2.1.1 Calculation of the equivalent atomic number (Z_eq)

The equivalent atomic number, Z_eq, is an energy-dependent parameter describing the properties of a composite material in terms of equivalent elements. We started by extracting the Compton partial mass attenuation coefficient (μ/ρ)_Compton and total mass attenuation coefficient (μ/ρ)_Total for all elements with Z = 4–92 and for the chosen dosimetric materials in an energy range of 0.015–15 MeV using the WinXCom computer program [17] (initially developed as XCOM). Then, to interpolate Z_eq when the ratio (μ/ρ)_Compton/(μ/ρ)_Total for a given human organ or tissue lies between two successive ratios of elements, the following formula was employed [18]:

where Z₁ and Z₂ are the atomic numbers of the two elements, R₁ and R₂ are the ratios (μ/ρ)_Compton/(μ/ρ)_Total for these elements, and R is the corresponding ratio for a given human organ or tissue at a given energy that lies between R₁ and R₂ (the nearest neighbors of R).

2.1.2 Calculation of the GP fitting parameters

Taking the GP fitting parameters for pure elements from the standard reference, ANSI/ANS-6.4.3-1991, and using the previously calculated Z_eq, we interpolated the corresponding GP fitting parameters (a, b, c, d, and Xk) for the buildup factors using the following formula [19-21]:

where P, P₁, and P₂ refer to the same GP parameters for the selected organ or tissue and both neighbors (which correspond to Z₁ and Z₂).

2.1.3 Calculation of the buildup factors

For a given photon energy and shielding material, we used two different sets of GP parameters to calculate the EABF and EBF. Thus, we first tabulated all the parameters from ANS-6.4.3-1991 and proceeded to calculate the buildup factors for a given incident photon energy E and penetration depth (source–detector distance) x, in mfp, as follows [2, 6, 22-24]:

where, for x ≤ 40 mfp:

and for x ≥ 40 mfp:

where $\Phi =\frac{K_{40}-1}{K_{35}-1}$, $\xi (x)=\frac{{(x/35)}^{0.1}-1}{{(40/35)}^{0.1}-1}$, K₃₅=K(E,35), and K₄₀=K(E,40). For deep penetration (greater than 40 mfp), an extrapolation of the dose multiplication factor K(E,x) was proposed by Harima et al. [24] and supported by Sakamoto and Trubey [25].

3.2.1 Incident photon energy dependence

Figure 1 shows the energy dependence of the buildup factors at selected penetration depths from 1 to 100 mfp for water. All the human tissues and organs behaved similarly to water (as shown in Table 2 and the Annex) owing to their similarity in terms of chemical composition (the major elements are H, C, N, O, Na, P, S, Cl, and K, as shown in Table 1). For all the plots, the buildup factor curves increased to a maximum value and then decreased more slowly. This fact confirms the predictions from theory and the literature, as the dominance of the photoelectric effect increased from a low energy up to a specific energy at which it equals the Compton effect, namely, E_pe. Moreover, if E_pp is the energy at which the Compton and pair production cross sections are equal, the buildup factors reach their maximum at an energy E_pe le; E le; E_pp, because Compton scattering results in more multiply scattered photons.

3.2.2 Equivalent atomic number dependence

Figure 2 shows the buildup factor values as a function of Z_eq for some tissues and organs at photon energies of 0.015, 0.15, 1.5, and 15 MeV and penetration depths of 5, 40, and 100 mfp. This figure demonstrates that the buildup factors were inversely proportional to Z_eq at photon energies of 0.015 and 0.15 MeV and independent of Z_eq at 1.5 and 15 MeV for all penetration depths. Thus, for low-energy photons, the buildup factor decreased with increasing Z_eq (chemical compounds), whereas for high-energy photons, it remained insensitive to the chemical composition of the medium. Here, we confirmed the conclusions reported elsewhere [2, 18, 21, 22, 28].

Fig. 2.Full-screen View

Calculated EABF of Z_eq for photon energies of 0.015, 0.15, 1.5, and 15 MeV and penetration depths of 5, 40, and 100 mfp for studied tissues.

In addition, we calculated and plotted Z_eq as a function of photon energy for the standard grid for some tissue substitutes commonly used in experimental radiation physics [16]. Figure 3 shows that the chemical compositions of the three tissues (adipose, lung, and muscle tissue) are equivalent to those of their substitute materials (AP6, LN1, and M3, respectively), especially at lower energies (up to 1 MeV). However, we see that M3 cannot be used as a muscle substitute, and AP6 can replace adipose tissue moderately well, at higher energies (above 1 MeV). This can be explained by the fact that the pair production effect is proportional to the square of the atomic number. In Table 1, the difference in elemental composition between M3 and muscle tissue is clear, and that between AP6 and adipose tissue is moderately clear.

Fig. 3.Full-screen View

Z_eq values of some tissues (adipose tissue, lung, muscle) and their substitutes (AP6, LN1, M3, respectively) as a function of photon energy.

3.2.3 Penetration depth dependence

From the plots of EABF as a function of penetration depth for selected tissues in Fig. 4, it can be seen that at a lower energy of 0.015 MeV, bone and breast have the lowest and highest buildup factors, respectively, for penetration depths up to 100 mfp. This effect can be explained by the differences in chemical composition of different tissues, as adipose tissue and bone have Z_eq values of 6.389 and 10.59, respectively. Further, at a photon energy of 0.15 MeV, the buildup values become more similar until they coincide to form a single dataset at a photon energy of 1.5 MeV for all penetration depths. Table 3 illustrates the good agreement between our results and those of Olarinoye for two human tissues for deep penetration (up to 100 mfp). Moreover, at a photon energy of 15 MeV, all the tissues have equal buildup values up to 20 mfp, where the data for bone tissue alone started to increase up to 100 mfp, and all the other tissues continued to exhibit similar behavior. In addition, the higher magnitude of the buildup factors at a photon energy of 0.15 MeV compared to the other cases can be explained by the important increase in multiply scattered photons, remembering that the buildup factor was defined as the ratio of scattered to primary photons.

Fig. 4.Full-screen View

Variation in the EABF of the penetration depth for selected human tissues at various photon energies.

3.2.4 Comparison of EABF and EBF

Figure 5 shows a comparison of the buildup factors for breast (adipose) and bone tissues at photon energies of 0.015–15 MeV and selected penetration depths of 1, 20, 40, and 80 mfp. The plotted ratios of the EABF and EBF indicate that they are equivalent for both materials at low energies (le; 40 keV) and high energies (ge; 3 MeV). However, at an intermediate energy, we can see that the EABF was lower than the EBF for breast tissue and higher than the EBF for bone tissue at all penetration depths. This behavior confirms the theory of gamma-ray exposure in air and media. It can be explained by comparing their Z_eq values to that of air, as the equivalent number for air was greater than Z_eq for breast tissue and lower than Z_eq for bone tissue in the entire studied energy range.

Fig. 5.Full-screen View

Comparison of EABF and EBF for breast and bone tissues as a function of photon energy for selected penetration depths.

3.2.5 Geant4 benchmark

We compared our results to those published in the literature [30] for the EBF of water for a point isotropic source of photons with an energy of 0.1 MeV and penetration depths up to 10 mfp. Figure 6 shows that the current computations are in good agreement with the literature. During our simulations, we followed the energy discrimination method with an energy window (ϵ) of 10%, as described elsewhere [28]. This method assumed that a gamma ray having an energy between E₀ and E₀-є can be considered to be uncollided; otherwise, it will be counted as collided. However, the small discrepancies observed, especially for deeper penetration, can be corrected either by using a supercomputer (not laptops, as in our study) or by using particle splitting as a variance reduction technique [30, 31]. These phenomena can be explained by the decrease in photon detection probability at large distances from the source.

Fig. 6.Full-screen View

Geant4-simulated EBF of water (for 0.1 MeV photons) as a function of penetration depth compared to data extracted from the work of Hirayama [30].