AAA Photon Dose Calculation Model in Eclipse™

Janne Sievinen1, Waldemar Ulmer2,3, Wolfgang Kaissl3

1
Varian Medical Systems Finland Oy, Helsinki, Finland
2
Max-Planck-Institut für Biophysik, Göttingen, Germany
3
Varian Medical Systems Imaging Laboratory GmbH, Baden, Switzerland

Abstract
A new photon dose calculation model, the Analytical Anisotropic Algorithm (AAA), has been
implemented in Eclipse™ Integrated Treatment Planning. The AAA model provides a fast and
accurate dose calculation for clinical photon beams even in regions of complex tissue
heterogeneities.

The AAA dose calculation model is a 3D pencil beam convolution-superposition algorithm that
has separate modeling for primary photons, scattered extra-focal photons, and electrons scattered
from the beam limiting devices. Functional forms for the fundamental physical expressions in
AAA allow analytical convolution, thus reducing significantly the computation times usually
required by these types of algorithms. Tissue heterogeneities are accounted for anisotropically in
the full 3D neighborhood by the use of 13 lateral photon scatter kernels. The final dose
distribution is obtained by superposition of the doses from the photon and electron convolutions.

Configuration of the AAA model is based on Monte-Carlo-determined basic physical parameters

that are adapted to measured clinical beam data. These in turn are used to construct a phase space
defining the fluence and energy spectrum of the clinical beam specific to each treatment unit.
Beam modifying accessories including blocks, hard wedges, dynamic wedges, compensators,
MLCs (static and dynamic) are fully supported in the dose calculation.
Introduction
Modern treatment techniques in clinical external beam radiotherapy are posing increasing
demands on the accuracy and speed of the dose calculation algorithms. Recent introduction of
the highly modulated beams used in IMRT techniques also necessitates accurate dose calculation
especially in regions of complex tissue heterogeneities. At the same time, resource constraints in
a real clinical environment dictate that any arbitrarily high accuracy cannot be afforded at the
expense of lengthy computation times. The AAA photon dose calculation model has been
developed to meet both of these clinical expectations, providing a fast Monte-Carlo-based 3D
convolution/superposition algorithm for accurate heterogeneity-corrected photon dose
calculation for all types of external beam treatments.
2,3 3
Originally developed by Drs. Waldemar Ulmer and Wolfgang Kaissl , AAA has a long history
culminating in the publication of the triple Gaussian photon kernel model in 1995 [2, 3, 4]. The
principal ideas of AAA have already been applied to stereotactic radiation therapy planning [5]
where the application of the triple Gaussian kernel formalism with appropriate correction
mechanisms for tissue heterogeneities has yielded excellent results. The AAA dose calculation
model presented in this article is a continuation of the work presented in these previous
publications.

The AAA dose calculation model is comprised of two main components, the configuration
algorithm and the actual dose calculation algorithm. The configuration algorithm is used to
determine the basic physical parameters used to characterize the fluence and energy spectra of
the photons and electrons present in the clinical beam and their fundamental scattering properties
in water equivalent medium. Although some of the parameters used in the dose calculation
algorithm could be deduced with reasonable accuracy from simple measurements of depth dose
and lateral dose profiles in a water-equivalent phantom, an experimental determination of all
parameters is practically impossible. This is resolved in the AAA model by pre-computing all the
parameters using Monte Carlo simulations and then modifying these parameters to match with
the actual measured clinical beam data during the beam data configuration phase. This approach
ensures a quick and highly accurate determination of all the important basic physical parameters
required for the AAA dose calculation. After the treatment-unit-specific fitting procedures in the

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beam configuration phase are completed, all the parameters are stored and later retrieved for the
actual dose calculation.

The dose calculation is based on separate convolution models for primary photons, scattered
extra-focal photons, and electrons scattered from the beam-limiting devices. The clinical broad
beam is divided into small, finite-sized beamlets to which the convolutions are applied. The final
dose distribution is obtained by the superposition of the dose calculated with photon and electron
convolutions for the individual beamlets.

Functional forms of the fundamental physical expressions in the AAA enable analytical
convolution, which significantly reduces the computational time required in the dose calculation.
Attenuations of the photons and electrons present in the clinical beam are modeled with the
energy deposition density functions I, and the dose deposition characteristics with scatter kernels
K that are composed of Gaussian functions.

AAA accounts for tissue heterogeneity anisotropically in the entire three-dimensional

neighborhood of an interaction site. This is performed by the use of radiological scaling of the
dose deposition functions and the electron density based scaling of the photon scatter kernels
independently in four lateral directions.

This article concentrates on the description of the dose calculation algorithm in the AAA model.
Descriptions of the algorithms developed for the beam data configuration to determine the basic
physical parameters required in the dose calculation are largely omitted and will be presented in
detail in a separate article.

Methods

Physical Parameters in Dose Calculation

AAA uses fundamental physical parameters that have been pre-computed with Monte Carlo
simulations. These parameters are modified during beam data configuration so that the resulting
calculated beam characteristics match the measured clinical beam data for each treatment unit.
This is a fast and accurate method of determining all the important physical parameters for the

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dose calculation. The parameters specific to the clinical beam are stored in the database and
retrieved for the patient dose distribution calculation.

The fundamental physical parameters used to model clinical beams are pre-computed for a set of
average beam energies from 6 to 23 MV. A Varian Clinac© 2300 C/D was used as the reference
accelerator for the determination of the beam fluence and energy phase space. All model
parameters for AAA are computed in a water-equivalent medium. During the dose distribution
calculation, these parameters are scaled according to the densities of actual patient tissues.

Photon Energy Spectrum

AAA derives the scatter kernels K needed for the dose calculation from the energy spectrum,
which is determined during the configuration process. The initial photon spectrum is
determined from Monte Carlo simulations of the Bremsstrahlung spectrum of the electrons
impinging on the target. Figure 1 shows an example of an initial photon spectrum for a 6 MV
beam.
Spectrum
Relative particle fluence

Energy [MeV]

Figure 1: Example of a 6 MV photon spectrum.

Another important parameter that affects the energy spectrum used by AAA is the mean energy
as a function of the radius from the beam central axis. An example of the mean radial energy for
a 6 MV beam is given in Figure 2. This curve is used by AAA to determine the beam hardening
effect of the flattening filter on the photon spectrum. Based on the mean energy curve and the

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user-specified flattening filter material, AAA determines the energy spectrum of the beam at any
radius from the beam central axis.

Mean Radial Energy

Mean Energy [MeV]

Radial Distance [mm]

Figure 2: Example of the mean energy as a function of distance
from the beam central axis of a 6 MV photon beam.

Intensity Profile
The flattening filter also causes the intensity of the photon beam to vary across the clinical
treatment field. The varying photon fluence is modeled with the help of a parameter called the
intensity profile curve. The intensity profile is computed as the photon energy fluence (number ×
energy of photons) as a function of the radial distance from the beam central axis. An example of
the intensity profile for a 6 MV beam is shown in Figure 3.

Fluence Map
Intensity

Radius [mm]

Figure 3: Example of intensity profile of an 18 MV photon beam.

Scatter Kernels
In addition to the phase space parameters, the fundamental physical parameters also include the
photon and electron scatter kernels and their depth dependence in a water-equivalent medium.
These scatter kernels describe the phantom-scatter effects for different beam qualities. The

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EGSnrc Monte Carlo code [6] was used to compute scatter kernels for monoenergetic pencil
beams in various media. A polyenergetic scatter kernel is constructed as a weighted sum of the
monoenergetic scatter kernels. During the 3D dose calculation these parameters are scaled
according to the densities of the actual patient tissues determined from the CT images.

Clinical Beam Modeling

The clinical beam model has three main components: primary photon energy fluence, extra-focal
photon energy fluence, and contaminating electron fluence. These are characterized by a number
of parameters.

Monte Carlo Simulations for the Clinical Beam Model

The Monte Carlo simulations used to create the initial phase-space model for the clinical beam
include the geometrical structure and exact material composition of the accelerator head
(including source, target, primary collimator, flattening filter, monitor ionization chamber, and
jaws), and the beam modifiers. The parameters derived by Monte Carlo simulation are modified
during beam data configuration so that the resulting calculated beam characteristics match the
clinical beam data for each treatment unit [1]. The simulations were performed mainly using the
EGSnrc program [6], which is well-suited for modeling physical interactions in clinical external
beam radiotherapy, such as Compton scattering with outer and inner-shell electrons,
Bremsstrahlung, photoelectric effect, and electron-positron pair production and annihilation.

The broad clinical beam is divided into finite-sized beamlets β, as illustrated in Figure 4.
Additionally, the clinical beam is divided into separate photon and electron components, each
with a beamlet intensity Φβ. The photons are divided into:

• Primary photons, originating from the target. Separate energy spectra for every fanline of
the broad beam are derived from the mean energy curve (see Figure 2).

• Extra-focal photons, scattered in the flattening filter or the beam limiting devices. Extra-
focal photons are assumed to be uniformly distributed across the broad beam, and they
are modeled with a secondary source having a configurable intensity and location below
the primary source collimated by the field aperture [7].

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 Figure 4: Treatment unit components, broad beam division.

Beam Modifiers in the Clinical Beam Model

Most beam-modifying accessories affect only the beam fluence used in the AAA dose
calculation. Blocks and MLCs employ a user-defined transmission factor to model the change in
the fluence when a beam is shaped by the accessory. Compensators, Dynamic Wedges and
IMRT fields also modify the fluence of the beam. The head scatter effects are taken into account
by convolving the fluence distribution with the Gaussian shape of the secondary source to

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produce varying amounts of extra-focal radiation, depending on the field shaping. The
contribution of the contaminating electrons also depends on the photon beam fluence shape.

Hard wedges modify the fluence and the spectral characteristics of the beam. The configuration
program determines the beam hardening effect from the appropriate lateral profile measured. The
user-defined wedge material is used to determine the effect of beam hardening on the configured
open field energy spectrum.

Volumetric Dose Calculation

For volumetric dose distribution calculation, the patient body volume is divided into a matrix of
3D calculation voxels based on the selected calculation grid (see Figure 4.) The geometry of the
calculation voxel grid is divergent, aligning the coordinate system with the beam fanlines. Every
calculation voxel is associated with the mean electron density ρ that is computed from the patient
CT images according to a user-specified calibration curve.

The 3D dose distribution is calculated from separate convolutions for each of the primary
photons, extra-focal photons and contaminating electrons. The convolutions are performed for all
finite-sized beamlets that comprise the clinical broad beam. The final dose distribution is
obtained by a simple superposition of the individual beamlet contributions.

Beamlets
Figure 5 shows the geometrical definitions of the coordinates referring to a single beamlet β on
the X–Z plane, with the Y-axis pointing outwards from the paper. The coordinates are defined in
two coordinate systems: patient and beamlet. The coordinates of the calculation point (P) in the
figure are (X̃ ,Ỹ ,Z̃ ) in the patient coordinate system, and (x, y, z) in the beamlet coordinate system.
The depth coordinate z is measured from the intersection point of the central fanline and the skin
in the beamlet coordinate system.

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 Figure 5: Coordinates in patient coordinate system and
beamlet coordinate system on X–Z plane.

The broad clinical beam is divided into finite-size beamlets β. The cross-sectional area of a
beamlet corresponds to the resolution of the calculation voxel.

The dose calculation is based on the convolutions over the beamlet cross-sections separately for
the primary photons, extra-focal photons (second source), and for electrons contaminating the
primary beam. The dose is convolved by using the basic physical parameters defined for every
beamlet β.

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All depth-dependent functions used in the beamlet convolutions are computed along the central
fanline of the beamlet using the depth coordinate z that defines the actual distance traveled from
the surface of the patient body (Figure 5.) Lateral dose scattering due to photons and electrons is
defined on the level perpendicular to the central fanline of the beamlet.

The dose to an arbitrary calculation point (X̃,Ỹ ,Z̃ ) in the patient is obtained by summing up the
dose contributions of all individual beamlets β of the broad beam in the final global
superposition.

Photon Dose Calculation

The photon beam attenuation is modeled with an energy deposition density function Iβ(z,ρ). The
photon scatter is modeled with a scatter kernel Kβ(x,y,z,ρ) that defines the lateral dose scattering.
Both functions I and K are defined individually for each beamlet β. The primary and extra-focal
photons are calculated in the same way, with the exception of their spectral composition and the
position and size of the focal spot.

The dose distribution resulting from an arbitrary beamlet β due to photons in a sufficiently large
homogenous neighborhood is calculated by the following convolution:

Dβ ,ph (X̃,Ỹ ,Z̃) = Φβ × I β (z,ρ) ×

∫ ∫ Kβ (u – x,v – y ,z;ρ)dudv (1)
(u,v) ∈ Area ( β )

In the convolution, the calculation point (X̃ ,Ỹ ,Z̃ ) is represented by (x,y,z) relative to the origin of
the beamlet coordinate system. The photon fluence Φβ is assumed to be uniform over the small
cross-section of beamlet β.

The energy deposition density function Iβ(z,ρ) denotes the area integral of the dose over the
transverse plane of the pencil beam at depth z, normalized to a single incident photon. The poly-
energetic function Iβ(z,ρ), based on the photon beam spectrum, is constructed from the
superposition of pre-calculated monoenergetic energy deposition density functions.

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The energy deposition density function I(z,ρ) accounts for tissue heterogeneity by employing the
concept of radiological scaling. This is performed by setting I(z,ρ) = I(z’), where the radiological
depth z’ is defined as

Z
ρ(t)
Z' =
∫ ρ------------------
water
- dt
0

where ρ is the electron density.

The photon scatter kernel Kβ(x,y,z,ρ) is composed of the weighted sum of four Gaussian
functions as shown in the following equation:

3
2 2
1 x +y
K β (x ,y ,z ,ρ) =
∑ c k ( z ) ----------------------- exp – --------------------
2
πσ k ( z )
2
σk (z)
(2)
k=0

The Gaussian kernels are characterized with the standard deviations σk (Figure 6.) The factors ck
define the weights for the four Gaussian kernels and ensure the unity normalization of the total
kernel energy. The above parameters of the polyenergetic scatter kernel Kβ(x,y,z,ρ) are
determined using the Monte Carlo calculated monoenergetic scatter kernels and the spectrum of
the photon beam.

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 Figure 6: Density scaling for photon scatter sigmas.

Density scaling for the photon scatter kernels is performed separately along the four main lateral
directions and according to the average density in these directions.

Scatter kernels σk and their depth dependencies are determined at the time of configuration for a
water-equivalent medium. In heterogeneous media, the values σk in Equation 2 are replaced by
d
density scaled values σ˜ ( k ≠ 0 ). During the dose distribution calculation, σk when traveling
k

along the fanline from depth z0 to zn, is evaluated in a manner that allows the density change to
gradually affect the scatter parameters. The short range effect σ0 is not scaled.

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The density scaling also accounts for the heterogeneity in the neighboring voxels. Therefore, the
σk are individually scaled for the density in four main lateral directions. This is denoted by the
labels d ∈ {x+, x- ,y+,y-} separately (Figure 6.) Scatter kernels σk are scaled according to the

average density ρd computed over the distance of the effective range of the σk along the main
lateral directions d as in the following equation:

⎛ ρ water⎞
σ̃kd ( z n ) = σ k ( z n ) × ⎜ -------------------⎟ for d ∈ {x+, x- ,y+ ,y-} (3)
⎝ ρ d ( z )⎠
n

Hence, as a whole, the photon scatter in heterogeneous surroundings is anisotropically modeled

by scaling the scatter kernels in all directions.

An essential feature of the Gaussian kernel is that its definitive integral form is conveniently
expressed as a sum of error functions [2, 5]. This allows convolution in Equation 1 to be
performed analytically, which significantly decreases the computation time required for dose
distribution calculation.

Contaminating Electrons
The primary photon beam is contaminated with electrons originating mainly in the flattening
filter, ion chamber, collimating jaws and air. If beam modifiers are used, the modifiers may
absorb most of the electrons in the open beam, but the modifier itself becomes a secondary
source of contaminating electrons. In general, electron contamination depends strongly on the
beam energy and the field size.

The dose distribution resulting from an arbitrary beamlet β due to the contaminating electrons is
calculated by the following convolution:

D cont ,β (X̃,Ỹ ,Z̃) = Φ cont ,β × I cont ,β (z,ρ) ×

∫ ∫ Kcont ,β (u – x,v – y ,z ,ρ)dudv (4)
(u,v) ∈ Area ( β )

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In the convolution, the calculation point (X̃ ,Ỹ ,Z̃ ) is represented by (x,y,z) relative to the origin of
the beamlet coordinate system. The electron fluence Φcont,β and the energy deposition density
Icont,β are assumed to be uniform over the cross-section of beamlet β.

