Ioannis Sgouralis

A mathematical model of the myogenic response to systolic pressure in the 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 Ca(2+) 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.

Impact of renal medullary three-dimensional architecture on oxygen transport

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.

Autoregulation and conduction of vasomotor responses in a mathematical model of the rat afferent arteriole

We have formulated a mathematical model for the rat afferent arteriole (AA). Our model consists of a series of arteriolar smooth muscle cells and endothelial cells, each of which represents ion transport, cell membrane potential, and gap junction coupling. Cellular contraction and wall mechanics are also represented for the smooth muscle cells. Blood flow through the AA lumen is described by Poiseuille flow. The AA models representation of the myogenic response is based on the hypothesis that changes in hydrostatic pressure induce changes in the activity of nonselective cation channels. The resulting changes in membrane potential then affect calcium influx through changes in the activity of the voltage-gated calcium channels, so that vessel diameter decreases with increasing pressure values. With this configuration, the model AA maintains roughly stable renal blood flow within a physiologic range of blood flow pressure. Model simulation of vasoconstriction initiated from local stimulation also agrees well with findings in the experimental literature, notably those of Steinhausen et al. (Steinhausen M, Endlich K, Nobiling R, Rarekh N, Schütt F. J Physiol 505: 493-501, 1997), which indicated that conduction of vasoconstrictive response decays more rapidly in the upstream flow direction than downstream. The model can be incorporated into models of integrated renal hemodynamic regulation.

Theoretical assessment of renal autoregulatory mechanisms.

A mathematical model of renal hemodynamics was used to assess the individual contributions of the tubuloglomerular feedback (TGF) mechanism and the myogenic response to glomerular filtration rate regulation in the rat kidney. The model represents an afferent arteriole segment, glomerular filtration, and a short loop of Henle. The afferent arteriole model exhibits myogenic response, which is activated by hydrostatic pressure variations to induce changes in membrane potential and vascular muscle tone. The tubule model predicts tubular fluid and Cl(-) transport. Macula densa Cl(-) concentration is sensed as the signal for TGF, which acts to constrict or dilate the afferent arteriole. With this configuration, the model afferent arteriole maintains stable glomerular filtration rate within a physiologic range of perfusion pressure (80-180 mmHg). The contribution of TGF to overall autoregulation is significant only within a narrow band of perfusion pressure values (80-110 mmHg). Model simulations of ramp-like perfusion pressure perturbations agree well with findings by Flemming et al. (Flemming B, Arenz N, Seeliger E, Wronski T, Steer K, Persson PB. J Am Soc Nephrol 12: 2253-2262, 2001), which indicate that changes in vascular conductance are markedly sensitive to pressure velocity. That asymmetric response is attributed to the rate-dependent kinetics of the myogenic mechanism. Moreover, simulations of renal autoregulation in diabetes mellitus predict that, due to the impairment of the voltage-gated Ca(2+) channels of the afferent arteriole smooth muscle cells, the perfusion pressure range in which single-nephron glomerular filtration rate remains stable is reduced by ~70% and that TGF gain is reduced by nearly 40%, consistent with experimental findings.

Renal hemodynamics, function, and oxygenation during cardiac surgery performed on cardiopulmonary bypass: a modeling study

Acute kidney injury, a prevalent complication of cardiac surgery performed on cardiopulmonary bypass (CPB), is thought to be driven partly by hypoxic damage in the renal medulla. To determine the causes of medullary hypoxia during CPB, we modeled its impact on renal hemodynamics and function, and thus oxygen delivery and consumption in the renal medulla. The model incorporates autoregulation of renal blood flow and glomerular filtration rate and the utilization of oxygen for tubular transport. The model predicts that renal medullary oxygen delivery and consumption are reduced by a similar magnitude during the hypothermic (down to 28°C) phase of CPB. Thus, the fractional extraction of oxygen in the medulla, an index of hypoxia, is increased only by 58% from baseline. However, during the rewarming phase (up to 37°C), oxygen consumption by the medullary thick ascending limb increases 2.3‐fold but medullary oxygen delivery increases only by 33%. Consequently, the fractional extraction of oxygen in the medulla is increased 2.7‐fold from baseline. Thus, the renal medulla is particularly susceptible to hypoxia during the rewarming phase of CPB. Furthermore, autoregulation of both renal blood flow and glomerular filtration rate is blunted during CPB by the combined effects of hemodilution and nonpulsatile blood flow. Thus, renal hypoxia can be markedly exacerbated if arterial pressure falls below its target level of 50 mmHg. Our findings suggest that tight control of arterial pressure, and thus renal oxygen delivery, may be critical in the prevention of acute kidney injury associated with cardiac surgery performed on CPB.

