A bone composition model for Monte Carlo x-ray transport simulations
Hu Zhou,a! Paul J. Keall, and Edward E. Graves
Department of Radiation Oncology and Department of Molecular Imaging Program at Stanford,
Stanford University, Stanford, California 94305
!Received 5 May 2008; revised 18 November 2008; accepted for publication 18 December 2008;
published 25 February 2009"
In the megavoltage energy range although the mass attenuation coefficients of different bones do
not vary by more than 10%, it has been estimated that a simple tissue model containing a singlebone composition could cause errors of up to 10% in the calculated dose distribution. In the
kilovoltage energy range, the variation in mass attenuation coefficients of the bones is several times
greater, and the expected error from applying this type of model could be as high as several hundred
percent. Based on the observation that the calcium and phosphorus compositions of bones are
strongly correlated with the bone density, the authors propose an analytical formulation of bone
composition for Monte Carlo computations. Elemental compositions and densities of homogeneous
adult human bones from the literature were used as references, from which the calcium and phosphorus compositions were fitted as polynomial functions of bone density and assigned to model
bones together with the averaged compositions of other elements. To test this model using the
Monte Carlo package DOSXYZnrc, a series of discrete model bones was generated from this
formula and the radiation-tissue interaction cross-section data were calculated. The total energy
released per unit mass of primary photons !terma" and Monte Carlo calculations performed using
this model and the single-bone model were compared, which demonstrated that at kilovoltage
energies the discrepancy could be more than 100% in bony dose and 30% in soft tissue dose.
Percentage terma computed with the model agrees with that calculated on the published compositions to within 2.2% for kV spectra and 1.5% for MV spectra studied. This new bone model for
Monte Carlo dose calculation may be of particular importance for dosimetry of kilovoltage radiation beams as well as for dosimetry of pediatric or animal subjects whose bone composition may
differ substantially from that of adult human bones. © 2009 American Association of Physicists in
Medicine. #DOI: 10.1118/1.3077129$
Key words: Mass attenuation coefficient, elemental composition, total energy released per unit
mass, dose distribution, Monte Carlo simulation
I. INTRODUCTION
The development of efficient computation code and the advances in computer processor technology in recent years
have significantly enabled applications of Monte Carlo
method in radiation therapy planning systems.1 Because the
interactions of x rays and ! rays with materials can be modeled in detail, the Monte Carlo algorithm can provide very
accurate dose predictions provided proper source and tissue
models are available. The critical parameters in a model for
dose calculation include the source characteristics, the geometries of the patient body and the target, and the distributions
of tissue densities and tissue elemental compositions in the
body. In a modern radiation therapy clinic, the source parameters can be extracted from the machine design, the control
settings of the treatment system, and the device calibration.
CT imaging technology provides information concerning the
subject geometry and tissue density at high spatial resolution.
However there is no direct source from which we can extract
the tissue elemental compositions,1 although the development of dual-energy CT imaging techniques could offer
more information.2 Without this information, the determination of subject composition relies on an empirical model.
Bone tissues provide significant challenges to dose calcu1008
Med. Phys. 36 „3…, March 2009
lation. Bones contain the most x-ray absorbing elements in
the body, their densities are much higher than other tissues,
and their elemental composition can vary significantly between different skeletal locations.3–5 Comparatively, the elemental compositions and densities of soft tissues and lungs
generally do not vary to the same degree.4 It is therefore
expected that variation in bone composition could introduce
complications into the computation of x-ray dose deposition.
Reports from experiments and simulations have shown that
in the 6 – 15 MV energy range, the error in dose distributions
arising from using a model containing single-bone composition could reach a few percent for low density tissues and
more than 10% for high density bones in some cases.6 Therefore the use of conversion techniques based purely on mass
density is discouraged within Monte Carlo simulations because these methods ignore dependencies of particle interactions on the materials, which can lead to notable discrepancies in high atomic number materials. The conversions
should include the use of both mass density and elemental
compositions of the materials.1
In the kilovoltage !kV" x-ray energy range the importance
of an accurate bone model is increased relative to the megavoltage !MV" energy range. This can be understood by ex-
0094-2405/2009/36„3…/1008/11/$25.00
© 2009 Am. Assoc. Phys. Med.
1008
Mas
ss Attenua
ation Coeffficient (cm
m²/g)
1009
10
10
Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
2
1
Cortical Bone
10
10
10
0
-1
Inner Bone
-2
10
-2
10
-1
10
0
Photon Energy (MeV)
10
1
FIG. 1. The mass attenuation coefficients of the inner bone and cortical bone
tissues calculated based on the data from Ref. 3.
amining the variation in the mass attenuation coefficients of
bones of different compositions as a function of photon energy. Figure 1 plots the mass attenuation coefficients of two
adult human bone tissues as a function of photon energy,
calculated using the data provided in Ref. 3 the highestdensity bone !cortical bone tissue, density of 1.9, containing
22.18% calcium, mainly consisting of osteogenic cells layers" and the lowest-density bone !inner bone, density of 1.12,
containing 4.97% of calcium, introduced by the literature as
an average composition of a mixture of hard bone and red
marrow found in trabecular bone structures". Within the photon energy range in MV x-ray therapy, from
200 keV to 20 MeV, the difference in the mass attenuation
coefficients of the two tissues is within 10%. However, over
the kV x-ray range from 10 to 200 keV the mass attenuation
coefficients of the cortical bone are several times higher than
those of the inner bone. A much higher error in the dose
distribution computed for a kV x-ray beam therefore would
be expected if an improper bone composition is assumed in
the dose computation.
Errors in dose calculation caused by tissue modeling must
be carefully investigated. Dutreix reported that differences in
response to radiotherapy are clinically detectable for dose
differences of as small as 7%.7 Other studies suggested that a
5% radiation dose change could result in a 10%–20% change
in tumor control probability and up to 20%–30% change in
normal tissue complication probabilities.8–10 As pointed out
by Verhaegen and Devic, errors in assignment of material
properties can lead to dose errors of 10%–30% for 6 – 18 MV
electron beams and 40% for 250 kVp photon beams.11 The
clinical importance of bone dose was emphasized by Montemaggi et al., who noted that bone irradiation can lead to
complications such as infection, fracture, and bone
necrosis.12 Fajardo commented that bones receiving therapeutic doses of radiation are susceptible to osteoblast and
osteocyte damage, and that the observed effects are most
detrimental in children.13 These observations are supported
Medical Physics, Vol. 36, No. 3, March 2009
1009
by reported measurements of bone density and composition
in children, which vary with age and are significantly different than those of adults.14 Accurate tissue composition models are also of relevance to emerging techniques for applying
conformal radiotherapy to small animal models of disease.15
For these small subjects it is also critical to have a biologically relevant tissue model for dose calculation.
