1 Introduction

Nuclear medicine is based on the administration of radionuclides into the human body for diagnostic or therapeutic applications. According to the International Commission on Radiological Protection (ICRP) regulations, it is intended to limit the occupational and man-made radiation exposure. It is necessary to know the internal dose to better assess the risk of early and late effects in irradiated organs during treatment. Improved dosimetry tools have been used for this purpose. However, direct measurement of the absorbed dose to the human body is not possible. The Medical Internal Radiation Dosimetry (MIRD) committee has published a systematic formalism, the MIRD schema, for assessing internal doses to overcome this problem [1]. The specific absorbed fraction (SAF) and S values are useful parameters of internal dose assessment derived from this approach. Hence, many publications have been added to the literature on this issue, often using Monte Carlo (MC) simulation methods in conjunction with computational human phantoms. However , to perform an accurate absorbed dose calculation, the choice of a more realistic anatomical model is desirable, given the progression of computational human phantoms. Two voxelized phantoms are commonly used:

  • The original Zubal phantom [2] was developed in 1994 by Yale University. It is a digital model of an adult male obtained from CT images. The phantom has been segmented into approximately 50 different types of tissues. The original dataset consists of \(243\times 128\times 128\) voxels with an 8-bit depth. The voxels of each organ or tissue have a unique index number that distinguishes them from other structures segmented in the phantom. Improvements to the original phantom were made using MRI scan data of a human brain. Users can freely download the original data [3].

  • The ICRP133 report [4] includes reference SAF values for photon and electron emitters derived from MC simulations in the adult reference male defined in ICRP110, with the anatomical and physiological characteristics of the reference male defined in ICRP89 [4]. The voxelized male phantom consisted of nearly 1.95 million voxels. The number of slices is 220, and each has a thickness of 8 mm. The height and weight of the ICRP133 phantom are 176 cm and 73 kg, respectively. The number of individually segmented structures is 136 in the male and female phantoms, and 53 different tissue compositions have been assigned to them [4].

Although SAF computations have been carried out by many authors, such as Asl et al. [5], Lamart et al. [6], and Momennezhad et al. [7], none of them have assessed the self- and cross-absorption SAF for the High-Definition Reference Korean-man (HDRK-man) [8]. We only found HDRK-man dosimetric studies done for external exposure, such as that of Park et al. [9], who developed a web-based system that allows the calculation of the effective dose for external exposure in real time. Therefore, in this work, we chose to implement the HDRK-man voxel phantom in the MC transport toolkit Geant4 [10] to calculate the S and SAF values for internal exposure. It was compared with both phantoms cited above for photon and electron beams with energies ranging from 15 to 3 MeV. The variability effect of the organ mass and the relative position of the target–source pair were also studied. The new monoenergetic photon and electron of the S and SAF values, for self- and cross-absorption in 44 organs forming the HDRK-man, can be considered to add value and provide new consistent dose estimates for radiopharmaceuticals for different applications. The application of these calculations to study internal absorptions for I-131, In-111, and Lu-177 isotope-based radiopharmaceuticals for certain target–source combinations was also presented.

2 Material and methods

2.1 Monte Carlo simulation

To mimic the interaction of particles with matter, we generally resort to MC simulation techniques. Currently, there are different codes available, such as MCNP [11, 12], EGSnrc [13], PENELOPE [14], and Geant4 [15,16,17], which was used in this study. Geant4 is a MC simulation toolkit written in the C++ programming language [18]. It has been proved to accurately simulate the passage of particles through matter for a wide range of applications, including high-energy physics, nuclear physics experiments, radiation shielding, gamma irradiator design, radiation processing, medical physics, radiation therapy, space dosimetry, and radiation protection [19]. The provided physics models cover a comprehensive set of hadronic, electromagnetic, and optical processes over a wide range of energy (from a few electronvolts to 1 TeV). Moreover, large material sets, physical parameters, and basic geometries are also available to the user. Geant4 comes with a complete range of functionalities including tracking, geometry, physical models, visualization, and user interfaces. Here, we used the physical processes describing the photoelectric effect, Compton and Rayleigh scattering for photons, and Bremsstrahlung, ionization, and multiple scattering models for electrons. The atomic de-excitation processes were also activated during the simulation. From the existing suite of physical packages, we preferred to use PhysListEm-Standard, as it is adequate for this type of application. The maximum step size (\(100\,\upmu {\text {m}}\)) and energy threshold for secondary particle production (990 eV) were used in this study to optimize tracking and to enable a sufficiently precise treatment of energy loss.

