In this study we present a mathematical model describing the transport of sodium in a fluid circulating in a counter-current tubular architecture, which constitutes a simplified model of Henle's loop in a kidney nephron. The model explicitly takes into account the epithelial layer at the interface between the tubular lumen and the surrounding interstitium. In a specific range of parameters, we show that explicitly accounting for transport across the apical and basolateral membranes of epithelial cells, instead of assuming a single barrier, affects the axial concentration gradient, an essential determinant of the urinary concentrating capacity. We present the solution related to the stationary system, and we perform numerical simulations to understand the physiological behaviour of the system. We prove that when time grows large, our dynamic model converges towards the stationary system at an exponential rate. In order to prove rigorously this global asymptotic stability result, we study eigen-problems of an auxiliary linear operator and its dual.

中文翻译：

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上皮在肾小管钠交换模型中的作用。

在这项研究中，我们提出了一个数学模型，描述了在逆流管状结构中循环的液体中钠的运输，该模型构成了肾肾中Henle环的简化模型。该模型明确考虑了管腔和周围间质之间界面上的上皮层。在特定的参数范围内，我们表明明确考虑跨上皮细胞的顶膜和基底外侧膜的转运，而不是假设单个屏障，会影响轴向浓度梯度，这是尿液浓缩能力的重要决定因素。我们提出与固定系统有关的解决方案，并执行数值模拟以了解系统的生理行为。我们证明，随着时间的流逝，我们的动态模型以指数速率朝固定系统收敛。为了严格证明这一全局渐近稳定性结果，我们研究了辅助线性算子及其对偶的本征问题。