## 1 Introduction  
Worldwide, the most common causes of death by cancer have had minor changes in the last four decades. Lung, liver, and stomach cancers have been the three leading causes of death by cancer since 1975 [1, 2, 3, 4, 5, 6]. Liver cancer is the sixth most diagnosed malignant cancer in the world, being the third most common cause of death by cancer [4, 5, 6]. Surgery is the first approach with curative intent in primary tumors. Nevertheless, for many patients, there are no curative surgical solutions, resorting in these cases to systemic, local, and regional treatments with palliative intent, such as ablation and chemoembolization [7]. The therapeutic intent in metastatic liver tumors is usually palliative, with surgery and chemotherapy being the main options. Radioembolization (RE) with glass or resin microspheres (MS) loaded with Yttrium-90 (\({}^{90}\)Y)--more recently also produced with Poly L-lactic acid (PLLA) loaded with Holmium-166 (\({}^{166}\)Ho)--has emerged as a local therapeutic approach to primary and metastatic tumorsin the liver [8; 9; 10; 11; 12; 13; 14; 15]. RE combines the embolic action of the MS in the hepatic vasculature with the radiation emitted by the radionuclide in the MS. Thus, MS with a diameter of approximately 20 \(\upmu\)m to 40 \(\upmu\)m are ideal for blocking the blood supply to tumors, providing them with a highly absorbed radiation dose and simultaneously having a tolerable effect on normal liver tissues. In current clinical practice, the RE procedure follows the guidelines of the glass MS manufacturers [10; 16; 17]. In preparation for RE treatment, patients are always evaluated to minimize any significant risk of undesirable extrahepatic radiation. For this reason, all patients undergo diagnostic angiography aiming to study the liver arterial vasculature. During this session, Technetium-99m (\({}^{99\text{m}}\)Tc)-labeled macroaggregated albumin (MAA) is injected into the patient's liver via hepatic arteries to assess the hepatopulmonary shunt fraction (LSF). After injection, patients are immediately submitted to a full-body planar scan by a gamma camera. In some clinical centers, single-photon emission computed tomography (SPECT) images of the voxelized distribution of MAA deposition in the liver are also acquired. Post-treatment positron emission tomography (PET) images acquired from the patients after the administration of the MS loaded with \({}^{99\text{Y}}\) are frequently acquired for activity distribution verification. The calculation of the \({}^{90\text{Y}}\) activity to be administered to the patient's liver, including the planning tumor volume (PTV) and part of the normal liver volume (NLV), follows the manufacturer's protocol, which considers the organ-based (i.e., one compartment) formalism of the Medical Internal Radiation Dose (MIRD) committee of the Society of Nuclear Medicine and Molecular Imaging (SNMMI). It considers only the prescribed absorbed dose (AD) and liver volumes, considering uniform deposition of the MS in the liver [16; 18]. Thus, it does not consider the real specific characteristics of each patient that may influence the deposition of MS in the liver. Therefore, the simple application of the MIRD calculation model is not tailored to the patient; this is the fundamental problem under investigation in this work. Moreover, the European Council Directive 2013/59/EURATOM of 5 December 2013, regarding the optimization of radiotherapy-apeutic exposures, states that the PTV shall be individually planned and their irradiation appropriately verified, taking into account that doses to NLV shall be as low as reasonably achievable and consistent with the intended radiotherapeutic exposure of the PTV [19]. This research work is a step forward in that direction.