Home

Search

Collections

Journals

About

Contact us

My IOPscience

Efficient simulation of voxelized phantom in GATE with embedded SimSET multiple photon history generator

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 Phys. Med. Biol. 59 6231 (http://iopscience.iop.org/0031-9155/59/20/6231) View the table of contents for this issue, or go to the journal homepage for more

Download details: IP Address: 129.105.215.146 This content was downloaded on 11/02/2015 at 19:11

Please note that terms and conditions apply.

Institute of Physics and Engineering in Medicine Phys. Med. Biol. 59 (2014) 6231–6250

Physics in Medicine & Biology doi:10.1088/0031-9155/59/20/6231

Efficient simulation of voxelized phantom in GATE with embedded SimSET multiple photon history generator Hsin-Hon Lin1, Keh-Shih Chuang1, Yi-Hsing Lin2, Yu-Ching Ni3, Jay Wu4 and Meei-Ling Jan3 1

  Department of Biomedical Engineering & Environmental Sciences, National Tsing-Hua University, Hsinchu, Taiwan 2   Health Physics Section, Kuosheng Nuclear Power Plant, Taiwan Power Company, New Taipei, Taiwan 3   Health Physics Division, Institute of Nuclear Energy Research, Atomic Energy Council, Taoyuan, Taiwan 4   Department of Biomedical Imaging and Radiological Science, China Medical University, Taichung, Taiwan E-mail: [email protected] Received 2 February 2014, revised 26 May 2014 Accepted for publication 26 August 2014 Published 26 September 2014 Abstract

GEANT4 Application for Tomographic Emission (GATE) is a powerful Monte Carlo simulator that combines the advantages of the general-purpose GEANT4 simulation code and the specific software tool implementations dedicated to emission tomography. However, the detailed physical modelling of GEANT4 is highly computationally demanding, especially when tracking particles through voxelized phantoms. To circumvent the relatively slow simulation of voxelized phantoms in GATE, another efficient Monte Carlo code can be used to simulate photon interactions and transport inside a voxelized phantom. The simulation system for emission tomography (SimSET), a dedicated Monte Carlo code for PET/SPECT systems, is well-known for its efficiency in simulation of voxel-based objects. An efficient Monte Carlo workflow integrating GATE and SimSET for simulating pinhole SPECT has been proposed to improve voxelized phantom simulation. Although the workflow achieves a desirable increase in speed, it sacrifices the ability to simulate decaying radioactive sources such as non-pure positron emitters or multiple emission isotopes with complex decay schemes and lacks the modelling of time-dependent processes due to the inherent limitations of the SimSET photon history generator (PHG). Moreover, a large volume of disk storage is needed to store the huge temporal photon history file produced by SimSET that must be transported to GATE. In this work, we developed a multiple photon emission history generator (MPHG) 0031-9155/14/206231+20$33.00  © 2014 Institute of Physics and Engineering in Medicine  Printed in the UK & the USA

6231

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

based on SimSET/PHG to support a majority of the medically important positron emitters. We incorporated the new generator codes inside GATE to improve the simulation efficiency of voxelized phantoms in GATE, while eliminating the need for the temporal photon history file. The validation of this new code based on a MicroPET R4 system was conducted for 124I and 18F with mouse-like and rat-like phantoms. Comparison of GATE/MPHG with GATE/ GEANT4 indicated there is a slight difference in energy spectra for energy below 50 keV due to the lack of x-ray simulation from 124I decay in the new code. The spatial resolution, scatter fraction and count rate performance are in good agreement between the two codes. For the case studies of 18F-NaF (124I-IAZG) using MOBY phantom with 1  ×  1 × 1 mm3 voxel sizes, the results show that GATE/MPHG can achieve acceleration factors of approximately 3.1 × (4.5 ×), 6.5 × (10.7 ×) and 9.5 × (31.0 ×) compared with GATE using the regular navigation method, the compressed voxel method and the parameterized tracking technique, respectively. In conclusion, the implementation of MPHG in GATE allows for improved efficiency of voxelized phantom simulations and is suitable for studying clinical and preclinical imaging. Keywords: Monte Carlo simulation, GATE, simSET, preclinical PET (Some figures may appear in colour only in the online journal) 1. Introduction Monte Carlo (MC) simulation in emission tomography is an essential tool to hasten the development of PET and SPECT imaging systems and the corresponding image reconstruction and correction methods, as well as the optimization of data acquisition and processing protocols (Zaidi 1999,Buvat and Lazaro 2006). Among these MC codes in emission tomography (Buvat and Castiglioni 2002), GEANT4 Application for Tomographic Emission (GATE) (Jan et al 2004) is a powerful Monte Carlo simulator that combines the advantages of the generalpurpose GEANT4 simulation code (Agostinelli et al 2003, Allison et al 2006) and the specific software tool implementations dedicated to emission tomography (Pietrzyk et al 2012). The platform can model time-dependent processes such as decay kinetics, dead time and movement, while benefitting from the same versatility and support as that of the general-purpose simulation codes. However, the detailed physical modelling of GEANT4 is very computationally demanding, especially when tracking particles through voxelized phantoms. The underlying GEANT4 behind GATE offers three methods for volume arrangement in voxelized phantom for tracking particle: G4VPVParameterised, G4VNestedParameterisation, and G4PhantomParameterisation classes (Schümann et al 2012). In GATE, the general particle tracking algorithm (called parameterized tracking) is developed based on the G4VPVParameterised class. As the method considers each voxel in a voxelized phantom as a large cuboidal volume set, it needs to spend a great deal of time on determining the next voxel from the total volume set and updating the paths of the particle at each voxel boundary encountered, resulting in low efficiency for simulation. To tackle the inefficient simulation in voxelized phantoms, two approaches, the compressed voxels techniques (Taschereau and Chatziioannou 2008) and improved voxel navigation (called a regular navigation algorithm) (Rehfeld et al 2009), have been proposed for GATE. The compressed voxels technique optimizes the original parameterized tracking method by combining adjacent voxels with identical 6232

