[Truncated abstract. Formulae and special characters in this field can only be approximated. See PDF version for accurate reproduction.] Magnetic resonance imaging (MRI) has been developed over the past two and a half decades to enable non-invasive assessment of soft tissues in the human body. MRI provides images of the tissues in the body with intensities weighted by nuclear magnetic relaxation properties of the tissue. Recent advances have utilised MRI as a quantitative tool with the nuclear magnetic relaxation rates in tissues being accurately quantified. One clinical application of quantitative MRI has been in the quantification of body iron stores in the management of iron overload diseases. MR images also contain information about the spatial variations of relaxation rates, which could be clinically useful. In the quantification of liver iron concentrations, proton transverse relaxation rate (R2) maps have been used not only to quantify iron concentrations but also to visualise the spatial variations. The work in this thesis addresses the use of spatial information from proton transverse relaxation rate maps in clinical practice. The quantitative spatial information contained in these maps is analysed in two clinically important settings, namely the non-invasive assessment of liver fibrosis and the assessment of magnetic susceptibility artefacts in cardiac proton transverse relaxometry. Spatial distributions of liver R2 maps were quantified using texture measures based on grey-tone spatial dependence (GTSD) matrices. Some of these measures gave a statistically significant distinction between patients with minimal or no fibrosis and those with fibrosis or cirrhosis. Distinction of fibrosis using this technique was enhanced in subjects with iron overload diseases, suggesting that iron is required as a contrast agent for sufficient sensitivity of image texture to fibrosis. In subjects with low tissue iron concentrations, tissue hydration was observed to also have an influence on R2. In patients with end stage liver disease, a model combining tissue iron concentration and tissue hydration gave a better prediction of R2 than iron concentration alone. A model combining several of the texture measures was developed using logistic regression and was found to improve distinction of high-grade fibrosis from low-grade fibrosis. For the distinction of F0 and F1 fibrosis stages (as assessed by the METAVIR system) from F2 and above the area under the receiver-operating characteristic (ROC) curve was 0.75. As this model was developed using a cohort of subjects with varying pathologies, the performance of the model is expected to improve if only iron-loaded subjects are considered.