# American Institute of Mathematical Sciences

December  2012, 7(4): 989-1018. doi: 10.3934/nhm.2012.7.989

## Analysis of a simplified model of the urine concentration mechanism

 1 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France, France 2 Univ Paris 06, Univ Paris 05, INSERM UMRS 872, and CNRS ERL 7226, Laboratoire de génomique, physiologie et physiopathologie rénales, Centre de Recherche des Cordeliers, 75006, Paris, France 3 UMR 7598 Laboratoire J.-L. Lions, UPMC Univ Paris 06, Paris, F-75005

Received  September 2011 Revised  October 2012 Published  December 2012

We study a nonlinear stationary system of transport equations with specific boundary conditions describing the transport of solutes dissolved in a fluid circulating in a countercurrent tubular architecture, which constitutes a simplified model of a kidney nephron. We prove that for every Lipschitz and monotonic nonlinearity (which stems from active transport across the ascending limb), the dynamic system, a PDE which we study through contraction properties, relaxes toward the unique stationary state. A study of the linearized stationary operator enables us, using eigenelements, to further show that under certain conditions regarding the nonlinearity, the relaxation is exponential. We also describe a finite volume scheme which allows us to efficiently approach the numerical solution to the stationary system. Finally, we apply this numerical method to illustrate how the countercurrent arrangement of tubes enhances the axial concentration gradient, thereby favoring the production of highly concentrated urine.
Citation: Magali Tournus, Aurélie Edwards, Nicolas Seguin, Benoît Perthame. Analysis of a simplified model of the urine concentration mechanism. Networks and Heterogeneous Media, 2012, 7 (4) : 989-1018. doi: 10.3934/nhm.2012.7.989
##### References:
 [1] G. Allaire, "Numerical Analysis and Optimization," Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2007. An introduction to mathematical modelling and numerical simulation, Translated from the French by Alan Craig. [2] F. Bouchut, "Non Linear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well Balanced Schemes for Sources," Birkhaüser-Verlag, 2004. doi: 10.1007/b93802. [3] C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 2010. doi: 10.1007/978-3-642-04048-1. [4] L. C. Evans, "Partial Differential Equations," 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. [5] J. Garner, K. Crump and J. Stephenson, Transient behaviour of the single loop solute cycling model of the renal medulla, Bulletin of Mathematical Biology, 40 (1978), 273-300. doi: 10.1007/BF02461602. [6] J. B. Garner and R. B. Kellogg, Existence and uniqueness of solutions in general multisolute renal flow problems, Journal of Mathematical Biology, 26 (1988), 455-464. doi: 10.1007/BF00276373. [7] E. Godlewski and P. A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws," 118 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996. [8] M. A. Katsoulakis and A. E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations, 22 (1997), 195-233. doi: 10.1080/03605309708821261. [9] A. T. Layton and H. E. Layton, A semi-Lagrangian semi-implicit numerical method for models of the urine concentrating mechanism, SIAM Journal on Scienfic Computing, 23 (2002), 1526-1548. doi: 10.1137/S1064827500381781. [10] H. Layton and E. Pitman, A dynamic numerical method for models of renal tubules, Bulletin of Mathematical Biology, 56 (1994), 547-565. [11] H. E. Layton, Distribution of henle's loops may enhance urine concentrating capability, Biophysical Journal, 49 (1986), 1033-1040. [12] H. E. Layton, Existence and uniqueness of solutions to a mathematical model of the urine concetrating mechanism, Mathematical Biosciences, 84 (1987), 197-210. doi: 10.1016/0025-5564(87)90092-7. [13] H. E. Layton, E. Bruce Pitman and Mark A. Knepper, A dynamic numerical method for models of the urine concentrating mechanism, SIAM J. Appl. Math., 55 (1995), 1390-1418. [14] R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. [15] L. C. Moore and D. J. Marsh, How descending limb of Henle's loop permeability affects hypertonic urine formation, Am J Physiol Renal Physiol, 239 (1980), F57-71. [16] R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Communications on Pure and Applied Mathematics, 49 (1996), 795-823. doi: 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3. [17] B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007. [18] D. Serre, "Matrices," 216 of Graduate Texts in Mathematics. Springer, New York, second edition, 2010. Theory and applications. doi: 10.1007/978-1-4419-7683-3. [19] J. L. Stephenson, "Urinary Concentration and Dilution: Models," Oxford University Press, New-York, 1992. [20] K. Werner and B. Hargitay, The multiplication principle as the basis for concentrating urine in the kidney(with comments by Bart Hargitay and S. Randall Thomas), J. Am. Soc. Nephrol., 12 (2001), 1566-1586.

