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Tetrahedral-mesh-based computational human phantom for fast Monte Carlo dose calculations
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Institute of Physics and Engineering in Medicine Phys. Med. Biol. 59 (2014) 3173–3185
Physics in Medicine and Biology
doi:10.1088/0031-9155/59/12/3173
Tetrahedral-mesh-based computational human phantom for fast Monte Carlo dose calculations Yeon Soo Yeom 1 , Jong Hwi Jeong 2 , Min Cheol Han 1 and Chan Hyeong Kim 1 1
Department of Nuclear Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Korea 2 Center for Proton Therapy, National Cancer Center, 111 Jungbalsan-ro, Ilsandong-gu, Goyang-si, Gyonggi-do 410-769, Korea E-mail: [email protected] Received 22 October 2013, revised 24 April 2014 Accepted for publication 2 May 2014 Published 27 May 2014 Abstract
Although polygonal-surface computational human phantoms can address several critical limitations of conventional voxel phantoms, their Monte Carlo simulation speeds are much slower than those of voxel phantoms. In this study, we sought to overcome this problem by developing a new type of computational human phantom, a tetrahedral mesh phantom, by converting a polygonal surface phantom to a tetrahedral mesh geometry. The constructed phantom was implemented in the Geant4 Monte Carlo code to calculate organ doses as well as to measure computation speed, the values were then compared with those for the original polygonal surface phantom. It was found that using the tetrahedral mesh phantom significantly improved the computation speed by factors of between 150 and 832 considering all of the particles and simulated energies other than the low-energy neutrons (0.01 and 1 MeV), for which the improvement was less significant (17.2 and 8.8 times, respectively). Keywords: computational phantom, tetrahedron, Monte Carlo simulation, computation speed (Some figures may appear in colour only in the online journal)
1. Introduction Computational human phantoms are widely used for Monte Carlo dose calculations in many different fields, including radiation protection, radiation therapy, medical imaging, and others. 0031-9155/14/123173+13$33.00
© 2014 Institute of Physics and Engineering in Medicine Printed in the UK & the USA 3173
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The first-generation phantoms, which were mathematical phantoms, also called stylized phantoms, were proposed by Snyder et al (1969). Composed of simplified organ models defined by mathematical equations, they had serious limitations in representing the real anatomy of a human body. In the 1980s, Gibbs and Pujol (1982) first introduced a voxel-based computational human phantom constructed using tomographic images of a human body, which provided much more anatomical realism than mathematical phantoms. Acknowledging the advantages of voxel phantoms, the International Commission on Radiological Protection (ICRP) officially adopted a set of male and female voxel phantoms as the ICRP reference phantoms (ICRP 2007). The voxel phantoms themselves, however, have several limitations. First, they show stairstepped surfaces, causing the so-called ‘voxel effect’ that results in inaccurate dose calculations (Rajon et al 2000). Secondly, due to the finite voxel resolution, it is difficult to model very thin structures such as skin, oral mucosa, or the extrathoracic region, which are very important structures for effective dose calculation. Lastly, the voxel phantoms are rigid, making it impossible to adjust them to model non-upright postures (e.g. that of a radiation worker sitting on a chair-type whole-body counter). Moving or deforming the internal organs of a voxel phantom during Monte Carlo simulation is also impossible. Recently, several investigators (Segars et al 2001, Lee et al 2007, Zhang et al 2009, Cassola et al 2010, Kim et al 2011) have developed surface-based computational human phantoms based on non-uniform rational B-spline (NURBS) or polygonal mesh surfaces. These surface phantoms successfully addressed the aforementioned limitations of the voxel phantoms. That is, they allowed for the modeling of very thin and complicated organs with smooth surfaces as well as posture adjustment for moving or deformation of internal organs as necessary. On this basis, a variety of deformed-surface phantoms have been constructed for modeling of various ages (Lee et al 2008, 2010, Hurtado et al 2012, Matthew et al 2011), body sizes (Johnson et al 2009, Na et al 2010, Ding et al 2012, Cassola et al 2011), and postures (Han et al 2010, Su et al 2012). Unfortunately however, the surface phantoms have their own limitations. The NURBSbased surface phantoms currently cannot be implemented in a Monte Carlo code and, therefore, need to be converted to voxel phantoms, by what is known as the re-voxelization process, before it is used in a Monte Carlo code. As for polygonal surface phantoms, even though they can be implemented in a Monte Carlo code, their computation speed normally is much slower than voxel phantoms. Kim et al (2011) showed that for the purposes of photon simulations in Monte Carlo dose calculations, a polygonal surface phantom was about 70–150 times slower than a corresponding voxel phantom. Neither can surface phantoms represent inhomogeneous density distribution in organs or tissues, a critical drawback given the very accurate dose calculations necessary in radiation therapy. Recently, Han et al (2013a) developed a new Geant4 solid class based on the direct accelerated geometry for the Monte Carlo library, DagSolid, for fast Monte Carlo simulation in polygonal surface geometry. The main purpose of their study, undertaken to remedy the problem of the polygonal surface phantoms’ repetitive use of computationally expensive raytracing functions to determine geometrical step length, was to utilize a binary search method to minimize the calculation time for geometrical step length calculation. Using the developed DagSolid class, the computation speed of the polygonal surface phantom (PSRK-Man) was significantly improved, by a factor in fact of 34–50, but the improvement was not sufficient, the polygonal surface phantom remaining significantly slower than the corresponding voxel phantom (HDRK-Man) (Han et al 2013b). To address this computation-speed problem, in this study, we developed a new type of computational human phantom, a tetrahedral mesh phantom, by converting a polygonal 3174
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surface phantom to tetrahedral mesh geometry. In principle, the tetrahedral mesh geometry can significantly improve computation speeds due to the fact that a considerable number of facets to be checked by the ray-tracing functions are reduced to just four facets in tetrahedral mesh geometries. Note that several authors have already demonstrated that the tetrahedral mesh geometry is much faster than the polygonal surface geometry in Monte Carlo particle transport simulation (Barker et al 2008a, 2008b, Shen and Wang 2010, Fang 2010, Poole et al 2012). We then implemented the constructed tetrahedral mesh phantom in the Geant4 code for application to Monte Carlo dose calculations. The calculated values of the organ doses and computation speeds were compared with those of the original polygonal surface phantom to evaluate the improvement for different source particles and energies. 2. Material and methods 2.1. PSRK-Man and TetGen code
For constructing a tetrahedral mesh phantom, in this study, a polygonal surface phantom developed by Kim et al (2011), PSRK-Man (figure 1), was converted to the tetrahedral mesh geometry. The PSRK-Man is a high-quality polygonal surface phantom, constructed based on a high-resolution voxel phantom called the HDRK-Man (Kim et al 2008). The PSRK-Man is composed of 26 organs and tissues for effective dose calculation. The PSRK-Man was tetrahedralized using the TetGen code developed by a research group at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS). The code is a C++ program for the generation of a high-quality tetrahedral mesh geometry from a 3D polygonal surface geometry using the Delaunay triangulation algorithm. A detailed description of the TetGen code can be found elsewhere (Si 2006). 2.2. Construction of tetrahedral mesh phantom
In this study, the TetGen code was used to convert the PSRK-Man polygonal surface phantom to a tetrahedral mesh phantom. In preparation, the data of the polygonal surface phantom was first transferred to the TetGen code in the PLY file format. Then, the transferred phantom was tetrahedralized using the TetGen code to construct a primitive tetrahedral mesh phantom according to the ‘pYY’ option, which minimizes the number of generated tetrahedrons while maintaining the original structure of the polygonal surface phantom. Figure 2 (left) shows the constructed primitive tetrahedral mesh phantom. The primitive tetrahedral mesh phantom has no organ region information, making individual-organ identification (ID) impossible. In order to make this possible, in this study, the organ ID numbers were assigned to the tetrahedrons of the phantom via a smesh file (∗ .smesh), one of the output files of the TetGen code. To that end, the smesh file of the primitive tetrahedral mesh phantom was modified by adding both arbitrary point coordinates within each of the independent organs along with corresponding organ ID numbers. Then, with the modified smesh file and according to the ‘pYYA’ option, the tetrahedralization was again performed using the TetGen code so that the organ ID numbers could be automatically assigned to the corresponding tetrahedrons as shown in figure 2 (right). The detailed work flow of the construction of the tetrahedral mesh phantom is illustrated in figure 3. 2.3. Monte Carlo simulations with Geant4 code
To enable comparison of the computation speeds of the tetrahedral mesh phantom and the original polygonal surface phantom, both phantoms were implemented in the Geant4 code. 3175
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Figure 1. Polygonal surface reference Korean male phantom, PSRK-Man.
