A modular description for collimator geometry in EGS simulation tasks Alessandro
Bevilacqua, Dante Bollini, Renato Campanini, Mirko Gombia, Student Nice Lanconelli, Student Member, IEEE, Alessandro Riccardi
Abstract-EGS is a very common Monte Carlo code, used in the simulation of Nuclear Medicine devices. Simulation techniques are particularly useful, in order to optimize collimator configuration and camera design in Single Photon Emission studies. Using the EGS code, users must define the geometry where particles are transported. This can be both a very hard task and a source of inefficiency, especially in case of complex geometries as, for instance, hexagonal hole collimators or pixellated detectors. In this paper we present a modular description for such geometries. Regions are seen as a basic cell repeated in the space. Our method allows the computation of the region to which a point belongs to in a few steps; thus we are able to calculate this region in a reduced number of operations, independently from the collimator and detector dimensions. We validated the modular description, by performing the characterization of two different collimators: one with square holes and one with hexagonal holes. With modular description we can reduce the computational time up to 25%, with respect to a “traditional” geometric description. It is also possible to simulate a breast phantom for different configurations: each run (one phantom relative to a 10 min acquisition with full photons and electrons transport) would take almost 24 hours on a cluster of four PI11 800 MHz processors. lar
KeywordsGeometry.
Monte
Carlo,
Nuclear
Medicine,
EGS,
IEEE,
efficient and simple description of such geometry. A “traditional” way to describe regions in EGS code is to consider them as bounded by the intersection of different planes [3]. In Monte Carlo simulation tasks the high computational time is often a drawback: realistic calculation can take several days of CPU time; hence improved calculating efficiency is desirable. Four main groups of methods exist, which aim to reduce computational time: variance reduction techniques, physical processes approximation, efficient description of the geometry and code parallelization. All these approaches can be used together, saving a large amount of time. This work is a part of a bigger project regarding the development of a Germanium detector for scintimammography [4]. In this paper we present a modular description for collimators and detectors for EGS code: our procedure allows both an easy way to characterize the geometric scenario and an efficient description of the various regions. We also distribute the computational load among four processors, by means of message-passing libraries.
Modu-
II. A.
I. INTRODUCTION
T
HE use of Monte Carlo simulation techniques is widespread in the development of Nuclear Medicine apparatus [l]. They are particularly useful when experimental measurements are impractical or to answer questions which can not be addressed by experimental investigations. Simulations can also be helpful in the choice of the camera design in SPECT studies: parameters such as collimator configuration and detector pixel size are often first optimized by means of simulated data and then experimentally tested. EGS is a well-known code (in its versions EGS4 and EGSnrc, the latest one) [a], which permits the transport of photons and electrons in various media; here the user must provide a code which describes the geometry: this is the most challenging programming aspect. Unfortunately, for complex geometries, this could be a very hard part to achieve. This is the case of one of the most common type of collimators used in SPECT: the hexagonal hole in hexagonal lattice collimator. It is indeed not so easy to give an D. Bollini, R. Campanini, N. Lanconelli with the Department of Physics, University Bologna, Italy. E-mail:
[email protected]. A. Bevilacqua is with DEIS, University Bologna, Italy. M. Gombia is with DIENCA, University Italy.
Member,
and A. Riccardi are of Bologna, and INFNof Bologna,
and
INFN-
of Bologna,
and
INFM,
Geometric
METHODS
modular description
Users adopting EGS code must perform themselves a couple of tasks: the description of the regions which constitute the geometry of the apparatus and the scoring of outputs. The most difficult section is often, giving a point with coordinates (z, y, z), the computation of the region to which the point belongs to. This is particularly true in case of complex geometries such as hexagonal collimators. This task is crucial because it is called several times during the transport of a particle; therefore any inefficiency here is propagated through the entire carriage and could dramatically increase the execution time. In order to reduce the computational time and to give a clear description of the collimator (or detector), we characterize it in a modular way: essentially, the collimator is seen as a basic cell repeated in the space. Luckily, the collimator can easily be described in a modular way: both in case of square and hexagonal holes. The basic cell in case of hexagonal holes is depicted in figure 1: r-hole represents the radius of the hole, septa the septal thickness of the parallel collimator and xcell the side of the cell. Each cell is constituted by six triangular regions, each one subdivided in an “air” region and a “lead” region (respectively, the transparent and the shadow regions in figure 1). By repeating the basic cell, we can obtain the hexagonal 2D lattice shown in figure 2. There is also depicted the coordinate system (~1, UZ, us) chosen to describe the points
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xcellf2
TABLE FEATURES
I
OF THE TWO SIMULATED
Hole
diameter [mm] Septa [mm] Length [mm] Sensitivity [counts/min/&i] Spatial Resolution @ 10 cm [mm]
B. Simulated
Fig.
1. The basic cell, which is repeated in a modular way; r-hole is the radius of the collimator holes, septa is the septal thickness and zcell is the side of the cell.
Fig.
