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Dose perturbations at high-Z interfaces in kilovoltage photon beams: comparison with Monte Carlo simulations and measurements
Radiation Physics and Chemistry 64 (2002) 173–179
Dose perturbations at high-Z interfaces in kilovoltage photon beams: comparison with Monte Carlo simulations and measurements Indra J. Dasa,*, Vadim P. Moskvinb, Alireza Kassaeea, Tatsuo Tabatac, Frank Verhaegend a
Department of Radiation Oncology, University of Pennsylvania, 2 Donner Building, 3400 Spruce Street, Philadelphia, PA 19104, USA b Department of Radiation Oncology, Indiana University School of Medicine, Indianapolis, IN, USA c Osaka Prefecture University, Sakai, Osaka, Japan d Royal Marsden Hospital, Institute of Cancer Research, London, UK Received 21 August 2000; accepted 4 July 2001
Abstract Dose perturbations, backscatter dose perturbation factor (BSDF), and forward dose perturbation factor (FDPF) near high-Z material interfaces have been reported to be intense and significant in kilovoltage photon beams. Validity of estimation of dose perturbations is critical in the evaluation of radiation effects associated with high-Z interfaces. The magnitude of dose perturbations has been debated due to limitations in the measuring devices (mainly window thickness and chamber perturbations). Monte Carlo (MC) simulations have been proposed for the interface effects but poor statistics in small spatial bins (1 mm) near the interface makes MC data questionable even with a well designed code. A moving-window least-square polynomial fit (MLPF) method is proposed to smooth MC simulated data. This method is shown to be useful in achieving reasonable accuracy from statistically poor MC data obtained within a reasonable computation time. The EGS4 and PENELOPE MC simulated data at the interfaces between high-Z materials and a water phantom have been used to calculate the BSDF and FDPF which differ significantly from the measurements suggesting the necessity of further refinements in MC approaches and better measuring devices at interface effects. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Interface dosimetry; Dose perturbation; Monte Carlo; PENELOPE; High-Z; Moving-window least-square polynomial fit
1. Introduction Dose perturbations at interfaces between high and low atomic number (Z) materials exposed to kilovoltage photon beams are known to be significant (Das, 1997; Das and Chopra, 1995). The reported dose perturbations near interfaces, however, differ widely among various investigators (Das and Chopra, 1995; Regulla et al., 1998; Verhaegen and Seuntjens, 1995). Regulla *Corresponding author. Tel.: +1-215-662-6472; fax: +1215-349-5978. E-mail address: [email protected] (I.J. Das).
et al. (1998) showed that the dose perturbation for a gold interface in kilovoltage beam may be as high as 100-fold using an exothermic electron emission detector. However, in a similar condition but with an ion chamber, Das, 1997 and Das and Chopra, 1995 showed only 20-fold increase. Thin window detectors with minimal perturbations are generally needed to measure doses in the vicinity of an interface. However, availability of such detectors is rare, and hence, Monte Carlo (MC) simulations have been suggested. MC simulations also differ in their approaches in handling interface effect, and differences in results are clearly visible even for the monoenergetic high energy 60Co beam. Nilsson
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et al. (1992) and Li et al. (1999) reported backscatter dose perturbation factor (BSDF) values of 2.2 and 1.52, respectively, at a Pb interface which are significantly different (45%). Similarly, Kooy et al. (1988) and Li et al. (1999) reported forward dose perturbation factor (FDPF) values of 1.0 and 0.9, respectively, which are also different (11%). These differences get more pronounced for the lower energies and high-Z: To study the dose perturbations close to high-Z material interfaces by MC simulation, high precision is necessary which is ensured by the choice of small depth bins to score deposited energy near the interface. The small depth bins in MC simulation require a considerable increase of computational time to achieve good statistics. Therefore, data computed within a reasonable time still have a considerable degree of statistical fluctuations. To study the behavior of dose perturbations, BSDF and FDPF, near the inhomogeneity in a target volume, it is necessary to reduce the effect of statistical fluctuations before processing MC simulated data. The variance reduction MC methods, e.g. forced interactions, based on a priori knowledge have been theoretically studied (Sobol, 1994). However, such methods have significant limitations and as a rule have been developed only for a certain application (Michailov, 1995). The conventional MC method without any variance reduction method is therefore used for a wide array of applications, especially for the case of radiation passage through inhomogeneous media. General-purpose codes like EGS4, PENELOPE, ITS, GEANT, MCNP are applied analogue techniques that are widely used for radiation transport calculations and as a rule, variance reduction techniques are used in specialized MC codes designed mainly for solving specialized problems. Therefore, it is important to study implementation of smoothing technique in Monte Carlo simulation that is an alternative to the variance reduction MC methods. The published data on dose perturbation near interfaces have been noted to increase dramatically with a large gradient. Therefore, a special technique, applicable for the functional dependency of the large gradient is needed for smoothing the MC data. Tabata (1997) discussed that the commonly known techniques used for reducing statistical fluctuations are not generally applicable in large gradient region and proposed a movingwindow least-square polynomial fit (MLPF) method to process data on energy and charge-deposition distributions of electrons near the target-vacuum interface. It has been shown that the MLPF method removes significant statistical fluctuations and adequately yields the values of deposited charge density close to the targetvacuum interface. In the present work, we have applied the MLPF method for processing the large gradient dose perturbation data at high-Z material interfaces in photon beams
simulated by the use of the MC codes, EGS; (Nelson et al., 1985) and PENELOPE (Baro! et al., 1995). The MC simulated and smoothed results are compared with ion chamber measured data at high-Z interfaces for low energy (p660 keV) photon beams.
