## 3 Results
Significant deviations were found between MC calculated dose values and those obtained using the analytic formulation when the lesion diameter drops below 2 cm (Figure 4, inset). Figure 4: Absorbed dose per unit administered activity (GBq), for \(\rho=1.00\) g/cm\({}^{3}\) (water density). Comparison between MC approach (MCNP) and the analytical formulation (AA) described by Equation (4). Inset: detail in the range 0 cm–1 cm lesion size. Figure 3: Absorbed dose per unit administered activity (GBq), for \(\rho=1.05\) g/cm\({}^{3}\) (liver density). Comparison between MC approach (MCNP) and the analytical formulation (AA) described by Equation (4). Inset: detail in the range 0 cm–1 cm lesion size. Ultimately, Tables 2 and 3 compare absorbed dose values per GBq of administered activity obtained with MC calculations (\(D_{MCNP}\)) and with the analytical approach reported in Equation (4), (\(D_{AA}\)). The percentage differences between the two methods (last column of both tables, \(\Delta\)) is also reported, calculated as \(100\cdot(D_{MCNP}-D_{AA})/D_{AA}\). The difference in absorbed dose values is within \(\sim\)10% as long as the diameter of the lesion exceeds 2 cm. The two calculation approaches deviate significantly when the lesion size drops below 2 cm, due to significant energy deposition outside the sphere. This is consistent with the maximum particle range in tissue for \({}^{90}\)Y (about 11 mm). In this case, for water (liver) density MC calculations provide absorbed doses \(-9.3\%\) (\(-9.6\%\)), \(-27.8\%\) (\(-26.7\%\)), \(-56.7\%\) (\(-55.4\%\)) lower than those obtained using the analytic formulation described by Equation (4) for tumor diameter of 2 cm, 1 cm and 0.5 cm, respectively (Tables 2 and 3). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Lesion Diameter** & **Mass (kg)** & **Dose/Particle (Gy/p)** & **MCNP (Gy/GBq)** & **AA (Gy/GBq)** & \(\Delta\) \\ \hline
20 cm & 4.40 & 3.38 \(\times 10^{-14}\) & \(1.12\times 10^{1}\) & \(1.14\times 10^{1}\) & \(-1.8\%\) \\ \hline
10 cm & \(5.50\times 10^{-1}\) & \(2.68\times 10^{-13}\) & \(8.91\times 10^{1}\) & \(9.10\times 10^{1}\) & \(-2.1\%\) \\ \hline
8.0 cm & \(2.81\times 10^{-1}\) & \(5.21\times 10^{-13}\) & \(1.73\times 10^{2}\) & \(1.78\times 10^{2}\) & \(-2.5\%\) \\ \hline
6.0 cm & \(1.19\times 10^{-1}\) & \(1.23\times 10^{-12}\) & \(4.08\times 10^{2}\) & \(4.21\times 10^{2}\) & \(-3.1\%\) \\ \hline
5.0 cm & \(6.87\times 10^{-2}\) & \(2.11\times 10^{-12}\) & \(7.01\times 10^{2}\) & \(7.28\times 10^{2}\) & \(-3.7\%\) \\ \hline
4.0 cm & \(3.52\times 10^{-2}\) & \(4.08\times 10^{-12}\) & \(1.36\times 10^{3}\) & \(1.42\times 10^{3}\) & \(-4.2\%\) \\ \hline
3.0 cm & \(1.48\times 10^{-3}\) & \(9.51\times 10^{-12}\) & \(3.16\times 10^{3}\) & \(3.34\times 10^{3}\) & \(-6.2\%\) \\ \hline
2.0 cm & \(4.40\times 10^{-3}\) & \(3.09\times 10^{-11}\) & \(1.03\times 10^{4}\) & \(1.14\times 10^{4}\) & \(-9.6\%\) \\ \hline
1.0 cm & \(5.50\times 10^{-4}\) & \(2.01\times 10^{-10}\) & \(6.67\times 10^{4}\) & \(9.10\times 10^{4}\) & \(-26.7\%\) \\ \hline
0.5 cm & \(6.87\times 10^{-5}\) & \(9.77\times 10^{-10}\) & \(3.25\times 10^{5}\) & \(7.28\times 10^{5}\) & \(-55.4\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Dose per unit activity calculated with MCNP4C for spherical lesions of different size uniformly filled with \({}^{90}\)Y. The lesions are assumed to have a density of \(\rho=1.05\) g/cm\({}^{3}\) (liver density). The same quantity (dose per unit activity) has been calculated using the analytic approach (AA) described by Equation (4). The last column of the table (\(\Delta\)) shows the percentage deviation between the two methods, calculated as \(100\cdot(D_{MCNP}-D_{AA})/D_{AA}\).