Kidney Modeling: Status and Perspectives

Mathematical models have played an essential role in elucidating various functions of the kidney, including the mechanism by which the avian and mammalian kidney can produce a urine that is more concentrated than blood plasma, quasi-isosmotic reabsorption along the proximal tubule, and the control and regulation of glomerular filtration by the myogenic and tubuloglomerular feedback mechanisms. This review includes a brief description of relevant renal physiology, a summary of the contributions of mathematical models at various levels and describes our recent work toward the Renal Physiome.

[1]  W. Wang,et al.  Effects of Anastomoses on Solute Transcapillary Exchange in Countercurrent Systems , 1997, Microcirculation.

[2]  A. Weinstein.  A mathematical model of rat distal convoluted tubule. I. Cotransporter function in early DCT. , 2005, American journal of physiology. Renal physiology.

[3]  R. Jamison,et al.  An in vivo study of the concentrating process in the descending limb of Henle's loop. , 1974, Kidney international.

[4]  Sheldon Weinbaum,et al.  A dual-pathway ultrastructural model for the tight junction of rat proximal tubule epithelium. , 2003, American journal of physiology. Renal physiology.

[5]  A. Weinstein.  A mathematical model of the inner medullary collecting duct of the rat: pathways for Na and K transport. , 1998, American journal of physiology. Renal physiology.

[6]  S. R. Thomas,et al.  Cycles and separations in a model of the renal medulla. , 1998, American journal of physiology. Renal physiology.

[7]  Andrew Lonie,et al.  Visualization tools for the Renal Physiome , 2004 .

[8]  S. Thomas.  Inner medullary lactate production and accumulation: a vasa recta model. , 2000, American journal of physiology. Renal physiology.

[9]  A. Weinstein,et al.  Mathematical models of tubular transport. , 1994, Annual review of physiology.

[10]  J. Atherton,et al.  Acute effects of lysine vasopressin injection (Single and continuous) on urinary composition in the conscious water diuretic rat , 2004, Pflügers Archiv.

[11]  Alan M Weinstein,et al.  A mathematical model of rat collecting duct. III. Paradigms for distal acidification defects. , 2002, American journal of physiology. Renal physiology.

[12]  H. Chang,et al.  A numerical model of acid-base transport in rat distal tubule. , 2001, American journal of physiology. Renal physiology.

[13]  G. Saidel,et al.  Role of inner medullary collecting duct NaCl transport in urinary concentration. , 1985, The American journal of physiology.

[14]  M. Hai,et al.  The time-course of changes in renal tissue composition during lysine vasopressin infusion in the rat , 2004, Pflügers Archiv.

[15]  J. L. Stephenson.  Concentration of urine in a central core model of the renal counterflow system. , 1972, Kidney international.

[16]  A. Layton,et al.  Two modes for concentrating urine in rat inner medulla. , 2004, American journal of physiology. Renal physiology.

[17]  D. Marsh,et al.  How descending limb of Henle's loop permeability affects hypertonic urine formation. , 1980, The American journal of physiology.

[18]  P. Lory.  Effectiveness of a salt transport cascade in the renal medulla: computer simulations. , 1987, The American journal of physiology.

[19]  Alan M Weinstein,et al.  A mathematical model of rat collecting duct. II. Effect of buffer delivery on urinary acidification. , 2002, American journal of physiology. Renal physiology.

[20]  R. Tewarson,et al.  Quantitative analysis of mass and energy balance in non-ideal models of the renal counterflow system. , 1974, Proceedings of the National Academy of Sciences of the United States of America.

[21]  K. Ullrich,et al.  Glucose- und Milchsäurekonzentration an der Spitze des vasculären Gegenstromsystems im Nierenmark , 1961, Pflüger's Archiv für die gesamte Physiologie des Menschen und der Tiere.

[22]  Anita T Layton,et al.  Multistability in tubuloglomerular feedback and spectral complexity in spontaneously hypertensive rats. , 2006, American journal of physiology. Renal physiology.

[23]  James T. McCarthy.  Renal Physiology: Principles, Structure, and Function , 1987 .

[24]  K. Lemley,et al.  Cycles and separations: the histotopography of the urinary concentrating process. , 1987, Kidney international.

[25]  T. Pannabecker,et al.  Three-dimensional functional reconstruction of inner medullary thin limbs of Henle's loop. , 2004, American journal of physiology. Renal physiology.

