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An Efficient Numerical Method and Parametric Study for Electrolyte Transport in the Renal Medulla

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Mathematical models of the mamalian urine concentrating mechanism consist of a large system of coupled, nonlinear and stiff equations. An efficient numerical orthogonal collocation method was employed to solve the steady-state formulation of urine concentrating mechanism. This method was used to solve the stiff and high order equations of electrolyte transport in a central core, single nephron model of the renal outer medulla. The presented results were in good agreement with implicit finite difference method’s results, but this new method was faster and more stable. Due to the greater stability and larger convergence domain of collocation method over Newton’s method, a parametric study on concentrated urine was investigated. The results showed that this model was sensitive to non-ideal countercurrent exchange between medulla interstitium and vasa recta. Although distal tubule lies in the cortex interstitium, it affects on the inflow to the collecting duct. This study showed that the effect of changing membrane transport properties of distal tubule wall on properties of outflow from the outer medulla collecting duct was considerable.
Cite this paper
Saadatmand, M. , Abdekhodaie, M. , Pishvaie, M. and Farhadi, F. (2014) An Efficient Numerical Method and Parametric Study for Electrolyte Transport in the Renal Medulla. Journal of Biomedical Science and Engineering, 7, 682-697. doi: 10.4236/jbise.2014.79068. 
 

[1] Layton, H.E. (2002) Mathematical Models of the Mammalian Urine Concentrating Mechanism, Membrane Transport and Renal Physiology. 129, Springer-Verlag, New York.
[2] Stephenson, J.L., Zhang, Y., Eftekhari, A. and Tewarson, R. (1987) Electrolyte Transport in a Central Core Model of the Renal Medulla. American Journal of Physiology—Renal Physiology, 253, F982-F997.
[3] Stephenson, J.L. (1972) Concentration of Urine in a Central Core Model of the Renal Counterflow System. Kidney International, 2, 85-94.
http://dx.doi.org/10.1038/ki.1972.75
[4] Wexler, A.S., Kalaba, R.E. and Marsh, D.J. (1991) Three-Dimensional Anatomy and Renal Concetrating Mechanism. Modeling Results. American Journal of Physiology—Renal Physiology, 260, F368-F383.
[5] Thomas, S.R., Layton, A.T., Layton, H.E. and Moor, L.C. (2006) Kidney Modeling: Status and Perspectives. Proceedings of the IEEE, 94, 740-752.
http://dx.doi.org/10.1109/JPROC.2006.871770
[6] Weinstein, A.M. (1994) Mathematical Models of Tubular Transport. Annual Review of Physiology, 56, 691-709.
http://dx.doi.org/10.1146/annurev.ph.56.030194.003355
[7] Bykova, A.A. and Regirer, S.A. (2005) Mathematical Models in Urinary System Mechanics (Review). Fluid Dynamics, 40, 1-19.
http://dx.doi.org/10.1007/s10697-005-0039-y
[8] Pannabecker, T.L., Dantzler, W.H., Layton, H.E. and Layton, A.T. (2008) Role of Three-Dimensional Architecture in the Urine Concentrating Mechanism of the Rat Renal Inner Medulla. American Journal of Physiology—Renal Physiology, 295, F1271-F1285.
http://dx.doi.org/10.1152/ajprenal.90252.2008
[9] Wexler, A.S. and Marsh, D.J. (1991) Numerical Methods for Three-Dimensional Models of the Urine Concentrating Mechanism. Applied Mathematics and Computation, 45, 219-240.
http://dx.doi.org/10.1016/0096-3003(91)90082-X
[10] Layton, A.T., Layton, H.E., Dantzler, W.H. and Pannabecker, T.L. (2009) The Mammalian Urine Concentrating Mechanism: Hypotheses and Uncertainties. Physiology, 24, 250-256.
http://dx.doi.org/10.1152/physiol.00013.2009
[11] Stephenson, J.L., Zhang, Y. and Tewarson, R. (1989) Electrolyte, Urea, and Water Transport in a Two-Nephron Central Core Model of the Renal Medulla. American Journal of Physiology—Renal Physiology, 257, F399-F413.
