Nuclear Instruments and Methods in Physics Research B 269 (2011) 189–196
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B
journal homepage: www.elsevier.com/locate/nimb
Depth-dose distribution of proton beams using inelastic-collision
cross sections of liquid water
C. Candela Juan, M. Crispin-Ortuzar, M. Aslaninejad ⇑
Imperial College London, Blackett Lab, High Energy Physics Department, Prince Consort Road, London, United Kingdom
a r t i c l e
i n f o
Article history:
Received 23 July 2010
Received in revised form 18 September 2010
Available online 27 October 2010
Keywords:
Bragg peak
Proton therapy
a b s t r a c t
Complete physical processes contributing to the shape of the depth-dose distribution and the so-called
Bragg peak of proton beams passing through liquid water are given. Height, width and depth of the Bragg
peak through investigation of collisions of the projectile, nuclear processes and stochastic phenomena are
discussed. Different analytical models for stopping cross sections which lead to Bragg peak are contrasted. Results are also compared with GEANT4 package simulation.
Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction
One of the most important types of cancer treatments to control
malignant cells is radiotherapy. A certain amount of ionising radiation is applied to the patient body, which deposits energy into the
tissues. This may result in the disruption of DNA and the subsequent
death of the cells. Radiotherapy research strives for higher precision
and biological effectiveness of the applied dose. This minimises the
damage to the healthy tissues. In the developed countries, every
year about 20,000 oncological patients are treated with high-energy
X-rays photons [1]. Usually, radiotherapists use electron linear
accelerators (linacs) as sources of X-ray radiation. Despite this being
so widely used, the absorbed dose actually reaches a maximum very
rapidly after hitting the body and decreases slowly in an exponential manner. Thus, using X-rays is not an efficient way to reach the
deep seated tumours. In addition, not only at the entrance of the
body, but also after the tumour, healthy tissues are irremediably
damaged, as the photons traverse our organs.
To quantify the effects of ionising radiation it is important to
know how its energy is distributed linearly. The linear energy
transfer (LET) is a measure of the energy transferred to a material
per unit length as an ionising particle travels through it. For photons, it shows a poor distribution, but protons and other charged
particles show a very different depth-dose curve. They produce
the so-called Bragg peak, a sharp increase in the deposited energy
in the very last region of their trajectory, where they lose their
whole energy and the deposited dose falls to zero. Hadron therapy
uses collimated beams of hadrons to improve targeting and
⇑ Corresponding author.
E-mail addresses: [email protected], [email protected], [email protected] (M. Aslaninejad).
0168-583X/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.nimb.2010.10.013
achieve a larger effectiveness. When protons pass through a medium, they experience Coulomb interactions with the orbiting electrons and nuclei of the atoms in the medium. Electrons may be
released from the atomic shells and the atoms become ionised.
The freed electrons damage the DNA of the cells and can sometimes ionise other neighbouring atoms. As the electrons mass is
small compared to that of protons, the protons lose in this process
a very small amount of energy in a non-linear manner. In addition,
protons may be involved in other inelastic processes, such as excitation of the medium or charge transfer. All these processes should
be addressed in detail to gain a precise knowledge about the
behaviour of proton beam in interaction with the patient body.
The rate of change of energy as a function of traversed material
is given by the Bethe-Bloch formula, which according to Groom
and Klein [2] is:
"
#
1 dE
2me c2 b2 c2 T max
dðbcÞ
2Z 1 1
2
¼ Kz
ln
b A b2 2
2
q dx
I2
ð1Þ
where z shows the charge of the incident particle, Z and A are the
atomic number and the atomic mass of the absorber, respectively.
I is the mean excitation energy, dðbcÞ is the density effect correction
to ionisation energy loss and b and c are the usual relativistic factors. T max is the maximum kinetic energy which can be imparted
to a free electron in a single collision. It is important to note that
the deposited energy is inversely proportional to the square of the
velocity of the particle b. Upon entering the medium, the protons
have maximum energy and therefore, maximum velocity, but as
they traverse the medium they interact with the orbiting electrons,
lose energy and velocity and deposit more and more dose. Physically, the dependence on v arises from the time needed for a
Coulomb interaction to take place. If the particle moves faster, there
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C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 189–196
is less time for the electric fields of the projectile and the atoms in
the medium to interact, and thus less energy is deposited.
