Bifurcation analysis of kidney pressure and flow regulation
Erik Mosekilde
Center for Chaos and Turbulence Studies,
Department of Physics, The Technical University of Denmark
2800 Lyngby, Denmark
By controlling the excretion of water and salts, the kidneys play an important role in regulating the blood pressure. To protect their own function the kidneys also dispose of mechamisms that can compensate for variations in the blood pressure. It has long been known that this ability partly rests with controls in the individual nephron, the functional unit of the kidney. Particularly important is the so-called tubuloglomerular feedback. This is a negative feedback that regulates the incoming blood flow independence of the chloride or sodium concentration in preurine leaving the nephron.
Experiments by Leyssac and Holstein-Rathlou at the Dept. of Medicine, University of Copenhagen, have demonstrated that the feedback regulation can become unstable and produce self-sustained oscillations in the tubular pressure with a period of 30-40 sec. While for normal rats the oscillations have the typical appearance of a limit cycle, highly irregular oscillations are observed for spontaneously hypertensive rats. It has also been found that irregular oscillations can develop in the nephron pressure for normal rats if the arterial pressure is increased by ligating the blood supply to the other kidney.
By integrating existing but hitherto separate physiological descriptions of glomerular filtration, tubular dynamics, and tubuloglomerular feedback with a detailed description of the reaction of the afferent arteriole we have formulated a model that accounts for the experimental observations. It is shown how a Hopf bifurcation leads the system to perform self-sustained oscillations if the arterial pressure or the feedback gain become high enough. Further increase of these parameters produces a complicated structure of overlaying period-doubling cascades. We have investigated this structure by means of one- and two-dimensional continuation techniques. The lecture will discuss the experimental background, the formulation of a physiologically based nonlinear dynamic model and the continuation results. The possible effects of including a more detailed description of the so-called myogenic response of the afferent arteriole will be analysed, and the role that this type of response could have for coupling neighboring nephrons will be discussed.