Journal of the ICRU Vol 4 No 2 (2004) Report 72 Oxford University Press
4
DETERMINATION OF DOSE DISTRIBUTIONS
4.1 GENERAL FORMALISM FOR DOSE DISTRIBUTIONS 4.1.1
DOI: 10.1093/jicru/ndh028
Introduction
4.1.2
Formalism for seed and wire sources
The formalisms for dose distributions of beta-ray and photon sources are very similar. The polarcoordinate (r, y) system as shown in Fig. 4.1 is used. It is assumed that the source dimensions are small compared with the mean-free path of the emitted radiations. The absorbed-dose rate, D_ ðr, yÞ, at a point (r, y) is given for beta-ray and short-wire sources by Gðr, yÞ D_ ðr, yÞ ¼ D_ ðr0 , y0 Þ ð4:1Þ gr ðrÞF ðr, yÞ, Gðr0 , y0 Þ 0 and for photon seed sources by Gðr, yÞ D_ ðr, yÞ ¼ Kd Lr0 gr ðrÞF ðr, yÞ: Gðr0 , y0 Þ 0
ð4:2Þ
In Eq. (4.2), D_ ðr0 , y0 Þ is the measured absorbeddose rate at the reference point (r0, y0). For betaray seed and wire sources, r0 ¼ 2 mm and y0 ¼ p/ 2 (Nath et al., 1999; see Section 3.1.1). In Equation 4.2, K_ d is the reference air-kerma rate, and Lr0 is a dimensionless constant, called the dose-rate constant, defined as absorbed-dose rate at a specified
ª International Commission on Radiation Units and Measurements 2004
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As described in Section 3, brachytherapy beta-ray sources are calibrated in terms of the reference absorbed-dose rate, and low-energy photon sources in terms of the reference air-kerma rate. For betaray seed and wire sources and for photon seed sources, the complete relative dose distribution around a single source can be determined a priori for each source design of each radionuclide, as a characteristic of the source design. If the sources are symmetrical and uniform within the limits recommended, a two-dimensional distribution is usually considered sufficient. These dose distributions can then be presented in a consistent formalism for all such beta-ray and photon sources. The formalism is an approach wherein the various effects are taken into consideration by appropriate factors and functions. It includes explicitly the reference absorbed-dose rate or reference air-kerma rate, as the starting point for the calculation of the complete dose distribution in absolute terms (in units of gray per hour). The formalism can also be conveniently applied to calculate the dose distribution from a given configuration of seed sources, e.g., a linear array of sources (a source train). The above approach has been proposed by the Interstitial Collaborative Working Group (ICWG, 1990), and subsequently adopted by the American Association of Physicists in Medicine (Nath et al., 1995; Rivard et al., 2004) for 125I, 103Pd and 192Ir seed sources. Recently, the same approach has been adopted also for beta-ray seed and wire sources used in catheter-based systems for intravascular radiotherapy (Nath et al., 1999). This ICRU Report adopts the ICWG approach. A traditional approach, for photons with energy more than 100 keV, is to incorporate dosimetric reference data into analytical expressions, whereby relatively simple formulae are available to calculate the dose distribution. Several authors have introduced additional terms into the simplest equations, to account for the additional dose variation with
distance. Some have proposed separate terms to account for buildup and attenuation (Berger, 1968; Meisberger et al., 1968), while another has included a general radial-dose variation term (Dale, 1983). For low-energy photons, however, the increased attenuation and scatter in the medium surrounding the source modify the dose rate substantially, even within a distance of 20 mm from the source. Moreover, low-energy photons are significantly absorbed in a spatially non-uniform way by the source encapsulation, leading to highly anisotropic photon emission. The effect of source encapsulation could be estimated by a Sievert-integral approach that accounts for the attenuation of the photons by their slant penetration through the material, but------given the complexity of the source structures and subsequent photon scattering by the medium------accurate results are difficult to obtain (Williamson et al., 1983). Instead, the approach introduced by the ICWG is recommended for low-energy photon sources.
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS
simple line-source approximation, assuming uniform activity distribution, with the result Gðr, yÞ ¼
b , Lr sin y
ð4:4Þ
where b is the angle subtended by the effective active source length, L, with respect to the point (r, y) (see Fig. 4.1). An example of such a calculation for a source of length 2.5 mm is given in Table 6.6 (Section 6). Note that
r b ¼ r2 , r!1 Lr sin y
L
lim
ð4:5Þ
Figure 4.1. Linear source geometry. Formalism for the calculation of absorbed-dose rate at point P is to be presented using the polar coordinates (r, y). b is the angle subtended by the effective active-source length L, with respect to point P. Symmetry about the z-axis is assumed, so there is no dependence on azimuthal angle j.
g r0 ð r Þ ¼
D_ ðr, y0 ÞGðr0 , y0 Þ : D_ ðr0 , y0 ÞGðr, y0 Þ
ð4:6Þ
In the point-source approximation, the radial-dose function is .h i gr0 ðrÞ ¼ D_ ðrÞr2 D_ ðr0 Þr20 : ð4:7Þ
reference point per unit reference air-kerma rate. The value of Lr0 is considered characteristic of a source design. The product of K_ d Lr0 then gives the absorbed-dose rate at the reference point (r0, y0). For photon seed sources in conventional brachytherapy, the reference point is often specified at 10 mm from the source in a perpendicular bisecting plane, i.e., r0 ¼ 10 mm, y0 ¼ p/2 (Nath et al., 1995). In Equation 4.2, G(r, y) is a geometry factor that describes the variation of relative dose due only to the spatial distribution of activity within the source, and ignores absorption and scattering in the source and surrounding medium. Division by G(r0, y0) removes to first order the source-geometry effects from the dose rate at the reference point to yield an effective point-source value. Multiplication by G(r, y) restores the effect of source geometry at the point of calculation. The general equation for the geometry factor G(r, y) is i R Vh 0 2 0 r ð r Þ=jr rj dV 0 0 Gðr, yÞ ¼ , ð4:3Þ RV 0 0 0 rðr Þ dV
Finally, in Equation 4.2, F(r, y) is the anisotropy function that accounts for the anisotropy of the dose distribution around the source in the plane containing the seed axis, including the effects of absorption and scatter in the medium. It is defined by F ðr, yÞ ¼
D_ ðr, yÞGðr, y0 Þ : D_ ðr, y0 ÞGðr, yÞ
ð4:8Þ
The anisotropy function F(r, y) is normalized to 1.0 at y ¼ p/2 for all r. In addition, the anisotropy function can be integrated over solid angle to produce a point-source approximation: Z p 1 fan ðrÞ ¼ F ðr, yÞGðr, yÞsinðyÞdy, ð4:9Þ 2Gðr, p=2Þ 0 where cylindrical symmetry has been assumed and fan(r) is called the anisotropy factor. The anisotropy factor averaged over distances r from 10 to 50 mm . As has been yields the anisotropy constant, f an reported for photon sources (Weaver et al., 1996; Weaver, 1998), F(r, y) is not a uniquely defined function, as random variances in seed construction can produce seed-to-seed variations in the angular-dose
