Medical Imaging

Over the past fifty years the processes and techniques of medical imaging have undergone a veritable explosion, calling into service increasingly sophisticated mathematical tools. Mathematics provides a language to describe the measurement processes that lead, eventually, to algorithms for turning the raw data into high-quality images. There are four principal modalities in wide application today: x-ray computed tomography (x-ray CT), ultrasound, magnetic resonance imaging (MRI), and emission tomography (positron emission tomography (PET) and single-photon emission computed tomography (SPECT)). Each modality uses a different physical process to produce image contrast: xray CT produces a map of the x-ray attenuation coefficient, which is strongly correlated with density; ultrasound images are produced by mapping absorption and reflection of acoustic waves; in their simplest form, magnetic resonance images show the density of water protons, but the subtlety of the underlying physics provides many avenues for producing clinically meaningful contrasts in this modality; PET and SPECT give spatial maps of the chemical activity of metabolites, which are bound to radioactive elements. It has recently been found useful to merge different modalities. For example, a fused MRI/PET image shows metabolic activity produced by PET, at a fairly low spatial resolution, against the background of a detailed anatomic image produced by MRI. Figure 1 shows a PET image, a PET image fused with a CT image, and the CT image as well. In this article we consider mathematical aspects of PET, whose underlying physics we briefly explain. Positron emission is a mode of radioactive decay stemming from the reaction proton → neutron + positron + neutrino + energy. (1)