## 1 Introduction  
Nowadays, liver cancer still represents a global disease burden due to its extremely aggressive nature and poor survival rate, remaining an important public health issue worldwide [1, 2, 3]. Radioembolization (RE) with Yttrium-90 labelled glass or resin microspheres (\({}^{90}\)Y-\(\upmu\)s) is a well-established therapeutic modality for treating unresectable, chemo-refractory primary and secondary liver malignancies [4, 5]. Patients with liver disease usually have multiple areas of metastatic spread that make difficult defining the tumor and normal parenchymal compartments [6]. The main advantage of RE with \({}^{90}\)Y-\(\upmu\)s is the fact that the radio-labelled particles are injected directly into the target area leading to tumor cells death with the attempt of sparing normal parenchyma. However, microspheres can be transported into the vascular territory of the gastrointestinal organs, resulting in severe damage in up to 5% of RE cases [7]. Thus, this treatment requires accurate planning to ensure a good therapeutic response along with as few side effects as possible [8; 9]. Along these lines, the absorbed dose to the tumor and to the normal liver tissue should be evaluated prior to commencing therapy [10; 11]. In fact, an individualized dosimetry-based treatment planning is expected to significantly improve the clinical efficacy and cost-effectiveness [12; 13]. Response to treatment is no doubt strongly related to the delivered dose distribution and target volume but also on cellular radio-sensitivity and in particular on the amount of hypoxic areas [14]. The current European legislation (EC directive 2013/59 [15]) and the requirement of the US FDA for dosimetry for new radiotherapics has also driven a resurgence in a field that has clinically been considered unnecessary, impractical and is often poorly understood. In the clinical practice, dosimetric studies are generally performed using pre-treatment Tecnettium-99m-macroaggregated albumin (\({}^{99\text{m}}\)Tc-MAA) images, potentially useful for determining the \({}^{90}\)Y microspheres activity to be administered. Indeed, pre-treatment angiographic evaluation combined with \({}^{99\text{m}}\)Tc-MAA is required in order to assess the vascular anatomy of the target lesions and normal liver. Given the similarities in sizes of \({}^{90}\)Y microspheres (20 to 40 \(\upmu\)m) and \({}^{99\text{m}}\)Tc-MAA (10 to 50 \(\upmu\)m, average particle size 35 \(\upmu\)m), the pattern of \({}^{99\text{m}}\)Tc-MAA deposition is assumed to be representative of the microsphere biodistribution during \({}^{90}\)Y-RE. When used in conjunction with dosimetric models, \({}^{99\text{m}}\)Tc-MAA biodistribution images provide absorbed doses estimates in tumor lesions and normal liver parenchyma. In fact, absorbed doses can be mathematically determined by converting quantitative \({}^{99\text{m}}\)Tc-MAA SPECT data to the absolute \({}^{90}\)Y activity maps. These maps, in turn, can be converted to absorbed doses using the available dose calculation algorithms (e.g., MIRD [16], partition [13], kernel convolution [17] or local energy deposition [17; 18; 19] models). Even if the computational improvements of quantitative software related to SPECT image reconstruction process have been adopted, one of the principal limitations of SPECT/CT acquisition is its resolution, and consequently the correctness of dosimetric evaluations that can be obtained. \({}^{90}\)Y-PET/CT represents the best choice for accurate patient-specific post-treatment dosimetry purposes, thanks to its superior resolution than SPECT/CT system and to the \({}^{90}\)Y decay characteristics. At present,the predictive value of \({}^{99\text{m}}\)Tc-MAA SPECT based dosimetry is still subject of debate in the scientific literature [20]. Of note, recent research reported patient population tumor dose constraints [21] providing support that dose-volume histograms in \({}^{90}\)Y SIRT could play a role in evaluative dosimetry.