The issue of extracting material characteristics from CT
images has long been a topic of interest.16 In early studies of
using Monte Carlo computation in radiation therapy, the environment and body were modeled by air and three tissues:
lung, skeletal muscle, and bone.14,17–19 In these models a
single tissue composition and single-bone composition were
assumed, but their densities were variable. In more recent
years models have been developed to include more materials,
as the tissue model in VMC"",20,21 which was used in the
electron beam Monte Carlo treatment planning system for
clinical applications.22 In the studies by Schneider and coworkers the radiation-tissue interaction was modeled through
multiparameter approximations based on CT Hounsfield
numbers, with parameters derived from biological
measurements.23,24 Kanematsu et al. modeled soft tissues and
bones as muscle-air, muscle-fat, and muscle-bone mineral
binary mixtures and related the ratios in the mixtures with
Hounsfield numbers through the effective electron densities
calculated using a stoichiometric model.25 There has been
increasing use of the Monte Carlo simulation package
VMC"" !Refs. 20 and 21" for electron beam radiation
therapy and XVMC !Ref. 26" for photon beam radiation
therapy because of their accuracy and execution speed. In
these packages the mass scattering and stopping powers of
tissues are fitted to a piecewise continuous function of media
mass over the energy range of MV beams. Siebers et al.
calculated water-to-material stopping power ratios of soft
bone and cortical bone by Monte Carlo simulations over the
spectra of 6 and 18 MV x-ray beams and used the ratios for
the dose-to-medium to dose-to-water correction.27 In a subsequent publication this group noted that the uncertainties in
absolute material selection can lead to substantial errors in
the determination of the absorbed dose to water and the dose
to the material, and the obvious remedy for this is to incorporate many materials in the CT-to-material conversion
table.28 The method of stopping power ratios was applied in
IMRT treatment planning using approximately 50 tissues
whose densities distributed over the range from
0.3 to 1.9 g / cm3.29 To study skeletal dosimetry, Kramer et
al. constructed adult male and female phantoms including
major organs and bones according to the standard data given
in ICRP89 and simulated external irradiation to these phantoms over photon energy range from 10 keV to 10 MeV using Monte Carlo methods.30 In that study CT images of
bones from different parts of the body were segmented into
cortical bone, spongiosa, bone marrow, and cartilage, after
which the elemental compositions and densities of these tissues were tabulated and the equivalent doses per kerma in air
for different bone components were obtained as function of
photon energies from 10 keV to 10 MeV. Multiple-bone
models for dose computation have also been studied and
1010
Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
compared to experimental measurements in the MV energy
range.6 Bitar et al.31 segmented CT images into organs and
tissues and assigned them predefined elemental compositions, computing radiotherapy dose distribution using the
32
MCNP Monte Carlo simulation package.
The focus of our work was to create a representation of
bones for Monte Carlo simulations that is both computationally efficient and accurate in both the kV and MV photon
energy ranges. In considering this issue, the elemental composition data of bone tissues from various literature
sources3–5 were investigated. We found that there exist strong
correlations between the mass densities of these tissues and
the compositions of some elements, which prompted us to
generate a model containing variable bone compositions.
In our work, an analytical formula to generate a general
bone composition model was derived spanning the kV and
MV energy ranges for Monte Carlo calculations. Discrete
materials with varying elemental compositions related to the
physical density were generated from the formula to satisfy
the input requirement of DOSXYZnrc !Refs. 33 and 34" for
evaluation. Unlike previous studies, the focus of this work
was to create a model applicable to photon transport in the
kV range that is also applicable in the less sensitive MV
range. Instead of studying the standard adult human bony
compositions as in the work of Kramer et al.,30 our model
stands on the analyses of CT images of specific cases so that
the variations from different individuals are automatically
taken into consideration. Because most Monte Carlo codes
require explicit specification of the elemental composition
within each patient voxel to obtain interaction data for dose
calculations,6 our model is directly applicable. In the work of
Kramer et al. the study units were the bone tissues, and bony
organs were treated as combinations of bone tissues. In our
model, the study units were elements whose physical characteristics are well known, and all bony structures were
treated as combinations of elements. The material modeling
algorithm used in VMC and XVMC was optimized to speed up
dose computation for MV-beam radiation therapy, in which
the attenuation coefficient # was decomposed into a Compton scattering term #C and a pair-production term # P.26
Variations in these parameters are modeled by either a piecewise continuous functions or alternately using a ramp of 16
materials from selected ICRU data.21 In the kV energy range
the contribution from photoelectric interaction to the attenuation coefficient is significant and is highly material dependent; therefore, the ramp-interpolation method applied to tissue composition would be sensitive to sample material
selections. In our work an averaged tissue composition
model is proposed. This analytical model is conceptually intuitive and mathematically simple. To evaluate this new algorithm, the compositions of modeled bones were compared
with those tabulated from the literature,3–5 and the Monte
Carlo simulated dose distributions produced by this model
were compared with those by the single-bone model.
II. METHODS AND MATERIALS
The densities and elemental compositions of a series of
adult human bones were taken from the literature3–5 for use
Medical Physics, Vol. 36, No. 3, March 2009
1010
as sample bone standards. In these articles the bone tissue
data as those of spongiosa and cortical bone were obtained
from fresh wet samples, and the bone organ data of the humerus and femur were given as the weighted mean over their
tissue compositions. The same data were widely cited in Ref.
35 for tissue substitutes and in Refs. 14 and 30 for radiationtissue interaction data. In Ref. 36, these ICRU data were
listed as physical reference data.
