Perfusion is a bodily function that describes the transport of blood and nutrients to an organ. If characteristics of the blood flow to an organ can be measured, clinically useful quantitative data about the organ can be obtained. This is particularly useful in the diagnosis of tumours as their rapid and poorly-formed capillary network is unique compared to healthy tissue. Tumour perfusion information can be obtained by performing Dynamic Contrast-Enhanced (DCE)-MRI. Using DCE-MRI, blood flow into a tumour from a nearby artery can be computed by measuring the concentration of an injected contrast agent as a function of time. This measurement, known as the arterial input function (AIF), can be computed based on the change of either the intensity or the phase of the MR signal from the blood, due to the presence of a contrast agent. While both methods can be used to acquire the AIF, the change in phase is preferred due to its superior accuracy. Even so, the method conventionally used to obtain the phase-derived AIF has deficiencies that lead to an AIF intensity overestimation. The goal of this thesis was to determine a better method to obtain AIF data for patients with brain tumours by using a combination of MR signal phase and accurate T1 relaxation measurements. This was done by first characterizing and deriving equations for phase measurement errors and simulating these effects on the AIF using realistic, clinical parameters. T1 measurement methodology was developed and validated in static-water phantoms and applied to a flowing-water phantom system on which phase AIF data were also acquired. The novel AIF method was tested on this system before it was applied clinically to patients diagnosed with high-grade gliomas. The differences between the quantitative perfusion parameters from the new method and previous AIF methods were compared. The novel AIF measurement method presented in this thesis was designed to be less prone to experimental error than current clinical methods. Theoretical predictions, computed simulations, experimental work with MRI test objects, and clinical results all show that the new method for measuring the AIF is significantly superior to procedures currently used clinically.