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

physical properties into larger voxels. This method can reduce the memory and central processing unit requirements for high resolution objects. Since GEANT4.9.1, a new navigation algorithm based on G4PhantomParameterisation classes (Arce et al 2008), was developed for the tracking of particles in voxelized volumes. The regular navigation algorithm optimizes the voxel navigation by searching only the neighbouring voxel, which eliminates the need for large memory overhead and skips the steps when the material of the next voxel is the same. The improved navigation was introduced in GATE and increased the speed up to approximately 2~7 × (dependent on voxel sizes) compared to the compressed voxels technique. To circumvent the relatively slow simulation of voxelized phantoms in GATE, other efficient Monte Carlo codes can be used to simulate photon interactions and transport inside a voxelized phantom. Simulation system for emission tomography (SimSET) (Harrison et al 1993), a dedicated Monte Carlo code for PET/SPECT systems, is well-known for its efficiency in the simulation of voxel-based objects (Barret et al 2005). It uses variance reduction techniques such as forced detection (Haynor et al 1990) and important sampling techniques (Haynor et al 1991) to improve efficiency. An efficient Monte Carlo workflow integrating GATE and SimSET for simulating pinhole SPECT was previously proposed (Chen et al 2008, Mok et al 2010) and significantly increased computational speed by about 10 fold was achieved by the workflow. Although the workflow achieved a desired increase in speed, it sacrificed the ability to simulate decaying radioactive sources such as non-pure positron emitters or multiple emission isotopes with complex decay schemes (Tang et al 2009, Zhu and El Fakhri 2009) and lacked the modelling of time-dependent processes due to the inherent limitation of the SimSET photon history generator (PHG). Moreover, high volumes of disk storage capacity for SimSET-GATE workflow are often needed to store the huge temporal photon history file produced from SimSET. However, the history file can be useful for efficient study or optimization of different detector configurations to save simulation time within the phantom. To overcome these problems, we developed a multiple photon emission history generator (MPHG) based on SimSET/PHG to support most of the medically important positron emitters and incorporated the new generator codes inside GATE to improve the simulation efficiency of voxelized phantoms in GATE. The Monte Carlo modeming of MPHG is described in the first part of this article, followed by the technical aspects of embedding MPHG in GATE. In the second part, we report the validation and efficiency of this new code based on different voxelized phantoms using various positron emitters in comparison to GATE. Finally, we discuss some of the advantages and limitations of the new code. 2.  Materials and methods 2.1.  Multiple photon emission history generator

The basic architecture of the MPHG code is identical to that of SimSET/PHG. Original PHG can be divided into three parts: (1) radioactive isotope simulation, (2) physical modelling, and (3) photon tracking in voxelized phantoms. In MPHG, major changes were made in part (1) to extend the capability of simulation of decaying radioactive sources with complex decay schemes, while preserving the efficient method of dedicated physical modelling and photon tracking. (i) Decay scheme and isotope definition A new input parameter file for isotope definition was needed for the MPHG codes. The file contains the half-life, atomic number and decay scheme of each isotope involved. For each particle 6233

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

Figure 1.  Full cascaded lines of decay scheme with 3 layers and the corresponding

finite state machine.

in the decay scheme, its type (positron or photon), energy and branching ratio from high-state to low-state transition are defined. The decay scheme is modelled as a finite state-machine (Laedermann and Décombaz 2000), as shown in figure 1. The model consists of a set of states and a transition function between those states and actions. Computation begins in the start state with an input state symbol of ‘0’. It continues changing to a new state depending on the corresponding transition function until reaching the final state. The transition function determines the next state sampled from a uniform distribution, which is apportioned pro rata among the subsequent states. The actions associated with state transition will generate the photon or positron, and then, the following cascade or annihilated photons are tracked through the attenuation distribution using the native photon tracking algorithm of SimSET. It is worth noting that once the action is electron captured, there is no particle generated but it descends to a new state. (ii) Photon cross section table The photon cross section table in SimSET was based on PETSIM’s parametric model (Picard et al 1992) incorporated with the Evaluated Photon Data Library 94 version (EPDL94) (Kaplan et al 1997). The data files were trimmed to fit the main needs of nuclear medicine, spanning energies from 1 keV to 1 MeV. However, some of cascade gamma ray’s energies were higher than 1 MeV for positron emitters with complex decay schemes. To accommodate the transport of cascade gammas with high energy, we built a new photon cross section table for energies ranging from 1 eV to 4 MeV based on the most recent EPDL97 version (Cullen et al 1997). The EPDL database developed by the Lawrence Livermore National Laboratory (LLNL) includes cross section data for photoelectric absorption, coherent and incoherent scatterings, and pair production for energies ranging from 1 eV to 100 GeV. It has been recommended to serve as a standard database and supersedes earlier cross section libraries used in Monte Carlo codes to simulate medical imaging system (Zaidi 2000). In addition, for most non-pure positron emitters, the energy of emitted gamma is lower than 4 MeV and the cross section of pair production under the energy level is considerably small so that probability of pair production is negligible. Comparisons of attenuation coefficients (along with individual contributions from photoelectric absorption, Compton scattering, and coherent scattering) in water as simulated by SimSET and the EPDL97 are plotted in figure 2. (iii) Positron range Positron range in SimSET (Harrison et al 1999) is distributed according to an empirical model given by Palmer and Brownell (Palmer and Brownell 1992). This model assumes that 6234

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

Figure 2.  Cross sections of photon interactions in water as calculated from EPDL97 (lines) and SimSET (symbols). The curves represent the total linear attenuation coefficient, photoelectric absorption (PE), Compton scattering (CS) and Rayleigh scattering (RS).

the equilibrium particle density resulting from sampling of beta-decay energy spectra can be represented by a 3D Gaussian distribution centred at the origin. The SimSET package includes several positron range tables for common nuclear medicine isotopes (18F, 11 C, 82Rb and 68Ge). To extend the variety of isotopes, especially for non-pure isotopes of high atomic number, we adopted an analytical model for positron emission energy density N(E). For a positron from an isotope of the atomic number Z with endpoint energy Emax, the probability density N(E) is given by N (E ) = (Emax − E )2 WpF (Z , W )

(2.1)

where E is the emission energy in keV, W = 1 + E/511, p (= W2 − 1 ) is the momentum, and F(Z,W) is the Fermi function. SimSET neglects the relativistic correction factor and employs a non-relativistic approximation form for the Fermi function, which is valid for conventional positron emitters of low Z, given by: F  (Z , W ) =

2πη 1 − exp (−2πη)

(2.2)

where η = −αZW / p, α (= 1 / 137) is the fine structure constant. However, the effect of the Coulomb field may significantly distort the beta spectrum for isotopes of high atomic number. 6235

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

18 F and 124I with (MPHG) and without (SimSET) relativistic correction. The probability curves are normalized to have equal areas under the curve.

