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 second-order 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 compliance to the tubule increased the delay around the loop of Henle. This additional delay elicited sustained oscillations.

The computationally expensive PDE model of 3 was simplified to an ordinary differential equation (ODE) model in Chapter 4 by assuming a spatial profile. This model exhibits much of the same qualitative behaviour as the PDE model including sustained oscillations for similar ranges of parameter space. Compliance was also found to be important in the generation of sustained oscillations in agreement with the PDE tubule model. This model is less computationally expensive than the PDE model and allows analysis that was unfeasible with the PDE model.

Significant natural and artificial blood pressure fluctuation occur in experimental rat models. Chapter 5 examines the effect of inlet pressure forcing on a nonoscillatory and an oscillatory model. The inherently nonoscillatory noncompliant model becomes oscillatory with a physiologically realistic pressure forcing. The oscillatory compliant model remains oscillatory with the addition of a inlet pressure forcing. Pressure fluctuations were hypothesised to contribute to sustained oscillations and could be validated experimentally.

Two extensions to the single nephron TGF models are presented in Chapter 6. A realistic juxtaglomerular delay is added to the single nephron models with both the ODE and PDE tubular models. Physiologically realistic juxtaglomerular delays induce sustained oscillations in the otherwise nonoscillatory noncompliant models. The remainder of this chapter presents a different model for a variable interstitial sodium chloride concentration profile. This model demonstrates experimentally observed function of the countercurrent mechanism by which a concentration gradient is set up and maintained in the interstitium.

Two single nephron models with ODE tubular models are coupled in Chapter 7. The coupling is modelled through the effect on the resistance of their neighbouring nephron's afferent arteriole resistance. The coupled nephron model exhibits entrainment as observed experimentally. Inhibiting the oscillation in one nephron reduces the amplitude of the oscillation in its neighbour. This result compares well with experiments where the TGF mechanism in one nephron is blocked by the administration of furosemide.