Chapter 3: Simulation study of the proposed design for triple-energy
3.2. Quantitative indices
We evaluated the filtered spectra as quantitative indices of mean energy ratio, contrast variation ratio, and exposure efficiency. In this study, the quantity of photons of filtered and unfiltered X-ray beam was set to 3.8×106 for evaluating the mean energy ratio, contrast variation ratio, and exposure efficiency. Mean energy was calculated as the ratio of post-object mean energy to pre-object mean energy as follows:
mean energy when X-ray photons pass through the matter in the proposed method.
Another way to characterize the effect of contrast when filtered X-ray beam is used was to compare the contrast of unfiltered X-ray beam. Contrast (C) is defined with the following equation:
obtained in the beam-filtered case to contrast in the unfiltered beam case at the same tube operating potential with the following equation:
contrast when X-ray beam is unfiltered at same tube potential.We used exposure efficiency to evaluate the influence of X-ray beam and tube potential, which is assumed to be related to more desirable dose efficiency quantitative index. The exposure efficiency is defined as:
osure
respectively. N is the noise (standard deviation) in the background region.Figure 3.6 shows results of mean energy ratio of comparing various filter materials at 7 HVL through a 2 cm PMMA. Choice of an appropriate operating range for tube potential is dependent on the filter material, with a wider range of tube potentials for the higher Z materials (40–80 kV) than for the lower Z materials (40–50 kV), as indicated by the values of kV for which the plot remains close to unity. However, filters having low K-edge energy out of spectral energy such as Al and Cu are nearly the invariant mean energy ratio in all tube potentials because their K-edge energy does not contribute to shaping the ray beam. Mean energy ratio obtained with filtered X-ray beam is lower than that acquired with unfiltered X-X-ray beam for all tube potentials.
Thus, filtered X-ray beam can reduce beam hardening by shaping a broad spectrum to a narrow beam. The mean energy of the narrow beam is maintained after the X-ray beam is through the object.
Figure 3.7 compares various filter materials at 7 HVL through a 0.5 cm Al. Mean energy ratio of filtered beam is reduced with comparison to unfiltered beam at the same tube potential. The mean energy ratio of I filter rapidly increases from 50 to 70 kV, and reduces from 70 to 90 kV. The mean energy ratio of Ba filter increases from 60 to 80 kV, and reduces from 80 to 90 kV. The mean energy ratio of Ce filter increases from 60 to 90 kV. The mean energy ratio of Gd increases from 80 to 90 kV. The mean energy of Al, Cu, Er, and W filters are almost equal. It is thought that K-edge energy of Al and Cu is influenced by tube potential. Considering K-edge energy of Er and W, the mean energy ratio of Er and W is expected to increase at more than 90 kV. The result indicated the same trend in case of mean energy ratio of PMMA in figure 3.6.
Figure 3.6 Results of mean energy ratio comparing various filter materials at 7 HVL through a 2 cm PMMA.
Figure 3.7 Results of mean energy ratio of comparing various filter materials at 7 HVL through a 0.5 cm aluminum filter.
The geometry of simulation was illustrated for image acquisition with the design X-ray beam, as shown in figure 3.8. The initial beam is emitted to the detector through the filter and phantom. The phantom image was used to evaluate contrast variation ratio and exposure efficiency.
Figure 3.8 Illustration of geometry to acquire phantom image with designed X-ray beam by using initial X-ray source, filter, and a-Se detector.
In figure 3.9, the phantom images were acquired to evaluate contrast variation ratio and exposure efficiency. Incident photon number was 3.8×106 for each simulation condition. Aluminum is located at the center (white) as a signal, and the peripheral part is PMMA as a background (black). The first row is the image when using Al filter with 7 HVL thicknesses. The images by using unfiltered and filtered X-ray beams with Cu, I, Ba, Ce, Gd, Er, and W are displayed. The last row is the images made by using no filtration from 40 to 90 kV. The results for contrast variation ratio and exposure efficiency are illustrated in figures 3.9 and 3.10, respectively.
