SENSITIVITY OF THE BRAGG PEAK CURVE TO THE AVERAGE
IONIZATION POTENTIAL OF THE STOPPING MEDIUM
J. SOLTANI-NABIPOUR1, D. SARDARI2, GH. CATA-DANIL3
1
Physics Department, University Politehnica of Bucharest, Romania
E-mail: jsoltani@physics.pub.ro
2
Faculty of Engineering, Science and Research Campus, Islamic Azad University, Tehran, Iran
E-mail: d.sardari@seiau.ir
3
Physics Department, University Politehnica of Bucharest and “Horia Hulubei” National Institute for
Physics and Nuclear Engineering, Bucharest-Magurele, Romania
E-mail: cata-danil@physics.pub.ro
Received August 12, 2008
Heavy ions accelerated up to energies of hundreds MeV/A are suitable for
treatment of malignant tissues that are deep seated, located close to critical organs and
poorly responding to conventional therapies such as those based on photons and
electrons beams. The accurate estimation of the Bragg peak position of these charged
particles is important for evaluation of the spatial distribution of the energy deposition
in biological targets. The human body contains soft (muscles and fat) and hard tissues
(bones). In the present study we modeled them by water and calcium targets
respectively. One of the parameters required for a precise calculation of the Bragg
peak position is the mean ionization potential (<I>-value) of the stopping medium. In
this paper, we performed a numerical study of the sensitivity of the relative depthdose profile to the mean ionization potential of the stopping medium. It is provided a
numerical estimate of the precision required for the <I>-values. As a Monte Carlo
simulation tool we employed the computer code FLUKA.
Key words: mean ionization potential, hadrontherapy, Monte Carlo simulation,
FLUKA code.
1. INTRODUCTION
The occurrence of the Bragg peak makes heavy charged particle beams
useful tools for treatment of the deep-seated tumors in living organisms [1]. The
advantages of the therapies based on proton and heavy ion radiation over the
conventional photon and electron ones are due to a better physical dose
distributions achievable by inversed depth dose profile at the end of their range,
tumor-conform treatment, and the radiobiological characteristics of heavy ions [1].
Rom. Journ. Phys., Vol. 54, Nos. 3–4, P. 321–330, Bucharest, 2009
J. Soltani-Nabipour, D. Sardari, Gh. Cata-Danil
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2
Since the human body contains mostly water, the knowledge of depth dose
profile of charged particles in this stopping medium is of great importance for an
accurate treatment planning. It is known that the range of energetic ions depends on
the mean ionization potential (<I>-value) of the stopping medium [2]. It is the goal of
the present paper to perform a systematic study on the sensitivity of the Bragg curve
to different <I>-values reported in the literature. As a working tool we employed
numerical Monte Carlo simulations based on the FLUKA computer code [3].
We developed detailed stopping simulations in water and calcium thick
targets, selected as simple models for soft and hard biological tissues, respectively.
As bombarding projectiles we considered ions of 12C and 16O with energies of
290 MeV/A. The default FLUKA values are <I>=75eV for water and <I>=191eV
for calcium. In our calculations we changed the <I>-values in order to cover the
range of the experimental uncertainties (whenever these are known) and to obtain a
relevant sensitivity curve for the Bragg peak position. Based on these curves we
estimated the accuracy required for the expected determination of the <I>-values.
2. STOPPING POWER OF THE HEAVY CHARGED PARTICLES AND
THE <I>-VALUES
Charged particles interact dominantly via inelastic collision with the electrons of
the stopping media. The energy loss versus distance is described by the BetheBloch formula [4, 5, 6]:
−

dE 4πe 4 Z 2 2  2mν 2
=
Z1 ln
− ln (1 − β2 ) − β2  + ( relativistic term )
2
dx
me ν
 <I >

(1)
where me is the rest mass of the electron, v the velocity of projectile, β = v/c, Z1
the atomic number of the projectile, Z2 the atomic number of the target and <I>
the mean ionization potential of the target atoms. The Bethe-Bloch analytical
formula gives a rough estimation of the energy deposition by charged particles in
matter and can be used to provide an estimation of the Bragg peak position. Precise
predictions of the entries Bragg curve can be obtained by numerical Monte Carlo
simulation as for example those reported in ref. [7].
