## 2 Materials and Methods
### BUF Computation  
The computation of the buildup factor followed the three steps of (i) the calculation of the equivalent atomic number(\(Z_{eq}\)), (ii) the interpolation of the fitting parameters from existing data and (iii) the derivation of the BUF value.  
Based on the total and the Compton attenuation coefficients, (\(\mu_{T}/\rho\)) and (\(\mu_{C}/\rho\)) (both given in cm\({}^{2}\)/g) provided by the NIST website [19], for a photon energy from 0.15 to 15 MeV (17 standard values) and an atomic number from 1 to 92, we calculated \(Z_{eq}\) according to the following equation [20]:  
\[Z_{eq}=\frac{Z_{1}(\log R_{2}-\log R)+Z_{2}(\log R-\log R_{1})}{\log R_{2}- \log R_{1}} \tag{1}\]  
where \(R\) represents the fraction (\(\mu_{C}/\mu_{T}\)) of the compound in consideration and \(R_{1}\) and \(R_{2}\) are the fractions of the two nearest elements with atomic numbers \(Z_{1}\) and \(Z_{2}\). Since the atomic number for composite materials is not represented by a single number across an entire energy range similarly to pure elements, they can be represented by \(Z_{eq}\).  
On the other hand, the five GP-fitting parameters (\(a,b,c,d\) and \(X_{k}\)) are dependent on \(Z_{eq}\) and photon energy. Hence, we calculated each parameter using the data provided by the ANSI/ANS-6.4.3-1991 report for 23 elements and 25 photon energies using the following logarithmic interpolation formula:  
\[F=\frac{F_{1}(\log Z_{2}-\log Z_{eq})+F_{2}(\log Z_{eq}-\log Z_{1})}{\log Z_{2} -\log Z_{1}} \tag{2}\]  
where \(F_{1}\) and \(F_{2}\) represents the values of GP-fitting parameters corresponding to \(Z_{1}\) and \(Z_{2}\), respectively. \(F\) can be any of the five parameters. Eventually, the GP form of the BUF as a function of depth, expressed in mean free paths \(X\) (defined as the inverse of the linear attenuation coefficient), can be expressed as follows:  
\[\frac{B(X)-1}{b-1}=\left\{\begin{array}{ccc}\frac{K^{X}-1}{K-1}&if&K\
eq 1\\ X&if&K=1\end{array}\right. \tag{3}\]  
where \(K\) represents the geometric progression term. Here, we can see the reason for the GP formula nomination, as the second term of Equation (3) represented the sum of \(K^{n}\) for \(n\) varying from 0 to \(X\). For \(X\leq 40\) mfp, \(K\) can be written in the following way:  
\[K(X)=cX^{a}+d\frac{\tanh(X/X_{k}-2)-\tanh(-2)}{1-\tanh(-2)} \tag{4}\]  
For a fixed material and photon energy, the BUF computation procedure was based on the calculation of \(Z_{eq}\), using the Equation (1), followed by the interpolation of the GP-fitting parameters (\(a,b,c,d\) and \(X_{k}\)) calculated using Equation (2) and plugged into Equations (3) and (4).  
During this work, we computed BUF for 5 tissues and 30 TMM candidates. Table 1 illustrated the elemental composition of the studied materials and tissues. For each stud  
Figure 1: Typical schematic view of PDD calculation for a linear accelerator photon beam in a water phantom. The different parts of the head Linac and the voxels used to score the dose within the box are shown.  
ied case, the photon energy range of 0.15-15 MeV and the penetration depth range of 0.5-40 mfp were represented by 17 standard energies and 17 standard depths, resulting in 289 values of BUF. More explanations were given within our previous work [13].