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A 3D photon superposition/convolution algorithm and its foundation on results of Monte Carlo calculations
This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2005 Phys. Med. Biol. 50 1767 (http://iopscience.iop.org/0031-9155/50/8/010) View the table of contents for this issue, or go to the journal homepage for more
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INSTITUTE OF PHYSICS PUBLISHING Phys. Med. Biol. 50 (2005) 1767–1790
PHYSICS IN MEDICINE AND BIOLOGY
doi:10.1088/0031-9155/50/8/010
A 3D photon superposition/convolution algorithm and its foundation on results of Monte Carlo calculations* W Ulmer1,2, J Pyyry3 and W Kaissl1 1 VARIAN Medical Systems International AG, Taefernstr. 7, CH-5405, Baden-Daettwil, Switzerland 2 Max-Planck-Institute of Physics, Goettingen, Germany 3 VARIAN Medical Systems, Helsinki, Finland
E-mail:
[email protected]
Received 2 July 2004, in final form 17 December 2004 Published 6 April 2005 Online at stacks.iop.org/PMB/50/1767 Abstract Based on previous publications on a triple Gaussian analytical pencil beam model and on Monte Carlo calculations using Monte Carlo codes GEANTFluka, versions 95, 98, 2002, and BEAMnrc/EGSnrc, a three-dimensional (3D) superposition/convolution algorithm for photon beams (6 MV, 18 MV) is presented. Tissue heterogeneity is taken into account by electron density information of CT images. A clinical beam consists of a superposition of divergent pencil beams. A slab-geometry was used as a phantom model to test computed results by measurements. An essential result is the existence of further dose build-up and build-down effects in the domain of density discontinuities. These effects have increasing magnitude for field sizes 5.5 cm2 and densities 0.25 g cm−3, in particular with regard to field sizes considered in stereotaxy. They could be confirmed by measurements (mean standard deviation 2%). A practical impact is the dose distribution at transitions from bone to soft tissue, lung or cavities. S This article has associated online supplementary data files (Some figures in this article are in colour only in the electronic version)
1. Introduction The past decade of therapy planning algorithms is characterized by an increased importance of convolution methods based on pencil beams. These algorithms have to be based on Monte Carlo calculations (Ahnesj¨o et al 1992, 1999, Bortfeld et al 1990, Boyer et al 1986, Brahme 1988, Mackie et al 1990, Miften et al 2000, Sharp et al 1993, Ulmer et al 1996, * This work has partially been presented at WC 2003, Sydney. 0031-9155/05/081767+24$30.00 © 2005 IOP Publishing Ltd Printed in the UK
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W Ulmer et al Table 1. Materials for slab-geometry phantom.
Abbreviation
Material
RW3 (PTW) SD PB CaP CaP + Sugar CaC
Water-equivalent (WEM) Styrodur Paste-board CaO.H2O pulverized CaS CaO crystalline
HU (Hounsfield Unit) and density (g cm−3) 0 (1.0) −900 (0.1) −850→ −760 (0.15–0.24) 700 (1.7) 250 (1.25) 1200 (2.2)
Dosimetric purpose Reference system (water) Lung Lung Bone Bone (margin) + soft tissue Bone
1998, 1999, Webb 1993) and/or an analysis of measurement data (Ceberg et al 1996, Storchi et al 1996, 1999). Essential aspects of modern therapy planning are as follows: inclusion of density-dependent dose distributions based on CT images and intensity modulations (IMRT) by static/dynamic wedges/multileaf collimators (MLC), micromultileaf collimators (MMLC), and stereotactic (SRT)/conformal radiotherapy (CRT). At the present level of SRT the restriction to solely circular fields is abandoned and the progress of the MMLC technique permits the inclusion of IMRT and CRT as a modality of SRT (Ahlswede et al 2000). The extension of SRT to the neck region and body SRT (Blomgren et al 1998, Nakagawa et al 2000, Uematsu et al 2000) requires algorithms suitable to take account of rather different heterogeneity areas. Therefore an algorithm has to be applicable to the computation of dose distributions of irregularly shaped fields (CRT, SRT), inclusive heterogeneity and fluence modulation by IMRT (Boman et al 2003, Tervo et al 2000). These requests imply that calculation models have to be based on convolution/pencil beam methods and on Monte Carlo calculations. A further request is to speed up the computation time. Scatter effects, electron transport and energy phase space prevent a rigorous processing of the heterogeneity problem by pure two-dimensional (2D) calculations. The calculation procedure consists of a convolution and superposition of voxels realizing every irregular beam. The results show that in particular for small field sizes commonly used in SRT density jumps between layers lead to second build-up and build-down effects of the dose distributions (Ulmer et al 1999). Since the algorithm presented in this study has been implemented in a preliminary version of ECLIPSE, the purpose of the paper is to present its principles starting with Monte Carlo calculations of the phase space (beam line), the determination of broad beams by superposition mutual scatter contributions of tilted rays (pencils) in a homogeneous system (water) and some configurations of slab-geometries (heterogeneities). A particular feature is the ‘translation’ of Monte Carlo results into a 3D superposition/convolution algorithm of pencil beams. 2. Methods and materials In order to reduce the length of the paper, a Laboratory Report has been added on-line for readers who are interested in some details of Monte Carlo calculations and their underlying physical principles, tables (except table 1, which is of common interest), some results and figures. This Laboratory Report, which is referred to as LR, is available as a PDF file from stacks.iop.org/PMB/50/1767. 2.1. Measurement data Radiation beams of 6 and 18 MV photons of a VARIAN Clinac 2300 C/D (golden beam data, Varian Medical Systems, Palo Alto, CA, USA) and a 2300 EX (Strahlentherapie
A 3D photon superposition/convolution algorithm
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Relative energy fluence
1 0.8 18 MV
0.6 0.4 0.2
6 MV
0 0
3
6
9
12
15
18
Energy/MeV
Figure 1. Relative spectral energy fluence f (E) of 6 MV and 18 MV at r = 0 cm (central ray) calculated by GEANT (samples) and the best adaptation by the theoretical formula (1). Mean standard deviations: 0.1% (6 MV) and 0.14% (18 MV).
