## 2 Materials and Methods
### Buildup Factors  
For nine standard photon energies, the equivalent atomic number (\(Z_{\rm{eq}}\)), the five GP fitting parameters, the energy absorption buildup factor (EABF), and the exposure buildup factor (EBF) related to the considered materials were given for photon energy of 30, 80 and  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & & & & & & & & \\ \hline
**Material** & **d (g/cm\({}^{3}\)/g)** & **30** & **80** & **150**  
150 keV in Table 3. With seventeen standard values, the EBF and EABF were approximated to photon penetration depths up to 40 mfp. These parameters were calculated in three stages: (i) calculating the equivalent atomic number, \(Z_{eq}\); (ii) evaluating the GP fitting parameters; and (iii) estimating the EABF and EBF for all tissue and phantom materials evaluated. The atomic number of an element is synonymous with the \(Z_{eq}\) of a composite material. It describes the material's property in the same way that the atomic number describes an element's property in terms of radiation interaction. It is a weighted average of the number of electrons per atom in a multi-element material. To assess \(Z_{eq}\) of the composite materials under consideration, the Total and Compton partial interaction coefficients, \((\mu_{T}/\rho)\) and \((\mu_{C}/\rho)\) (both given in cm\({}^{2}\)/g), were calculated using the WinXCom computer code for the same photon energy range previously described. The ratio \(R=(\mu_{C}/\mu_{T})\) of each material is then calculated and matched to the corresponding ratio of elements up to the heaviest element in composite materials at standard energies. The value of R for all tissue and phantom materials considered in the study, however, did not match that of any element, but rather fell between ratios of two successive elements. As a result, the following expression was used to interpolate their \(Z_{eq}\)[5,14]:  
\[Z_{eq}=\frac{Z_{1}(Log(R_{2})-Log(R))+Z_{2}(Log(R)-Log(R_{1}))}{Log(R_{2})-Log( R_{1})} \tag{2}\]  
\(R_{1}\) and \(R_{2}\) are the ratios of the two successive elements of atomic numbers \(Z_{1}\) and \(Z_{2}\), respectively, within which R falls at corresponding energy.  
The GP method requires five fitting parameters to evaluate photon buildup factors. These coefficients (\(b\), \(c\), \(a\), \(X_{k}\), and \(d\)) are dependent on \(Z_{eq}\) and photon energy. These coefficients are provided in the ANSI report [10] for 23 elements and 25 standard photon energies. However, because the \((\mu_{C}/\mu_{T})\) for the materials considered in this study did not correspond to any of the 23 elements, their GP fitting coefficients were interpolated using the logarithmic interpolation formula given below:  
\[F(a,b,c,d,X_{k})=\frac{F_{1}(Log(Z_{2})-Log(Z_{eq}))+F_{2}(Log(Z_{eq})-Log(Z_{ 1}))}{Log(Z_{2})-Log(Z_{1})} \tag{3}\]  
where \(F_{1}\) and \(F_{2}\) are the values of GP fitting parameters obtained from ANS data base corresponding to the atomic numbers \(Z_{1}\) and \(Z_{2}\), respectively.  
According to the GP fitting method, the buildup factor can be expressed in the following way [15; 16; 17]:  
\[\frac{B(X)-1}{b-1}=\left\{\begin{array}{ccc}\frac{K^{X}-1}{K-1}&if&K\
eq 1\\ X&if&K=1\end{array}\right. \tag{4}\]  
where \(X\) and \(K\) are the depth expressed in mean free paths (1 mfp = \(1/(\mu\times\rho)\), with \(\mu\) and \(\rho\) correspond to the linear attenuation coefficient and the atomic density, respectively) and the geometric progression term.  
For \(X\leq 40\) mfp, we have [17]:  
\[K(E,x)=cx^{a}+d\frac{\tanh(x/X_{k}-2)-\tanh(-2)}{1-\tanh(-2)} \tag{5}\]  
where, \(a\), \(b\), \(c\), \(d\), and \(X_{k}\) are the five fitting parameters.