Analysis of a simplified model of the urine concentration mechanism

Magali Tournus (1, 2) , Aurélie Edwards (2) , Nicolas Seguin (1) , Benoît Perthame (1, 3, 4)

1 LJLL - Laboratoire Jacques-Louis Lions
2 CRC (UMR_S 872) - Centre de Recherche des Cordeliers
3 BANG - Nonlinear Analysis for Biology and Geophysical flows
4 IUF - Institut universitaire de France

We study a non linear stationary system describing the transport of solutes dissolved in a fluid circulating in a counter-current 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 counter-current arrangement of tubes enhances the axial concentration gradient, thereby favoring the production of highly concentrated urine.

Mots clés: countercurrent; non linear transport equation; ion transport; relaxation toward steady state; contraction property

Domaines: Equations aux dérivées partielles [math.AP]

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), pp.989-1018. ⟨10.3934/nhm.2012.7.989⟩. ⟨hal-00605109v2⟩