[26]  M. Knepper,et al.  In vitro perfusion of chinchilla thin limb segments: urea and NaCl permeabilities. , 1993, The American journal of physiology.

[27]  S. Hebert,et al.  Ionic conductance pathways in the mouse medullary thick ascending limb of Henle. The paracellular pathway and electrogenic Cl- absorption , 1986, The Journal of general physiology.

[28]  Aurélie Edwards,et al.  A multiunit model of solute and water removal by inner medullary vasa recta. , 1998, American journal of physiology. Heart and circulatory physiology.

[29]  L. Bankir,et al.  Papillary plasma flow in rats , 1982, Pflügers Archiv.

[30]  E. Bruce Pitman,et al.  A dynamic numerical method for models of renal tubules , 1994 .

[31]  J H Stein,et al.  Studies on the mechanism of reduced urinary osmolality after exposure of renal papilla. , 1978, The Journal of clinical investigation.

[32]  R. Greger,et al.  Presence of luminal K+, a prerequisite for active NaCl transport in the cortical thick ascending limb of Henle's loop of rabbit kidney , 1981, Pflügers Archiv.

[33]  F. Stewart,et al.  Progression rate of radiation damage to the mouse kidney: a quantitative analysis of experimental data using a simple mathematical model of the nephron. , 1997, International journal of radiation biology.

[34]  A Edwards,et al.  Transport of plasma proteins across vasa recta in the renal medulla. , 2001, American journal of physiology. Renal physiology.

[35]  D E Oken,et al.  A network thermodynamic model of glomerular dynamics: application in the rat. , 1981, Kidney international.

[36]  J. Atherton,et al.  The time course of changes in renal tissue composition during mannitol diuresis in the rat , 1968, The Journal of physiology.

[37]  W. Niesel,et al.  Möglichkeiten der Konzentrierung von Stoffen in biologischen Gegenstromsystemen , 1963, Pflüger's Archiv für die gesamte Physiologie des Menschen und der Tiere.

[38]  Aurélie Edwards,et al.  Theoretical effects of UTB urea transporters in the renal medullary microcirculation. , 2003, American journal of physiology. Renal physiology.

[39]  E. Pitman,et al.  Bifurcation analysis of TGF-mediated oscillations in SNGFR. , 1991, The American journal of physiology.

[40]  T. Pannabecker,et al.  Three-dimensional lateral and vertical relationships of inner medullary loops of Henle and collecting ducts. , 2004, American journal of physiology. Renal physiology.

[41]  Harold E. Layton,et al.  Mathematical Models of the Mammalian Urine Concentrating Mechanism , 2002 .

[42]  S. Weinbaum,et al.  A hydrodynamic mechanosensory hypothesis for brush border microvilli. , 2000, American journal of physiology. Renal physiology.

[43]  Rainer Oberbauer,et al.  Reduction in Mean Glomerular Pore Size Coincides with the Development of Large Shunt Pores in Patients with Diabetic Nephropathy , 2000, Nephron Experimental Nephrology.

[44]  E A Trowbridge,et al.  A mathematical model of the ultra-filtration process in a single glomerular capillary. , 1974, Journal of theoretical biology.

[45]  R. Kalaba,et al.  Three-dimensional anatomy and renal concentrating mechanism. II. Sensitivity results. , 1991, The American journal of physiology.

[46]  A H Oien,et al.  A multinephron model of renal blood flow autoregulation by tubuloglomerular feedback and myogenic response. , 1991, Acta physiologica Scandinavica.

[47]  T. Dwyer,et al.  Concentration of solutes in the renal inner medulla: interstitial hyaluronan as a mechano-osmotic transducer. , 2003, American journal of physiology. Renal physiology.

[48]  R. Greger,et al.  Cellular mechanism of the action of loop diuretics on the thick ascending limb of Henle's loop , 1983, Klinische Wochenschrift.

[49]  S. Hebert,et al.  ADH increases apical Na+, K+, 2Cl- entry in mouse medullary thick ascending limbs of Henle. , 1987, The American journal of physiology.

[50]  E. Pitman,et al.  Limit-cycle oscillations and tubuloglomerular feedback regulation of distal sodium delivery. , 2000, American journal of physiology. Renal physiology.