Although it is claimed that proton therapy is very efficient, this
is only true if the position and the dose can be calculated very
accurately. The greatest advantage of proton therapy, that is, the
concentration of the deposited energy on a small region of the
space, may be lethal if there is even a very slight shift in the position of the Bragg peak. Despite the fact that, one can obtain the
right position of the Bragg peak via Monte Carlo based simulation
packages like GEANT4, the full understanding of the whole processes contributing to the right height, width and position of the
depth-dose distribution requires analytical and/or semi-empirical
models. The present study investigates the physical processes involved in the passage of protons through matter, to study the accuracy in the determination of the position of the Bragg peak. We
follow the same techniques used by Dingfelder et al. [3] to study
the cross section for inelastic interactions of energetic protons with
liquid water. These cross sections are used for determining the linear energy transfer of ions against the depth of their penetration
according to the approach introduced by Obolensky et al. [4] and
Surdutovich et al. [5]. While, in Refs. [4,5], the main study is on carbon ions, we focus on proton ions. Armed with inelastic cross section models and data through Ref. [3], we can in detail account for
the different processes to which protons contribute, and study the
contribution and effects of each process. We expect to establish a
stable set of physical contributions which describes the depth-dose
distribution of a certain projectile. Previous works in this direction
have provided different models for the cross section terms, without
obtaining the final curve for the linear energy transfer. Moreover,
the literature does not combine the semi-empirical models for proton inelastic processes with effects like straggling and nuclear fragmentation, which we do here. In the current study, we analyse the
effects of the addition of each contribution to the position, height
and width of the Bragg peak, and compare with the Monte Carlo results obtained using GEANT4 [7] v.9.3, with the standard QGSP BIC
physics package. To summarise, we try to provide a deep and thorough analysis of the depth-dose distribution of protons through
liquid water through the basic physical phenomena.
The paper is organised as follows. In Section 2 we review and
reproduce the inelastic processes that contribute to the total cross
section for protons passing through liquid water. To this end, we
follow the approach and use data mainly from Refs. [3,6] to obtain
cross sections for processes like ionisation, excitation, stripping
and charge transfer. Through calculation of the total stopping cross
section and the linear energy transfer (LET) [4,5] for these phenomena we attempt to find the right position of the Bragg peak. On the
other hand, particle fragmentation and stochastic processes like
straggling, which characterise the height and the width of the peak
are also discussed in this section. In Section 3, we compare the calculated stopping cross section using the foregoing Rudd model [6]
with a more precise result obtained from Bethe-Bloch equation (1)
for high energies. Then we discuss the validity of different
approaches to find the right position of the peak. The results of
our study are contrasted with GEANT4 simulations. Finally, we
present a brief conclusion of our study.
2. The passage of protons through liquid water
2.1. Cross section of inelastic processes
We first review the passage of a single proton through liquid
water in detail. When the energetic ion enters the medium, inelastic cross sections are typically small, but they grow to reach a
maximum, namely the Bragg Peak, where the ion loses its energy
at the highest rate. For our analysis, we follow mainly the approach
Table 1
Inelastic interactions of the projectile with the medium resulting in different
contributions to the stopping cross section [3].
Contribution
Reaction
Excitation by the projectile
Ionisation by the projectile
Charge transfer
Stripping of Hydrogen
Ionisation by Hydrogen
p þ H2 O ! p þ H2 O
p þ H2 O ! p þ H2 Oþ þ e
p þ H2 O ! H þ H2 Oþ
H þ H2 O ! p þ e þ H2 O
H þ H2 O ! H þ H2 Oþ þ e
from Dingfelder et al. [3]. We initially consider impact ionisation,
which is the dominant process contributing to the energy loss
and resulting in the production of the secondary electrons, and
then add several other terms; namely excitation, charge transfer,
stripping and ionisation by hydrogen. Table 1 gives an outline of
the next steps.
The secondary electrons are not produced in the excitation process. The charge transfer phenomenon consists of the transfer of
electrons from the water molecules to the projectile in such a
way that it ends up becoming neutral. When the projectile is a proton, which is the case being considered in this study, charge transfer involves the transfer of one single electron. The hydrogen atom
formed can also ionise the medium or can become stripped, in
which case it loses the previously absorbed electron.
2.1.1. Ionisation by protons
The singly differentiated cross section, SDCS, is the basic quantity in the analysis of the ionisation process. It is the total ionisation cross section differentiated with respect to the energy, W, of
the ejected electrons and is dependent on the kinetic energy of
the projectile, T. We use a semi-empirical expression introduced
by Rudd and co-workers [6], which was derived by adjusting the
experimental data to Rutherford cross section, among other analytical models,
X 4pa2 Ni R 2
drðW; TÞ
0
¼ z2
dW
Bi
B
i
i
F 1 ðv i Þ þ F 2 ðv i Þxi
ð1 þ xi Þ3 ð1 þ expðaðxi xmax
Þ=v i Þ
i
ð2Þ
where the sum is taken over the five electron shells of the water
molecule, a0 ¼ 0:0529 nm is the Bohr radius, R ¼ 13:6 eV is the
Rydberg constant, Ni is the shell occupancy, Bi is the ionisation potential of the sub-shell i for water vapour, xi ¼ W=Bi is the dimensionless normalised kinetic energy of the ejected electron. The
dimensionless normalised velocity, v i , is given by [5]
vi
sffiffiffiffiffiffiffiffiffiffi
mV 2
¼
2Bi
ð3Þ
where m is the mass of the electron and V is the velocity of the projectile, which in the relativistic case is given by bc,
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffi v
!2
u
u
1
Mc2
t
b¼ 1 2 ¼ 1
c
Mc2 þ T
ð4Þ
M being the mass of the projectile.