where r(r 0 ) represents the density of radioactivity at the point (r0 ) ¼ (x0 , y0 , z0 ) within the source, V denotes the integration over the source core, and dV 0 is a volume element located at r0 in the source. As an example for seed sources, the geometry factor G(r, y) is calculated using the much more tractable, 46
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i.e., in the point-source approximation, the geometry factor reduces to the simple inverse-square factor. The quantity gr0 ðrÞ is the radial-dose function and accounts for the effects of absorption and scattering in the medium, along the transverse axis of the source. It does not include geometrical effects, and is normalized to 1.0 at the reference point, i.e., gr0 ðr0 Þ ¼ 1. It is defined as
DETERMINATION OF DOSE DISTRIBUTIONS
(a)
X Position, mm
4.1.3 Application of the formalism for seed and wire sources
Y position, mm
The application of the formalism for seed and wire sources inherently requires that values of all the factors be available for each radionuclide and source design. Such dosimetric reference data can be determined both by theoretical calculations and by measurements. Different methods of calculation and measurement are discussed in Sections 4.2----4.5. It is generally recommended (IAEA, 2003; Williamson et al., 1998) that each new brachytherapy source design should have at least two independent, published studies giving the parameters of the dose-distribution formalism; one study should be experimental, the other could be theoretical, e.g., Monte Carlo calculations. Results of such determinations are presented in Section 6. It should be noted that none of the factors in the formalism has to be determined explicitly from the internal structure of the source. Factoring out the geometry term G(r, y) minimizes the spatial dependence of gr0 ðrÞ and F(r, y), even for low-energy sources, and allows them to be fit with slowly varying functions. The factors can also be accurately interpolated when discrete values are stored in lookup tables for computer calculations. A Practical Example. An example of the use of the formalism, Equation 4.1, is given in the following for the case of a linear array (train) of beta-ray seed sources (Fig. 4.2). The dose rate at a distance of 2 mm in water above the central seed in the train is the linear sum of the dose rates of all the seeds in the train. Thus, if each seed has a dose rate at 2 mm of unity, the dose rate at a point 2 mm above the center of the central seed from each neighboring seed can be calculated using Equation 4.1 with the data from
(b)
Figure 4.2. Isodose-rate contour maps of a train of (a) 9 seeds, and (b) a single seed of 90Sr----90Y, both at a distance of 2 mm in water from a plane containing the source axis, calculated using the recommended formalism (Eq. 4.1). Contour intervals are 0.2, relative to the maximum dose rate from a single seed.
Tables 6.4----6.6 (Section 6). This is shown in Table 4.1. From the last column in Table 4.1, it can be seen that only the two seeds on both sides of the central seed, and the central seed itself, significantly influence the dose rate above the central seed. For seeds with equal reference absorbed-dose rates, the 47
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distribution. These variations are especially pronounced at points near the seed axis, where measured values of F(r, y) can vary by as much as 80% for individual 125I seeds. Accordingly, average values for F(r, y) are used. The formalism outlined above is based on the assumption that the source dimensions are small compared with the mean-free path of the emitted radiations. For applications in which at least one source dimension cannot be considered small, as might be the case for long beta-ray line sources used in intravascular applications, this formalism leads to certain complications. For the description of dose-rate distributions about relatively long beta-ray line sources, therefore, an adapted formalism, conceptually similar to the ICWG approach, with another coordinate system has been proposed (Schaart et al., 2001).
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS Table 4.1. Example of the use of the formalism (Eq. 4.1) to calculate dose-rate enhancement due to neighbouring seeds for a point 2 mm in water above the center of the central seed. Seed location from center of train
r (mm)
y ( )
gr0(r)
G(r, y)
G(r0, y0)
F(r, y)
D(r, y)
3 2 1 0 1 2 3
7.76 5.39 3.20 2.00 3.20 5.39 7.76
14.9 21.8 38.7 90.0 38.7 21.8 14.9
0.039 0.282 0.756 1.000 0.756 0.282 0.039
0.0170 0.0361 0.1042 0.2234 0.1042 0.0361 0.0170
0.2234 0.2234 0.2234 0.2234 0.2234 0.2234 0.2234
0.997 0.861 0.876 1.000 0.876 0.861 0.997
0.003 0.039 0.309 1.000 0.309 0.039 0.003
X Position, mm
(a) 25
20
4.1.4 Formalism for planar and concave sources
15
10
(b)
ð4:10Þ
where (z0, r0) is the reference location, z is the depth in water-equivalent medium measured along the central axis of the planar or concave source, and r is the radial distance from this axis. The reference location is r0 ¼ 0 mm and z0 ¼ 1 mm (Section 3.1.1).
0
0
5
10
5
10
10
5
4.2 CALCULATION OF DOSE DISTRIBUTIONS FOR BETA RAYS 4.2.1
Y Position, mm
5
D_ ðz, rÞ ¼ D_ ðz0 , r0 Þ½D_ ðz, r0 Þ=D_ ðz0 , r0 Þ · ½D_ ðz, rÞ=D_ ð z, r0 Þ ,
A’
A
Absorbed-dose distributions for planar and concave beta-ray applicators can also be conveniently presented by analytical formalism. The absorbeddose distribution in this case can be broken down into the product of: (1) the reference absorbed-dose rate, D_ ðz0 , r0 Þ, (2) the relative central-axis depthdose function, D_ ð z, r0 Þ=D_ ðz0 , r0 Þ, and (3) the relative off-axis dose function, D_ ð z, rÞ=D_ ð z, r0 Þ. Thus, the absorbed-dose rate at any point, D_ ð z, rÞ, can be expressed as
0
0
Figure 4.3. Measured isodose-rate contour maps of (a) a train of 9 seeds, and (b) a single seed of 90Sr----90Y, both at a distance of 1.98 mm in A150 plastic from a plane containing the source axis, obtained from measurements using radiochromic film. Contour intervals are 5 mGy s 1. The dashed circle indicates the position of the 1 mm collecting electrode during measurements with the extrapolation chamber. The line AA’ is discussed in Section 4.4.3.
Introduction
Because ideal point-like detectors are not available for the accurate measurement of relative-dose distributions, calculations of dose distributions are carried out to supplement or verify the data from 48
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enhancement factor is 1.70. For comparison purposes, radiochromic film was exposed to nine of the seeds loaded end-to-end in A150 plastic at a depth of 1.98 mm. The scan of this film is shown in Fig. 4.3a. The dose rate averaged over the central circle of diameter 1 mm (Fig. 4.3a) is 73 mGy s1, while that of the central seed, which is shown alone in Fig. 4.3b, is 45 mGy s1; the ratio of these, 1.62, is in fair agreement with that predicted by the calculation above.
DETERMINATION OF DOSE DISTRIBUTIONS measurements with finite-size detectors. This section deals with the calculation of the spatial distributions of absorbed dose from beta-ray sources with activity distributed in area or volume. A number of methods have been applied to the calculation of dose distributions from beta rays:
The solutions of the electron transport problem by most ‘‘deterministic’’ methods, such as analytical or numerical solutions of the transport equations (e.g., Spencer, 1955, 1959) are limited either to unbounded homogeneous media or to very simple geometrical shapes and boundaries. Moreover, such methods cannot easily handle stochastic processes such as range straggling, and therefore significantly underestimate doses at distances from the source that are comparable with the range of the electrons. Analytical and numerical methods can be used that are based on beta-ray point-source dose functions (the point kernels), obtained from measurements or calculations. The point-source dose function gives the absorbed-dose distribution from an ideal point source emitting beta rays isotropically in a uniform medium, and is then a function of only a single spatial variable, the distance from the point source. Integration of the point-source dose function over the surface or volume containing the beta activity will give the dose distributions in the vicinity of the extended source. These analytical and numerical methods are fast and easy to apply. However, they are strictly valid only in a homogeneous medium of uniform density. The effects of air gaps, exit windows, backscatter from the applicator, or other source-encapsulation issues are not taken into account. This can introduce errors, which can be important at the level of clinical applications (see Section 4.2.3.3). In contrast to the above, Monte Carlo methods can handle complex geometrical arrangements, multiple materials, and the coupling of electron and photon histories. The accuracy of the results is often as good as measurements, or even better. Many former limitations in statistical accuracy are disappearing with the rapid increases in computer speed and availability.
4.2.2 Monte Carlo methods for electron transport This section gives a brief overview of the most widely used Monte Carlo codes for solving electrontransport problems. ICRU Report 56 (ICRU, 1997b) gives a more extensive discussion. There are also many reviews in the literature (Andreo, 1991; Berger, 1963; Raeside, 1976; Rogers and Bielajew, 1990; Turner et al., 1985), including the full text of a course on the subject given in 1987 (Jenkins et al., 1988). Monte Carlo calculations for electron transport are usually done differently from those for indirectly ionizing particles such as photons or neutrons, for which the simulated-particle history treats each interaction separately. For electrons, such a detailed history is usually not practical except at low energies (e.g., below about 100 keV), because the very large number of Coulomb collisions would require a prohibitively long computing time. Instead, a ‘‘condensed history’’ is used, in which short path segments are chosen and the net energy loss and angular deflection in each step are selected randomly from the results of multiple-scattering theories. The choice of step size is important and is a compromise. It must be small enough that in most of the steps the electron does not cross a boundary between different materials, that the interaction cross sections do not change significantly within a step, and that approximations made in treating the deviations from the straight line path that occur within a step are appropriate. It must be large enough to contain sufficient collisions to justify using
In view of the limitations of the analytical and numerical methods based on integrating the betaray point-source function, Monte Carlo methods of calculation (Section 4.2.2) are recommended in this Report for comparisons with measurements for practical sources. However, analytical or numerical methods based on the beta-ray point-source dose functions or point kernels are often applied in different clinical situations, and can be reasonably 49
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accurate for some particular cases such as for radioactive stents (Coffey and Duggan, 1996; Duggan et al., 1998; Janicki et al., 1997; Prestwich et al., 1995). Moreover, Monte Carlo methods are not always available for daily practice in the clinic, and a simple empirical calculation might still be a practical way to study the dose distributions for various clinical situations. Therefore, the most useful beta-ray point-source dose functions and their application for the calculation of dose distributions are discussed in detail in Appendix C. The important limitations of the analytical and numerical methods are discussed further in Section 4.2.3.3. Dose distributions in water are calculated as they most closely represent clinical situations and because measurements are often made in water. If doses in another medium are required, a convenient scaling procedure will allow, for homogeneous situations, the conversion from one low-Z medium to another (Section 4.5).
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS
4.2.2.1
ETRAN, ITS and MCNP codes
The current ETRAN code (Seltzer, 1991) is a descendant of versions developed in the 1960s (Berger, 1963; Berger and Seltzer, 1968a, b, c), which have been repeatedly improved over the last three decades (Berger and Seltzer, 1982; Seltzer and Berger, 1985, 1986; Seltzer, 1988). It was developed originally for applications involving electrons up to 100 MeV but now covers electron----photon cascades from giga electronvolt energies down to 1 keV. The geometries treated are mainly homogeneous bodies with simple shapes. The variety of geometrical shapes that could be handled was greatly extended in the SANDYL code (Colbert, 1974) and in the ITS of codes (Halbleib, 1980, 1988; Halbleib and Mehlhorn, 1984, 1986). ITS codes form a unified package of codes, each treating a different geometry: TIGER for multiple plane slabs, CYLTRAN for geometries with cylindrical symmetry, and ACCEPT, which uses threedimensional combinatorial geometry. All ITS codes 50
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use ETRAN to provide the radiation-transport physics. The current version of ITS, ITS 3.0 (Halbleib et al., 1992a, b), was distributed in 1992. It includes all the cross sections and other physical data required. Among the changes from the previous version (ITS 2.1) that significantly affect the transport of electrons of up to a few mega electronvolts, the most important are the use of revised collision stopping powers (ICRU, 1984; 2 % smaller for water than earlier values), improved bremsstrahlung production cross sections, and an improved method of calculating energy-loss straggling (Seltzer, 1991). These codes have been widely used for electron and beta-ray dosimetry problems in radiation protection and medical physics. These include calculation of electron point kernels in water for energies from 0.02 up to 4 MeV (Cross et al., 1992a) and up to 20 MeV (Seltzer, 1991), dose distributions from isotropic point or plane sources of more than 140 beta emitters (Cross et al., 1992b), distributions from mono-directional beams of electrons incident on water either normally or at various angles (Grosswendt and Chartier, 1994; Seltzer, 1993a), a study of the effects of air backscattering for sources on the skin (Faw, 1992), and a study of beta spectra emitted from practical covered sources (Faw et al., 1990). A significant limitation of the standard ITS codes is that, although they can treat a wide variety of geometrical target shapes, they provide for only point or thin-disc sources. Other more general source geometries require non-trivial coding changes. This is of importance in applications to radiation therapy of the eye, where sources are often distributed over part of a spherical surface. A concave source can be simulated by combining the doses calculated for a number of point sources at carefully chosen locations, but at the cost of considerably increased effort. The MCNP code (Briesmeister, 1996; RSIC, 1996), originally developed for neutron-transport problems, has been extended to cover electron transport by incorporating algorithms from ITS. The geometrical package permits complex source and target shapes, and, in particular, sources can be distributed over a wide variety of surfaces, volumes, and thick shells. It is capable of treating a concave source, such as that of a beta-ray eye applicator. The code is more complex to use than ITS, but does not require the user to write any subroutines. It can however require extensive input of scoring volumes to capture the complete spatial distribution of absorbed dose from sources such as those considered here. The current version (Brown et al., 2003) is MCNP-5, but most reported calculations have been done with MCNP-4B and MCNP-4C.
multiple-scattering theory, and to keep the computing time within acceptable limits. This choice is made in different ways in different Monte Carlo codes and can have a significant effect on the accuracy of the results (Bielajew and Rogers, 1987). Of all the electron-transport Monte Carlo codes available, two groups have been more widely used than others------those (ETRAN, Integrated Tiger Series (ITS) codes, MCNP) developed from the original ETRAN code of Berger (1963) and Berger and Seltzer (1968a, b, c) and the electron gamma shower (EGS) code originally developed by Ford and Nelson (1978). Only these two groups will be discussed here. Rogers and Bielajew (1989, 1990) and Andreo (1991) have compared these codes. Seltzer (2002) reviewed Monte Carlo codes and calculations (ETRAN, ITS, MCNP, EGS, and the more recent PENELOPE code) and presented some comparisons in brachytherapy applications with emphasis on intravascular beta brachytherapy, including point-kernel approaches. From the user’s point of view, an important difference between codes is the amount of work necessary to set up a problem. In the ITS family of codes the user needs only to specify parameters selecting the geometry, source properties, and output requirements, from those already contained in the code. In EGS, the user incorporates auxiliary subroutines describing these quantities. This permits considerably greater flexibility in the problems that can be treated, but at the cost of a much greater investment of user time. Some geometrical packages are now distributed with the main EGS4 code.
DETERMINATION OF DOSE DISTRIBUTIONS 4.2.2.2 EGS codes
Figure 4.4. Comparison of scaled dose distributions in water, calculated by different Monte Carlo codes, for a point-isotropic source of 1 MeV electrons. All doses at distance x are scaled by multiplying by 4px2x0, where x0 is the CSDA range. Solid curve: ACCEPT 3.0 (Cross, calculations for this Report), circles: EGS4PRESTA (Simpkin and Mackie, 1990), and triangles: ETRAN (Seltzer, 1991).
particularly for intravascular beta sources, that have indicated some deficiencies in the MCNP-4C implementation of the electron-transport algorithms (Schaart et al., 2002; Seltzer, 2002; Mourtada et al., 2003). When the results of ETRAN-based and EGS4 codes are compared, it is important to be aware of what auxiliary programs have been used with EGS4, as the latter is not a unique package. Differences in results can be caused by differences in the userwritten parts of the program. Applications of EGS4 to the transport of electrons of energies up to a few mega electronvolt include the study of electron and beta-ray point kernels in water (Simpkin and Mackie, 1990), distributions from beams of monoenergetic electrons incident normally on water (Rogers, 1984a, b; Hirayama, 1994), and beta rays from hot particles (Busche et al., 1991). Improved physics, both in cross-section information and in pertinent algorithms, has been grafted onto the EGS structure by Kawrakow (2000). Detailed information on the most recent version can be found at http://www.irs.inms.nrc.ca/inms/ irs/EGSnrc/EGSnrc.html. A treatment of electron transport, based on EGS4, has been added to the MCNP code; this code is referred to as MCNP-BO (Guaraldi et al., 1990; Gualdrini et al., 1994; Ferrari et al., 1992). 4.2.2.3
Other Monte Carlo codes
A number of other codes developed for electrontransport problems are described briefly in ICRU Report 56 (ICRU, 1997b), which also outlines some novel approaches to Monte Carlo problems. The 51
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The EGS code was originally developed to treat high-energy (giga electronvolt) electron----gamma cascade showers but has been extended down to 1 keV in a widely used version, EGS4 (Nelson et al., 1985; Nelson and Rogers, 1988). A more recent version of EGS4 includes an improved method for calculating the electron spatial-step size (EGS4PRESTA) (Bielajew and Rogers, 1987; Rogers, 1984a; Bielajew and Kawrakow, 1997) and other refinements (Rogers and Bielajew, 1990). Since EGS4 was made available, it has been under continual modification (Bielajew et al., 1994; Kawrakow, 2000). Various photon- and electron-interaction cross sections have been revised. Different versions can be used with either Windows or various Unix platforms. A number of auxiliary codes are available, including geometrical packages for calculations in Cartesian (DOSXYZ) or cylindrical (DOSRZ) coordinates. The most recent version is EGSnrc (Kawrakow, 2000) includes a number of improvements, including a robust boundary-crossing algorithm for electron transport. An important difference between EGS and ITS codes is that EGS has a modular structure that requires users to develop extensions to suit their own applications. In particular, this permits arbitrary source distributions and a wide variety of output quantities. It requires more understanding of the code than does ITS. Jeraj et al. (1999) have summarized the differences in electron transport between EGS4 and MCNP codes. In MCNP-4B, the transport algorithms of either ITS 1.0 (‘‘default’’ value) or ITS 3.0 can be chosen. In comparisons of depth doses in water from beams of 6, 10, and 20 MeV electrons (Love et al., 1998; Jeraj et al., 1999), agreement with EGS4 results was obtained only when the ITS 3.0 algorithms were selected. For electrons of moderate energies, the most important differences between EGS4 and ETRANbased codes are the use of different multiplescattering distributions from which angular deflections are sampled, and the way in which electrons scattered with large energy losses are treated. Andreo (1991) has discussed further differences. Extensive comparisons of these two codes (e.g., Rogers and Bialajew, 1986, 1989, 1990) originally showed some significant differences in results, but, after corrections to both codes, these differences have largely disappeared. The majority of these comparisons apply to beams of high energy (e.g., 10----30 MeV) electrons (Andreo, 1991; Jeraj et al., 1999), but some have also been made for beta radiation (Fig. 4.4). More recently, comparisons have been presented for brachytherapy applications,
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS
4.2.2.4
Accuracy
The uncertainties of Monte Carlo results contain statistical and non-statistical components. Statistical accuracy depends on the numberpof ffiffiffiffiffi histories N and varies approximately as 1= N . In dosedistribution calculations it typically varies a great deal from one point to another, depending on what fraction of source electrons reach that point. While statistical accuracy has in the past been an important part of overall accuracy, it is becoming of less importance as computing speed increases and problems can be run for many hours (or even days) on personal computers. In dose-distribution calculations it is now common to run 107 or more electron histories, often to achieve statistical errors 10 keV (Cullen et al., 1989). 57
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2.5
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS
particular for concave beta-ray applicators. These requirements rule out all the other systems available except radiochromic film and the smallestvolume scintillator systems for these types of measurements. Hitherto, the most commonly used dosimeters for measuring the dose around low-energy photonbrachytherapy sources have been thermoluminescent dosimeters (TLDs) and silicon diodes. Ion chambers are difficult to use for this application: large chambers have inadequate spatial resolution, and small chambers often have inadequate sensitivity and leakage currents too large for measurements of long duration. Silver-halide film has an atomic number much larger than tissue, and is generally thought to be unreliable for accurate measurements. In the following sections a few methods are described that are available for the determination of the dosimetric reference data as presented in this Report (Section 6). The methods include those that have been used to obtain the dosimetric reference data for the beta-ray sources (Sections 5.2 and 6.2). Only brief descriptions of the most important characteristics are given; the user may refer to the quoted references for more information. A summary of detector applicability is given by the IAEA (2003).
4.4 MEASUREMENT OF DOSE DISTRIBUTIONS 4.4.1
General considerations
Some fundamental differences in the measurement requirements for beta rays and low-energy photons must be taken into consideration when selecting the methods of measurement. Betaparticle fields have much larger spatial dose-rate gradients than photon sources, except very near small photon sources where the gradients are similar, both being dominated by inverse-square dependence. These gradients dictate that relative measurements in beta-particle fields need to be made with detectors that approximate point detectors as closely as possible. In addition, water equivalence is different for photons and electrons: a detector that is water-equivalent for beta particles might not be water-equivalent for low-energy photons and vice versa. For the analysis of water equivalence, mass-energy absorption coefficients and stopping powers for the various detector materials to be discussed are given in Appendix A. For the measurement of relative central-axis depth-dose distributions of beta sources, detector thinness is the most important requirement. Thus, all of the reasonably small detectors described below are usually sufficient for these measurements, particularly for planar and concave beta-ray applicators for which the lateral-dose gradient is not large. For off-axis measurements of such plaques, and for all measurements of brachytherapy beta sources close (
4
DETERMINATION OF DOSE DISTRIBUTIONS
4.1 GENERAL FORMALISM FOR DOSE DISTRIBUTIONS 4.1.1
DOI: 10.1093/jicru/ndh028
Introduction
4.1.2
Formalism for seed and wire sources
The formalisms for dose distributions of beta-ray and photon sources are very similar. The polarcoordinate (r, y) system as shown in Fig. 4.1 is used. It is assumed that the source dimensions are small compared with the mean-free path of the emitted radiations. The absorbed-dose rate, D_ ðr, yÞ, at a point (r, y) is given for beta-ray and short-wire sources by Gðr, yÞ D_ ðr, yÞ ¼ D_ ðr0 , y0 Þ ð4:1Þ gr ðrÞF ðr, yÞ, Gðr0 , y0 Þ 0 and for photon seed sources by Gðr, yÞ D_ ðr, yÞ ¼ Kd Lr0 gr ðrÞF ðr, yÞ: Gðr0 , y0 Þ 0
ð4:2Þ
In Eq. (4.2), D_ ðr0 , y0 Þ is the measured absorbeddose rate at the reference point (r0, y0). For betaray seed and wire sources, r0 ¼ 2 mm and y0 ¼ p/ 2 (Nath et al., 1999; see Section 3.1.1). In Equation 4.2, K_ d is the reference air-kerma rate, and Lr0 is a dimensionless constant, called the dose-rate constant, defined as absorbed-dose rate at a specified
ª International Commission on Radiation Units and Measurements 2004
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As described in Section 3, brachytherapy beta-ray sources are calibrated in terms of the reference absorbed-dose rate, and low-energy photon sources in terms of the reference air-kerma rate. For betaray seed and wire sources and for photon seed sources, the complete relative dose distribution around a single source can be determined a priori for each source design of each radionuclide, as a characteristic of the source design. If the sources are symmetrical and uniform within the limits recommended, a two-dimensional distribution is usually considered sufficient. These dose distributions can then be presented in a consistent formalism for all such beta-ray and photon sources. The formalism is an approach wherein the various effects are taken into consideration by appropriate factors and functions. It includes explicitly the reference absorbed-dose rate or reference air-kerma rate, as the starting point for the calculation of the complete dose distribution in absolute terms (in units of gray per hour). The formalism can also be conveniently applied to calculate the dose distribution from a given configuration of seed sources, e.g., a linear array of sources (a source train). The above approach has been proposed by the Interstitial Collaborative Working Group (ICWG, 1990), and subsequently adopted by the American Association of Physicists in Medicine (Nath et al., 1995; Rivard et al., 2004) for 125I, 103Pd and 192Ir seed sources. Recently, the same approach has been adopted also for beta-ray seed and wire sources used in catheter-based systems for intravascular radiotherapy (Nath et al., 1999). This ICRU Report adopts the ICWG approach. A traditional approach, for photons with energy more than 100 keV, is to incorporate dosimetric reference data into analytical expressions, whereby relatively simple formulae are available to calculate the dose distribution. Several authors have introduced additional terms into the simplest equations, to account for the additional dose variation with
distance. Some have proposed separate terms to account for buildup and attenuation (Berger, 1968; Meisberger et al., 1968), while another has included a general radial-dose variation term (Dale, 1983). For low-energy photons, however, the increased attenuation and scatter in the medium surrounding the source modify the dose rate substantially, even within a distance of 20 mm from the source. Moreover, low-energy photons are significantly absorbed in a spatially non-uniform way by the source encapsulation, leading to highly anisotropic photon emission. The effect of source encapsulation could be estimated by a Sievert-integral approach that accounts for the attenuation of the photons by their slant penetration through the material, but------given the complexity of the source structures and subsequent photon scattering by the medium------accurate results are difficult to obtain (Williamson et al., 1983). Instead, the approach introduced by the ICWG is recommended for low-energy photon sources.
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS
simple line-source approximation, assuming uniform activity distribution, with the result Gðr, yÞ ¼
b , Lr sin y
ð4:4Þ
where b is the angle subtended by the effective active source length, L, with respect to the point (r, y) (see Fig. 4.1). An example of such a calculation for a source of length 2.5 mm is given in Table 6.6 (Section 6). Note that
r b ¼ r2 , r!1 Lr sin y
L
lim
ð4:5Þ
Figure 4.1. Linear source geometry. Formalism for the calculation of absorbed-dose rate at point P is to be presented using the polar coordinates (r, y). b is the angle subtended by the effective active-source length L, with respect to point P. Symmetry about the z-axis is assumed, so there is no dependence on azimuthal angle j.
g r0 ð r Þ ¼
D_ ðr, y0 ÞGðr0 , y0 Þ : D_ ðr0 , y0 ÞGðr, y0 Þ
ð4:6Þ
In the point-source approximation, the radial-dose function is .h i gr0 ðrÞ ¼ D_ ðrÞr2 D_ ðr0 Þr20 : ð4:7Þ
reference point per unit reference air-kerma rate. The value of Lr0 is considered characteristic of a source design. The product of K_ d Lr0 then gives the absorbed-dose rate at the reference point (r0, y0). For photon seed sources in conventional brachytherapy, the reference point is often specified at 10 mm from the source in a perpendicular bisecting plane, i.e., r0 ¼ 10 mm, y0 ¼ p/2 (Nath et al., 1995). In Equation 4.2, G(r, y) is a geometry factor that describes the variation of relative dose due only to the spatial distribution of activity within the source, and ignores absorption and scattering in the source and surrounding medium. Division by G(r0, y0) removes to first order the source-geometry effects from the dose rate at the reference point to yield an effective point-source value. Multiplication by G(r, y) restores the effect of source geometry at the point of calculation. The general equation for the geometry factor G(r, y) is i R Vh 0 2 0 r ð r Þ=jr rj dV 0 0 Gðr, yÞ ¼ , ð4:3Þ RV 0 0 0 rðr Þ dV
Finally, in Equation 4.2, F(r, y) is the anisotropy function that accounts for the anisotropy of the dose distribution around the source in the plane containing the seed axis, including the effects of absorption and scatter in the medium. It is defined by F ðr, yÞ ¼
D_ ðr, yÞGðr, y0 Þ : D_ ðr, y0 ÞGðr, yÞ
ð4:8Þ
The anisotropy function F(r, y) is normalized to 1.0 at y ¼ p/2 for all r. In addition, the anisotropy function can be integrated over solid angle to produce a point-source approximation: Z p 1 fan ðrÞ ¼ F ðr, yÞGðr, yÞsinðyÞdy, ð4:9Þ 2Gðr, p=2Þ 0 where cylindrical symmetry has been assumed and fan(r) is called the anisotropy factor. The anisotropy factor averaged over distances r from 10 to 50 mm . As has been yields the anisotropy constant, f an reported for photon sources (Weaver et al., 1996; Weaver, 1998), F(r, y) is not a uniquely defined function, as random variances in seed construction can produce seed-to-seed variations in the angular-dose
where r(r 0 ) represents the density of radioactivity at the point (r0 ) ¼ (x0 , y0 , z0 ) within the source, V denotes the integration over the source core, and dV 0 is a volume element located at r0 in the source. As an example for seed sources, the geometry factor G(r, y) is calculated using the much more tractable, 46
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i.e., in the point-source approximation, the geometry factor reduces to the simple inverse-square factor. The quantity gr0 ðrÞ is the radial-dose function and accounts for the effects of absorption and scattering in the medium, along the transverse axis of the source. It does not include geometrical effects, and is normalized to 1.0 at the reference point, i.e., gr0 ðr0 Þ ¼ 1. It is defined as
DETERMINATION OF DOSE DISTRIBUTIONS
(a)
X Position, mm
4.1.3 Application of the formalism for seed and wire sources
Y position, mm
The application of the formalism for seed and wire sources inherently requires that values of all the factors be available for each radionuclide and source design. Such dosimetric reference data can be determined both by theoretical calculations and by measurements. Different methods of calculation and measurement are discussed in Sections 4.2----4.5. It is generally recommended (IAEA, 2003; Williamson et al., 1998) that each new brachytherapy source design should have at least two independent, published studies giving the parameters of the dose-distribution formalism; one study should be experimental, the other could be theoretical, e.g., Monte Carlo calculations. Results of such determinations are presented in Section 6. It should be noted that none of the factors in the formalism has to be determined explicitly from the internal structure of the source. Factoring out the geometry term G(r, y) minimizes the spatial dependence of gr0 ðrÞ and F(r, y), even for low-energy sources, and allows them to be fit with slowly varying functions. The factors can also be accurately interpolated when discrete values are stored in lookup tables for computer calculations. A Practical Example. An example of the use of the formalism, Equation 4.1, is given in the following for the case of a linear array (train) of beta-ray seed sources (Fig. 4.2). The dose rate at a distance of 2 mm in water above the central seed in the train is the linear sum of the dose rates of all the seeds in the train. Thus, if each seed has a dose rate at 2 mm of unity, the dose rate at a point 2 mm above the center of the central seed from each neighboring seed can be calculated using Equation 4.1 with the data from
(b)
Figure 4.2. Isodose-rate contour maps of a train of (a) 9 seeds, and (b) a single seed of 90Sr----90Y, both at a distance of 2 mm in water from a plane containing the source axis, calculated using the recommended formalism (Eq. 4.1). Contour intervals are 0.2, relative to the maximum dose rate from a single seed.
Tables 6.4----6.6 (Section 6). This is shown in Table 4.1. From the last column in Table 4.1, it can be seen that only the two seeds on both sides of the central seed, and the central seed itself, significantly influence the dose rate above the central seed. For seeds with equal reference absorbed-dose rates, the 47
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distribution. These variations are especially pronounced at points near the seed axis, where measured values of F(r, y) can vary by as much as 80% for individual 125I seeds. Accordingly, average values for F(r, y) are used. The formalism outlined above is based on the assumption that the source dimensions are small compared with the mean-free path of the emitted radiations. For applications in which at least one source dimension cannot be considered small, as might be the case for long beta-ray line sources used in intravascular applications, this formalism leads to certain complications. For the description of dose-rate distributions about relatively long beta-ray line sources, therefore, an adapted formalism, conceptually similar to the ICWG approach, with another coordinate system has been proposed (Schaart et al., 2001).
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS Table 4.1. Example of the use of the formalism (Eq. 4.1) to calculate dose-rate enhancement due to neighbouring seeds for a point 2 mm in water above the center of the central seed. Seed location from center of train
r (mm)
y ( )
gr0(r)
G(r, y)
G(r0, y0)
F(r, y)
D(r, y)
3 2 1 0 1 2 3
7.76 5.39 3.20 2.00 3.20 5.39 7.76
14.9 21.8 38.7 90.0 38.7 21.8 14.9
0.039 0.282 0.756 1.000 0.756 0.282 0.039
0.0170 0.0361 0.1042 0.2234 0.1042 0.0361 0.0170
0.2234 0.2234 0.2234 0.2234 0.2234 0.2234 0.2234
0.997 0.861 0.876 1.000 0.876 0.861 0.997
0.003 0.039 0.309 1.000 0.309 0.039 0.003
X Position, mm
(a) 25
20
4.1.4 Formalism for planar and concave sources
15
10
(b)
ð4:10Þ
where (z0, r0) is the reference location, z is the depth in water-equivalent medium measured along the central axis of the planar or concave source, and r is the radial distance from this axis. The reference location is r0 ¼ 0 mm and z0 ¼ 1 mm (Section 3.1.1).
0
0
5
10
5
10
10
5
4.2 CALCULATION OF DOSE DISTRIBUTIONS FOR BETA RAYS 4.2.1
Y Position, mm
5
D_ ðz, rÞ ¼ D_ ðz0 , r0 Þ½D_ ðz, r0 Þ=D_ ðz0 , r0 Þ · ½D_ ðz, rÞ=D_ ð z, r0 Þ ,
A’
A
Absorbed-dose distributions for planar and concave beta-ray applicators can also be conveniently presented by analytical formalism. The absorbeddose distribution in this case can be broken down into the product of: (1) the reference absorbed-dose rate, D_ ðz0 , r0 Þ, (2) the relative central-axis depthdose function, D_ ð z, r0 Þ=D_ ðz0 , r0 Þ, and (3) the relative off-axis dose function, D_ ð z, rÞ=D_ ð z, r0 Þ. Thus, the absorbed-dose rate at any point, D_ ð z, rÞ, can be expressed as
0
0
Figure 4.3. Measured isodose-rate contour maps of (a) a train of 9 seeds, and (b) a single seed of 90Sr----90Y, both at a distance of 1.98 mm in A150 plastic from a plane containing the source axis, obtained from measurements using radiochromic film. Contour intervals are 5 mGy s 1. The dashed circle indicates the position of the 1 mm collecting electrode during measurements with the extrapolation chamber. The line AA’ is discussed in Section 4.4.3.
Introduction
Because ideal point-like detectors are not available for the accurate measurement of relative-dose distributions, calculations of dose distributions are carried out to supplement or verify the data from 48
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enhancement factor is 1.70. For comparison purposes, radiochromic film was exposed to nine of the seeds loaded end-to-end in A150 plastic at a depth of 1.98 mm. The scan of this film is shown in Fig. 4.3a. The dose rate averaged over the central circle of diameter 1 mm (Fig. 4.3a) is 73 mGy s1, while that of the central seed, which is shown alone in Fig. 4.3b, is 45 mGy s1; the ratio of these, 1.62, is in fair agreement with that predicted by the calculation above.
DETERMINATION OF DOSE DISTRIBUTIONS measurements with finite-size detectors. This section deals with the calculation of the spatial distributions of absorbed dose from beta-ray sources with activity distributed in area or volume. A number of methods have been applied to the calculation of dose distributions from beta rays:
The solutions of the electron transport problem by most ‘‘deterministic’’ methods, such as analytical or numerical solutions of the transport equations (e.g., Spencer, 1955, 1959) are limited either to unbounded homogeneous media or to very simple geometrical shapes and boundaries. Moreover, such methods cannot easily handle stochastic processes such as range straggling, and therefore significantly underestimate doses at distances from the source that are comparable with the range of the electrons. Analytical and numerical methods can be used that are based on beta-ray point-source dose functions (the point kernels), obtained from measurements or calculations. The point-source dose function gives the absorbed-dose distribution from an ideal point source emitting beta rays isotropically in a uniform medium, and is then a function of only a single spatial variable, the distance from the point source. Integration of the point-source dose function over the surface or volume containing the beta activity will give the dose distributions in the vicinity of the extended source. These analytical and numerical methods are fast and easy to apply. However, they are strictly valid only in a homogeneous medium of uniform density. The effects of air gaps, exit windows, backscatter from the applicator, or other source-encapsulation issues are not taken into account. This can introduce errors, which can be important at the level of clinical applications (see Section 4.2.3.3). In contrast to the above, Monte Carlo methods can handle complex geometrical arrangements, multiple materials, and the coupling of electron and photon histories. The accuracy of the results is often as good as measurements, or even better. Many former limitations in statistical accuracy are disappearing with the rapid increases in computer speed and availability.
4.2.2 Monte Carlo methods for electron transport This section gives a brief overview of the most widely used Monte Carlo codes for solving electrontransport problems. ICRU Report 56 (ICRU, 1997b) gives a more extensive discussion. There are also many reviews in the literature (Andreo, 1991; Berger, 1963; Raeside, 1976; Rogers and Bielajew, 1990; Turner et al., 1985), including the full text of a course on the subject given in 1987 (Jenkins et al., 1988). Monte Carlo calculations for electron transport are usually done differently from those for indirectly ionizing particles such as photons or neutrons, for which the simulated-particle history treats each interaction separately. For electrons, such a detailed history is usually not practical except at low energies (e.g., below about 100 keV), because the very large number of Coulomb collisions would require a prohibitively long computing time. Instead, a ‘‘condensed history’’ is used, in which short path segments are chosen and the net energy loss and angular deflection in each step are selected randomly from the results of multiple-scattering theories. The choice of step size is important and is a compromise. It must be small enough that in most of the steps the electron does not cross a boundary between different materials, that the interaction cross sections do not change significantly within a step, and that approximations made in treating the deviations from the straight line path that occur within a step are appropriate. It must be large enough to contain sufficient collisions to justify using
In view of the limitations of the analytical and numerical methods based on integrating the betaray point-source function, Monte Carlo methods of calculation (Section 4.2.2) are recommended in this Report for comparisons with measurements for practical sources. However, analytical or numerical methods based on the beta-ray point-source dose functions or point kernels are often applied in different clinical situations, and can be reasonably 49
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accurate for some particular cases such as for radioactive stents (Coffey and Duggan, 1996; Duggan et al., 1998; Janicki et al., 1997; Prestwich et al., 1995). Moreover, Monte Carlo methods are not always available for daily practice in the clinic, and a simple empirical calculation might still be a practical way to study the dose distributions for various clinical situations. Therefore, the most useful beta-ray point-source dose functions and their application for the calculation of dose distributions are discussed in detail in Appendix C. The important limitations of the analytical and numerical methods are discussed further in Section 4.2.3.3. Dose distributions in water are calculated as they most closely represent clinical situations and because measurements are often made in water. If doses in another medium are required, a convenient scaling procedure will allow, for homogeneous situations, the conversion from one low-Z medium to another (Section 4.5).
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS
4.2.2.1
ETRAN, ITS and MCNP codes
The current ETRAN code (Seltzer, 1991) is a descendant of versions developed in the 1960s (Berger, 1963; Berger and Seltzer, 1968a, b, c), which have been repeatedly improved over the last three decades (Berger and Seltzer, 1982; Seltzer and Berger, 1985, 1986; Seltzer, 1988). It was developed originally for applications involving electrons up to 100 MeV but now covers electron----photon cascades from giga electronvolt energies down to 1 keV. The geometries treated are mainly homogeneous bodies with simple shapes. The variety of geometrical shapes that could be handled was greatly extended in the SANDYL code (Colbert, 1974) and in the ITS of codes (Halbleib, 1980, 1988; Halbleib and Mehlhorn, 1984, 1986). ITS codes form a unified package of codes, each treating a different geometry: TIGER for multiple plane slabs, CYLTRAN for geometries with cylindrical symmetry, and ACCEPT, which uses threedimensional combinatorial geometry. All ITS codes 50
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use ETRAN to provide the radiation-transport physics. The current version of ITS, ITS 3.0 (Halbleib et al., 1992a, b), was distributed in 1992. It includes all the cross sections and other physical data required. Among the changes from the previous version (ITS 2.1) that significantly affect the transport of electrons of up to a few mega electronvolts, the most important are the use of revised collision stopping powers (ICRU, 1984; 2 % smaller for water than earlier values), improved bremsstrahlung production cross sections, and an improved method of calculating energy-loss straggling (Seltzer, 1991). These codes have been widely used for electron and beta-ray dosimetry problems in radiation protection and medical physics. These include calculation of electron point kernels in water for energies from 0.02 up to 4 MeV (Cross et al., 1992a) and up to 20 MeV (Seltzer, 1991), dose distributions from isotropic point or plane sources of more than 140 beta emitters (Cross et al., 1992b), distributions from mono-directional beams of electrons incident on water either normally or at various angles (Grosswendt and Chartier, 1994; Seltzer, 1993a), a study of the effects of air backscattering for sources on the skin (Faw, 1992), and a study of beta spectra emitted from practical covered sources (Faw et al., 1990). A significant limitation of the standard ITS codes is that, although they can treat a wide variety of geometrical target shapes, they provide for only point or thin-disc sources. Other more general source geometries require non-trivial coding changes. This is of importance in applications to radiation therapy of the eye, where sources are often distributed over part of a spherical surface. A concave source can be simulated by combining the doses calculated for a number of point sources at carefully chosen locations, but at the cost of considerably increased effort. The MCNP code (Briesmeister, 1996; RSIC, 1996), originally developed for neutron-transport problems, has been extended to cover electron transport by incorporating algorithms from ITS. The geometrical package permits complex source and target shapes, and, in particular, sources can be distributed over a wide variety of surfaces, volumes, and thick shells. It is capable of treating a concave source, such as that of a beta-ray eye applicator. The code is more complex to use than ITS, but does not require the user to write any subroutines. It can however require extensive input of scoring volumes to capture the complete spatial distribution of absorbed dose from sources such as those considered here. The current version (Brown et al., 2003) is MCNP-5, but most reported calculations have been done with MCNP-4B and MCNP-4C.
multiple-scattering theory, and to keep the computing time within acceptable limits. This choice is made in different ways in different Monte Carlo codes and can have a significant effect on the accuracy of the results (Bielajew and Rogers, 1987). Of all the electron-transport Monte Carlo codes available, two groups have been more widely used than others------those (ETRAN, Integrated Tiger Series (ITS) codes, MCNP) developed from the original ETRAN code of Berger (1963) and Berger and Seltzer (1968a, b, c) and the electron gamma shower (EGS) code originally developed by Ford and Nelson (1978). Only these two groups will be discussed here. Rogers and Bielajew (1989, 1990) and Andreo (1991) have compared these codes. Seltzer (2002) reviewed Monte Carlo codes and calculations (ETRAN, ITS, MCNP, EGS, and the more recent PENELOPE code) and presented some comparisons in brachytherapy applications with emphasis on intravascular beta brachytherapy, including point-kernel approaches. From the user’s point of view, an important difference between codes is the amount of work necessary to set up a problem. In the ITS family of codes the user needs only to specify parameters selecting the geometry, source properties, and output requirements, from those already contained in the code. In EGS, the user incorporates auxiliary subroutines describing these quantities. This permits considerably greater flexibility in the problems that can be treated, but at the cost of a much greater investment of user time. Some geometrical packages are now distributed with the main EGS4 code.
