4D image reconstruction for emission tomography

An overview of the theory of 4D image reconstruction for emission tomography is given along with a review of the current state of the art, covering both positron emission tomography and single photon emission computed tomography (SPECT). By viewing 4D image reconstruction as a matter of either linear or non-linear parameter estimation for a set of spatiotemporal functions chosen to approximately represent the radiotracer distribution, the areas of so-called 'fully 4D' image reconstruction and 'direct kinetic parameter estimation' are unified within a common framework. Many choices of linear and non-linear parameterization of these functions are considered (including the important case where the parameters have direct biological meaning), along with a review of the algorithms which are able to estimate these often non-linear parameters from emission tomography data. The other crucial components to image reconstruction (the objective function, the system model and the raw data format) are also covered, but in less detail due to the relatively straightforward extension from their corresponding components in conventional 3D image reconstruction. The key unifying concept is that maximum likelihood or maximum a posteriori (MAP) estimation of either linear or non-linear model parameters can be achieved in image space after carrying out a conventional expectation maximization (EM) update of the dynamic image series, using a Kullback-Leibler distance metric (comparing the modeled image values with the EM image values), to optimize the desired parameters. For MAP, an image-space penalty for regularization purposes is required. The benefits of 4D and direct reconstruction reported in the literature are reviewed, and furthermore demonstrated with simple simulation examples. It is clear that the future of reconstructing dynamic or functional emission tomography images, which often exhibit high levels of spatially correlated noise, should ideally exploit these 4D approaches.