Eq. (3) uses the kinetic energy of an electron having the same
velocity as the projectile, normalised by the corresponding binding
energy. The expressions for the functions in Eq. (2) are
lnð1 þ v 2 Þ
C 1 v D1
þ
2
2
B1 =v þ v
1 þ E1 v D1 þ4
2
A2 v þ B 2
F 2 ðv Þ ¼ C 2 v D2
C 2 v D2 þ4 þ A2 v 2 þ B2
R
¼ 4v 2i 2v i wmax
i
4Bi
F 1 ð v Þ ¼ A1
ð5Þ
ð6Þ
ð7Þ
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C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 189–196
Table 2
Parameter sets for the SDCS of protons on water (from Refs. [3,6]).
Parameter
Outer shells
A1
B1
C1
D1
E1
A2
B2
C2
D2
a
Inner shell
Liquid
Vapour
K-shell
1.02
82.0
0.45
0.80
0.38
1.07
14.6
0.60
0.04
0.64
0.97
82.0
0.40
0.30
0.38
1.04
17.3
0.76
0.04
0.64
1.25
0.50
1.00
1.00
3.00
1.10
1.30
1.00
0.00
0.66
eter set for liquid water by modifying the water vapour set until
the equations reproduced the recommended ICRU stopping cross
section for the desired material [8]. It must be noted that Rudd
and co-workers developed their model for vapour water, and took
the binding energies for this phase as parameters. Hence, even
though our approach is for liquid water, the binding energies used
in the SDCS (Eqs. (2), (3) and (7)) are those from water vapour.
By integrating the SDCS over the secondary electron energy, W,
the total cross section of impact ionisation by the ion with kinetic
energy T is obtained:
rðTÞ ¼
Z
W max
drðW; TÞ
dW
dW
0
ð10Þ
The maximum kinetic energy is
Eqs. (5)–(7) show original non relativistic expressions for Rudd
model. V 2 is calculated classically as 2T=M. The parameter set for
the protons SDCS used in this work can be found in Table 2 and
were calculated by Dingfelder and co-workers [3]. Surdutovich
et al. [5] introduce some modifications on F 1 which account for
the relativistic effects due to the high energies reached (up to
b 0:6),
F rel
1 ð v Þ ¼ A1
ln
1þv 2
1b2
b2
B 1 =v 2 þ v 2
þ
C 1 v D1
1 þ E1 v D1 þ4
ð8Þ
When we expand the Rudd’s model to relativistic energies, we
note that F 1 and F 2 tend to A1 lnðv 2 Þ=v 2 and A2 =v 2 , respectively.
Since F 2 approaches zero faster, F 1 absorbs the relativistic corrections. This results from the fact that as A1 A2 , for high velocity
ðv 1Þ, F 1 is always higher than F 2 . In particular, the new F rel
1
introduced by Surdutovich and co-workers [5] reproduces the
same asymptotic behaviour as Bethe-Bloch formula. In this work,
the Rudd model with relativistic corrections has been employed
to account for the ionisation process.
The parameter set for the SDCS of protons on water vapour is
also given in 1992 by Rudd et al. [6] and are also shown on Table
2. The division into inner and outer shells that Rudd suggests for
choosing one or another set of parameters has been accepted.
But, we followed the approach from Ref. [3], which consists of
separating the K-shell from the rest, as seen on Table 3.
In their investigation, Rudd and co-workers reviewed the
different approaches for the total cross sections and suggested this
semi-empirical approach. According to Ref. [3], the latter does not
reproduce accurately the stopping cross section for liquid water
due to the lack of a careful consideration of the water molecule’s
sub-shells. To overcome this, they suggest adding a partitioning
factor Gj which adjusts the contributions of the sub-shells to the
results from the Born approximation.
Note that in this case, the binding energy corresponds to that of
liquid water.
2.1.2. Excitation by protons
For the case of excitation of the molecules of liquid water, we
use a semi-empirical model which relates proton excitation cross
section to electron excitation cross section, developed by Miller
and Green [9]. The cross section for a single excitation of an electron in sub-shell k is given by the expression:
rexc;k ðTÞ ¼
r0 ðZaÞX ðT Ek Þm
ð12Þ
J Xþm þ T Xþm
where r0 ¼ 1020 m2 , Z = 10 is the number of electrons in the target
material and Ek is the excitation energy for the state k. a, X, J and m
are parameters (shown in Table 4), but they have physical meaning:
a(eV) and X (dimensionless) represent the high energy limit and J
(eV) and m (dimensionless) the low energy limit. We have taken
the values recommended by Dingfelder and his group, who in turn
used the original suggestions in Ref. [9].
2.1.3. Charge transfer
There exists the possibility that an electron from the water joins
the projectile, the proton in this case. A neutral hydrogen atom is
then formed either in a bound state or in a state in which the electron travels with the proton at the same velocity. This phenomenon is known as charge transfer. Again following Dingfelder’s
approach [3], who showed the contribution of the charge-transfer
cross section by an analytical formula:
r10 ðTÞ ¼ 10YðXÞ
ð13Þ
where X ¼ logðTÞ with T measured in eV and
þ ða1 X þ b1 ÞHðX x1 Þ
ð9Þ
In this work, we reproduce the previous expressions for liquid
water from Dingfelder et al. [3] directly, who obtained the paramTable 3
Ionisation energies (eV) for the different sub-shells in the water vapour (Bj ) and liquid
water (Ij ), the partitioning function Gj and the number of electrons N j . The data are
taken from Ref. [3].
Ij
Bj
Gj
Nj
ð11Þ
YðXÞ ¼ ½a0 X þ b0 c0 ðX x0 Þd0 HðX x0 ÞHðx1 XÞ
dr X drj
Gj
¼
dw
dW j
allj
Shell
W max;i ¼ T Ii
Outer shells
ð14Þ
where HðxÞ is the Heaviside step function and the other parameters
can be found in Table 5.
These parameters were obtained by considering the experimental data on water vapour [10–12] as a starting point and adjusting
Table 4
Set of parameters for the excitation cross section for liquid water, taken from [3].
Inner shell (K-shell)
k
Excited state
2a1
1b2
3a1
1b1
1a1
1
32.30
32.20
0.52
2
16.05
18.55
1.11
2
13.39
14.73
1.11
2
10.79
12.61
0.99
2
539.00
539.70
1.00
2
2
e 1 B1
A
e 1 A1
B
3
4
5
Ryd A + B
Ryd C + D
Diffuse bands
t
Ek (eV)
a (eV)
J (eV)
X
8.17
876
19820
0.85
1
10.13
2084
23490
0.88
1
11.31
12.91
14.50
1373
692
900
27770
30830
33080
0.88
0.78
0.78
1
1
1
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C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 189–196
Table 5
Parameter set for the charge transfer cross section, taken from [3].
−18
a0
b0
c0
d0
a1
b1
x0
x1
0.180
18.22
0.215
3.550
3.600
1.997
3.450
5.251
them to fit the contribution of the charge transfer to the total cross
section in the recommended values for liquid water. The subscripts
10, in the charge transfer cross section mean that the projectile
goes from the state 1 to the state 0, where 1 corresponds to a proton and 0 to a neutral hydrogen.
2.1.4. Stripping of neutral hydrogen
The neutral hydrogen atom can undergo stripping or electronloss. As before, not many experimental data exist for liquid water.
Hence we decided to use a semi-empirical formula given by Rudd
et al. [13] for the cross section. In this expression the harmonic
mean of a low energy and a high energy part is considered. Parameters used are those obtained by Dingfelder and his group to fit the
recommended values for liquid water.
r01 ðTÞ ¼
1
rlow
þ
1
1
rhigh
ð15Þ
:
The two parts of the stopping cross section may be calculated as
rlow ðTÞ ¼ 4pa20 C
sD
ð16Þ
Rh
s i
2 R
A ln 1 þ
rhigh ðTÞ ¼ 4pa0
þB
s
R
ð17Þ
where A, B, C and D are fitted parameters as given by Table 6.
R = 13.6 eV is the Rydberg constant and s is the kinetic energy of
an electron travelling at the same speed as the proton.
2.1.5. Ionisation by neutral hydrogen
The last effect to be taken into account is the ionisation by neutral hydrogen. Despite of the lack of direct experimental information on this process, Bolorizadeh and Rudd [14] concluded that
the ratio of the SDCS for hydrogen impact to the SDCS for proton
impact as a function of the energy of the secondary electron is
independent on this parameter when the projectile energy is fixed.
We can then use the SDCS for protons as the starting point and correct it by a factor gðTÞ that depends only on the particle energy T
(eV), that is:
total cross section (m2)
10
Ionisation (p)
Excitation (p)
Charge transfer (p)
Ionisation (H)
Stripping (H)
−19
10
−20
10
−21
10
2
3
10
10
ð18Þ
where the correcting function is adjusted by Dingfelder and his
group to fit the values for liquid water provided by ICRU 49 [8].
1
logðTÞ 4:2
gðTÞ ¼ 0:8 1 þ exp
þ 0:9
0:5
ð19Þ
5
10
6
10
7
10
Fig. 1. Total cross sections of the considered inelastic processes of protons in liquid
water: ionisation by protons and by neutral hydrogen, excitation by protons, charge
transfer and stripping.
higher for low energies of the projectile but it falls to zero very fast
for energies around 0.1 MeV. It is then expected that only for low
energies, the probability to find the projectile as a neutral hydrogen becomes important.
2.2.1. Probabilities of charge state and stopping cross section
The combination of the contributions to the total stopping cross
section has to be done in such a way that it takes into account the
state of the projectile. In the case of a proton, we have just considered two possible charge states: neutral hydrogen (H, i = 0) and
proton (p, i = 1). The total stopping cross section will be the sum
over all charged states i of the particle [3],
X
rst ¼
Ui ðrst;i þ rij T ij Þ
ð20Þ
i;j;i–j
where the term T ij denotes the energy loss in the charge changing
process, from state i to state j. Only one electron will be ripped
out from the water molecule in the charge transfer process, and
so we can suppose the most probable sub-shell to undergo the reaction is that with the smallest binding energy. This way we can
establish a lower bound for the sum T 01 þ T 10 as the ionisation energy of water I0 plus the kinetic energy of the electron,
T 01 þ T 10 ¼ I0 þ
dr
dr
¼ gðTÞ
dE hydrogen
dE proton
4
10
Incident particle energy, T (eV)
mv 2
2
ð21Þ
and the Ui is the probability that the projectile is in state i. This is
determined by the charge transfer and the stripping cross section:
r10
r01 þ r10
r01
U1 ¼
r01 þ r10
U0 ¼
ð22Þ
ð23Þ
Eq. (20) may now be rewritten as
2.2. From the total cross section to the LET
rst ¼ U0 rst;0 þ U1 rst;1 þ rst;CC
The total cross sections for the different processes considered
above for protons in liquid water are plotted in Fig. 1. From this figure, it can be observed that the charge transfer cross section is
where
rst;CC ¼
r01 r10
ðI þ sÞ
r01 þ r10 0
ð24Þ
ð25Þ
The stopping cross sections are calculated as
Table 6
Parameter set for the stripping total cross section, taken from [3].
rst;i ¼
A
B
C
D
2.835
0.310
2.100
0.760
Z
Emax
E
0
dri ðE; TÞ
dE
dE
ð26Þ
where Emax is the maximum energy loss by the projectile in the corresponding inelastic process.
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C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 189–196
results from the convolution of the previously CSDA LET with this
Gaussian range straggling distribution. The concept is known as
‘‘depth scaling’’ and was introduced by Carlsson et al. [19]. For each
point x in the trajectory, it is assumed that every particle in the
beam has an effective penetrated depth x0 . These effective depths
are distributed as a Gaussian centred on the actual penetrated
depth x, which coincides with the CSDA range,
−18
2
Stopping cross sections (eV m )
x 10
Total
Ionisation (p)
Charge changing
Ionisation (H)
Excitation (p)
PSTAR values
2.5
2
1.5
1
ðx0 xÞ2
pðx; x0 Þ ¼ pffiffiffiffiffiffiffi
exp 2
2rstrag ðxÞ
2prstrag ðxÞ
1
0.5
0 2
10
3
10
4
10
5
10
6
10
7
10
8
10
Kinetic energy of the projectile, T (eV)
Fig. 2. Total stopping cross section of protons through liquid water.
dT
¼ nrst ðTÞ
dx
ð27Þ
where n ¼ 3:343 1028 molecules=m3 is the number density of
water molecules.
The position of the projectile as a function of its kinetic energy
is obtained by integrating the inverse of LET [4,5],
xðTÞ ¼
Z
T
T0
0
dT
0
jdT =dxj
ð29Þ
where rstrag ðxÞ is the variance, which may be obtained from a phenomenological formula given by Chu et al. [20] and depends in general on x. Nevertheless the shape of the depth-dose distribution
allows us to assume that it is approximately constant, with
rðxÞ rðR0 Þ [21], since most of the curve is approximately flat
and will not be significantly affected by the convolution. The
expression for rstrag ðxÞ is then
rstrag ¼ 0:012x0:951 A0:5
Fig. 2 shows that from all the inelastic processes considered, for
high particle energies, only proton ionisation is noticeable. Approximately below 0.3 MeV contributions from the other processes appear and for 20 keV, ionisation by neutral hydrogen becomes the
most important process, followed by the charge changing process.
Note also that the energy loss by excitation is nearly negligible.