2. Method of experiment and computation 2.1. Experimental method A Stabilipan Siemens X-ray unit was used for beam energies in the range of 100–300 kVp. A 137Cs teletherapy unit (Shephard Model 81-14R) was used for Cs gamma rays. Measurements of the BSDF and the FDPF were performed in a polystyrene phantom with a thin window (5 mm) parallel-plate ionization chamber as described elsewhere (Das, 1997; Das and Chopra, 1995). The field size was maintained to be 10 10 cm2. High-Z materials of high purity and precision thickness were used for creating interfaces with tissue like material. Interface was created at a depth of 5 mm from the surface of a 30 30 20 cm3 polystyrene phantom. Thin polystyrene films (3–12 mm) were used for the measurement of interface effect away from interface. The statistical uncertainty with repeated measurements was within 71.5%. 2.2. Monte Carlo simulations The data on dose perturbation near the high-Z material interfaces presented in this paper have been computed by the use of the EGS4 and PENELOPE MC codes. EGS4 (Nelson et al., 1985) is a class II MC code for coupled electron–photon transport in the energy region from 1 keV to several GeV. The photon interactions that are modeled in the energy region of interest are Compton scatter and photo-electric effect. The emission angle of photo-electrons is sampled from the Sauter distribution. Rayleigh scatter was included in the simulations. Relaxation of excited atoms is modeled for K and L-shells (Namito et al., 1998). Auger electron emission and bound Compton effect is not modeled in the EGS4 version used in this study, however, electron transport is performed using the PRESTAII algorithm. Photon and electron transport were followed down to an energy cut-off of 1 keV. For the kilovoltage beam interface simulation, the photon spectrum generated by Verhaegen et al. (1999) based on target material, target angle and thickness, and material of added filter that matched the experimental X-ray beam characteristics in terms of half-value-layer was used. The simulation was performed exactly as the experimental condition outlined above except polystyrene was replaced with water. The limitation of EGS4 for interface effect is well known due to condensed history and boundary crossing
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algorithm (Jeraj et al., 1999; Kawrakow, 2000). A new EGS code, EGSnrc (Kawrakow, 2000) could eliminate some of the existing problems, however, such code is not yet widely available. In this regard a new MC code; PENELOPE was attempted which was initially designed to simulate electron and positron transport in the energy range from 100 eV to about 1 GeV (Baro! et al., 1995). Photon transport was introduced at a later stage (Salvat et al., 1996; Sempau et al., 1997). The algorithm of PENELOPE belongs to the Class II of MC schemes, according to Berger’s classification (Adler et al., 1963). In the PENELOPE algorithm, hard inelastic collisions are described using an optical data model (Ferna! ndezVarea, 1998), where Born differential cross-section was obtained from the generalized oscillator strength model, with the inclusion of the density effect correction. One of the features of the physical model involves the simulation of angular deflection in inelastic collisions that is not implemented in EGS4 code. The physical model is realized in PENELOPE in a form of mixed algorithm to simulate electron–positron transport. Photons are traced by a direct way in a conventional MC scheme (Sempau et al., 1997). The coherent and incoherent scattering, and photoelectric absorption are taken into consideration in the simulation of photon interactions. To minimize computational time, differential cross-sections of photon interactions are written in an analytical form, which sufficiently increases the performance of the simulation. To reduce the time required for PENELOPE simulation, the following strategies were realized. Photon trajectories were traced along the volume of interest. A region of small size near the interface was clearly defined, where BSDF and FDPF were studied. All secondary electrons created inside this region were traced. Secondary electrons created outside the outlined region were traced only if their range is larger than the distance to the limit of the outlined region so that they have a chance to deposit part of their energy in the region of interest. This technique allows several-fold reduction of simulation time. In the present simulation, the size of a region close to the interface was chosen to be equal to 1 cm, which was subdivided further into thin regions of 1 mm. The mixed algorithm is controlled inside the PENELOPE by the use of two parameters, C1 and C2 ; prescribed in an input file. The parameter C1 represents an average angular deflection produced by multiple elastic scattering along a path length equal to the mean free path between hard elastic events, and C2 defines the maximum average fractional energy loss between consecutive hard elastic events. Notations in this paper are given according to those used in the PENELOPE manual (Salvat et al., 1996) and related publications (Sempau et al., 2001). The maximum allowed values of C1 and C2 are limited to 0.2 and 0.1, respectively. The
