The Circle of Willis is a ring-like structure of blood vessels found beneath the hypothalamus at the base of the brain. Its main function is to distribute oxygen-rich arterial blood to the cerebral mass. A one-dimensional model of the Circle of Willis has been created to simulate a series of possible clinical scenarios such as occlusions in afferent arteries, absent or string-like circulus vessels, or arterial infarctions. The model captures cerebral haemodynamic auto-regulation by using a Proportional-Integral-Derivative (PID) controller to modify efferent resistances and maintain optimal efferent flowrates for a given circle geometry and afferent blood pressure. Results match limited clinical data and results obtained in prior studies to within 6%. In addition, a set of boundary conditions and geometry is presented for which the auto-regulated system cannot provide the necessary efferent flowrates and perfusion, representing a condition with increased risk of stroke and highlighting the importance of modelling the haemodynamics of the Circle of Willis. The system model created is computationally simple so it can be used to identify at-risk cerebral arterial geometries and conditions prior to surgery or other clinical procedures.