Image reconstruction method

In order to improve the image anisotropicity which results from the limited-angle problem in typical Compton camera measurements, we propose a multi-angle data acquisition method19. In the multi-angle measurement in this study, the Compton camera was rotated around the image region and data was collected from multiple angles as shown in Fig. 6(a). We reconstructed 3-D images using the following algorithm, which is based on list-mode MLEM20:

$${{\lambda }}_{j}^{n}={{\lambda }}_{j}^{n-1}\,\sum _{i=1}\frac{1}{{s}_{j}^{l}}\frac{{t}_{ij}{v}_{i}}{{\sum }_{k}{t}_{ik}{{\lambda }}_{k}^{n-1}}$$ (1)

where l is the suffix corresponding to a certain data acquisition angle (l = 0, 1, 2, …, 11), \({\lambda }_{j}^{n}\) is the reconstructed image value after n th iteration, \({s}_{j}^{l}\) is the probability that a photon emitted from image voxel j is detected at a certain acquisition angle l, v i is the probability that an event i comes from the image space21, and t ij is the system matrix where a photon emitted from imaging voxel j will be measured as event i. Figure 6(b) shows the schematic parameters used in the 3-D reconstruction. We incorporated the solid angle of the imaging voxel and the reaction probability of each gamma ray in system matrix t ij :

$${t}_{ij}=2\pi (1-\frac{d}{\sqrt{{d}^{2}+{a}^{2}}})\times \exp \,[-\frac{1}{2}{(\frac{|{{\rm{\Theta }}}_{j}|-|{\theta }_{k}|}{\sigma })}^{2}]\times \frac{1}{\sin \,{\theta }_{k}}\times f({E}_{1},{E}_{2},{\theta }_{k},{x}_{s},{x}_{a})$$ (2)

$$f({E}_{1},{E}_{2},{\theta }_{k},{x}_{s},{x}_{a})=\exp \,\{-{\sigma }_{t}\,({E}_{1}+{E}_{2})\cdot {x}_{s}\}\cdot \frac{d{\sigma }_{C}}{d{\rm{\Omega }}}\cdot \exp \,\{-{\sigma }_{t}({E}_{2})\cdot {x}_{a}\}\cdot {\sigma }_{p,C}$$ (3)

where a and d are the half size of the imaging voxel and the distance between the detector and the voxel, respectively; Θ j is the angle between the cone axis and the direction of the interested image voxel j; θ k is the scattering angle calculated by the energy information; and σ is the angular resolution of the camera. The first term of equation (2) represents the effect of the solid angle of the voxel seen from the detector; the second term represents a Gaussian distribution from the uncertainty of the cone; and the third term is a weighting factor of each event. The last term f is the reaction probability of a photon: \(\exp \{-{\sigma }_{t}\,({E}_{1}+{E}_{2})\cdot {x}_{s}\}\,{\rm{and}}\,\exp \{-{\sigma }_{t}\,({E}_{2})\cdot {x}_{a}\}\) are the transportation probabilities of a photon in the scintillator (x s and x a are the path length in the scintillator); \(\frac{d{\sigma }_{C}}{d{\rm{\Omega }}}\) is the Compton scattering probability of a photon which has the energy E 1 + E 2 in the scattering angle θ; and σ p,C is the interaction probability of a photon of energy E 2 by either photoelectric absorption or Compton scattering.

Figure 6 (a) Configuration of the multi-angle data acquisition measurement, (b) diagram of the 3-D MLEM reconstruction, and (c) configuration of the measurement of the plane source. Full size image

To obtain the 3-D images in both imaging experiments, data was collected for 12 angles, which was equal to a 30° pitch. The image region was defined as 8 × 8 × 8 cm3 through the experiments. Our calculation includes both dead-time and decay correctons which correspond to the measurement.

3-D imaging of uniform plane source

We utilized a plane source phantom of 30 mm × 30 mm × 3 mm as the uniform plane source, which was filled with 137Cs solution. The total intensity of the source was 2 MBq. For the target source, we rotated the medical Compton camera as shown in Fig. 6(c). The data acquisition configuration was 12 angles. Each took 20 min, making the total integration time 4 h.

In this plane source measurement, we evaluated the edge delineation performance and image intensity uniformity as the parameters that defined imaging capability. To evaluate the edge delineation in the Y and Z directions (i.e., the parallel direction to the source plane), the error function was fitted to the 1-D slice of the reconstructed image, and the spatial resolution was calculated from the fitting results. On the other hand, in the X direction, which is equal to the thickness direction, the response R(x) can be expressed by the following convolution:

$$R(x)=\int LSF\,(x-x^{\prime} )\cdot e(x^{\prime} )dx^{\prime} $$ (4)

$$e(x)=\{\begin{array}{ll}1 & (0\le x\le \mathrm{3)}\\ 0 & (x < 0,x > 3)\end{array}$$ (5)

where the line spread function is denoted by LSF. Hence, we calculated various responses R(x) as a function of σ of the LSF, and the resolution was estimated based on the FWHM value of the R(x).

Furthermore, in order to evaluate image uniformity, we obtained the 2-D ROI which was determined by eliminating the 2σ region of the spatial resolution from the reconstructed edge position. For the ROI, we evaluated the fluctuation of the all voxels based on the averaged value.

To determine of the number of iteration, we optimized in terms of both the uniformity and spatial resolution. Figure 7 (left) and (right) show the uniformity and spatial resolution as a function of the iteration number, respectively. These results suggest that uniformity had the best value when the iteration number was approximately 20–30. On the other hand, the spatial resolution improved with increasing iteration number in the range of less than 40. This indicates that increasing the number of iterations can improve the spatial resolution, which is close to the values obtained by point source measurements. However, because the amount of data was insufficient, the fluctuations of each voxel stand out and the performance of the uniformity declined. Based on abovementioned reasons, we adopted the iteration number of 30 for the imaging of the plane source.

Figure 7 (Left) Uniformity (1σ) and (right) spatial resolution as a function of the number of iteration. The blue, green, and red plots in the right figure indicate the spatial resolution in the X, Y, and Z directions, respectively. Full size image

3-D in vivo imaging with a living mouse

For the in vivo imaging, we studied an eight-week-old male mouse (39.9 g), which fed a low iodine diet for two weeks and was injected via tail vein under inhalation anesthesia (isoflurance/air mixture). Three radioactive tracers were utilized: (1) 131I-NaOH (4.0 MBq) was injected two days before the imaging experiment; (2) 85SrCl 2 -HCL (1.12 MBq) was injected one day before the imaging experiment; and (3) 65ZnCl 2 -saline (0.93 MBq) was injected 1 h before the imaging experiment. The features of these tracers are listed in Table 1. These tracers showed the in vivo behavior of accumulation in specific areas: dissociated 131I accumulated in the thyroid, 85Sr accumulated in the bone, and 65Zn accumulated in the liver22. The mouse was treated with inhalation anesthesia and placed on the rotation stage in an upright position. The medical Compton camera was rotated around the mouse in the plane which was perpendicular to the body axis of the mouse. The measurement time for each position was 10 min, thus the total integration time was 2 h. All of the animal experiments were approved by the animal ethics committees of Osaka University and were performed according to the institutional guidelines.