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values of C1 and C2 used in the present simulation have been half of the above values, i.e. 0.1 and 0.05, respectively. The cutoff parameters, i.e. energy cutoff of photon, EABS (2), and electron trajectories tracing EABS (1), cutoff energy loss, WCC, for hard inelastic collisions of electrons and the cutoff energy loss, WCR, by electrons on hard bremsstrahlung emission are all assigned to be equal to 1 keV in this simulation. The continuous slowing down approximation range of 1 keV electrons in water is equal to 4.911 106 cm. This allows us to choose the variable depth bin width of 1 mm near an interface (up to the depth of 10 mm), 5 mm (for depths from 10 to 100 mm), 30 mm (for depths from 100 to 1000 mm) and 50 mm (at depths greater than 1000 mm). Thus the residual range of electrons is considerably less than the minimum size of a depth bin in the chosen grid. This grid is called the ‘‘basic grid’’. The proposed scheme of depth bin choice allows us to study the dependence of BSDF and FDPF on the depth near the interface where these quantities have high gradients. The kilovoltage photon beams spectra were the same as used in EGS4 code simulation (Verhaegen et al., 1999). To study the efficacy of the smoothing method to reduce the statistical fluctuations for a small depth bin size, we also consider a grid with the constant depth bin size of 1 mm along the selected region. In this study smoothing and interpolation of the data within a basic grid is investigated by the use of MLPF method, and compared with the results of simulation to the basic grid. 2.3. Smoothing and interpolation It is known that the standard deviation, s; of the results of energy deposition computed by the conventional MC method is proportional to the reciprocal of the square root of the number of histories used. When the contribution from secondary electrons due to photon irradiation in a small spatial region is simulated, we have a small number of events per photon trajectory in a limited spatial region, due to relatively large mean free path of photons. Therefore, contribution of a given photon trajectory to secondary electron production, and consequently to energy deposition in a given depth bin of a small size, has large statistical fluctuations. To use the available data effectively by reducing the statistical fluctuations, the smoothing and interpolation are often required at the same time. A combination of the smoothing and interpolation can be made by moving-window least-squares polynomial fits (also called polynomial regression (Simonoff, 1996)). A FORTRAN subroutine subprogram LESINT for this algorithm has been explicitly given and discussed by Tabata (Tabata, 1997). The central part of the algorithm of LESINT finds a smoothed and interpolated (or
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extrapolated) value yout of the dependent variable for an input value xm of independent variable. A polynomial of an arbitrary degree m is fitted by the method of least squares to n data points, which is present in a window taken from a total set of N data points. The number n should be greater than (m þ 1), and the position of the window is chosen so that xm is located as nearly as possible at the center of the window.
of identically distributed random variables with the finite dispersion have the sample mean that tends to be the normal distribution. According to the central limit theorem, we can write the standard deviation s of the sampling in the form of sffiffiffiffiffiffiffiffiffiffiffi DzjðkÞ s¼ ; ð4Þ N
2.4. Monte Carlo data on dose perturbation
where DzjðkÞ is the variance of zjðkÞ :
The contribution ei;ðkÞj by a given event i of energy loss DE at a trajectory j to scoring deposited energy at a given depth bin Bk is determined in the conventional MC method as: ( DEi;j ; if ~ rABk ðkÞ ; ð1Þ ei; j ¼ 0; if ~ reBk where r is the radius-vector of a particle when an event i happens on its trajectory. During the trajectory tracing, contributions ei;ðkÞj from a set of events determine a contribution zj of a given trajectory to the energy deposited in a given depth bin B zjðkÞ ¼
H X
ei;ðkÞj ;
ð2Þ
i¼1
where H is the amount of inelastic events on a given trajectory j; and N is the number of simulated of a given trajectory trajectories. The contribution zðkÞ j to deposited energy, Dk (depth–dose distribution) is the random variable, which has an expected value, mean, Mzk ¼ Dk : Thus the scoring of the deposition energy is determined by DðkÞ E ¼
N X H 1 X eðkÞ : N j¼1 i¼1 i; j
ð3Þ
It is known that the conventional scoring technique, described above, has a finite dispersion. The sequences
3. Results 3.1. Statistical analysis Table 1 illustrates the statistical uncertainties of the results of depth–dose simulation near the high-Z interface of a 0.1 mm lead slab placed in a water phantom, irradiated by a 300 kV photon beam. To highlight the necessary computational time of simulation, we have evaluated the performance on two computers: one is a computer with 586 CPU, 133 MHz, as a lower limiting case, and another with Intel Pentium II CPU, 233 MHz. The speed of simulation by the use of PENELOPE package was 4.23E+01 and 2.00E+02 photon–electron showers per second, respectively. To ensure statistics of 10%, 4 and 0.8 h were spent to simulate 600,000 photon–electron showers for a 300 kVp photon beam incident on water phantom with the 0.1 mm thick lead interface. The simple evaluation shows that to reduce statistical uncertainty down to 1.5%, the 17,700,000 photon–electron showers should be simulated; which will require 116 and 25 h of computation time for CPU 586, 133 MHz and Pentium II, 233 MHz, respectively. With the availability of faster computers the statistical issues may be insignificant; however, the use of MC methods in its conventional form will be extremely inefficient from the point of view of the length of time required for application such as patient treatment
Table 1 Statistical uncertainties of Monte Carlo depth–dose data near the 0.1 mm lead interface in a water phantom, irradiated by 300 kV photon beams Number of photon histories
Statistical uncertainty, 3s=DðkÞ E BSDF
6.0E+05 5.0E+05 3.0E+05 2.0E+05 1.0E+05 5.0E+04
FDPF
1 mm
10 mm
1 mm
10 mm
8.1E02 8.9E02 1.2E01 1.4E01 2.0E01 2.8E01
1.2E01 1.3E01 1.6E01 2.0E01 2.8E01 4.0E01
9.4E02 1.0E01 1.3E01 1.6E01 2.3E01 3.3E01
1.3E01 1.5E01 1.9E01 2.3E01 3.2E01 4.6E01
