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Reduction of a model for sodium exchanges in kidney nephron

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  • This work deals with a mathematical analysis of sodium's transport in a tubular architecture of a kidney nephron. The nephron is modelled by two counter-current tubules. Ionic exchange occurs at the interface between the tubules and the epithelium and between the epithelium and the surrounding environment (interstitium). From a mathematical point of view, this model consists of a 5$ \times $5 semi-linear hyperbolic system. In literature similar models neglect the epithelial layers. In this paper, we show rigorously that such models may be obtained by assuming that the permeabilities between lumen and epithelium are large. We show that when these permeabilities grow, solutions of the 5$ \times $5 system converge to those of a reduced 3$ \times $3 system without epithelial layers. The problem is defined on a bounded spacial domain with initial and boundary data. In order to show convergence, we use $ {{{\rm{BV}}}} $ compactness, which leads to introduce initial layers and to handle carefully the presence of lateral boundaries. We then discretize both 5$ \times $5 and 3$ \times $3 systems, and show numerically the same asymptotic result, for a fixed meshsize.

    Mathematics Subject Classification: Primary: 35L04, 35L60, 35L81; Secondary: 35Q92, 92C42.


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  • Figure 1.  Simplified model of the loop of Henle. $ q_1 $, $ q_2 $, $ u_1 $ and $ u_2 $ denote solute concentration in the epithelial layer and lumen of the descending/ascending limb, respectively

    Figure 2.  Numerical error estimates

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