Modèles d'échanges ioniques dans le rein: théorie, analyse asymptotique et applications numériques

Abstract : This thesis of applied mathematics deals with theoretical, numerical and asymptotic questions in transport, motivated by the renal physiology. More specifically, the purpose is to understand and quantify solute exchanges in physiological and pathological cases and to explain why nephrocalcinosis, i.e. the deposition of calcium salts in kidney tissue, arise. The manuscript is divided in two parts. The first part describes the development and the mathematical analysis of a simplified kidney model. It is a system of $3$ hyperbolic PDE's with constant velocities, coupled by a non-linear source term and with specific boundary conditions. This model can be considered in the framework of kinetic models with a finite number of velocities and reflexion boundary conditions. We prove that the system is well posed and that it relaxes toward the unique stationary state for large time with an exponential rate of convergence. Thanks to a spectral analysis, we prove that the rate of convergence is exponential. We study the role of two parameters through an asymptotic analysis. One of these analyses is formulated in the framework of hyperbolic relaxation toward a scalar conservation law with an heterogeneous flux on a bounded domain. The second part describes the development and the numerical analysis of a realistic kidney model. It is an hyperbolic system of 27 hyperbolic partial differential equations whose velocities are solutions to 8 non linear differential equations, all coupled by their source term. The boundary conditions are also very specific. We then interpret the results from a physiological point of view, by predicting calcium concentration profiles in the kidney, under normal conditions and in some specific pathological cases.
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https://tel.archives-ouvertes.fr/tel-00845333
Contributor : Magali Tournus <>
Submitted on : Tuesday, July 16, 2013 - 6:42:21 PM
Last modification on : Wednesday, December 9, 2020 - 3:10:33 PM
Long-term archiving on: : Thursday, October 17, 2013 - 4:19:47 AM

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• HAL Id : tel-00845333, version 1

Citation

Magali Tournus. Modèles d'échanges ioniques dans le rein: théorie, analyse asymptotique et applications numériques. Equations aux dérivées partielles [math.AP]. Université Pierre et Marie Curie - Paris VI, 2013. Français. ⟨tel-00845333⟩

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