Mathematic Models And Algorithms In Positron Emission Tomography

PET (positron emission tomography) is the most advanced equipment in nuclear medical imaging, which can show glucose metabolism. PET can noninvasively detect tumors, cardiovascular and cerebrovascular diseases in an early phase, so it has been paid more and more attention. In PET imaging, tracer is injected into patient to generate a pair of y photons, then, the photon pairs can be reconstructed to obtain the distribution of tracer. For a patient, every PET image involves two scans-emission and transmission, where the latter is similar to CT. Although both PET and CT can reconstruct image by using filtered back-projection and iterative reconstruction algorithm, PET need more critical data corrections, which is main difference between them. Another difference is that filtered back-projection is mainly used by CT, but iterative reconstruction technique is more and more focused by PET. We investigate the mathernatic models and algorithms in PET reconstruction as follows:1. Single-rod source system spends a lot of time on a transmission scan, which makes patient uncomfortable and restrict availability. Although the more rod sources can save time, they bring a lot of random coincidences to degrade image quality. The noise reduction technique of single-rod system is not appropriate for multi-rod sources system. By studying the specificity of multi-rod sources system, we design a rod-restricted randoms correction algorithm. It can reduce noise artifacts caused by more rod sources. The algorithm was evaluated by a cylinder phantom and a NEMA phantom. Experimental results showed that the proposed algorithm could obtain the high image quality just like what can be obtained from single-rod source with half of the scan time saved.2. In general, attenuation correction need acquire100M data in a position, but segmented attenuation correction need only1/3number of the data. For the latter, image segmentation is a critical step. Gaussian mixture model (GMM) has been widely used in attenuation image segmentation. In this paper, we focus on the generalization of GMM. which called GEMM. We first derive an algorithm for one-dimensional GEMM. then, it will be generalized to the multi-dimensional case. We tested some special examples of GEMM on simulation data and PET attenuation image. The results showed that some selected GEMMs had better clustering performance than GMM. It is worth noting that we design an effective algorithm for Laplacian mixture model. It is rarely used in image processing, which, however, is very useful in sound signal processing.3. Fuzzy c-means (FCM) is another segmentation tool in PET. This paper investigates the convergence properties of it. The known work showed the sub-sequence convergence of FCM by using fixed point theory under an assumption. The proof has two unsolvable problems:one is that the assumption may not be tenable; the other is that the sub-sequence convergence shows that every convergent sub-sequence will converge to a stationary point, but nobody can judge the complete sequence convergence. We overthrow the wide-used fixed, point theory:instead, we provide a new idea to prove the complete sequence convergence, moreover, the assumption is not needed. The algorithmic derivation of FCM can also interpret GEMM, furthermore, this paper shows the similar convergence properties of FCM. and GEMM.4. PET data include much noise and iterative reconstruction algorithm can suppress them better than analytical approach. This paper presents a general iterative reconstruction framework, which includes ML-EM, Anderson’s WLS. Zhu’s WLS, Byrne’s SMART, Liu and Wang’s algorithm, De Pierro’s Bayesian algorithm and Fessler’s PWLS, and so on. Furthermore, we prove a global convergence proof. In fact, except for ML-EM and Fessler’s PWLS, few convergence proofs can be given. In short, we provide a general analytic method for several iterative reconstruction algorithms and prove the global convergence, under which we can derive many new updating algorithms and prove convergence for them.5. It is a challenge to recognize the relationship between tracer activity and time from dynamic PET data. Non-negative matrix factorization (NMF) technique can complete the task. NMF is a new approach in pattern recognition, which factorizes a non-negative matrix into the product of two other non-negative matrices with low rand and dimension. Although NMF and image reconstruction have different physics backgrounds and applications, they have the same update rule. We design a new algorithm for NMF to suppress noise and we provide a convergence proof. For the similar ESMART algorithm, the convergence depends on the penalty parameter. If selecting a bigger one, the ESMART will diverge. On a dynamic brain PET data set, the proposed algorithm and the ESMART are used to extract the activity-time curve of tracer from arteries. The results showed the new algorithm obtain low noise basis images and high accuracy.