Analysis of a simplified model of the urine concentration mechanism

Magali Tournus; Aurélie Edwards; Nicolas Seguin; Benoît Perthame; Magali Tournus; Aurélie Edwards; Nicolas Seguin; Benoît Perthame

doi:10.3934/nhm.2012.7.989

Networks and Heterogeneous Media

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

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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.
- Countercurrent,
- nonlinear transport equation,
- solute transport,
- relaxation toward steady state,
- contraction property
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

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Abstract

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.

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