Parametric PET Image Reconstruction via Regional Spatial Bases and Pharmacokinetic Time Activity Model
Abstract
:1. Introduction
 Noise model:A Poisson distribution model is employed. The distance between the given measured sinogram and the projection of the reconstructed PET image is measured by using the KL divergence.
 Spatial model:The nonnegativity of voxel values is assumed. In addition, it is assumed that the foreground/background regions and the reference region in an image are known in advance. The voxel values in the background are set to zero.
 Temporal model:A compartment model is employed. It is assumed that, given the ligand, one can identify the number of compartments appropriate for the representation of the kinetics. In this paper, we assume a threecompartment model, which is appropriate for a variety of ligands, such as a [${}^{11}\mathrm{C}$]carfentanil and a [${}^{18}\mathrm{F}$]fludeoxyglucose [23,24].
 A compartment modelbased constraint is explicitly introduced in order to constrain the tTACs to the solution space in which the relationships between the tTACs are consistent with the compartment model, while retaining the KLdivergence of data fidelity as a convex function.
 A constraint of a target region in an image is introduced to restrict the pixel values of the background to be zero.
 The dependency of the solutions on the initial values is discussed and is experimentally shown.
2. Basic Materials
2.1. PET Image Reconstruction
2.2. Compartment Model
3. Proposed Method
3.1. Notation
3.2. Description of Proposed Method
3.2.1. The Gradient Step, $\mathcal{G}$
3.2.2. The Projection Step, $\mathcal{P}$
Algorithm 1 Description of proposed method. 

4. Experimental Results
4.1. Evaluation with Simulated Data
4.2. Practical PET Images Reconstruction from Clinical Sinograms
5. Discussion
5.1. Optimization Strategy
5.2. Sensitivity to Initialization
5.3. Related Works
6. Conclusions
Supplementary Materials
Acknowledgments
Author Contributions
Conflicts of Interest
Appendix A
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Notations of Methods  Problems to Be Optimized 

NC  ${\mathrm{min}}_{\mathit{X}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{D}_{\mathrm{KL}}(\tilde{\mathit{Y}}\left\right\mathit{PX}).$ 
NC + spatial bases  ${\mathrm{min}}_{\mathit{Z}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{D}_{\mathrm{KL}}(\tilde{\mathit{Y}}\left\right\mathit{P}\mathbf{\Psi}\mathit{Z}).$ 
TR  ${\mathrm{min}}_{\mathit{X}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{D}_{\mathrm{KL}}(\tilde{\mathit{Y}}\left\right\mathit{PX})+\frac{\lambda}{2}{\sum}_{f=1}^{F1}\left\right{\mathit{x}}_{f+1}{\mathit{x}}_{f}{\left\right}_{2}^{2}.$ 
TR + spatial bases  ${\mathrm{min}}_{\mathit{Z}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{D}_{\mathrm{KL}}(\tilde{\mathit{Y}}\left\right\mathit{P}\mathbf{\Psi}\mathit{Z})+\frac{\lambda}{2}{\sum}_{f=1}^{F1}\left\right\mathbf{\Psi}({\mathit{z}}_{f+1}{\mathit{z}}_{f}){\left\right}_{2}^{2}.$ 
Proposed  ${\mathrm{min}}_{\mathit{Z}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{D}_{\mathrm{KL}}(\tilde{\mathit{Y}}\left\right\mathit{P}\mathbf{\Psi}\mathit{Z})\phantom{\rule{4pt}{0ex}},\mathrm{s}.\mathrm{t}.\text{}\mathit{Z}\in S,\text{}\mathit{Z}\ge \mathit{0}.$ 
Data  Methods  ROI #1  ROI #2  ROI #3  ROI #4  ROI #5  ROI #6  ROI #7 

PET Data $\#1$  TR ($\lambda =0.005$)  0.507  0.663  0.515  0.645  0.660  0.486  0.647 
Proposed  0.174  0.302  0.219  0.236  0.214  0.240  0.304  
PET Data $\#2$  TR ($\lambda =0.0025$)  0.859  1.01  0.864  0.795  1.166  0.745  0.752 
Proposed  0.136  0.245  0.164  0.124  0.134  0.165  0.153  
PET Data $\#3$  TR ($\lambda =0.005$)  0.710  0.812  0.640  0.616  0.618  0.616  0.772 
Proposed  0.240  0.295  0.251  0.199  0.240  0.221  0.351 
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Kawamura, N.; Yokota, T.; Hontani, H.; Sakata, M.; Kimura, Y. Parametric PET Image Reconstruction via Regional Spatial Bases and Pharmacokinetic Time Activity Model. Entropy 2017, 19, 629. https://doi.org/10.3390/e19110629
Kawamura N, Yokota T, Hontani H, Sakata M, Kimura Y. Parametric PET Image Reconstruction via Regional Spatial Bases and Pharmacokinetic Time Activity Model. Entropy. 2017; 19(11):629. https://doi.org/10.3390/e19110629
Chicago/Turabian StyleKawamura, Naoki, Tatsuya Yokota, Hidekata Hontani, Muneyuki Sakata, and Yuichi Kimura. 2017. "Parametric PET Image Reconstruction via Regional Spatial Bases and Pharmacokinetic Time Activity Model" Entropy 19, no. 11: 629. https://doi.org/10.3390/e19110629