Numéro 
Radioprotection
Volume 54, Numéro 2, April–June 2019



Page(s)  125  132  
DOI  https://doi.org/10.1051/radiopro/2019014  
Publié en ligne  27 mai 2019 
Article
Dose calculations in heterogeneous volumes with the GATE Monte Carlo software for radiological protection
^{1}
Université de Strasbourg,CNRS, IPHC UMR 7178,
67000
Strasbourg, France
^{2}
ALARA Expertise,
7 allée de l’Europe,
67960
Entzheim, France
^{*} Corresponding author: thomas.deschler@iphc.cnrs.fr
Received:
24
April
2018
Accepted:
18
April
2019
Monte Carlo methods have become widespread in the field of radiation protection and in particular in medical physics where the use of voxelized volumes for the reconstruction of dosimetric quantities is increasing. Changing the resolution of a dose map can be useful to compare dosimetric results coming from voxelized volumes with different resolutions, or to reduce computation time. This can be done by superimposing a dosel grid with a different resolution than that of the voxelized volume. In this case, each dosel will cover several voxels, leading the Monte Carlo code to calculate the dose in heterogeneous volumes. Two algorithms are available in GATE to perform these calculations, the VolumeWeighting (VW) and the MassWeighting (MW) algorithms, the latter being the subject of this work. In a general way, the MW algorithm tends to reconstruct a higher dose than that the VW one. In dosels involving heavy and lightweight materials (airskin, bonetissue), the MW reconstructed dose is better estimated than the VW one (up to 10% better at the airskin interface). Moreover, the statistical uncertainty of the MW dose can be up to 80% lower than the VW one at airskin interfaces. These results show that the MW algorithm is more suitable for radiological protection applications and must be preferentially used in GATE for dose calculations in heterogeneous volumes.
Key words: Monte Carlo / dosimetry / voxel phantoms / radiation / medical / radiation protection
© EDP Sciences 2019
1 Introduction
The use of voxelized volumes (CT scans, anthropomorphic phantoms) with Monte Carlo (MC) simulations for dosimetric purposes is widespread in medical physics (Abella et al., 2010; CandelaJuan et al., 2013; Figueira et al., 2013).
The dosimetric results of MC simulations can be expressed through dose maps (3D matrices), where the voxelized volume is divided following a scoring grid with a userdefined resolution. The dose scoring element of the grid is generally called dosel by analogy with the term voxel (Sarrut and Guigues, 2008). The deposited energy is tracked inside each dosel to compute the absorbed dose.
In order to perform precise comparisons among voxelized volumes, it can be useful to modify the dosel resolution of each resulting absorbed dose map to the same value regardless of the original resolution of the voxelized volumes. In addition to voxelized volume comparisons, another challenge of MC dosimetry is to reduce simulation computing time. Since the dosel statistical uncertainty is correlated with its volume, decreasing the dosel resolution obviously decreases the dose uncertainty. A reduction of dose map resolution can thus provide a notable gain in simulation computing time.
Resampling methods can be applied at three different levels to obtain a dose map with a different resolution than that of the original voxelized volume:

presimulation: resampling the voxelized volume before the simulation;

persimulation: generating the dose map following a defined scoring grid resolution during the simulation;

