Model single nephron

The above discussion completes our description of the single-nephron model. In total we have six coupled ordinary differential equations, each representing an essential physiological relation. Because of the need to numerically evaluate Ce in each integration step, the model cannot be brought onto an explicit form. The applied parameters in the single-nephron model are specified in Table 12.1. They have all been adopted from the experimental literature, and their specific origin is discussed in Jensen et al. [13]. 

Figure 12.5 shows an example of a one-dimensional bifurcation diagram for the single-nephron model obtained by varying the slope a of the open-loop response... 

For T = 16 s, the single nephron model undergoes a supercritical Hopf bifurcation at a = 11 (outside the figure), fn this bifurcation, the equilibrium point loses its stability, and stable periodic oscillations emerge as the steady-state solution. For a = 19.5, at the point denoted PDla 2 in Fig. 12.5, this solution undergoes a period-... 

Figure 12.6a shows the temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model for a = 12 and T = 16 s. All other parameters attain their standard values as listed in Table 12.1. Under these conditions the system operates slightly beyond the Hopf bifurcation point, and the depicted pressure variations represent the steady-state limit cycle oscillations reached after the initial transient has died out For physiologically realistic parameter values the model reproduces the observed self-sustained oscillations with characteristic periods of 30-40 s. The amplitudes in the pressure variation also correspond to experimentally observed values. Figure 12.6b shows the phase plot Here, we have displayed the normalized arteriolar radius r against the proximal intratubular pressure. Again, the amplitude in the variations of r appears reasonable. The motion... 

Fig. 12.6 (a) Temporal variation of the proximal tubular pressure Pt as obtained from the single-nephron model fora = 12 and T = 16 s. (b) Corresponding phase plot. With the assumed parameters the model displays self-sustained oscillations in good agreement with the behavior observed for normotensive rats. The unstable equilibrium point falls in the middle of the limit cycle, and the motion along the cycle proceeds in the clockwise direction. 

Fig. 12.9 Two-dimensional bifurcation diagram for the single-nephron model. The diagram illustrates the complicated bifurcation structure in the region of 1 1, 1 2, and 1 3 resonances between the arteriolar dynamics and the TGF-mediated oscillations. In the physiologically interesting regime around T = 16 s, another set of complicated period-doubling and saddle-node bifurcations occur. Here, we are operating close to the 1 4...

Fig. 12.11 Two-mode oscillatory behavior in the single nephron model. Black colored regions correspond to a chaotic solution. The figure shows different regions in which 1 4, 1 5 and 1 6 synchronization occurs in the interaction between the fast myogenic oscillations...

Fig. 12.12 Internal rotation number as a function of the parameter a calculated from the single-nephron model. Inserts present phase projections for typical regimes. Note how the intra-nephron synchronization is maintained through a complete period-doubling cascade to chaos.

Under normal conditions, GFR is submaximal because adaptive increases in single nephron GFR follow loss of damaged nephrons. Sensitized animal models that mimic risk factors commonly found in patients with drug-induced acute renal failure are advisable. This need should stimulate research in the field of safety pharmacology. The choice of the species, strain, and sex of test animals should take into account physiological and/or pharmacotoxicological specificities. 

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