ABSTRACT
Institute for Computational and Applied Mathematics, University of Mu¨nster, Mu¨nster, Germany
Thomas Ko¨sters
European Institute for Molecular Imaging, University of Mu¨nster, Mu¨nster, Germany
Frederic Lamare
Medical Research Council Clinical Sciences Centre, Imperial College London, Hammersmith Campus, London, United Kingdom
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 9.2 Parameter identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
9.2.1 Compartment modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 9.2.2 4D methods incorporating linear parameter
identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 9.2.3 4D methods incorporating nonlinear parameter
identification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 9.3 Combined reconstruction and motion correction . . . . . . . . . . . . . . . . 192
9.3.1 The advantages of the list mode format . . . . . . . . . . . . . . . . . 193 9.3.2 Motion correction during an iterative
reconstruction algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 9.3.2.1 Approaches based on a rigid or affine motion
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 9.3.2.2 Approaches based on a non-rigid motion
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 9.4 Combination of parameter identification and
motion estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
In the beginning of PET reconstruction in the middle of the 1970s the well-known reconstruction algorithms for CT were applied since both imaging techniques rely on the efficient recovery of a function from its line integrals [40]. It turned out that the resulting images were good enough that these algorithms were use over years until new algorithms especially designed for PET were introduced [54]. As presented in Chapter 3 the filtered-backprojection type algorithms (FBP) were more and more replaced by new model-based, iterative algorithms that take into account the Poisson statistical properties [57, 22]. The new algorithms-mainly originating from the maximumlikelihood expectation-maximization (ML-EM) algorithm-show good performance even in the case of low statistic measurements where the old reconstruction algorithms may fail. Next to the statistical properties, these algorithms can easily be extended with any other corrections, such as resolution modeling, the use of different forward and backward projectors, or even higher dimensional reconstruction approaches with the use of suitable basis functions. These extensions are possible since the model-based reconstruction algorithms are not limited to the X-ray projection geometry of the FBP type algorithms. We mention that there are, as well, approaches to extend the analytical algorithms, such as the motion compensated local tomography of Katsevich [26], but usually the ML-EM-type algorithms are used for extensive modeling. In general the statistical model is used and extended for the necessary correction, whereas the resulting linear (in some cases even non-linear) equation system is solved afterwards. Due to the enormous speed increase of CPUs and porting of reconstruction algorithms to clusters or even GPUs [52] we are now able to work with extensive PET models. In this chapter we will focus on two interesting, recently growing approaches:
• Parameter estimation during 4D reconstructions using compartment models to investigate physiological parameters.
