Modelling nephron dynamics and tubuloglomerular feedback

Abstract

The kidneys are amazingly versatile organs that perform a wide range of vital bodily functions. This thesis provides an analysis into a range of mathematical models of the tubuloglomerular feedback (TGF) mechanism. The TGF mechanism is an autoregulatory mechanism unique to the kidney that maintains approximately constant blood flow to the organ despite wide fluctuations in pressure. Oscillations in pressure, flow, and sodium chloride concentration have been attributed to the action of the TGF mechanism through a number of experimental studies. These oscillations appear spontaneously or in response to a natural or artificial pressure step or microperfusion. The reason for sustained oscillatory behaviour in nephrons is not immediately clear. Significant research has gone into experimentally determining the signal to the TGF mechanism, but the physiological significance is not mentioned in the literature. Considerable modelling of the oscillations attributed to the TGF mechanism has also been undertaken. However, this modelling uses models that are inherently oscillatory, such as a second-order differential equation or delay differential equations. While these models can be fitted to closely approximate the experimental results they do not address the physiological factors that contribute to sustained oscillations. This thesis aims to determine the contributing factors to the sustained oscillations. By understanding these factors a better hypothesis of the physiological role of the oscillations should be possible. Chapter 3 presents a mathematical model by Holstein-Rathlou and Marsh [28] that uses a partial differential equation (PDE) model for the tubule and a second-order differential equation for the TGF feedback. The remainder of this chapter shows that oscillations occur without an inherently oscillatory secondorder differential equation due to the delays in the system. Tubular compliance was also shown to be necessary for sustained oscillations. Sustained oscillations were not exhibited in the TGF model with a noncompliant tubule. Although damped oscillations were exhibited for a wide range of parameter space. Adding