The construction of the HDRK-man phantom was performed using high-resolution photographic anatomical images [8]. The nomination of the voxel model “High-Definition Reference Korean-man” is owing to its high precision in the representation of organs and tissues. The height and weight of the HDRK-man are 171 cm and 68 kg, respectively. The size of each voxel is \(1.981\times 1.981\times 2.0854\,\text {mm}^3\), and the voxel model has an array size of \(247\times 141\times 850\) (29,602,950 voxels). The “G4NestedParameterization” class was used to implement this model. The computing platform used in this work to carry out the simulations includes the Geant4 code version 10.1 installed on a Linux operating system (Ubuntu 14.04) and two personal workstations. Each workstation has 40 cores and is equipped with an Intel Xeon E5-2680v2 CPU at 185 2.80 GHz and 256 GB of RAM. One hundred million photon/electron histories were tracked per source region for each case study to calculate the S and SAF values. All the calculations were carried out with a low statistical error (\(<3\%\)).

2.2 SAF and S values

To estimate the internal absorbed doses during either diagnostic or therapeutic processes in nuclear medicine, it was necessary to calculate the SAF values as essential assessment parameters for the internal irradiation scenario. SAF is defined as the ratio of the fraction of emitted energy by the source organ, which is absorbed in the source organ itself (self-irradiation) or in other target organs (cross-irradiation), to the mass of the target organ. On the contrary, the S value is simply calculated based on the particle yields for a given radiopharmaceutical disintegration scheme. The SAF and S values can be evaluated using the following formulas, based on the MIRD formalism [20, 21]:

$$\begin{aligned} {\text {SAF}}(r_{\text {target}} \leftarrow r_{\text {source}})[\text {kg}^{-1}] =\frac{E_{\text {target}}/E_{\text {source}}}{m_{\text {target}}} \end{aligned}$$
(1)

and

$$\begin{aligned}&S(r_{\text {target}} \leftarrow r_{\text {source}})\left[\frac{\text {mGy}}{\text {Bq} \, \text {s}}\right] =1.6\times 10^{-13} \nonumber \\&\quad \times \sum (E_{{\text {source}}_{i}}\times Y_{i}\times {\text {SAF}}((r_{\text {target}} \leftarrow r_{{\text {source}}})_{i}) \end{aligned}$$
(2)

where \(r_{\text {target}}\) is a target organ, \(r_{\text {source}}\) is a source organ, \(E_{\text {target}}\) is the energy absorbed in \(r_{\text {target}}\), \(E_{\text {source}}\) is the energy emitted in \(r_{\text {source}}\), \(m_{\text {target}}\) is the mass of the target organ, and \(Y_{i}\) is the yield of the \(i\text {th}\) radiation emitted during the disintegration.

In this study, the SAF and S values were calculated from MC simulations, which were carried out separately for photons and electrons, using Eqs. (1) and (2) for nineteen initial photon and electron energies (15, 20, 30, 40, 50, 60, 80, 100, 150, 200, 300, 400, 500, 600, 800, 1000, 1500, 2000, and 3000 keV). We also studied the self- and cross-absorbed doses for all possible organ combinations, considering the 44 organs forming the HDRK-man phantom.

2.3 Radiopharmaceuticals case study

As an immediate application of the calculated SAF values corresponding to mono-energetic photon or electron beams, we studied the internal absorptions (S calculation) of I-131, In-111, and Lu-177 isotope-based radiopharmaceuticals for certain target–source combinations (liver \(\leftarrow\) liver, spleen \(\leftarrow\) liver, and spleen \(\leftarrow\) spleen).

3 Results and discussion

This section describes the comparison of the calculated SAF values for mono- and polyenergetic particles using different couples of MC code and voxelized phantom: Geant4 (HDRK-Man) used in this work, MCNPX (ICRP133), and MCNP4B (Zubal) taken from the literature [4, 22] for selected target–source combinations. These data are divided into two groups according to the absorption types, either photons or electrons. The last subsection provides a comparative study of the S values corresponding to I-131, In-111, and Lu-177 with those from the literature.

