## 2 Materials and Methods
### Dosimetry with \({}^{90}\)Y  
\({}^{90}\)Y disintegrates by \(\beta\) emission mainly (99.983% [16]) to the stable \({}^{90}\)Y ground state level. A weak beta branch occurs to the 1760 keV excited level which decays by a E0 gamma transition [16]. This 0\({}^{+}\)-0\({}^{+}\) transition is followed by the emission of two gammas, or an electron-positron pair, or internal conversion. The adopted half life of \({}^{90}\)Y ground state is 64.041(31) h or 2.6684(13) days [16]. Among the radionuclides used in clinical practice, \({}^{90}\)Y has attractive physical and radiobiologic features that make this radionuclide suitable for a loco-regional therapeutic option. The high-energy \(\beta\)-particles (maximum energy 2278.7(16) keV, average energy 926.7(8) keV [16]) and their penetration depth (maximum particle range in tissue, 11 mm; range in tissue after which 50% of the energy particles is transferred, 4 mm) allows high radiation doses to be selectively delivered to the target area, while sparing surrounding tissues and normal organs. In particular, the penetration depth of the high-energy \(\beta\)-particles is a key element of this radionuclide's success in radioembolization, allowing for high-dose deposition into the tissues between embolized capillaries. In the traditional dose calculation formalism (after locoregional administration of \({}^{90}\)Y) two important simplifying assumptions are generally made:  
* \(\beta\) radiation released from \({}^{90}\)Y within a given organ is fully absorbed by that organ. In most cases, this assumption is supported by the average 4 mm \({}^{90}\)Y \(\beta\) range in tissue. * permanence of \({}^{90}\)Y in the area where they have been delivered (i.e., no migration of the radiopharmaceutical outside the tumor region). Combining these two assumptions allows for easy calculation of average absorbed dose to an organ of interest on a macroscopic scale. The calculation, carried out using most up-to-date nuclear data for \({}^{90}\)Y [16], is illustrated below and it is generally referred to as the MIRD approach [17]:  
\[E_{agg}=\int_{0}^{\infty}E\varphi(E)dE=926.7\;\mathrm{keV}=1.4847\times 10^{-13} \;\mathrm{J} \tag{1}\]  
\[E_{tot}=A_{0}E_{avg}\int_{0}^{\infty}e^{-\lambda t}dt=\frac{A_{0}}{\lambda}(1. 4847\times 10^{-13}\;\mathrm{J})=A_{0}\cdot k \tag{2}\]  
where \(E_{avg}\) is the average energy released per decay of \({}^{90}\)Y based on the probability density function \(\varphi(E)\) for emission, \(\lambda\) is the \({}^{90}\)Y decay constant based on the half-life of 64.041(31) hours, and \(k\) a constant term. \(A_{0}\) is the activity present in the organ in GBq and \(E_{tot}\) is the total energy released by \(A_{0}\) from the time that it is infused until it has fully decayed. Assuming that all of the energy of the \(\beta^{-}\) decay is absorbed in the volume where the decay occurs, the constant term, \(k\), can be calculated taking the given physical values and their statistical uncertainties:  
\[k=\frac{1.4847\times 10^{-13}\,\mathrm{J}}{\lambda}=\frac{1.4847\times 10^{-13} \,\mathrm{J}\times 230547\,\mathrm{s}}{0.69315}\times\frac{10^{9}\,\mathrm{ dis/s}}{\mathrm{GBq}}=49.38(5)\ (\mathrm{J/GBq}) \tag{3}\]  
The constant factor 49.38 (J/GBq) is the energy released per unit activity of \({}^{90}\)Y. The adopted uncertainties on the nuclear data [16] reported in Equation (4) lead to a relative standard uncertainty of 0.1% on the constant term, i.e., 49.38(5) (J/GBq), in line with [18]. Finally, the absorbed dose \(D\) (expressed in Gy) can be obtained by dividing the delivered energy, \(E_{tot}\), by the mass of the target region \(m\) (expressed in kg):  
\[D_{avg}\ (\mathrm{Gy})=\frac{A_{0}\ (\mathrm{GBq})\times 49.38(5)\ (\mathrm{J/GBq})}{m\ ( \mathrm{kg})} \tag{4}\]  
Of note, the same formula with a slightly different constant term, \(k\), is reported in other publications (e.g., 49.98 [17], 49.67 [19]).