postsimulation: resampling the dose map after the simulation.
A simple presimulation solution consists of resampling the Hounsfield units (Schneider et al., 2000) (for a CT scan) or index numbers (for voxelized phantoms). In the case of a CT scan, averaging the Hounsfield number of a group of voxels gives interesting results (Mora et al., 2001; AiDong et al., 2005). Applying these methods to the index numbers^{1} of voxelized phantoms will not work at interfaces between different index numbers because they are not related to the material density of the voxel (like the Hounsfield units). It will thus skew the reconstruction of the materials in the simulation. It should be noted that more complex solution can be used to resample voxelized phantom, like assigning the majority index value between the resampled voxels to the new voxel.
Postsimulation methods can be easier to implement, as they can consist of resampling a dose map. According to the definition of the absorbed dose (i.e., deposited energy over the mass of the medium), these methods will work only with materials of similar density. If the material compositions are too different, the dose resulting from the resampling can diverge significantly from the absorbed dose in each of the materials.
Concerning persimulation methods, superimposing a scoring grid over the geometrical voxel grid of the volume allows to generate dose maps with no limitation of dosel resolution. This method preserves all the physical interactions by using the original geometrical information of the voxelized volume, allowing it to work with any type of voxelized volumes. It will bring MC codes to compute absorbed dose inside dosels which contain more than one material. Applying the definition of the absorbed dose to such heterogeneous volumes is a complex issue and the method used to reconstruct the dose in such volumes will necessarily impact its accuracy.
In the present work, we propose a comprehensive study of two persimulation algorithms allowing the calculation of the heterogeneous dose based on weighting the dose in the voxel by either its volume or mass fraction. The first of the two algorithms is the default dose calculation of the GATE Monte Carlo software (v8.0), a GEANT4 framework dedicated to medical physics (Sarrut et al., 2014), and the second algorithm is proposed by the authors. A large number of publications used the GATE software for dosimetric and radiological protection calculations (Perrot et al., 2014; Bouzid et al., 2015; Marcatili et al., 2015). As the choice of a given algorithm can strongly affect the computed dose values, it is interesting to compare the two algorithms in terms of bias and uncertainty on the reconstructed doses in heterogeneous voxelized volumes.
2 Materials and methods
Two methods for calculating the absorbed dose in a heterogeneous medium are described in this paper. They are based either on weighting of the absorbed dose in each of the dosel materials by its volume or mass fraction and they will be referred to as VolumeWeighting (VW) and MassWeighting (MW) algorithms respectively. The VolumeWeighting algorithm is the one used by default in the MC simulation code GATE. The MassWeighting has been recently added to the GATE code by the authors of the present work.
Two grid types are present in the MC simulation, the voxel geometrical grid of the volume and the dosel grid that scores the dose, each having its own resolution. The scoring grid is superimposed on the geometrical grid. GATE uses a system called DoseActor to compute the absorbed dose (Sarrut et al., 2014). This system handles the scoring grid and its resolution to store the absorbed dose in a 3D matrix called a dose map. All the simulations performed in this work used the builtin physics list emlivermore with the photon and electron cuts set to a range of 1 mm. The compositions and densities of the air (G4_AIR) and vacuum (G4_Galactic) materials came from the GEANT4 NIST material database. The compositions and densities of the tissues and organs of the voxelized phantom are taken from the publication 110 of ICRP (ICRP, 2009).
2.1 VolumeWeighting algorithm
To compute the dose inside a dosel (D_{dosel}) the VW algorithm weights the absorbed dose (D_{i}) of each of the M voxels contained in the dosel by their respective volume fraction (V_{i}/V_{dosel}). (1) with V_{i} the portion of volume of the voxel i contained in the dosel. Using the definition of the absorbed dose, equation (1) can be expressed in terms of voxel deposited energy E_{i} and mass m_{i}. (2)
2.2 MassWeighting algorithm
The MW algorithm takes into account the heterogeneity of the dosel by using the absorbed dose definition. At the initialization step of the simulation, the MW algorithm computes the mass of each dosel (m_{dosel}) by taking into account the portion of mass (m_{i}) of each of the M voxels contained in it. (3) with ρ_{i} the density of the voxel i contained in the dosel. The computation takes less than one minute on a full voxelized anthropomorphic phantom and should only be performed once per voxelized volume. Then, during the simulation, the algorithm divides the deposited energy in each dosel (E_{dosel}) by their respective masses to get the dosel absorbed dose (D_{dosel}). (4)
Equation (4) can be written in an equivalent form where the voxel dose is weighted by its mass fraction: (5)
If the dosel and voxel resolutions are equal, a unique material is attached to each dosel (homogeneous case) and so m_{dosel} = m_{i} and V_{dosel} = V_{i}. Hence, the two algorithms follow the absorbed dose definition and compute the same dose values.