[12] Stephenson, J.L., Zhang, Y. and Tewarson, R. (1976) Electrolyte, Urea, and Water Transport in a Two-Nephron Central Core Model of the Renal Medulla. Proceedings of the National Academy of Sciences, 73, 252-258.
http://dx.doi.org/10.1073/pnas.73.1.252
[13] Mejia, R. and Stephenson, J.L. (1984) Solution of a Multinephron, Multisolute Model of the Mammalian Kidney by Newton and Continuation Methods. Mathematical Biosciences, 68, 279-298.
http://dx.doi.org/10.1016/0025-5564(84)90036-1
[14] Tewarson, R.P., Stephenson, J.L., Garcia, M. and Zhanc, Y. (1985) On the Solution of Equation for Renal Counterflow Models. Computers in Biology and Medicine, 15, 287-295.
http://dx.doi.org/10.1016/0010-4825(85)90012-5
[15] Wexler, A.S., Kalaba, R.E. and Marsh, D.J. (1991) Three-Dimensional Anatomy and Renal Concetrating Mechanism. Sensitivity Results. American Journal of Physiology—Renal Physiology, 260, F384-F394.
[16] Layton, A.T. and Layton, H.E. (2002) A Semi-Lagrangian Semi-Implicit Numerical Method for Models of the Urine Concentrating Mechanism. SIAM Journal on Scientific Computing, 23, 1526-1548.
http://dx.doi.org/10.1137/S1064827500381781
[17] Layton, H.E., Pitman, E.B. and Knepper, A.M. (1995) A Dynamic Numerical Method for Models of the Urine Concentrating mechanism. SIAM Journal on Applied Mathematics, 55, 1390-1418.
http://dx.doi.org/10.1137/S0036139993252864
[18] Layton, A.T. and Layton, H.E. (2003) An Efficient Numerical Method for Distributed-Loop Models of the Urine Concentrating Mechanism. Mathematical Biosciences, 181, 111-132.
http://dx.doi.org/10.1016/S0025-5564(02)00176-1
[19] Layton, A.T., Pannabecker, T.L., Dantzler, W.H. and Layton, H.E. (2004) Two Modes for Concentrating Urine in Rat Inner Medulla. American Journal of Physiology—Renal Physiology, 287, F816-F839.
http://dx.doi.org/10.1152/ajprenal.00398.2003
[20] Jator, S. and Sinkala, Z. (2007) A High Order B-Spline Collocation Method for Linear Boundary Value Problems. Applied Mathematics and Computation, 191, 100-116.
http://dx.doi.org/10.1016/j.amc.2007.02.027
[21] Guyton, A.C. and Hall, J.E. (2006) Textbook of Medical Physiology. 11th Edition, Elsevier Saunders, Amsterdam.
[22] Sands, J.M. and Layton, H.E. (2009) The Physiology of Urinary Concentration: An Update. Seminars in Nephrology, 29, 178-195.
http://dx.doi.org/10.1016/j.semnephrol.2009.03.008
[23] Kedem, O. and Katchalsky, Q. (1963) Permeability of Composite Membranes. Transactions of the Faraday Society, 59, 1931-1953.
http://dx.doi.org/10.1039/tf9635901931
[24] Weinstein, A.M. (2002) A Mathematical Model of Rat Collecting duct 1.Flow Effects on Transport and Urinary Acidification. American Journal of Physiology—Renal Physiology, 283, F1237-F1251.
[25] Rice, R.G. and Do, D.D. (1995) Applied Mathematics and Modeling for Chemical Engineers. John Wiley and Sons, Inc., Hoboken.
[26] Schafer, J.A. (2000) Interaction of Modeling and Experimental Approaches to Understanding Renal Salt and Water Balance. Annals of Biomedical Engineering, 28, 1002-1009.
http://dx.doi.org/10.1114/1.1308497
[27] Schafer, J.A. (2004) Experimental Validation of the Countercurrent Model of Urinary Concentration. American Journal of Physiology—Renal Physiology, 287, F861-F863.
http://dx.doi.org/10.1152/classicessays.00020.2004         eww141103lx
[28] (2001) CRC Handbook of Chemistry and Physics. 82nd Edition, CRC, Cleveland.
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