To quantitatively analyze the errors from improper bone
composition assignments, we used the concept of “total energy released per unit mass” of the primary photons !terma",
introduced by Ahnesjö et al.37 Terma at a point r is defined
as
Terma!r" =
%
dE!#/$"!E,r"%!E,r",
!2.1"
E
where # / $ is the mass attenuation coefficient !cm2 / g" at
location r as a function of photon energy E, and % is the
energy fluence of primary photons !cm−2". Terma was applied in 3D dose calculations38–42 using the convolution algorithm in homogeneous media
Dose!r" =
%
d3r! Terma!r!"A!r − r!",
!2.2"
and the superposition algorithm in inhomogeneous media
Dose!r" =
%
d3r! Terma!r!"$!r!"H!r − r!, $av"/$av , !2.3"
where $!r!" is the medium density at r! and $av is the averaged density from r! to r. The functions A!r − r!" and H!r
− r! , $av" are the energy deposition kernels for these two
cases, respectively, obtained from simulation or measurement.
Terma provides a convenient way to evaluate our model
because of its analytical formulation. In our analysis, the
percentage of the energy released in tissues is used in addition to the absolute value; therefore we defined percentage
terma of x rays through a homogeneous bone of thickness x
and density $ as
percentage terma!x" =
=
=
terma!x"
' 100 %
terma!&"
&x0d(&EdE!#/$"!E"Ef!E"e−!#/$"!E"$(
&&0 d(&EdE!#/$"!E"Ef!E"e−!#/$"!E"$(
' 100 %
&EdEEf!E"#1 − e−!#/$"!E"$x$
' 100 % ,
&EdEEf!E"
!2.4"
where f!E" is the x-ray spectrum.
The percentage terma of three representative situations
were used as our test cases: an x-ray beam from a 120 kVp
source of tungsten target through a 2.5 mm Al filtration !calculated using a MATLAB program Spektr43" through a homogeneous bone 5 mm thick !a typical bone size of small experimental animals such as mice" and those by x-ray beams
with spectra of 6 and 15 MV linear accelerators44 passing
through a bone 5 cm thick !a typical bone size in human
1011
Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
1011
these sample bones, remove calcium and phosphorus
from the composition and normalize the sum of the rest
elements to 100%,
COMPnormalized!element"
=
COMPsample!element"
.
1 − COMPsample!Ca" − COMPsample!P"
!2.5"
Hereafter “element” represents H, C, N, O, Na, Mg,
S, Cl, K, and Fe.
!2" Average elemental compositions:
Average the normalized elemental compositions over
all the sample bones,
COMPav!element"
=
FIG. 2. Flow chart of the process to generate bone composition model. See
text for more details.
body". The significant level of difference in the error analysis
was set to 2%. Since no uncertainties were given for the
source data in the original literature, we set this level by
considering the biological effect and the uncertainties from
other sources, such as the radiation therapy system commissioning confidence level.1
In order to preliminarily estimate the upper limit of the
error caused by improper composition assumptions, the compositions of the bones of the highest-density !cortical bone"
and the lowest-density !inner bone" were swapped while
their densities were maintained. The percentage termas calculated for these bones were compared with those using the
correct compositions.
To determine the elements that are most responsible for
these deviations, we varied the composition of each of the 12
major tissue elements found in bone !H, C, N, O, Na, Mg, P,
S, Cl, K, Ca, and Fe" by a small amount individually and
calculated the corresponding percentage terma change. The
result of this examination, as discussed in Sec. III A, showed
that among these elements the calcium composition variation
created the largest percentage terma change, followed by
phosphorus. This assessment is supported by the high atomic
numbers of these elements and their substantial concentrations in bones and is consistent with the observations of
Schneider et al.24 This observation led us to develop a bone
model of “density-varying averaged composition” as a mapping algorithm from bone density to bone composition. Using the bone data from the literature as our standard bone
samples, the model was defined in following steps !Fig. 2".
!1" Normalize elemental compositions:
Normalize the compositions of elements other than
calcium and phosphorous in sample bones. For each of
Medical Physics, Vol. 36, No. 3, March 2009
'sample
bonesCOMPnormalized!element"
No. of sample bones
.
!2.6"
!3" Create a model bone for each of the sample bones:
For each sample bone, the corresponding model bone
is assigned with the same density and the calcium and
phosphorus compositions. The other compositions of the
model are assigned as the scaled average compositions
of the rest of elements,
Densitymodel = Densitysample
COMP!Ca" = COMPmodel!Ca" = COMPsample!Ca",
COMP!P" = COMPmodel!P" = COMPsample!P",
!2.7"
COMPmodel!element"
= COMPav!element"#1 − COMP!Ca" − COMP!P"$.
!4" Verify model bones:
Compute the mass attenuation coefficients for the
model bones according to their compositions. Compute
the percentage terma of each model bone and compared
with its sample bone. The terma of a successful model
should not differ more than 2% from its sample in any of
the three cases.
!5" Create model series:
!a"
!b"
Fit the calcium and phosphorus compositions versus
the bone density over all the sample bones to polynomials.
Assign the first model in the series with a density a
little lower than that of the inner bone, calculate the
calcium and phosphorus compositions from the fitting polynomials obtained by step 5!a", and apply
the last equation in Eq. !2.7" for the rest of the elements.
1012
Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
1012
TABLE I. Densities and elemental compositions !% wt" of adult human bones.