Figure 3.  Simulated positron spectra of

In this work, MPHG used a more accurate approximation form with the Coulomb correction factor (Venkataramaiah et al 1985): F  (Z , W ) =

2πη [W2(1 + 4γ 2) − 1]s 1 − exp (−2πη)

(2.3)

where γ = αZ and S = (1 − α 2Z2 )1/2 − 1. Figure 3 shows the normalized positron spectra corresponding to 18F and 124I, with and without relativistic correction. It can be seen that the difference relativistic correction makes is larger for 124I due to its higher atomic number. (iv) Temporal information and decay generation Although the native photon history generator in SimSET did not support the temporal simulation, it did track the given simulated decays and scaled the particle with a weighting factor calculated by the given simulated decay numbers, activity distribution and length of scan time. Therefore, it is not capable of describing some physical mechanisms in terms of time-dependent phenomena. In addition, since the decay generation in SimSET/PHG is voxel-by-voxel in sequence and uses pre-calculated decay maps, it may conflict with the randomized decay generation in GATE and affect the simulation results of time-dependent process, especially in coincidence detections of PET. Hence, additional options for randomized decay generation are implemented in MPHG for connecting it seamlessly with GATE. To enable temporal information in MPHG, while keeping the synchronization with the time clock of GATE, we 6236

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

imported the decay time generated from GATE into MPHG when decay occurs; the subsequent time of photon flight is calculated based on the travel length of emitted photons inside the object (Harrison et al 2004). 2.2.  Implementation of GATE/MPHG

To embed the MPHG in GATE, we developed the GateSourceVoxelSimSETInserter program inherited from the GateSourceVoxellized class in the GATE code. The GateSourceVoxelSimSETInserter reads the SimSET parameters to initialize the MPHG engine. Once initialized, the routine GateSourceVoxellized::GeneratePrimaries (G4Event* event) evokes MPHG to generate a cascaded decay process. The generated particles of each decay transport within the voxelized phantom as governed by SimSET. Once the tracked photons leave the object, the photons are projected and recorded onto a bounding cylinder (referred to as a target cylinder in SimSET) and the cascade process of the event is finished in MPHG. Synchronously, GATE will produce the propagating photons at the same positions on the bounding cylinder that inherits all the photon information of the event from MPHG. Then, GATE tracks the propagated photons through the collimator and detector level according to the user’s instructions. The output file supports the LMF, ASCII, ECAT and Root formats according to GATE data output streams. The implementation flowchart of GATE/MPHG is illustrated in figure 4. At the time of this development, the MPHG is based on the version of SimSET/PHG 2.9.1 and GATE is 6.1. 2.3.  Positron emitters

In this work, two representative isotopes of pure and non-pure positron emitters were simulated. 18 F is the most commonly used emitter in PET for its attractive properties, one of which is that it can be easily substituted into biomolecules. 18F is an almost pure positron emitter with 97% positron emission and 3% electron capture. 124I radionuclide has been reported to be useful for imaging applications in oncology as a surrogate for 131I prior to radioimmunotherapy. 124I has a complex decay scheme, including 24 electron capture decay lines and five beta plus decay lines, resulting in 81 cascade gamma emissions (Bhat 1992). As an acceptable compromise between computing efficiency and accuracy, we modelled the emitted particles if their absolute emission probabilities were more than 0.01 %. A total of 66 cascade gamma emissions were modelled, and the resulting gamma abundance of 124I was 99.53% in our simulations. 2.4.  Simulation models of MicroPET R4 scanners and physics

The scanner geometry of the MicroPET R4 system (Knoess et al 2003, Lartizien et al 2007) consists of four rings with 96 detector modules, each coupled to an 8   ×   8 array of 2.1  ×  2.1  ×  10 mm3 LSO crystals resulting in 32 rings with 192 elements. The 96 modules were arranged in four rings of 24 modules each, with a diameter of 7.4 cm. The crystal pitches were 2.423 mm and 2.426 mm in the axial and transverse directions, respectively. The external lead shields were modelled with an opening of 120 mm at the front and 134.5 mm at the rear. The simulated energy resolution for each crystal was randomly sampled from a uniform distribution over the interval [17% 35%] and a 0.91 quantum efficiency was applied. A back compartment was also simulated to account for the back scattering induced by the light collection system. To compare object simulation between GATE/MPHG and GATE/GEANT4 directly, the dead time models at different levels of the electronic read-out and transfer bandwidth were neglected in this study. 6237

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

Create an event

Generate a decay (excited state)

Attach the decay time

Cascade down the next state

Generate a particle

Photon Tracking in voxelized phantom

No

Reach the ground state ? M PH G Yes Transfer the event of MPHG to GATE

No

Do photons hit collimators/detectors ? Yes Photons transport in collimators/detectors Yes

No

Reach the scan time ? GAT E Yes Output file (ROOT/LMF/ASCII/Sinogram)

Figure 4.  Flowchart of the main steps of embedding MPHG into the GATE program.

2.5.  Validation of GATE/MPHG with the GATE/GEANT4 model

The underlying GEANT4 engine behind GATE is a general purpose Monte Carlo simulation package (Agostinelli et al 2003). Three physics models in GEANT4 (Geant4.9.3.p02) are available for electromagnetic processes. The standard process model for photon interactions 6238

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

is employed in this study for its efficiency and to ensure sufficient accuracy in PET simulation. To avoid infrared divergence, some electromagnetic processes (ex: gammas and electrons) require production thresholds below which no secondary particles will be generated. This threshold is defined as a range cut-off, which is internally converted to energy for each individual material. The range cut-off for all GATE system is set to 1 mm by default. In addition, GATE codes offers three options of source types for PET: ion source, particle source and back-to-back source. The ion source was used since it afforded the description of complex decay schemes with the most realistic manner of simulating a radionuclide among these options. To validate the accuracy of the proposed MPHG model, the simulation results of energy spectra, spatial resolution, scatter and cascade gamma fractions and count rate performance were compared with the GEANT4 model. 2.5.1.  Energy spectra.  Energy spectra were acquired for the MicroPET R4 system using a 1 mm diameter 124I line source in the centre of a cylindrical water phantom (5 cm in diameter and 10 cm in height). The acquisition of energy window and coincidence time window setting were set 1-1000 keV and 6 ns, respectively. The energy spectra of detected photons with energy between 1 keV and 1000 keV were extracted from data from both MC simulations for comparison.