2.3 Relative statistical uncertainty
In GATE Monte Carlo simulations, the relative statistical uncertainty ϵ_{k} of the absorbed dose in the dosel k is computed using (Chetty et al., 2006): (6) with N the number of primary particles and d_{k,j} is the absorbed dose contribution of the primary particle j in the dosel k.
3 Applications
The introduced algorithms were compared in two simulation setups. First, a simple geometry with two voxels of different size was considered to study the reconstructed heterogeneous dose with respect to the absorbed dose in each material (Fig. 1a). In a second setup, the algorithms were applied to an anthropomorphic voxelized phantom to analyze the effect of dosel resolution on the algorithm’s dose reconstruction (Fig. 1b).
Fig. 1 Illustration of the two applications studied. 
3.1 Simple geometry
The simple geometry consists of a cube of size 10 × 10 × 10 cm^{3} containing two voxels with different materials irradiated tomographically with a square beam (with dimensions greater than the dimensions of the cube) of 100 keV Xrays at eight angles separated by 45°. A beam of 10^{7} primary Xrays was simulated in each simulation. The pairs of materials considered in this study are: watervacuum, waterbone and waterair. To study the dose variation in the entire cube reconstructed with the algorithms, several MC simulations with different proportions of the two materials were generated.
3.1.1 Results
3.1.1.1 Watervacuum case
Figure 2a illustrates the absorbed dose in each of the materials and the reconstructed dose in the whole cube as a function of the volume ratio of each material. Knowing that the only mass contribution comes from the water volume, it is expected that the MW dose will follow the water dose. Thus, it should remain the same regardless of the material proportions in the volume. The shape of the VW dose is determined by the proportion of water volume in the simulation. Thus, the resulting dose corresponds to the water dose weighted by the fraction of the water volume. Figure 2b shows that the statistical uncertainties of the two algorithms follow those of the water since there is no contribution from the vacuum.
Fig. 2 (a) Absorbed dose in water and vacuum and heterogeneous doses computed from the MassWeighting and VolumeWeighting algorithms for the watervacuum case. (b) Relative statistical uncertainty on dose (statistical uncertainty of vacuum is not represented). : water; : vacuum; : MassWeighting; : VolumeWeighting. 
3.1.1.2 Waterbone case
Figure 3a displays the absorbed dose in water and bone^{2} and the heterogeneous dose reconstructed by the algorithms. With this combination of materials, the MW dose is always higher than the one given by VW. It is explained by the significant contribution of absorbed dose in bone due to its higher density (1.92 g/cm^{3}). However, the difference between the two algorithms is significantly lower than in the watervacuum case because of the smaller difference between the material densities. Figure 3b shows that the relative uncertainty of dose in water and bone are of the same order of magnitude. But for a bone volume ratio lower than about onethird of the total volume, the number of interactions becomes higher in the water volume than in the bone volume. As the MW algorithm gives more weight to the bone in the dose calculation, this results in a slightly higher MW statistical uncertainty at low bone volume ratio compared to the VW one (up to 4.8% for a ratio of 0.3).
Fig. 3 (a) Absorbed dose in water and bone and heterogeneous doses computed from the MassWeighting and VolumeWeighting algorithms for the waterbone case. (b) Relative statistical uncertainty on dose. : water; : vacuum; : MassWeighting; : VolumeWeighting. 
3.1.1.3 Waterair case
Figure 4a shows the results for a volume containing water and air. As for the waterbone case, the dose reconstructed by the MW algorithm is higher than the one of VW (up to 17%). Due to the very low density of air (1.20479 mg/cm^{3}), the MW dose receives almost exclusively the contribution of the absorbed dose in water, while the VW dose also receives a contribution of the absorbed dose in air that depends on its volume fraction. The difference of statistical uncertainty seen on Figure 4b reflects the contribution of air in each of the algorithms. Because of the low density of air, the small number of particles interacting leads to a high statistical uncertainty on its absorbed dose and thus on the uncertainty of the VW dose. While having almost no contribution coming from the air, the MW relative uncertainty follows that of the water and can be up to 90% lower than the one of the VW algorithm.
Fig. 4 (a) Absorbed dose in water and air and heterogeneous doses computed from the MassWeighting and VolumeWeighting algorithms for the waterair case. (b) Relative statistical uncertainty on dose. : water; : vacuum; : MassWeighting; : VolumeWeighting. 
3.1.2 Discussion
The two algorithms present some significant discrepancies on the reconstructed dose depending on the materials involved at the interface, especially for materials with a large difference of density. Knowing that there is more deposited energy in heavier materials, higher dose values are computed by the MW algorithm. The behavior of the MW algorithm is particularly useful at airtissue interfaces, providing a more accurate estimation of absorbed dose in tissue and a statistical uncertainty not influenced by the air. These basic setups show that the reconstructed dose for a heterogeneous dosel always falls between the absorbed dose of each separate material.