Sample name
Density
H
C
N
O
Na
Mg
P
S
Cl
K
Ca
Fe
Inner bone tissue
Spongiosa
Male sternum
Sacrum male
D6L3 male with cartilage
Femur whole column
Male vertebral column
D6L3 male no cartilage
Humerus spherical head
Femur spherical head
C4 male with cartilage
Femur conical trochanter
Sacrum female
Humerus whole specimen
Male ribs 2 and 6
Male pelvic bones
C4 male no cartilage
Femur whole specimen
Female pelvic bones
Humerus total bone
Male clavicle scapula
Humerus cylindrical shaft
Male ribs 10
Cranium
Mandible
Femur cylindrical shaft
Cortical bone
1.12
1.18
1.25
1.29
1.30
1.32
1.33
1.33
1.33
1.34
1.37
1.37
1.38
1.39
1.41
1.41
1.42
1.43
1.45
1.46
1.46
1.49
1.51
1.60
1.66
1.75
1.90
8.680
8.556
7.799
7.403
7.308
7.089
7.098
6.987
7.084
6.941
6.645
6.735
6.547
6.514
6.339
6.330
6.280
6.176
5.980
6.004
5.994
5.719
5.569
4.934
4.586
4.194
3.387
13.014
29.090
31.686
30.240
26.677
25.960
25.792
28.706
37.813
24.873
24.456
24.314
27.086
23.743
26.335
26.288
26.108
22.840
25.010
31.279
31.232
21.627
23.493
21.195
19.892
20.372
15.488
3.604
2.380
3.615
3.646
3.519
3.540
3.599
3.673
2.594
2.892
3.578
2.953
3.713
3.023
3.724
3.724
3.734
3.123
3.754
2.902
2.902
3.263
3.782
3.845
3.867
3.795
3.967
66.473
48.434
44.148
44.156
47.759
47.669
47.186
44.106
34.122
47.213
47.519
47.032
44.092
46.838
44.105
44.099
44.099
46.538
44.099
37.312
37.336
46.128
44.072
44.087
44.090
41.442
44.065
0.080
0.079
0.050
0.050
0.040
0.040
0.100
0.050
0.100
0.020
0.050
0.020
0.050
0.020
0.050
0.050
0.050
0.030
0.050
0.040
0.040
0.030
0.050
0.060
0.060
0.100
0.060
0.060
3.232
0.080
0.090
0.090
0.100
0.100
0.100
0.100
0.070
0.050
0.080
0.120
0.080
0.120
0.120
0.120
0.100
0.130
0.140
0.140
0.120
0.150
0.160
0.170
0.200
0.210
2.433
0.139
3.964
4.525
4.541
4.853
5.098
5.100
5.587
5.455
5.473
5.737
5.729
6.026
6.020
6.040
6.110
6.476
6.530
6.894
6.905
7.086
7.126
8.000
8.502
9.287
10.192
0.461
0.456
0.120
0.130
0.241
0.241
0.300
0.150
0.200
0.189
0.261
0.200
0.170
0.209
0.180
0.180
0.180
0.220
0.200
0.230
0.230
0.230
0.210
0.240
0.260
0.300
0.310
0.000
0.000
0.000
0.000
0.000
0.000
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.100
0.230
0.228
0.110
0.100
0.080
0.080
0.100
0.090
0.090
0.090
0.070
0.070
0.080
0.080
0.080
0.070
0.070
0.070
0.070
0.070
0.070
0.070
0.080
0.040
0.030
0.030
0.030
4.965
7.406
8.378
9.620
9.704
10.398
10.497
10.899
12.172
12.117
11.767
12.731
12.275
13.328
12.918
12.969
13.119
14.299
14.048
14.999
15.021
15.599
15.329
17.319
18.433
20.172
22.182
0.000
0.000
0.050
0.040
0.040
0.030
0.030
0.040
0.040
0.040
0.030
0.030
0.040
0.040
0.030
0.030
0.030
0.030
0.030
0.030
0.030
0.030
0.040
0.020
0.010
0.010
0.010
!d"
Step up the density for the next model bone, calculate the elemental compositions by the polynomials
for calcium and phosphorus, and averaged compositions for other elements. Calculate the percent terma
of this model. Adjust the model density until the
terma difference from the previous model is less
than 2%.
Repeat step 5!c" to continue the series generation
until the calcium composition of the last model exceeds that of the cortical bone.
!6" Generate radiation-tissue interaction cross-section data:
The cross-section data of each model bones in the
series is calculated using PEGS4 for future Monte Carlo
computation.
The bone densities and compositions in this algorithm, as
in the data from the original literature, represented the mean
values over the volumes of the bone organs. Stated alternately, a bone organ was treated as a homogeneous entity.
When the model is applied in the conversion from the Hounsfield value of a CT voxel, the density and composition represent the mean values over the volume of the voxel. A similar concept was applied by Kramer et al.,30 in which the
bones were modeled as variable volumes of CT voxel clusters so that a volume in some cases may contain just one type
of tissue and in other cases may contain a mixture of marrow
and osteogenic cells layers.
Medical Physics, Vol. 36, No. 3, March 2009
Ref. 8
Ref. 10
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 9 and
Ref. 10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
10
The model bone composition proposed by this algorithm,
in principle, is a continuous analytical function of the bone
density. Discrete points from this model are created in steps
5!b" to 5!d" for the inputs of DOSXYZnrc. A C"" program
BoneModel was developed to build this bone model following the above procedure. In this program the compositions
and the densities of the sample bones from the literatures are
25
20
C
Composition
n (%)
!c"
Source
Ca
15
5
P
10
5
0
1.1
1.2
1.3
1.4
1.5
1.6
Bone Density (g/cm3)
1.7
1.8
1.9
FIG. 3. Correlation of the calcium and phosphorus compositions with the
bone density, and their polynomial fittings as given in Eq. !3.1".
1013
Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
1013
TABLE II. Percentage terma through typical bones. Inner: inner bone; Cortical: cortical bone.
5 mm, 120 kVp
Actual composition !%"
Swapped composition !%"
Error !%"
5 cm, 6 MV
5 cm, 15 MV
Inner
Cortical
Inner
Cortical
Inner
Cortical
13.26
18.99
43.21
29.28
21.29
−27.29
22.54
21.74
−3.53
33.83
34.94
3.27
15.95
15.83
−0.72
25.22
25.36
0.56
loaded. The mass attenuation coefficients of these sample
bones are computed using XCOM,45 and these data are then
saved into a data file as the sample bone library. For each
sample bone, a model bone of the averaged composition as
defined by Eqs !2.5"–!2.7" is generated and evaluated. Then a
series of models is created as described in step 5 of the
procedure so that the calcium compositions in the series
bracket the range of those of the sample bones. The mass
attenuations of these models are computed, and the data are
saved as the model series library. Following step 6, the elemental compositions of each model in the series are composed into an input file and PEGS4 is invoked to generate the
cross-section data.
To investigate the dose computation error related to the
modeling error, the photon-tissue-interaction cross sections
of the single-bone model and our analytical bone model were
applied in a Monte Carlo simulation using DOSXYZnrc. In
the simulation a mouse brain was irradiated by a radiotherapy plan consisting of 90 radially convergent beams from
a 120 kVp x-ray spectrum with tungsten target and 2.5 mm
Al filtration. The isocenter was placed at the target center and
the source was 354 mm away. All beams were 1 mm in diameter at the isocenter and the same weight, uniformly distributed over 360° in a transverse plane passing through the
brain, as is achievable using a micro-CT-based small animal
radiotherapy system.15 The two bone models were used to
generate the Monte Carlo input phantom, in conjunction with
the corresponding cross-section data in the dose calculation.
The resulting dose distributions were compared.
In the default configuration, DOSXYZnrc allows the user
to specify up to seven media. A region of interest in the CT
image must be converted to a phantom through voxel rebinning and air/tissue assignment. The air/tissue assignment is
performed according a predefined composition-densityHounsfield number lookup table. Each voxel in the phantom
is labeled by a digit from 1 to 9 as the air/tissue index. We
modified the DOSXYZnrc code to allow it to read more than
one digit for each voxel and set the maximum number of
media to a higher number so that more bone entities could be
accepted. Correspondingly in the phantom file more than one
digit was used to represent a tissue or air. EGSnrc includes a
program ctcreate that performs the Hounsfield-composition
mapping. In order to gain more process control and improved memory management under WINDOWS, we developed
our own phantom generation procedure in C"", which is part
of a larger C"" package DoseCalc developed by the authors.
DoseCalc combines the source model, phantom generation,
DOSXYZnrc input file generation, and dose calculation with
a graphical user interface.
III. RESULTS
III.A. Sample bones
A total of 27 sample human bones was taken from the
literature. Their compositions and densities are listed in
Table I, sorted by their densities. Some bone data from the
literature were reported with elemental compositions not totaling 100%, in which cases we normalized the compositions
to the reported cumulative level.
A strong correlation of calcium and phosphorus compositions with bone density was observed for these samples, as
shown in Fig. 3. The calcium compositions !Ca%" can be
fitted to a cubic function and the phosphorus compositions
!P%" can be fitted to a quadratic function of the bone density
$ !g / cm3",
Ca % = − 4.796$3 + 7.761$2 + 32.051$ − 33.934,
!3.1"
P % = − 11.089$2 + 44.597$ − 34.716.
The standard deviations of the fits, defined as the deviations
of the fit values from the sample values, were 0.48% for
Ca% and 0.54% for P%, respectively. The significance of the
fitting error was evaluated by the percentage terma difference
between each sample bone and its model bone created using
these fittings. The results of this experiment are discussed
below in Sec. III B.
The percentage terma errors generated by swapping the
compositions of the inner bone !density of 1.12, calcium
4.97%" and the cortical bone !density of 1.9, calcium
TABLE III. The percentage change of terma through female pelvic bones produced by varying the composition of each element by a factor of 1%, respectively.
120 kVp
6 MV
15 MV
H
C
N
O
Na
Mg
P
S
Cl
K
Ca
Fe
0.005
0.044
0.040
−0.122
−0.018
−0.025
−0.015
−0.004
0.000
−0.240
−0.036
−0.045
0.000
0.000
0.000
0.000
0.000
0.000
0.020
−0.004
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.000
0.240
−0.004
0.015
0.000
0.000
0.000
Medical Physics, Vol. 36, No. 3, March 2009
1014
Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
1014
TABLE IV. The normalized percent compositions of an averaged bone excluding calcium and phosphorus, as the COMPave in Eqs. !2.6" and !2.7".
H
C
N
O
Na
Mg
S
Cl
K
Fe
7.839
31.28
4.271
55.74
0.065
0.270
0.288
0.098
0.101
0.035
22.18%", but keeping the bone densities unchanged, are
listed in Table II. We see that in the MV range the percentage
terma errors did not exceed 4%. In the kV range, however,
the errors could be as high as 27% underestimated or 43%
overestimated.
The terma changes induced by elemental composition
variations in female pelvic bone are listed in Table III in
order to identify the element or elements most responsible
for the percentage terma error. The density of 1.45 g / cm3
and calcium composition of 14% of this bone are near the
average of our samples. In the analysis we increased the
composition of each element by a factor of 1%, respectively,
and scaled down the compositions of other elements so that
the sum of all the element compositions remained 100%.
From the table we see that in the kV spectrum range, the
oxygen and calcium composition variations are most associated with the percentage terma changes, and the influences
from carbon and phosphorus are the next. All other elements,
either because of their small atomic numbers or their low
compositions, do not significantly contribute to the change.
We noticed that an increase in the oxygen composition results in a decrease in the percentage terma. This suggests that
this percentage terma change is more likely due to the decrease in other elements. In an additional study we increased
the oxygen composition by a factor of 1% but kept the calcium unchanged, and scaled down other elements, the percentage terma change in the 120 kVp spectrum range was
reduced from 0.24% to 0.01%. The same experiment applied
to carbon leads to similar conclusion. These investigations
clarified that the main causes of the percentage terma change
were the composition variations in calcium and phosphorus.
The number of bones in the discrete model can be estimated by the data in Table II. We see that the ratio of the
percentage termas of a 120 kVp x-ray beam through the cortical bone and the inner bone is 29.28% : 13.26% = 2.21. To
cover this range and keep the percentage terma difference
between the adjacent points within 2%, the model will need
at least 41 composition-density-Hounsfield points, that is, a
series containing at least 41 model bones.
III.B. Averaged bone model
An “averaged bone” was generated for each of the 27
sample bones according to eqs. !2.5"–!2.7". The “averaged
compositions,” excluding the calcium and phosphorus, after
normalization to 100%, are listed in Table IV. The percentage terma of the sample bones is calculated and plotted in
Fig. 4, together with the differences of those of the models
from their samples.
Because higher bone density is associated with higher absorption, it is expected that the percentage terma is increasing function of the bone density over all the three spectrum
ranges, as shown in Fig. 4. From the plot we can see that the
percentage terma differences of most of the model bones
from their sample bones are within 1.5%, below our significant level of 2%. Although the percentage terma differences
FIG. 4. Percentage termas of sample bones !left" and percentage terma differences of model bones from their sample bones !right": 120 kVp x-ray spectrum
through a 5 mm bone, 6 MV x ray through a 5 cm bone, and 15 MV x ray through a 5 cm bone, respectively.
Medical Physics, Vol. 36, No. 3, March 2009
1015
Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
1015
TABLE V. Densities, elemental compositions by weight, and percentage terma !%" of model bone series.
Series name
Density
H
C
N
O
Na
Mg
P
S
Cl
K
Ca
Fe
5 mm
120 kV
5 cm
6 MV
5 cm
15 MV
Model Series 0480
1.1149
7.3673
29.399
4.0146
52.389
0.0612
0.2543
1.2215
0.2710
0.0927
0.0953
4.8000
0.0334
12.726
22.200
15.662
Model Series 0514
1.1257
7.3239
29.226
3.9909
52.080
0.0609
0.2528
1.4355
0.2694
0.0922
0.0947
5.1400
0.0332
12.980
22.383
15.806
Model Series 0548
1.1366
7.2806
29.053
3.9673
51.772
0.0605
0.2513
1.6481
0.2678
0.0916
0.0941
5.4800
0.0330
13.236
22.566
15.950
Model Series 0582
1.1476
7.2374
28.881
3.9437
51.464
0.0602
0.2498
1.8593
0.2662
0.0911
0.0936
5.8200
0.0328
13.493
22.749
16.095
Model Series 0617
1.1590
7.1930
28.703
3.9196
51.149
0.0598
0.2483
2.0754
0.2646
0.0905
0.0930
6.1700
0.0326
13.761
22.939
16.245
Model Series 0652
1.1704
7.1488
28.527
3.8954
50.834
0.0594
0.2468
2.2899
0.2630
0.0900
0.0924
6.5200
0.0324
14.030
23.129
16.396
Model Series 0688
1.1822
7.1033
28.346
3.8707
50.511
0.0590
0.2452
2.5091
0.2613
0.0894
0.0918
6.8800
0.0322
14.309
23.326
16.552
Model Series 0724
1.1942
7.0581
28.165
3.8460
50.189
0.0587
0.2437
2.7267
0.2596
0.0888
0.0913
7.2400
0.0320
14.591
23.523
16.709
Model Series 0761
1.2065
7.0117
27.980
3.8207
49.859
0.0583
0.2421
2.9488
0.2579
0.0882
0.0907
7.6100
0.0318
14.882
23.727
16.871
Model Series 0798
1.2190
6.9654
27.795
3.7955
49.530
0.0579
0.2405
3.1692
0.2562
0.0877
0.0901
7.9800
0.0316
15.176
23.932
17.035
Model Series 0835
1.2315
6.9192
27.611
3.7704
49.202
0.0575
0.2389
3.3880
0.2545
0.0871
0.0895
8.3500
0.0314
15.472
24.138
17.199
Model Series 0873
1.2445
6.8719
27.422
3.7446
48.866
0.0571
0.2372
3.6109
0.2528
0.0865
0.0889
8.7300
0.0312
15.778
24.350
17.369
Model Series 0911
1.2577
6.8248
27.234
3.7189
48.531
0.0567
0.2356
3.8321
0.2510
0.0859
0.0882
9.1100
0.0310
16.087
24.564
17.540
Model Series 0950
1.2713
6.7766
27.042
3.6927
48.188
0.0563
0.2339
4.0573
0.2493
0.0853
0.0876
9.5000
0.0307
16.407
24.785
17.717
Model Series 0989
1.2850
6.7285
26.850
3.6665
47.846
0.0559
0.2323
4.2805
0.2475
0.0847
0.0870
9.8900
0.0305
16.729
25.007
17.896
Model Series 1029
1.2993
6.6794
26.654
3.6397
47.496
0.0555
0.2306
4.5076
0.2457
0.0841
0.0864
10.290
0.0303
17.063
25.237
18.080
Model Series 1069
1.3137
6.6304
26.458
3.6130
47.148
0.0551
0.2289
4.7326
0.2439
0.0835
0.0857
10.690
0.0301
17.399
25.469
18.267
Model Series 1109
1.3282
6.5815
26.263
3.5864
46.801
0.0547
0.2272
4.9555
0.2421
0.0828
0.0851
11.090
0.0298
17.738
25.702
18.455
Model Series 1149
1.3430
6.5328
26.069
3.5598
46.454
0.0543
0.2255
5.1765
0.2403
0.0822
0.0845
11.490
0.0296
18.081
25.937
18.645
Model Series 1189
1.3579
6.4843
25.875
3.5334
46.110
0.0539
0.2239
5.3953
0.2385
0.0816
0.0838
11.890
0.0294
18.426
26.175
18.837
Model Series 1229
1.3731
6.4360
25.683
3.5071
45.766
0.0535
0.2222
5.6120
0.2367
0.0810
0.0832
12.290
0.0292
18.775
26.415
19.032
Model Series 1269
1.3884
6.3878
25.490
3.4808
45.423
0.0531
0.2205
5.8266
0.2350
0.0804
0.0826
12.690
0.0290
19.127
26.657
19.228
Model Series 1309
1.4040
6.3398
25.299
3.4546
45.082
0.0527
0.2189
6.0389
0.2332
0.0798
0.0820
13.090
0.0288
19.483
26.903
19.427
Model Series 1349
1.4198
6.2920
25.108
3.4286
44.742
0.0523
0.2172
6.2491
0.2314
0.0792
0.0814
13.490
0.0285
19.842
27.151
19.629
Model Series 1389
1.4359
6.2443
24.918
3.4026
44.403
0.0519
0.2156
6.4570
0.2297
0.0786
0.0807
13.890
0.0283
20.205
27.402
19.833
Model Series 1429
1.4522
6.1968
24.728
3.3767
44.065
0.0515
0.2139
6.6625
0.2279
0.0780
0.0801
14.290
0.0281
20.572
27.656
20.041
Model Series 1469
1.4689
6.1496
24.540
3.3510
43.729
0.0511
0.2123
6.8657
0.2262
0.0774
0.0795
14.690
0.0279
20.944
27.914
20.251
Model Series 1509
1.4859
6.1025
24.352
3.3253
43.394
0.0507
0.2107
7.0664
0.2245
0.0768
0.0789
15.090
0.0277
21.319
28.176
20.465
Model Series 1549
1.5032
6.0556
24.164
3.2998
43.061
0.0503
0.2090
7.2646
0.2227
0.0762
0.0783
15.490
0.0275
21.699
28.443
20.683
Model Series 1589
1.5209
6.0089
23.978
3.2743
42.729
0.0499
0.2074
7.4602
0.2210
0.0756
0.0777
15.890
0.0273
22.084
28.713
20.904
Model Series 1629
1.5389
5.9624
23.793
3.2490
42.398
0.0496
0.2058
7.6531
0.2193
0.0750
0.0771
16.290
0.0270
22.474
28.989
21.130
Model Series 1669
1.5575
5.9161
23.608
3.2238
42.069
0.0492
0.2042
7.8431
0.2176
0.0745
0.0765
16.690
0.0268
22.870
29.271
21.361
Model Series 1709
1.5765
5.8701
23.424
3.1987
41.742
0.0488
0.2026
8.0303
0.2159
0.0739
0.0759
17.090
0.0266
23.272
29.559
21.597
Model Series 1749
1.5960
5.8243
23.242
3.1738
41.416
0.0484
0.2011
8.2143
0.2142
0.0733
0.0753
17.490
0.0264
23.681
29.853
21.839
Model Series 1789
1.6161
5.7788
23.060
3.1489
41.093
0.0480
0.1995
8.3950
0.2126
0.0727
0.0747
17.890
0.0262
24.097
30.156
22.088
Model Series 1829
1.6369
5.7336
22.879
3.1243
40.771
0.0477
0.1979
8.5723
0.2109
0.0722
0.0741
18.290
0.0260
24.521
30.467
22.343
Model Series 1869
1.6585
5.6886
22.700
3.0998
40.451
0.0473
0.1964
8.7458
0.2093
0.0716
0.0736
18.690
0.0258
24.954
30.787
22.607
Model Series 1909
1.6809
5.6439
22.522
3.0755
40.134
0.0469
0.1948
8.9153
0.2076
0.0710
0.0730
19.090
0.0256
25.398
31.119
22.880
Model Series 1949
1.7043
5.5996
22.345
3.0513
39.819
0.0465
0.1933
9.0804
0.2060
0.0705
0.0724
19.490
0.0254
25.853
31.464
23.164
Model Series 1989
1.7288
5.5557
22.170
3.0274
39.506
0.0462
0.1918
9.2406
0.2044
0.0699
0.0718
19.890
0.0252
26.321
31.824
23.460
Model Series 2029
1.7548
5.5123
21.996
3.0037
39.197
0.0458
0.1903
9.3952
0.2028
0.0694
0.0713
20.290
0.0250
26.806
32.202
23.772
Model Series 2069
1.7823
5.4693
21.825
2.9803
38.891
0.0455
0.1888
9.5435
0.2012
0.0688
0.0707
20.690
0.0248
27.310
32.603
24.102
Model Series 2109
1.8120
5.4269
21.656
2.9572
38.590
0.0451
0.1873
9.6842
0.1996
0.0683
0.0702
21.090
0.0246
27.838
33.031
24.454
Model Series 2148
1.8436
5.3863
21.494
2.9350
38.301
0.0448
0.1859
9.8123
0.1981
0.0678
0.0696
21.480
0.0244
28.383
33.484
24.827
Model Series 2185
1.8768
5.3486
21.343
2.9145
38.033
0.0445
0.1846
9.9233
0.1967
0.0673
0.0692
21.850
0.0243
28.938
33.958
25.216
Model Series 2220
1.9125
5.3139
21.205
2.8956
37.787
0.0442
0.1834
10.015
0.1955
0.0669
0.0687
22.200
0.0241
29.511
34.464
25.631
Model Series 2252
1.9509
5.2836
21.084
2.8791
37.571
0.0439
0.1824
10.082
0.1944
0.0665
0.0683
22.520
0.0240
30.100
35.004
26.073
in the models of the two least dense bones from their sample
are slightly over this level, we accept this model because the
uncertainties from other sources1 are not well defined.
III.C. Discrete model series
The discrete model series generation resulted in a total 47
model bones that covers the calcium composition from
4.80% to 22.65%, bracketing the calcium composition range
Medical Physics, Vol. 36, No. 3, March 2009
of the sample bones from 4.965% for the inner bone to
22.18% for the cortical bone. The calcium and phosphorus
compositions of these bones were calculated using polynomials in eq. !3.1" from their densities. The normalized compositions of other elements, listed in Table IV, were assigned
to these 47 bones after scaling. The names of these models
reflect their calcium compositions, as model series 0480 indicates that the calcium in this model bone is 4.80%. Table V
1016
Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
1016
TABLE VI. The mouse skull mass attenuation coefficients # / $ at the mean
energy of the 120 kVp spectrum, percentage termas, and the relative dose,
by the single-bone and the 47-bone models.
Percentage
terma
Near scalp
Mid head
0.339
0.292
16%
2.742%
2.249%
20.10%
162%
100%
62%
212%
100%
121%
Single-bone model
47-bone model
Difference
(a)
(b)
(c)
3.5
x 10
4
target
3
Do
ose (arb unit)
47-bone Model
skull
2.5
Single-bone
Single
bone Model
2
1.5
skull
1
maxilla
0.5
0
(d)
0
20
40
60
Pixel
tooth
80
100
120
140
FIG. 5. CT image of a mouse head used in the dose distribution calculation
!a"; simulated dose distribution of a 120 kVp x-ray radiation targeted to a
mouse brain using single-bone model !b" and the 47-bone model !c"; and the
dose profiles predicted by the single-bone model and the 47-bone model !d"
along the white line shown in !b" and !c".
lists these models. The photon-tissue-interaction crosssection data of each model bone in the series were calculated
by invoking PEGS4 from the C"" program BoneModel over
the photon energy range from 1 kV to 100 MV.
III.D. Dose distribution simulations
A micro-CT image acquired from a mouse brain #Fig.
5!a"$ was used to generate Monte Carlo phantom files using
Medical Physics, Vol. 36, No. 3, March 2009
Relative skull dose
# / $ !cm2 / g"
at 57.65 kV
the single-bone model and the 47-bone model, respectively.
The photon-tissue-interaction cross-section data generated
from these two models were used in the computation of delivered dose. The results are shown in Figs. 5!b" and 5!c".
Following a pair of rays irradiating the target from opposite
directions, through the center of the target and a tooth, as
shown in Figs. 5!b" and 5!c", a profile of the dose distribution is extracted for each of the two cases and plotted together in Fig. 5!d".
From these figures we find that in this case the dose prediction by the single-bone model in soft tissues is about 70%
of the dose by the 47-bone model. In the brain target it is
72%. In the skull near the scalp it is 162%, and in the skull
of the center of the head it is more than 200%. The 47-bone
model predicted a more detailed structure of the dose distribution in the tooth. In this examination the mouse skull density is 1.41 and the skull thickness is 0.5 mm, according to
the CT image. The single-bone model assigns to the skull the
composition of the cortical bone whose density is 1.92. The
average energy of the 120 kVp spectrum is 57.65 kV. The
mass attenuation coefficients # / $ at this energy, the percentage termas, and the relative dose depositions for each simulation are listed in Table VI.
In this case, although the same bone densities were used
as in the 47-bone model, the single-bone model assumes
higher calcium and phosphorus compositions in the mouse
skull, and therefore predicts higher mass attenuation coefficient and higher percentage terma in the bony structure. The
resulting prediction of doses to the bones and teeth are
higher, and the doses to the soft tissue near the bones are
lower. We notice that the percentage terma is calculated with
the radiation beam perpendicular to the surface of a bone, in
the direction of the minimum thickness. In a dose calculation
the beam can be incident from different angles thus longer
paths would generate more significant effects. Therefore it is
not a surprise to see that, in our example a 27.3% of percentage terma error in the bones, by 16% of # / $ error, can cause
more than 60% of overestimate in the bony dose and near
30% underestimate of dose to the adjacent soft tissue.
IV. DISCUSSION
In the MV range, the single-bone model in the x-ray absorption computation, in general, would not create percentage terma error more than 4% for the bone thickness about
5 cm. While in kV range, in an extreme case, an error of up
to 43% can result when calculating the percentage terma of a
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Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
120 kVp x-ray beam through a 5 mm bone. The level of this
error emphasizes that accurate tissue models are critical for
kV dosimetry, more than for MV dosimetry. The error from
applying a single-bone model is mainly caused by ignoring
variations in the calcium and phosphorus compositions in
different bones. An algorithm was proposed in this work to
correct this error in which the elemental compositions of the
bones were modeled as an analytical function of their densities, based on the observation that the calcium and phosphorus contents are strongly correlated with the bone density,
and the effects of variations in other elements on the mass
attenuation coefficients are relatively small.
Compared with the single-bone model, the elemental
compositions assigned to our model bones are much closer to
those of the standard samples from the literatures. Using this
algorithm, different parts of a bone as well as bones from
different locations in a body can be assigned different elemental compositions according to their densities. When the
model was discretized to conform to the input format of
DOSXYZnrc, we defined a significance level of 2% in the
percentage terma errors by the consideration of the biology
effect and the uncertainties from other sources such as the
radiation therapy system commission confidence level. The
series generated contained 47 model bones of averaged compositions, separated by 2% percentage terma difference,
which guarantees the percentage terma error will be below
1.5% in the MV spectrum range. In the kV range this series
guarantees the percentage terma error will be less than 2.5%,
for most of cases less than 0.5%.
The percentage terma error of the inner bone model from
its sample is slightly larger than the 2% significance level
!Fig. 4". This can be improved if we take the average by
grouping the bones according to their densities so that within
each group the model covers a smaller density range. However we chose to leave the model in the original simpler
format because of uncertainties in both the source data and
the averaging algorithm. The range of variability of the
sample bone compositions and densities were not specified
in the literature. Bone compositions could vary from person
to person. The Hounsfield numbers of the CT image voxels
contain 1%–2% uncertainties due to noise and beam hardening artifacts.25 In addition, the averaging algorithm contains
some arbitrariness owing to the fact that the bone data collected from the literature were neither uniformly distributed
over the bone density range nor over the bone volumes in a
body. This potential sampling bias leads to different density
ranges having different weights in the average. Therefore the
2% significance level used in this study is likely to be within
experimental variations.
Although many factors other than tissue elemental compositions influence dose calculation errors, it can be expected
that an overestimate in the terma of a large structure is, in
general, associated with an overestimation of the dose deposition in the structure and an underestimation of dose deposition behind the structure. This error could be dramatically
large. In the 6 – 15 MV x-ray spectrum range, although the
percentage terma error produced by the single-bone model
would never exceed 4% for a bone 5 cm thick !Table II", a
Medical Physics, Vol. 36, No. 3, March 2009
1017
10% dose computation error can occur in high-density
bones.6 Our Monte Carlo simulation in the kV range demonstrates the level of dose correction required for a target near
a large bony structure due solely to assigning the bones improper elemental compositions. In our example, if two
mouse brain targets are given the same doses according to
the single-bone model and the 47-bone model, respectively,
the dose to the skull calculated by the single-bone model
would be more than twice as that by the 47-bone model.
In the MV spectrum range in radiation therapy, which
extends from 200 kV to 18 MV, the mass attenuation coefficient # / $ of hydrogen varies from 0.1 to 0.2 cm2 / g. The
# / $ of all other elements are roughly equivalent to each
other and are about half of that of hydrogen, reflecting the
fact that the electron density per mass of hydrogen is twice
that of other elements. The average energy of the 6 MV
spectrum is about 2 MeV and that of the 15 MV spectrum is
about 4 MeV. In the vicinity of these average energies the
# / $ of all the elements except hydrogen do not change significantly with respect to the photon energy. This fact makes
the # / $ of different bones in the MV range much less sensitive to their compositions than in the kV range, so that the
bone density becomes the dominant factor in the percentage
terma. We notice that in the kV range the mass attenuation
coefficient is, in general, an increasing function of bone density because the calcium and phosphorus compositions increase with density. However in most of the MV range, the
mass attenuation coefficient is a slightly decreasing function
of bone density because the higher-density bones generally
have less hydrogen. In this range the hydrogen composition
plays an important role, as pointed out by Vanderstraeten
et al.6 Treating hydrogen composition with a similar technique as that applied for calcium and phosphorus in this
work will further reduce the modeling error.
V. CONCLUSION
In this work we propose a simple analytical algorithm to
create a bone composition model for kV radiotherapy Monte
Carlo calculation. With a setting of 2% terma tolerance, this
algorithm produced a discrete model containing 47 bones
with artificial compositions. We showed that one, in the kV
energy range this model can bring a percentage terma correction by 40% relative to a single-bone model; and two, a
20% percentage bone terma overestimate could result in
more than a 100% overestimate in the bone dose and a 30%
underestimate of the dose in the adjacent soft tissue. We
believe that higher accuracy in kV radiation dose computations will substantially improve the quality of therapies in
this energy range, including experimental treatment of small
animals, human superficial therapies, and human brachytherapy. This development may be particularly important in
the context of pediatric radiation therapy, because the bone
compositions of children are known to be different from
adults and change significantly with age.14
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Zhou, Keall, and Graves: Bone model for Monte Carlo x-ray transport simulations
ACKNOWLEDGMENT
The authors thank Dr. Blake Walters for his technical support on our Monte Carlo simulation using the EGSnrc. Upon
request the authors will be glad to provide the tissueradiation-interaction cross-section data for the 47-bone
model series.
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