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planning. Therefore, a method of data processing applicable to the results of MC simulation for radiation passage through inhomogeneous medium may be quite important as a possible way to reduce statistical uncertainty of MC simulation within a reasonable time. 3.2. Application of MLPF method We have applied the MLPF method to process PENELOPE MC data on depth–dose curves in a water phantom with high-Z inhomogeneity irradiated by kilovoltage photon beams. All initial MC data were computed with statistical uncertainty of 10%, in the sense of 3s: The MC data on the depth–dose curve computed for a grid with the depth bin size of 1 mm were smoothed and interpolated at the depths of the basic grid. To check the applicability of the method, the dose perturbation factors were calculated from the direct MC data and the data processed by the use of the MLPF method. To process the data, the degree m of the polynomial was chosen to be equal to 2 (parameter NTERMS=3 in LESINT subroutine), and the number n of the data points in the moving window was 10 (NPTS=10). We tested the case of m ¼ 3; and noted the better smoothing of the MC data. However, in this case it is necessary to manipulate the window size, because the smoothed curve started to oscillate far from interface. In addition, these two groups of the data were compared with the results of MC simulation for the basic grid. Good agreement was obtained among all the sets of the data. Fig. 1 shows the results of processing of the original MC data (symbols) by the MLPF method (lines). The data shows depth–dose profile computed for the depth bin of 1 mm for the case of a 0.1 mm lead interface irradiated by 137Cs gamma rays in the forward direction compared to depth–dose in water phantom. It is seen from this figure that the data are well smoothed in the region of interest close to the interface and in homogeneous medium. Fig. 2 shows the dose perturbation factors, BSDF and FDPF at the 0.1 mm thick lead interface irradiated by 137 Cs and 200 kV photon beams. The BSDF and FDPF calculated from MC simulation are shown by points, and the results of applying the MLPF method are shown by lines. The latter are seen to represent excellent smoothing of the MC data in a given interval of depths. Thus the use of MLPF reasonably reduces statistical fluctuations in MC data and makes simulation near the interfaces obtainable within reasonable computational time. 3.3. Comparison with the experiment and Monte Carlo We compared the results of smoothing of initial PENELOPE MC data with the set of experimental data
Fig. 1. Smoothing of depth–dose distributions of the 137Cs photon beam in a homogeneous (water) medium and near the interface created by a 0.1 mm Pb slab and water phantom. The Pb interface data is for the forward dose perturbation. Points represent the Monte Carlo data computed with 1 mm bins, and solid lines, the results of smoothing by MLPF.
and calculations by the use of EGS4 code. The data on BSDF for 137Cs gamma rays are qualitatively in agreement with measured data and with the simulation by EGS4 in the backscatter direction. This is illustrated in Fig. 3. It is seen that even with the use of the basic grid for scoring in MC simulation, the fluctuations are relatively large, while the smoothing and interpolation produce data with a relatively lower level of fluctuations. It is to be noted that some discrepancies are observed among all the sets of data for the FDPF (Fig. 3). The comparisons of PENELOPE results with the experimental data for kilovoltage photon beams are shown in Fig. 4, for 100, 200 and 300 kVp beams. The agreement is better for the high-energy beams, between experiment and simulation by PENELOPE. With the lower energies, PENELOPE yields values of dose perturbation factors much higher than those obtained by ion chamber measurements. These differences may be due to the window thickness used in experiment, differences in the stopping power ratio, which has been discussed elsewhere (Das et al., 1999) and possible perturbations caused by the ion chamber near high-Z interface. A comparison of simulations by PENELOPE and EGS4 for a lower Z medium (Sn) was also performed showing that the PENELOPE and EGS4 results differ considerably. The PENELOPE implementation of
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Fig. 2. Smoothing of dose perturbation factor vs. distance from a 0.1 mm thick Pb slab in a water phantom for (a) 137Cs and (b) 200 kV photon beams. Points represent the Monte Carlo data and lines are the results of smoothing by the MLPF.
Fig. 3. Comparison of Monte Carlo data for BSDF and FDPF with experiment for the 137Cs photon beam at 0.1 mm thick Pb in a water phantom; PENELOPE (histogram), EGS4 (- - - - and W) and experiment (K).
advanced description of low-energy electron transport provides a detailed description of low-energy photons compared to the last updated version of EGS4. On the other hand, the PENELOPE model of low-energy electron transport can allow us to describe the secondary electron field close to the interface and study the behavior of dose perturbation in this region. Berger (1998) showed that accurate simulation can be performed in a high-Z medium with a condensed history approach. However, in this study, large deviations from experimental values and simulation are observed be-
Fig. 4. Comparison of BSDF data between measurement (symbols) and PENELOPE simulation (lines) for a Pb interface for various kilovoltage beams. Discrepancies are clearly visible at lower energies.
tween high- and low-Z interfaces. This is clearly explained by Kawrakow’s statement (Kawrakow, 2000) that EGS4 model has limitations at low energies due to the boundary crossing algorithm and condensed history, and he proposed a new code EGSnrc to rectify some of these anomalies.
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4. Conclusions The small depth bins, necessary to map the large gradients of dose perturbations at interfaces, produce large statistical uncertainties in Monte carlo simulation. A MLPF method has been shown to be powerful enough to process any large gradient data with a relatively low level of statistical uncertainty. The MLPF method saves a significant amount of time and increases accuracy in MC simulated data. This method can be applied to the MC study of the behavior of dose perturbation close to the interface. The magnitudes of dose perturbation have been found to differ among EGS4 and PENELOPE MC codes and the experiment. Even though PENELOPE data in some cases are closer to the measured ion chamber data, there are significant differences, which needs further elucidation. Additional study is underway to compare ultra thin window ion chamber (1 mm) measurements with the revised EGSnrc MC code.
