In order to select the features of the ICS beams that best fit the requirements of the analysed imaging tasks, we first focused on CEDEM, which is the simplest case since it involves phantoms composed of two materials only. Conversely, coronary angiography is a more complicated task to face, due to the presence of the bone. Indeed, the signal of the bone competes with the signal of the small blood vessels perfused with the contrast medium. Furthermore, coronary angiography is much more demanding in terms of X-ray beam intensities, due to the significant attenuation of surrounding tissues in the chest [
54]. A careful setting of the ICS X-ray beams is mandatory to obtain a correct reconstruction of the mass thickness of the iodinated structures, a sufficient
in the contrast image and a good rejection of the bone signal (coronary angiography). In this section, we show the simulated images and the relative figures of merit as a function of the bandwidth and the peak energy of the two beams bracketing the iodine K-edge.
5.1. Contrast-Enhanced Dual-Energy Mammography (CEDEM)
As previously pointed out, when dealing with the X-rays emitted from an ICS source, it is desirable to set a wide collimation angle, in order to maximize the beam intensity and the size of the radiation field at a given distance from the interaction point, but this leads also to a larger energy bandwidth. As shown in
Figure 4a, if the bandwidth of the photon beams increases, the separation
between the two peaks has to be increased in order to correctly estimate the contrast medium signal with the KES technique. For a given value of
, that is symmetric with respect to the contrast medium K-edge, a better estimation of the signal is instead obtained by choosing beams with narrow bandwidth. This effect is due to the overlap of the two spectra, namely due to the fact that even if there are photons with energy higher (lower) than the K-edge in the LE (HE) image, they are treated as their energy was lower (higher), since we simply apply the KES algorithm using the mean energy of the X-ray beams. One might think then to keep the beams well separated by setting, for instance,
= 8 keV to increase the relative energy bandwidth up to 5% RMS. Unfortunately, the noise in the contrast image increases monotonically if the peak separation increases. This effect is shown is
Figure 4b, which reports the results of a calculation carried out using Equations (
13) and (
14), where
= 4.5
.
The effects described above were obtained with Gaussian beams, but they hold true also for ICS beams. However, ICS spectra feature a peculiar asymmetric shape with a rapid fall-off at energies higher than the peak value and a longer tail at lower energies. This suggests that a symmetric separation of the peaks with respect to the contrast medium K-edge may not be the optimal choice. As a consequence, the most appropriate approach to identify the most convenient irradiation conditions consists in setting a-priori a desired bandwidth and calculating the figures of merit, in particular the
, for different couples of peak energies (
,
).
Figure 5 shows the values of the signal-to-noise ratio per pixel for a iodinated detail irradiated with the ICS beams whose specifications are listed in
Table 3. In particular, we considered a bandwidth of 3% RMS, which is small enough to not cause significant distortions in the signal reconstruction and at the same time is wide enough to have a maximum beam divergence that guarantees a sufficient beam intensity and radiation field.
Such figure suggests that the couple of beams with the peak energy of 32 keV and 36 keV, respectively, is an optimal choice to maximize the signal-to-noise ratio of the details. As expected, the optimal peak energies are not equally spaced with respect to iodine K-edge. In particular, the HE spectrum is more distant, to avoid that its tail crosses the K-edge.
Here, we reported the case of a detail with a thickness of 5 mm filled with 2 mg mL of iodine, namely a single value of mass thickness. For different mass thicknesses, the figures of merit change, in particular the signal scales linearly with both and . Nonetheless, the optimal couple of peak energies does not depend on these variables. The same concept holds true also for the detail transversal size (number of pixel n) and the number of impinging photons (). Indeed, the of a given detail scales with the square root of both n and ; however the optimal couple of peak energies does not change.
Once we defined the bandwidth and the peak value of the ICS spectra, we simulated the irradiation of a phantom with the significant features in a CEDEM examination, namely details of various size embedded in bulk of variable thickness to simulate the anatomical background. The mean thickness of the bulk was set to 5 cm as in the previous calculations, while the thickness fluctuation was obtained by subdividing the bulk in horizontal (
x) stripes with random vertical (
y) size and thickness (
t). The features of the phantom are reported in
Table 2, where the number after the symbol ± in the
column indicates the standard deviation of the random variables used to simulate the anatomical background. We considered 9 details placed as a 3 by 3 matrix. The details share the same thickness in the horizontal direction and the same transversal size in the vertical one. Each detail represents the vascular bed of a small tumoral mass.
To make our simulation more realistic, we considered the spatial variation of the mean energy, the local bandwidth and the beam intensity, as described in
Section 4.2. Since the divergence angle was
= 2.8 mrad, the source-to-phantom distance was set to 20.2 m to irradiate the whole phantom. We simulated 5
photons, namely a fluence of about 5
ph mm
impinging on the phantom subdivided in voxels with a transversal size of 300 × 300
m
. Therefore, the mean number of photons per voxel was 4.5
, as in the previous calculations. By using the normalized glandular dose coefficients calculated by Boone [
55], it is possible to calculate that the considered fluence (LE + HE exposure) would correspond to a mean glandular dose (MGD) of 0.5 mGy in a 5 cm-thick breast with a glaundularity of 50%.