The total stopping cross section is compared with the PSTAR
values [15]. The reference data base for Stopping power and range
tables for protons (PSTAR) is provided by the National Institute of
Standard and Technology (NIST) of the US department of Commerce. These values are obtained, for high energies, through
Bethe’s stopping formula including shell corrections, Barkas and
Bloch corrections [16,17] and density effect corrections. The mean
excitation energy, I, used in PSTAR is taken from ICRU 37 [18],
corresponding to I = 75 eV for liquid water. For low energies,
experimental data in PSTAR are fitted to empirical formulas.
By definition, the linear energy transfer is proportional to the
stopping cross section,
!
ð30Þ
where A denotes the ion’s mass number [21]. Straggling has two
important consequences on the depth-dose distribution. Firstly, as
fewer particles arrive to the same final position, the dose in the peak
is considerably reduced. In addition, different particles have different ranges, and thus the peak is broadened.
Nuclear fragmentation accounts for the possibility that when
the projectile hits a target, it undergoes fragmentation. Two different effects need to be considered. The first one is the reduction of
the beam fluence, lowering the height of the depth-dose distribution at the peak. On the other hand, fragments can be produced
in the same direction of the original beam. If lighter elements are
formed, as having a longer range, they produce a tail in the
depth-dose distribution which goes clearly beyond the position
of the Bragg peak maxima. For protons, fragments are not important and so a tail is not observed. Hence, in this work we only include fragmentation by considering the reduction in the number
of particles of the beam, N. This model deals with a pencil beam
with a single projectile, so the concept of fluence loses its meaning
within this context (it would be a Dirac delta function). Instead, the
fragmentation effect is considered through the decay of the total
number of particles. If the projectile has probability p to react with
a water nucleus, then:
dNðxÞ
¼ pNðxÞ
dx
x
NðxÞ ¼ N0 epx ¼ N0 ek :
ð31Þ
ð32Þ
ð28Þ
where T 0 is the initial energy of the projectile. The parametric representation of LET versus depth is known as the depth-dose
distribution.
This process, used to calculate the energy transfer, is known as
Continuous Slowing Down Approximation (CSDA). In this approximation, the projectile is assumed to lose energy in a continuous
way. However, energy loss by a proton is a stochastic process,
which produces variations in the range of the particles. In addition,
the projectile might react with a water molecule and become fragmented. Then, for a real model we need to include two important
contributions: straggling and nuclear fragmentation.
2.2.2. Straggling and fragmentation
Straggling accounts for the stochastic fluctuations in the range
of the particles which traverse the medium. The straggling model
considered here claims that the ranges of the projectiles follow a
Gaussian distribution. The so-called broad-beam depth-dose curve
where k is called mean free path length and N 0 is the initial number
of incident protons. From the prediction of Ref. [22], we have
k ¼ 435 mm, for protons in liquid water.
The broad-beam depth-dose curve considering fragmentation
and straggling is therefore calculated as
Z R0
dT
dT
ðxÞ ¼ NðxÞ
pðx; x0 Þ 0 dx0
dx
dx
0
ð33Þ
The results of adding fragmentation and straggling are illustrated on Fig. 3. Different depth-dose distributions for various initial energies have been compared on Fig. 4. It can be observed that
not only the height of the peak, but also the entrance dose depends
on the initial kinetic energy of the projectile. Protons with higher
initial energies deposit less dose in the entrance since the cross
section increases as the velocity of the projectile decreases. In addition, projectiles with a longer range suffer more straggling and nuclear fragmentation and hence their Bragg peaks are shorter and
wider.
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C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 189–196
7
6
CSDA
CSDA + straggling
CSDA + straggling + fragmentation
5
GEANT4
This work (using PSTAR cross section)
This work (using the simulated cross section)
6
5
LET (AU)
LET (AU)
4
3
2
3
2
1
1
0
0
75
80
85
Depth, x (mm)
Fig. 3. Depth-dose distribution of protons in liquid water for T = 100 MeV,
considering the CSDA and adding straggling and nuclear fragmentation.
6
Linear Energy Transfer, LET (AU)
4
80 MeV
100 MeV
150 MeV
200 MeV
5
4
3
2
1
0
0
50
100
150
200
250
300
Depth, x (mm)
Fig. 4. Depth-dose distribution of protons in liquid water with initial energies of 80,
100, 150 and 200 MeV.
There is one additional process which has not been considered,
multiple scattering, which accounts for the divergence of the beam
as it travels along the matter. It has been shown that multiple scattering produces changes on the shape of the Bragg peak [23], especially on the amplitude and the peak to entrance dose ratio,
although not on the range. Usually, hadron therapy simulations
merely apply normalisation procedures to fit the experimental
data [24], as the effect is only due to a reduction of the fraction
of primary particles. To compare with experimental data, the current model normalises the depth-dose distribution to the experimental entrance dose and so scattering is not included.
