Analysis of a simplified model of the urine concentration mechanism

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.