I and 18F were measured using an ideal point source (10 MBq) embedded in a 10 mm acrylic cube phantom. Simulations were performed with the point source located at radial distances of 0, 10 and 25 mm from the centre. Data were acquired with an energy window of 350–750 keV and a coincidence time window of 6 ns. The simulated list mode data at each location were histrogramed into sinogram data sets with span 3 and maximum ring difference 31. The 3D sino grams were reconstructed by FORE + 2D FBP using a ramp filter with cut-off at the Nyquist frequency and a zoom factor of 5 to achieve a reconstructed voxel size of 0.162  ×  0.162  ×  1.21 mm3. Data were not corrected for normalization, scatter, attenuation or source dimension. The response function was formed by summing all one-dimensional profiles that were parallel to the radial, tangential and axial directions. According to the NEMA procedure (NEMA 2008), a parabolic fit of the peak point and its two nearest neighbouring points was used to determine the maximum value of the response function. Linear interpolation between adjacent pixels was used to determine the positions of 50% and 10% of the maximum intensity for measurement of full width at half maximum (FWHM) and full width at one-tenth maximum (FWTM). 2.5.2. Spatial resolution.  The spatial resolutions for

124

2.5.3.  Scatter and cascade gamma fractions.  A 50 mm diameter, 150 mm-long polyethyl-

ene (HDPE) phantom (density 0.96 g cm−3) with a hole at 17.5 mm distance from the axis and a 25 mm diameter, 70 mm long HDPE phantom with a hole at 10 mm distance from the axis were used to represent objects t size of a rat (referred to as the “rat-like phantom”) and a mouse (referred to as the “mouse-like phantom”), respectively. The line source with diameter of 1.0 mm was inserted in the drilling of each phantom. Both 18F and 124I were simulated on the rat-like and mouse-like phantoms. The total activity for each simulation was set as 10 μCi, which was low enough to ignore the random coincidence (below 0.5% of the true event). Data were acquired in three energy window settings of 350–650 keV, 350–600 keV and 425–650 keV. The simulated list-mode data were binned into 2D sinograms by the single slice rebinning SSRB method (Daube-Witherspoon and Muehllehner 1987) and sinograms of true, scatter, gamma and random components were stored separately. The gamma component was specific to non-pure positron emitters and was defined as a double coincidence involving at least one cascaded gamma originating from the same decay. For each simulation, all 6239

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

63 slices in the sinograms were summed first and the total counts of the summed sinograms for each component were scored separately. The gamma fraction was calculated as the ratio of the gamma counts to the total counts. However, to cross validate the real impact of scatter coincidence at the same phantom between 18F and 124I, the scatter fraction was calculated by taking the ratio of scattered coincidences to the sum of scatter and true coincidence events. It is worth noting that to separate the coincidence events involving cascaded gammas and annihilated gammas, the analysis module of the GATE code was modified to tag the single events originating from annihilated or cascaded gamma ray photons (De Beenhouwer et al 2009). 2.5.4.  Count rate performance.  A count rate experiment was performed using the two aforementioned phantoms (rat-like and mouse-like phantom) to validate GATE/MPHG against GATE/GEANT4. These two phantoms with line source filled with 18F and 124I were simulated for activities varying from 0.2 to 8 mCi. Data were acquired with an energy window of 350– 650 keV and two settings of coincidence time window (6 and 10 ns). The simulated list-mode data were rebinned into sinograms of true, scatter, gamma and random components separately. The count rates of prompts (total coincidence) and random coincidences were calculated as a function of the activity concentration. 2.6.  Efficacy of GATE/MPHG model

To demonstrate the efficacy of GATE/MPHG in the efficient simulation of voxelized phantoms with reasonable accuracy, realistic simulations of biological distributions in MicroPET were performed. The current GATE platform supports three options to simulate the voxelized phantom, including general parameterized tracking (‘parameterized’), the compressed voxel method (‘compressed’), and the regular navigation method (‘regular’) for imaging applications. Another nested parameterized method based on the G4VNestedParameterisation class was used for radiotherapy applications in GATE at the time of this development. A new version of GATE (version 7.0), released on 12 May 2014, has unified the navigation engine for radiation therapy and imaging applications (OpenGATE collaboration, 2014). The mouse whole-body MOBY voxelized phantom (Segars et al 2004) was used to compare the efficacy among GATE/MPHG and the above mentioned three methods offered in GATE/GEANT4. All the parameter settings for each method were under the same system configuration and acquisition protocols. It is worth noting that the tracking of electrons in GEANT4 enabled the calculation of radiation dose and produced more accurate results with more cost efficiency. In contrast, SimSET is fast but cannot be used for dose estimation (Chuang et al 2014). So it would be comparable with GATE/MPHG, which ignores the simulation of electrons, the electron range cut-off in the phantom region of GATE/GEANT4 was set to 500 mm. This range cutoff setting is sufficiently large to suppress the production of electrons in phantoms. The settings of range cut-off outside the phantom remained 1 mm. All simulations were performed on four 64 bit Fedora Linux 15 computers (AMD Phenom™ II X6 CPU @2.8 GHz, 8 GB). Simulations of each method were assigned to one core in the computers and the CPU time was recorded. As shown in figure 5(a), the input attenuation distribution of the MOBY phantom was segmented into six materials including air, lung, spine, rib, skull and soft tissue. Two simulations of different radiopharmaceuticals distribution using 18F-NaF and 124I-IAZG are presented in this study (see figures 5(b) and (c). The first simulation protocol consisted of a static 5 min acquisition of the MOBY phantom corresponding to a radiotracer distribution of 10 µCi of 18 F-NaF in the bones. The activity distribution in bone regions was assumed uniform. In the 6240

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

(a)

(b)

(c)

Figure 5. (a) Coronal views of the attenuation map, (b) sagittal views of the activity map for 18F-NaF and (c) coronal views of the activity map for 124I-IAZG, including a spherical lesion in right hind leg, generated using the MOBY software.

