Proton Range Calculations: Monte Carlo Simulations and Analytical
2.1 Proton Range Calculations with Monte Carlo Simu- Simu-lations
2.1.4 Comparison of the Results from the Different Simulations
The values for proton range, range straggling, transverse beam spread and the fraction of nuclear interactions have been obtained through simulations with the three MC programs
2.1 Proton Range Calculations with Monte Carlo Simulations 31
Figure 2.2:The relationship between the proton beam’s energy loss (light gray), and the protons’ final stopping positionR(green). Shown is also the range stragglingσRand protons undergoing nuclear interactions. MC simulation results from a 200 MeV beam in water, MC simulated using GATE.
for the three different geometries: they are listed below.
Proton Ranges
Table 2.3 lists MC simulatedRvalues for a few selected initial primary proton energies, as well as the correspondingRPSTARvalues. In Fig. 2.4 (left), the range deviation∆(R) is shown: For water and aluminum,∆(R)m,i =Rm,i−RPSTARis shown, wheremis the medium,iis the MC program and “PSTAR” is the corresponding projected range from the PSTAR database. For the detector geometry, the deviation from the average results,
∆(R)D,i=RD,i−∑
jRD,j/3is shown, as no accurate experimental values are available.
The largest deviation for water,∆(R)w, is less than 1.7 mm (0.5% of the range) and
∆(R)Alis below 0.2 mm (0.13% of the range). For the detector geometry,∆(R)Dis below 0.2 mm (0.15% of the range). While FLUKA and GATE match each other quite well in water, and their∆(R)wvalues both are below 0.5 mm, results from MCNP6 show a larger range deviation as a function of increasing initial proton energy.
The Value of the Ionization Potential for Water
In Fig. 2.4, it is observed that the deviation∆(R)w,MCNP6is large compared to the∆(R) values of GATE and FLUKA. A possible cause for this divergence is the ionization
po-32 2. Proton Range Calculations: Monte Carlo Simulations and Analytical Models
Water 100 77.0 76.8 77.0 77.1
150 157.3 156.9 157.3 157.6
230 328.7 327.4 328.6 329.1
50 10.8 10.8 10.9 10.8
Aluminum 100 37.0 36.9 37.1 37.0
150 75.0 75.0 75.3 75.1
230 155.8 156.1 156.3 156.0
50 11.1 11.1 11.1
-Detector 100 37.9 38.0 37.9
-150 76.8 76.8 77.1
-210 137.0 137.2 137.3
-Table 2.3:The proton ranges from MC simulations and PSTAR data for 50, 100, 150 and 230 MeV primary proton energies in water, in aluminum and in the detector geometry.
The maximum energy applied for the simulations with the detector geometry is 210 MeV.
tential of water,Iw, which is an important parameter in estimating the range of protons in lowZmaterials (Newhauser and Zhang, 2015). Five separate GATE simulations with varyingIwvalues were performed, and the resulting ranges are compared to the range predicted by MCNP6: Figure 2.3 shows different curves forRw,GATE−Rw,MCNP6. Where applicable, MCNP6 uses the ICRU49-recommended values for theIof a material (Wyck-off, 1993; ICRU, 2016). For composite materials, MCNP6 uses the Bragg Additivity rule (Thwaites, 1983) to calculateI. The deviationRw,GATE−Rw,MCNP6is smallest for Iw=73 eV.
Proton Range Straggling
The obtained results for the range stragglingσRfor some selected primary proton energies are listed in Table 2.4 and the complete MC simulation results are displayed in Fig. 2.4 (right). The values are compared to Janni (available for water and aluminum), and to each other.
The results from all three MC programs show a similar amount of range straggling, with a maximum difference between the MC programs for water at 0.48 mm (12.5% of σR), and for aluminum of 0.08 mm (4.5% ofσR). The largest deviation compared to Janni is seen with the MCNP6 result, perhaps due to theIwvalue as discussed above.
The MC results in water agree well with the PSTAR values (within 0.1% on average),
2.1 Proton Range Calculations with Monte Carlo Simulations 33
Initial energy [MeV]
50 100 150 200 250
Range deviation MCNP - GATE [mm]
−1
−0.5 0 0.5 1 1.5
75 eV
74 eV
73 eV
72 eV
71 eV
Figure 2.3:Range deviation between MCNP6 and five separate GATE simulations. The GATE simulations are made using different values of the ionization potential in water, varying between 71 eV and 75 eV.
and within 4% on average in aluminum. For the detector geometry, the largest difference between the MC programs is 0.24 mm (13.7% ofσR). A higher variation in the range straggling is observed in the detector geometry, this is perhaps due to its longitudinal structure with different materials, with varying densities and material composition along the longitudinal axis.
Transverse Proton Beam Spread
The obtained results for the transverse beam spread, σx/R, are listed in Table 2.5 for some selected primary proton energies. Curves for σx/Rare shown in Fig. 2.5 (left).
There is a good agreement between MCNP6 and FLUKA. However, the results from the GATE simulations are 5%–20% lower compared to the average results from the other MC programs. This can also be seen in Fig. 2.6, where the lateral beam profiles of a 120 MeV proton beam incident on water are compared with respect to the three MC programs.
34 2. Proton Range Calculations: Monte Carlo Simulations and Analytical Models
Figure 2.4:The range deviation in different materials (left). *Range deviation: In water (top left) and aluminum (middle left), the range deviation is the deviation between MC and the PSTAR data. In the detector geometry (bottom left), it is the deviation from the average results from the three MC programs. The range straggling is also shown (right), with corresponding values from Janni for water and aluminum.
2.1 Proton Range Calculations with Monte Carlo Simulations 35
Water 100 0.87 0.93 0.91 0.91
150 1.70 1.90 1.79 1.79
230 3.36 3.85 3.57 3.57*
50 0.16 0.15 0.17 0.14
Aluminum 100 0.45 0.47 0.48 0.44
150 0.87 0.93 0.92 0.86
230 1.73 1.78 1.81 1.70*
50 0.16 0.14 0.17
-Detector 100 0.48 0.42 0.49
-150 0.88 1.02 0.98
-210 1.53 1.64 1.60
-Table 2.4: The range straggling values,σR, from MC results and data from Janni. *The 230 MeV values from Janni are interpolated using a spline approach. The maximum energy applied in the simulations of the detector geometry is 210 MeV.
Material Energy
Table 2.5: The transverse beam spreadσx/Rfrom the MC programs. The maximum energy applied in simulations of the detector geometry is 210 MeV.
36 2. Proton Range Calculations: Monte Carlo Simulations and Analytical Models
xxx
Figure 2.5: The transverse beam spreadσx/R, calculated in water (top left), in alu-minum (middle left) and in the detector geometry (bottom left). The fractions of nuclear interactions,fNI, are displayed in the right figures for the same geometries together with the data from Janni.
2.1 Proton Range Calculations with Monte Carlo Simulations 37
Proton stopping position in x [mm]
−2 −1.5 −1 −0.5 0 0.5 1 1.5 2
Number of protons
10 100 1000
GATE MCNP FLUKA
Figure 2.6:The lateral profile of the Bragg Peak for the three MC programs. The lateral beam spread obtained from simulations with GATE is less compared to the other MC programs, a similar similar to the trend seen in Fig. 2.5 (left).
Fraction of Nuclear Interactions
The simulated results for the fraction of nuclear interactions,fNI, for some selected en-ergies as well as corresponding data from Janni are collected in Table 2.6 and curves for all energies are shown in Fig. 2.5 (right).
The largest relative deviations offNIwhen compared to the data from Janni are 7.5%
for water and 6.9% for aluminum. The MC results are on average 6% higher than Janni in water, and 1.3% lower than Janni in aluminum. For the detector geometry, the largest deviation between the MC results is 6.2%.