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Dose perturbations at high-Z interfaces in kilovoltage photon beams: comparison with Monte Carlo simulations and measurements Indra J. Dasa,*, Vadim P. Moskvinb, Alireza Kassaeea, Tatsuo Tabatac, Frank Verhaegend a
Department of Radiation Oncology, University of Pennsylvania, 2 Donner Building, 3400 Spruce Street, Philadelphia, PA 19104, USA b Department of Radiation Oncology, Indiana University School of Medicine, Indianapolis, IN, USA c Osaka Prefecture University, Sakai, Osaka, Japan d Royal Marsden Hospital, Institute of Cancer Research, London, UK Received 21 August 2000; accepted 4 July 2001
Abstract Dose perturbations, backscatter dose perturbation factor (BSDF), and forward dose perturbation factor (FDPF) near high-Z material interfaces have been reported to be intense and significant in kilovoltage photon beams. Validity of estimation of dose perturbations is critical in the evaluation of radiation effects associated with high-Z interfaces. The magnitude of dose perturbations has been debated due to limitations in the measuring devices (mainly window thickness and chamber perturbations). Monte Carlo (MC) simulations have been proposed for the interface effects but poor statistics in small spatial bins (1 mm) near the interface makes MC data questionable even with a well designed code. A moving-window least-square polynomial fit (MLPF) method is proposed to smooth MC simulated data. This method is shown to be useful in achieving reasonable accuracy from statistically poor MC data obtained within a reasonable computation time. The EGS4 and PENELOPE MC simulated data at the interfaces between high-Z materials and a water phantom have been used to calculate the BSDF and FDPF which differ significantly from the measurements suggesting the necessity of further refinements in MC approaches and better measuring devices at interface effects. r 2002 Elsevier Science Ltd. All rights reserved. Keywords: Interface dosimetry; Dose perturbation; Monte Carlo; PENELOPE; High-Z; Moving-window least-square polynomial fit
1. Introduction Dose perturbations at interfaces between high and low atomic number (Z) materials exposed to kilovoltage photon beams are known to be significant (Das, 1997; Das and Chopra, 1995). The reported dose perturbations near interfaces, however, differ widely among various investigators (Das and Chopra, 1995; Regulla et al., 1998; Verhaegen and Seuntjens, 1995). Regulla *Corresponding author. Tel.: +1-215-662-6472; fax: +1215-349-5978. E-mail address: [email protected] (I.J. Das).
et al. (1998) showed that the dose perturbation for a gold interface in kilovoltage beam may be as high as 100-fold using an exothermic electron emission detector. However, in a similar condition but with an ion chamber, Das, 1997 and Das and Chopra, 1995 showed only 20-fold increase. Thin window detectors with minimal perturbations are generally needed to measure doses in the vicinity of an interface. However, availability of such detectors is rare, and hence, Monte Carlo (MC) simulations have been suggested. MC simulations also differ in their approaches in handling interface effect, and differences in results are clearly visible even for the monoenergetic high energy 60Co beam. Nilsson
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et al. (1992) and Li et al. (1999) reported backscatter dose perturbation factor (BSDF) values of 2.2 and 1.52, respectively, at a Pb interface which are significantly different (45%). Similarly, Kooy et al. (1988) and Li et al. (1999) reported forward dose perturbation factor (FDPF) values of 1.0 and 0.9, respectively, which are also different (11%). These differences get more pronounced for the lower energies and high-Z: To study the dose perturbations close to high-Z material interfaces by MC simulation, high precision is necessary which is ensured by the choice of small depth bins to score deposited energy near the interface. The small depth bins in MC simulation require a considerable increase of computational time to achieve good statistics. Therefore, data computed within a reasonable time still have a considerable degree of statistical fluctuations. To study the behavior of dose perturbations, BSDF and FDPF, near the inhomogeneity in a target volume, it is necessary to reduce the effect of statistical fluctuations before processing MC simulated data. The variance reduction MC methods, e.g. forced interactions, based on a priori knowledge have been theoretically studied (Sobol, 1994). However, such methods have significant limitations and as a rule have been developed only for a certain application (Michailov, 1995). The conventional MC method without any variance reduction method is therefore used for a wide array of applications, especially for the case of radiation passage through inhomogeneous media. General-purpose codes like EGS4, PENELOPE, ITS, GEANT, MCNP are applied analogue techniques that are widely used for radiation transport calculations and as a rule, variance reduction techniques are used in specialized MC codes designed mainly for solving specialized problems. Therefore, it is important to study implementation of smoothing technique in Monte Carlo simulation that is an alternative to the variance reduction MC methods. The published data on dose perturbation near interfaces have been noted to increase dramatically with a large gradient. Therefore, a special technique, applicable for the functional dependency of the large gradient is needed for smoothing the MC data. Tabata (1997) discussed that the commonly known techniques used for reducing statistical fluctuations are not generally applicable in large gradient region and proposed a movingwindow least-square polynomial fit (MLPF) method to process data on energy and charge-deposition distributions of electrons near the target-vacuum interface. It has been shown that the MLPF method removes significant statistical fluctuations and adequately yields the values of deposited charge density close to the targetvacuum interface. In the present work, we have applied the MLPF method for processing the large gradient dose perturbation data at high-Z material interfaces in photon beams
simulated by the use of the MC codes, EGS; (Nelson et al., 1985) and PENELOPE (Baro! et al., 1995). The MC simulated and smoothed results are compared with ion chamber measured data at high-Z interfaces for low energy (p660 keV) photon beams.
