Rong Guo1, Pei Han1, Yicheng Chen2, Jinsong Ouyang3, Georges El Fakhri3, and Kui Ying1
1Engineering Physics, Tsinghua University, Beijing, China, People's Republic of, 2UC Berkeley-UCSF Graduate Program in Bioengineering, University of California, Berkeley, Berkeley, CA, United States, 3Center for Advanced Radiological Sciences, Massachusetts General Hospital, Boston, MA, United States
Synopsis
The maximum likelihood activity and attenuation (MLAA) method usually utilizes
time-of-flight (TOF) information to solve the problem of attenuation
correction. 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. TARGET AUDIENCE
People who are interested in PET-MRI and
attenuation correction.
PURPOSE
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.
METHOD
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 Mjk is the Markov prior between pixel j and its adjacent pixel k, i.e., the squares of deviation. The ωjk is a factor of weight. To encourage the similarity
between MRI and PET activity image, ωjk is defined as: $$\omega_{jk}=\begin{cases}1 & k\in N_{j}\\0 & otherwise\end{cases}\cdot\cdot\cdot\cdot\cdot\cdot(6)$$ Here, Nj 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.
RESULTS
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.
CONCLUSION
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.
Acknowledgements
No acknowledgement found.References
[1] R. Boellaard, etc.
Accurate PET/MR Quantification Using Time
of Flight MLAA Image Reconstruction. Mol Imaging Biol (2014) 16:469-477.
[2]
Kathleen Vunckx, etc. Heuristic
Modification of an Anatomical Markov Prior Improves its Performance.
Nuclear Science Symposium Conference Record (NSS/MIC), 2010 IEEE: 3262 – 3266
[3] openGate: http://www.opengatecollaboration.org
[4] D.Kroon. http://www.mathworks.com/matlabcentral/fileexchange/21451-multimodality-non-rigid-demon-algorithm-image-registration