The current study investigated absorbed doses to organs and metastatic regions in nine-mCRPC patients. GATE Monte Carlo simulations served as a reference for comparing dose estimates from IDAC and OLINDA software. The calculated liver dose of 0.09 Gy/GBq demonstrated strong agreement with both OLINDA and IDAC (< 1% relative difference). This finding is consistent with those of Xue et al. [33], machine learning based mean liver dose of 0.067 ± 0.035 Gy (range: 0.019–0.151). A study by Rosar et al. [23] using IDAC2.1 reported a mean liver dose of 0.10 ± 0.05 Gy/GBq. In comparison, Shozo Okamoto and colleagues using the IDAC2.1 computational program, the average computational dose of liver was 0.10 ± 0.05 Gy/GBq and in other studies conducted by Okamoto et al. [7] and Prive et al. [34] based on MIRD calculations, liver doses were determined as 0.12 ± 0.06 Gy/GBq and 0.8 Gy (range 0.6–1.1 Gy), respectively. According to Peters et al. [35] study, based on the MIRD formulation, the liver dose was obtained in the range of 0.06–0.14 Gy/GBq. Moreover, Violet et al. [13] computed the dose of different organs at the voxel level with the average liver dose 0.1 Gy/GBq. A review of various studies indicates consistent liver dose estimates, likely attributed to the liver's large size, uniform tissue composition, and minimal density fluctuations.

In the present study, regarding the absorbed dose of the kidneys, the average dose calculated by IDAC and OLINDA are in agreement with each other and with GATE. In the study of Xue et al. [33], the calculated kidney dose is in the range of 0.236–1.041 Gy/GBq, and the average kidney dose is higher than that in this study. Rosar et al. [23] reported a mean kidney dose of 0.54 ± 0.28 Gy/GBq using IDAC2.1. Okamoto et al. [7] and Prive et al. [34], employing MIRD calculations, determined renal doses of 0.72 ± 0.21 Gy/GBq and 4.3 Gy (range: 3.1–6.1 Gy), respectively. According to the study of Peters et al. [35], the kidney dose has been obtained in the range of 0.21–0.88 Gy/GBq. Also, in the study of Violet et al. [13], the average kidney dose has been determined to be 0.39 Gy/GBq, which is close to the present study. Kidney dose estimates varied across studies, potentially influenced by kidney size, function, and imaging/analysis techniques. Lacrimal gland doses, determined using the spherical model in IDAC and OLINDA, showed better agreement between IDAC (2.91 Gy/GBq) and GATE (2.82 Gy/GBq) compared to OLINDA (3.22 Gy/GBq). The latter aligned more closely with MIRD-based estimates from Okamoto et al. [7] (3.8 Gy/GBq) and Violet et al. [13] (3.78 Gy/GBq).

In the study of Violet et al. [13], using the Voxel base method, the calculated dose was about ten times lower (0.36 Gy/GBq-By expanding contours around of lacrimal gland pluse1to 2 cm) than the MIRD method. As mentioned before, the difference between the results of OLINDA, GATE, and IDAC can be due to the difference in density determination and gland shape. Tables 4 and 5 present better agreement between IDAC and GATE for parotid gland dose estimates using the spherical model and for submandibular gland dose estimates using the phantom standard model. This correlation suggests similarities in mass densities and shapes between the models and actual gland anatomy. In this regard, the estimated dose of parotid and submandibular glands by Rosar et al. [23] (2021), which used CT images to determine tissue density in the IDAC spherical model, is 0.81 Gy/GBq and 0.72 Gy/GBq, respectively. The ICRP phantom's salivary gland models were used to estimate parotid and submandibular gland doses. Results for the submandibular gland showed better agreement with the reference (GATE) data, possibly due to the closer anatomical correspondence between the phantom and patient anatomy for this gland. Prive et al. [34] reported salivary gland doses ranging from 1.2–5.9 Gy using MIRD, aligning with our GATE results (2.7–5.9 Gy). Okamoto et al. [7] calculated a parotid gland dose of 0.55 Gy/GBq using OLINDA/EXM, consistent with our findings. However, the same study reported a submandibular gland dose of 0.64 Gy/GBq, higher than our GATE results (RD: 25%) but closer to OLINDA estimates (RD: 9%). Peters et al. [35] reported a median salivary gland dose of 0.50 (0.15–1.28) Gy/GBq using MIRD formalism.

Violet et al. [13], study using SPECT imaging calculated mean doses of 0.58 Gy/GBq and 0.64 Gy/GBq for parotid glands and 0.44 Gy/GBq and 0.67 Gy/GBq for submandibular glands using voxel technique and MIRD sphere model methods, respectively. These results align with GATE and OLINDA findings. Also in the study of Xue et al. [33], the salivary gland dose was calculated 0.57(0.15–1.87) Gy/GBq, which is in agreement with the average calculated dose of GATE and IDAC for the parotid gland and that of OLINDA for the submandibular gland (RD < 1%).

The most significant discrepancy in absorbed dose for metastatic regions occurred between GATE and IDAC calculations for patient one's lung metastases, likely due to differences in lung density input. In bone metastases (58 metastases), the results of GATE(3.42 Gy/GBq) were less than IDAC (3.81 Gy/GBq-RD = 12.23%) with a better agreement with OLINDA (4.37 Gy/GBq-RD = 29.58%). IDAC bone metastasis dose estimates are more closely aligned with GATE due to its use of a higher tissue density (1.108 g/cm3) compared to OLINDA (1.0 g/cm3). Lenz et al. [36] reported a − 21% to + 56% dose difference between GATE 8.2 and OLINDA spheres for bone lesions in mCRPC patients, consistent with our findings of -14.55% to + 62.97% between GATE 9.0 and OLINDA.

The average dose of bone metastases calculated by Rosar et al. [23]using the IDAC spherical model was equal to 1.68 Gy/GBq, which is lower than the present study. Okamoto et al. [7] calculated an average dose of 3.2 Gy/GBq for 93 metastases, aligning with our GATE result of 3.21 Gy/GBq. The study further divided metastases into bone (74 areas), lung (3 areas), and lymphatic (8 areas), reporting doses of 3.4 Gy/GBq, 1.7 Gy/GBq, and 3.2 Gy/GBq, respectively. While bone and lymph metastasis doses aligned with GATE (3.42 Gy/GBq and 2.51 Gy/GBq), lung metastasis dose was closer to IDAC (1.08 Gy/GBq).

In the Maffey-Steffan et al. [37] study, the dose of bone and lymphatic metastases in the first treatment cycle was calculated as 4.01 (1.10–13.00) Gy/GBq and 3.12 (0.7–8.7) Gy/GBq, respectively. For bone metastases, OLINDA results (4.37 Gy/GBq) aligned more closely with GATE than lymphatic metastases, in which discrepancies were larger, potentially due to sample size limitations. Peters et al. [35] reported a median absorbed dose of 2.07 Gy/GBq (range: 0.30–16.40 Gy/GBq) for 40 metastatic areas, highlighting the dose variability in these regions. In radionuclide therapy with Lu-177, there are few studies to determine the cross-absorbed dose, which can be due to its small amount compared to the self-absorbed dose (Sandström et al. [38]). The current study observed significant discrepancies between OLINDA, IDAC, and GATE results, potentially influenced by the limited sample size. Grime et al. [39] reported -40 ± 35% and -30 ± 31% differences between Monte Carlo and OLINDA calculations for kidney-to-liver and liver-to-kidney cross-doses, respectively, in a study of six patients using an approximated Lu-177 biological half-life. Limitations of this study included workflow disruptions, a small patient cohort, and heterogeneous metastatic distribution.