3.2 Voxelized phantom
The algorithms were studied on the female version of the ICRP 110 anthropomorphic voxelized phantom (ICRP, 2009). A 10 pixels slice, located at the thoracic level, was used in the MC simulations. Its resolution is 299 × 137 × 10 for a voxel size of 1.775 × 1.775 × 4.84 mm^{3}. The slice was placed in air and irradiated tomographically by 80 keV Xrays at eight angles. 8 × 10^{9} Xrays were simulated in order to have an average statistical uncertainty on the dose inferior to 1% at the original voxel resolution. The dose maps were computed with the two algorithms for the following dosel resolutions: 299 × 137 × 10, 150 × 68 × 5 and 75 × 34 × 3, corresponding respectively to a dosel volume of 15.1, 121 and 817 mm^{3} and a resampling factor respectively equal to 1, 8 and 64. Each of the following dose and uncertainty maps were generated in the same simulation.
3.2.1 Results
3.2.1.1 299 × 137 × 10 dosel resolution
Figure 5 shows dose and relative statistical uncertainty maps at the dosel resolution 299 × 137 × 10 (same as the original voxel resolution). No matter which algorithm is used, the generated maps are exactly the same. This was expected because each dosel contains only one voxel and thus one material. The average relative statistical uncertainty on Figure 5b is presented in Table 1 and is equal to 0.97%.
Fig. 5 Dosel resolution 299 × 137 × 10 (same resolution as the ICRP phantom). 
Evolution of the mean relative statistical uncertainty () as a function of the three studied dosel resolutions and the VolumeWeighting (VW) and MassWeighting (MW) algorithms. The time gain factor (f_{t}) is calculated using the proportionality between the statistical uncertainty (σ) and the inverse square root of the number of primary particles in the Monte Carlo simulation. Knowing that the computation time (t) is proportional to the number of primary particles: .
3.2.1.2 150 × 68 × 5 dosel resolution
Figures 6a and 6b show the dose maps obtained with the VW and MW algorithms for a dosel resolution eight times lower than the original voxel resolution. In order to show the discrepancies between the two algorithms, Figures 7a and 7c display respectively the ratio of reconstructed dose and the ratio of their relative uncertainties. Figure 7a shows that the VW reconstructed dose is lower at heavylight material interfaces (bonetissue, tissueair) compared to the MW algorithm. One can see in Table 2 that the dose difference can be up to 49% (3.6% on average). As seen in Table 1, at the dosel resolution 150 × 68 × 5 the mean relative statistical uncertainty on the reconstructed dose is 36% lower with the MW algorithm. Figure 7c shows that the MW algorithm gives a smaller uncertainty than the VW algorithm at airtissue and lungtissue interfaces, but the MW uncertainty is slightly higher than the VW one at the bonetissue interface. Table 2 shows that the MW relative uncertainty can be up to 88% lower at the airtissue interface and 9% higher than the VW one at the bonetissue interface, which is consistent with the heterogeneous algorithms behavior described in the simple geometry application.
Fig. 6 Dose maps from VolumeWeighting (VW) and MassWeighting (MW) algorithms for scaling factors of 8 and 64. Dosel resolution of Figures (a) and (b): 150 × 68 × 5. Dosel resolution of Figures (c) and (d): 75 × 34 × 3. 
Fig. 7 Ratio of dose maps and relative uncertainty maps from VolumeWeighting (VW) and MassWeighting (MW) algorithms for scaling factors of 8 (×8) and 64 (×64). Dosel resolution of Figures (a) and (c): 150 × 68 × 5. Dosel resolution of Figures (b) and (d): 75 × 34 × 3. 
Comparison of mean absorbed dose and statistical uncertainty ratio between the dose maps obtained with VolumeWeighting (VW) and MassWeighting (MW) algorithms at 150 × 68 × 5 and 75 × 34 × 3 dosel resolutions.
3.2.1.3 75 × 34 × 3 dosel resolution
Figures 6c and 6d show that at a 64 times smaller dosel resolution (75 × 34 × 3) the discrepancies between the two algorithms are more pronounced. Table 2 shows that, in this case, the dose reconstructed with the VW algorithm can be up to 43% lower than the MW one (5.4% on average). Table 1 shows that the mean relative statistical uncertainty on the dose map obtained with the MW algorithm is 49% smaller compared to the VW one. At this dosel resolution, to get a mean relative statistical uncertainty on dose of about 1%, the MW and VW algorithms need respectively about 35 times and 9 times less computation time compared to the MC simulation at the original resolution.
3.2.1.4 Comparison with the original resolution dose map
In order to compare the dose maps reconstructed with the algorithms with the dose map at the original resolution, each dose map of 150 × 68 × 5 and 75 × 34 × 3 dosels was resampled to the 299 × 137 × 10 resolution using the nearest neighbor interpolation. This allows to highlight the biases induced by the algorithms in comparison to the reference dose map at the voxel original resolution. Table 3 presents the mean dose ratio between the original resolution dose map and the resampled one from the VW and MW algorithms for different materials. Figure 8 shows the ratio between the original resolution dose map and the mass and volume weighted resampled ones. The main discrepancies between the heterogeneous algorithms and the original dose map are located at the tissuebone interface. A comparison shows that the VW algorithm underestimates the dose on the entire phantom slice by 0.6% at 150 × 68 × 5 and 0.4% at 75 × 34 × 3 on the average. However, when the skin region is compared, the VW algorithm underestimates