Reduction of a model for sodium exchanges in kidney nephron

Marta Marulli; Vuk Miliši$\grave{\rm{c}}$; Nicolas Vauchelet; Marta Marulli; Vuk Miliši$\grave{\rm{c}}$; Nicolas Vauchelet

doi:10.3934/nhm.2021020

Networks and Heterogeneous Media

2021, Volume 16, Issue 4: 609-636. doi: 10.3934/nhm.2021020

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Reduction of a model for sodium exchanges in kidney nephron

1.
University of Bologna, Department of Mathematics, Piazza di Porta S. Donato 5, 40126 Bologna, Italy
2.
Université Sorbonne Paris Nord, Laboratoire Analyse Géométrie et Applications (LAGA), CNRS UMR 7539, F-93430, Villetaneuse, France

Received: 01 June 2020 Revised: 01 April 2021 Published: 09 July 2021
Primary: 35L04, 35L60, 35L81; Secondary: 35Q92, 92C42

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 $3 systems, and show numerically the same asymptotic result, for a fixed meshsize.
- Hyperbolic systems,
- relaxation limit,
- characteristics method,
- boundary layers,
- ionic exchanges
Citation: Marta Marulli, Vuk Miliši$\grave{\rm{c}}$, Nicolas Vauchelet. Reduction of a model for sodium exchanges in kidney nephron[J]. Networks and Heterogeneous Media, 2021, 16(4): 609-636. doi: 10.3934/nhm.2021020

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Abstract

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.

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