Figure 2. Constructed tetrahedral mesh phantom without organ region information (left) and with organ region information (right).
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Figure 3. Work flow for construction of tetrahedral mesh phantom.
In the tetrahedral mesh phantom implementation, the G4Tet solid class in the Geant4 code was used. The phantom materials were defined for the G4LogicalVolume class via the organ ID numbers assigned to each of the tetrahedrons in the phantom. Due to the unavailability of the dedicated physical volume class optimized for the tetrahedral mesh geometry in the Geant4 code, each tetrahedron in the phantom was positioned individually according to the G4PVPlacement class, which is the general physical volume class in the Geant4 code. The PSRK-Man polygonal surface phantom was implemented in the Geant4 code on the basis of the G4TessellattedSolid class. Detailed information on the implementation process of a polygonal surface phantom can be found elsewhere (Kim et al 2011). The physics library G4EmLivermorePhysics was used for photon and electron simulations; for neutron and proton simulations, the G4EmLivermorePhysics, HadronPhysicsQGSP_BIC_HP, G4EmExtraPhysics, G4Hadron-ElasticPhysics, G4QStoppingPhysics, and G4IonBinaryCascadePhysics libraries were referenced. For the range cut for all particles in all regions, the default value of 1 mm was used. The simulations were performed using a single core of the AMD OpteronTM6176 (@ 2.3 GHz and 64GB memory) with Geant4 9.5. Prior to the computation-speed comparison, the calculated dose values of the tetrahedral mesh phantom were compared with those of the polygonal surface phantom in order to confirm that the tetrahedral mesh phantom provided the correct dose values. This involved calculation of the organ doses for broad parallel photon beams in the anterior–posterior (AP) direction as well as for seven photon energies (0.01, 0.1, 1, 10, 100, 1000, and 10 000 MeV). The number of primary photons simulated was varied between 106 and 109 to limit the statistical errors to less than 10%. 3177
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Figure 4. Comparison of PSRK-Man polygonal surface phantom (left) and constructed
tetrahedral mesh phantom (right). (a) (b) (a) (b)
(c) (c)
Figure 5. Detailed view of tetrahedral mesh phantom with enlarged views of thin and
complicated organs: (a) oral mucosa, (b) skin, and (c) small intestine.
In the computation-speed comparison, computation times were calculated for four particles (photons, neutrons, electrons, and protons) and four energies (0.01, 1, 100, and 10 000 MeV) in broad AP-direction beams. Because we were interested only in computation speed, not calculated dose values, only 104 primary particles were simulated in all cases. The average computation time and standard deviation were determined by repeating the same calculation ten times. 3178
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Table 1. Comparison of numerical data between the PSRK-Man and tetrahedral mesh phantom.
Mass (g)
Organ Large intestine Lungs Stomach Breast Gonads (Male) Bladder Oesophagus Liver Thyroid Bone Brain Salivary glands Skin Adrenals Gall bladder Heart wall Kidneys Blood Oral mucosa Pancreas Prostate Small intestine Spleen Thymus Eyes Lenses Soft tissue Total
Density (g cm−3) 1.03 0.296 1.05 0.92 1.04 1.04 1.04 1.06 1.05 1.3 1.03 1.03 1.09 1.02 1.03 1.03 1.05 1.06 1.03 1.05 1.03 1.04 1.06 1.03 1.03 1.079 0.976
PSRK-Man
Tetrahedral mesh phantom
330.00 1,123.00 140.00 22.00 29.00 40.00 40.00 1,438.00 15.00 9,649.00 1,522.00 82.00 2,400.00 14.00 13.00 380.00 338.00 118.08 3.73 130.00 12.00 590.00 170.00 40.00 20.00 0.40 49 340.80 68 000.00
330.00 1,123.00 140.00 22.00 29.00 40.00 40.00 1,438.00 15.00 9,649.00 1,522.00 82.00 2,400.00 14.00 13.00 380.00 338.00 118.08 3.73 130.00 12.00 590.00 170.00 40.00 20.00 0.40 49 340.80 68 000.00
Number of polygons
Number of tetrahedrons
PSRK-Man
Tetrahedral mesh phantom
8,000 4,600 1,000 600 200 300 300 1,500 200 41 650 7,000 800 30 000 200 400 1,600 800 3,300 1,600 300 300 15 000 300 300 500 100 – 120 850
14 184 7,262 1,807 843 274 494 708 2,372 257 62 429 11 734 1,147 56 058 250 710 2,801 1,281 4,690 2,400 442 433 27 520 456 421 682 104 202 249 404 008
3. Results and discussion 3.1. Whole-body tetrahedral mesh phantom
In this study, a whole-body tetrahedral mesh phantom was successfully constructed by tetrahedralizing the PSRK-Man polygonal surface phantom using the TetGen code. Figure 4 shows the constructed tetrahedral phantom along with the original polygonal surface phantom. Figure 5 shows a more detailed view, which indicates that the phantom maintains the original structure of the PSRK-Man not only for simple organs, but also for thin or complicated organs, such as the small intestine, skin, and oral mucosa. Table 1 summarizes the numerical data on the constructed tetrahedral mesh phantom and the original PSRK-Man. As in the PSRK-Man, the tetrahedral mesh phantom, for effective dose calculation, also consists of 26 organs. The masses of the organs, moreover, exactly match those of the original PSRK-Man. The constructed tetrahedral mesh phantom is composed of 404 008 tetrahedrons, the minimum number for minimization of computation time and memory usage. It incorporates 120 858 boundary facets, just slightly more than PSRK-Man’s 120 850. These additional eight facets, unavoidably created in the tetrahedralization process, do not in any way distort the phantom’s shape. 3179
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1000
1000
20
2
Organ dose per fluence (pGy cm )
20
2
Large intestine 10
1 5 0.1 0 0.01 -5 1E-3
Polygon phantom Tetrahedron phantom Dose difference
1E-4 1E-5
-10 -15 -20
1E-6 0.01
0.1
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1000
10000
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AP
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Liver 10
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1 5 0.1 0 0.01 -5 1E-3
Polygon phantom Tetrahedron phantom Dose difference
1E-4 1E-5
0.1
1
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1000
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Organ dose per fluence (pGy cm )
15
Oral mucosa 10
10
1 5 0.1 0 0.01 -5 1E-3
Polygon phantom Tetrahedron phantom Dose difference
1E-4 1E-5
-10
Dose difference (%)
2
Organ dose per fluence (pGy cm )
20
AP
-15 -20
0.1
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10
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Bone 10