2. Method of experiment and computation 2.1. Experimental method A Stabilipan Siemens X-ray unit was used for beam energies in the range of 100–300 kVp. A 137Cs teletherapy unit (Shephard Model 81-14R) was used for Cs gamma rays. Measurements of the BSDF and the FDPF were performed in a polystyrene phantom with a thin window (5 mm) parallel-plate ionization chamber as described elsewhere (Das, 1997; Das and Chopra, 1995). The field size was maintained to be 10 10 cm2. High-Z materials of high purity and precision thickness were used for creating interfaces with tissue like material. Interface was created at a depth of 5 mm from the surface of a 30 30 20 cm3 polystyrene phantom. Thin polystyrene films (3–12 mm) were used for the measurement of interface effect away from interface. The statistical uncertainty with repeated measurements was within 71.5%. 2.2. Monte Carlo simulations The data on dose perturbation near the high-Z material interfaces presented in this paper have been computed by the use of the EGS4 and PENELOPE MC codes. EGS4 (Nelson et al., 1985) is a class II MC code for coupled electron–photon transport in the energy region from 1 keV to several GeV. The photon interactions that are modeled in the energy region of interest are Compton scatter and photo-electric effect. The emission angle of photo-electrons is sampled from the Sauter distribution. Rayleigh scatter was included in the simulations. Relaxation of excited atoms is modeled for K and L-shells (Namito et al., 1998). Auger electron emission and bound Compton effect is not modeled in the EGS4 version used in this study, however, electron transport is performed using the PRESTAII algorithm. Photon and electron transport were followed down to an energy cut-off of 1 keV. For the kilovoltage beam interface simulation, the photon spectrum generated by Verhaegen et al. (1999) based on target material, target angle and thickness, and material of added filter that matched the experimental X-ray beam characteristics in terms of half-value-layer was used. The simulation was performed exactly as the experimental condition outlined above except polystyrene was replaced with water. The limitation of EGS4 for interface effect is well known due to condensed history and boundary crossing
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algorithm (Jeraj et al., 1999; Kawrakow, 2000). A new EGS code, EGSnrc (Kawrakow, 2000) could eliminate some of the existing problems, however, such code is not yet widely available. In this regard a new MC code; PENELOPE was attempted which was initially designed to simulate electron and positron transport in the energy range from 100 eV to about 1 GeV (Baro! et al., 1995). Photon transport was introduced at a later stage (Salvat et al., 1996; Sempau et al., 1997). The algorithm of PENELOPE belongs to the Class II of MC schemes, according to Berger’s classification (Adler et al., 1963). In the PENELOPE algorithm, hard inelastic collisions are described using an optical data model (Ferna! ndezVarea, 1998), where Born differential cross-section was obtained from the generalized oscillator strength model, with the inclusion of the density effect correction. One of the features of the physical model involves the simulation of angular deflection in inelastic collisions that is not implemented in EGS4 code. The physical model is realized in PENELOPE in a form of mixed algorithm to simulate electron–positron transport. Photons are traced by a direct way in a conventional MC scheme (Sempau et al., 1997). The coherent and incoherent scattering, and photoelectric absorption are taken into consideration in the simulation of photon interactions. To minimize computational time, differential cross-sections of photon interactions are written in an analytical form, which sufficiently increases the performance of the simulation. To reduce the time required for PENELOPE simulation, the following strategies were realized. Photon trajectories were traced along the volume of interest. A region of small size near the interface was clearly defined, where BSDF and FDPF were studied. All secondary electrons created inside this region were traced. Secondary electrons created outside the outlined region were traced only if their range is larger than the distance to the limit of the outlined region so that they have a chance to deposit part of their energy in the region of interest. This technique allows several-fold reduction of simulation time. In the present simulation, the size of a region close to the interface was chosen to be equal to 1 cm, which was subdivided further into thin regions of 1 mm. The mixed algorithm is controlled inside the PENELOPE by the use of two parameters, C1 and C2 ; prescribed in an input file. The parameter C1 represents an average angular deflection produced by multiple elastic scattering along a path length equal to the mean free path between hard elastic events, and C2 defines the maximum average fractional energy loss between consecutive hard elastic events. Notations in this paper are given according to those used in the PENELOPE manual (Salvat et al., 1996) and related publications (Sempau et al., 2001). The maximum allowed values of C1 and C2 are limited to 0.2 and 0.1, respectively. The