3. Results and discussion
GEANT4 provides us with data for the depth-dose distribution
of protons in liquid water. For an initial energy of 100 MeV, which
is within the usual range in proton therapy, the depth-dose distributions found through the different methods are shown on Fig. 5.
The solid line shows the result from GEANT4. The position, height
and width of the Bragg peak obtained with our approach, but using
PSTAR values for the stopping cross section, is in excellent agreement with the result from GEANT4. However, using the values
for the stopping cross sections obtained semi-empirically gives rise
to a shift of the curve. The height of the peak is correct, as the straggling and fragmentation contributions remain the same, but the
0
10
20
30
40
50
60
70
80
90
100
Depth, x (mm)
Fig. 5. Bragg peaks obtained with different methods (GEANT4, our program and our
program using PSTAR values) compared. The curves have been normalised to the
entrance dose for a better comparison.
peak appears around 5 mm further away. This distance increases
to around 1 cm if Rudd model without relativistic corrections is
used.
A first look at the stopping cross section distribution on Fig. 2
shows some differences between the PSTAR and the calculated values for energies below 0.5 MeV, which could be considered in principle to be the cause of the shift. However, a physical analysis of
the whole picture of protons going through the matter shows the
real problem. As we have already mentioned, highly energetic protons have a lower probability to interact with the matter and hence
the energy per unit length they deposit is also lower. When their
energy decreases (approximately below 1 MeV) the linear energy
transfer increases considerably. They lose their whole remaining
energy in a very short distance, usually only a few millimetres.
Therefore, the final range of the projectile is mostly determined
by the distance it travels before reaching the 1 MeV threshold.
From Fig. 5, we observe a difference about 5 mm in the position
of the peaks. This shift is too large to result from an error in the
stopping cross section for energies below 1 MeV. Neither Rudd
equations for the ionisation cross section nor the corrections introduced by Refs. [3] or [4] give the precision required in the high energy limit (see Eq. (8)) of the stopping cross section to obtain the
right position of the Bragg peak.
As discussed on Section 1, Bethe-Bloch equation is believed to
give the right values in the high energy limit. As this equation includes ionisation and excitation by the projectile, and these are the
only contributions for energies above 1 MeV (see Fig. 2), it is worth
using Bethe-Bloch equation for that range, keeping the semiempirical models introduced in Section 2 for energies below
1 MeV. Fig. 6 shows the stopping cross section using this remedy,
which appears to be very similar to Fig. 2, as expected. However,
a closer look for high energies shown on Fig. 7, presenting the total
stopping cross section from the combined semi-empirical and
Bethe-Bloch model using a linear scale, proves otherwise.
We observe that the Rudd equation with relativistic corrections,
which seemed precise on Fig. 2, turns out to deviate from the
PSTAR results by about 5%. The discrepancy is even higher (around
17%) if no relativistic correction is considered. What we now show
on Fig. 7 is a more precise version of the stopping cross sections
plots. The observed error in the position of the Bragg peak is now
understood through comparison with Bethe-Bloch formula in this
limit. Note that only the first two terms in Eq. (1) are needed in this
approach to obtain the desired precision in the final results.
Fig. 8 shows the new Bragg peak, obtained by using the semiempirical model for energies below 1 MeV and the Bethe-Bloch
correction above this energy. It is compared with the Bragg peak
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C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 189–196
−18
x 10
2
T. Bortfeld
This work
2
0
2.5
Range, R (mm)
Stopping cross sections (eV m )
300
PSTAR values
Total (p)
Ionisation (p)
Charge transfer
Ionisation (H)
Excitation (p)
1.5
150
100
0.5
50
80
3
4
10
5
10
6
10
7
10
2
PSTAR values
Semi−empirical model
Semi−empirical (T<1 MeV)
& Bethe−Bloch formula (T>1 MeV)
2
3
4
5
6
7
8
9
10
7
x 10
Kinetic energy of the projectile, T (eV)
Fig. 7. Stopping cross section of protons in liquid water using two different models
(Bethe-Bloch formula and the semi-empirical formulae) compared with the PSTAR
results.
7
GEANT4
Semi−empirical
Semi−empirical + Bethe−Bloch approach
6
5
4
3
2
1
0
0
10
20
30
40
50
60
70
80
90
160
180
200
220
Fig. 9. Range of protons in liquid water as a function of its initial kinetic energy. The
range given using the semi-empirical model and the Bethe-Bloch formula is
compared with the range given in Ref. [26], which fits experimental results.
R0 ¼ aT p0
0.5
0
1
140
I = 78 eV is used in our simulation, which is the one employed by
GEANT4 version 9.3 and recommended at the revised tables of
ICRU 73 [25].
Through applying this new approach for the total stopping cross
section, the right position of the depth-dose distribution of protons
in the liquid water in the energy range for proton therapy (from 70
to 250 MeV) is achieved.
Bortfeld [26] shows that the relation between the range, R0 , and
the initial kinetic energy of the projectile, T 0 , is given by:
−19
x 10
1
120
0
10
Fig. 6. Stopping cross section of protons in liquid water considering the semiempirical model for T < 1 MeV and Bethe-Bloch formula for T > 1 MeV.
100
Initial kinetic energy of the projectile, T (MeV)
8
10
Kinetic energy of the projectile, T (eV)
Stopping cross sections (eV m )
200
1
0 2
10
LET (AU)
250
100
Depth, x (mm)
Fig. 8. Depth-dose distribution of protons in liquid water for T 0 ¼ 100 MeV,
considering the semi-empirical model, the Bethe-Bloch equation for T > 1 MeV and
GEANT4. The curves have been normalised to the entrance dose for a better
comparison.
obtained by the semi-empirical formulae in the whole energy
range. These plots are also compared with the results of the simulation from GEANT4. Note that Bethe-Bloch formula uses the mean
excitation energy of the medium as a parameter (see Eq. (1)). In order to have consistency in the following comparisons, the value
ð34Þ
where a ¼ 0:022 mm MeVp and p is around 1.77 for the 70–250
energy range.
Eq. (34) is constructed so as to fit the experimental results. Fig. 9
shows the range of protons in liquid water for different initial kinetic energies using Eq. (34) and the current model. We observe
a great agreement for the range of protons in liquid water.
We have therefore obtained an explanation for the passage of
protons through the matter which reproduces accurately the results from Monte Carlo simulations. In order to further address
the physics behind the Bragg peak features, we now analyse the
individual contribution of the different inelastic processes to the
depth-dose curve.
For a typical proton beam with initial kinetic energy of
100 MeV, it is observed that if we neglect stripping, charge transfer
and also hydrogen ionisation terms, the change produced in the
position of the peak is lower than 1 mm; a negligible distance. In
addition, the linear energy transfer deposited in the peak is reduced around 0.005%. If only the excitation term is removed and
the other contributions are all kept, we observe a shift in the position of the peak of around 1.5 mm, whereas, the change in the
amount of energy deposited per unit length is negligible. Finally,
removing just proton ionisation makes the Bragg peak vanish.
This shows that, from all the physical processes taking place in
this situation, only proton ionisation and excitation (the latter in a
smaller scale) contribute to the properties of the Bragg peak. Thus,
the other terms, affecting only low energies, produce all together
solely a very small variation in the height and width of the peak.
4. Conclusion
In this study, we explain qualitatively and quantitatively, the
processes which affect the passage of a proton beam through liquid
water. Different effects contributing to the total stopping cross section including ionisation and excitation of the medium, charge
transfer, stripping and ionisation by the neutral projectile, have been
taken into account. We tried to find a consistent semi-analytical
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C. Candela Juan et al. / Nuclear Instruments and Methods in Physics Research B 269 (2011) 189–196
approach to obtain a good numerical approximation. The role of the
most relevant processes which contribute to the total cross section
for protons is explained. From the stopping cross section, we
obtained several depth-dose distributions and compared them with
the results from GEANT4 Monte Carlo simulations, which have been
shown to be in agreement with experimental data. We obtain a
realistic height for the peak via phenomena such as straggling
and fragmentation. We show that the small error in the values of
stopping cross section, which has been overlooked in other studies
and phenomena leads to a considerable shift in the position of the
Bragg peak. This means that Rudd model with relativistic corrections for high energy protons is not precise enough to obtain the
right position of the Bragg peak in the therapeutic energy range.
For high energies, corrections from the Bethe-Bloch equation gives
rise to more precise results for the stopping cross section.
The PSTAR data for the stopping cross section and the correction
from Bethe-Bloch equation result in the right position of the Bragg
peak. We note that the position of the peak is mainly determined
by the stopping cross section for high energies. Low energies also
contribute to the position of the peak, but in a very small scale.
Changes in the stopping cross section for small energies only produce variation in the height of the peak of some eV/mm. For hadron therapy precise knowledge about the different processes
contribution to the height, width and the position of the peak is
of vital importance.
The model presented here for the stopping cross section (BetheBloch corrections for high energy terms like ionisation and excitation, and low energy semi-empirical contributions for charge
transfer, stripping and ionisation by Hydrogen) gives the right position for the Bragg peak. The right values for the height and the
width of the peak are also achieved from fragmentation and straggling phenomena.
The present work tries to continue the path towards the full
semi-empirical derivation of the characteristics of the Bragg peak
without having to use Monte Carlo methods such as GEANT4, whose
simulations take a considerably long time. This would allow medical
physicists to perform individual simulations for each patient, taking
into account the particular inhomogeneities of each case, which are
the main sources of divergence with the average results.
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