Mathematical model for ionic exchanges in renal tubules: the role of epithelium.

This thesis deals with a mathematical model for a particular component of the kidney, the loop of Henle. We focus our attention on the ionic exchanges that take place in the tubules of the nephron, the functional unit of this organ. The model explicitly takes into account the epithelial layer at the interface between the tubular lumen and the surrounding environment (interstitium) where the tubules are immersed. The main purpose of this work is to understand the impact of the epithelium (cell membrane) on the mathematical model, how its role influences it and whether it provides more information on the concentration gradient, an essential determinant of the urinary concentrating capacity. In the first part of this transcript, we describe a simplified model for sodium exchanges in the loop of Henle, and we show the well-posedness of problem proving the existence, uniqueness and positivity of the solution. This model is an hyperbolic system 5x5 with constant speeds, a 'source' term and specific boundary conditions. We present a rigorous passage to the limit for this system 5x5 to a system of equations 3x3, representing the model without epithelial layers, in order to clarify the link between them. In the second part, thanks to the analysis of asymptotic behaviour, we show that our dynamic model converges towards the stationary system with an exponential rate for large time. In order to prove rigorously this global asymptotic stability result, we study eigen-elements of an auxiliary linear operator and its dual. We also perform numerical simulations on the stationary system solution to understand the physiological behaviour of ions concentrations.