DETERMINATION OF DOSE DISTRIBUTIONS 4.2.2.2 EGS codes
Figure 4.4. Comparison of scaled dose distributions in water, calculated by different Monte Carlo codes, for a point-isotropic source of 1 MeV electrons. All doses at distance x are scaled by multiplying by 4px2x0, where x0 is the CSDA range. Solid curve: ACCEPT 3.0 (Cross, calculations for this Report), circles: EGS4PRESTA (Simpkin and Mackie, 1990), and triangles: ETRAN (Seltzer, 1991).
particularly for intravascular beta sources, that have indicated some deficiencies in the MCNP-4C implementation of the electron-transport algorithms (Schaart et al., 2002; Seltzer, 2002; Mourtada et al., 2003). When the results of ETRAN-based and EGS4 codes are compared, it is important to be aware of what auxiliary programs have been used with EGS4, as the latter is not a unique package. Differences in results can be caused by differences in the userwritten parts of the program. Applications of EGS4 to the transport of electrons of energies up to a few mega electronvolt include the study of electron and beta-ray point kernels in water (Simpkin and Mackie, 1990), distributions from beams of monoenergetic electrons incident normally on water (Rogers, 1984a, b; Hirayama, 1994), and beta rays from hot particles (Busche et al., 1991). Improved physics, both in cross-section information and in pertinent algorithms, has been grafted onto the EGS structure by Kawrakow (2000). Detailed information on the most recent version can be found at http://www.irs.inms.nrc.ca/inms/ irs/EGSnrc/EGSnrc.html. A treatment of electron transport, based on EGS4, has been added to the MCNP code; this code is referred to as MCNP-BO (Guaraldi et al., 1990; Gualdrini et al., 1994; Ferrari et al., 1992). 4.2.2.3
Other Monte Carlo codes
A number of other codes developed for electrontransport problems are described briefly in ICRU Report 56 (ICRU, 1997b), which also outlines some novel approaches to Monte Carlo problems. The 51
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The EGS code was originally developed to treat high-energy (giga electronvolt) electron----gamma cascade showers but has been extended down to 1 keV in a widely used version, EGS4 (Nelson et al., 1985; Nelson and Rogers, 1988). A more recent version of EGS4 includes an improved method for calculating the electron spatial-step size (EGS4PRESTA) (Bielajew and Rogers, 1987; Rogers, 1984a; Bielajew and Kawrakow, 1997) and other refinements (Rogers and Bielajew, 1990). Since EGS4 was made available, it has been under continual modification (Bielajew et al., 1994; Kawrakow, 2000). Various photon- and electron-interaction cross sections have been revised. Different versions can be used with either Windows or various Unix platforms. A number of auxiliary codes are available, including geometrical packages for calculations in Cartesian (DOSXYZ) or cylindrical (DOSRZ) coordinates. The most recent version is EGSnrc (Kawrakow, 2000) includes a number of improvements, including a robust boundary-crossing algorithm for electron transport. An important difference between EGS and ITS codes is that EGS has a modular structure that requires users to develop extensions to suit their own applications. In particular, this permits arbitrary source distributions and a wide variety of output quantities. It requires more understanding of the code than does ITS. Jeraj et al. (1999) have summarized the differences in electron transport between EGS4 and MCNP codes. In MCNP-4B, the transport algorithms of either ITS 1.0 (‘‘default’’ value) or ITS 3.0 can be chosen. In comparisons of depth doses in water from beams of 6, 10, and 20 MeV electrons (Love et al., 1998; Jeraj et al., 1999), agreement with EGS4 results was obtained only when the ITS 3.0 algorithms were selected. For electrons of moderate energies, the most important differences between EGS4 and ETRANbased codes are the use of different multiplescattering distributions from which angular deflections are sampled, and the way in which electrons scattered with large energy losses are treated. Andreo (1991) has discussed further differences. Extensive comparisons of these two codes (e.g., Rogers and Bialajew, 1986, 1989, 1990) originally showed some significant differences in results, but, after corrections to both codes, these differences have largely disappeared. The majority of these comparisons apply to beams of high energy (e.g., 10----30 MeV) electrons (Andreo, 1991; Jeraj et al., 1999), but some have also been made for beta radiation (Fig. 4.4). More recently, comparisons have been presented for brachytherapy applications,
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS
4.2.2.4
Accuracy
The uncertainties of Monte Carlo results contain statistical and non-statistical components. Statistical accuracy depends on the numberpof ffiffiffiffiffi histories N and varies approximately as 1= N . In dosedistribution calculations it typically varies a great deal from one point to another, depending on what fraction of source electrons reach that point. While statistical accuracy has in the past been an important part of overall accuracy, it is becoming of less importance as computing speed increases and problems can be run for many hours (or even days) on personal computers. In dose-distribution calculations it is now common to run 107 or more electron histories, often to achieve statistical errors 10 keV (Cullen et al., 1989). 57
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2.5
DOSIMETRY OF BETA RAYS AND LOW-ENERGY PHOTONS FOR THERAPEUTIC APPLICATIONS
particular for concave beta-ray applicators. These requirements rule out all the other systems available except radiochromic film and the smallestvolume scintillator systems for these types of measurements. Hitherto, the most commonly used dosimeters for measuring the dose around low-energy photonbrachytherapy sources have been thermoluminescent dosimeters (TLDs) and silicon diodes. Ion chambers are difficult to use for this application: large chambers have inadequate spatial resolution, and small chambers often have inadequate sensitivity and leakage currents too large for measurements of long duration. Silver-halide film has an atomic number much larger than tissue, and is generally thought to be unreliable for accurate measurements. In the following sections a few methods are described that are available for the determination of the dosimetric reference data as presented in this Report (Section 6). The methods include those that have been used to obtain the dosimetric reference data for the beta-ray sources (Sections 5.2 and 6.2). Only brief descriptions of the most important characteristics are given; the user may refer to the quoted references for more information. A summary of detector applicability is given by the IAEA (2003).
4.4 MEASUREMENT OF DOSE DISTRIBUTIONS 4.4.1
General considerations
Some fundamental differences in the measurement requirements for beta rays and low-energy photons must be taken into consideration when selecting the methods of measurement. Betaparticle fields have much larger spatial dose-rate gradients than photon sources, except very near small photon sources where the gradients are similar, both being dominated by inverse-square dependence. These gradients dictate that relative measurements in beta-particle fields need to be made with detectors that approximate point detectors as closely as possible. In addition, water equivalence is different for photons and electrons: a detector that is water-equivalent for beta particles might not be water-equivalent for low-energy photons and vice versa. For the analysis of water equivalence, mass-energy absorption coefficients and stopping powers for the various detector materials to be discussed are given in Appendix A. For the measurement of relative central-axis depth-dose distributions of beta sources, detector thinness is the most important requirement. Thus, all of the reasonably small detectors described below are usually sufficient for these measurements, particularly for planar and concave beta-ray applicators for which the lateral-dose gradient is not large. For off-axis measurements of such plaques, and for all measurements of brachytherapy beta sources close (