1 5 0.1 0 0.01 -5 1E-3
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1E-4 1E-5 1E-6 0.01
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0.1
1
10
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1000
10000
A
1000
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Salivary glands 10
10
1 5 0.1 0 0.01 -5 1E-3
Polygon phantom Tetrahedron phantom Dose difference
1E-4 1E-5 1E-6 0.01
-10
Dose difference (%)
2
2
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Organ dose per fluence (pGy cm )
A
Organ dose per fluence (pGy cm )
10
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1E-6 0.01
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1 5 0.1 0 0.01 -5 1E-3
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1E-4 1E-5 1E-6 0.01
-10
Dose difference (%)
Organ dose per fluence (pGy cm )
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-15 -20
0.1
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1000
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A
A
Figure 6. Comparison of organ doses per photon fluence between PSRK-Man and tetrahedral mesh phantom.
3.2. Comparison of calculated dose values
Figure 6 compares the tetrahedral mesh phantom’s and PSRK-Man’s calculated dose values (i.e., the organ-averaged absorbed doses per photon fluence) for six selected organs (large intestine, liver, bone, salivary glands, oral mucosa, and skin) and seven photon energies. The results clearly show that there was no significant dose discrepancy between the phantoms: the average value of the differences ( = the absolute percentile differences between the calculated organ dose values) was as low as 1.2% considering all of the simulations. The results show, in other words, that the constructed tetrahedral mesh phantom provides, within statistical uncertainties, dose values identical to those of the original polygonal surface phantom. 3.3. Comparison of computation speed
The improvement of the computation speed due to the use of the tetrahedral mesh phantom was evaluated by comparing its computation times with those of the PSRK-Man polygonal 3180
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Figure 7. Comparison of particle steps between polygonal surface geometry and tetrahedral mesh geometry for penetrative uncharged particles and non-penetrative charged particles.
surface phantom for 16 simulations each testing one particle (photon, neutron, electron or proton, respectively) with four different energies (0.01, 1, 100, and 10 000 MeV). In addition, the computation times of the tetrahedral phantom were compared with those of the voxelized PSRK-Man with the voxel resolution (1.301 × 1.301 × 1.301 mm3). The voxel phantom was implemented in Geant4 code using G4VNestedParameterisation class. Table 2 summarizes the results. The data show that the tetrahedral mesh phantom was indeed much faster than the original polygonal surface phantom in all of the simulations. The improvement of computation speed was within the range of 150–832 times considering all of the particles and energies other than the low-energy neutrons (0.01 and 1 MeV), for which the improvement was less significant (17.2 and 8.8 times, respectively). Furthermore, it can be seen that computation speeds of the tetrahedral phantom were even faster than those of the voxelized phantom for some cases: the maximum computation time ratio (voxel/tetrahedron) was 6.8 for the proton (0.01 MeV). In general, the computation speed was better improved for charged particles (electrons and protons) than for uncharged particles (photons and neutrons). The relatively less significant improvement of computation speed for the uncharged particles was due mainly to their long mean free path in the medium. To transport a particle as a step in the Geant4 code, both the physical step length and the geometrical step length ( = the particle-direction distance from the current location of the particle to a facet) are calculated, and the shorter one is selected as the particle step length. Therefore, if the geometrical step length is shorter than the physical step length, which is the usual case for uncharged particles in tetrahedral mesh geometries, the particle is moved by the geometrical step length and stopped at the facet in front of the particle. Note that the conversion of polygonal surfaces to tetrahedral mesh significantly increases the number of facets in the simulation geometry, which results in more frequent stopping and calculation of the geometrical step length for uncharged particles (see figure 7), but not so much for charged particles of which the physical step lengths are usually much less than the size of the tetrahedrons. In this study correspondingly, the computation speed typically was relatively less improved for low-energy neutrons than for high-energy neutrons and the other particles. This was due 3181
Computation time (sec)
Ratio
Particle
Energy (MeV)
Polygonal surface phantom (A)
Tetrahedral mesh phantom (B)
Voxelized phantom (C)
A/B
C/B
Gamma
0.01 1 100 10 000 0.01 1 100 10 000 0.01 1 100 10 000 0.01 1 100 10 000
214.4 ( ± 68.3) 1491.3 ( ± 487.1) 13 017.0 ( ± 3767.5) 24 917.2 ( ± 6579.6) 716.2 ( ± 189.7) 737.6 ( ± 73.9) 16 493.7 ( ± 3064.5) 74 457.4 ( ± 16 970.9) 352.5 ( ± 93.3) 46 10.4 ( ± 1055.2) 82 624.0 ( ± 14 317.1) 103 518.9 ( ± 17 826.2) 68.3 ( ± 16.0) 2492.7 ( ± 746.7) 197 032.4 ( ± 15 929.1) 154 536.0 ( ± 31 712.8)
0.6 ( ± 0.1) 4.7 ( ± 0.3) 27.6 ( ± 2.2) 43.3 ( ± 4.2) 41.7 ( ± 1.7) 84.4 ( ± 2.3) 91.7 ( ± 8.1) 268.8 ( ± 28.3) 0.7 ( ± 0.1) 5.5 ( ± 0.4) 136.8 ( ± 7.6) 174.0 ( ± 9.5) 0.5 ( ± 0.1) 3.9 ( ± 0.1) 1025.1 ( ± 34.4) 460.9 ( ± 19.8)
2.4 ( ± 0.1) 7.4 ( ± 0.3) 30.4 ( ± 1.6) 45.8 ( ± 3.2) 87.6 ( ± 2.8) 165.0 ( ± 7.1) 85.0 ( ± 3.2) 246.0 ( ± 14.3) 3.1 ( ± 0.1) 7.5 ( ± 0.1) 141.3 ( ± 3.1) 174.9 ( ± 6.2) 3.4 ( ± 0.1) 5.9 ( ± 0.1) 863.1 ( ± 10.1) 403.4 ( ± 15.0)
375.4 ( ± 124.1) 314.9 ( ± 104.8) 471.9 ( ± 141.5) 575.3 ( ± 68.3) 17.2 ( ± 4.6) 8.8 ( ± 0.9) 179.9 ( ± 37.0) 277.0 ( ± 69.5) 532.5 ( ± 163.9) 831.6 ( ± 198.0) 603.8 ( ± 109.8) 595.1 ( ± 107.5) 149.4 ( ± 38.3) 645.3 ( ± 193.6) 192.2 ( ± 16.8) 335.3 ( ± 70.3)
4 ( ± 0.7) 1.6 ( ± 0.1) 1.1 ( ± 0.1) 1.1 ( ± 0.1) 2.1 ( ± 0.1) 2.0 ( ± 0.1) 0.9 ( ± 0.1) 0.9 ( ± 0.1) 4.4 ( ± 0.6) 1.4 ( ± 0.1) 1.0 ( ± 0.1) 1.0 ( ± 0.1) 6.8 ( ± 1.4) 1.5 ( ± 0.0) 0.8 ( ± 0.0) 0.9 ( ± 0.0)
Neutron 3182
Electron
Proton
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Table 2. Computation times of tetrahedral mesh phantom, PSRK-Man polygonal surface phantom, and voxelized PSRK-Man.