second simulation protocol, the radiotracer distribution of the MOBY phantom emulated a 124 I-labeled iodo-azomycin-galactoside (124I-IAZG) uptake at 3 h post injection in a tumourbearing mouse (Zanzonico et al 2004). A spherical-shaped tumour with 2.5 mm radius was inserted in the right hind leg. The activity concentration ratios of the 450 µCi extended source distribution were 51:9:10:8:18:11:24:46:255:8 for tumour, blood pool, heart, lung, liver, spleen, kidney, stomach, intestine and background, respectively. Various image sets with voxel sizes of 0.125  ×  0.125  ×  0.125, 0.25  ×  0.25  ×  0.25, 0.5  ×  0.5  ×  0.5 and 1  ×  1 × 1 mm3 (corresponding to voxel dimensions of 256  ×  256  ×  900, 128  ×  128  ×  450, 64  ×  64  ×  225 and 32  ×  32  ×  113) were generated from MOBY software to investigate the influence of the voxel number on simulation time. 3. Results 3.1.  Validation of GATE/MPHG with the GATE/GEANT4 model 3.1.1.  Energy spectra.  The energy spectra of detected photons for 124I are plotted in figure 6.

Due to the limited energy resolution of the simulated PET detector, there is no apparent energy peak at 603 keV, where an abundance of cascaded gamma with energy of 603 keV can be 6241

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

Figure 6.  Comparisons of energy spectra of detected photons of 124I line source inside

a cylindrical water phantom obtained from GATE/MPHG and GATE/GEANT4 Monte Carlo simulation results.

detected. The results shows good agreement between the GATE/GEANT4 and GATE/MPHG models except for energies below 50 keV. The discrepancy at energies below 50 keV is mainly due to the simulation of x-ray emissions for 124I in GATE/GEANT4, but not in GATE/MPHG. 3.1.2.  Spatial resolution.  The FWHM’s and FWTM’s of the point source profiles at different radial offsets from GATE/MPHG and GATE/GEANT4 simulations are listed in table 1. Good agreement in both FWHM and the FWTM values in all three directions between GATE/ MPHG and GATE/GEANT4 can be seen. Although the discrepancy for 124I between the two models is larger than that for 18F, most errors in FWHM between the two models remain below 5%. Since the transport of photons in MicroPET scanner for both models is based on the original GATE/GEANT4 codes, the cause of this difference in spatial resolution is mainly due to the different positron models employed in MPHG and GEANT4. The modelling of positron range in MPHG is based on the analytical modelling proposed by Palmer and Brownell (1992), while GATE modelling is based on the Monte Carlo simulations using cross-sections for the annihilation process initially described in Heitler (1954). Recent reports have indicated the difference in positron range simulation between the two models is greater for isotopes with higher positron range (Lehnert et al 2011). 3.1.3.  Scatter fraction and cascade gamma fraction.  Scatter and/or gamma fractions of the

line source inside HDPE phantoms for 18F and 124I with various energy window settings are listed in table 2 and table 3. For both isotopes, there is excellent agreement between the scatter fractions for the two phantoms at different energy window settings with GATE/MPHG and GATE/GEANT4. Additionally, the gamma fractions of 124I on both phantoms agree well 6242

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

Table 1. Comparisons of FWHMs and FWTMs from GATE/GEANT4 and GATE/ MPHG simulation results for different isotopes.

Isotopes 18

F

124

I

Radial resolution (FWHM/FWTM)

Tangential resolution (FWHM/FWTM)

Axial resolution (FWHM/FWTM)

Radial offset (mm)

GEANT4

MPHG

GEANT4

MPHG

GEANT4

MPHG

0 10 25 0 10 25

1.854/4.020 2.135/4.352 3.177/6.086 3.563/8.600 3.731/8.669 4.581/9.391

1.851/4.157 2.138/4.461 3.190/6.163 3.335/8.498 3.547/8.841 4.444/9.301

1.852/4.048 2.211/4.427 2.411/5.280 3.574/8.692 3.777/8.902 4.172/9.324

1.865/4.207 2.209/4.573 2.406/5.475 3.308/8.685 3.541/8.781 3.991/9.243

1.941/4.582 3.029/8.123 3.489/9.994 4.120/9.155 4.941/10.46 4.752/12.48

1.914/4.606 2.991/8.196 3.437/9.978 3.876/9.008 4.704/10.41 4.688/12.37

Table 2. Comparison of scatter fractions from GATE/GEANT4 and GATE/MPHG simulation results for 18F.

Scatter fraction (%) Phantoms Mouse-like Rat-like

Energy window (keV)

GEANT4

MPHG

350–600 350–650 425–650 350–600 350–650 425–650

10.16 9.79 5.79 23.46 22.66 13.20

10.09 9.71 5.81 23.09 22.35 13.34

Table 3. Comparisons of scatter and gamma fractions from GATE/GEANT4 and

GATE/MPHG simulation results for 124I.

Phantoms

Mouse-like Rat-like

Energy window (keV)

Scatter fraction (%)

Gamma fraction (%)

GEANT4

MPHG

GEANT4

MPHG

350–600 350–650 425–650 350–600 350–650 425–650

10.04 9.65 5.74 23.98 23.12 13.56

9.96 9.59 5.87 23.16 22.42 13.20

19.24 25.29 23.51 30.58 37.80 35.32

19.26 24.88 23.02 30.45 37.12 34.92

between GATE/MPHG and GATE/GEANT4. The agreement also indicates that the neglect of pair production and non-essential cascade gammas at the energy window of typical PET in the simulation is practical and justified. 3.1.4.  Count rate performance.  Figure 7 shows simulated prompt and random coincidence

count rates of 18F on the MicroPET R4 system for the mouse-like and rat-like phantoms as a function of the total activity at 6 and 10 ns coincidence time windows. The figure shows that prompt and random count rate curve with various activities between GATE/MPHG and GATE/GEANT4 agree well. Figure 8 shows similar results for 124I. There is a minor systematic underestimation of count rates from GATE/MPHG compared with those from GATE/GEANT4. The slight discrepancy might be due to the fact that the positron range is only modelled in the object in MPHG. In other words, positrons leaving the object are not propagated into the collimator and detector of GATE. The detection of the escaping positrons or their associated annihilated gammas increases additional detector signal and 6243

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

(a)

(b)

(c)

(d)

18 F from GATE/MPHG and GATE/GEANT4. Upper row: 6 ns coincidence time window for (a) mouse-like and (b) rat-like phantom. Lower row: 10 ns coincidence time window for (c) mouse-like and (d) rat-like phantom.