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values of C1 and C2 used in the present simulation have been half of the above values, i.e. 0.1 and 0.05, respectively. The cutoff parameters, i.e. energy cutoff of photon, EABS (2), and electron trajectories tracing EABS (1), cutoff energy loss, WCC, for hard inelastic collisions of electrons and the cutoff energy loss, WCR, by electrons on hard bremsstrahlung emission are all assigned to be equal to 1 keV in this simulation. The continuous slowing down approximation range of 1 keV electrons in water is equal to 4.911 106 cm. This allows us to choose the variable depth bin width of 1 mm near an interface (up to the depth of 10 mm), 5 mm (for depths from 10 to 100 mm), 30 mm (for depths from 100 to 1000 mm) and 50 mm (at depths greater than 1000 mm). Thus the residual range of electrons is considerably less than the minimum size of a depth bin in the chosen grid. This grid is called the ‘‘basic grid’’. The proposed scheme of depth bin choice allows us to study the dependence of BSDF and FDPF on the depth near the interface where these quantities have high gradients. The kilovoltage photon beams spectra were the same as used in EGS4 code simulation (Verhaegen et al., 1999). To study the efficacy of the smoothing method to reduce the statistical fluctuations for a small depth bin size, we also consider a grid with the constant depth bin size of 1 mm along the selected region. In this study smoothing and interpolation of the data within a basic grid is investigated by the use of MLPF method, and compared with the results of simulation to the basic grid. 2.3. Smoothing and interpolation It is known that the standard deviation, s; of the results of energy deposition computed by the conventional MC method is proportional to the reciprocal of the square root of the number of histories used. When the contribution from secondary electrons due to photon irradiation in a small spatial region is simulated, we have a small number of events per photon trajectory in a limited spatial region, due to relatively large mean free path of photons. Therefore, contribution of a given photon trajectory to secondary electron production, and consequently to energy deposition in a given depth bin of a small size, has large statistical fluctuations. To use the available data effectively by reducing the statistical fluctuations, the smoothing and interpolation are often required at the same time. A combination of the smoothing and interpolation can be made by moving-window least-squares polynomial fits (also called polynomial regression (Simonoff, 1996)). A FORTRAN subroutine subprogram LESINT for this algorithm has been explicitly given and discussed by Tabata (Tabata, 1997). The central part of the algorithm of LESINT finds a smoothed and interpolated (or
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extrapolated) value yout of the dependent variable for an input value xm of independent variable. A polynomial of an arbitrary degree m is fitted by the method of least squares to n data points, which is present in a window taken from a total set of N data points. The number n should be greater than (m þ 1), and the position of the window is chosen so that xm is located as nearly as possible at the center of the window.
of identically distributed random variables with the finite dispersion have the sample mean that tends to be the normal distribution. According to the central limit theorem, we can write the standard deviation s of the sampling in the form of sffiffiffiffiffiffiffiffiffiffiffi DzjðkÞ s¼ ; ð4Þ N
2.4. Monte Carlo data on dose perturbation
where DzjðkÞ is the variance of zjðkÞ :
The contribution ei;ðkÞj by a given event i of energy loss DE at a trajectory j to scoring deposited energy at a given depth bin Bk is determined in the conventional MC method as: ( DEi;j ; if ~ rABk ðkÞ ; ð1Þ ei; j ¼ 0; if ~ reBk where r is the radius-vector of a particle when an event i happens on its trajectory. During the trajectory tracing, contributions ei;ðkÞj from a set of events determine a contribution zj of a given trajectory to the energy deposited in a given depth bin B zjðkÞ ¼
H X
ei;ðkÞj ;
ð2Þ
i¼1
where H is the amount of inelastic events on a given trajectory j; and N is the number of simulated of a given trajectory trajectories. The contribution zðkÞ j to deposited energy, Dk (depth–dose distribution) is the random variable, which has an expected value, mean, Mzk ¼ Dk : Thus the scoring of the deposition energy is determined by DðkÞ E ¼
N X H 1 X eðkÞ : N j¼1 i¼1 i; j
ð3Þ
It is known that the conventional scoring technique, described above, has a finite dispersion. The sequences
3. Results 3.1. Statistical analysis Table 1 illustrates the statistical uncertainties of the results of depth–dose simulation near the high-Z interface of a 0.1 mm lead slab placed in a water phantom, irradiated by a 300 kV photon beam. To highlight the necessary computational time of simulation, we have evaluated the performance on two computers: one is a computer with 586 CPU, 133 MHz, as a lower limiting case, and another with Intel Pentium II CPU, 233 MHz. The speed of simulation by the use of PENELOPE package was 4.23E+01 and 2.00E+02 photon–electron showers per second, respectively. To ensure statistics of 10%, 4 and 0.8 h were spent to simulate 600,000 photon–electron showers for a 300 kVp photon beam incident on water phantom with the 0.1 mm thick lead interface. The simple evaluation shows that to reduce statistical uncertainty down to 1.5%, the 17,700,000 photon–electron showers should be simulated; which will require 116 and 25 h of computation time for CPU 586, 133 MHz and Pentium II, 233 MHz, respectively. With the availability of faster computers the statistical issues may be insignificant; however, the use of MC methods in its conventional form will be extremely inefficient from the point of view of the length of time required for application such as patient treatment
Table 1 Statistical uncertainties of Monte Carlo depth–dose data near the 0.1 mm lead interface in a water phantom, irradiated by 300 kV photon beams Number of photon histories
Statistical uncertainty, 3s=DðkÞ E BSDF
6.0E+05 5.0E+05 3.0E+05 2.0E+05 1.0E+05 5.0E+04
FDPF
1 mm
10 mm
1 mm
10 mm
8.1E02 8.9E02 1.2E01 1.4E01 2.0E01 2.8E01
1.2E01 1.3E01 1.6E01 2.0E01 2.8E01 4.0E01
9.4E02 1.0E01 1.3E01 1.6E01 2.3E01 3.3E01
1.3E01 1.5E01 1.9E01 2.3E01 3.2E01 4.6E01