[51]  W. Deen,et al.  Structural determinants of glomerular permeability. , 2001, American journal of physiology. Renal physiology.

[52]  B. Myers,et al.  Pathophysiology of proteinuria in immune glomerular injury. , 1990, American journal of nephrology.

[53]  Alan M Weinstein,et al.  A mathematical model of the inner medullary collecting duct of the rat: acid/base transport. , 1998, American journal of physiology. Renal physiology.

[54]  C. Peskin.  The immersed boundary method , 2002, Acta Numerica.

[55]  A. Weinstein.  Mathematical models of renal fluid and electrolyte transport: acknowledging our uncertainty. , 2003, American journal of physiology. Renal physiology.

[56]  W. Deen,et al.  Analysis of pulsatile pressures and flows in glomerular filtration. , 1991, The American journal of physiology.

[57]  N. Holstein-Rathlou,et al.  A dynamic model of the tubuloglomerular feedback mechanism. , 1990, The American journal of physiology.

[58]  K. Chon,et al.  Nonlinear interactions in renal blood flow regulation. , 2005, American journal of physiology. Regulatory, integrative and comparative physiology.

[59]  R. Kalaba,et al.  Three-dimensional anatomy and renal concentrating mechanism. I. Modeling results. , 1991, The American journal of physiology.

[60]  Anita T Layton,et al.  A numerical method for renal models that represent tubules with abrupt changes in membrane properties , 2002, Journal of mathematical biology.

[61]  A. Weinstein.  A mathematical model of the outer medullary collecting duct of the rat. , 2000, American journal of physiology. Renal physiology.

[62]  E. Mosekilde,et al.  Bifurcation analysis of nephron pressure and flow regulation. , 1996, Chaos.

[63]  A. Weinstein,et al.  A mathematical model of rat cortical collecting duct: determinants of the transtubular potassium gradient. , 2001, American journal of physiology. Renal physiology.

[64]  A S Wexler,et al.  The effect of solution non-ideality on membrane transport in three-dimensional models of the renal concentrating mechanism. , 1994, Bulletin of mathematical biology.

[65]  A S Wexler,et al.  Outer medullary anatomy and the urine concentrating mechanism. , 1998, American journal of physiology. Renal physiology.

[66]  W. Kuhn,et al.  The multiplication principle as the basis for concentrating urine in the kidney. , 2001, Journal of the American Society of Nephrology : JASN.

[67]  B. Brenner,et al.  Three-dimensional reconstructed glomerular capillary network: blood flow distribution and local filtration. , 1992, The American journal of physiology.

[68]  A S Wexler,et al.  The effects of collecting duct active NaCl reabsorption and inner medulla anatomy on renal concentrating mechanism. , 1996, The American journal of physiology.

[69]  A. Wexler,et al.  Inner medullary external osmotic driving force in a 3-D model of the renal concentrating mechanism. , 1995, The American journal of physiology.

[70]  Anita T Layton,et al.  A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. II. Parameter sensitivity and tubular inhomogeneity. , 2005, American journal of physiology. Renal physiology.

[71]  S. R. Thomas,et al.  Effect of varying salt and urea permeabilities along descending limbs of Henle in a model of the renal medullary urine concentrating mechanism , 1991, Bulletin of mathematical biology.

[72]  Kayne M. Smith,et al.  Advective transport of nitric oxide in a mathematical model of the afferent arteriole. , 2003, American journal of physiology. Renal physiology.

[73]  M. Imai,et al.  Transition of permeability properties along the descending limb of long-loop nephron. , 1988, The American journal of physiology.

[74]  Anita T Layton,et al.  An efficient numerical method for distributed-loop models of the urine concentrating mechanism. , 2003, Mathematical biosciences.

[75]  T. Fujita,et al.  A kinetic model of the thiazide-sensitive Na-Cl cotransporter. , 1999, American journal of physiology. Renal physiology.

[76]  R. Jamison,et al.  Urinary concentration in the papillary collecting duct of the rat. Role of the ureter. , 1982, The Journal of clinical investigation.

[77]  Kevin V Lemley,et al.  Podocytopenia and disease severity in IgA nephropathy. , 2002, Kidney international.

[78]  A H Oien,et al.  Myogenic vasoconstriction in the rat kidney elicited by reducing perirenal pressure. , 1992, Acta physiologica Scandinavica.