Klinikum Hannover, Germany) have been recorded with PTW dose monitor systems: Farmer chamber (0.6 cm3) and Mephysto water phantom (depth–dose curves and transverse profiles for comparison with phase space calculations). Diamond detector and Markus chamber have been used for measurements (Hannover) in phantoms with slab-geometry (slabs with different densities). Each measurement point in slab-geometry is determined by absolute dosimetry: 3 times 100 MUs and SAD = 100 cm. Table 1 presents materials used for slab-geometry. Hounsfield values have been obtained by the measurement of mass densities and absorption coefficients. The energy fluence at SAD = 100 cm was determined by measurements of absorbed dose in air with a brass cap. The thickness dbc of the brass cap (Wiezorek et al 2000) was chosen according to the energy of the contamination electrons, 6 MV: dbc = 3 mm; 18 MV: dbc = 7 mm. 2.2. Monte Carlo calculations 2.2.1. EGSnrc/BEAMnrc and GEANT. EGS4 has often been described and applied to problems of clinical radiation physics (Ahnesj¨o et al 1992, 1999, Mackie et al 1990, Mohan et al 1985, Sheikh-Bagheri and Rogers 2002a, 2002b, Ulmer et al 1996). EGSnrc is an improved version of EGS4. A detailed description of the code is given in a reference manual (Kawrakow et al 2000). The version BEAMnrc supports the simulation of the beam line in an accelerator by suitable component modules. Although the Monte Carlo simulation of the accelerator beam line is a very tedious task in the Monte Carlo code GEANT (GEANT is an open system, component modules such as in BEAM/BEAMnrc are not supported), we have made use of it to study the importance of neutron production (18 MV) and the influence of higher order Feynman diagrams with respect to ‘bremsstrahlung’ and Compton scatter (see also LR). Monte Carlo calculations have been performed for three different tasks: (1) Determination of the energy spectrum and photon fluence of 6 MV/18 MV according to figures 1–3 (comparison between GEANT and BEAMnrc) with respect to some jaw positions (field sizes of 2 × 2 cm2 up to 40 × 40 cm2). The phase space for each jaw position was simulated with 2 × 109 primary electron histories (GEANT and BEAMnrc). (2) Determination of the broadening and attenuation of mono-energetic pencil beams in water by 2 × 106 histories (from 0.015 MeV up to 21 MeV). The calculations have
W Ulmer et al
Relative energy fluence
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1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
GEANT solid curves BEAMnrc samples r = 0 cm r = 10 cm r = 20 cm
18 MV
0
3
6
9
12
15
18
Energy/MeV
Relative photon energy fluence
Figure 2. Relative spectral energy fluences f (E) of 18 MV at r = 0 cm (central ray, Emax = 4.78 MeV), r = 10 cm (Emax = 3.92 MeV) and r = 20 cm (Emax = 3.35 MeV) calculated by GEANT (solid curves) and by BEAMnrc (samples).
GEANT solid curves BEAMnrc samples r = 0 cm
1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
r = 10 cm
0
1
2
3
4
5
6
Energy/MeV
Figure 3. Relative spectral energy fluences f (E) of 6 MV at r = 0 cm (central ray, Emax = 1.74 MeV), r = 10 cm (Emax = 1.49 MeV) and r = 20 cm (Emax = 1.33 MeV) calculated by GEANT (solid curves) and by BEAMnrc (samples).
been performed between 0.015 MeV and 0.1 MeV in intervals of 0.005 MeV; between 0.1 MeV and 21 MeV in intervals of 0.05 MeV (GEANT) and of 0.1 MeV (EGSnrc). (3) Model calculations of slabs and boxes with different associations, field sizes and materials according to table 1 in order to determine the attenuation/absorption, scatter behaviour of photons and the transport of electrons under disequilibrium conditions (GEANT). It should be noted that GEANT consumes much more time for a calculation run than BEAMnrc/EGSnrc (time factor ca 40–50). Since our particular goal is the development of a 3D superposition/convolution algorithm based on Monte Carlo calculations, we treat the corresponding results in section 2.3. 2.2.2. Spectral distribution of energy. The knowledge of the energy phase space of an incident photon beam at surface z = 0 by Monte Carlo calculations is a principal tool for dose calculation models. The radial dependence of the energy spectrum provided by the beam line has been obtained by two different Monte Carlo codes (GEANT and BEAMnrc). Therefore a comparison between the results is indicated. Before we proceed in this way, we should
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1771 Maximum 6 MV Average 6 MV Maximum 18 MV Average 18 MV
8
Energy/MeV
7 6 5 4 3 2 1 0 0
5
10 Radial distance r/cm
15
20
Figure 4. Radial dependence of Emax (r) and of the average energy Eaverage(r) of 18 MV (upper curves: dashes) and of 6 MV (lower curves: solid lines).
mention that with the help of a Laplace transform of Poisson distribution functions we could verify a formula we guessed for the spectral distribution function of the energy (see LR, a derivation will be published separately), f (E) = (1 − e−αE/E0 )p e−βE
2
/E02
(1 − E/E0 )q
(1)
where E0 refers to the upper limit energy, which is 6 MeV or 18 MeV, respectively, but it may also slightly vary. The formula is suitable to describe the energy spectrum in the central ray and for tilted rays, and we apply it as an adaptation formula of Monte Carlo data (least-squares fit with respect to the parameters of formula (1)). The average energy Eaverage of f (E) is calculated by E0 E0 Eaverage = f (E)E dE f (E) dE. (2) 0
0
The following adaptation parameters depending on the radial distance r from the central ray are available: p, q, α and β. However, the limit energy E0 may depend on the installation and be rather different for linacs of other manufacturers. With regard to the considered cases of a VARIAN 2300EX the following radial dependences are valid: α = α0 + α r,
q = q0 + q r.
(3)
All parameters necessary to evaluate the above spectral formula are stated in table 1 (LR). In particular, it is not evident that the above relationship is linear over the complete distance. A piecewise application of formulae (1) and (3) is therefore preferred. The remaining parameters p and β might also depend on the radial distance r. The Monte Carlo spectrum (GEANT) of the central ray (6 MV, 18 MV) is presented in figure 1 (samples), solid lines refer to the best adaptations (maximal mean standard deviation of all distribution functions between r = 0 and r = 20 cm is 0.21%). Figures 2–3 show the analytical adaptations of the spectra at different distances from the central ray (GEANT: solid lines, BEAMnrc: samples). With respect to 6 MV both Monte Carlo codes show satisfactory agreement. But for 18 MV BEAMnrc provides spectra shifted slightly to higher energies than GEANT. The reason may be that BEAMnrc ignores creation of low-energy photons by Compton and ‘bremsstrahlung’ effects of higher order and release of neutrons. A comparison between the maximum values of the energy spectra Emax (r) and the average values Eaverage according to equation (2) is given in figure 4.
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4.00E-14 1.77 MeV 1.47 Mev 1.30 MeV
3.50E-14
Dose/Gray
3.00E-14 2.50E-14 2.00E-14 1.50E-14 1.00E-14 5.00E-15 0.00E+00 0
5
10
15
20
25
30
35
40
45
50
z/cm Figure 5. Depth–dose curves of mono-energetic pencil beams (E = 1.77, 1.47 and 1.30 MeV) obtained by GEANT. 4.00E-14 4.8 MeV 3.93 MeV 3.33 MeV
3.50E-14
Dose/Gray
3.00E-14 2.50E-14 2.00E-14 1.50E-14 1.00E-14 5.00E-15 0.00E+00 0
5
10
15
20
25
30
35
40
45
50
z/cm
Figure 6. Depth–dose curves of mono-energetic pencil beams (E = 4.80, 3.93 and 3.33 MeV) obtained by GEANT.
