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A three tube kidney model with water permeability: steady state formulation and analysis
Dissertation   Open access

A three tube kidney model with water permeability: steady state formulation and analysis

Anya Pant
Doctor of Philosophy (Ph.D.), Drexel University
Sep 2026
DOI:
https://doi.org/10.17918/00011587
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Abstract

We study a steady state model of the renal countercurrent system consisting of two descending tubes and one ascending tube, all of which exchange solute and water with a common interstitium. Because the tube walls are permeable to water, the axial flows are not constant but instead respond to differences in solute concentration. This introduces a nonlinear coupling between solute transport and water flow that is absent from the constant flow models on which the present formulation is based. A main goal of this work is to understand how this additional coupling affects the steady state problem. In each tube, the axial derivative is oriented in the direction of flow, so the countercurrent geometry is incorporated directly into the equations rather than being represented only through the boundary conditions. The interstitial flux balance can be solved explicitly, giving the interstitial concentration as a bounded combination of the three tubular concentrations and the pump flux, independently of the flows. The flows can then be recovered through explicit integral formulas. This allows the full system to be reduced to a fixed point problem involving only the tubular concentrations. The model also satisfies a conservation law for the net axial solute flux, providing a useful consistency check for both the analytical results and the numerical computations. We prove the existence of an admissible steady state under a consistency condition on the flow bounds, with no contraction condition, and establish its uniqueness within an explicit invariant set. The uniqueness result follows from a contraction argument under smallness conditions relating the tube length and wall permeabilities to the prescribed inflow. Numerical computations are used to approximate the steady state, verify the conservation law and the derived a priori bounds, and identify the parameter regimes in which the iteration breaks down. We also examine the corresponding time-dependent problem and show numerically that its solutions relax toward the computed steady state.

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