Search Advanced

Networks and Heterogeneous Media

2012, Volume 7, Issue 4: 989-1018. doi: 10.3934/nhm.2012.7.989
Previous Article Next Article

Analysis of a simplified model of the urine concentration mechanism

  • 1.
    UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris
  • 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
  • 3.
    UMR 7598 Laboratoire J.-L. Lions, UPMC Univ Paris 06, Paris, F-75005
  • Received: 01 September 2011 Revised: 01 October 2012

  • 35L60, 92C35, 34B45.

  • 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[J]. Networks and Heterogeneous Media, 2012, 7(4): 989-1018. doi: 10.3934/nhm.2012.7.989

    Related Papers:

    [1] 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
    [2] Qinglan Xia, Shaofeng Xu . On the ramified optimal allocation problem. Networks and Heterogeneous Media, 2013, 8(2): 591-624. doi: 10.3934/nhm.2013.8.591
    [3] Jinhu Zhao . Natural convection flow and heat transfer of generalized Maxwell fluid with distributed order time fractional derivatives embedded in the porous medium. Networks and Heterogeneous Media, 2024, 19(2): 753-770. doi: 10.3934/nhm.2024034
    [4] José Antonio Carrillo, Yingping Peng, Aneta Wróblewska-Kamińska . Relative entropy method for the relaxation limit of hydrodynamic models. Networks and Heterogeneous Media, 2020, 15(3): 369-387. doi: 10.3934/nhm.2020023
    [5] Xu Yang, François Golse, Zhongyi Huang, Shi Jin . Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks and Heterogeneous Media, 2006, 1(1): 143-166. doi: 10.3934/nhm.2006.1.143
    [6] Toyohiko Aiki, Kota Kumazaki . Uniqueness of solutions to a mathematical model describing moisture transport in concrete materials. Networks and Heterogeneous Media, 2014, 9(4): 683-707. doi: 10.3934/nhm.2014.9.683
    [7] Nastassia Pouradier Duteil . Mean-field limit of collective dynamics with time-varying weights. Networks and Heterogeneous Media, 2022, 17(2): 129-161. doi: 10.3934/nhm.2022001
    [8] Edoardo Mainini . On the signed porous medium flow. Networks and Heterogeneous Media, 2012, 7(3): 525-541. doi: 10.3934/nhm.2012.7.525
    [9] Graziano Guerra, Michael Herty, Francesca Marcellini . Modeling and analysis of pooled stepped chutes. Networks and Heterogeneous Media, 2011, 6(4): 665-679. doi: 10.3934/nhm.2011.6.665
    [10] Hantaek Bae, Rafael Granero-Belinchón, Omar Lazar . On the local and global existence of solutions to 1d transport equations with nonlocal velocity. Networks and Heterogeneous Media, 2019, 14(3): 471-487. doi: 10.3934/nhm.2019019
  • 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.


    [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.

  • This article has been cited by:

    1. Nicolas Seguin, Magali Tournus, Asymptotic preserving discretization of a Jin–Xin model with implicit equilibrium manifold on a bounded domain, 2020, 40, 0272-4979, 530, 10.1093/imanum/dry089
    2. Marta Marulli, Vuk Miliši$\grave{\rm{c}}$, Nicolas Vauchelet, Reduction of a model for sodium exchanges in kidney nephron, 2021, 16, 1556-1801, 609, 10.3934/nhm.2021020
    3. Debora Amadori, Laurent Gosse, 2015, Chapter 4, 978-3-319-24784-7, 45, 10.1007/978-3-319-24785-4_4
    4. Debora Amadori, Laurent Gosse, Stringent error estimates for one-dimensional, space-dependent 2 × 2 relaxation systems, 2016, 33, 0294-1449, 621, 10.1016/j.anihpc.2015.01.001
  • Reader Comments
  • © 2012 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索
Networks and Heterogeneous Media
1.3 1.9

Metrics

Article views(3893) PDF downloads(74) Cited by(4)

Preview PDF
Download XML
Export Citation
Article outline
Abstract
References

Other Articles By Authors

Related pages

Tools

Copyright © AIMS Press

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog