F1 F6 F11 F16
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F21 F26 F31 F36
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Figure 4.4: Transversal slices number 40 and central coronal views of the selected frames 1, 6, 11, 16, 21, 26, 31 and 36 of a respiratory-gated helical CT acquisition. The study covered an axial length of 27 cm with a FOV of 35 cm. The first four and last four pairs of frames correspond to inhalation and exhalation, respectively. Pulled by contraction of the thoracic diaphragm, organs move out of the selected transversal slice during the inhalation phase.
I0 I1+ I2+ I3+
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I0 I1− I2− I3−
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Figure 4.5: Transversal slices number 40 and coronal central views of the initial FBP reconstruction and the corrected images after one, two and three iterations. The white squares show the boundaries of close-up views. The last row emphasizes on the difference with the reference frame 21 from the dynamic phantom. This frame corresponds to the time point in the middle of the respiratory cycle.
F1 I20+ F21 I20−
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Figure 4.6: Side-by-side comparisons of the two possible image correction scenarios. The corrected images are shown after 20 iterations of the iterative motion compensation frame-work. As suggested by the analyses in figure 4.8, the images get closer to one of the two extreme motion states: The beginning of exhalation (F1) when the position of the diaphragm is low or the beginning of inhalation (F21) when the position of the diaphragm is high.
I0−F1 I20+ −F1 I0−F21 I20−−F21
TransversalCoronal
Figure 4.7: Transversal slices number 40 and coronal central views of the two selected framesF1 andF21 of the dynamic phantom compared to the initial FBP imageI0 and the two corrected images I20+ and I20− obtained after 20 iterations. F1 corresponds to the start of exhalation (full inspiration) motion state. F21 corresponds to the other extreme start of inhalation (full exhalation) motion state. The differences are significantly reduced with motion compensation.
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Figure 4.8: Quantitative analyses of images for the initial FBP reconstructed image and corrected images after 1, 2 and 3 iterations when the correction is applied from the positive (left) and negative (right) part of the decomposed differences. The plots show the mean average error (MAE) in Hounsfield units (HU) between the reconstructed image and each of the original 40 gates shown in figure 4.4. The analysis considers only voxels in regions significantly compensated after the first iteration.
At the first frame, the sequence starts at full exhalation, where the diaphragm is low. During the next 20 frames, the inspiration takes place. At frame 21, the maximum inhalation is reached and the expiration begins. One can remark that the images still contain some slight blurring due to helical reconstruction artifacts and residual motion within each gate of the breathing cycle.
The forward projection relies on the approximation of line integrals through the digital image using a fast ray-tracing algorithm (Schretter, 2006). The acquisition time was 12 seconds, matching the typical rotation speed of a C-arm system. This experiment validates the method on a worst case scenario when the patient moves during the whole acquisition instead of holding the breath. Although the simulated breathing motion is naturally periodic, only about 3 cycles can be observed for 12 seconds. Selected transversal and coronal slices from the dynamic phantom are shown in figure 4.4.
Interpretation
The initial FBP reconstruction and the corrected images after one, two, and three iterations are shown in figure 4.5. The strong reconstruction artifacts due to in-consistencies among projections can be seen in the initial FBP images of the first column. Image resolution is mainly limited by motion-blur artifacts. The liver and stomach appear to be semitransparent in the transversal view, as can be seen in the selected close-up views. When using motion correction, the image quality improves progressively with the number of iterations for some of the frames of the dynamic phantom.
It can be observed that in the first two rows of figure 4.5, the corrected image converges towards the motion state at begin-inhale (end-exhale) when correcting with the positive part of differences. In the last two rows, the corrected image converges toward the motion state at end-inhale (begin-exhale) when correcting with the negative part of differences. In all images, the window is centered to
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Figure 4.9: Analysis of the convergence toward one specific motion state. The “pos”
curve plots the mean absolute error (MAE) in Hounsfield units (HU) between F1 and the corrected imageIi+ obtained after iterationi∈[0,20]. The reference frameF1 corresponds to the motion state at the beginning of exhalation. The “neg” curve plots the MAE between F21 and the corrected imageIi− obtained after iterationi∈[0,20]. The reference frameF21 corresponds to the motion state at the beginning of inhalation. In both cases, a reduction of 50% in terms of the mean absolute error can be observed already after the fourth iteration.
the attenuation value of water (0 HU) and the window width equals 2000 HU. In difference images, the window center is set to zero.
One of the two extreme motion states (where the diaphragm is the lower or the highest) is selected as an asymptotic state of the iterative framework. The convergence depends on the choice of the positive or negative part of the difference projections. Indeed, when the diaphragm is low (or high), then the lungs are filled (or empty) and the total mass of the reconstructed image is minimized (or maximized).
In figure 4.6, side-by-side comparisons of images obtained after convergence (after 20 iterations) illustrate the end results of the experiment with the two possible image correction scenarios. Image quality improvements can be assessed visually from the difference images in figure 4.7.
Quantitative Analyses
Figure 4.8 shows quantitative analyses of the reconstructed volumetric image. As the number of iterations increases, the algorithm converges to a particular motion state. The mean absolute error (MAE) with respect to frames close to this optimum motion state decreases with the number of iterations.
The plots in figure 4.9 demonstrate the early convergence of the method. After only four iterations, a significant improvement in mean absolute error above 50%
can be appreciated. The analyses only consider voxels that belong to a region of interest (ROI) which is the collection of voxels that are significantly compensated after the first iteration. A voxel belongs to the ROI if its absolute value of correction exceeds 100 HU.