the dose by 15 and 17% on average respectively for the 150 × 68 × 5 and 75 × 34 × 3 dosel resolutions. On the other hand, the MW algorithm overestimates the dose by a factor of 1.2 and 2.4% on average respectively for the 150 × 68 × 5 and 75 × 34 × 3 dosel resolutions. Considering only the skin region, the MW algorithm underestimates the original dose by a mean factor of 4.1 and 7.0% for the respective dosel resolutions 150 × 68 × 5 and 75 × 34 × 3. The presence of air in the bronchi leads also the VW algorithm to underestimate the bronchial dose at airbronchi interfaces, while this problem does not occur with the MW algorithm. Despite a slight overestimation of the dose in the whole phantom slice and in the majority of the structures, the MW algorithm gives a better estimation of the dose in the skin and a constant lower standard deviation of the mean dose ratio than that obtained with the VW algorithm.
Comparison of the mean dose ratio between the original resolution dose map (D_{orig}) and the resampled dose map obtained with the VolumeWeighting and MassWeighting algorithms (D_{VW} or D_{MW}).
Fig. 8 Ratio of the dose maps at original resolution (D_{orig}) and the resampled one with the VolumeWeighting and MassWeighting algorithms (D_{VW} or D_{MW}). Figures (a) and (b) are resampled from the resolution 150 × 68 × 5 to 299 × 137 × 10. Figures (c) and (d) are resampled from the resolution 75 × 34 × 3 to 299 × 137 × 10. 
3.2.2 Discussion
The simulation setups described previously show that the two proposed methods to calculate dose in heterogeneous dosels produce different results depending on the densities of the materials.
These methods provide an effective solution to reduce statistical variance (up to an average factor of 2.7 with the MW algorithm with a resampling factor of 8) and a good reconstruction of the absorbed dose in heterogeneous dosels (less than 2% underestimation on average with the MW algorithm with a resampling factor of 8) but at the expense of a loss in the spatial accuracy of dose deposition. The VW algorithm gives a good estimation of the mean dose in a whole slice of the phantom. However, VW can locally underestimate the reconstructed dose, especially at materials interface. Table 3 shows that for all tissues the MW algorithm gives a higher dose value than the VW one. This behavior makes the MW algorithm particularly suitable for radiation protection purposes, which always prefer the most unfavorable estimation with respect to the ALARA principle^{3}. Nevertheless, results on the twovoxels cube have shown that the MW algorithm can lead to neglecting the dose in the more abundant material of the dosel if its density is too low compared to the other components.
The ability of the MW algorithm to compute skin dose with a low underestimation (about 7% on average at dosel resolution resampled by a factor of 64) compared to the VW algorithm (which produces about 10% more underestimation than MW) is of particular interest for skin dosimetry in radiology procedures.
In addition to the statistical variance reduction caused by the increase of the dosel size, the main dose uncertainty reduction between algorithms occurs at materials interfaces, where the MW algorithm provides a relative uncertainty which can be up to 80% lower than computed by the VW algorithm.
It should finally be noted that in the case where only part of a heterogeneous dosel is irradiated, for example in microbeam radiotherapy, the difference between the dose reconstructed by the algorithms and the absorbed dose in the homogeneous voxels can be much larger than those presented in this work.
4 Conclusions
The comprehensive study of the two available algorithms for dose calculation in the GATE v8.0 Monte Carlo software allows to evaluate the bias and uncertainties in the dose maps that contain heterogeneous voxels. At material interfaces like skinair, the VolumeWeighting method can lead to an underestimation of the dose as high as 20%. The MassWeighting method, integrated recently into GATE, enables to limit the bias in the dose reconstruction at material interfaces, and to reduce the uncertainty up to 80% depending on the voxel size. This algorithm should therefore be favored for radiological protection studies based on dose calculations in heterogeneous volumes.
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Corresponding to the mineral bone material from the ICRP publication 110 (ICRP, 2009).
Cite this article as: Deschler T, Arbor N, Carbillet F, Nourreddine A. 2019. Dose calculations in heterogeneous volumes with the GATE Monte Carlo software for radiological protection. Radioprotection 54(2): 125–132
All Tables
Evolution of the mean relative statistical uncertainty () as a function of the three studied dosel resolutions and the VolumeWeighting (VW) and MassWeighting (MW) algorithms. The time gain factor (f_{t}) is calculated using the proportionality between the statistical uncertainty (σ) and the inverse square root of the number of primary particles in the Monte Carlo simulation. Knowing that the computation time (t) is proportional to the number of primary particles: .
Comparison of mean absorbed dose and statistical uncertainty ratio between the dose maps obtained with VolumeWeighting (VW) and MassWeighting (MW) algorithms at 150 × 68 × 5 and 75 × 34 × 3 dosel resolutions.
Comparison of the mean dose ratio between the original resolution dose map (D_{orig}) and the resampled dose map obtained with the VolumeWeighting and MassWeighting algorithms (D_{VW} or D_{MW}).
All Figures
Fig. 1 Illustration of the two applications studied. 