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mainly to the fact that the low-energy neutrons do not generate many high-energy secondary charged particles, which have long ranges in the medium. Note also that the computationspeed improvement was less significant for uncharged particles than for charged particles. Additionally, for neutrons, a significant fraction of the total computation time was the calculation time for the physical step length, due to the massive neutron cross-section data, which situation was not improved using the tetrahedral mesh geometry in the simulation. In contrast to the low-energy neutrons, for the high-energy neutrons (100 MeV), the computation speed was significantly improved, by a factor in fact of 179–276, due mainly to the high-energy neutrons’ generation of many high-energy secondary charged particles (ions and electrons) of long range, for which the improvement of computation speed by means of the tetrahedral mesh geometry is more effective. In addition, the high-energy neutrons (>20 MeV) are transported using theoretical physics models, not using massive evaluated cross-section data, which partially contributes to improvement of computation speed. 4. Conclusions In this study, we developed a new type of computational human phantom, a tetrahedral mesh phantom, by conversion of an existing polygonal surface phantom to tetrahedral mesh geometry. The external shape and the organs of the tetrahedral mesh phantom were identical to those of the original polygonal surface phantom. The dose values calculated with the tetrahedral phantom, likewise, were identical to those of the original polygonal surface phantom (within statistical uncertainties). Use of the tetrahedral mesh phantom significantly improved the computation speed, specifically by a factor of 150–832 considering all of the particles and simulated energies other than the low-energy neutrons (0.01 and 1 MeV), for which the improvement was less significant (17 and 8.7 times, respectively). In addition to improved computation speed, the tetrahedral mesh phantom offers many other advantages. Indeed, it combines the advantages of the voxel phantom (fast computation speed and inhomogeneous density representation) and the surface phantom (deformability, flexibility, and surface smoothness). Note that the density distribution in an organ or tissue can easily be modeled using a fine tetrahedral mesh. With regard to the tetrahedral mesh geometry’s easy deformability, many investigators have already developed tetrahedral-mesh-based deformable objects for many interactive virtual reality applications such as surgery simulators and video-games. Considering especially this deformability, we believe that the tetrahedral mesh geometry is currently the most suitable for 4D computational human phantoms in which the organs and tissue move and deform during Monte Carlo particle transport simulation. Acknowledgments This research was supported by the General Researcher Program, the National Nuclear R&D Program and Global PhD Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (nos NRF-20110025496, NRF-2012M2A8A5026057, NRF-2011-Global PhD Fellowship Program). References Barker T, Bird A, Thetford R and Cooper A 2008a CAD import for MONK and MCBEND by converting to tetrahedral mesh format Trans. Am. Nucl. Soc. 99 570–1 3183
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Barker T, Bird A, Thetford R and Cooper A 2008b Use of tetrahedral mesh geometry to import a converted CAD file for shielding and criticality calculations with MONK and MCBEND 11th Int. Conf. on Radiation Shielding (ICRS-11) and 14th Topical Meeting on Radiation Protection and Shielding (RPS-2008), (Pine Mountain, Georgia, USA) pp 13–18 Cassola V F, Lima V J D, Kramer R and Khoury H J 2010 FASH and MASH: female and male adult human phantoms based on polygon mesh surfaces: I. Development of the anatomy Phys. Med. Biol. 55 133–62 Cassola V F, Milian F M, Kramer R, Lira C A B D and Khoury H J 2011 Standing adult human phantoms based on 10th, 50th and 90th mass and height percentiles of male and female Caucasian populations Phys. Med. Biol. 56 3749–72 Ding A, Mille M M, Liu T, Caracappa P F and Xu X G 2012 Extension of RPI-adult male and female computational phantoms to obese patients and a Monte Carlo study of the effect on CT imaging dose Phys. Med. Biol. 57 2441–59 Fang Q 2010 Mesh-based Monte Carlo method using fast ray-tracing in Pl¨ucker coordinates Biomed. Opt. Express 1 165–75 Gibbs S and Pujol J 1982 A Monte Carlo method for patient dosimetry from diagnostic x-ray Dentomaxillofac. Radiol. 11 25–33 Han B, Zhang J Y, Na Y H, Caracappa P F and Xu X G 2010 Modelling and Monte Carlo organ dose calculations for workers walking on ground contaminated with Cs-137 and Co-60 gamma sources Radiat. Prot. Dosim. 141 299–304 Han M C, Kim C H, Jeong J H, Yeom Y S, Kim S, Wilson P P H and Apostolakis J 2013a DagSolid: a new Geant4 solid class for fast simulation in polygon-mesh geometry Phys. Med. Biol. 58 4595–609 Han M C, Yeom Y S, Kim C H, Jeong J H, Kim S, Wilson P P H and Apostolakis J 2013b Development of Geant4 class for fast Monte Carlo simulation of polygonal surface based phantom 4th Int. workshop on computational phantoms for radiation protection, imaging and radiotherapy (CP2013), (Zurich, Switzerland) pp 66–67 Hurtado J L, Lee C, Lodwick D, Goede T, Williams J L and Bolch W E 2012 Hybrid computational phantoms representing the reference adult male and adult female: construction and applications for retrospective dosimetry Health Phys. 102 292–304 ICRP 2007 The 2007 Recommendations of the International Commission on Radiological Protection ICRP Publication 103 (Oxford: Elsevier) Johnson P, Lee C, Johnson K, Siragusa D and Bolch W E 2009 The influence of patient size on dose conversion coefficients: a hybrid phantom study for adult cardiac catheterization Phys. Med. Biol. 54 3613–29 Kim C H, Choi S H, Jeong J H, Lee C and Chung M S 2008 HDRK-Man: a whole-body voxel model based on high-resolution color slice images of a Korean adult male cadaver Phys. Med. Biol. 53 4093–106 Kim C H, Jeong J H, Bolch W E, Cho K W and Hwang S B 2011 A polygonal surface reference Korean male phantom (PSRK-Man) and its direct implementation in Geant4 Monte Carlo simulation Phys. Med. Biol. 56 3137–61 Lee C, Lodwick D, Hasenauer D, Williams J L, Lee C and Bolch W E 2007 Hybrid computational phantoms of the male and female newborn patient: NURBS-based whole-body models Phys. Med. Biol. 52 3309–33 Lee C, Lodwick D, Hurtado J, Pafundi D, LWilliams J and Bolch W E 2010 The UF family of reference hybrid phantoms for computational radiation dosimetry Phys. Med. Biol. 55 339–63 Lee C, Lodwick D, Williams J L and Bolch W E 2008 Hybrid computational phantoms of the 15-year male and female adolescent: applications to CT organ dosimetry for patients of variable morphometry Med. Phys. 35 2366–82 Matthew R M, John W G, John P A, Roger Y S and Wesley B 2011 The UF family of hybrid phantoms of the developing human fetus for computational radiation dosimetry Phys. Med. Biol. 56 4839 Na Y H, Zhang B Q, Zhang J Y, Caracappa P F and Xu X G 2010 Deformable adult human phantoms for radiation protection dosimetry: anthropometric data representing size distributions of adult worker populations and software algorithms Phys. Med. Biol. 55 3789–811 Poole C M, Cornelius I, Trapp J V and Langton C M 2012 Fast tessellated solid navigation in GEANT4 IEEE Trans. Nucl. Sci. 59 1695–701 Rajon D A, Jokisch D W, Patton P W, Shah A P and Bolch W E 2000 Voxel size effects in threedimensional nuclear magnetic resonance microscopy performed for trabecular bone dosimetry Med. Phys. 27 2624–35 3184
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Segars W P, Tsui B M, Lalush D S, Frey E C, King M A and Manocha D 2001 Development and application of the new dynamic nurbs-based cardiac-torso (NCAT) phantom J. Nucl. Med. 42 7P-P Shen H and Wang G 2010 A tetrahedron-based inhomogeneous Monte Carlo optical simulator Phys. Med. Biol. 55 947–62 Si H 2006 A quality tetrahedral mesh generator and three-dimensional delaunay triangulator version 1.4 User’s Manual http://tetgen.berlios.de/index.html Snyder W S, Ford M R, Warner G G and Fisher H L Jr 1969 Estimation of absorbed fractions for monoenergetic photon sources uniformly distributed in various organs of a heterogeneous phantom J. Nucl. Med. 10 (Suppl. 3) 7–52 Su L, Han B and Xu X G 2012 Calculated organ equivalent doses for individuals in a sitting posture above a contaminated ground and a PET imaging room Radiat. Prot. Dosim. 148 439–43 Zhang J Y, Na Y H, Caracappa P F and Xu X G 2009 RPI-AM and RPI-AF, a pair of mesh-based, size-adjustable adult male and female computational phantoms using ICRP-89 parameters and their calculations for organ doses from monoenergetic photon beams Phys. Med. Biol. 54 5885–908
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Tetrahedral-mesh-based computational human phantom for fast Monte Carlo dose calculations
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Institute of Physics and Engineering in Medicine Phys. Med. Biol. 59 (2014) 3173–3185
Physics in Medicine and Biology
doi:10.1088/0031-9155/59/12/3173
Tetrahedral-mesh-based computational human phantom for fast Monte Carlo dose calculations Yeon Soo Yeom 1 , Jong Hwi Jeong 2 , Min Cheol Han 1 and Chan Hyeong Kim 1 1
Department of Nuclear Engineering, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul 133-791, Korea 2 Center for Proton Therapy, National Cancer Center, 111 Jungbalsan-ro, Ilsandong-gu, Goyang-si, Gyonggi-do 410-769, Korea E-mail: [email protected] Received 22 October 2013, revised 24 April 2014 Accepted for publication 2 May 2014 Published 27 May 2014 Abstract
Although polygonal-surface computational human phantoms can address several critical limitations of conventional voxel phantoms, their Monte Carlo simulation speeds are much slower than those of voxel phantoms. In this study, we sought to overcome this problem by developing a new type of computational human phantom, a tetrahedral mesh phantom, by converting a polygonal surface phantom to a tetrahedral mesh geometry. The constructed phantom was implemented in the Geant4 Monte Carlo code to calculate organ doses as well as to measure computation speed, the values were then compared with those for the original polygonal surface phantom. It was found that using the tetrahedral mesh phantom significantly improved the computation speed by factors of between 150 and 832 considering all of the particles and simulated energies other than the low-energy neutrons (0.01 and 1 MeV), for which the improvement was less significant (17.2 and 8.8 times, respectively). Keywords: computational phantom, tetrahedron, Monte Carlo simulation, computation speed (Some figures may appear in colour only in the online journal)
1. Introduction Computational human phantoms are widely used for Monte Carlo dose calculations in many different fields, including radiation protection, radiation therapy, medical imaging, and others. 0031-9155/14/123173+13$33.00
© 2014 Institute of Physics and Engineering in Medicine Printed in the UK & the USA 3173
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The first-generation phantoms, which were mathematical phantoms, also called stylized phantoms, were proposed by Snyder et al (1969). Composed of simplified organ models defined by mathematical equations, they had serious limitations in representing the real anatomy of a human body. In the 1980s, Gibbs and Pujol (1982) first introduced a voxel-based computational human phantom constructed using tomographic images of a human body, which provided much more anatomical realism than mathematical phantoms. Acknowledging the advantages of voxel phantoms, the International Commission on Radiological Protection (ICRP) officially adopted a set of male and female voxel phantoms as the ICRP reference phantoms (ICRP 2007). The voxel phantoms themselves, however, have several limitations. First, they show stairstepped surfaces, causing the so-called ‘voxel effect’ that results in inaccurate dose calculations (Rajon et al 2000). Secondly, due to the finite voxel resolution, it is difficult to model very thin structures such as skin, oral mucosa, or the extrathoracic region, which are very important structures for effective dose calculation. Lastly, the voxel phantoms are rigid, making it impossible to adjust them to model non-upright postures (e.g. that of a radiation worker sitting on a chair-type whole-body counter). Moving or deforming the internal organs of a voxel phantom during Monte Carlo simulation is also impossible. Recently, several investigators (Segars et al 2001, Lee et al 2007, Zhang et al 2009, Cassola et al 2010, Kim et al 2011) have developed surface-based computational human phantoms based on non-uniform rational