Estimation of and correction for finite motion sampling errors in small animal PET rigid motion correction

In this study, we used the Siemens Inveon microPET (Siemens Medical Solutions, Inc., Knokville, USA) scanner. The scanner consists of 80 rings (16 cm diameter) of 320 lutetium oxyorthosilicate (LSO) scintillation crystals with a size of 1.5 × 1.5 × 1.0-mm³ with a reconstructed image resolution of 1.5 mm [3, 25]. The scanner’s bore diameter is 12 cm.

For motion tracking during the PET imaging, a commercially available stereo optical tracking device (MicronTracker Sx60, Claron Technology Inc., Toronto, Canada) was used. The MicronTracker measures the pose of specially designed checkerboard markers in a reference frame relative to the camera position. Four markers of 2.4 × 2.4 mm² and four of 1.2 × 1.2 mm² were pasted on the scanner bore to be used as reference ([12]; Fig. 1). The motion of the subject marker was measured relative to these reference markers. The maximum tracking frame rate is 48 Hz, and the reported tracking accuracy by the manufacturer is 0.25 mm root mean square.

The spatial calibration between the MicronTracker and the PET scanner coordinate system is established by measuring the spatial location of a radioactive point source, attached to the center of a checkerboard marker in PET space, and the corresponding pose of the checkerboard marker in camera space for 20 different positions [13]. The transformation matrix between both coordinate systems is then determined using least squares regression. The same transformation matrix was used for all the experiments.

The time synchronization between the MicronTracker and the PET scanner is established using an Arduino Uno (Arduino SA) microcontroller to generate a regular square wave signal that triggers the MicronTracker, which in turn sends a signal to the microPET gating input. A few seconds before the end of the scan, the frequency of the gating signal changes with a predetermined pattern. This pattern is then used to relate each gate tag with its corresponding pose. The frequency of the square wave signal can be adjusted by the user to change the tracking frame rate. The tracking frame rate was set to 31.2 Hz in all experiments. Although a maximum tracking frame rate of 48 Hz is possible, 31.2 Hz was chosen due to the fact that at increased frame rate the camera exposure time reduces, increasing the noise in the images and therefore reducing the tracking accuracy [8].

For the motion corrected reconstructions, we have implemented the subsetized list-mode expectation maximization (OSEM) algorithm [18] with calculation of the sensitivity image by interpolation in the image space. The bore size of the Siemens Inveon microPET is substantially smaller than in previous implementations of awake small animal brain PET (12 cm compared to 22 cm for the microPET Focus 220 [12]). This poses challenges on the motion tracking; however, it does not impact the motion correction method as such. Briefly, in this method, each measured pose X_k (k = 1, …, K poses) is assigned to the group of LORs that are within the sampling interval centered around the time t_k, the time that the pose was measured (Fig. 2a). These LORs, i.e., the lines connecting two coincidence events, are then moved to a reference pose X_ref by applying the transformation \( M={X}_{\mathrm{ref}}{X}_k^{-1} \). The LORs in the reference pose are finally reconstructed using the OSEM algorithm without attenuation, scatters, or random correction. Resolution modeling was implemented in the image space [20] with a 1.2-mm Gaussian point spread function. These motion corrected images will then be deconvolved using our proposed residual motion kernel (see Section 2.2).

We compared our proposed deconvolution procedure to motion correction using pose interpolation of the motion tracking data [27]. The interpolated poses are calculated from the measured poses to assign a single position to each LOR (within the scanner time resolution of 0.2 ms), thus reducing the finite sampling interval. The interpolation of the pose position was obtained from cubic interpolation, while the pose orientation was calculated from cubic interpolation of the Euler angles [23].

To calculate the residual motion kernel, the range of the residual motion that exists within each of the sampling intervals is estimated in image space. Each reference voxel location is transformed to the current measured position. The position uncertainty is then estimated by considering a position uncertainty distribution spanning from the voxel position that lie halfway between the measured voxel position in the current frame and previous frame and between the current and next frames respectively. To discretize the uncertainty distribution, a number of points are distributed over the uncertainty positions span. Finally, these points are transformed back to the reference orientation. Once this has been done for all frames, a kernel is fitted to the resulting point cloud. The detailed procedure is presented in the next paragraph.

For each measured pose X_k (k = 2, …, K − 1), the intermediate pose \( {X}_k^{-} \) halfway between X_k and X_k − 1, and the intermediate pose \( {X}_k^{+} \) between X_k and X_k + 1, are calculated (Fig. 2b). The location (x, y, z) of the intermediate pose is calculated using linear interpolation while for the orientation the mean rotation matrix is obtained through the singular value decomposition of the sum of the rotation matrices corresponding to the neighboring poses [7].

Then, to estimate the corresponding blurring in image space that the range of residual motion within the sampling interval causes, the intermediate poses \( {X}_k^{-} \) and \( {X}_k^{+} \) from the previous step are used to move the centers of the image voxels as follows:

The use of points to represent the motion path of the voxel due to the residual motion was preferred over a ray intersection with the image voxels to reduce calculation time.

The deconvolution kernel K_{N, j} of size N is truncated and normalized if the residual motion is wider. Therefore, to evaluate the impact of the kernel size, three types of kernels, with kernel size N = 3, 5, and 7, were calculated. These kernels will be referred to as K₃, K₅, and K₇, respectively.

The kernel can be calculated in a single voxel of interest for a fast assessment of the resolution loss in a particular position. If the kernels are however calculated for each voxel in the region of interest, they can be used in a 3D Richardson-Lucy (RL) deconvolution algorithm [14] to deconvolve the motion corrected reconstructions:where W_{j, r} is the image value of the jth voxel in the deconvolved image at the rth iteration, K_{j, l} is the contribution of the kernel in the jth position to the lth voxel, and U_l is the lth voxel value of the motion corrected reconstruction before deconvolution. In the RL deconvolution, a Poisson noise model is assumed. We will refer to the deconvolution with the residual motion kernel as residual motion deconvolution (RMD).

In our experiments, the kernel calculation and deconvolution were performed only on the region of interest that contained the object. The kernel calculation and the deconvolution were implemented in MATLAB Release 2012b (The Mathworks, Inc., Natick, USA).

In order to asses to which degree the residual motion blurring affects the regional brain quantification in two animal conditions with different levels of locomotion, a numerical mouse brain phantom experiment was performed. Motion data from a head motion tracking experiment in naïve and memantine treated mice placed inside the PET scanner was used to calculate the residual motion kernels. Memantine administration has been shown to significantly increase mouse locomotion in comparison with a naïve condition [5]. Then, to estimate the effect of the residual motion in the two conditions on brain quantification, the kernels were used to blur a mouse brain numerical phantom. The numerical mouse brain phantom was based on a template in Waxholm space [11]. Increased striatal uptake was simulated with a 1.65 to 1 ratio (striatum versus rest of the brain). To quantify the effect, average regional striatal uptake after blurring with the residual motion kernel was calculated in the reference image as well as in both animal conditions blurred images.

A simulated dataset was generated by combining list mode data of a static (i.e., motion-free) PET scan of a microDerenzo phantom and previously measured motion data of a manually moved microDerenzo phantom. The LORs in the list mode data were then moved according to this motion to generate simulated PET data that was affected by subject motion. The microDerenzo phantom (diameter 30 mm, height 13 mm) had six rod sections with an inner diameter of 1.25, 1.5, 2, 2.5, and 3 mm respectively. The scan time was 10 min, and the activity was 12.9 MBq of [¹⁸F]-FDG. The motion data was previously measured using the MicronTracker in a phantom experiment where the phantom was moved manually. The motion was sampled every 32 ms, and the average measured motion speed was 74 mm/s. The list mode data has a temporal resolution of 0.2 ms, and to simulate continuous motion, the original motion was interpolated (using cubic interpolation) to obtain 1 ms sampling intervals. The list mode data of the motion-free scan was moved according to the interpolated motion, i.e., with 1 ms sampling intervals. Then, these displaced LORs were corrected for motion according to the original measured motion, i.e., with bin intervals of 32 ms. Thus, all the LORs in the 32 ms interval that have been moved with different poses (every 1 ms) are being corrected using a single pose corresponding to the central pose of the interval. A second simulated dataset was generated by considering sampling intervals of 64 ms for the motion correction. This would equally correspond to the situation where the motion speed doubles.

The errors present in this motion corrected simulation is caused uniquely by the difference between the actual pose of the LOR and the central pose of the corresponding interval that is used for the motion correction. All the other factors that can affect the spatial resolution in the motion correction technique (e.g., calibration error, tracking accuracy, synchronization delay) are not simulated.

To assess the different kernel sizes, the kernels K_N (N = 3, 5, 7) were calculated for the 32 ms sampling interval data. The different kernels were then used in the residual motion deconvolution of the motion corrected reconstruction. The average FWHM, taken in tangential direction, and the average peak-to-valley ratio (PVR) of all the rods of 2, 2.5, and 3 mm were measured in the deconvolved images for eight iterations of the residual motion deconvolution. The FWHM was calculated from a Gaussian fit to the profiles through the individual rods of the same size and was then averaged to obtain the average FWHM. The FWHM and PVR values were then compared to the average FWHM and PVR of the rods in the reference reconstruction of the motion-free scan. The data for the 64 ms sampling interval simulation was analyzed similarly as aforementioned.

To further evaluate the performance of the deconvolution on real measured data, two microDerenzo phantom experiments whereby the phantom was moved manually at two different speeds were considered. The same phantom as described in Section 2.3 was used. The phantom was filled with 12.9 MBq of [¹⁸F]-FDG, and scan duration was 10 min. During the first scan, the phantom was moved more slowly while during the second scan, the phantom was moved with a faster speed. The motion was measured with the MicronTracker device at a frame rate of 31.2 Hz (32 ms bins) with a checkerboard marker of 36 × 30 mm². At the end of the two motion experiments, a third reference static scan, i.e., without any motion, was made for evaluation purposes.

The reconstructions after motion correction were deconvolved for residual motion using kernel K₅ with eight RMD iterations. The average FWHM and PVR values for the 2, 2.5, and 3 mm rods were determined and compared to the values of the motion-free scan.

In addition, a motion correction of the slow and fast motion scan was performed using interpolated poses as detailed in Section 2.1.

The region of interest where the deconvolution was performed contained 45 × 43 × 24 voxels, resulting in 46,440 kernels.

We quantify the loss of spatial resolution due to the residual motion blurring through the FWHM and PVR of the microDerenzo phantom rods, with the values of the motion-free case as the reference values.

In addition, to quantify the loss of spatial resolution of the motion corrected reconstructions in comparison with the motion-free reconstructions, a scale parameter [2] was calculated as follows. A mask covering the region where the loss of spatial resolution is to be assessed is defined by using an activity threshold in both motion corrected and motion-free images. Then, the motion-free image is filtered with a spatially invariant Gaussian filter with a scale parameter (σ²) ranging from 0 to 2.88 mm². The image correlation between the filtered motion-free and motion corrected images is calculated. Finally, the scale parameter value of the filtered motion-free image with maximum correlation with the motion corrected image is selected as the corresponding scale parameter for that motion corrected image. Although the loss of spatial resolution can vary over the image, the scale parameter serves as a metric of the average loss of spatial resolution.

To investigate the shape of the estimated deconvolution kernel K₅, the principal axes of the average kernel were calculated for all experiments as follows. The deconvolution kernels for all voxels in the region of interest of each experiment were summed. Then, the central image moments [10] of the summed kernels \( {\overline{K}}_5 \) were calculated to create the image covariance matrix:where μ^′ is the image central moment where the subindex indicates the order of the moment in the x, y, and z direction respectively. Finally, from the covariance matrix, the eigenvalues and eigenvectors were calculated to obtain the magnitude (standard deviation) and direction of the kernel principal axes.

The reconstruction of the motion-free and motion corrected rod together with the FWHM measured along the z axis is shown in Fig. 3. The measured FWHM in the motion corrected reconstruction increases as a function of the distance from the origin. At a distance of 3.11, 21.0, and 41.1 mm from the origin, the FWHM error with respect to the motion-free scan is 0, 0.84, and 1.24 mm respectively. The FWHM of the estimated deconvolution kernel K₅ follows the measured FWHM of the motion corrected reconstruction. At the origin of the coordinate system, the shape of the deconvolution kernel is an impulse, as there is no residual motion to correct for in that location (zero tangential speed). As we move away from the origin along the y axis direction, the tangential speed in the z axis direction increases and the shape of the kernels becomes wider in that direction. Along the x and y directions, the shape of the kernel is an impulse, as the tangential speed is approximately zero in the y direction and equals zero in the x direction.

The measured average speed of the mouse in the naïve condition was 1.55 cm/s, while for the memantine challenge condition, it was 4.57 cm/s. Figure 4 shows the numerical mouse brain phantom images after blurring by the residual motion kernels in all conditions.

The average striatum activity in the reference image is 1.65. After residual motion blurring, the average striatum activity reduces to 1.64 and 1.52 for the naïve and memantine challenge conditions respectively, i.e., a difference of − 0.61% and − 7.9% with respect to the reference. The difference between naïve and challenge conditions is 7.3%.

In Fig. 5, the reconstruction of the reference motion-free scan is shown together with the reconstructions of the uncorrected simulated data and the reconstruction after motion correction using both motion sampling intervals. The motion corrected reconstructions show degradation of the spatial resolution and contrast compared to the motion-free reconstruction. As can be seen from the profiles, there is a higher degradation for the 64 ms sampling interval case compared to the 32 ms case, as was expected. The average peak-to-valley ratio of the 2.5 mm diameter rods in the motion-free scan is 5.75, in comparison with 4.61 and 3.74 for the corrected reconstructions with 32 and 64 ms sampling intervals respectively. As the simulation with the 64 ms bin intervals also corresponds to a sampling at 32 ms but with the motion speed doubled, this shows that the performance of the motion correction without additional residual motion deconvolution is motion dependent.

Figure 6a shows the FWHM measured for the 2, 2.5, and 3 mm rod sizes for the case of the 32 ms sampling interval as a function of the RMD iteration number. For all kernels, the FWHM converges after eight iterations for all three rod sizes. Depending on the kernel size, the deconvolution overcompensates or undercompensates the resolution loss. After eight iterations, for the 2.5 mm rods, the difference with the motion-free reference scan FWHM is 3.53%, 2.35%, and 7.84% for kernels K₃, K₅, and K₇ respectively. For the 2, 2.5, and 3 mm rods, kernel K₅ results in a FWHM that most closely resembles the FWHM in the reference scan with a 2.5%, 2.35%, and 0.66% difference respectively. For all rod sizes, K₇ results in the most severe overestimation.

Figure 6b shows the comparison of the average FWHM of the rods after RMD with kernels K₃, K₅, and K₇ for the 64 ms sampling intervals. Kernel K₇ results in the FWHM value that is closest to the FWHM of the motion-free reconstruction for the 2 and 3 mm rod sizes. For the 2.5 mm rods, the optimal kernel is K₅. The difference with the motion-free reference scan was 1.7%, 2.39%, and 0.77% for the 2, 2.5, and 3 mm rod sizes using the optimal kernel in each case. For all three rod sizes, kernel K₃ underestimates the FWHM by more than 5% for the 64 ms sampling interval case.

As was the case for the FWHM, the PVR values converged after eight iterations. Table 1 summarizes the average PVR for the 2, 2.5, and 3 mm rod sizes for the motion correction simulation deconvolutions with sampling intervals of 32 ms at iteration 8. For the 2 and 2.5 mm rods, kernel K₃ results in the PVR that was closest to the value in the motion-free case, while for the 3 mm rods, kernels K₅ result in a PVR that is closest to the reference motion-free value.