3.1 Self-absorption

3.1.1 Photon beam

Figure 1 presents the SAF values for photon self-irradiation, i.e., when the source and target organs are the same. The three organ cases studied were the kidneys, liver, and spleen for the photon energy range from 0.015 to 3 MeV. A visual evaluation of the curves reveals that the SAF for organs in the HDRK-man phantom has a similar trend to those obtained for the ICRP133 reference and Zubal phantoms. However, the average ratios of the SAF of the HDRK-man to those of the ICRP133 and Zubal phantoms were, respectively, 1.13 and 1.27 for the kidneys, 1.42 and 1.23 for the liver, and 1.19 and 1.51 for the spleen. It is notable that HDRK-man has a better correlation with the ICRP reference data series than the Zubal phantom. This figure also shows that SAF values decrease with increasing photon energy from 15 to 100 keV, remain nearly constant until approximately 500 keV, and thereafter continue to slightly decrease. This behavior of photon self-absorption can be explained by the increased probability of the leakage of scattered photons with increased energy [23], and it confirms the results from the literature [5].

Fig. 1
figure 1

Photon specific absorbed fractions (\({\text {kg}}^{-1}\)) for self-irradiation of kidneys, liver, and spleen for energy range (15–3000 keV) in HDRK-man, ICRP133, and Zubal phantoms

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The curves in Fig. 1 show that the SAF values for the HDRK-man phantom are significantly higher than those of the other models for the same considered organs, which can be explained by the fact that the largest organ mass has the smallest SAF. This is confirmed by the following comparison:

  • spleen: \(m_{\text {HDRK-man}}< m_{\text {ICRP133}} < m_{\text {Zubal}}\) \({\text{SAF}}_{\text {HDRK-man}}> {\text{SAF}}_{\text {ICRP133}} > {\text{SAF}}_{\text {Zubal}}\)

  • kidneys: \(m_{\text {HDRK-man}}< m_{\text {ICRP133}} < m_{\text {Zubal}}\) \({\text{SAF}}_{\text {HDRK-man}}> {\text{SAF}}_{\text {ICRP133}} > {\text{SAF}}_{\text {Zubal}}\)

  • liver: \(m_{\text {HDRK-man}}< m_{\text {Zubal}} < m_{\text {ICRP133}}\) \({\text{SAF}}_{\text {HDRK-man}}> {\text{SAF}}_{\text {Zubal}} > {\text{SAF}}_{\text {ICRP133}}\)

For photon self-absorption, it can be concluded that the photon energy and organ mass have important influences on the estimation of the SAF, especially for energies below 100 keV.

An additional study of this mass dependency was performed through the calculation of the absorbed fractions (AFs), which are less dependent on the target organ mass. The AF was calculated as the SAF multiplied by the known mass of the target organ. In other words, it is the energy absorbed in the target organ divided by the energy emitted in the source organ. Figure 2 shows the AF(kidneys \(\leftarrow\) kidneys), AF(liver  \(\leftarrow\) liver), and AF(spleen \(\leftarrow\) spleen) values for photons in the HDRK-man and ICRP133 phantoms. These discrepancies were found to be due to the differences in geometric shape and volume, which directly affect the energy deposition and the radiation transport.

Fig. 2
figure 2

Photon absorbed fractions (AFs) for self-irradiation of kidneys, liver, and spleen for energy range (15–3000 keV) in the HDRK-man and ICRP133 phantoms

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3.1.2 Electron beam

Figure 3 shows the electron SAF values for self-absorption of the liver, kidneys, and spleen. The above phantoms were considered with energy ranging from 15 keV to 3 MeV. As previously, a visual assessment of the trend reveals a good correlation among the three data series.

From the comparison of all the SAF ratios (HDRK-man/ICRP and Zubal/HDRK-man) for the three organs considered, kidneys (1.12 and 73.8), liver (1.53 and 77.31), and spleen (1.21 and 5.07), a large difference between the Zubal and HDRK-man phantoms can be observed. The main factors that may have caused these discrepancies were:

  • The voxel size: The voxelization is not identical for the two phantoms (Zubal: \(4\times 4\times 4\,\text {mm}^3\); HDRK-man: \(1.981\times 1.981\times 2.0854\,{\text {mm}}^3\)), which may affect the geometry and shape of the organs. This defines the geometric effect due to different shapes of the borders of the organs in each phantom, which is more appreciable for electrons.

  • The calculation of the kidneys SAF, for the Zubal phantom, was carried out by assuming both the right and left kidneys (\(r_{\text {right-kidneys}}\) and \(r_{\text {left-kidneys}}\)) as one organ, such that the SAF includes self- (\(r_{\text {right-kidneys}} \leftarrow r_{\text {right-kidneys}}\) and \(r_{\text {left-kidneys}} \leftarrow r_{\text {left-kidneys}}\) and cross-absorption (\(r_{\text {right-kidneys}} \leftarrow r_{\text {left-kidneys}}\) and \(r_{\text {left-kidneys}} \leftarrow r_{\text {right-kidneys}}\)), contrary to those for the ICRP133 and HDRK-man. To avert such errors, the counterpart organs should be not treated jointly; rather, they should be considered as two separate organs. Moreover, it is known that SAF values depend on the voxel size, distance between the source and target voxels, and the beam particles. Thus, the assumption of the inverse dependency of the SAF on the mass cannot be applied to such organs (kidneys), for the Zubal calculations.

