Analysis of a simplified model of the urine concentration mechanism

Previous Article
NHM Home
This Issue
Next Article
Liouville-type theorems for elliptic Schrödinger systems associated with copositive matrices

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

Abstract
Related Papers

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 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.

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.
[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.

[1]	Youcef Mammeri, Damien Sellier. A surface model of nonlinear, non-steady-state phloem transport. Mathematical Biosciences & Engineering, 2017, 14 (4) : 1055-1069. doi: 10.3934/mbe.2017055
[2]	Zhiming Chen, Weibing Deng, Huang Ye. A new upscaling method for the solute transport equations. Discrete and Continuous Dynamical Systems, 2005, 13 (4) : 941-960. doi: 10.3934/dcds.2005.13.941
[3]	Tehuan Chen, Chao Xu, Zhigang Ren. Computational optimal control of 1D colloid transport by solute gradients in dead-end micro-channels. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1251-1269. doi: 10.3934/jimo.2018052
[4]	Nassif Ghoussoub. A variational principle for nonlinear transport equations. Communications on Pure and Applied Analysis, 2005, 4 (4) : 735-742. doi: 10.3934/cpaa.2005.4.735
[5]	Thomas Lorenz. Partial differential inclusions of transport type with state constraints. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1309-1340. doi: 10.3934/dcdsb.2019018
[6]	Proscovia Namayanja. Chaotic dynamics in a transport equation on a network. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3415-3426. doi: 10.3934/dcdsb.2018283
[7]	Ioana Ciotir, Nicolas Forcadel, Wilfredo Salazar. Homogenization of a stochastic viscous transport equation. Evolution Equations and Control Theory, 2021, 10 (2) : 353-364. doi: 10.3934/eect.2020070
[8]	Wolfgang Wagner. Some properties of the kinetic equation for electron transport in semiconductors. Kinetic and Related Models, 2013, 6 (4) : 955-967. doi: 10.3934/krm.2013.6.955
[9]	Jorge Clarke, Christian Olivera, Ciprian Tudor. The transport equation and zero quadratic variation processes. Discrete and Continuous Dynamical Systems - B, 2016, 21 (9) : 2991-3002. doi: 10.3934/dcdsb.2016083
[10]	Alexander Bobylev, Raffaele Esposito. Transport coefficients in the $2$-dimensional Boltzmann equation. Kinetic and Related Models, 2013, 6 (4) : 789-800. doi: 10.3934/krm.2013.6.789
[11]	Atul Kumar, R. R. Yadav. Analytical approach of one-dimensional solute transport through inhomogeneous semi-infinite porous domain for unsteady flow: Dispersion being proportional to square of velocity. Conference Publications, 2013, 2013 (special) : 457-466. doi: 10.3934/proc.2013.2013.457
[12]	Thi-Thao-Phuong Hoang. Optimized Ventcel-Schwarz waveform relaxation and mixed hybrid finite element method for transport problems. Discrete and Continuous Dynamical Systems - S, 2022, 15 (10) : 2945-2964. doi: 10.3934/dcdss.2022060
[13]	Gerhard Rein, Christopher Straub. On the transport operators arising from linearizing the Vlasov-Poisson or Einstein-Vlasov system about isotropic steady states. Kinetic and Related Models, 2020, 13 (5) : 933-949. doi: 10.3934/krm.2020032
[14]	Karthik Elamvazhuthi, Piyush Grover. Optimal transport over nonlinear systems via infinitesimal generators on graphs. Journal of Computational Dynamics, 2018, 5 (1&2) : 1-32. doi: 10.3934/jcd.2018001
[15]	Stefan Possanner, Claudia Negulescu. Diffusion limit of a generalized matrix Boltzmann equation for spin-polarized transport. Kinetic and Related Models, 2011, 4 (4) : 1159-1191. doi: 10.3934/krm.2011.4.1159
[16]	Fang Liu. An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2395-2421. doi: 10.3934/cpaa.2018114
[17]	Yohan Penel. An explicit stable numerical scheme for the $1D$ transport equation. Discrete and Continuous Dynamical Systems - S, 2012, 5 (3) : 641-656. doi: 10.3934/dcdss.2012.5.641
[18]	Roman Romanov. Estimates of solutions of linear neutron transport equation at large time and spectral singularities. Kinetic and Related Models, 2012, 5 (1) : 113-128. doi: 10.3934/krm.2012.5.113
[19]	Martin Frank, Thierry Goudon. On a generalized Boltzmann equation for non-classical particle transport. Kinetic and Related Models, 2010, 3 (3) : 395-407. doi: 10.3934/krm.2010.3.395
[20]	John C. Schotland, Vadim A. Markel. Fourier-Laplace structure of the inverse scattering problem for the radiative transport equation. Inverse Problems and Imaging, 2007, 1 (1) : 181-188. doi: 10.3934/ipi.2007.1.181

2021 Impact Factor: 1.41

Tools

Metrics

Other articles
by authors

on AIMS
Magali Tournus Aurélie Edwards Nicolas Seguin Benoît Perthame
on Google Scholar
Magali Tournus Aurélie Edwards Nicolas Seguin Benoît Perthame

[Back to Top]