## 4 Discussion  
Molecular radiotherapy with \({}^{90}\)Y has received much attention in the past two decades and the dosimetry of small lesions remains a key issue. The maximum range of \({}^{90}\)Y \(\beta\)-particles is 11 mm in tissue, while its average energy \(\beta\)-particles has a range of about 2.5 mm. It is worth noting that the penetration depth of the high-energy \({}^{90}\)Y \(\beta\)-particles is a critical component of this radionuclide's success in liver radioembolization, allowing for high dose delivery into the tissue between embolized capillaries. Previous work has been dedicated to the assessment of absorbed fractions for electrons and beta particles in spheres [11; 12; 13] and ellipsoidal volumes [14] of various sizes. The present study focused on the assessment of the absorbed dose per unit activity in a scenario of percutaneous ablation of HCC through the intratumoral injection of \({}^{90}\)Y in lesions of varying size. A simplified model tumor areas was implemented into MCNP4C MC code with the aim to determine the absorbed dose to the lesion when the tumor mass is uniformly filled with \({}^{90}\)Y. Spherical lesions of different size (diameter in the range 0.5-20 cm) were simulated for two different densities: \(\rho=1.00\) g/cm\({}^{3}\) (water density) and \(\rho=1.05\) g/cm\({}^{3}\) (liver density). In both scenarios, lesions were assumed to be immersed in a semi-infinite medium with the same density of the lesion. The problem of beta dosimetry in small lesions is not new [11; 12; 13]. Absorbed fractions calculated in this study are hardly comparable with those assessed in [11; 13] as the authors performed calculations assuming monoenergetic electrons, while in this study the full \({}^{90}\)Y spectrum was implemented. Furthermore, in [13] the authors used MCNP4B, while in this work MCNP4C was used instead. The widely used analytic approach described by Equation (4) and MCNP calculations provide results in close agreement (within 10%, no matter the density of the lesion) as long as the lesion diameter exceeds 2 cm. When the lesion diameter drops below 2 cm, significant differences were obtained between MC calculations and the analytic approach (i.e., deviations \(>\)10%). As a general rule, the analytic described by Equation (4) tends to overestimate the absorbed dose in small lesions, as the basic assumption of the model is that \(\beta\) radiation is fully absorbed by the tumor or tissue where the decay occurs. When the radius of the tumor is smaller than the maximum range of the \(\beta\) radiation in the medium, a significant amount of the energy is delivered out of the lesion, thus providing smaller absorbed dose values. Presently, despite the availability of different dose algorithms, the analytic algorithm described by Equation (4) is still widely used to assess the absorbed dose in tumor and in the liver compartment at the clinical level. As a general rule, the size of normal liver is large and Equation (4) provides accurate dose estimates (provided that accurate input parameters are introduced, among which the fractional uptake of the target). However, when this approach is applied to assess the absorbed dose to small tumor masses (i.e., approximately below 2 cm diameter) inaccurate dose estimates can be obtained. In addition, the analytic algorithm is safely used for treatment planning with glass microspheres. The foundational principle is based on Equation (4), which describes the average dose in a tissue volume as a function of \({}^{90}\)Y activity. During treatment planning, this equation can be solved for the treatment activity \(A_{0}\). The results obtained in the present study raise questions as to whether Equation (4) should be used to assess the prescribed \({}^{90}\)Y activity in order to achieve a given tumoricidal endpoint in small liver lesions. This is especially true when intratumoral injection of \({}^{90}\)Y is performed. For example, for HCC 120 Gy should be considered to be a reasonable minimum target dose [21; 27; 28].