Figure 7.  Comparisons of the prompt and random coincidence count rate of

coincidence. This effect will be amplified when the positron rage of an isotope is large and the object size is small such as in the case of mouse-like phantom with 124I as shown in figure 8 (a) and 8 (c). 3.2.  Efficacy of GATE/MPHG model

The summed sinograms of 18F-NaF and 124I-IAZG for the four voxelized tracking methods are shown in figure 9 and figure 10, respectively. Figure 11 shows the profiles summed over all projection angles of the sinograms for the four methods. The profiles of the sinogram show good agreement among MPHG, the compressed voxel technique, regular navigation, and the parameterized method. There is a larger average relative error (~8%) in the profiles of figure 11(b) for 124I-IAZG between MPHG and other tracking methods. This slightly larger discrepancy 6244

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

(a)

(b)

(c)

(d)

Figure 8.  Comparisons of the prompt and random coincidence count rate of

124

I from GATE/MPHG and GATE/GEANT4. Upper row: 6 ns coincidence time window for (a) mouse-like and (b) rat-like phantom. Lower row: 10 ns coincidence time window for (c) mouse-like and (d) rat-like phantom.

(a)

(b)

(c)

(d)

Figure 9.  Summed sinograms of the full 3D sinogram simulated with (a) MPHG, and (b) regular, (c) compressed, and (d) parameterized methods for 18F-NaF acquisitions of MOBY phantom with voxel sizes of 0.5 mm. The raw sinograms of prompt coincidence were summed to increase the statistics. 6245

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

(a)

(b)

(c)

(d)

Figure 10.  Summed sinograms of the full 3D sinogram simulated with (a) MPHG, and

(b) regular, (c) compressed, and (d) parameterized methods for 124I-IAZG acquisitions of MOBY phantom with voxel sizes of 0.5 mm.

(a)

(b)

Figure 11.  Comparisons of the profiles in the sinogram summed over the projection

angles (figures 9 and 10) for (a) 18F-NaF acquisitions and (b) 124I-IAZG acquisitions of the MOBY phantom, respectively.

of MPHG found in the profile of regions outside the phantom is due to the negligence of coincidence caused by the escaping positrons as mentioned in the previous section. The CPU execution times as a function of voxel numbers in phantoms of 18F-NaF and 124I-IAZG are compared in figures  12(a) and 12(b), respectively. The results indicate that the simulation time using the parameterized and compressed voxel methods increased more quickly with voxel grid size than with the MPHG and regular navigation methods. These trends conform to those presented in Rehfeld et al (2009), although the scanners studied are different. The ratios of improvements in efficiency of GATE/MPHG as compared to the regular navigation method for 18F-NaF (124I-IAZG) under different voxel grid sizes stay almost constant and are 3.1 × (4.5 ×) on average. By contrast, the speedup ratios using GATE/MPHG in comparison to the compressed voxels and parameterized tracking methods for 18F-NaF (124I-IAZG) cases improve from 6.5 × to 21.7 × (10.7 × to 30.0×) and 9.5 × to 35.9 × (31.0 × to 170.3×) when the matrix sizes increase from 32  ×  32  ×  113 to 256  ×  256  ×  900. The improvement factors of GATE/MPHG for 124I are greater than those for 18F. This can be attributed to the efficient 6246

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

(a)

(b)

Figure 12.  Total CPU time needed for (a) 5 min simulations of 10 µCi 18F-NaF and (b) 10 s simulations of 450 µCi 124I-IAZG for the four different tracking methods. The matrix dimensions were 32  ×  32  ×  113, 64  ×  64  ×  225, 128  ×  128  ×  450 and 256  ×  256  ×  900 (corresponding to voxel sizes of 1  ×  1 × 1 mm3, 0.5  ×  0.5  ×  0.5 mm3, 0.25  ×  0.25  ×  0.25 mm3 and 0.125  ×  0.125  ×  0.125 mm3), respectively.

simulation of the decay scheme in MPHG in comparison with the time-consuming simulation process of the radioactive source in GATE. 4.  Discussion and conclusions The work presented here demonstrates a newly developed MPHG program based on SimSET/ PHG to model the simplified decay process of radionuclides and the subsequent photon transport in objects. Although no neutrinos or de-excitation process (Auger effect or x-ray emission) were simulated, the program can meet most nuclear medicine simulation needs for a wide variety of radionuclides while preserving the efficiency of the simulator. We compared our simulations with the well-validated GATE/GEANT4 simulation package for imaging 18F and 124I isotopes with MicroPET R4. They are in good agreement in terms of the energy models. The main discrepancy of the 124I energy spectra was observed in the energy range below 50 keV due to the lack of simulation in the de-excitation process. Fortunately, this did not affect the coincidence acquisitions within the PET energy window. A comparison of spatial resolution reveals that the GATE/MPHG positron range is comparable with the GATE/GEANT4 positron range for 18F, but is slightly lower than the GATE/GEANT4 positron range for 124I. Despite the underestimation of the positron range for 124I in MPHG, the relative error of averaged spatial resolution in terms of FWHM and FWTM between the two models remained below 5%. MPHG uses an analytical model to calculate the positron range instead of directly simulating the transport of positrons. Only photons leaving the object will be transferred from MPHG to GATE; no positrons (or other particles) will be transferred. As a result, the photons from positrons that might escape from the object and interact with the tomograph will not be simulated by GATE/MPHG, resulting in a slight underestimation of count rate compared with GATE/GEANT4. However, it is reasonable to neglect the escaping positrons because they will be blocked by the outer casing of PET in real-world scenarios. 6247

