Nonlinear Models of Nephron Autoregulation
O. Sosnovtseva, D. Postnov, N.-H. Holstein-Rathlou, D. Marsh, and E. Mosekilde
Arterial hypertension is a common condition in our modern societies and is known to represent one of the main risk factors in the development of cadiovascular diseases. By controlling the extracellular fluid volume, the kidneys play a central role in regulating the blood pressure, and there is strong evidence to suggest that a deficiency in the ability of the kidneys to excrete water and salts can be the cause of hypertension.
Two mechanisms are responsible for the pressure and flow regulation in the individual nephron, the functional unit of the kidney. The first is the myogenic response associated with the ability of the vascular smooth muscle cells to respond to changes in arteriolar pressure, producing vasoconstriction when the pressure is increased. The second is the tubuloglomerular feedback which is a negative feedback mechanism that translates flow dependent changes in the composition of the tubular fluid into changes in the afferent arteriolar resistance. Recent experiments have shown that both mechanisms can become unstable, producing self-sustained oscillations with characteristic frequencies of 100-200 mHz and 30-50 mHz, respectively. Under certain conditions period-doubling, chaos, and other highly nonlinear dynamic phenomena have been observed.
We have previously developed a detailed model of the nonlinear dynamic phenomena associated with the tubuloglomerular feedback. A couple of years ago the model was amended by the introduction of a detailed description of the nonlinear reaction of the arteriolar wall to change in the transmural pressure. This reaction consists of an active (myogenic) component in parallel with a passive (elastic) component. A detailed bifurcation analysis of the extended model has shown that it reproduces the experimentally observed dynamics with physiologically realistic parameters. In particular, the model shows resonance phenomena between the fast and the slow regulatory components in the physiologically interesting regime where the feedback delay in the turbuglomerular coupling is of the order of 16 sec.
The single nephron model has subsequently been extended by introducing a coupling to neighboring nephrons. The nephrons are typically arranged in couples or triplets with a common afferent arteriole, and they are known to interact via electrochemical signals that propagate along the arteriolar wall. We have shown that a model of two coupled nephrons can reproduce the experimentally observed in-phase and antiphase synchronization for neighboring nephrons in normotensive rats as well as the observed chaotic phase synchronization in hypertensive rats. The phenomenon of chaotic phase synchronization is revealed by using a Hilbert transformation of the observed pressure signals to define the instantaneous phase of each nephron.
We are presently trying to build a model of a larger nephronic network, accounting for the various coupling mechanisms, for the variation in the operational conditions across the nephrons, as well as for the anatomical structure of the arteriolar network.
For further details see our recent publications.