Mathematical Modeling of Renal Hemodynamics in Physiology and Pathophysiology

In addition to the excretion of metabolic waste and toxin, the kidney plays an indispensable role in regulating the balance of water, electrolyte, acid-base, and blood pressure. For the kidney to maintain proper functions, hemodynamic control is crucial. In this review, we describe representative mathematical models that have been developed to better understand the kidneys autoregulatory processes. We consider mathematical models that simulate glomerular filtration, and renal blood flow regulation by means of the myogenic response and tubuloglomerular feedback. We discuss the extent to which these modeling efforts have expanded the understanding of renal functions in health and disease.

Conduction of feedback-mediated signal in a computational model of coupled nephrons

The nephron in the kidney regulates its fluid flow by several autoregulatory mechanisms. Two primary mechanisms are the myogenic response and the tubuloglomerular feedback (TGF). The myogenic response is a property of the pre-glomerular vasculature in which a rise in intravascular pressure elicits vasoconstriction that generates a compensatory increase in vascular resistance. TGF is a negative feedback response that balances glomerular filtration with tubular reabsorptive capacity. While each nephron has its own autoregulatory response, the responses of the kidneys many nephrons do not act autonomously but are instead coupled through the pre-glomerular vasculature. To better understand the conduction of these signals along the pre-glomerular arterioles and the impacts of internephron coupling on nephron flow dynamics, we developed a mathematical model of renal haemodynamics of two neighbouring nephrons that are coupled in that their afferent arterioles arise from a common cortical radial artery. Simulations were conducted to estimate internephron coupling strength, determine its dependence on vascular properties and to investigate the effect of coupling on TGF-mediated flow oscillations. Simulation results suggest that reduced gap-junctional conductances may yield stronger internephron TGF coupling and highly irregular TGF-mediated oscillations in nephron dynamics, both of which experimentally have been associated with hypertensive rats.

Bladder urine oxygen tension for assessing renal medullary oxygenation in rabbits: experimental and modeling studies

Oxygen tension (Po2) of urine in the bladder could be used to monitor risk of acute kidney injury if it varies with medullary Po2 Therefore, we examined this relationship and characterized oxygen diffusion across walls of the ureter and bladder in anesthetized rabbits. A computational model was then developed to predict medullary Po2 from bladder urine Po2 Both intravenous infusion of [Phe(2),Ile(3),Orn(8)]-vasopressin and infusion of N(G)-nitro-l-arginine reduced urinary Po2 and medullary Po2 (8-17%), yet had opposite effects on renal blood flow and urine flow. Changes in bladder urine Po2 during these stimuli correlated strongly with changes in medullary Po2 (within-rabbit r(2) = 0.87-0.90). Differences in the Po2 of saline infused into the ureter close to the kidney could be detected in the bladder, although this was diminished at lesser ureteric flow. Diffusion of oxygen across the wall of the bladder was very slow, so it was not considered in the computational model. The model predicts Po2 in the pelvic ureter (presumed to reflect medullary Po2) from known values of bladder urine Po2, urine flow, and arterial Po2 Simulations suggest that, across a physiological range of urine flow in anesthetized rabbits (0.1-0.5 ml/min for a single kidney), a change in bladder urine Po2 explains 10-50% of the change in pelvic urine/medullary Po2 Thus, it is possible to infer changes in medullary Po2 from changes in urinary Po2, so urinary Po2 may have utility as a real-time biomarker of risk of acute kidney injury.

Renal medullary and urinary oxygen tension during cardiopulmonary bypass in the rat

Abstract Renal hypoxia could result from a mismatch in renal oxygen supply and demand, particularly in the renal medulla. Medullary hypoxic damage is believed to give rise to acute kidney injury, which is a prevalent complication of cardiac surgery performed on cardiopulmonary bypass (CPB). To determine the mechanisms that could lead to medullary hypoxia during CPB in the rat kidney, we developed a mathematical model which incorporates (i) autoregulation of renal blood flow and glomerular filtration rate, (ii) detailed oxygen transport and utilization in the renal medulla and (iii) oxygen transport along the ureter. Within the outer medulla, the lowest interstitial tissue P\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}

Modeling Blood Flow and Oxygenation in a Diabetic Rat Kidney

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