2.2.3. Mono-energetic profiles. Some results of GEANT (mono-energetic pencil beams) are presented in figures 5–7. The selected mono-energies are those of some Emax (r) values of 6 and 18 MV. The transverse profiles always give an indication that we can represent beam broadening by Gaussians (see Monte Carlo descriptions of EGSnrc and GEANT). As an example, we present profiles of 4.8 MeV photons in figure 7. Together with formula (1) mono-energetic depth dose and transverse profiles represent the basis for the algorithm of this work. 2.2.4. Energy absorption coefficients µ(E) of mono-energetic depth–dose curves. An analytic representation of the energy absorption coefficients (mono-energy) of calculated
A 3D photon superposition/convolution algorithm
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4.00E-14 z = 5 cm
3.50E-14
z = 10 cm
4.8 MeV
z = 20 cm
Dose/Gray
3.00E-14
z = 30 cm
2.50E-14 2.00E-14 1.50E-14 1.00E-14 5.00E-15 0.00E+00 0
2.5
5
7.5
10
12.5
15
Radial distance r/cm
Figure 7. Transverse profiles of a mono-energetic pencil beam (4.8 MeV) at z = 5, 10, 20 and 30 cm calculated by GEANT. The profiles tend to Gaussian shapes. Data Seltzer
4 3.5
Monte Carlo (GEANT)
µE-2/cm
-1
3 2.5 2 1.5 1 0.5 0 0.1
2.1
4.1
6.1
8.1
10.1
12.1
14.1
16.1
18.1
20.1
Energy/MeV
Figure 8. Energy absorption coefficient µ(E) for mono-energetic photons in water obtained by GEANT and adapted by formulae (5)–(6) and comparison with the data of Seltzer (samples). Energy range: 0.1 MeV E 20.6 MeV.
depth–dose curves (GEANT) is given by the formulae (4)–(6). Figures 8 and figure 4 (LR) show calculated results and a comparison with synchrotron data (Seltzer 1993). Unfortunately, it is not possible to use one formula for all domains. Since the physical interaction processes are quantitatively different in the considered energy domains, we have to distinguish between the domains (parameters of the formulae (4)–(6) are stated in table 2 (LR)). With respect to poly-energetic depth–dose curves and transverse profiles, we take account of the spectral distributions (1). It is also useful to apply analytical representations of the absorption coefficient µ(E): 0.015 MeV E 0.1 MeV µ(E) = µ11 e−E1 /E + µ12 e−E
2
/16E12
(4)
0.1 MeV E 0.6 MeV µ(E) = µ21 e−E2 /E + µ22 e−(E−0.5)
2
/E32
e−E/E4
(5)
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0.6 MeV E 21 MeV µ(E) =
3
µ3k e−(E−0.6)/E3k .
(6)
k=1
2.3. Theoretical methods 2.3.1. Deconvolution of profiles. We perform the following general task: ρ(x) is a given source (input) function, then via a Gaussian kernel K(s, u − x) one obtains the image (output) function ϕ(x) with the help of the integral transform (convolution): 1 2 2 K(s, u − x) = √ e−(u−x) /s ∞ s π ϕ(x) = K(s, u − x)ρ(u) du.
(7)
−∞
The inverse problem of deconvolution arises if one wants to obtain the source function ρ(x) from the image function ϕ(x). This is accomplished via the integral equation: ∞ ρ(x) = K −1 (s, u − x)ϕ(u) du. (8) −∞
The solution of the task is the derivation of the inverse kernel K−1 from the kernel K. According to a previous study (Ulmer et al 2003), this kernel is given by K −1 (s, u − x) =
∞ (−1)n s 2n 0
2n n!
H2n (s, u − x)K(s, u − x).
(9)
Here H2n (s, u − x) are Hermite polynomials of even order; details are given in a previous publication (Ulmer et al 2003). If ρ(x) is the fluence distribution before collimation, then the corresponding fluence ϕ(x), obtained by a convolution, represents the corresponding fluence distribution (rounded profiles) behind the collimators (e.g., incident plane), which can be measured. Via the inverse problem the initial fluence distribution can be calculated. Formulae (7)–(9) have to be extended to two dimensions: ∞ ∞ K(s, u − x, v − y)ρ(u, v) du dv (10) ϕ(x, y) = −∞ −∞ ∞ ∞ ρ(x, y) = K −1 (s, u − x, v − y)ϕ(u, v) du dv. (11) −∞
−∞
The kernels K(s, u − x, v − y) and K −1 (s, u − x, v − y) are obtained by the products The removal of the K(s, u − x) K(s, v − y) and K −1 (s, u − x) K −1 (s, v − y). finite size of measurement chambers/detectors is a further application of deconvolution. The finite size of the detector (see figure 3 (LR)) changes the measured profiles (depth dose, transverse profiles) in the region with significant profile gradients. Detectors/chambers do not represent ideal passive systems, which only receive signals from the environment. Due to scatter at the walls, transport and stopping power properties of electrons correction factors are commonly used, if a detector is calibrated under a certain standard condition (e.g., the water-energy dose of Co60 at z0 = 5 cm). We have compared Monte Carlo calculations of some profiles of 6 MV/18 MV photons with measured curves. We denote measured depth dose by Dm (z), transverse profiles by Dm (x, y, z0 ) and those profiles recorded from an ideal passive point detector by Dp (z) and
A 3D photon superposition/convolution algorithm
1775 Calculation 40 x 40 cm 2
1 40 x 40 cm
2
Measurement 40 x 40 cm2
Relative dose
0.8
Calculation 4 x 4 cm2 Measurement 4 x 4 cm2
0.6 4 x 4 cm
0.4
2
0.2 0 0
4
8
12
16 z/cm
20
24
28
32
Figure 9. Calculated and measured (samples) depth–dose curves of 6 MV photons (field sizes: 4 × 4 cm2 and 40 × 40 cm2). 1
40 x 40 cm2 Measurement 40 x 40 cm2 Calculation
0.8
Relative dose
4 x 4 cm2 Measurement 4 x 4 cm2 Calculation 0.6
0.4
0.2
0 0
5
10
15
20
25
30
z/cm
Figure 10. Calculated and measured (samples) depth–dose curves of 18 MV photons (field sizes: 4 × 4 cm2 and 40 × 40 cm2).
Dp (x, y, z0 ). The profiles can be calculated by the deconvolution procedure: ∞ Dp (z) = Dm (z) · K −1 (sd (E), w − z) dw 0
Dp (x, y, z0 ) =
∞ −∞
∞ −∞
K −1 (sp (E), u − x, v − y) · Dm (z0 , u, v) du dv.
(12) (13)
The above integrals are evaluated according to the grid size of the recorded data (Farmer chamber) within finite boundaries (see figures 8–10 (LR)). The parameters sd (E) and sp (E) are stated in table 3 (LR). 2.3.2. Analytical representation of pencil beams. In the first step, we consider 0 = 2 × 106 mono-energetic photons mentioned in section 2.2.1 with a certain energy E located in a point at the surface of a phantom (z = 0) and propagating in z-direction without beam divergence (see also figures 5–7). The dose deposition Dp,mono and beam widening of this pencil beam can be described by Dp,mono (r, z) = 0 I E (z)HE (r, z).
