## 4 Discussion
Therefore, when treating an HCC patient with \({}^{90}\)Y \(\beta\)-particles, one may wish to set \(D_{tumor}\) to a minimum of 120 Gy. Equation (5) can be rearranged to derive the prescribed treatment activity: \(A(\text{GBq})=D_{tumor}\cdot m_{tumor}(\text{kg})/49.38\) (J/GBq) \(\cdot FU_{tumor}\). Assuming, for example, a tumor mass, \(m_{tumor}\), of 0.52 g (diameter 1 cm), \(FU_{tumor}=1\) and \(D_{tumor}=120\) Gy, Equation (5) would yield a treatment activity of 1.21 MBq (considering \(\rho=1.05\) g/cm\({}^{3}\)). On the other hand, if Equation (7) is used instead of Equation (4) a prescribed activity of 1.62 MBq is obtained. As previously outlined, the cause of this difference is a result of a significant energy deposition outside the sphere (about 26% of the \(\beta\)-particles energy is delivered outside the sphere, as reported in Figure 1). As a consequence,a therapeutic activity of 1.21 MBq would actually correspond to an absorbed dose of about 90 Gy, well below the therapeutic endpoint. As mentioned, the intratumoral injection of \({}^{90}\)Y is likely to pose specific treatment planning issues related to the possibility of treating very small lesions very selectively. In his seminal paper [29], [PERSON] reported the first interstitial use of \({}^{90}\)Y microspheres for the treatment of a rhabdomyosarcoma. A nodule measuring 1.5 cm in diameter was successfully treated with interstitial injection of 185 MBq of microspheres. In another study [30], \({}^{90}\)Y-glass microspheres were injected into predetermined tumor sites using an ultrasound-guided procedure. Tumor size ranged from 1.9 to 8.8 cm, with most lesions being less than 5 cm in diameter [30]. More recently, [PERSON] and co-workers assessed the absorbed doses to small-volumes brain necovatities and surrounding tissues after local \({}^{90}\)Y-DOTATOC injection [5]. A recent review of the literature on the intratumoral treatment with radioactive beta-emitting microparticles can be found in [31]. The evidence from this study suggests that caution must be taken when planning the treatment of very small lesions with \({}^{90}\)Y, implementing the standard analytic approach. This is particularly true when intratumoral administration of \({}^{90}\)Y is performed, as this approach allows sub-centimeters tumors to be selectively treated. In such a scenario, the use of the analytic approach to calculate the therapeutic activity needed to achieve a given tumoricidal endpoint may result in important dose underestimations. On the other hand, Equation (7) can be usefully employed in the treatment planning of spherical lesions of small size (down to 0.5 cm diameter) providing dose estimates in close agreement with Monte Carlo calculations (maximum deviation below 0.5%). Finally, two potential limitations need to be considered. The first is that in the present study tumor masses were modelled as perfect spheres. The extent to which it is possible to extend the present results to irregular spheres (e.g., oblate spheroids) should be further investigated. Another limitation is that in this study we considered a uniform distribution of the radionuclide into the spheres. As a consequence, the findings might not be representative of situations where the radionuclide is not homogeneously distributed in the tumor region.