## 1 Introduction  
Hepatocellular carcinoma (HCC) is the most common primary liver malignancy and today multiple treatment modalities exist [1]. Due to the lack of effective systemic therapies for HCC, researchers have been investigating the use of locoregional tumor control with \({}^{90}\)Y radioembolization since the 1960s. Today radioembolization (or Selective Internal Radio Therapy, SIRT) is an established and effective treatment for liver malignancies based on trans-arterial infusion of \({}^{90}\)Y-laden microspheres [2; 3; 4]. At present, there are two clinically available microsphere devices in which \({}^{90}\)Y is incorporated: one with microspheres made of glass (TheraSphere; MDS Nordon, Ottawa, ON, Canada) and the other with microspheres made of resin (SIR-Spheres; Sirtex Medical, Sydney, Australia). Once administered, the spheres remain in the liver as a permanent implant. Radiation dose distributions arising from intrahepatic arterial infusion of \({}^{90}\)Y microspheres have been investigated by a number of authors in the past. In recent years a number of studies have addressed the problem of dosimetry in therapies based on the use of intratumoral administration of \({}^{90}\)Y-conjugates by percutaneous puncture. This technique has been successfully applied to patients treated with \({}^{90}\)Y-labeled [DOTA\({}^{0}\)-D-Phe\({}^{1}\)-Tyr\({}^{3}\)]octreotide (\({}^{90}\)Y-DOTATOC) for malignant gliomas [5; 6; 7]. In fact, based on the clinical experience gained in liver radioembolization, it is reasonable to assume that percutaneous ablation of HCC through the intratumoral injection of an appropriate activity of \({}^{90}\)Y has the potential to reduce drastically the chances of local recurrence. In this context, there is growing interest in the development of new intratumoral procedures for HCC throughout a localized administration of \({}^{90}\)Y in the form of microspheres mixed with biocompatible compounds [8]. As a general rule, intratumoral administration of radionuclides raises questions about the dosimetry of small lesions as this approach allows sub-centimeters tumors to be selectively treated [9; 10]. To date, there is no simple way to assess exactly the absorbed dose to tumours and normal liver when \({}^{90}\)Y is administered. This is because \({}^{90}\)Y only emits pure beta radiation with limited penetration range in tissue. As a consequence, the delivered dose is highly dependent on the distribution of the radionuclide and the tumor mass. Calculation of the radiation dose to tumors may require consideration of the losses of electron energy. In the traditional macrodosimetric approach, all of the electron energy emitted in a source region is assumed to be absorbed in that region. This is quite reasonable for most situations because the range of most electrons in body tissues is small compared to the size of most source regions. However, when the tumor size is small this assumption is no longer true and the absorbed fraction may be significantly less than unity. Some preliminary work dedicated to the assessment of the absorbed fractions for beta particles in spherical regions of different sizes was carried out in the early 1990s by [PERSON] [11] and [PERSON] [12]. In a major advance in 2000 [PERSON] and [PERSON] [13] reevaluated and updated absorbed fractions in spherical regions comparing different Monte Carlo codes, while in 2010 [PERSON] and colleagues [14] calculated absorbed fractions for electrons in ellipsoidal volumes. The aim of the present paper is to assess the absorbed dose of \({}^{90}\)Y after percutaneous administration in small spherical lesions proposing a practical analytical formula that incorporates the absorbed fractions for \({}^{90}\)Y. Of note, in [13] absorbed fractions were assessed for monoenergetic electrons using MCNP version 4B, while in this study absorbed dose fractions were calculated implementing the full \({}^{90}\)Y spectrum and using an upgraded version of MCNP (version 4C).