Figure 6 shows the image obtained irradiating the phantom with the beam peaking at 32 keV (LE image), the one obtained with the beam peaking at 36 keV (HE image), the contrast image and the tissue image. In the LE image none of the details is visible. In the HE image, the details of larger size are visible, while the detectability of smaller details is limited by the striped structure, namely the random thickness of the soft tissue bulk. The structure mimicking the anatomical background is present in the tissue image but disappears in the contrast image, which shows the iodinated structures only, even if the noise is higher than the LE and HE images.
Table 4 reports the figures of merit calculated for the contrast image. The mass thickness value reconstructed from the contrast image is, in general, affected by the mean energy and the bandwidth of the two X-ray beams. Negative values of reconstructed mass thickness could be obtained for thinner iodinated details or in the background, as described in [
53]. For ICS beams, the mean energy and the local energy spread vary as a function of the emission angle, hence pixel-by-pixel in the image. Conversely, the energies used for the reconstruction are fixed. As a consequence, the iodine mass thickness reconstruction for pixels that are more distant from the centre of the image is less accurate. This explains the values of
and
reported in
Table 4. Nonetheless, the signal
is in good agreement (within 10%) with the actual iodine mass thickness of each detail. Moreover, the values of
confirm the expectation, i.e., larger details feature higher values of
.
5.2. Coronary Angiography
In the case of coronary angiography, the presence of the bone and the goal of minimizing its visibility in the contrast image impose to follow a slightly different approach than the case of mammography. Since the contrast image will encompass regions where the vessels filled of iodinated solution cross the bone structures and regions where they do not, a more appropriate figure of merit to consider for the selection of the optimal couple of peak energies is the mass thickness of the reconstructed detail
.
Figure 7 shows the values of
for a vessel irradiated with the ICS beams whose specifications are listed in
Table 3. The vessel thickness was 5 mm and the iodine concentration was 10 mg mL
, thus the actual value of mass thickness of the detail was 5
g cm
. However, through the application of the KES method for monochromatic beams with a two-material basis, the reconstructed values of
are significantly smaller than the expected one due to the masking effect of the third material. Indeed, the spurious term due to the bone in Equation (
5) is negative in the energy range of interest and its absolute value increases with
. For high values of
, this term can overcome the contribution of the iodine actually present in the detail so as that the estimated
results negative.
Nonetheless,
Figure 7 suggests that the optimal couple of peak energies is about
= 32.5 keV and
34.5 keV. These energy values are closer to the iodine K-edge with respect to the previous case. This is consistent with the result obtained with monochromatic beams, which give best results when using energies as close as possible to the K-edge [
6]. Obviously, it is not possible to fulfil this requirement when spectra with finite bandwidth are used, to avoid their overlap.
As in the previous case, once we defined the peak values of the ICS spectra, we simulated the irradiation of a proper phantom. The phantom was composed of 3 vessels of different diameter/thickness filled with contrast medium and embedded in a 20 cm-thick bulk of soft tissue. In the central region, the vessels crossed a slab of bone. Furthermore, 3 mm of the central vessel featured a 50% stenosis. The anatomical background was simulated as before. It is worth noting that the overall size of considered phantom is very small if compared to the actual chest region of interest for coronary angiography. The phantom size was limited to avoid to simulate an excessive number of photons. Indeed, we simulated 1.94 × 10
photons which led to a fluence of 1
ph mm
on the phantom positioned at a distance of 8.8 m. The considered fluence (LE + HE exposure) leads to skin dose of about 1.3 mGy, evaluated according to Sarnelli et al. [
16]. The nominal numbers of photons impinging per voxel (transversal size 300 × 300
m
) was then 9.07
, namely about twice than the CEDEM case, where phantom attenuation was smaller. As in the CEDEM case, we considered the spatial variation of the features of the X-ray beams.
Figure 8 shows the simulated images, while
Table 5 reports the figures of merit calculated on the contrast image. The strips used to account for anatomical background make difficult to discern the size of the vessels in the LE and HE images. Conversely, all the vessels are visible in the contrast image, even if the smaller vessel and the stenosis in the central one have a small
at the considered fluence. Indeed, the contrast image is quite noisy and higher fluence values are required to enhance the visibility of the smaller details.
The values reported in
Table 5 were obtained by considering a ROI with size proportional to the vessel diameter. The fluctuation in signal values related to the region of each vessel with or without the superposition of the bone are of statistical nature. Yet, the higher values of the noise in the region of the vessels with the superposition of the bone are consistent with the value calculated using Equation (
15). Furthermore, the bigger the vessel the higher the
, as expected. The vessel signal is reconstructed with an error of about 20% due to the choice of the peak energy of the X-ray beams. Better signal reconstruction could be obtained by increasing
, but at the expense of an increased visibility of the bone. For the sake of comparison, we report the contrast images obtained with two different couples of spectra in
Figure 9.