In previous work we calculated the stopping of a 12C ions beam with
290 MeV/A in liquid H2O and natCa and we found penetration depths of 16.28 cm
and 13.22 cm respectively [7]. These estimations considered mean ionization
potentials for water 75 eV and for calcium 191 eV, the default <I>-values of
FLUKA. The <I>-value of 75eV for water was suggested by the publications ICRU
37 and 49 and are accepted for long time [8, 9, 10].
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Ionization potential of the stopping medium
323
As shows in (1), the energy loss by charged particles has reverse correlation
with <I>-value of the medium. The theoretical calculation of the <I>-values is
presented in several review papers [11–13]. The average ionization potential is
defined [10] as the average value of the excitation energies over all atomic states
( Ei ) weighted by their transfer probability to continuum ( fi ):
ln I = ∑ f i ln Ei
(2)
i
where fi is the optical dipole oscillator strength for 0 → Ei transition and Ei is the
n-th level energy. The probabilities fi are unknown for most materials other than
hydrogen. The following approximate empirical formulas can be used to estimate
the <I>-value (in eV) for an element with atomic number Z [15]:
I ≈ 19.0
Z =1
I ≈ 11.2 + 11.7 ⋅ Z
I ≈ 52.8 + 8.71 ⋅ Z
2 ≤ Z ≤ 13
Z > 13
(4)
For a compound or a mixture as water, the <I>-value can be calculated by adding
the separate contributions from the individual constituent elements:
n ln I = ∑ N i Z i ln I i
(5)
i
where Zi is the atomic number for the ingredient “i” in compound, Ni is its relative
atomic number in compound, n is the total number of electrons in the compound
n = ∑ i N i Z i and Ii is the ionization potential of the ingredient “i” gives by (4).
By applying Eqs. (4) and (5) the average ionization potential for water was 74.6 eV, a
value closed to the ICRU recommendation (75eV). Table 1 gives an overview of
available (measured or calculated) <I>-values for water. The value <I>=75 eV
from ICRU reports 37 and 49 has been used as a norm for many years [8, 9]. <I>value of the new ICRU report 73 for water is 67.2 eV which is rather low in
comparison to all other recommendations made before and this <I>-value was
obtained from oscillator strength spectra [16]. More recent measurements made by
Bichsel et al. [17] provide a value of 79.7 ± 2 eV, and the evaluation of dielectric
response functions by Dingfelder et al. measured a <I>-value of 81.8 eV [18].
Based on Bragg peak position, Kramer et al. [19] suggested a <I>-value of 77 eV.
Emfietzoglou et al. [20] suggested different <I>-value in the range 80–85 eV.
Based on the most recent experimental determinations, H. Paul et al. [10]
suggested a newer <I>-value for water at 80.8±2 eV.
(
)
J. Soltani-Nabipour, D. Sardari, Gh. Cata-Danil
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Table 1
Survey of the water <I>-values reported in the literature
I(eV)
75.0±3
67.2
79.7±2
81.8
77
80–85
80.8±2
References
ICRU Report 37and 49
ICRU Report 73,effectively as used for stopping tables
Bichsel et al. [17], based on stopping experiment
Dingfelder et al. [18], based on optical absorption
Krämer et al. [19], based on Bragg peak position
Emfietzoglou et al. [20], based on optical loss functions
H.Paul et al. [10], based on Bichsel et al. [17] and Dingfelder et al. [18]
As shown in Table 1, the differences on <I>-values reported for water are
quite large and should be properly considered in high accuracy dosimetry
calculations.
