A Reduced Model for Nephron Flow Dynamics Mediated by Tubuloglomerular Feedback

Previously we developed a “minimal” mathematical model of the tubuloglomerular feedback (TGF) system in a short-looped nephron of the mammalian kidney. In that model, a hyperbolic partial differential equation (PDE) represented the advection and transepithelial transport of chloride in the thick ascending limb (TAL). The feedback response was represented by an empirical relationship that determined the glomerular filtration rate as a function of time-delayed TAL luminal chloride concentration alongside the macula densa. This PDE model system with feedback and a time delay presents analytical and computational challenges. In this report, we derive a reduced model that is based on the minimal model. The reduced model, which is formulated as an integral equation in time, is easier to study than the PDE model. As in the case of the minimal model, analysis of the reduced model suggests that sustained oscillations in nephron fluid flow arise from a Hopf bifurcation, with delay time and system gain as bifurcation parameters. Both analysis and numerical calculations indicate that the principal bifurcation locus predicted by the reduced model coincides with the analogous locus obtained from the minimal model. Near the principal bifurcation locus, numerical solutions of the two models nearly coincide. For bifurcation parameters that differ sufficiently from the principal bifurcation locus, the numerical solutions to the two models differ somewhat. The reduced TGF model has the potential to facilitate simulation and analysis of interactions among TGF systems in multiple nephrons.