Scatter Fraction. The scatter fraction is an important parameter included in both National Electrical Manufacturers Association (NEMA) and International Electrotechnical Commission (IEC) standards, and measured in every performance evaluation report of PET scanners. The scatter fractions of PET scanners from 3 major manufactures are listed in Table 1  [28, 40, 87].

Multiple Scatter Coincidence Events. The gamma photons can be scattered multiple times before recorded. Monte Carlo simulation is the best tool to study the Compton scattering during PET acquisition. We have done similar studies with our new PET scanner, HAMAMATSU SHR74000, we retrieve all the data, and analyze the composition of simulated measurement data according to the number of scattering of each photon in the coincidence photon pairs. The first data set is analyzed with the standard NEMA phantom, a 70 cm long, 20 cm diameter cylinder, the composition of scatter coincidences is summarized in Table 2. Furthermore, we have a lot of over-sized and over-weighted patients, especially in the western countries, and these patients usually take higher risk of many diseases than other patients. For these patients, their size makes more photons scattered inside their body, we correspondingly analyze the composition of scatter coincidence photons with a 70 cm long, 35 cm diameter cylinder, the results are summarized in Table 3. The number of total scatter coincidence events increase by about 20 %. If the tumor is near the body surface, the scatter fractions shown in Table 4 turn to be similar as in Table 2.

Scatter Coincidence from Activity Concentrations outside of FOV. A whole body PET scan generally needs 4–5 bed positions to complete, then gamma photons from outside of FOV will also have the possibility to be misrecorded by the detector system. Sossi et al. show the scatter coincidence events need to be accounted for when the amount of radioactivity outside of FOV is comparable to the radioactivity inside the FOV in [81]. In the case of line sources, the scatter fraction increases by about 15 % when the source is extended 4 cm outside of the FOV and 20 % when it is extended 8 cm outside of the FOV. Spinks et al. also show for a full 3D PET, the scatter fraction increased from 40 to 45 % with the scatter photons from outside of FOV, after introducing a fitted brain shielding, the scatter fraction decreases to be 41 % [82]. A recent paper by Ibaraki et al. show the use of the neck-shield to suppress scatters from outside FOV can improve the SNR by 8 and 19 % for and , respectively [38].

Scatter Correction Methods. Bailey et al. [2] and Bentourkia et al. [7] present a Convolution-Subtraction (CS) scatter correction technique for 3D PET data. The scatter distribution is estimated by iteratively convolving the photo-peak projections with a mono-exponential kernel. The method is based on measuring the scatter fraction and the scatter function at different positions in a cylinder. The method performs well on 2D measurement data and also accounts for the 3D acquisition geometry and nature of scatter by performing the scatter estimation on 2D projections.

The method is easy to set up and still applies to a lot of animal studies, where the scatter correction are usually considered not necessary. Lubberink et al. propose a non-stationary CS scatter correction with a dual-exponential scatter kernel for scatter correction in both emission and transmission data of studies of conscious monkeys using Hamamatsu SHR7700 PET scanner [54] . Kitamura et al. implement a hybrid scatter correction method, which estimates scatter components with a dual energy acquisition using a CS to estimate the true coincidence events in the upper energy window for their four layer Depth of Interaction (DOI)—PET scanner [45]. Naidoo-Variawa et al. suggest that scatter correction methods based on spatially invariant scatter functions, such as CS, may be suitable for non-human primate brain imaging in [65].

Single Scatter Simulation (SSS) is one of the most important methods for PET scatter correction, and commercial PET scanners from Siemens are using SSS derived scatter correction methods. Ollinger [67] and Watson [98] first introduced the concept of model based scatter coincidence estimation. The single scatter approximation is defined with the well accepted formulation based on Klein-Nishina equation as
where

is the sampled scatter volume, is the scatter point, and are sampled detectors, so is the coincidence scatter rate in detectors and , and are the detector cross sections presented to the rays and , and are the distances from to and . and are the total and differential Compton scattering cross sections, is the solid angle, is the energy of the non-scattered photon, is the energy of the scattered photon, and are the approximated detector efficiencies for gamma rays which are incident along and , is the linear attenuation coefficient at the energy and position , and is the estimated activity distribution from reconstructed images.