2. The hexagonal
coordinate lattice.
system
chosen
for the description
1
Square hole 1.8 0.2 20 1160 10.8
Hexagonal hole 1 0.2 10 1115 11.0
apparatus
The Monte Carlo code used is EGSnrc, the latest version of EGS family. Our simulations include all the physical processes available with EGS: Compton and Rayleigh scattering, photoelectric absorption with emission of fluorescence photons or Auger electrons. In addition, we fully transport photons and electrons (the lower energy cut-off is equal to 5 keV). These assumptions hold in the entire simulated apparatus; it is worth remarking that there is no restrictions or approximations regarding physical aspects. The breast phantom consists of a 10 x 6 x 5 cm3 box made of breast-equivalent tissue with background activity of 80 nCi/cm3. The total number of 140 keV photons emitted is 475.106, relative to a 10 minute acquisition time. The simulated camera includes a lead collimator and a Ge pixellated detector, both surrounded by a lead shielding. We considered two high-sensitivity parallel hole collimators: one with square holes and one with hexagonal holes; their features are displayed in Table 1. The pixellated detector consists of an array of 2 x 2 x 11 mm3 pixels incorporated in a 1.5 mm-thick Aluminum housing. III.
of the 2D
of the space ((z, y) is the usual Cartesian system) [5]. The transformation between the two systems is clearly:
In that way each triangular region the basic cell is univocally determined (ulcell, uacell, uscell) given by:
COLLIMATORS
which constitutes by three indexes
ulcell = (id)(*) u2ceZZ = (id)(*) uscell = (id)(*)
It is worth noting that the computation of these three operations is sufficient to calculate the region of any points of the space, regardless of the number and the dimension of the holes and the septa. Another significant issue is that the only information we have to know is the collimator dimensions (width, height and length), the septal thickness and the hole radius.
RESULTS
In figure 3 there is shown an example of the spatial distribution of photons emitted by a parallel beam source exiting from a hexagonal hole collimator: this is a qualitative way to test that the geometric modular description of the collimator is correct. In order to validate the modular description in a quantitative way, we performed the characterization of the two collimators used. We calculated their sensitivity and spatial resolution and compared them with the theoretical predictions. The conditions to achieve these results are as follows: we set up two different simulations with a point source in air located at various distances from the collimator. The first simulation, called “geometric”, neglects septal penetration, scattering or any other physical effect inside the collimator (a photon is immediately discarded when it impinges on the lead of the collimator). The second kind of simulation, called “full simulation”, includes all the physical effects available with EGS. Sensitivity is computed as the number of photons detected per minute and PCi. Spatial resolution is equal to the FWHM of the collimator-detector response to the point source. Some results are depicted in figure 4 and figure 5: we can notice that the “geometric” sensitivity is in good agreement
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Fig.
3. Example of the spatial distribution the parallel hexagonal collimator.
Sensitivity
of photons
exiting
from
of the two collilnators Hexagonal-full silnulation Square-full silnulation Square-geometric Hexagonal-geometric
-- - -- ----~ -
with theoretical values in the entire range of distances, whilst the “full simulation” values agree only for distances greater than 40 cm. At smaller distances we get a sensitivity greater than the theoretical one because the latter does not take into account septal penetration: at small distances the number of photons which can reach the detector after having penetrated the collimator septa is rather high; whereas, for photons emitted at far distances this effect is less significant for geometric reasons. Septal penetration plays an important role also in the spatial resolution of the collimators: indeed, once again, “geometric” results agree with theoretical ones, whilst “full simulation” FWHM is always greater than the theoretical prediction. The simulation of different collimator-detector configurations for a breast phantom can be hampered by the huge computational time required. In our study we distribute the computational load between four PI11 800 MHz processors (two dual processor machines), by means of PVM (Parallel Virtual Machine) message-passing libraries, following a master-slave paradigm. In practice, the master program sends a fraction of events to slave processes, collects and store the outcome of detected events. Slaves are self-scheduled, asking master new events to process, as their task ends. With modular description it is possible to reduce the computational time up to 25%, with respect to a “traditional” geometric description [5]. That allows the simulation of the breast phantom for different configurations: each run (one phantom relative to a 10 min acquisition with full photons and electrons transport) takes almost 24 hours of CPU time on the ensemble of four PI11 800 MHz processors. IV.
1000
Fig.
0
I 10
4. Characterization simulation” sensitivity Table 1).
I
I I 20 30 Source-collimator
I I 40 50 distance (cm)
I 60
70
of the collimators: “geometric” and “full (theoretical predictions are displayed in
Spatial resolution I
of the hexagonal collilnators I I
I
CONCLUSIONS
In this paper we presented a geometric description for collimators and pixellated detectors in EGS code. With this method we achieve the computation of the region to which a point belongs to in a few steps, giving rise to a reduction in computational time up to 25%, with respect to a “traditional” geometric description. That allows the simulation of about 500 millions 140 keV photons in almost 24 hours on a cluster of four PI11 800 MHz processors, with full transport of photons and electrons. The execution time is not affected neither by the collimator and detector dimensions, nor by the shape of the holes (square or hexagonal). REFERENCES [I] [2]
0
20
Fig.
4 6 8 Source-collixnator distance (cm)
5. Characterization of the collimators: metric” and theoretical spatial resolution tion only values of hexagonal collimator
10
[3] 12
[4]
“full simulation”, “geo(for a better visualizaare shown).
[5]
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Monte Carlo calculations in nuclear medicine: applications in diagnostic imaging, ed. M. Ljungbert et al., IoP Publishing, Bristol, UK, 1998. I. Kawrakow, Accurate condensed history Monte Carlo simulation of electron transport. Part I: EGSnrc, the new EGS4 version, Med. Phys., 27(4), 2000. W. E. Snyder, H. Qi and W. Sander, A coordinate system for hexagonal pixels, Proc. SPIE! the International Society for Optical Engineering, Vol. 3661, 1999. M. Gombia, A. B. Brill, D. Bollini and A. Del Guerra, A simulation and modeling study comparing the performance of a Germanium Orthogonal Strip Detector and an Anger camera, to appear on IEEE NSS & MIC Symposium, Conference Record, 2001. A. F. Bielajew, HOWFAR and HOWNEAR: geometry modeling for Monte Carlo particle transport , National Research Council of Canada Report, PIRS-0341, 1995.