[79]  W. Schütz,et al.  Pelvic urine composition as a determinant of inner medullary solute concentration and urine osmolarity , 2004, Pflügers Archiv.

[80]  B. Schmidt-nielsen,et al.  Peristaltic flow of urine in the renal capillary collecting ducts of hamsters. , 1981, Kidney international.

[81]  A. Ramel,et al.  Aktiver Salztransport als möglicher (und wahrscheinlicher) Einzeleffekt bei der Harnkonzentrierung in der Niere , 1959 .

[82]  R. Loutzenhiser,et al.  Renal Myogenic Response: Kinetic Attributes and Physiological Role , 2002, Circulation research.

[83]  V Dzodic,et al.  Web-based tools for quantitative renal physiology. , 2004, Cellular and molecular biology.

[84]  N. Holstein-Rathlou,et al.  Analysis of interaction between TGF and the myogenic response in renal blood flow autoregulation. , 1995, The American journal of physiology.

[85]  A. W. Cowley,et al.  Countercurrent exchange in the renal medulla. , 2003, American journal of physiology. Regulatory, integrative and comparative physiology.

[86]  A. Weinstein.  Insights from Mathematical Modeling of Renal Tubular Function , 1998, Nephron Experimental Nephrology.

[87]  J. P. Gassee,et al.  Determination of glomerular intracapillary and transcapillary pressure gradients from sieving data , 1976, Pflügers Archiv.

[88]  Aurélie Edwards,et al.  Oxygen transport across vasa recta in the renal medulla. , 2002, American journal of physiology. Heart and circulatory physiology.

[89]  W. Kuhn,et al.  Das Multiplikationsprinzip als Grundlage der Harnkonzentrierung m der Niere , 1951, Zeitschrift für Elektrochemie und angewandte physikalische Chemie.

[90]  C. Michel,et al.  Theoretical Studies of Steady‐State Transcapillary Exchange in Countercurrent Systems , 1996, Microcirculation.

[91]  A. Rich,et al.  Ascending myogenic autoregulation: interactions between tubuloglomerular feedback and myogenic mechanisms. , 1994, Bulletin of mathematical biology.

[92]  A. Weinstein.  A mathematical model of rat distal convoluted tubule. II. Potassium secretion along the connecting segment. , 2005, American journal of physiology. Renal physiology.

[93]  B. Kristal,et al.  Structural basis for reduced glomerular filtration capacity in nephrotic humans. , 1994, The Journal of clinical investigation.

[94]  S. R. Thomas,et al.  Inner medullary lactate production and urine-concentrating mechanism: a flat medullary model. , 2003, American journal of physiology. Renal physiology.

[95]  A. Layton,et al.  A region-based model framework for the rat urine concentrating mechanism , 2003, Bulletin of mathematical biology.

[96]  D. J. Marsh,et al.  A dynamic model of renal blood flow autoregulation , 1994 .

[97]  Roman M. Zaritski,et al.  Feedback-mediated dynamics in two coupled nephrons , 2004, Bulletin of mathematical biology.

[98]  A. Layton,et al.  A region-based mathematical model of the urine concentrating mechanism in the rat outer medulla. I. Formulation and base-case results. , 2005, American journal of physiology. Renal physiology.

[99]  B. Schmidt-nielsen,et al.  The renal pelvis: machinery that concentrates urine in the papilla. , 2003, News in physiological sciences : an international journal of physiology produced jointly by the International Union of Physiological Sciences and the American Physiological Society.

[100]  F. Rector,et al.  Countercurrent multiplication system without active transport in inner medulla. , 1972, Kidney international.

[101]  G. Saidel,et al.  Mathematical model of mass transport throughout the kidney: Effects of nephron heterogeneity and tubular-vascular organization , 1981, Annals of Biomedical Engineering.

[102]  P. Leyssac,et al.  An oscillating intratubular pressure response to alterations in Henle loop flow in the rat kidney. , 1983, Acta physiologica Scandinavica.

[103]  A. Weinstein.  A mathematical model of rat collecting duct. I. Flow effects on transport and urinary acidification. , 2002, American journal of physiology. Renal physiology.

[104]  A Edwards,et al.  Facilitated transport in vasa recta: theoretical effects on solute exchange in the medullary microcirculation. , 1997, The American journal of physiology.