The tendency of electron SAF values is approximately constant as the energy increases; nevertheless, it begins to decrease slightly with energy greater than 800 keV. Therefore, above this value, electrons have the possibility to leave the source owing to their physical stopping power characteristics.

Fig. 3
figure 3

Electron specific absorbed fractions (\({\text {kg}}^{-1}\)) for self-irradiation of kidneys, liver, and spleen for energy range (15–3000 keV) in HDRK-man, ICRP133, and Zubal phantoms

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The electron self-AF values in kidneys, liver, and spleen are presented in Fig. 4. The self-AF is almost constant, followed by a drop-off of the values with the increase in electron energy. Given the weak penetration of electrons, when the organ volume is smaller (e.g., spleen), more electrons can escape from the borders of the source organ. Therefore, the slight decrease in the AF values is much more notable when the electron energy increases for smaller organ volumes.

Fig. 4
figure 4

Electron absorbed fractions (AFs) for self-irradiation of kidneys, liver, and spleen for energy range (15–3000 keV) in HDRK-man and ICRP133 phantoms

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3.2 Cross-absorption

3.2.1 Photon beam

The photon SAF values were calculated for cross-irradiation and were compared to the results of the ICRP133 and Zubal phantoms with three target–source combinations: liver \(\leftarrow\) kidneys, spleen \(\leftarrow\) kidneys, and liver \(\leftarrow\) spleen for energy ranging from 15 to 3000 keV.

Figure 5 shows that there is a good agreement of the trend curves among the three datasets, but the higher values correspond to HDRK-man, as explained previously. The average ratios of the SAF of the HDRK-man to those of the ICRP133 and Zubal phantoms were, respectively, 1.23 and 1.36 for liver \(\leftarrow\) kidneys, 1.54 and 1.47 for spleen \(\leftarrow\) kidneys, and 3.45 and 1.32 for liver \(\leftarrow\) spleen. This figure also reveals that the SAF values increase proportionally with the photon energy up to a maximum of 50 keV, whereas above this value, a slight decrease is observed. These discrepancies in the photon cross-irradiation SAF values can be explained by the differences in organ geometry (shape and size), density, and distance between the source and target distance [24]. These factors have significant impact at low energy (\(< 50\) keV).

Fig. 5
figure 5

Photon specific absorbed fractions (\({\text {kg}}^{-1}\)) for cross-irradiation of (liver \(\leftarrow\) kidneys), (spleen \(\leftarrow\) kidneys), and (liver \(\leftarrow\) spleen) combinations for the energy range (15–3000 keV) in HDRK-man, ICRP133, and Zubal phantoms

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3.2.2 Electron beam

In Fig. 6, the SAF values of electron cross-irradiation for the HDRK-man, ICRP133, reference, and Zubal phantoms are plotted for the three (target \(\leftarrow\) source) combinations. There is a clearly observable proportional increase in the SAF values with the electron energy, over the entire studied range. Given that the self-absorption SAF remains approximately constant, the observed parameter increase as a function of the energy for the cross-absorption can be explained by the fact that electrons are more likely to cross organ boundaries at high energies, allowing them to deposit their energy locally in the target organ. In addition, the electron irradiation of adjacent organs cannot always be neglected above 1 MeV, despite their weak penetration, contrary to the hypothesis that electrons are completely absorbed in the source organ itself. It can be concluded that when the distance between the target and the source is greater, the SAF values are smaller than those of the nearer organs, because of the short electron range. Therefore, the inter-organ distances have an important impact on the electron SAF, which can be expressed using chord length distributions (CLDs).