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

GATE is highly object-oriented software; therefore, MPHG can be readily embedded inside GATE codes without changing the original architecture of GATE. The new MPHG engine offers an alternative option for the simulation of voxel phantoms. In comparison to the regular navigation method, the speedup ratios using MPHG can achieve approximately 2.2 × (3.5 ×) in spatial resolution studies of 18F (124I), where only the point source in air was simulated and approximately 3.1 × (4.4 ×) in scatter and gamma fraction studies of 18F (124I). Using realistic MOBY phantoms with 18F-NaF and 124I-IAZG radiopharmaceuticals distribution as input, the efficiency gain for a complete MicroPET simulation is about three-fold for 18F-NaF and fourfold for 124I-IAZG as compared to the regular navigation method, which is the fastest method currently available in GATE. The acceleration of GATE/MPHG is even more significant when compared with compressed voxel and parameterized tracking methods, especially for 124I. The computation speedup makes the GATE/MPHG method very useful for tasks requiring large object simulations such as realistic simulations of biological distributions, the validation of reconstruction algorithms and studies of cardiac or respiratory motions in dynamic PET. When compared to other tracking methods for voxel phantoms, the main benefits of this method are its efficiency in photon tracking and the simplification of decay processes while still retaining most imaging characteristics of the radionuclide. However, the underlying SimSET/PHG behind MPHG can only simulate the photon tracking in voxelized phantoms and modelling of the positron range. However, in nuclear medicine, most imaging instruments such as PET, SPECT and gamma camera are employed to capture the gamma ray. Hence, GATE/MPHG can meet most needs for simulations of imaging applications in nuclear medicine. On the other hand, when charge particle simulations are required, e.g. for dosimetry applications in radionuclide therapy (Chuang et al 2014), other tracking methods with careful verification of the transportation step size should be considered such as compressed voxel or parameterized tracking (Taschereau and Chatziioannou 2008). Although we only demonstrated the feasibility of the proposed method with MicroPET imaging of 18F and 124I, the method is versatile and can also be applied to other radioisotopes for diagnostic nuclear medicine by inputting a user-defined decay scheme. We preformed preliminary tests, not detailed above, of an 111In line source inside HDPE phantoms (5 cm radius and 15 cm length) with the SPECT detector built-in the SPECT benchmark of GATE. The MPHG method achieved an approximately 5.3 fold speedup when compared with the regular tracking method. In addition, the detection probability for the cascaded gammas with higher energy or lower abundance within the energy window is usually small. Hence, it is possible to further optimize the simulated cascaded gamma lines to accelerate the simulation by only modelling the major cascaded gammas. Despite the lack of time modelling in the native mode of SimSET/PHG, SimSET can order and stamp the list mode file with detection times through SimSET’s post-processing utilities (Harrison et al 2005). This function requires huge storage space for the list mode files for reassigning the decay and detection times according to their decay and photon information. However, this is not very pertinent to our developed program as one of our goals was to avoid list mode files. By contrast, the most distinctive feature of GATE apart from other MC codes is the modelling of time information throughout the simulation. However, this information also makes it difficult to apply the variance reduction techniques for time-dependent phenomena such as time-of-flight, detector dead time, random coincidence rate and time-based coincidence detection. As a result, most variance reduction techniques were developed for SPECT (De Beenhouwer et al 2008, Descourt et al 2010), while few variance reduction techniques have been introduced for PET in GATE. Therefore, it is important to improve the efficiency for voxelized phantom in PET, which we have done here using MPHG. On the other hand, as SimSET/PHG is equipped with variance reduction techniques, further work could investigate applying these variance reduction techniques in GATE/MPHG. 6248

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

We have established a new MC simulation tool to improve the simulations of voxelized phantom in GATE by embedding our MPHG program based on SimSET/PHG. The development should be of interest to the user communities of both SimSET and GATE for studying clinical and preclinical imaging. We also plan to release the source code to the communities as open source software in the future. Acknowledgements The authors would like to thank Cynthia Chuang for reading this manuscript and to Robert Harrison, University of Washington Medical Center, for help with the SimSET package. We also would like to thank Dr Uwe Pietrzyk and Michaela Gaens, Institute of Neurosciences and Medicine (INM), Research Center Juelich, Germany, for their assistance with ion source management in GATE. Our thanks also goes to the National Science Council of Taiwan for financially supporting this research under Contract No. NSC102-NU-E-007-007-NU and NSC101-2211-E-007-012-MY3. References Agostinelli  S et al 2003 GEANT4—a simulation toolkit Nucl. Instrum. Methods Phys. Res. A 506 250–303 Allison J et al 2006 Geant4 developments and applications IEEE Trans. Nucl. Sci. 53 270–8 Arce P, Apostolakis J and Cosmo G 2008 A technique for optimised navigation in regular geometries IEEE Nuclear Science Symp. Conf. Record (Dresden, Germany, 19–25 October 2008) pp 857–9 Barret O, Carpenter T A, Clark J C, Ansorge R E and Fryer T D 2005 Monte Carlo simulation and scatter correction of the GE Advance PET scanner with SimSET and Geant4 Phys. Med. Biol. 50 4823–40 Bhat M 1992 Evaluated nuclear structure data file (ENSDF) Nuclear Data for Science and Technology (Berlin: Springer) pp 817–21 Buvat I and Castiglioni I 2002 Monte Carlo simulations in SPET and PET QJ Nucl. Med. 46 48–61 Buvat I and Lazaro D 2006 Monte Carlo simulations in emission tomography and GATE: an overview Nucl. Instrum. Methods Phys. Res. A 569 323–9 Chen C L, Wang Y, Lee J J S and Tsui B M W 2008 Integration of SimSET photon history generator in GATE for efficient Monte Carlo simulations of pinhole SPECT Med. Phys. 35 3278–84 Chuang K S, Lu J C, Lin H H, Dong S L, Yang H J, Shih C T, Lin C H, Yao W J, Ni Y C, Jan M L and Chang S J 2014 Improvements on a patient-specific dose estimation system in nuclear medicine examination Radiat. Prot. Dosim. 158 1–7 Cullen D E, Hubbell J H and Kissel L 1997 EPDL97: The evaluated photon data library,’97 version Lawrence Livermore National Laboratory Report UCRL-50400 6 Daube-Witherspoon M E and Muehllehner G 1987 Treatment of axial data in three-dimensional PET J. Nucl. Med. 28 1717–24 De Beenhouwer  J, Staelens  S, Vandenberghe  S and Lemahieu I 2008 Acceleration of GATE SPECT simulations Med. Phys. 35 1476–85 De Beenhouwer  J, Staelens  S, Vandenberghe  S, Verhaeghe  J, Van Holen  R, Rault  E and Lemahieu  I 2009 Physics process level discrimination of detections for GATE: assessment of contamination in SPECT and spurious activity in PET Med. Phys. 36 1053–60 Descourt  P, Carlier  T, Du  Y, Song  X, Buvat  I, Frey  E, Bardies  M, Tsui  B and Visvikis  D 2010 Implementation of angular response function modelling in SPECT simulations with GATE Phys. Med. Biol. 55 N253–66 Harrison R L, Alessio A M, Kinahan P E and Lewellen T K 2004 Signal to noise ratio in simulations of time-of-flight positron emission tomography IEEE Nuclear Science Symp. Conf. Record (Rome, 16–22 October 2004) pp 4080–3 Harrison R L, Gillispie S B, Alessio A M, Kinahan P E and Lewellen T K 2005 The effects of object size, attenuation, scatter, and random coincidences on signal to noise ratio in simulations of timeof-flight positron emission tomography IEEE Nuclear Science Symp. Conf. Record (Puerto Rico, 23-29 October 2005) pp 1900–4 6249