The calculation is repeated through all the sampled scatter points from activity distribution and sampled detector blocks. The final result is the distribution of single scatter coincidence events, and will be scaled and then subtracted from the measurement data to perform scatter correction. Figure 6 demonstrates the procedure of SSS based scatter correction for clinical PET scanner.

Dynamic PET imaging is a combination of short interval PET scans and reflects the dynamic metabolism of injected radiotracers. For example, a dynamic acquisition can consist of 85 frames in all: 15  0.2 min, 20  0.5 min, 40 1 min, and 10  3 min. The series of acquisitions can be used to estimate the kinetic parameters which represent the metabolism of radiotracers in vivo. The difficulties in estimating kinetic parameters arise from the low count rates in the first several time frames and the low SNR. To obtain the kinetic parameters, a typical approach is first to reconstruct the activity distributions from the dynamic PET data, and then to fit the calculated time activity curve (TAC) to a predefined kinetic model. Samples of TAC curves in different organs are shown in Fig. 7. The accuracy of this kind of approaches relies on the reconstructed activity distributions. The complicated statistical noise properties, especially in the low-count dynamic PET imaging, and the uncertainties introduced by various PET data corrections will affect the activity reconstruction and lead to a suboptimal estimation of kinetic parameters [29]. There are also many efforts that try to estimate the kinetic parameters from PET projection data directly and achieve better bias and variance including both linear and nonlinear models [83, 91, 103]. One problem is that the optimization algorithms are very complicated. Kamasak et al. apply the coordinate descent algorithm for optimization but it is still limited to specific kinetic models [61]. Wang et al. apply a generalized algorithm for reconstruction of parametric images [96], however, it still lacks of estimation and analysis of individual kinetic parameter. Exactly, every kinetic parameter has its own physical meaning like radiotracer transport rate, phosphorylation rate and dephosphorylation rate (it is very sensitive to the system [35]) in FDG study, which will be critical to clinical research, drug discovery and drug development [15, 92].

The accuracy of quantitative dynamic PET studies depends on various factors including kinetic models, quantitative methods and the approximation of arterial input function from blood sampling. The most general kinetic models used are compartment model with assumptions that physiological process and molecular interactions are not influenced by injected radioligand. Current clinical adopted quantitative methods are actually semi-quantitative methods, which include methods using reference regions or calculating Standard Uptake Value (SUV). Methods using reference regions are easy to implement but have several drawbacks, e.g. the reference tissue is hard to define and has low SNR due to the low resolution of PET and SPECT scans, and the uptake of the reference tissue may change after the radiotherapy. SUV now is included in every clinical study, which is calculated as a ratio of tissue radioactivity concentration and injected dose divided by body weight, the advantage of SUV in clinical study is that the blood sampling is not required. However, the full quantitative analysis requires both dynamic PET scans and tracer concentration in the arterial blood plasma. The gold standard of blood sampling is serial arterial sampling of a superficial artery, and clinical alternative methods include venous blood sampling, image derived input function and population based input function. The drawback of the full quantitative method is only one FOV/bed position can be taken into consideration. For metastasized disease, not all lesions can be quantified simultaneously.

Compartment models are used in many fields including pharmacokinetics, biology, engineering etc. Compartment models are the type of mathematical models to describe the way materials (radiotracers and their metabolite in PET and SPECT scan) are transmitted among the compartments (different organs and tissues). Inside each compartment, the concentration of radiotracers is assumed to be uniformly equal. Due to their simplicity and plausibility, compartment models are widely used in the dynamic PET scans to describe the tracer/drug kinetics.