(14)
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The radius r is taken perpendicular to the z-axis with r 2 = x 2 + y 2 . Dose deposition results from HE (r, z). 0 is the relative from IE (z) and beam widening by scatter photon fluence at 0 δ(x)δ(y) dx dy = x = y = z = 0 obeying the condition = 0 HE (r, z = 0) dx dy = 0 , i.e. HE (r, z) assumes the shape HE (r, z) = δ(x)δ(y) at z = 0 and is normalized at each depth z by 2π ∞ ∞ ∞ HE (r, z) dx dy = HE (r, z)r dr dϕ = 1. (15) −∞
−∞
0
0
In the second step, we replace the mono-energetic pencil beam by a poly-energetic one according to formula (1) and energy intervals available by mono-energetic Monte Carlo calculations: Dp (r, z) = 0
E0
f (E)I E (z)HE (r, z) = 0 I (z)H (r, z).
(16)
E=0
Relation (15) remains valid for the poly-energetic case, if HE (r, z) ⇒ H (r, z). The dimension of IE (z) and I (z) is Gy cm2. A broad beam might be described by boundaries (17) D(x, y, z) = (1 + z/SAD)−2 Dp z, (x − u)2 + (y − v)2 du dv. field
Due to the phase space properties of broad beams, I (z) and 0 have to change as a function of the distance r from the central ray. The applicability of formula (17) is restricted to small field sizes, which do not exceed 4 × 4 cm2, where changes in the phase space can be ignored. Based on Monte Carlo calculations (EGS4) we have previously specified H (r, z) as a triple Gaussian kernel (Ulmer et al 1996, 1998, 2003): H (r, z) =
2
ck (z)K(σk (z), r)
k=0 2
(18)
ck (z) = 1.
k=0
A further reason to consider kernel (18) was a previously proposed calculation model (Ahnesj¨o et al 1992, 1999), where the scatter kernel H (r, z) is adapted by H (r, z) =
ε(z) −ε(z)r e /r. 2π
(19)
The convolution integration is facilitated if we model kernel (19) by Gaussians kernels (7) and (18). It should be noted again that by taking into account the angular spread of inner shell transitions in Compton scatter, the improved Monte Carlo codes GEANT and EGSnrc give some preference to model H (r, z) via formula (18), since kernel (19) may lead to overly long scatter ranges. Using kernel (18) the integration of formula (17) provides the analytical expression: D(x, y, z) =
1 0 (1 + z/SAD)−2 I (z) 4
2 a2 − x a1 + x b2 − y b1 + y × − erf erf − erf . ck erf σk (z) σk (z) σk (z) σk (z) k=0 (20)
A 3D photon superposition/convolution algorithm
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The field-size restriction has to be kept in mind, and the field parameters in equation (20) = a1,2 (1 + z/SAD) and b1,2 = b1,2 · (1 + z/SAD). The have to satisfy a1 < a2 , b1 < b2 , a1,2 error function erf(s) is defined by s 2 2 erf(s) = √ e−u du. (21) π 0 2.3.3. Determination of I (z), σ k(z) and contamination particles. The complete depth dose behaviour of Dp (r, z) determined by Monte Carlo inclusive build-up region is represented by a modification of the ‘ansatz’ (Laughlin 1954): Dp (r, z) = 0 I (z)H (r, z) − Eel (z)Kel (r, z) I (z) = Imax [a0 e− µ0 z − a1 e− µ1 z ] −µel z
Eel (z) = Eel max e
(22)
.
The term Eel (z) in formula (22) reflects the fact that between the photon fluence and dose deposition the equilibrium is not yet reached (build-up region). The kernel Kel (r, z) of Compton electrons is adapted by a Gaussian: 1 2 2 Kel (r, z) = e− r /σel (z) . (23) π σel (z)2 According to Monte Carlo calculations the spectral distribution of Compton electrons yields a decrease of the scatter function σel (z): σel (z) = σel,0 + σel,1 e− z/zel .
(24)
The range parameter µel also results from large angle scattering. It must be inverted to the dimension of a length and can be determined by a convolution. The result is given by
a2 a1 b2 b1 1 −1 −1 erf − erf erf − erf . (25) µ = µ − µ−1 el el,0 4 el,1 S0 S0 S0 S0 It is well known that the dose in the build-up region is field-size dependent. This fact results from two different influences: Compton electrons released at surface and contamination by incident particles. Both factors include large angle scattering. Equation (22) does not yet model the build-up region. Due to contamination particles (mainly electrons, but also positrons and neutrons (18 MV)) the build-up effect is reduced. The sources of these contamination particles obeying large angle scatter are flattening filter, primary collimator, monitor chamber, air and jaws. The fluence el,cont of these contamination particles is strongly dependent on the jaw positions and can be modelled with a convolution of a Gaussian distribution:
a1 b2 b1 a2 1 − erf erf − erf . (26) erf cont = cont max 4 S0 S0 S0 S0 The dose contribution of the electron (positron/neutron) contamination is accounted for by an additional kernel Eel,cont: Eel,cont = cont e− µel z Kel (r, z).
(27)
Kernel Kel (r, z) reflects the situation that in the case of small field sizes the jaws absorb contamination particles with large scatter angles. Then Dp (r, z) subjected to convolution is given by Dp (r, z) = 0 I (z)H (r, z) − Eel,max e−µel z Kel (r, z) + Eel,cont .
(28)
The pure photon term I (z) contains two absorption coefficients µ0 and µ1. µ1 results from a correction of the energy spectrum and the basic parameters of formulae (22)–(28) are presented
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in tables 4–5 (LR). An analysis of fluence distributions of incident photons (figure 5 (LR)) and electrons el,cont is presented in LR. The depth dependence of the poly-energetic scatter functions σk (k = 0, 2) valid for water can be computed by an adaptation formula of Monte Carlo data: σ0 (z) = σ00 + σ01 z/α σ1 (z) = σ10 + σ11 tanh(z/β) + σ 12 e−(z−zs)
2
/β12
σ2 (z) = σ20 + σ21 tanh(z/γ ) + σ 22 e−(z−zs) /β2 E0 c0 = c00 − tanh(z/δ) (29) Emax (r)π 2 E0 c1 = c10 + tanh(z/δ) q(E0 )/e Emax (r)π 2 E0 c0 = c20 + tanh(z/δ) (1 − q(E0 )/e). Emax (r)π 2 The tanh terms in formula (29) represent saturation due to increasing photon scatter effects of energy with depth z. The Gaussian parts σ12 exp −(z−zs)2 β12 and σ22 exp −(z−zs)2 β22 in σ1 (z) and σ2 (z) refer to scatter of low-energy photons and annihilation of incident positrons in the build-up region. They vanish rapidly, and already in the domain of the dose maximum their importance is negligible. The radial decrease of Emax (r) implies a modification of σ1 and σ2 ; σ0 remains unaffected by the change of the energy spectrum: Emax (r = 0) − Emax (r) σ1 (z, r) = σ1 (z, r = 0) E0 (30) Emax (r = 0) − Emax (r) σ2 (z, r) = σ2 (z, r = 0) . E0 Information on the calculation of σk and ck is given in tables 6–8 (LR). In particular, σ0 is related to the primary Compton effect (scatter angle ϑ ≈ 0, order is up to ca 0.5 cm). σ1 (order is up to ca 7 cm) and σ2 (order is up to ca 20 cm) result from single Compton with ϑ ≈ π /2, multiple scatter, inner shell Compton effects and pair creation/annihilation (γ ⇒ e+ + e− + γ , e+ + e− ⇒ 2γ ). A statistical analysis of all medium-dependent scatter influences gave an excellent adaptation of the scatter data generated by Monte Carlo. The ρ-correction in a homogeneous medium (ρ = 1) of the σk -values referring to ρ = 1 is obtained by a scaling transformation: σk (z, ρ) = σk (z, ρ = 1)ρ (31) σel (z, ρ) = σel (z, ρ = 1)ρ. Please note that the density correction in formula (31) must be regarded as ρ/ρ water. This is valid for all formulae in which the density correction ρ appears. Since the saturation of scatter is also ρ-dependent, the restricted applicability of formula (31) and its relevance to dose calculations in radiotherapy is discussed in section 4 (LR) and table 13 (LR). With regard to patient models it is convenient to compute the local increment σk (z) of σk (z) by σk (z + z) = σk (z) + z(σk (z)/z), i.e. tan ϕk (z) = σk (z)/z: σk (z + z) = σk (z) + σk (z) = z tan ϕk (z) (32) k = 0, 2. If the heterogeneity problem is given as a sequence of different slabs, the ‘history’ of scatter processes is taken into account by σk (z + z, ρ(z + z)) = σk (z, old) + z tan ϕk (z)ρ(z + z) (32a) k = 0, 2. 2
2
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The heterogeneous medium has to consist of slabs, ρ is constant within each slab. If the photon pencil beam passes through a homogeneous medium with ρ = 1, then according to Monte Carlo calculations the terms of Dp (r, z) have to be corrected by scaling transformations: Dp (z, ρ) = 0 I (z, ρ)H (r, z, ρ) + (el,cont − Eel, max (ρ)) e−µel ρz Kel (r, ρ, z) p p I (z, ρ) = I max (ρ) a0 e−µ0 zρ − a1 e−µ1 zρ .
(33)
Equation (33) is based on Monte Carlo calculations and results from a modification of formulae (14) and (16): Dp (r, z, ρ) = 0
E0
f (E)IE (z, ρ)HE (r, z, ρ) = 0 I (z, ρ)H (r, z, ρ).
(33a)
E=0
Since the exponential functions according formula (22) have to be dimensionless, ρ and ρ p have to be understood as the density ρ divided by the unit density 1 of water. The power p is slightly energy dependent: 6 MV ⇒ p = 0.28 and for 18 MV, p = 0.31 is valid. We point out that the maximum value and position zmax of Dp (z, ρ) essentially depends—in addition to the energy spectrum—on the density of the medium. Some preliminary values of Dp (z, ρ) based on EGS4 have been published in a previous paper (Ulmer et al 1999). Since the photon absorption coefficients in formula (33) are not linearly scaled with the density ρ, the question of the origin of the power p arises, which is of the order 1.3. The main reason is Compton scatter with inner shell transitions. The restriction to the outermost valence electrons would support p = 1, i.e. a proportionality to ρ would hold in agreement with the Klein–Nishina formula. Conservation of energy of a pencil beam defined by equation (32a) yields that the following relation must hold: ∞ ∞ ∞ ∞ ∞ ∞ Dp (r, z, ρ) dx dy dz = Dp (r, z, ρ = 1) dx dy dz. z=0
x=−∞
y=−∞
z=0
x=−∞
y=−∞
(33b) Since H (r, z, ρ = 1) and H (r, z, ρ) are normalized kernels, the integration of relation (33b) yields Imax (ρ) = ρ p−1 Imax (ρ = 1).
(33c)
The reference values of I (ρ = 1, z = 0) = Imax and Imax (ρ) may be rather different, therefore the use of a normalized function I (z, ρ = 1) with Imax = 1 implies that this normalization also changes the relative height of I (z, ρ) at depth z, i.e. Imax (ρ) = ρ p−1 . This behaviour must be accounted for with respect to electron transport released at interfaces of borders with different ρ-values, which we shall consider in section 2.3.6. Monte Carlo calculations of pencil beams with respect to ρ-values stated in table 1 have provided agreement with the results of formula (33a), the mean standard deviation amounted to 0.6% and the maximal deviation to 1.3%. 2.3.4. Extension to a 3D superposition/convolution algorithm and voxel scanning. Twodimensional convolution models evidently exhibit the principal limits: the separation of the z-coordinate from the convolution problem does not hold if the tissue heterogeneity is considered within a patient contour and parts of a broad beam scan different tissue densities. We partition the field size of a broad beam at the patient surface according to a desired calculation grid (voxel size). The real broad beam actually consists of a superposition of divergent pencil rays, and each of them passes through tissue with varying density, which in general is rather different for each pencil. At patient surface, we define x = y = z = 0 (central ray) and x = xβ , y = yβ , z = zβ as initial coordinates of the pencil beams outside the central
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beam (zβ is only zero for z = 0, if the photon beam is perpendicular to a phantom plane). The scaled coordinates of a pencil at depth z are given by yβ = yβ (1 + z/SAD) xβ = xβ (1 + z/SAD) zβ = zβ 1 + xβ2 + yβ2 SAD2 .
(34)
Thus zβ corresponds to the fan-line. A 3D-generalized and ρ-dependent pencil beam model based on evaluation of CT images requires the determination of I (zβ , ρ) and the scatter functions σk (zβ , ρ(xβ , yβ , zβ )) for divergent pencils (rays). 2.3.5. Photon scatter functions σk (z) in inhomogeneous media. According to the above relation (32a) a density correction of the scatter functions σk has to be carried out to include the modified scatter range. Since σ0 does not at all exceed 0.5 cm and the local increment σ0 is extremely slight, a local correction according to formula (32a) is absolutely justified. This is not true for σ1 and σ2 . Within the increment of the scatter range the density of voxels may significantly vary, and there is also a difference between the possible environmental directions of every position of a (usually) tilted pencil at zβ . The range length increments σ1 and σ2 are compared to those of water, but the varying density within this increment only permits an average value, which has to be calculated to correct the scatter functions, i.e. we , xβ− , yβ+ , yβ− ) directions: ρav (xβ+ ), ρav (xβ − ), have to compute average values in four (xβ+ ρav (yβ+ ), ρav (yβ− )- This provides four different incremental values for both σ 1 and σ 2. For each tilted pencil these 2 × 4 scatter functions now have to be determined. The density dependence of these scatter functions requires their recalculation according to the phase space dependence: σkx− = σkx− (old, zβ − zβ ) + σkx− (ρav x− (zβ )) σkx+ = σkx+ (old, zβ − zβ ) + σkx+ (ρav x+ (zβ )) σky− = σky− (old, zβ − zβ ) + σky− (ρav y− (zβ )) σky+ =
σky+ (old, zβ
−
zβ )
+
(35)
σky+ (ρav y+ (zβ ))
k = 1, 2. This set of scatter functions has to be used in the convolution of each voxel along the tilted ray (fan-line). The convolution integration taken over a voxel at the position zβ now has to be carried out with regard to four directions. The depth dose functions I (ρ, zβ ), Eel,max(ρ) and exp(−µel zβ ρ) according to equation (22) have to be scaled locally. Since the absolute values of I (ρzβ ) and Eel,max(ρ) may differ considerably with varying ρ, a computation of I (ρ, zβ ) simultaneously requires the computation of the local fluence β (xβ , yβ , zβ ). β is initialized at the surface (phantom, patient) with a certain value the attenuation of 0,β has to be determined by: − zβ1 , ρβ0 )β2 (zβ2 − zβ1 , ρβ1 ) β (xβ , yβ , zβ ) = 0,β β1 (zβ0 · · · βn (zβn − zβn−1 , ρ βn−1 ).