B-spline (NURBS) or polygonal mesh surfaces. These surface phantoms successfully addressed the aforementioned limitations of the voxel phantoms. That is, they allowed for the modeling of very thin and complicated organs with smooth surfaces as well as posture adjustment for moving or deformation of internal organs as necessary. On this basis, a variety of deformed-surface phantoms have been constructed for modeling of various ages (Lee et al 2008, 2010, Hurtado et al 2012, Matthew et al 2011), body sizes (Johnson et al 2009, Na et al 2010, Ding et al 2012, Cassola et al 2011), and postures (Han et al 2010, Su et al 2012). Unfortunately however, the surface phantoms have their own limitations. The NURBSbased surface phantoms currently cannot be implemented in a Monte Carlo code and, therefore, need to be converted to voxel phantoms, by what is known as the re-voxelization process, before it is used in a Monte Carlo code. As for polygonal surface phantoms, even though they can be implemented in a Monte Carlo code, their computation speed normally is much slower than voxel phantoms. Kim et al (2011) showed that for the purposes of photon simulations in Monte Carlo dose calculations, a polygonal surface phantom was about 70–150 times slower than a corresponding voxel phantom. Neither can surface phantoms represent inhomogeneous density distribution in organs or tissues, a critical drawback given the very accurate dose calculations necessary in radiation therapy. Recently, Han et al (2013a) developed a new Geant4 solid class based on the direct accelerated geometry for the Monte Carlo library, DagSolid, for fast Monte Carlo simulation in polygonal surface geometry. The main purpose of their study, undertaken to remedy the problem of the polygonal surface phantoms’ repetitive use of computationally expensive raytracing functions to determine geometrical step length, was to utilize a binary search method to minimize the calculation time for geometrical step length calculation. Using the developed DagSolid class, the computation speed of the polygonal surface phantom (PSRK-Man) was significantly improved, by a factor in fact of 34–50, but the improvement was not sufficient, the polygonal surface phantom remaining significantly slower than the corresponding voxel phantom (HDRK-Man) (Han et al 2013b). To address this computation-speed problem, in this study, we developed a new type of computational human phantom, a tetrahedral mesh phantom, by converting a polygonal 3174
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surface phantom to tetrahedral mesh geometry. In principle, the tetrahedral mesh geometry can significantly improve computation speeds due to the fact that a considerable number of facets to be checked by the ray-tracing functions are reduced to just four facets in tetrahedral mesh geometries. Note that several authors have already demonstrated that the tetrahedral mesh geometry is much faster than the polygonal surface geometry in Monte Carlo particle transport simulation (Barker et al 2008a, 2008b, Shen and Wang 2010, Fang 2010, Poole et al 2012). We then implemented the constructed tetrahedral mesh phantom in the Geant4 code for application to Monte Carlo dose calculations. The calculated values of the organ doses and computation speeds were compared with those of the original polygonal surface phantom to evaluate the improvement for different source particles and energies. 2. Material and methods 2.1. PSRK-Man and TetGen code
For constructing a tetrahedral mesh phantom, in this study, a polygonal surface phantom developed by Kim et al (2011), PSRK-Man (figure 1), was converted to the tetrahedral mesh geometry. The PSRK-Man is a high-quality polygonal surface phantom, constructed based on a high-resolution voxel phantom called the HDRK-Man (Kim et al 2008). The PSRK-Man is composed of 26 organs and tissues for effective dose calculation. The PSRK-Man was tetrahedralized using the TetGen code developed by a research group at the Weierstrass Institute for Applied Analysis and Stochastics (WIAS). The code is a C++ program for the generation of a high-quality tetrahedral mesh geometry from a 3D polygonal surface geometry using the Delaunay triangulation algorithm. A detailed description of the TetGen code can be found elsewhere (Si 2006). 2.2. Construction of tetrahedral mesh phantom
In this study, the TetGen code was used to convert the PSRK-Man polygonal surface phantom to a tetrahedral mesh phantom. In preparation, the data of the polygonal surface phantom was first transferred to the TetGen code in the PLY file format. Then, the transferred phantom was tetrahedralized using the TetGen code to construct a primitive tetrahedral mesh phantom according to the ‘pYY’ option, which minimizes the number of generated tetrahedrons while maintaining the original structure of the polygonal surface phantom. Figure 2 (left) shows the constructed primitive tetrahedral mesh phantom. The primitive tetrahedral mesh phantom has no organ region information, making individual-organ identification (ID) impossible. In order to make this possible, in this study, the organ ID numbers were assigned to the tetrahedrons of the phantom via a smesh file (∗ .smesh), one of the output files of the TetGen code. To that end, the smesh file of the primitive tetrahedral mesh phantom was modified by adding both arbitrary point coordinates within each of the independent organs along with corresponding organ ID numbers. Then, with the modified smesh file and according to the ‘pYYA’ option, the tetrahedralization was again performed using the TetGen code so that the organ ID numbers could be automatically assigned to the corresponding tetrahedrons as shown in figure 2 (right). The detailed work flow of the construction of the tetrahedral mesh phantom is illustrated in figure 3. 2.3. Monte Carlo simulations with Geant4 code
To enable comparison of the computation speeds of the tetrahedral mesh phantom and the original polygonal surface phantom, both phantoms were implemented in the Geant4 code. 3175
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Figure 1. Polygonal surface reference Korean male phantom, PSRK-Man.
Figure 2. Constructed tetrahedral mesh phantom without organ region information (left) and with organ region information (right).
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Figure 3. Work flow for construction of tetrahedral mesh phantom.
In the tetrahedral mesh phantom implementation, the G4Tet solid class in the Geant4 code was used. The phantom materials were defined for the G4LogicalVolume class via the organ ID numbers assigned to each of the tetrahedrons in the phantom. Due to the unavailability of the dedicated physical volume class optimized for the tetrahedral mesh geometry in the Geant4 code, each tetrahedron in the phantom was positioned individually according to the G4PVPlacement class, which is the general physical volume class in the Geant4 code. The PSRK-Man polygonal surface phantom was implemented in the Geant4 code on the basis of the G4TessellattedSolid class. Detailed information on the implementation process of a polygonal surface phantom can be found elsewhere (Kim et al 2011). The physics library G4EmLivermorePhysics was used for photon and electron simulations; for neutron and proton simulations, the G4EmLivermorePhysics, HadronPhysicsQGSP_BIC_HP, G4EmExtraPhysics, G4Hadron-ElasticPhysics, G4QStoppingPhysics, and G4IonBinaryCascadePhysics libraries were referenced. For the range cut for all particles in all regions, the default value of 1 mm was used. The simulations were performed using a single core of the AMD OpteronTM6176 (@ 2.3 GHz and 64GB memory) with Geant4 9.5. Prior to the computation-speed comparison, the calculated dose values of the tetrahedral mesh phantom were compared with those of the polygonal surface phantom in order to confirm that the tetrahedral mesh phantom provided the correct dose values. This involved calculation of the organ doses for broad parallel photon beams in the anterior–posterior (AP) direction as well as for seven photon energies (0.01, 0.1, 1, 10, 100, 1000, and 10 000 MeV). The number of primary photons simulated was varied between 106 and 109 to limit the statistical errors to less than 10%. 3177
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Figure 4. Comparison of PSRK-Man polygonal surface phantom (left) and constructed
tetrahedral mesh phantom (right). (a) (b) (a) (b)
(c) (c)
Figure 5. Detailed view of tetrahedral mesh phantom with enlarged views of thin and
complicated organs: (a) oral mucosa, (b) skin, and (c) small intestine.
In the computation-speed comparison, computation times were calculated for four particles (photons, neutrons, electrons, and protons) and four energies (0.01, 1, 100, and 10 000 MeV) in broad AP-direction beams. Because we were interested only in computation speed, not calculated dose values, only 104 primary particles were simulated in all cases. The average computation time and standard deviation were determined by repeating the same calculation ten times. 3178
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Table 1. Comparison of numerical data between the PSRK-Man and tetrahedral mesh phantom.
Mass (g)
Organ Large intestine Lungs Stomach Breast Gonads (Male) Bladder Oesophagus Liver Thyroid Bone Brain Salivary glands Skin Adrenals Gall bladder Heart wall Kidneys Blood Oral mucosa Pancreas Prostate Small intestine Spleen Thymus Eyes Lenses Soft tissue Total
Density (g cm−3) 1.03 0.296 1.05 0.92 1.04 1.04 1.04 1.06 1.05 1.3 1.03 1.03 1.09 1.02 1.03 1.03 1.05 1.06 1.03 1.05 1.03 1.04 1.06 1.03 1.03 1.079 0.976
PSRK-Man
Tetrahedral mesh phantom
330.00 1,123.00 140.00 22.00 29.00 40.00 40.00 1,438.00 15.00 9,649.00 1,522.00 82.00 2,400.00 14.00 13.00 380.00 338.00 118.08 3.73 130.00 12.00 590.00 170.00 40.00 20.00 0.40 49 340.80 68 000.00
330.00 1,123.00 140.00 22.00 29.00 40.00 40.00 1,438.00 15.00 9,649.00 1,522.00 82.00 2,400.00 14.00 13.00 380.00 338.00 118.08 3.73 130.00 12.00 590.00 170.00 40.00 20.00 0.40 49 340.80 68 000.00
Number of polygons
Number of tetrahedrons
PSRK-Man
Tetrahedral mesh phantom
8,000 4,600 1,000 600 200 300 300 1,500 200 41 650 7,000 800 30 000 200 400 1,600 800 3,300 1,600 300 300 15 000 300 300 500 100 – 120 850
14 184 7,262 1,807 843 274 494 708 2,372 257 62 429 11 734 1,147 56 058 250 710 2,801 1,281 4,690 2,400 442 433 27 520 456 421 682 104 202 249 404 008
3. Results and discussion 3.1. Whole-body tetrahedral mesh phantom
In this study, a whole-body tetrahedral mesh phantom was successfully constructed by tetrahedralizing the PSRK-Man polygonal surface phantom using the TetGen code. Figure 4 shows the constructed tetrahedral phantom along with the original polygonal surface phantom. Figure 5 shows a more detailed view, which indicates that the phantom maintains the original structure of the PSRK-Man not only for simple organs, but also for thin or complicated organs, such as the small intestine, skin, and oral mucosa. Table 1 summarizes the numerical data on the constructed tetrahedral mesh phantom and the original PSRK-Man. As in the PSRK-Man, the tetrahedral mesh phantom, for effective dose calculation, also consists of 26 organs. The masses of the organs, moreover, exactly match those of the original PSRK-Man. The constructed tetrahedral mesh phantom is composed of 404 008 tetrahedrons, the minimum number for minimization of computation time and memory usage. It incorporates 120 858 boundary facets, just slightly more than PSRK-Man’s 120 850. These additional eight facets, unavoidably created in the tetrahedralization process, do not in any way distort the phantom’s shape. 3179
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Organ dose per fluence (pGy cm )
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-15 -20
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A
Figure 6. Comparison of organ doses per photon fluence between PSRK-Man and tetrahedral mesh phantom.
3.2. Comparison of calculated dose values
Figure 6 compares the tetrahedral mesh phantom’s and PSRK-Man’s calculated dose values (i.e., the organ-averaged absorbed doses per photon fluence) for six selected organs (large intestine, liver, bone, salivary glands, oral mucosa, and skin) and seven photon energies. The results clearly show that there was no significant dose discrepancy between the phantoms: the average value of the differences ( = the absolute percentile differences between the calculated organ dose values) was as low as 1.2% considering all of the simulations. The results show, in other words, that the constructed tetrahedral mesh phantom provides, within statistical uncertainties, dose values identical to those of the original polygonal surface phantom. 3.3. Comparison of computation speed
The improvement of the computation speed due to the use of the tetrahedral mesh phantom was evaluated by comparing its computation times with those of the PSRK-Man polygonal 3180
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Figure 7. Comparison of particle steps between polygonal surface geometry and tetrahedral mesh geometry for penetrative uncharged particles and non-penetrative charged particles.