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planning. Therefore, a method of data processing applicable to the results of MC simulation for radiation passage through inhomogeneous medium may be quite important as a possible way to reduce statistical uncertainty of MC simulation within a reasonable time. 3.2. Application of MLPF method We have applied the MLPF method to process PENELOPE MC data on depth–dose curves in a water phantom with high-Z inhomogeneity irradiated by kilovoltage photon beams. All initial MC data were computed with statistical uncertainty of 10%, in the sense of 3s: The MC data on the depth–dose curve computed for a grid with the depth bin size of 1 mm were smoothed and interpolated at the depths of the basic grid. To check the applicability of the method, the dose perturbation factors were calculated from the direct MC data and the data processed by the use of the MLPF method. To process the data, the degree m of the polynomial was chosen to be equal to 2 (parameter NTERMS=3 in LESINT subroutine), and the number n of the data points in the moving window was 10 (NPTS=10). We tested the case of m ¼ 3; and noted the better smoothing of the MC data. However, in this case it is necessary to manipulate the window size, because the smoothed curve started to oscillate far from interface. In addition, these two groups of the data were compared with the results of MC simulation for the basic grid. Good agreement was obtained among all the sets of the data. Fig. 1 shows the results of processing of the original MC data (symbols) by the MLPF method (lines). The data shows depth–dose profile computed for the depth bin of 1 mm for the case of a 0.1 mm lead interface irradiated by 137Cs gamma rays in the forward direction compared to depth–dose in water phantom. It is seen from this figure that the data are well smoothed in the region of interest close to the interface and in homogeneous medium. Fig. 2 shows the dose perturbation factors, BSDF and FDPF at the 0.1 mm thick lead interface irradiated by 137 Cs and 200 kV photon beams. The BSDF and FDPF calculated from MC simulation are shown by points, and the results of applying the MLPF method are shown by lines. The latter are seen to represent excellent smoothing of the MC data in a given interval of depths. Thus the use of MLPF reasonably reduces statistical fluctuations in MC data and makes simulation near the interfaces obtainable within reasonable computational time. 3.3. Comparison with the experiment and Monte Carlo We compared the results of smoothing of initial PENELOPE MC data with the set of experimental data
Fig. 1. Smoothing of depth–dose distributions of the 137Cs photon beam in a homogeneous (water) medium and near the interface created by a 0.1 mm Pb slab and water phantom. The Pb interface data is for the forward dose perturbation. Points represent the Monte Carlo data computed with 1 mm bins, and solid lines, the results of smoothing by MLPF.
and calculations by the use of EGS4 code. The data on BSDF for 137Cs gamma rays are qualitatively in agreement with measured data and with the simulation by EGS4 in the backscatter direction. This is illustrated in Fig. 3. It is seen that even with the use of the basic grid for scoring in MC simulation, the fluctuations are relatively large, while the smoothing and interpolation produce data with a relatively lower level of fluctuations. It is to be noted that some discrepancies are observed among all the sets of data for the FDPF (Fig. 3). The comparisons of PENELOPE results with the experimental data for kilovoltage photon beams are shown in Fig. 4, for 100, 200 and 300 kVp beams. The agreement is better for the high-energy beams, between experiment and simulation by PENELOPE. With the lower energies, PENELOPE yields values of dose perturbation factors much higher than those obtained by ion chamber measurements. These differences may be due to the window thickness used in experiment, differences in the stopping power ratio, which has been discussed elsewhere (Das et al., 1999) and possible perturbations caused by the ion chamber near high-Z interface. A comparison of simulations by PENELOPE and EGS4 for a lower Z medium (Sn) was also performed showing that the PENELOPE and EGS4 results differ considerably. The PENELOPE implementation of
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Fig. 2. Smoothing of dose perturbation factor vs. distance from a 0.1 mm thick Pb slab in a water phantom for (a) 137Cs and (b) 200 kV photon beams. Points represent the Monte Carlo data and lines are the results of smoothing by the MLPF.
Fig. 3. Comparison of Monte Carlo data for BSDF and FDPF with experiment for the 137Cs photon beam at 0.1 mm thick Pb in a water phantom; PENELOPE (histogram), EGS4 (- - - - and W) and experiment (K).
advanced description of low-energy electron transport provides a detailed description of low-energy photons compared to the last updated version of EGS4. On the other hand, the PENELOPE model of low-energy electron transport can allow us to describe the secondary electron field close to the interface and study the behavior of dose perturbation in this region. Berger (1998) showed that accurate simulation can be performed in a high-Z medium with a condensed history approach. However, in this study, large deviations from experimental values and simulation are observed be-
Fig. 4. Comparison of BSDF data between measurement (symbols) and PENELOPE simulation (lines) for a Pb interface for various kilovoltage beams. Discrepancies are clearly visible at lower energies.
tween high- and low-Z interfaces. This is clearly explained by Kawrakow’s statement (Kawrakow, 2000) that EGS4 model has limitations at low energies due to the boundary crossing algorithm and condensed history, and he proposed a new code EGSnrc to rectify some of these anomalies.
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4. Conclusions The small depth bins, necessary to map the large gradients of dose perturbations at interfaces, produce large statistical uncertainties in Monte carlo simulation. A MLPF method has been shown to be powerful enough to process any large gradient data with a relatively low level of statistical uncertainty. The MLPF method saves a significant amount of time and increases accuracy in MC simulated data. This method can be applied to the MC study of the behavior of dose perturbation close to the interface. The magnitudes of dose perturbation have been found to differ among EGS4 and PENELOPE MC codes and the experiment. Even though PENELOPE data in some cases are closer to the measured ion chamber data, there are significant differences, which needs further elucidation. Additional study is underway to compare ultra thin window ion chamber (1 mm) measurements with the revised EGSnrc MC code.
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