Hybrid PET/MR systems combine functional information obtained from positron emission tomography (PET) and anatomical information from magnetic resonance (MR) imaging. In spite of the advantages that such systems can offer, PET attenuation correction still represents one of the biggest challenges for imaging in the thorax. This is due to the fact that the MR signal is not directly correlated to gamma-photon attenuation. In current practice, pre-defined population-based attenuation values are used. However, this approach is prone to errors in tissues such as the lung where a variability of attenuation values can be found both within and between patients. A way to overcome this limitation is to exploit the fact that stand-alone PET emission data contain information on both the distribution of the radiotracer and photon attenuation. However, attempts to estimate the attenuation map from emission data only have shown limited success unless time-of-flight PET data is available. Several groups have investigated the possibility of using scattered data as an additional source of information to overcome re- construction ambiguities. This thesis presents work to extend the previous methods by using PET emission data acquired at multiple energy windows and incorporating prior information derived from MR. This thesis is organised as follows. We first cover both the literature and mathematical theory relevant to the framework. Then, we present and discuss results on the case of attenu- ation estimation from scattered data only, when the activity distribution is known. We then give an overview of several candidates for joint reconstruction, which reconstruct both the activity and attenuation from scattered and unscattered data. We present extensive results using simulated data and compare the proposed methods to state-of-the-art MLAA from a single energy window acquisition. We conclude with suggestions for future work to bring the proposed method into clinical practice