In the 64 ms case, RMD using kernel K₅ results in the PVR value that most closely resembles the PVR of the motion-free reference reconstruction for the 2 and 2.5 mm rod sizes and kernel K₇ for the 3 mm rods. The highest difference occurs in the 2.5 mm rods (11.9%) and the lowest in the 2 mm rods (1.3%).

Although the optimal kernel to correct for the residual motion loss of spatial resolution depends on the structure size (rod diameter) and the sampling interval, for the sampling interval used in the following experiments (32 ms), RMD using the kernel K₅ results in reconstructions that most closely resemble the motion-free reconstruction. For this reason we have selected this kernel for the phantom and the in vivo experiments.

The translational speed (including displacement caused by rotational motion) at the center of the phantom throughout the scan of the microDerenzo phantom is shown in Fig. 7 for the slow and fast motion cases respectively. For the slow motion scan, the average displacement speed was 20 mm/s (0.66 mm per 32 ms bin) with a maximum of 420 mm/s. For the fast motion scan, the average displacement speed was 74 mm/s (2.44 mm per 32 ms bin) with a maximum of 904 mm/s.

For a volume of 45 × 43 × 24 voxels, the calculation of the kernel took approximately 20 min on an Intel Core i7 2.5 GHz processor. Eight iterations of the RMD took approximately 1 min using the 5 × 5 × 5 voxel kernel.

In Fig. 8, the reconstructions of the slow and fast motion scans after motion correction and after eight iterations of the RMD with kernel K₅ are shown. The profiles through the 2.5 and 3 mm rods along the white lines in Fig. 8 are shown in Fig. 9. It can be seen that the loss of spatial resolution and contrast is more severe for the case of the fast motion scan in comparison with the slow motion scan.

The average FWHM of the 2, 2.5, and 3 mm rods for the different reconstructions is shown in Table 2. Compared to the motion-free reconstruction, the FWHM of the motion corrected reconstruction of the slow motion case is increased by 6.25% and 3.14% for the 2 and 2.5 mm rods respectively. For the 3 mm rods, there is no increase in FWHM. After RMD, the FWHM was increased with respect to the motion-free case by only 4.60% and 1.57% for the 2 and 2.5 mm rods respectively. For the fast motion case, the FWHM of the 2, 2.5, and 3 mm rods is increased by 18.7%, 15.3%, and 6.29% respectively for the motion corrected compared to the motion-free reconstruction. After RMD, the FWHM was increased with respect to the motion-free case by only 12.1%, 8.24%, and 2.45%. The difference between the FWHM of the fast and slow motion cases of the motion corrected reconstructions before RMD is 11.1%, 11.1%, and 6.8% for the 2, 2.5, and 3 mm rods respectively. After RMD, these differences are reduced to 6.15%, 6.36%, and 4.18% respectively.

Finally, the FWHM values for the three rod sizes of the motion corrected reconstructions using interpolated poses were similar (within the standard deviation) to the rods FWHM obtained using RMD.

The average PVR for the 2, 2.5, and 3 mm rods is summarized in Table 3. As could be expected, the PVR values after motion correction for the fast motion case are lower compared to the slow motion case. However, after deconvolution, the PVR values are closer to the PVR values of the motion-free reconstruction. In addition, the PVR values for the slow and fast motion are closer to each other after the deconvolution. Similar to what was found using the FWHM metric, the PVR of the motion corrected reconstructions using interpolated poses was similar (within the standard deviation) to the PVR obtained using RMD.

Table 4 shows the scale parameter and magnitude and direction of the average kernel \( {\overline{K}}_5 \) principal axes for all experiments. For the microDerenzo phantom experiments, the loss of spatial resolution is greater for the 64 ms motion sampling and the fast motion cases in comparison with 32 ms sampling and slow motion case respectively. For the real data experiments, factors other than the finite motion sampling error further degrade the spatial resolution.

For the kernel principal axes magnitude, the value is greater for the 64 ms motion sampling and fast motion cases in comparison with the 32 ms and slow motion cases respectively.