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WIREs Syst Biol Med
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Mathematical modeling of kidney transport

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In addition to metabolic waste and toxin excretion, the kidney also plays an indispensable role in regulating the balance of water, electrolytes, nitrogen, and acid–base. In this review, we describe representative mathematical models that have been developed to better understand kidney physiology and pathophysiology, including the regulation of glomerular filtration, the regulation of renal blood flow by means of the tubuloglomerular feedback mechanisms and of the myogenic mechanism, the urine concentrating mechanism, epithelial transport, and regulation of renal oxygen transport. We discuss the extent to which these modeling efforts have expanded our understanding of renal function in both health and disease. WIREs Syst Biol Med 2013, 5:557–573. doi: 10.1002/wsbm.1232 This article is categorized under: Analytical and Computational Methods > Computational Methods Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models Physiology > Physiology of Model Organisms

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An illustration of three nephrons, together with their glomeruli. (Reprinted with permission from Ref . Copyright 1972 Springer)
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Transport pathways across apical and basolateral membranes of the proximal convoluted cell of the rat. (Reprinted with permission from Ref . Copyright 1992 American Physiological Society)
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Schematic diagram of a cross section through the outer stripe, inner stripe, upper inner medulla (IM), mid‐IM, and deep IM, showing regions and relative positions of tubules and vessels. Decimal numbers indicate relative interaction weightings with regions. R1, R2, R3, and R4, regions in the outer medulla; R5, R6, and R7, regions in the IM. SDL, descending limbs of short loops of Henle. SAL, ascending limbs of long loops of Henle. LDL, descending limb of long loop of Henle. LAL, ascending limb of long loop of Henle. Subscripts ‘S,’ ‘M,’ and ‘L’ associated with a LDL or LAL denote limbs that turn with the first mm of the IM (S), within the mid‐IM (M), or reach into the deep IM (L). CD, collecting duct. SDV, short descending vasa recta. SAV3 and SAV4, two populations of short ascending vasa recta. LDV, long descending vas rectum. LAV1, LAV2, …, LAV7, populations of long ascending vasa recta. (Reprinted with permission from Ref . Copyright 2011 American Physiological Society)
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Schematic diagrams of tubular organization in the rat renal medulla. (a) Cross section through the inner stripe of outer medulla, where tubules appear to be organized around a vascular bundle. (b) Cross section through the upper inner medulla, where tubules and vessels are organized around a collecting duct cluster. Inset: schematic configuration of a collecting duct, ascending vasa recta (AVR), an ascending thin limb, and a nodal space. (Reprinted with permission from Ref . Copyright 2009 American Physiological Society).
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Schematic diagram of the central core model. (a) Tubules along spatial axis. DL, descending limb; AL, ascending limb; CD, collecting duct; CC, central core. Arrows, steady‐state flow directions. Heavy lines, water‐impermeable boundaries. (b) Cross section showing connectivity between CC and other tubules. (Reprinted with permission from Ref . Copyright 2002 Society of Industrial and Applied Mathematics)
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Top‐right: A flow chart that illustrates the steps by which periodic oscillations in cytosolic calcium concentration give rise to spontaneous vasomotion of the model afferent arteriole in Refs . Bottom‐right: Average vessel inner diameter as a function of steady‐state transmural pressure, with and without a myogenic response. (a) Oscillations in Ca2 + and K+ currents (denoted ICa and IK, respectively) and membrane potential v. (b) Oscillations in intracellular arteriolar inner diameter. (Reprinted with permission from Ref . Copyright 2011 American Physiological Society)
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(a) Steady‐state diameter at renal arteriolar pressure of 60 to 180 mmHg. (b) Percent change from basal (60 mmHg) diameter, fitted using a simple quadratic equation. (Reprinted with permission from Ref . Copyright 2002 American Heart Association)
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Schematic representation model tubuloglomerular feedback (TGF) system. Hydrodynamic pressure Po(t) = P(0,t) drives flow into loop entrance (x = 0) at time t. Oscillations in pressure result in oscillations in loop pressure P(x,t), flow rate Q(x,t), radius R(x,t), and tubular fluid chloride concentration C(x,t). (Reprinted with permission from Ref . Copyright 2012 Springer)
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Left: Bifurcation diagrams indicating observed behavior of tubuloglomerular feedback (TGF) model solutions. Four qualitatively different model solutions are possible: (1) a regime having one stable, time‐independent steady‐state solution (labeled ‘Steady State’); (2) a regime having one stable oscillatory solution only, with fundamental frequency f (‘1−f LCO’); (3) a regime having one stable oscillatory solution only, with frequency approximately 2f (‘2−f LCO’) and (4) a regime having two possible stable oscillatory solutions, of frequencies approximately f and approximately 2f (‘1,2−f LCO’). (Reprinted with permission from Ref . Copyright 2006 American Physiological Society) Right: Oscillations in single nephron glomerular filtration rate (SNGFR) and thick ascending limb (TAL) tubular fluid Cl concentration and at the macula densa obtained using τ and γ values denoted by the point B3.
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A schematic drawing of two short‐looped nephrons and their afferent arterioles (AA). The arterioles branch from a small connecting artery (unlabeled), which arises from a cortical radial artery (CRA). The nephron consists of the glomerulus (G) and a tubule having several segments, including: the proximal tubule (PT), the descending limb (DL), the thick ascending limb (TAL), and the distal convoluted tubule (DCT). Each nephron has its glomerulus in the renal cortex, and each short‐looped rat nephron has a loop that that extends into the outer medulla of the kidney. The axis on the TAL of the lower nephron corresponds to the spatial axis used in the model (vide infra, Section Glomerular Filtration); in this figure distance is indicated in terms of fractional (nondimensional) TAL length. Tubular fluid from the DL flows into the TAL lumen at x = 0; the chloride concentration of TAL luminal fluid is sensed by the macula densa (MD) at x = 1. The MD, a localized plaque of specialized cells, forms a portion of the TAL wall that is separated from the AA by a few layers of extraglomerular mesangial cells; in this figure, the MD is part of the short TAL segment that passes behind the AA. Fluid from the DCT enters the collecting duct system (not shown), from which urine ultimately emerges. Structures labeled on one nephron apply to both nephrons. (Reprinted with permission from Ref . Copyright 2004 Springer)
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Typical tubuloglomerular feedback (TGF) response, with operating point (Q,CMD) = (30,32).
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Physiology > Physiology of Model Organisms
Analytical and Computational Methods > Computational Methods
Models of Systems Properties and Processes > Organ, Tissue, and Physiological Models

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