In the text 
Fig. 2 (a) Absorbed dose in water and vacuum and heterogeneous doses computed from the MassWeighting and VolumeWeighting algorithms for the watervacuum case. (b) Relative statistical uncertainty on dose (statistical uncertainty of vacuum is not represented). : water; : vacuum; : MassWeighting; : VolumeWeighting. 

In the text 
Fig. 3 (a) Absorbed dose in water and bone and heterogeneous doses computed from the MassWeighting and VolumeWeighting algorithms for the waterbone case. (b) Relative statistical uncertainty on dose. : water; : vacuum; : MassWeighting; : VolumeWeighting. 

In the text 
Fig. 4 (a) Absorbed dose in water and air and heterogeneous doses computed from the MassWeighting and VolumeWeighting algorithms for the waterair case. (b) Relative statistical uncertainty on dose. : water; : vacuum; : MassWeighting; : VolumeWeighting. 

In the text 
Fig. 5 Dosel resolution 299 × 137 × 10 (same resolution as the ICRP phantom). 

In the text 
Fig. 6 Dose maps from VolumeWeighting (VW) and MassWeighting (MW) algorithms for scaling factors of 8 and 64. Dosel resolution of Figures (a) and (b): 150 × 68 × 5. Dosel resolution of Figures (c) and (d): 75 × 34 × 3. 

In the text 
Fig. 7 Ratio of dose maps and relative uncertainty maps from VolumeWeighting (VW) and MassWeighting (MW) algorithms for scaling factors of 8 (×8) and 64 (×64). Dosel resolution of Figures (a) and (c): 150 × 68 × 5. Dosel resolution of Figures (b) and (d): 75 × 34 × 3. 

In the text 
Fig. 8 Ratio of the dose maps at original resolution (D_{orig}) and the resampled one with the VolumeWeighting and MassWeighting algorithms (D_{VW} or D_{MW}). Figures (a) and (b) are resampled from the resolution 150 × 68 × 5 to 299 × 137 × 10. Figures (c) and (d) are resampled from the resolution 75 × 34 × 3 to 299 × 137 × 10. 

In the text 
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