surface phantom for 16 simulations each testing one particle (photon, neutron, electron or proton, respectively) with four different energies (0.01, 1, 100, and 10 000 MeV). In addition, the computation times of the tetrahedral phantom were compared with those of the voxelized PSRK-Man with the voxel resolution (1.301 × 1.301 × 1.301 mm3). The voxel phantom was implemented in Geant4 code using G4VNestedParameterisation class. Table 2 summarizes the results. The data show that the tetrahedral mesh phantom was indeed much faster than the original polygonal surface phantom in all of the simulations. The improvement of computation speed was within the range of 150–832 times considering all of the particles and energies other than the low-energy neutrons (0.01 and 1 MeV), for which the improvement was less significant (17.2 and 8.8 times, respectively). Furthermore, it can be seen that computation speeds of the tetrahedral phantom were even faster than those of the voxelized phantom for some cases: the maximum computation time ratio (voxel/tetrahedron) was 6.8 for the proton (0.01 MeV). In general, the computation speed was better improved for charged particles (electrons and protons) than for uncharged particles (photons and neutrons). The relatively less significant improvement of computation speed for the uncharged particles was due mainly to their long mean free path in the medium. To transport a particle as a step in the Geant4 code, both the physical step length and the geometrical step length ( = the particle-direction distance from the current location of the particle to a facet) are calculated, and the shorter one is selected as the particle step length. Therefore, if the geometrical step length is shorter than the physical step length, which is the usual case for uncharged particles in tetrahedral mesh geometries, the particle is moved by the geometrical step length and stopped at the facet in front of the particle. Note that the conversion of polygonal surfaces to tetrahedral mesh significantly increases the number of facets in the simulation geometry, which results in more frequent stopping and calculation of the geometrical step length for uncharged particles (see figure 7), but not so much for charged particles of which the physical step lengths are usually much less than the size of the tetrahedrons. In this study correspondingly, the computation speed typically was relatively less improved for low-energy neutrons than for high-energy neutrons and the other particles. This was due 3181
Computation time (sec)
Ratio
Particle
Energy (MeV)
Polygonal surface phantom (A)
Tetrahedral mesh phantom (B)
Voxelized phantom (C)
A/B
C/B
Gamma
0.01 1 100 10 000 0.01 1 100 10 000 0.01 1 100 10 000 0.01 1 100 10 000
214.4 ( ± 68.3) 1491.3 ( ± 487.1) 13 017.0 ( ± 3767.5) 24 917.2 ( ± 6579.6) 716.2 ( ± 189.7) 737.6 ( ± 73.9) 16 493.7 ( ± 3064.5) 74 457.4 ( ± 16 970.9) 352.5 ( ± 93.3) 46 10.4 ( ± 1055.2) 82 624.0 ( ± 14 317.1) 103 518.9 ( ± 17 826.2) 68.3 ( ± 16.0) 2492.7 ( ± 746.7) 197 032.4 ( ± 15 929.1) 154 536.0 ( ± 31 712.8)
0.6 ( ± 0.1) 4.7 ( ± 0.3) 27.6 ( ± 2.2) 43.3 ( ± 4.2) 41.7 ( ± 1.7) 84.4 ( ± 2.3) 91.7 ( ± 8.1) 268.8 ( ± 28.3) 0.7 ( ± 0.1) 5.5 ( ± 0.4) 136.8 ( ± 7.6) 174.0 ( ± 9.5) 0.5 ( ± 0.1) 3.9 ( ± 0.1) 1025.1 ( ± 34.4) 460.9 ( ± 19.8)
2.4 ( ± 0.1) 7.4 ( ± 0.3) 30.4 ( ± 1.6) 45.8 ( ± 3.2) 87.6 ( ± 2.8) 165.0 ( ± 7.1) 85.0 ( ± 3.2) 246.0 ( ± 14.3) 3.1 ( ± 0.1) 7.5 ( ± 0.1) 141.3 ( ± 3.1) 174.9 ( ± 6.2) 3.4 ( ± 0.1) 5.9 ( ± 0.1) 863.1 ( ± 10.1) 403.4 ( ± 15.0)
375.4 ( ± 124.1) 314.9 ( ± 104.8) 471.9 ( ± 141.5) 575.3 ( ± 68.3) 17.2 ( ± 4.6) 8.8 ( ± 0.9) 179.9 ( ± 37.0) 277.0 ( ± 69.5) 532.5 ( ± 163.9) 831.6 ( ± 198.0) 603.8 ( ± 109.8) 595.1 ( ± 107.5) 149.4 ( ± 38.3) 645.3 ( ± 193.6) 192.2 ( ± 16.8) 335.3 ( ± 70.3)
4 ( ± 0.7) 1.6 ( ± 0.1) 1.1 ( ± 0.1) 1.1 ( ± 0.1) 2.1 ( ± 0.1) 2.0 ( ± 0.1) 0.9 ( ± 0.1) 0.9 ( ± 0.1) 4.4 ( ± 0.6) 1.4 ( ± 0.1) 1.0 ( ± 0.1) 1.0 ( ± 0.1) 6.8 ( ± 1.4) 1.5 ( ± 0.0) 0.8 ( ± 0.0) 0.9 ( ± 0.0)
Neutron 3182
Electron
Proton
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Table 2. Computation times of tetrahedral mesh phantom, PSRK-Man polygonal surface phantom, and voxelized PSRK-Man.
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mainly to the fact that the low-energy neutrons do not generate many high-energy secondary charged particles, which have long ranges in the medium. Note also that the computationspeed improvement was less significant for uncharged particles than for charged particles. Additionally, for neutrons, a significant fraction of the total computation time was the calculation time for the physical step length, due to the massive neutron cross-section data, which situation was not improved using the tetrahedral mesh geometry in the simulation. In contrast to the low-energy neutrons, for the high-energy neutrons (100 MeV), the computation speed was significantly improved, by a factor in fact of 179–276, due mainly to the high-energy neutrons’ generation of many high-energy secondary charged particles (ions and electrons) of long range, for which the improvement of computation speed by means of the tetrahedral mesh geometry is more effective. In addition, the high-energy neutrons (>20 MeV) are transported using theoretical physics models, not using massive evaluated cross-section data, which partially contributes to improvement of computation speed. 4. Conclusions In this study, we developed a new type of computational human phantom, a tetrahedral mesh phantom, by conversion of an existing polygonal surface phantom to tetrahedral mesh geometry. The external shape and the organs of the tetrahedral mesh phantom were identical to those of the original polygonal surface phantom. The dose values calculated with the tetrahedral phantom, likewise, were identical to those of the original polygonal surface phantom (within statistical uncertainties). Use of the tetrahedral mesh phantom significantly improved the computation speed, specifically by a factor of 150–832 considering all of the particles and simulated energies other than the low-energy neutrons (0.01 and 1 MeV), for which the improvement was less significant (17 and 8.7 times, respectively). In addition to improved computation speed, the tetrahedral mesh phantom offers many other advantages. Indeed, it combines the advantages of the voxel phantom (fast computation speed and inhomogeneous density representation) and the surface phantom (deformability, flexibility, and surface smoothness). Note that the density distribution in an organ or tissue can easily be modeled using a fine tetrahedral mesh. With regard to the tetrahedral mesh geometry’s easy deformability, many investigators have already developed tetrahedral-mesh-based deformable objects for many interactive virtual reality applications such as surgery simulators and video-games. Considering especially this deformability, we believe that the tetrahedral mesh geometry is currently the most suitable for 4D computational human phantoms in which the organs and tissue move and deform during Monte Carlo particle transport simulation. Acknowledgments This research was supported by the General Researcher Program, the National Nuclear R&D Program and Global PhD Fellowship Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning (nos NRF-20110025496, NRF-2012M2A8A5026057, NRF-2011-Global PhD Fellowship Program). References Barker T, Bird A, Thetford R and Cooper A 2008a CAD import for MONK and MCBEND by converting to tetrahedral mesh format Trans. Am. Nucl. Soc. 99 570–1 3183
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