## 3 Results  
Figure 1 shows the fraction of \({}^{90}\)Y absorbed dose into the tumor as a function of the tumor size, obtained from MCNP simulations. The \({}^{90}\)Y beta spectrum implemented in the model is also reported in the same figure. Calculations were performed both for water spheres (\(\rho=1.00\) g/cm\({}^{3}\)) and for spheres made of liver tissue (\(\rho=1.05\) g/cm\({}^{3}\)). In both cases, when the lesion diameter drops below 2 cm, a great amount of the \(\beta\) particle energy is delivered outside the sphere. Consistently, the delivered energy per unit activity, \(k(r)\), shows the same trend (Figure 2) confirming that when the tumor size is small such term deviates significantly from the value 49.38 (J/GBq), considered in Equation (4). In order to use information reported in Figure 2 at the clinical level, \(k(r)\) data obtained from MC calculations were fitted with the following function:  
\[k(r)=k_{0}+A\cdot(1-exp(-\frac{r}{a}))+B\cdot(1-exp(-\frac{r}{b})) \tag{6}\]  
where \(r\) is the lesion radius in \(cm\) (assuming spherical tumors) and \(k_{0},A,a,B,b\) are parameters determined by the fit (reported in Table 1 both for water and liver density). An \(r^{2}\) = 0.999 was obtained form the fit, both for water and liver density. Furthermore, goodness-of-fit was also assessed through the analysis of residuals (Figure 2B), which shows maximum deviations below 0.2 between calculated and fitted data, confirming the accuracy of the fit. Based on the fitting function described in Equation (6), Equation (4) can be rewritten in the following form:  
\[D_{avg}(r)=\frac{A_{0}\cdot k(r)}{m} \tag{7}\]  
Figure 1: Fraction of \({}^{90}\)Y absorbed dose into the tumor as a function of the tumor size. Simulations were performed for \(\rho=1.00\) g/cm\({}^{3}\) (water density) and \(\rho=1.05\) g/cm\({}^{3}\) (liver density). Inset: \({}^{90}\)Y beta spectrum implemented in the model. For a given activity \(A_{0}\), Equation (7) can be used to accurately calculate the absorbed dose for very small lesions (down to 0.5 cm diameter). The absorbed dose to lesions calculated using Equation (7) provides results in good agreement with MC calculations (maximum deviation below 0.5%). As expected, when ideally \(r\rightarrow\infty\) Equation (7) reduces to Equation (4). Of note, the energy per unit activity, \(k(r)\), obtained from Equation (6) when \(x\rightarrow\infty\) is 49.35 (J/GBq), against the accepted value of 49.38 (J/GBq) derived from Equation (4) (0.06% deviation). Figure 3 compares absorbed doses per unit activity (Gy/GBq) calculated with Monte Carlo with those obtained using the analytic approach described by Equation (4), for spherical lesions of different size and for \(\rho=1.05\) g/cm\({}^{3}\). The same results for \(\rho=1.00\) g/cm\({}^{3}\) are shown in Figure 4. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & **ValueW** & **urelW\%** & **ValueL** & **urelL\%** \\ \hline \(k_{0}\) & \(-11.883\) & \(1.10\) & \(-12.999\) & \(1.00\) \\ \hline A & \(4.416\) & \(1.15\) & \(4.654\) & \(1.10\) \\ \hline a & \(2.190\) & \(0.90\) & \(1.909\) & \(1.00\) \\ \hline B & \(56.820\) & \(0.05\) & \(57.701\) & \(0.05\) \\ \hline b & \(0.290\) & \(0.15\) & \(0.271\) & \(0.10\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Fitting parameters of Equation (6) with relative standard uncertainties, for water and liver density (denoted with superscripts W and L, respectively). \(R^{2}=0.999\) in both cases. Figure 2: (**A**) Delivered energy per unit activity of \({}^{90}\)Y, \(k(r)\), calculated with MCNP4c as a function of the tumor diameter. Data were fitted with Equation (6). (**B**) Analysis of residuals. In addition Figure 4 reports absorbed doses calculated using the well established Olinda/EXM code, developed by the RAdiation Dose Assessment Resource (RADAR) Task Group of the Society of Nuclear Medicine [26]. As illustrated in Figure 4, absorbed dose values calculated with MC approach concur well with those obtained using Olinda/EMX.