Fig. 6
figure 6

Electron specific absorbed fractions (\({\text {kg}}^{-1}\)) for cross-irradiation of (liver \(\leftarrow\) kidneys), (spleen \(\leftarrow\) kidneys), and (liver \(\leftarrow\) spleen) combinations for the energy range (15–3000 keV) in HDRK-man, ICRP133, and Zubal phantoms

Full size image

To investigate the significant influence of inter-organ distances for photon and electron cross-irradiation on SAF values, we computed the CLD between two organs. This distribution is used to estimate the separation distance between them [25]. The CLD is a frequency histogram of the distances between several randomly selected points in each source–target combination [6]. The CLD values for the target–source pairs of the HDRK-man phantom are plotted in Fig. 7. The kidneys were chosen as an example of the source organ, and the spleen and liver were chosen as the target organs. The mean value and the standard deviation of the chord lengths for the spleen \(\leftarrow\) kidneys and liver \(\leftarrow\) kidneys are, respectively, (129.8 mm, 52.3) and (165.4 mm, 45.0). As shown by this comparison, the smallest source-to-target distance is observed in the spleen case. Therefore, it is the nearest target from the kidneys source that explains why spleen \(\leftarrow\) kidneys had the highest SAF values for photons and electrons compared with liver \(\leftarrow\) kidneys. Consequently, we can conclude that the SAF values for photon and electron cross-irradiation are inversely proportional to the inter-organ distance. If the target is closest to the source, it has the highest SAF values, because this results in the maximum energy absorption. However, if the target is far from the source, it has the smallest SAF values, because the attenuation increases in the crossed organs, which decreases the influence.

Table 1 Comparison of monoenergetic photon and electron SAF values for kidneys \(\leftarrow\) kidneys (\({\text {kg}}^{-1}\)) calculated in this work with those published for the ICRP133 [4]
Full size table
Fig. 7
figure 7

Chord length distributions CLDs (mm) for different target \(\leftarrow\) source combinations: spleen \(\leftarrow\) kidneys and liver \(\leftarrow\) kidneys. A schematic view of the CLD definition is shown (right) as an example for the liver and spleen organs with P and Q voxels, respectively

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Table 2 Comparison of I-131, In-111, and Lu-177 S values (mGy/Bq s) for different source–target combinations calculated in this work with those published by VIDA [23] and Lamart [27]
Full size table

3.3 Usefulness of the database

As another comparison, we tabulated the self-absorption SAF values for kidneys for monoenergetic photons and electrons, as shown in Table 1. The results were compared to ICRP133 data [4], showing average percentage differences of \(13.9 \pm 2.8\) and \(12.1 \pm 3.2\) for photons and electrons, respectively. These discrepancies can be explained by the different use of voxelized phantoms (ICRP133 and HDRK-man), codes (MCNPX and Geant4 ), and statistical errors. Moreover, we calculated the S values according to Eq. 2 and the decay scheme provided by ICRP[26] for three commonly used radio-isotopes. I-131 has specific yields of 1.008 photons per nuclear transition (nt) with an energy of 381.2 keV and 1.000 electron per nt with an energy of 181.9 keV. In-131 has a specific yield of 1.847 photons per nt with an energy of 386.1 keV. Lu-177 has specific yields of 0.1803 photons per nt with an energy of 31.56 keV and 1.000 electron per nt with an energy of 133.3 keV.

Table 2 shows a comparison of the S values (mGy/Bq s) calculated is this work with previously published data [23, 27]. As expected, we always observed lower and higher values for the spleen \(\leftarrow\) spleen and the liver \(\leftarrow\) liver, respectively, when comparing our results with those of VIDA [23]. In the case of the liver \(\leftarrow\) spleen, we observed higher values for I-131 and In-111, but lower values for Lu-177, owing to its low photon yield.

Supplementary databases for the simulation of the SAF values of monoenergetic photons (Appendix 1) and electrons (Appendix 2) in the energy range 15–3000 keV, based on Geant4 , for both cross- and self-absorption into the 44 organs forming the HDRK-man, are provided with this work, including a total of 88 tables.

4 Conclusion

The estimation of the S and SAF values using the Geant4 toolkit could be feasibly computed for different organs and tissues of the voxelized HDRK-man phantom for radionuclide emitters of \(\gamma\) and \(\beta -\). This feasibility has allowed us to construct an overall database that is essential for assessing the internal absorbed dose in the human body. After conducting a comparative study with the computational frameworks of (ICRP133, MCNPX) and (Zubal,MCNP4B), we observed a global similarity of the trends of the curves as functions of the particle energy. Additionally, we found average percentage differences of \(13.9\pm 2.8\) and \(12.1 \pm 3.2\) for photons and electrons, respectively, owing to differences in the calculated CLD between the organs studied: kidneys and spleen. In addition, it was confirmed that the SAF values for self-absorption and cross-absorption depend, respectively, on the energy and volume of the organs and the relative distance between source and target organs.