3. RESULTS AND DISCUSSION
For the simulations reported in the present work we considered the target as a
single block of water or calcium with a shape of rectangular parallel pipe (RPP
geometry). The FLUKA default value for the mean ionization potential of water is
75 eV as discussed in section 2. The energy threshold for scoring in the Monte
Carlo approach was set at 10 keV. To achieve good statistics the simulations were
run with 105 primaries for 12C and 16O beams. The binning was set to 1 with 2 cm
width in the X and Y axis, and 500 with 0.1 cm width in the Z direction.
3.1. BEAMS OF
12
C STOPPED IN WATER
We calculated the variation of the Bragg curve with the mean ionization
potential of water. In this calculation we chose 12C ions beam with 290MeV/A
energy impinging on a thick water target. In the current simulation approach we
changed the <I>-value of water from 70 eV to 85 eV in order to fully cover the
range of the experimental uncertanties. From Figure 1 it can be observed that the
position and the profile of the Bragg curve suffers significant modifications as a
result of these changes.
We obtained different Bragg profiles in the range of the exprimental
uncertantiy of <I>= ±3 eV [8,9] and the Bragg peak position at 16.2 cm by using
the default <I>-value of water used by FLUKA. By varying <I>-values from 72 eV
to 78 eV, the Bragg peak position shifted from 16.15 cm to 16.50 cm. As shown in
Figure 1, our calculations indicate a higher relative dose for <I>=78 eV than
<I>=72 eV. The highest relative dose occured at 70 eV corresponding to 16.1 cm
penetration depth of the 12C ions beam in water.
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Ionization potential of the stopping medium
325
Fig. 1. – Relative dose deposited in a thick target of water by a 290MeV/A 12C beam as a function of
the penetration depth for different mean ionization potential of water. Simulations were performed by
FLUKA code. The experimental values of <I> are indicated by open squares and the corresponding
uncertainty by horizontal lines.
Fig. 2. – The evolution of the Bragg peak position (average range) at stopping of 12C ions beam in
water for different <I>-values.
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J. Soltani-Nabipour, D. Sardari, Gh. Cata-Danil
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In Figure 2 is presented the variation of Bragg peak position (range) of
C ions beam with the average ionization potential of water. It can be observed
that by changing <I>-value from 70 eV to 90 eV the range increase from 15.8 to 16.6
cm. The calculations have been conducted for the <I>-values reported in the survey
from Table 1. From our calculatios can be concluded that a variation of ~1.5 mm in
range is obtained as a result of the present knowledge on the experimental
uncertainties of the <I>-values.
12
3.2. BEAMS OF
16
O STOPPED IN WATER
In order to reveal the features of the stopping process dependence on the
nature of the bombarding particle we performed simulation for the case of 16O ions
beam. The stopping medium was chosen also water. With the default <I>-value of
FLUKA foe water, we calculated the Bragg peak position for stopping the
290 MeV/A 16O ions beam at 12.3 cm. As shown in Figure 3 we changed <I>value of water from 60 eV to 90 eV in steps of 5 eV. By increasing the <I>-value
the position of the Bragg peak shifted to higher penetration depths but the relative
dose decreases. A prominent difference relative to the case discussed in section 3.1
is the abruptly decreases of the relative dose at 75±15 eV, while the Bragg peak
positions at 90 eV and 60 eV are 12.5 cm and 11.9 cm, respectively.
Fig. 3. – Relative dose deposited in water by a 290 MeV/A 16O ions beam, as a function of the
penetration depth at different <I>-values. Simulations were performed with the FLUKA code.
As shown in Figure 4 we extracted from our simulations the ranges of the 16O
ions beam in a thick water target by varying <I>-value from 60 eV to 90 eV.
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Ionization potential of the stopping medium
327
Fig. 4. – The evolution of the Bragg peak position (average range) at stopping in water for different
<I>-values of water.
The calculations show that the range of 16O ions beam changes from 11.7 cm
to 12.42 cm for <I>-values from 60 eV to 90 eV respectively. According to our
knowledge there are no experimentally data for this case reported in the literature,
our calculations having only a predictive character.
3.3. BEAMS OF
12
C AND
16
O IONS STOPPED IN NATURAL CALCIUM
As a primitive model for the hard tissues, we chose nat.Ca as a target. The
isotopes included in this structures are 40Ca(96.941%), 42Ca(0.647%), 43Ca(0.135%),
44
Ca(2.086%), 46Ca(0.004%), 48Ca(0.187%). By using the default <I>-value for
calcium used in the FLUKA code, we calculated depth dose profle at stopping of
290 MeV/A 12C ions in a thick calcium target. The results of these calculations,
presented in Figure 5, show the Bragg peak position at 13.22 cm depth. In order to
conduct a sensitivity analysis, we changed the <I>-value of calcium from 175 eV to
205 eV by steps of 5 eV. As shown in Figure 5, the relative dose for <I> in the range
175 eV to 205 eV is nearly constant. A 9% decrease and a widening of the Bragg
curve can be observed at <I>=200 eV. At interaction of the 12C ions beam with
calcium there is less variation in the hight of the relative dose as compared with the
calculation discussed in the previous section.
In order to reveal the influence of the projectile type, we changed the beam to
16
O ions keeping constant all other parameters. The results of these calculations are
presented in Figure 6. The calculations indicate the position of the Bragg peak at
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J. Soltani-Nabipour, D. Sardari, Gh. Cata-Danil
8
Fig. 5. – Relative dose in a calcium thick target due to a 290 MeV/A 12C ions beam, as a function of
the penetration depth at different mean ionization potentials of natCa. Simulations were performed by
FLUKA code.
Fig. 6. – Relative dose in calcium due to a 290 MeV/A 16O beam, as a function of the penetration depth at
different mean ionization potential of natCa. Simulations were performed with the FLUKA code.
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Ionization potential of the stopping medium
329
9.5 cm by using the default <I>-value for the natCa. By changing the <I>-value in
the range [185–195] eV, the Bragg peak positions of 16O ions beam reached at
penetration depths between 9.4 and 9.8 cm. A prominent differences from the case
presented in Figure 5 is due to the abruptly decrease on the peak relative dose at
<I> equal to 185 eV and 195 eV. At <I>=200 eV the decrease is 1.39 times larger
than in Figure 5. At the <I>-values equal 191±15 eV, the peak relative doses are
similar with the 12C case but the depth dose profile is different.
4. SUMMARY AND CONCLUSIONS
In the present study we investigated the sensitivity of the Bragg peak position
and the shape of the Bragg curve on the average ionization potential (<I>-value) of
water and calcium employed as primitive models for soft and hard body tissues
respectively. The study has been conducted for 12C and 16O bombarding projectiles
with the incident energies of 290 MeV/A. As mentioned in section 2, in the
literature there are reported different <I>-values for water which a survey of this
data is presented in Table 1. Correspondingly, we employed these <I>-values in the
FLUKA simulations in order to obtain the depth dose profiles represented in
Figures 1, 3, 5 and 6. The FLUKA default <I> value for water is 75 eV. Our
simulations cover a <I>-values range from 60 eV to 90 eV. A basic result from our
calculatios is that a variation of ~1.5 mm in range is obtained in the conditions of
the present uncertainties of the <I>-values. Further experimental and theoretical
work is required for improving the accuracy in the <I>-values required for
predicting the Bragg peak position in bio-materials, an important ingredient for
accurate treatment plannings. There are not known experimental <I>-values for
natural calcium. Therefore, in our calculations we changed the default <I>-value of
FLUKA in a relative range closed to those used in the water case. At some <I>value abrupt changes in the peak values and shape of the Bragg curve has been
observed mostly for 16O ions beam. In general a smooth behavior is observed for
12
C ions stopped in the natural calcium targets.
In conclusion, our Monte Carlo simulations indicate that more experimental
work is required in order to find solid <I> values required by the numerical
predictions of the heavy ions Bragg peak position in the body tissues with
accuracies better than mm.
Acknowledgements. We acknowledge Dr. Ana Maria Popovici for helpful discussions related
with this work.
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