H-H Lin et al

Phys. Med. Biol. 59 (2014) 6231

Harrison R L, Kaplan M S, Vannoy S D and Lewellen T K 1999 Positron range and coincidence noncollinearity in SimSET IEEE Nuclear Science Symp. Conf. Record (Seattle, WA, 24–30 October 1999 ) pp 1265–8 Harrison R L, Vannoy S D, Haynor D R, Gillispie S B, Kaplan M S and Lewellen T K 1993 Preliminary experience with the photon history generator module of a public-domain simulation system for emission tomography IEEE Nuclear Science Symp. Conf. Record (San Francisco, 31 October– 6 November 1993)pp 1154–8 Haynor D R, Harrison R L, Lewellen T K, Bice A N, Anson C P, Gillispie S B, Miyaoka R S, Pollard K R and Zhu J B 1990 Improving the efficiency of emission tomography simulations using variance reduction techniques IEEE Trans. Nucl. Sci. 37 749–53 Haynor D R, Harrison R L and Lewellen T K 1991 The use of importance sampling techniques to improve the efficiency of photon tracking in emission tomography simulations Med. Phys. 18 990–1001 Heitler W 1954 The Quantum Theory Of Radiation (London: Oxford University Press) Jan S et al 2004 GATE: a simulation toolkit for PET and SPECT Phys. Med. Biol. 49 4543–61 Kaplan M S, Harrison R L and Vannoy S D 1997 Coherent scatter implementation for SimSET IEEE Nuclear Science Symp. Conf. Record (Albuquerque, NM, 9-15 Nov 1997)pp 1303–7 Knoess C, Siegel S, Smith A, Newport D, Richerzhagen N, Winkeler A, Jacobs A, Goble R N, Graf R and Wienhard K 2003 Performance evaluation of the microPET R4 PET scanner for rodents Eur. J. Nucl. Med. Mol. Imag. 30 737–47 Laedermann J-P and Décombaz M 2000 Simulation of nuclear decay Appl. Radiat. Isot. 52 419–25 Lartizien C, Kuntner C, Goertzen A, Evans A and Reilhac A 2007 Validation of PET-SORTEO Monte Carlo simulations for the geometries of the MicroPET R4 and Focus 220 PET scanners Phys. Med. Biol. 52 4845–62 Lehnert  W, Gregoire  M-C, Reilhac  A and Meikle  S R 2011 Analytical positron range modelling in heterogeneous media for PET Monte Carlo simulation Phys. Med. Biol. 56 3313–35 Mok G S, Du Y, Wang Y, Frey E C and Tsui B M 2010 Development and validation of a Monte Carlo simulation tool for multi-pinhole SPECT Mol. Imag. Biol. 12 295–304 NEMA 2008 Performance measurements for small animal positron emission tomographs NEMA Standards Publication NU 4–2008, Technical Report (Rosslyn, VA: National Electrical Manufacturers Association) OpenGATE collaboration 2014 User’s Guide www.opengatecollaboration.org/UsersGuide Palmer  M R and Brownell  G L 1992 Annihilation density distribution calculations for medically important positron emitters IEEE Trans. Med. Imaging 11 373–8 Picard  Y, Thompson  C J and Marrett  S 1992 Improving the precision and accuracy of Monte Carlo simulation in positron emission tomography IEEE Trans. Nucl. Sci. 39 1111–6 Pietrzyk U, Zakhnini A, Axer M, Sauerzapf S, Benoit D and Gaens M 2012 EduGATE–basic examples for educative purpose using the GATE simulation platform Z Med. Phys. 23 65–70 Rehfeld N S, Stute S, Apostolakis J, Soret M and Buvat I 2009 Introducing improved voxel navigation and fictitious interaction tracking in GATE for enhanced efficiency Phys. Med. Biol. 54 2163–78 Schümann  J, Paganetti  H, Shin  J, Faddegon  B and Perl  J 2012 Efficient voxel navigation for proton therapy dose calculation in TOPAS and Geant4 Phys. Med. Biol. 57 3281–93 Segars W P, Tsui B M, Frey E C, Johnson G A and Berr S S 2004 Development of a 4D digital mouse phantom for molecular imaging research Mol. Imag. Biol. 6 149–59 Tang J, Rahmim A, Lautamäki R, Lodge M A, Bengel F M and Tsui B M 2009 Optimization of Rb-82 PET acquisition and reconstruction protocols for myocardial perfusion defect detection Phys. Med. Biol. 54 3161–71 Taschereau R and Chatziioannou A F 2008 Compressed voxels for high-resolution phantom simulations in GATE Mol. Imag. Biol. 10 40–7 Venkataramaiah P, Gopala K, Basavaraju A, Suryanarayana S and Sanjeeviah H 1985 A simple relation for the Fermi function J. Phys. G: Nucl. Phys. 11 359–64 Zaidi  H 1999 Relevance of accurate Monte Carlo modelling in nuclear medical imaging Med. Phys. 26 574–608 Zaidi H 2000 Comparative evaluation of photon cross-section libraries for materials of interest in PET Monte Carlo simulations IEEE Trans. Nucl. Sci. 47 2722–35 Zanzonico P, O’Donoghue J, Chapman J D, Schneider R, Cai S, Larson S, Wen B, Chen Y, Finn R and Ruan S 2004 Iodine-124-labeled iodo-azomycin-galactoside imaging of tumour hypoxia in mice with serial microPET scanning Eur. J. Nucl. Med. Mol. Imag. 31 117–28 Zhu  X and El Fakhri  G 2009 Monte Carlo modelling of cascade gamma rays in 86Y PET imaging: preliminary results Phys. Med. Biol. 54 4181–93 6250