Monte Carlo simulation

3.4.1 Beam selection

Based on the results of designed beam from simulated and measured spectra, linear attenuation coefficients of PMMA, Al, and I were simulated with Monte Carlo simulation prior to obtain the thickness density map of phantom containing three materials such as I, Al, and PMMA. The results of linear attenuation coefficients obtained with proposed TE X-ray beam were compared with the result of linear attenuation coefficients acquired from photon-counting method in simulation study.

The linear attenuation coefficients were used as matrix to solve material density map for I, Al, and PMMA.

Figure 3.16 illustrates the X-ray spectra by the number of photon of proposed TE monochromatic X-ray beams. TE monochromatic beams are generated by using I, Ba, and Gd filters at 50, 60, and 70 kV, respectively considering quantitative indices. The mean energies of the proposed TE monochromatic X-ray beams were 31.47, 35.38, and 46.37 keV, respectively. Spectral separations for TE imaging were observed by combinations of K-edge filter materials and tube potentials. Narrow spectra could be able to discriminate the information including various materials in an object. In the photon-counting method, three energy bins were selected to match the mean energy of each TE X-ray beams. Therefore, the energy is binned into 21–33, 34–41, and 42–50 keV from X-ray spectrum at 90 kV tube potential, and the mean energies of each bin

are 29.34, 37.57, and 45.87, respectively, in photon-counting mode, as shown in figure 3.17.

Figure 3.16 X-ray spectra for proposed TE monochromatic X-ray beam by generating I, Ba, and Gd filters with 50, 60, and 70 kV, respectively.

Figure 3.17 In photon-counting mode, energy binning was performed from 90 kV broad spectrum to match the energies of proposed TE monochromatic X-ray beam.

The information of the X-ray beam on TE X-ray beam and photon-counting method is given in table 3.2. In the proposed TE monochromatic X-ray method, E_1, E_2, and E_3 were energies at 50, 60, and 70 kV, respectively, with I, Ba, and Gd filters, respectively. The number of photons was 3.8×10⁶ for each beam in the proposed TE monochromatic X-ray. Since the photon number affects the image quality, such as by inducing noise, the incident photon number is set to same level for multi-energy imaging. In the photon counting method, the numbers of photons were 3.9×10⁶, 3.7×10⁶, and 3.8×10⁶ for bin 1, bin 2, and bin 3, respectively. The limitation of count rate of photon-counting system used in this study is 1.2×10⁶, though the photon-counting system can acquire the signal several times without effect of low-count rate.

Table 3.2 Proposed triple-energy X-ray beam and binning of photon-counting method.

Triple-energy X-ray beam Binning of photon-counting Tube

We try to discriminate three materials (I, Al, and PMMA) by density map when the three materials are overlapped. To obtain density map, linear attenuation coefficients for I, Al, and PMMA can be decided to calculate matrix in equation 3.6. GATE simulation tool was used to obtain linear attenuation coefficients of I, Al, and PMMA.

The X-ray imaging system was designed with a SID of 100 cm. The phantom consisted of I (100 mg/cm³), Al, and PMMA, as shown in figure 3.18. In the validation for

simulation of proposed method, the charge-integrating detector was modeled as described in chapter 2. The energy-resolved photon-counting detector was modeled as eV 2500, as described in chapter 2. The exposure condition is expressed for both proposed and photon-counting method in table 3.2.

Figure 3.18 The cubic phantom of I, Al, and PMMA is on the detector for obtaining linear attenuation coefficient and thickness density map.

3.4.3 Density map reconstruction

In case of three materials, the logarithmic intensity attenuation is described by the well-known Beer’s law for three component systems, as in equation 3.4 [31].

P P A A I

I

L L L

I I

T  ln(

/ )      

Equation 3.5 is assuming that the exposure is performed only at once to the object containing three materials. If the object is scanned with triple-energy X-ray beam, equation 3.5 can be extended to the following three-component system:

 

where, T(1), T(2), and T(3) is log measurement by using three X-ray beam with added filtration or three energy binning. _Iⁱ , ⁱ_A , and ⁱ_P is the linear attenuation coefficient for I, Al, and PMMA, respectively. LI, LA, and LP are density map. We can solve the system with matrix inversion. The density map reconstruction algorithm is used to calculate the thickness of I, Al, and PMMA for both simulation and experimental result.

3.4.4 Linear attenuation coefficients and mean energy

The linear attenuation coefficients and their mean energy of I, Al, and PMMA were obtained by using Monte Carlo simulation. In table 3.3, the linear attenuation coefficients and mean energies of I, Al, and PMMA for the values obtained with proposed TE monochromatic X-ray beams. K-edge energies of I, Al, and PMMA were used as a reference compared with simulated energies. As shown in table 3.3, the linear

attenuation coefficients and mean energies of I, Al, and PMMA are similar to those of reference mean energies. In table 3.4, the linear attenuation coefficients and mean energies of I, Al, and PMMA for photon-counting method. Reference energies are the energies as binned in the photon-counting system. The mean energies of I, Al, and PMMA are similar to those of binned mean energies. Resultant linear attenuation coefficient maps of I, Al, and PMMA for the proposed and photon-counting methods illustrated in figure 3.19. (a), (b), and (c) are the attenuation coefficients at 50, 60, and 70 kV, respectively, with I, Ba, and Gd filters, respectively. (d), (e), and (f) are the attenuation coefficients map at 29.34, 37.57, and 45.87 keV, respectively. In figure 3.19 (a) and (d), since the mean energies of I are below those of the K-edge energy of I, effective μ of I is lower than Al.

Linear attenuation coefficients of I, Al, and PMMA acquired with both TE X-tray beams and photon-counting method were used as a matrix into equation 3.6 ⁱ_I, ⁱ_A, and _Pⁱ for producing thickness density map. The obtained images of figure 3.19 (a), (b), and (c) and (d), (e), and (f) were used as a log measurement image into equation 3.6 T(1), T(2), and T(3) for proposed and photon-counting method, respectively.

Table 3.3 Linear attenuation coefficients and mean energies of I, Al, and PMMA with Monte Carlo simulation for proposed method. Reference energy is K-edge energies of I, Al, and PMMA.

Energy

Iodine Aluminum PMMA reference

Effective

Table 3.4 Linear attenuation coefficients and mean energies of iodine, aluminum, and PMMA with Monte Carlo simulation for photon-counting method. Reference energy is the energies as binned in the photon-counting system.

Energy

Iodine Aluminum PMMA reference

Effective

Figure 3.19 Linear attenuation coefficient maps of I, Al, and PMMA obtained with proposed TE X-ray beams and photon-counting method. (a), (b), and (c) are the attenuation coefficients at 50, 60, and 70 kV, respectively, with I, Ba, and Gd filters, respectively. (d), (e), and (f) are the attenuation coefficients map at 29.34, 37.57, and 45.87 keV, respectively.

Figure 3.20 (a), (b), and (c) are thickness density maps of I, Al, and PMMA acquired with TE X-ray beams. (d), (e), and (f) are thickness density maps of I, Al, and PMMA with the photon-counting method.

Figure 3.20 (a), (b), and (c) shows thickness density maps of I, Al, and PMMA, respectively, with the proposed method. (d), (e), and (f) are thickness density maps of I, Al, and PMMA, respectively, with the photon-counting method. The true values of thickness for I, Al, and PMMA are each 1.00. I, Al, and PMMA were well separated at each thickness density map, as shown in figure 3.20. The resultant thicknesses of I, Al, and PMMA were 1.00, 1.00, and 0.99, respectively, in proposed method. In the photon-counting method, thickness densities of I, Al, and PMMA were 1.00, 0.96, and 1.02, respectively. The evaluation of thickness density is illustrated in figure 3.21. The result indicated that the density map obtained with the proposed TE monochromatic X-ray beam was similar to that acquired with photon-counting method.

Figure 3.21 The results of thickness density maps for I, Al, and PMMA.