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

Analysis of a simplified model of the urine concentration mechanism

Magali Tournus 1, , Aurélie Edwards 2, , Nicolas Seguin 3, and  Benoît Perthame 1,

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.
Keywords: solute transport, nonlinear transport equation, contraction property., Countercurrent, relaxation toward steady state.
Mathematics Subject Classification: 35L60, 92C35, 34B4.
Citation: Magali Tournus, Aurélie Edwards, Nicolas Seguin, Benoît Perthame. Analysis of a simplified model of the urine concentration mechanism. Networks & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. C. Evans, "Partial Differential Equations," 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

H. Layton and E. Pitman, A dynamic numerical method for models of renal tubules, Bulletin of Mathematical Biology, 56 (1994), 547-565. Google Scholar

[11]

H. E. Layton, Distribution of henle's loops may enhance urine concentrating capability, Biophysical Journal, 49 (1986), 1033-1040. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[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.  Google Scholar

[19]

J. L. Stephenson, "Urinary Concentration and Dilution: Models," Oxford University Press, New-York, 1992. Google Scholar

[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. Google Scholar

show all references

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

L. C. Evans, "Partial Differential Equations," 19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, second edition, 2010.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

H. Layton and E. Pitman, A dynamic numerical method for models of renal tubules, Bulletin of Mathematical Biology, 56 (1994), 547-565. Google Scholar

[11]

H. E. Layton, Distribution of henle's loops may enhance urine concentrating capability, Biophysical Journal, 49 (1986), 1033-1040. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[14]

R. J. LeVeque, "Finite Volume Methods for Hyperbolic Problems," Cambridge University Press, 2002. doi: 10.1017/CBO9780511791253.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[17]

B. Perthame, "Transport Equations in Biology," Frontiers in Mathematics. Birkhäuser Verlag, Basel, 2007.  Google Scholar

[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.  Google Scholar

[19]

J. L. Stephenson, "Urinary Concentration and Dilution: Models," Oxford University Press, New-York, 1992. Google Scholar

[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. Google Scholar

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