Figure 3.9 The signal (aluminum) and background (PMMA) were obtained to evaluate contrast variation ratio and exposure efficiency by simulation study. The effect of filtered X-ray beam was shown with signal and noise of signal and background. The proton quantities of unfiltered and filtered X-ray beam are each 3.8×106.
As illustrated in figure 3.10, higher Z filters and higher tube potential appear to have less gain than lower Z filters because the designed X-ray beam by filter at high tube potential increases the transmission of the X-ray beam. Al and Cu are decreased with increasing tube potentials due to the too low K-edge energy. If one is interested in using the technique to reduce beam hardening without degrading contrast, then I, Ba, and Ce are appropriate for filter materials. Contrast of image obtained with filtered X-ray beam is higher compared with that acquired with unfiltered beam in this study.
Thus, filtered X-ray beam is efficient to enhance image contrast in a specific tube potential range.
Figure 3.10 Contrast of the image obtained with filtered X-ray beam can be higher than that acquired with unfiltered X-ray beam.
Mean energy only takes into account designed X-ray beam characteristics. Contrast results are impacted by beam hardening in part but also take into account detector characteristics. However, noise is not included in either of the above results. Thus, to more completely characterize the system response, we include X-ray noise, detector efficiency, and incident beam characteristics by examining the exposure efficiency.
The number of photons were 3.8×106 for each X-ray beam.
From the simulation results, exposure efficiency increased with increasing filtration for almost all tube potentials. Figure 3.11 indicates the exposure efficiency considering both signal-to-noise ratio and the number of photons (exposure) on images obtained with monochromatic X-ray beam with increasing Ba filter thicknesses. In case of Ba filter, exposure efficiency increases between 40 to 80 kV when using 2, 7, and 8 HVL filter thicknesses. The exposure efficiency with changing tube potential illustrates that more filtration for a given tube potential yields better exposure efficiency (SNR2/exposure) in figure 3.11. The exposure efficiency is considered a reasonable surrogate for dose efficiency, and is ultimately a more easily measured quantity. Al and Cu filter reduced the exposure efficiency. The range of exposure efficiency with filtered beam is higher than the conventional beam from 40 to 50 kV for all filters, as shown in figure 3.12. The exposure efficiency obtained with monochromatic X-ray beam with I, Ba, and Ce filters is increasing in the range of 50 to 60 kV compared to that acquired with unfiltered beam at the same exposure. Overall, monochromatic X-ray beam generated by I, Ba, Ce, and Gd filters were higher exposure efficiency than that acquired with unfiltered X-ray beam. The evaluated quantitative indices for mean
energy ratio, contrast variation ratio, and exposure efficiency are summarized in table 3.1.
Figure 3.11 Exposure efficiency by considering the SNR and the number of photons through 2 cm PMMA and 0.5 cm Al object for designed beam obtained with Ba filter at various filter thickness.
Figure 3.12 Exposure efficiency considering the SNR and the number of photons through 2 cm PMMA and 0.5 cm Al object for the designed beam obtained with all filters at 7 HVL filter thickness.
From the simulation results for mean energy ratio, contrast variation ratio, and exposure efficiency, the effect by using additional filter is summarized. As shown in table 3.1, quantitative indices obtained with I, Ba, and Ce filters outperform other filters at a certain range of tube potential. X-ray beam filtered by Al and Cu minimized mean energy ratio for the full tube potential range and maximized exposure efficiency from 40 to 50 kV. This means that mean energy of filtered spectra by using Al and Cu was only shifted to high energy without spectral shaping by K-edge of filter. Since Gd, Er, and W filters have high K-edge energies, the contrast is decreased compared to that with X-ray beam without filter.
Table 3.1 Tube operating range summary for different quantitative indices considering mean energy ratio, exposure efficiency, and contrast variation ratio.
Element
Mean energy ratio minimized (kV)
Exposure efficiency Maximized (kV)
Contrast variation ratio maximized
(kV)
Al 40–90 40–50 …
Cu 40–90 40–50 …
I 40–50 40–60 40–60
Ba 40–60 40–80 50–80
Ce 40–70 40–80 60–80
Gd 40–90 40–60 …
Er 40–90 40–50 …
W 40–90 40–50 …