(36)
and zβk . Here ρβk−1 is the density of the voxel between zβk−1
2.3.6. Electron transport due to voxels with different ρ-values and disequilibrium of electron release at borders. In homogeneous media, I (z, ρ) is characterized by electron transport equilibrium beyond the build-up region, and only in the build-up region is this equilibrium not yet established. This fact implies a correction of I (z, ρ). In inhomogeneous media this correction has to be performed at each position. In order to model this situation by
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Monte Carlo data we have performed calculations with GEANT under various conditions, where the borders of finite boxes touch with corresponding density jumps. An analysis of these Monte Carlo calculations has supported that this correction is proportional to the difference ), zβ1 ) − Iβ (ρ(zβ2 ), zβ2 ), zβ2 ), if xβ1 ⇒ xβ2 , y˙ β1 ⇒ yβ2 and zβ1 ⇒ zβ2 . If a density Iβ (ρ(zβ1 jump at the interface of the border does not occur, the latter condition implies that there is an = zβ2 . equilibrium of forward and backward travelling electrons storing their energy at zβ1 For ρ(zβ1 ) > ρ(zβ1 ) there is a surplus of electrons travelling to zβ2 and to zβ > zβ2 due to the ) < ρ(zβ1 ), the situation is reduced density ρ (forward scatter of Compton electrons). If ρ(zβ1 partially reversed, since the magnitude of electrons with backward scatter increases compared to the equilibrium condition, and a decrease of Compton electrons with forward scatter is a )) and I (ρ(zβ2 )) for ρ(zβ1 ) = further feature. In every case, the discontinuity between I (ρ(zβ1 ρ(zβ1 ) and zβ1 ⇒ zβ1 is proportional to the surplus of released electrons at zβ1 = zβ1 travelling either in forward or backward direction. By that Monte Carlo results obtain an illustrative description. The later ‘history’ of these electrons is also determined by multiple scatter, which and zβk has to be accounted for. It is necessary to scan along a fan-line all positions zβk−1 and the density values ρ1 (zβk−1 ) and ρ2 (zβk ). Since the magnitude of the released electron transport is given by the proportionality relation Eβ ≈ Iβ (zβ , ρ) − Iβ (zβ + zβ , ρ), we have to differentiate between the following cases. For ρ < 0, we have to include forward scatter with a factor fel,fw (ρ, zβ ) and missing backward scatter with a factor fel,bw (ρ, zβ ). For ρ > 0, the situation is reversed. The range decrease of electron transport along the fan-line can be calculated by two suitable exponential functions exp[−µel,fw ρ(w)(w − zβ − zβ )] and exp[−µel,bw ρ(w)(zβ − w)]; µel,fw and µel,bw represent the linear attenuation; w may assume every position of the fan-line related to zβ (see table 9 (LR)). Thus the correction of the local Iβ (zβ , ρ) results from the sum of the electron transport of all other positions w = zβ and the lateral scatter Krelease of these electrons, which is mainly large angle scattering according to Monte Carlo calculations. The correction function Eβ along the fan-line reads Eβ (zβ − w) = [Iβ (zβ ) − Iβ (zβ + zβ )] fel,fw (ρ(zβ ), zβ ) × exp(−µel,fw ρ(w) (w − zβ − zβ )) (if Eβ (zβ
− w) =
w zβ + zβ )
[Iβ (zβ )
Iβ (zβ
−
(if w
zβ ).
+
zβ )]fel,bw (ρ(zβ ), zβ ) exp(−µel,bw ρ(w)(w
(37) −
zβ ))
The convolution of the electron transport due to local disequilibrium of forward and backward scatter has to be a complete 3D calculation: Erelease (xβ , yβ , zβ ) = Eβ (zβ − w) Krelease (uβ − xβ , vβ − yβ , zβ ) duβ dvβ dw. (38) Krelease can be represented by a normalized Gaussian kernel (electron transport equation): 1 Krelease (uβ − xβ , vβ − yβ , zβ ) = exp(−[(uβ − xβ )2 + (vβ − yβ )2 ]/σel,sc (zβ )2 ). π σel,sc (zβ )2 (39) σel,sc (zβ ) is determined by linear combinations of σ 1 and σ 2: 6 MV: σel,sc (zβ ) = 0.855 σ1 (zβ ) + 0.145 σ2 (zβ ) 18 MV:
σel,sc (zβ ) = 0.751 σ1 (zβ ) + 0.249 σ2 (zβ ).
(40)
We should point out that the same density and direction dependence is also accounted for the transport of release electrons under disequilibrium conditions according to relation (37),
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i.e. formula (33) must be modified according to relations (35), (37)–(40). The complete dose distribution is obtained by a superposition over all pencils: −2 β (xβ , yβ , zβ ) (1 + zβ /SADβ ) D(x, y, z) = β
×
[Hβ (uβ − xβ , vβ − yβ , zβ ) Iβ (zβ , ρ(zβ )) − {Eel,β (zβ , ρ(zβ ))
− cont,β (zβ , ρ(zβ ))}Kel,β (uβ − xβ , vβ − yβ , zβ )] duβ dvβ
+ Erelease (xβ , yβ , yβ ) .
