People who are interested in PET-MRI and attenuation correction. The maximum likelihood activity and attenuation (MLAA) method usually utilizes time-of-flight (TOF) information to solve the problem of attenuation correction, which is necessary for quantitative reconstruction of PET image [1]. However, the application of TOF brings noise. In this work, we proposed a method, Maximum a Posteriori for simultaneous activity and attenuation reconstruction (MAPAA), which introduces MRI information as prior knowledge into MLAA to reduce noise.

Algorithm: To exploit the structural similarity among PET activity distribution, attenuation map and MRI image, MRI information was introduced into TOF-MLAA algorithm. As Fig.1 shows, each iteration of this method contains four steps as a typical TOF-MLAA algorithm. Each step can be expressed as one equation: $$a_i^m=e^{-\sum_jl_{ij}\mu_j^m}\cdot\cdot\cdot\cdot\cdot\cdot(1)$$ $$\lambda_j^{m+1}=\frac{\lambda_j^{m}}{\sum_{it}a_i^mc_{ijt}+\beta\frac{\partial P(\lambda)}{\partial \lambda_{j}}}\sum_{it}a_i^mc_{ijt}\frac{y_{it}}{\sum_ja_i^mc_{ijt}\lambda_j^m}\cdot\cdot\cdot\cdot\cdot\cdot(2)$$ $$\psi_i^m=a_i^m\sum_{jt}c_{ijt}\lambda_j^{m+1}\cdot\cdot\cdot\cdot\cdot\cdot(3)$$ $$\mu_j^{m+1}=\mu_j^{m}+\frac{\sum_il_{ij}(\psi_i^m-y_{i})}{\sum_il_{ij}\psi_i^m\sum_jl_{ij}}\sum_{it}a_i^mc_{ijt}\frac{y_{it}}{\sum_ja_i^mc_{ijt}\lambda_j^m}\cdot\cdot\cdot\cdot\cdot\cdot(4)$$

Equations 1, 3 and 4 are the same as TOF-MLAA algorithm [1]. In Eq.2, we added a partial derivative entry ($$$\beta\frac{\partial P(\lambda)}{\partial \lambda_{j}}$$$) on the denominator referring to quintessential Maximum a Posteriori (MAP) estimation. In this deviation entry, P(λ) is a prior function extracted from a MRI image and parameter β is used to adjust its weight. Here, an asymmetric Bowsher prior function [2] is used due to the simplicity of its implementation. $$P(\lambda)=\sum_j\sum_k\omega_{jk}M_{jk}\cdot\cdot\cdot\cdot\cdot\cdot(5)$$ Where M_jk is the Markov prior between pixel j and its adjacent pixel k, i.e., the squares of deviation. The ω_jk is a factor of weight. To encourage the similarity between MRI and PET activity image, ω_jk is defined as: $$\omega_{jk}=\begin{cases}1 & k\in N_{j}\\0 & otherwise\end{cases}\cdot\cdot\cdot\cdot\cdot\cdot(6)$$ Here, N_j represents the set containing pixels next to pixel j, which belong to the same tissue. Whether a pixel meets this condition can be determined from a MRI image by checking its pixel index and value. The general effect of this prior is to make PET activity of pixels in one tissue tends to be the same, i.e., to smoothen the image without eliminating high frequency signal (e.g. the edge of tissue). This is why the involvement of MRI can help to decrease noise. After one iteration cycle is completed, the updated estimation values of attenuation and activity become the initial values for the next iteration cycle. Through these iterations, the attenuation is corrected from the activity distribution map.

Simulation experiment: One slice of a NCAT torso phantom was used for simulation phantom study. PET data was acquired from an open-source simulation software, GATE. [3] In GATE, a Siemens Biograph mMR machine was constructed to simulate the physical process of PET scan. Then, the TOF information was calculated on MATLAB and MRI images were obtained from MRILAB, a MRI simulation software. [4] Finally, the results of MAPAA were compared with MLEM and MLAA.

The reconstructed images of MAPAA, MLEM and MLAA are shown in Fig.2 (MLEM as a reference). Fig.3 and Fig.4 display the line plots of these methods on myocardium and tumor, respectively. Besides, the variances of the ROIs with relative uniformity were calculated to evaluate the noise level of an image. Table 1 shows the variances of the selected ROIs for different methods. The results show that MAPAA helps to reduce the noise brought by utilization of TOF. However, the variance from MAPAA is higher than the variance from the reference MLEM, in which attenuation correction is not involved. We proposed a method that introduces MRI information in order to decrease noise in the TOF-based attenuation correction. The simulation results validate its feasibility.