## 2 Materials and Methods
### Patient Studies
#### 2.2.5 Proposed Methodology for Optimization of the Activity to Be Administered  
Ionizing radiation therapies such as external radiotherapy and RE aim to achieve treatment efficacy and safety. The goal is to deliver a sufficiently high AD to the PTV while controlling the AD to the NLV to avoid and minimize the possibility of hepatic toxicity. This methodology is based on the fact that the healthy liver tissue is very sensitive to radiation and because the radio-sensitivity of tumor cells (i.e., primary hepatic carcinoma or liver metastases) differs for different tumor types [10; 15; 29; 30; 31; 39; 40; 41; 42; 43; 44; 45]. Specifically, voxelized ADD should be optimized according to the ALARA (as low as reasonably achievable) principle. Minimizing the AD in the NLV around the PTV to be as low as reasonably possible minimizes the likelihood of side effects. In addition, the AD in the PTV should remain above a threshold value so that the patient can benefit from the therapy. The mean AD in the NLV should not be greater than 70 Gy and should preferably remain below 50 Gy [29; 30; 31]. The PTV generally requires a mean AD higher than 75 Gy for proper treatment [16; 29; 30]. Little or no benefit is gained by exceeding an AD threshold of 120 Gy in the PTV, especially because it may increase the unnecessary exposure of the  
Figure 3: Example of a patient’s liver ROI in a single CT slice. The red and green lines represent the PTV and the liver masks, respectively. The mask of the dose computation box is represented by the blue square. NLV [30, 46, 47]. However, patients with HCC and portal vein thrombosis were classified in recent research papers as good candidates for RE with an AD in the PTV greater than 205 Gy [39, 44]. Ultimately, the AD level needed to achieve a certain probability of tumor control depends on the specific radio-sensitivity of the tumor type and the type of MS used in the treatment [48, 49]. These variables were not considered in the present work. We have used only glass MS. On the other hand, in the presence of a very favorable tumor-to-normal liver perfusion ratio (highly selective distribution within the PTV), the PTV could significantly exceed the 120 Gy (thus improving expected efficacy probability) while exposing the NLV to doses much less than the mentioned 50 Gy (treatment reasonably safe). This will be better detailed later during the discussion. Currently, the calculation of the \({}^{90}\)Y activity administered in each RE treatment session followed the glass MS manufacturer's protocol [16], following the one-compartment MIRD-based formalism that considers the prescribed AD (i.e., typically from 80 Gy to 120 Gy) and liver volumes (total, lobar, or segmental). This calculation considers the previously determined LSF as well as the estimated value of the radioactive residue that is not administered to the patient and is left in the injector circuit. Moreover, these calculations assume that the deposition of MS in the liver is uniform. In this work, we propose a methodology and a computational solution to optimize the activity to be administered, considering the SPECT-MAA images acquired in the treatment planning stage. The optimization is based on the imposition of AD thresholds at the level of the PTV and NLV voxels. The optimal activity that should be administered into the patient's liver is obtained by maximizing Equation (1):  
\[OF(A)=\frac{\#PTV_{thr_{max}}(A)}{\#PTV_{var}}+\frac{\#NLV_{thr_{max}}(A)}{ \#NLV_{var}} \tag{1}\]  
where \(OF(A)\) is the proposed objective function, \(\#PTV_{var}\) and \(\#NLV_{var}\) are the total number of voxels in the PTV and NLV, respectively. \(\#PTV_{thr_{max}}(A)\) and \(\#NLV_{thr_{max}}(A)\) are the number of voxels of the SPECT-MAA-based ADD that comply with the AD threshold imposed to, respectively, the PTV and NLV, as a function of the \({}^{90}\)Y activity (\(A\)) to be administered to the liver.