## 2 Materials and Methods
### Image Segmentation, Dose Calculation and Data Comparison  
After SPECT/CT reconstruction of the AbdoMan phantom, images were exported in DICOM format from the xSPECT software and imported into the MIM software (v6.1.7 Software, Inc., Cleveland, OH, USA) where volume of interest (VOI) related to the liver compartment and spherical inserts were drawn. SPECT/CT images along with the contoured structures were exported in DICOM format and used as input for a home-made tool developed in MATLAB (R2014b, Natick, MA, USA) that converted reconstructed SPECT activity images into absorbed dose maps. Dose volume histograms (DVH) for the defined structures were also calculated. Dose calculation was performed using both the dose-kernel method (DKM) and the local energy deposition method (LDM). Concerning the DKM, the MATLAB tool tool allows loading the dose kernel generated using EGSnrc Monte Carlo code for the specific dimension of cubic voxels of investigated SPECT [27]. The convolution algorithm implemented in the MATLAB tool allows the calculation of the absorbed dose from \({}^{90}\)Y by the convolution of the 3D activity concentration matrix \(A(x,y,z)\) with the 3D dose kernel for \({}^{90}\)Y, \(K(x,y,z)\), according to the following equation:  
\[D(x,y,z)=\frac{1}{\lambda}[K(x,y,z)\otimes A(x,y,z)] \tag{1}\] \[=\frac{1}{\lambda}\int\int\int K(x-x^{\prime},y-y^{\prime},z-z^{ \prime})\cdot A(x^{\prime},y^{\prime},z^{\prime})dx^{\prime}dy^{\prime}dz^{\prime}\] \[\textit{discrete space}\rightarrow\frac{1}{\lambda}\sum_{x^{ \prime}}\sum_{y^{\prime}}\sum_{z^{\prime}}\left[K(x-x^{\prime},y-y^{\prime},z- z^{\prime})\cdot A(x^{\prime},y^{\prime},z^{\prime})dx^{\prime}dy^{\prime}dz^{ \prime}\right]\]  
where \(A(x,y,z)\) is the 3D activity concentration matrix obtained by product of the reconstructed SPECT phantom data and the SPECT calibration factor, \(K(x,y,z)\) is the Monte Carlo derived 3D dose-kernel and \(\lambda=ln(2)/T_{1/2}\) is the decay constant of \({}^{90}\)Y. \({}^{90}\)Y resin/glass microspheres are a permanent implant with a relative distribution that does not change after the infusion, having a negligible biological elimination. For the LDM, the energy released by charged particles is assumed to be locally absorbed within the same voxel as the decay. Moreover, the full width at half maximum of the point spread function is assumed be larger than the dose kernel. Hence, the dose at a single voxel can be calculated from the average energy (\(E_{avg}\)) released per decay of \({}^{90}\)Y based on the probability density function \(\Psi(E)\) for emission:  
\[E_{avg}=\int_{0}^{\infty}E\times\Psi(E)dE=0.935\textit{ MeV}=1.498\times 10^{-13}\textit{ J} \tag{2}\]  
In addition, the committed total energy released per unit volume, \(E_{com}\), by an initial activity concentration \(A_{0}\) [MBq/mL] in a voxel can be estimated as:  
\[E_{com}=A_{0}E_{avg}\int_{0}^{\infty}e^{-\lambda\cdot t}dt=\frac{A_{0}}{ \lambda}\times(1.498\times 10^{-13})(\frac{\textit{J}}{mL}) \tag{3}\]  
Being \(\lambda\) the decay constant of \({}^{90}\)Y since microspheres are a permanent implant and there is no biological removal of the radiopharmaceutical. The energy deposited into the voxel, \(E_{com}\) can be converted into the absorbed dose, \(D_{voxel}\), using the voxel density, \(\rho\) (kg/mL):  
\[D_{voxel}=\frac{E_{com}\ (\textit{J}/mL)}{\rho\ (kg/mL)} \tag{4}\]  
Equation (4) then becomes:\[D_{voxel}=\frac{A_{0}}{\lambda}\frac{(1.498\times 10^{-13})}{\rho}(Gy) \tag{5}\]  
And ultimately [28]:  
\[D_{voxel}(Gy)=A_{0}(\frac{Bq}{mL})\frac{(4.986\times 10^{-8})(J\cdot s)}{\rho(kg/ mL)} \tag{6}\]  
Using Equations (1) and (6), the MATLAB tool generates two dose matrixes in *.mat format as well as in RTDOSE-DICOM format. The obtained dose matrixes have been compared each other by means of the 3D \(\gamma\)-index method [29] using the 2%-2 mm, 1%-1 mm criteria and the one obtained by means of the DKM was used as reference in the comparison.