 2004 Royal Statistical Society 0035–9254/04/53475 Appl. Statist. (2004) 53, Part 3, pp. 475–493 Bayesian analysis of dynamic magnetic resonance breast images Francesco de Pasquale, Consiglio Nazionale delle Ricerche, Rome, Italy, and University of Plymouth, UK Piero Barone and Giovanni Sebastiani Consiglio Nazionale delle Ricerche, Rome, Italy and Julian Stander University of Plymouth, UK [Received May 2002. Final revision September 2003] Summary. We describe an integrated methodology for analysing dynamic magnetic resonance images of the breast. The problems that motivate this methodology arise from a collaborative study with a tumour institute. The methods are developed within the Bayesian framework and comprise image restoration and classification steps. Two different approaches are proposed for the restoration. Bayesian inference is performed by means of Markov chain Monte Carlo algorithms. We make use of a Metropolis algorithm with a specially chosen proposal distribution that performs better than more commonly used proposals. The classification step is based on a few attribute images yielded by the restoration step that describe the essential features of the contrast agent variation over time. Procedures for hyperparameter estimation are provided, so making our method automatic. The results show the potential of the methodology to extract useful information from acquired dynamic magnetic resonance imaging data about tumour morphology and internal pathophysiological features. Keywords: Bayesian methods; Classification; Dynamic magnetic resonance imaging; Hyperparameter estimation; Image analysis; Mammography; Restoration; Spatiotemporal models 1. Introduction The aim of this collaboration between radiologists at the Istituto Regina Elena in Rome and statisticiansistoimprovethediagnosticcapabilitiesofthedynamicmagneticresonanceimaging (DMRI) technique for some breast pathologies. This task is important because breast cancer is a serious public health problem and is the most common cancer in women (Heywang- Kobrunner and Beck, 1995). DMRI can offer considerable advantages over techniques such as X-ray mammography, ultrasonography, ‘transcutaneous’ core or needle biopsy and thermog- raphy(HighnamandBrady,1999).Itinvolvestheacquisitionofatemporalsequenceofimages of the breast acquired after the injection of a gadolinium salt contrast agent (Villringer etal., 1988). The contrast agent diffuses within the intravascular or interstitial spaces of tissue and modifies the MR image intensities. Different breast tissues (normal, benign and malignant tumorous)showdifferentpatternsofuptake(Hayton etal., 1999). In particular, DMRI signals Addressforcorrespondence: Francesco de Pasquale, Department of Mathematics and Statistics, University of Plymouth, Drake Circus, Plymouth, Devon, PL4 8AA, UK. E-mail: f.depasquale@plymouth.ac.uk