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##### References:
 [1] G. Allaire, "Numerical Analysis and Optimization," Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford, 2007. An introduction to mathematical modelling and numerical simulation, Translated from the French by Alan Craig. [2] F. Bouchut, "Non Linear Stability of Finite Volume Methods for Hyperbolic Conservation Laws and Well Balanced Schemes for Sources," Birkhaüser-Verlag, 2004. doi: 10.1007/b93802. [3] C. M. Dafermos, "Hyperbolic Conservation Laws in Continuum Physics," 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, third edition, 2010. doi: 10.1007/978-3-642-04048-1. [4] L. C. Evans, "Partial Differential Equations," 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010. [5] J. Garner, K. Crump and J. Stephenson, Transient behaviour of the single loop solute cycling model of the renal medulla, Bulletin of Mathematical Biology, 40 (1978), 273-300. doi: 10.1007/BF02461602. [6] J. B. Garner and R. B. Kellogg, Existence and uniqueness of solutions in general multisolute renal flow problems, Journal of Mathematical Biology, 26 (1988), 455-464. doi: 10.1007/BF00276373. [7] E. Godlewski and P. A. Raviart, "Numerical Approximation of Hyperbolic Systems of Conservation Laws," 118 of Applied Mathematical Sciences, Springer-Verlag, New York, 1996. [8] M. A. Katsoulakis and A. E. Tzavaras, Contractive relaxation systems and the scalar multidimensional conservation law, Comm. Partial Differential Equations, 22 (1997), 195-233. doi: 10.1080/03605309708821261. [9] A. T. Layton and H. E. Layton, A semi-Lagrangian semi-implicit numerical method for models of the urine concentrating mechanism, SIAM Journal on Scienfic Computing, 23 (2002), 1526-1548. doi: 10.1137/S1064827500381781. [10] H. Layton and E. Pitman, A dynamic numerical method for models of renal tubules, Bulletin of Mathematical Biology, 56 (1994), 547-565. [11] H. E. Layton, Distribution of henle's loops may enhance urine concentrating capability, Biophysical Journal, 49 (1986), 1033-1040. [12] H. E. Layton, Existence and uniqueness of solutions to a mathematical model of the urine concetrating mechanism, Mathematical Biosciences, 84 (1987), 197-210. doi: 10.1016/0025-5564(87)90092-7. [13] H. E. Layton, E. Bruce Pitman and Mark A. Knepper, A dynamic numerical method for models of the urine concentrating mechanism, SIAM J. Appl. Math., 55 (1995), 1390-1418. [14] R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253. [15] L. C. Moore and D. J. Marsh, How descending limb of Henle's loop permeability affects hypertonic urine formation, Am J Physiol Renal Physiol, 239 (1980), F57-71. [16] R. Natalini, Convergence to equilibrium for the relaxation approximations of conservation laws, Communications on Pure and Applied Mathematics, 49 (1996), 795-823. doi: 10.1002/(SICI)1097-0312(199608)49:8<795::AID-CPA2>3.0.CO;2-3. [17] B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007. [18] D. Serre, "Matrices," 216 of Graduate Texts in Mathematics. Springer, New York, second edition, 2010. Theory and applications. doi: 10.1007/978-1-4419-7683-3. [19] J. L. Stephenson, "Urinary Concentration and Dilution: Models," Oxford University Press, New-York, 1992. [20] K. Werner and B. Hargitay, The multiplication principle as the basis for concentrating urine in the kidney(with comments by Bart Hargitay and S. Randall Thomas), J. Am. Soc. Nephrol., 12 (2001), 1566-1586.
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