Drug kinetic models include simple drug transport model, which generally contains equal or less than three compartments and can be solved directly, and complicated biological models, which can contain up to twenty compartments and generally require prior knowledge to solve [30, 31]. Most of the complicated models with many compartments can usually be decomposed into a combination of simple models with less than four compartments. The most basic compartment models used in kinetic analysis shown in Fig. 8 include two compartment blood flow model (Model 1), standard two tissue three compartment Phelps 4K model with reversible target tissue (Model 2) and Sokoloff 3K model with irreversible target tissue (Model 3), three tissue five parameter bertoldo model (Model 4), standard three tissue four compartment model (Model 5 and Model 6). More complicated models with more compartments and parallel model with multiple injection can be extended from aforementioned standard models [24, 44].

All six models can be represented by a set of differential equations with corresponding kinetic parameters K = , where is the number of kinetic parameters. Here we utilize Model 2 as an example for demonstration. Model 2 can be represented by first-order differential equations

The measurement of dPET is the combination of radiotracer in plasma , non-specific binded radiotracer and specific binded radiotracer through

where is the blood volume fraction, Y is measured projection data , D is the system probability matrix, and is the noises during acquisition. Equation (5) can be represented by a more general time-dependent form for all models as
Simple models with less than three compartments generally can be solved directly, while more complicated models need simplifications and various numerical approximations.

The goal of mathematical modeling of data acquisition is to describe the transforms from spatial distribution of imaging objects to projection distributions on detector pairs in PET system. Denoting the spatial distribution of imaging object by a set of spatial variables , where is the total number of voxels, and the expected values (means) of projection bins by , where is the total number of bins, a mathematical expression of the transform can be obtained
where is the system response model giving the probability matrix of mapping the transform from to , and is the means of background noises. A block diagram of the above procedure is shown in Fig. 9a.

With the mathematical model specified above, the problem of PET image reconstruction is to find an estimation of the imaging object from the measurement data (projection bins) . The image reconstruction is an ill-posed inverse problem, so an intuitive solution is to apply statistical assumptions of measurement data as regularizations. The statistical formulation tries to find an optimized relationship between measurement data and imaging object (or expected values of projection bins ) by defining different objective functions. If denoting is the measurement data after all data corrections and assuming all the data correction are perfectly applied, is equal to the after all data corrections, however, in real clinical data acquisition, there will be some difference between and , which is accumulated from residuals of each data correction.

A block diagram of statistical image reconstruction framework is shown in Fig. 9b: the statistical properties of system noises or measurement data are first modeled based on certain statistical distributions (Gaussian, Poisson, their combination or other derivations) in block , then inputted into system block ; the system output is generated and compared with corrected measurement data based on the predefined criteria, when the system goes convergent, estimations of the imaging object will be obtained. Three major criteria can be used are Least Square based (LS), Maximum Likelihood based (ML), Maximum A Posterior based (MAP).

LS based methods try to obtain the estimation by minimizing the difference of fit between the predicted data by means of the modeling of the acquisition process and the measured data , the general objective function is
Extended algorithms are all based on the above equation and try to improve the performance by introducing different weights or penalization items.

ML based methods try to obtain the estimation by maximizing the likelihood functions which represent the goodness of fitting statistical assumptions to measurement data. The general objective function is based on the conditional probability density of the image object with known measurement data as
where represents the probability density, and the above equation is a function of . When a statistical model of measurement data or noises is defined, either Gaussian/Poisson or their combination, the objective likelihood function can be solved correspondingly. An improved method based on poisson assumption widely used is EM-SP (Shifted Possion) algorithm, which introduces two times the mean of randoms coincidences to the random precorrected data and models this sum as a Poisson random variable, and the objective function is still based on the above one [63]. Furthermore, a priori information can also be introduced in the form of statistical properties of the imaging object as constraints into the ML objective function.

Bayesian formulation is such a probabilistic approach, statistical information of the unknown imaging object is introduced by adopting the probability density of , which is the prior . Equation (9) can then be solved as follow: