## 1 Introduction  
TransArterial RadioEmbolization (TARE), also known as Selective Internal Radiation Therapy (SIRT), is a well-established technique used in order to treat unresectable primary and metastatic liver tumors. TARE involves the direct infusion of \({}^{90}\)Yttrium (\({}^{90}\)Y)-labelled microspheres into the tumor via a hepatic artery catheter [1]. \({}^{90}\)Y is preferentially taken up by hyper vascular tumors, such as hepatocellular carcinoma (HCC), which receive most of their blood supply from the hepatic artery, as opposed to normal liver parenchyma, which is supplied by the portal system [2]. Radioembolization consists of a planning procedure (using a surrogate gamma-emitting radionuclide) followed by a treatment procedure (using the radioactive microspheres) [3; 4]; Technetium-99m Macro-Aggregated Albumin (\({}^{99}\)mTcMAA) is used as an imaging surrogate of radioactive microspheres, and it is administered at the catheter intended for treatment with microspheres. Since arteriovenous shunts may provide a direct vascular pathway to the lungs for \({}^{90}\)Y microspheres, causing, in certain conditions in the lungs, irradiation [5], the distribution of \({}^{99}\)Tc-MAA is used to calculate the predicted Lung Shunt Fraction (LSF). Radiation-induced Pneumonia (RP) is in fact one of the major risks of radioembolization procedures. The recommendations, expressed as LSF, do not provide direct information on doses absorbed in the lungs. However, the threshold doses are based on a maximum dose of 30 Gy to a lung mass of 1.0 kg for a single treatment and 50 Gy for multiple treatments [6]. These values were set because patients who received a Lung Mean Dose (LMD) greater than 30 Gy for a single treatment or a cumulative lung dose greater than 50 Gy for multiple treatments developed RP [7; 8]. In order to calculate the absorbed dose, TARE uses the Medical Internal Radiation Dose (MIRD) formalism, whereby all activity is localized within the volume of the perfused liver and the administered activity is converted to the absorbed dose by assuming that 1 GBq of administered activity per kg of water-equivalent tissue (liver) mass gives an absorbed dose of 49.38 \(\pm\) 0.5 Gy, usually rounded up to 50 Gy (Equation (1)). \[\mathrm{D[Gy]}=50[\mathrm{J}/\mathrm{GBq}]\times\left(\frac{\mathrm{A}[ \mathrm{GBq}]}{\mathrm{m[kg]}}\right) \tag{1}\]  
In the presence of lung shunting, if the total activity is divided between the liver and lungs volumes, the dose to the lungs can be calculated directly from Equation (1), knowing the lungs' mass and the partitioned activity of the lungs (Equation (2)). \[\mathrm{D[Gy]}=50[\mathrm{J}/\mathrm{GBq}]\times\left(\frac{\mathrm{A_{total} [\mathrm{GBq}]}}{\mathrm{m_{lung}[kg]}}\right)\times\mathrm{LSF[\%]} \tag{2}\]  
Traditionally, the MIRD scheme has been applied to organs and sub-organs with the assumption of uniform activity distributions [9; 10]; within each organ or major component of an organ, the source was considered to be uniformly distributed. In general, both the tumor and the normal tissues acquire non-uniform activity distributions. More accurate and sophisticated methods to better assess the dose distribution in \({}^{90}\)Y-TARE have been based on this formalism. Several voxel-based dosimetry models have been developed using activity concentrations from treatment imaging including the Local Deposition Model (LDM) [11; 12], Dose Point Kernel convolution (DPK) [13; 14], Voxel S-Value (VSV) kernel convolution [15; 16], and collapsed cone convolution [17; 18]. However, all these methods assume that \({}^{90}\)Y microspheres release the dose in a water-equivalent medium (i.e., the liver). Since lung density is quite different from liver density, the lung's absorbed dose could vary considerably, especially at the liver/lungs interface.