(41)
SADβ now is the focus–surface distance of the pencil β. The initial values of β (xβ , yβ , zβ ) may be given as a matrix; the tacit assumption β ≈ 1 at surface may hold for open fields. 0 = constant is a constraint which only holds for static fields. Equation (41) represents a tool for the calculation of dose distributions induced by static/dynamic wedges; irregular shaped fields (CRT, MLC, MMLC, blocks, IMRT and SRT). 3. Results Information on contamination of positrons/neutrons and deconvolution of fluence and some profiles (depth–dose curves and transverse profiles) is given in LR. 3.1. Calculations of water-phantom data Although the verification of the algorithm with regard to profiles recorded in a water phantom does not provide significantly new information, it represents the base for more intricate and realistic applications. We briefly present here the depth–dose curves of 4 × 4 cm2 and 40 × 40 cm2 fields (6 MV: figure 9, 18 MV: figure 10). All measured curves (depth–dose curves and transverse profiles (figure 10 (LR)) have been refined by the stated deconvolution (finite size of the detector) before comparison with calculations. Some properties of the build-up region are presented in tables 10–11 (LR). The agreement with the refined measurement data is noteworthy (mean standard deviation 0.4%). 3.2. Calculations of slabs and boxes and comparison with measurements A profound test of the calculation model in the case of varying density (slab-geometry and boxes) is important with regard to clinical applications. Figure 11 makes evident the field-size dependence of depth–dose curves in a medium with rather low density (SD, 6 MV HU: −900). The disequilibrium in the domain of density jumps of the slabs at z = 2 cm and z = 9 cm, yielding also forward and backward scatter, is a noteworthy feature. In the case of 3 × 3 cm2, the dominant part of Compton electrons released in the initial slab (WEM) gets lost by large angle scatter reducing the dose in the central ray. The density properties of figure 12 are the base for figures 13–16 containing a slab with WEM (RW3, 18 MV: 3 cm, 6 MV: 2 cm thickness). A pressure of the slab to the succeeding phantom material (PB) could be avoided, but not that of the upper layers to the deeper ones. This leads to the increasing density in z-direction. The behaviour in the central ray clearly indicates the energy dependence of depth–dose curves. Thus 6 MV is less sensitive than 18 MV, where even for a 20 × 20 cm2 field a small build-down effect can be verified in the transition area. The penumbra of the profiles in three depths reflects the behaviour of large
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0 0.5
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Figure 11. Depth–dose curves (6 MV, 3 × 3 cm2 and 20 × 20 cm2) of a phantom with slabgeometry. Between 2 cm z 9 cm a slab (material: SD, ρ = 0.1 g cm−3, HU: −900) was inserted. Samples (triangles: 3 × 3 cm2, squares: 20 × 20 cm2) refer to measurements, solid curves to calculations. 1 Phantom 6 MV
Density g/cm3
0.8
Phantom18 MV
0.6
0.4
0.2
0 0
4
8
12
16 z/cm
20
24
28
32
Figure 12. Density dependence of phantoms containing PB. 6 MV: 2 cm WEM (RW3) at the top, which is kept fixed, then PB with increasing density due to the pressure of the upper layers. 18 MV: 3 cm WEM (RW3), followed by PB.
angle Compton electrons: the roundness of the profiles is significantly increased compared to well-known profile properties recorded in water. A further effect with identical reason is the increased long-range dose distribution far from the geometric field size due to the reduced stopping power of Compton electrons. Figures 17–18 (18 MV) present a configuration with material equivalent to bone. Although in practical applications this case does not appear with the same severity, it is suitable to study the scatter behaviour of slabs with density jumps not only in the z-direction, but also in x/y-direction. To do this, the field size has to be chosen sufficiently large (25 × 25 cm2). The box containing CaC is only 20 × 20 × 20 cm3 and symmetrically positioned between
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Lung box z > 2 cm 40 Reference: water (10x10 cm2 field)_____ 20
0 0
4
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Figure 13. Computed depth–dose curves (solid) of 4 × 4 cm2 and 10 × 10 cm2 fields and measured data (samples) of 6 MV photons in phantom according to figure 12. The depth–dose curve of the 10 × 10 cm2 field in water is included for comparison.
20 x 20 cm2 Calculation 20 x 20 cm2 Measurement 4.5 x 4.5 cm2 Calculation 4.5 x 4.5 cm2 Measurement
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60 Lung box z > 3 cm
40
Reference: water (20x20 cm2 field)_____
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Figure 14. Computed depth–dose curves (solid) of 4.5 × 4.5 cm2 and 20 × 20 cm2 fields and measured data (samples) of 18 MV photons (phantom: figure 12). The depth–dose curve of the 20 × 20 cm2 field in water is included for comparison.
z = 11 cm and z = 31 cm; the remaining environment is WEM. In figure 18 we can verify the backscatter from CaC to WEM, whereas the forward scatter at z = 31 cm from CaC to WEM cannot be recorded due to the remarkable attenuation of the beam in the CaC area. The transverse profiles recorded at the beginning/end of the CaC box and in its middle part at z = 20 cm reflect the scatter behaviour of photons from the region of high density to the environment with lower density. Since by increasing density the lateral photon scatter
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x/cm (one half) Figure 15. Transverse profiles (solid: computation, samples: measurement) of 10 × 10 cm2 and 4.5 × 4.5 cm2 fields (one half) of 6 MV photons at z = 7, 12 and 18 cm (phantom: figure 12). 100 20x20 cm2
90 80
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40
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10 0 17.5
20
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25
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Figure 16. Transverse profiles (solid: computation, samples: measurement) of 20 × 20 cm2 and 4.5 × 4.5 cm2 fields (one half) of 18 MV photons at z = 7, 12 and 18 cm (phantom: figure 12).
also increases, a disequilibrium exists at borders of media with lower density. It should be mentioned that in the plane z = 31 cm this behaviour is only of minor importance, because the inner box (CaC) has reached a notably reduced photon fluence. The increasing part of the profile beyond the box results from divergent rays, which only partially pass through the box. Figures 19–20 present depth–dose curves (6 MV photons) of slab geometries with different bone densities (middle part, CaP and CaS) and field sizes. In the case of figure 19, forward and backward scatter can be recognized at the borders of the slabs, whereas for figure 20 only
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25 x 25 cm2 Calculation Measurement WEM
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20
0 0
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z/cm
Figure 17. Depth–dose curve (18 MV, 25 × 25 cm2 field, solid: computation, samples: measurement) of a phantom containing box geometry. Between z = 11 and 31 cm a box is placed (material: CaC, HU: 1200), followed again by WEM (RW3). The lateral extension of the CaC box is only 18 cm (i.e., smaller than the field size), and the environment of this box also consists of WEM. Parts of the photon beam only pass through water or partially through the CaC box due to beam divergence. z = 10.5 cm Calculation z = 31 cm Calculation
90
z = 20 cm Calculation
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70 60 50 40
25 x 25 cm2
Bone HU: 1200
30 margin of the plane bone - water
20 10 0 19.6
22.6
25.6
28.6
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34.6
x/cm (one half) Figure 18. Transverse profiles (one half, 18 MV) of the box geometry according to figure 17 at z = 10.5 cm (before entrance to the box), z = 20 cm (box–water), and z = 31 cm (lower end of the box and WEM at the lateral end of the box). Samples: measurement.
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60 WEM
box: 40
bone HU: 700 20
0 0.5
5.5
10.5
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Figure 19. Computed (solid) and measured (samples) depth–dose curves of 4 × 4 cm2 and 10 × 10 cm2 fields of 6 MV photons of a slab-geometry (material: CaP, HU: 700 between z = 3 and 10 cm).
100 WEM
BOX HU: 250
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20
0 0
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z/cm Figure 20. Computed (solid) and measured (samples) depth–dose curve of a 20 × 20 cm2 field of 6 MV photons of a slab-geometry (material CaS, HU: 250 between z = 3 and 10 cm).
backward scatter is present due to the thickness of the CaS box. The behaviour is similar to that of figure 17.
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4. Discussion 4.1. Monte Carlo calculations of the beam line A precise knowledge of phase space properties is an essential tool of modern dose calculation algorithms. The use of Monte Carlo codes to describe the photon beam line from electron current creating ‘bremsstrahlung’ in the target to the surface of phantoms has a long history. Mohan et al (1985) have obtained valuable results with the EGS3 code, but the maximum values of spectral distributions turned out to be partially too high (6 MV: 1.92 MeV for the central ray and 1.5 MeV for a tilted ray at r = 20 cm). Recent publications (Ding 2002, Sheikh-Bagheri and Rogers 2002b) with EGS4 provide good results with respect to 6 MV, but with regard to 18 MV their results are not quite satisfactory. The reason might be that in order to speed up the computation time of the EGS4 code, higher order Feynman diagrams of Compton scatter and ‘bremsstrahlung’ yielding more than one scatter photon are not included in the code. These simplifications ignore the creation and scatter of low-energy photons. The cited authors also use the terminus ‘average energy’, whereas figure 4 and an examination of the published results show that they mean the maximum values of the spectral distributions. We have calculated Eaverage according to formula (2). Figure 4 tells us that the average values Eaverage are much higher than the maximum values Emax of the spectral distributions. BEAMnrc provides rather acceptable results compared to GEANT. The effect of neutrons (table 12 (LR)), which is not included in EGSnrc/BEAMnrc, is rather small even for 18 MV. Only due to RBE (factor ≈ 10) the neutron dose contamination might be taken into account (Ongaro et al 2000). 4.2. Phantoms with different materials and densities Preliminary calculations using ECLIPSE treatment planning system (VARIAN Medical Systems, Palo Alto, CA, USA) with implementation of the described algorithm have been given in figures 9–20. The solution of the heterogeneity problem in radiation therapy is an essential feature of accurate dose application. The present study agrees with the well-known knowledge that all effects resulting from varying density significantly depend on the field size and photon energy. The behaviour of some fields (length: 10 cm and 4.5 cm) at the border can be compared with published results (Ahnesj¨o et al 1999). These authors considered a similar case (length: 10 cm and 5 cm) and cork (ρ = 0.32 g cm−3) as lung phantom. The dose decrease at the border is smaller. We have re-examined the corresponding configuration and obtained almost the same results (deviation: ca 1.5%) as by Monte Carlo calculations (EGS4, Ahnesj¨o et al (1999)). These results support the findings that build-down effects at borders significantly depend on the change of local density and on the field size. The depth–dose curves clearly indicate the importance of the photon energy with regard to density jumps between different phantom slabs. Investigations (Boyer et al 1986, Mackie et al 1985, Rice et al 1988) have shown comparable effects for field sizes ≈5 × 5 cm2 (15 MV, phantom slab material: cork, ρ ≈ 0.32 g cm−3), whereas 6 MV photon beams require either smaller field sizes and/or densities of slabs to verify comparable effects. Due to the local disequilibrium of scatter electrons at borders one obtains build-up and build-down effects before and behind the heterogeneity with increased density. The experimental data generally follow the computed curves. With regard to critical heterogeneity domains, measurement data obtained by a diamond detector/Markus chamber only yielded maximal deviations of 4% (average 2%). It should be pointed out that the experimental configurations invoked by the density jumps and the local disequilibrium of electron transport may violate the Bragg–Gray conditions. According to estimations (Rice et al 1988) the
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violation may yield dosimetric errors of ca 2%. Build-up/build-down effects along photon beam propagation between slabs with noteworthy discontinuity need accurate algorithms. The results also show a practical impact, because the method presented allows the computation of the dose distribution of tumours embedded in tissue with different density. The results presented may also indicate the importance of field-size dependence of dose distributions in inhomogeneous areas treated in SRT/IMRT. The divergent voxel scanning method and superposition of convolutions can accurately calculate IMRT dose for arbitrary fluence distributions. The optimization of IMRT according to electron density data provided by CT requires a sufficiently accurate local dose calculation. 4.3. Remarks to the implementation in ECLIPSE The algorithm presented is supported by all modules, which are provided by ECLIPSE (image processing, IMRT, etc). All routines are written in the programming code C++ and the computer equipment is a DELL Precision workstation with Dual Intel 3.06 GHz Xeon processors and 2 GB memory. It is impossible to restrict dose calculations to one single image, as is possible with the present single pencil beam (SPB) model. Basis data referring to phase space information (energy spectra, scatter functions, etc) are stored in appropriate intervals in files. The time requested for the computation of some square fields (heterogeneous case, grid size 5 mm) is as follows: 6 cm length—4.1 s; 10 cm length—7.0 s; 20 cm length—14.1 s; 30 cm length—29.0 s. References Ahlswede J, Pfaender M, Grebe G, Steingraeter M, Sommer B, Scheffler D, Budach V and Wurm R 2000 IMRT, dynamic arcs and static beams with MMLC—a comparison Med. Phys. 27 4703–4 Ahnesj¨o A and Aspradakis M M 1999 Dose calculations for external photon beams in radiotherapy Phys. Med. Biol. 44 R99–R155 Ahnesj¨o A, Saxner M and Trepp P 1992 A pencil beam model for photon dose calculations Med. Phys. 19 263–73 Blomgren H, Lax I, G¨oranson H, Kraepelien T, Nilsson B, N¨aslund I, Svanstr¨om R and Tilikidis A 1998 Radiosurgery for tumors in body: clinical experience using a new method J. Radiosurg. 1 63–74 Boman E, Lyyra-Laitinen T, Kolmonen P, Jaatinen K and Tervo J 2003 Simulations for inverse radiation treatment planning using a dynamic MLC algorithm Phys. Med. Biol. 48 925–42 Bortfeld T, Birkelbach J, Boesecke R and Schlegel W 1990 Methods of image reconstruction from projections applied to conformation therapy Phys. Med. Biol. 35 1423–34 Boyer A L and Mok E C 1986 Calculation of photon dose distributions in an inhomogeneous medium using convolutions Med. Phys. 13 503–9 Brahme A 1988 Optimization of stationary and moving beam radiation therapy technique Radiother. Oncol. 12 129–40 Brusa D, Stutz G, Riveros J A, Fernandez-Varea J M and Salvat J M 1996 Fast sampling algorithm for the simulation of photon Compton scattering Nucl. Instrum. Methods A 379 167–75 Ceberg C P, Bjaerngard B E and Zhu T C 1996 Determination of the dose kernel in high-energy x-ray beams Med. Phys. 23 505–11 CERN Report 1998 GEANT Monte Carlo Code Description version 3.2 Ding G X 2002 Energy spectra, angular spread, fluence profiles and dose distributions of 6 and 18 MV photon beams: results of Monte Carlo simulations for a Varian 2100EX accelerator Phys. Med. Biol. 47 1025–46 Feynman R P 1962 Quantum Electrodynamics—A Lecture Note and Reprint Volume (New York: Benjamin) pp 91–127, 184 Kawrakow I and Rogers D W O 2000 The EGSnrc code system: Monte Carlo simulation of electron and photon transport NRCC Report PIRS-701 NRC Canada Kohn W 1995 Overview of Density Functional Theory (NATO Advanced Science Institute Series) (New York: Plenum) pp 3–10 Laughlin J S 1954 Physical Aspects of Betatron Therapy (Springfield: Charles C Thomas) pp 46–53 Lieb E H 1983 Density functional for Coulomb systems Int. J. Quantum Chem. 24 864–71
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