. Furthermore, simulations predict that owing to the Bohr effect, the presumed greater acidity of blood in the interbundle regions, where thick ascending limbs are located, relative to that in the vascular bundles, facilitates the delivery of O2 to support the high metabolic requirements of the thick limbs and raises NaCl reabsorption.

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

2011 ◽  
Vol 301 (5) ◽  
pp. F1047-F1056 ◽  
Author(s):  
Anita T. Layton ◽  
Harold E. Layton
Keyword(s):  
Alternative Explanation ◽  
Rat Kidney ◽  
Vascular Bundles ◽  
Outer Medulla ◽  
Anatomic Structure ◽  
Physical Separation ◽  
Nacl Absorption ◽  
Vasa Recta ◽  
Osmotic Water ◽  
Collecting Ducts

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., the physical separation of the descending limbs of short loops of Henle from contiguous ascending limbs), recent physiologic experiments (e.g., those that suggest that the thin descending limbs of short loops of Henle have a low osmotic water permeability), and mathematical modeling studies (e.g., those that predict that water-permeable descending limbs of short loops are not required for the generation of an axial osmolality gradient) suggest that countercurrent multiplication may be an incomplete, or perhaps even erroneous, explanation. We propose an alternative explanation for the axial osmolality gradient: we regard the thick limbs as NaCl sources for the surrounding interstitium, and we hypothesize that the increasing axial osmolality gradient along the outer medulla is primarily sustained by an increasing ratio, as a function of increasing medullary depth, of NaCl absorption (from thick limbs) to water absorption (from thin descending limbs of long loops of Henle and, in antidiuresis, from collecting ducts). We further hypothesize that ascending vasa recta that are external to vascular bundles will carry, toward the cortex, an absorbate that at each medullary level is hyperosmotic relative to the adjacent interstitium.

scholarly journals Architecture of the human renal inner medulla and functional implications

2015 ◽  
Vol 309 (7) ◽  
pp. F627-F637 ◽  
Author(s):  
Guojun Wei ◽  
Seymour Rosen ◽  
William H. Dantzler ◽  
Thomas L. Pannabecker
Keyword(s):  
Collecting Duct ◽  
Human Kidney ◽  
Vascular Bundles ◽  
Outer Medulla ◽  
Vasa Recta ◽  
Functional Features ◽  
Renal Inner Medulla ◽  
Myosin Heavy Chain Protein ◽  
Muscle Myosin ◽  
Solute Transporters

The architecture of the inner stripe of the outer medulla of the human kidney has long been known to exhibit distinctive configurations; however, inner medullary architecture remains poorly defined. Using immunohistochemistry with segment-specific antibodies for membrane fluid and solute transporters and other proteins, we identified a number of distinctive functional features of human inner medulla. In the outer inner medulla, aquaporin-1 (AQP1)-positive long-loop descending thin limbs (DTLs) lie alongside descending and ascending vasa recta (DVR, AVR) within vascular bundles. These vascular bundles are continuations of outer medullary vascular bundles. Bundles containing DTLs and vasa recta lie at the margins of coalescing collecting duct (CD) clusters, thereby forming two regions, the vascular bundle region and the CD cluster region. Although AQP1 and urea transporter UT-B are abundantly expressed in long-loop DTLs and DVR, respectively, their expression declines with depth below the outer medulla. Transcellular water and urea fluxes likely decline in these segments at progressively deeper levels. Smooth muscle myosin heavy chain protein is also expressed in DVR of the inner stripe and the upper inner medulla, but is sparsely expressed at deeper inner medullary levels. In rodent inner medulla, fenestrated capillaries abut CDs along their entire length, paralleling ascending thin limbs (ATLs), forming distinct compartments (interstitial nodal spaces; INSs); however, in humans this architecture rarely occurs. Thus INSs are relatively infrequent in the human inner medulla, unlike in the rodent where they are abundant. UT-B is expressed within the papillary epithelium of the lower inner medulla, indicating a transcellular pathway for urea across this epithelium.

The Impact of Light Polarisation on Light Field of Scenes with Multiple Reflections

2020 ◽  
pp. 108-115 ◽  
Author(s):  
Vladimir P. Budak ◽  
Anton V. Grimaylo
Keyword(s):  
Mathematical Model ◽  
Light Field ◽  
Global Illumination ◽  
Multiple Reflections ◽  
Software Environment ◽  
The Monte Carlo Method ◽  
Mueller Matrices ◽  
Volume Integration ◽  
The Impact

The article describes the role of polarisation in calculation of multiple reflections. A mathematical model of multiple reflections based on the Stokes vector for beam description and Mueller matrices for description of surface properties is presented. On the basis of this model, the global illumination equation is generalised for the polarisation case and is resolved into volume integration. This allows us to obtain an expression for the Monte Carlo method local estimates and to use them for evaluation of light distribution in the scene with consideration of polarisation. The obtained mathematical model was implemented in the software environment using the example of a scene with its surfaces having both diffuse and regular components of reflection. The results presented in the article show that the calculation difference may reach 30 % when polarisation is taken into consideration as compared to standard modelling.

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