The scatter kernel for contaminating electrons is modeled in a conventional way by a Gaussian
distribution function:

2 2
1 x +y
K cont ,β (x,y ,z ,ρ) = ----------- exp – -------------------- (5)
2 2
πσ σ
E E

where factor σE is a constant derived from measured data.

The fluence of the contaminating electrons is determined by convolving photon fluence with a
Gaussian kernel with σ = σcont. The energy deposition density function Icont,β(z,ρ) for the
contaminating electrons is determined from the measured data and tabulated as a function of the
depth z.

Superposition
The final dose D (X˜ ,Ỹ ,Z̃) at an arbitrary calculation point in the patient is obtained by a
superposition of the separate dose contributions from the primary photons (ph1) (Equation 1),
extra-focal photons (ph2) (Equation 1), and contaminating electrons (Equation 4) from all
individual beamlets denoted by index β:

D (X̃,Ỹ ,Z̃) = ∑ ( D ph 1 ,β (X̃,Ỹ ,Z̃) + D ph2 ,β (X̃,Ỹ ,Z̃) + D c on t ,β ( X̃ ,Ỹ ,Z̃ ) ) (6)
β

Essentially, the majority of the convolutions appearing in the superposition can be performed
analytically, because the scatter kernels have Gaussian shapes, and because the photon and

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electron fluences, as well as the energy deposition functions, can be treated as uniform across the
cross-sections of the beamlets without introducing an significant error in the final dose
distribution.

MU Calculation
Calculation of monitor units is based on the calibration measurements made with the smallest
and largest field sizes and a selection of rectangular field sizes in between. The head scatter
effects are Monte Carlo simulated for Varian accelerators using an extra-focal photon source.
Phantom scatter factors are calculated by photon transport. Backscatter effects to the MU
chamber are determined from the output factor table. Accelerators from other vendors are also
supported because the Monte Carlo simulated data merely serves as a starting point for the beam
configuration procedures that fit the beam parameters to the measured beam data.

Implementation in Eclipse
The new AAA algorithm is implemented as a dose calculation server in Eclipse. The AAA
algorithm is configured in the Eclipse Beam Configuration task. The resolution of the dose
calculation grid can be selected in the range of 2–10 mm during treatment planning in the Eclipse
External Beam Planning task.

A fundamental model of the radiation generated by the medical linear accelerator is first derived
using Monte Carlo simulations of the treatment head. Then for each clinical beam, the Monte
Carlo phase space parameters are modified to construct a customized phase space specific to the
clinical beam to be modeled. The customized phase space defines the particle fluence and energy
spectrum characteristic of the clinical beam. The dose calculation with the AAA supports the use
of beam modifiers, such as blocks, compensators, hard wedges, dynamic wedges, MLCs and
Intensity Modulated Radiation Therapy (IMRT) with Dynamic MLC (DMLC).

Required Beam Measurements

The basic physical parameters are predefined by Monte Carlo simulations and adapted to the
available beam data measured in a water equivalent medium. The minimum set of beam
measurements for AAA beam data configuration includes:

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 • Open field depth doses for three small square field sizes between 3×3 cm2 and 6×6 cm2
for 10×10 cm2 field and for the largest field size.

• Open field lateral dose profiles for five of the field sizes mentioned above. For the largest
field size, the diagonal dose profiles at the five depths are also required.

• Wedged field depth dose for the largest wedged square field size.

• Wedged field dose profile at the depth of dose maximum for the largest wedged square
field size.

• MU-to-dose calibration table for the user selected calibration depth and Source-to-
Phantom Distance (SPD). Measurements are required for the smallest and the largest
square fields and for the two extreme rectangular fields (for example, 3×40 cm2
and 40×3 cm2).

In general, the beam data required to configure the Pencil Beam Convolution model in Eclipse
(or the Single Pencil Beam model in CadPlan™) is sufficient for the configuration of the AAA
model. Therefore, there are usually no additional measurements required when implementing
AAA for treatment units that are already commissioned in Eclipse.

Results
A large number of experiments have been carried out to verify the accuracy of the
implementation of the AAA dose calculation model in Eclipse. Testing has included a full range
of field sizes from 3×3 to 40×40 cm2, both in homogeneous and inhomogeneous phantoms. The
measurements presented in this paper are part of the Golden Beam Data for Varian Clinacs for
6 and 18 MV photon energies. Additionally Monte Carlo calculations were conducted in test
phantoms in a slab geometry and compared to calculations with the AAA dose calculation
model.

The open field dose calculation in a homogeneous water-equivalent phantom is in excellent

agreement with measurement and is shown for 6 MV (Figures 7 – 9) and for 18 MV
(Figures 10 – 12). Deviations are well within ± 1 % of the dose maximum for all the field sizes
tested from 4×4 to 40×40 cm2 with 6 and 18 MV energies.

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Heterogeneity corrections were studied with small field sizes and low to medium densities, as
the influence of strong scattering effects on the dose distribution is most prominent under those
conditions. Figures 13 to 16 show depth-dose curves for various field sizes in water with
(1) a 8 cm thick Styrofoam layer embedded at 2 to 10 cm depth to simulate lung interfaces and
(2) a 2 cm thick bone slab embedded at 5 to 7 cm depth. The data are for 6 MV and 18 MV
photon energies and are compared to data generated by Monte Carlo calculations. The AAA
model accurately predicts both the dose build-down effect at the water-lung interface and the
dose build-up effect at the lung-water interface. Table 1 summarizes the results to a 95 %
confidence level.

Table 1: Summary of results

Criteria
Phantom Type
Dose Distance Region
<1% < 1 mm Beyond dmax
Homogeneous
<2% < 2 mm Below dmax
<3% < 2 mm Inside heterogeneity
Heterogeneous
<1% < 1 mm Outside heterogeneity

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160
3x3 cm AAA
3x3 cm Measurement
140 10x10 cm Measurement
10x10 cm Measurement
20x20 cm Measurement
120
20x20 cm Measurement
40x40 cm Measurement
100 40x40 cm Measurement

80

60

40

20
0 50 100 150 200 250 300 350

Figure 7: Percentage Depth Doses for 6 MV

100 3x3 cm AAA

3x3 cm Measured
10x10 cm AAA
80 10x10 cm Measured
20x20 cm AAA
20x20 cm Measured
40x40 cm AAA
60 40x40 cm Measured

40

20

0
0 50 100 150 200 250 300 350 400

Figure 8: Profiles for depth 10 cm for 6 MV

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100 3x3 cm AAA
3x3 cm Measured
10x10 cm AAA

80 10x10 cm Measured
20x20 cm AAA
20x20 cm Measured
40x40 cm AAA
60 40x40 cm Measured

40

20

0
0 50 100 150 200 250 300 350 400

Figure 9: Profiles for depth 30 cm for 6 MV

130

3x3 cm AAA
120
3x3 cm Measurement
10x10 cm Measurement
110
10x10 cm Measurement
20x20 cm Measurement
100
20x20 cm Measurement
40x40 cm Measurement
90
40x40 cm Measurement

80

70

60

50

40
0 50 100 150 200 250 300 350

Figure 10: Percentage Depth Doses for 18 MV

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100 3x3 cm AAA
3x3 cm Measured
10x10 cm AAA
10x10 cm Measured
80
20x20 cm AAA
20x20 cm Measured
40x40 cm AAA
60
40x40 cm Measured

40

20

0
0 50 100 150 200 250 300 350 400

Figure 11: Profiles for depth 10 cm for 18 MV

100 3x3 cm AAA

3x3 cm Measured
10x10 cm AAA

80 10x10 cm Measured
20x20 cm AAA
20x20 cm Measured
40x40 cm AAA
60
40x40 cm Measured

40

20

0
0 50 100 150 200 250 300 350 400

Figure 12: Profiles for depth 30 cm for 18 MV

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 6 MV
EGSnrc and AAA
Water with 8 cm Lung Slab (2 cm to 10 cm)
2.5E-11 5%

4%

2.0E-11 3%

2%

1.5E-11 1%

Difference
Dose

0%

1.0E-11 -1%

-2%
average error MC water w/.26 3x3
AAA water w/lung 3x3 MC water w/.26 4x4
5.0E-12 AAA water w/lung 4x4 MC water w/.26 5x5 -3%
AAA water w/lung 5x5 MC water w/.26 10x10
AAA water w/lung 10x10 MC water w/.26 20x20 -4%
AAA water w/lung 20x20

0.0E+00 -5%
0 5 10 15 20 25 30
Depth (mm)

Figure 13: Percentage Depth Doses for 6 MV with Lung

6 MV
EGSnrc and AAA
Water with 2 cm Bone Slab (5 cm to 7 cm)
2.5E-11 5%
average error MC water w/1.85 3x3
AAA water w/bone 3x3 MC water w/1.85 4x4
AAA water w/bone 4x4 MC water w/1.85 5x5 4%
AAA water w/bone 5x5 MC water w/1.85 10x10
AAA water w/bone 10x10 MC water w/1.85 20x20
2.0E-11 AAA water w/bone 20x20 3%

2%

1.5E-11 1%
Difference
Dose

0%

1.0E-11 -1%

-2%

5.0E-12 -3%

-4%

0.0E+00 -5%
0 5 10 15 20 25 30
Depth (mm)

Figure 14: Percentage Depth Doses for 6 MV with Bone

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 18 MV
EGSnrc and AAA
Water with 8 cm Lung Slab (2 cm to 10 cm)
4.50E-11 3%

3.75E-11 2%

3.00E-11 1%

Difference
Dose

2.25E-11 0%

1.50E-11 -1%

average error MC w/026 3x3

AAA w/lung 3x3 MC w/026 4x4
7.50E-12 AAA w/lung 4x4 MC w/026 5x5 -2%
AAA w/lung 5x5 MC w/026 10x10
AAA w/lung 10x10 MC w/026 20x20
AAA w/lung 20x20
0.00E+00 -3%
0 5 10 15 20 25 30
Depth (mm)

Figure 15: Percentage Depth Doses for 18 MV with Lung

18 MV
EGSnrc and AAA
Water with 2 cm Bone Slab (5 cm to 7 cm)
4.50E-11 3%
average error MC w/185 3x3
AAA w/bone 3x3 MC w/185 4x4
AAA w/bone 4x4 MC w/185 5x5
AAA w/bone 5x5 MC w/185 10x10
3.75E-11 2%
AAA w/bone 10x10 MC w/185 20x20
AAA w/bone 20x20

3.00E-11 1%
Difference
Dose

2.25E-11 0%

1.50E-11 -1%

7.50E-12 -2%

0.00E+00 -3%
0 5 10 15 20 25 30
Depth (mm)

Figure 16: Percentage Depth Doses for 18 MV with Bone.

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Conclusion
The AAA photon dose calculation model is a powerful tool for clinical radiotherapy planning
that provides the increased speed and accuracy required for complicated modern treatment
techniques. The AAA model also introduces a novel, physical method for modeling electron
disequilibrium in tissue heterogeneities. The dose calculation accuracy has been tested by
measurements in heterogeneous slab phantoms and the results demonstrate very good agreement
with calculations. In general the agreement is better than ± 1.5 % of the dose maximum or ± 2
mm lateral shift of isodose lines in high dose gradient regions.

References
1. Ulmer W and Kaissl W: The inverse problem of a Gaussian convolution and its application to
the finite size of the measurement chambers/detectors in photon and proton dosimetry, Phys.
Med. Biol. 48 (2003) 707-727

2. Ulmer W, Harder D: A Triple Gaussian Pencil Beam Model for Photon Beam Treatment
Planning, 2. Med. Phys. 5 (1995) 25-30

3. Ulmer W, Harder D: Applications of a Triple Gaussian Pencil Beam Model for Photon Beam
Treatment Planning, 2. Med. Phys. 6 (1996) 68-74

4. Ulmer W, Harder D: Corrected Tables of the Area Integral I(z) for the Triple Gaussian Pencil
Beam Model, Z. Med. Phys. 7 (1997) 192-193

5. Ulmer W, Brenneisen W: Application of an Analytical Pencil Beam Model to Stereotactic

Radiation Therapy Planning, Journal of Radiosurgery, Vol 1, No.3, 1998

6. NRCC Report PIRS-701: The EGSnrc Code System: Monte Carlo Simulation of Electron and
Photon Transport, I. Kawrakow and D.W.O. Rogers; Nov 7, 2003

7. Liu HH, Mackie TR, McCullough EC: A dual source photon beam model used in
convolution/superposition dose calculations for clinical megavoltage x-ray beams.
Med Phys. 24 (1997) 1960-1974.

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