3 Monte Carlo tools for proton dosimetry

Transcription

1 3 Monte Carlo tools for proton dosimetry 3.1 Introduction As is clear from the previous chapter the problems in proton radiation transport are difficult to solve analytically, because of the statistical fluctuations and the large number of interactions involved. Therefore already since the Manhattan project in World War II automatic calculating machines were of interest. The Monte Carlo (MC) technique which has been developed during this project has been applied during the last 5 decades in a broad range of fields. In the context of radiation transport, MC techniques are those which simulate the random trajectories of individual particles by using machine-generated (pseudo-)random numbers to sample from the probability distributions governing the physical processes involved. By simulating a large number of histories, information can be obtained about average values of macroscopic quantities such as energy deposition. It is also possible to use the MC technique to answer questions which cannot (easily) be addressed by experimental investigation, such as What fraction of the energy loss is caused by nuclear reaction products? or How big is the backscattering effect of B-electrons?. A large number of MC codes have been developed in the past for application in nuclear/high energy physics. The physics at low energies ( 250 MeV) was not treated in sufficient detail, however, to be of use in proton therapy application. In recent years, the interest in the lower energy region has increased, amongst others caused by growing use of protons for medical applications (other applications include: accelerator driven nuclear reactors for energy production and waste transmutation, the modelling of single event upsets in micro-electronics caused by cosmic protons). In this chapter we examine two existing nuclear/high energy physics MC codes which have been extended during the last years and claim now to be valid in the region 250 MeV: }jba 3.21 ( [49], developed at WjiA in the period ) and u,ý"b97 ( [40], developed in the period by various collaborations). In 1993 also a dedicated code for proton therapy / dosimetry became available: V iba [16]. It has been developed by one of the founding-fathers of the application of MC techniques to charged particle transport: Martin J. Berger from the US National Institute of Standards and Technology, who al-

2 54 Monte Carlo tools for proton dosimetry ready in 1963 described algorithms [15] which are still used in most present codes. We will present a comparison between experimental data, V iba, u,ý"b, }jba and the analytical model (2.28) of section In order to judge the agreement between MC simulations and experimental data there are two important issues: î the simulated initial beam phase space has to be related to the experimental beam. î the relation between dosimeter response and simulated quantity (energy loss) has to be determined. In order to determine the sensitivity for different beam parameters we use experimental data obtained at a number of proton beam facilities: î 177 MeV protons from the scanning beam gantry at PSI (Switzerland). Because at PSI the proton beam is downgraded from 600 MeV using absorbers and analyzing magnets, this beam has a relatively large initial energy spread. î 175 MeV protons from the therapeutic beam line at the TSL laboratory in Uppsala (Sweden). This is an example of beam which is broadened by passive scattering on a double foil [54]. This beam has a much lower initial energy spread. î 80 MeV protons at the therapeutic beam line at the cyclotron laboratory in Louvain-la-Neuve (Belgium). Here a double scatter foil with annulus beam spreading technique is used [32], which has a comparable relative energy spread as Uppsala. Because the protons start with a lower energy, they travel a shorter path before stopping which means that their energy spread in the Bragg peak is lower. The Bragg peak is therefore much sharper than the Uppsala Bragg peak (see also section 2.1.6). We have investigated two media: water because this is the standard medium in dosimetry and polystyrene, which is a very practical medium to perform fixed detector position measurements. We will show the influence of secondary particles on the shape of the Bragg peak, and conclude with some studies on dosimeter effects on the shape of the Bragg peak.

3 3.2 Description of the codes Description of the codes MC codes overview A Monte Carlo program which has been developed specifically for the application in dosimetry is the code V iba. It is a descendant of the electron transport program j iba [17] and is based on the condensed random walk method [15]. In this method the path of the primary particle is divided into a series of steps. The effects of the large number of individual interactions during one step are grouped together. One grouping accounts for the large number of deflections caused by elastic scattering (using the theory described section 2.1.4), another for the large number of small energy losses (using the theory described in section 2.1.3). Berger [15] divided the condensed random walk algorithms into two broad classes: î class I algorithms in which the effects on the primary particle of all interactions are grouped together for each condensed-history step. î class II algorithms in which only a subset of the interactions for each type are grouped and the remaining, catastrophic, interactions are treated individually. V iba (like j iba) is an example of an class I algorithm. The general advantage of class I algorithms is that they are fast, due to the possibility of calculating tables with energy loss and deflection angle distributions in advance. The disadvantage is that there is no correlation between the primary proton and the secondary particles. With respect to the B-electron production (section 2.1.3) this is not a problem as long as the medium is dense, and the range of the B-electrons is small. Since nuclear interactions occur much more seldom than electromagnetic interactions, the natural way to treat these catastrophic events is using a class II algorithm. In V iba, however, also the energy loss due to nuclear reactions is implemented as a class I algorithm, because of simplicity and speed. This has been realized in the following way: from all primary protons a certain, energy dependent, fraction undergoes a nuclear reaction. From this fraction the kinetic energy will be transferred to charged and neutral particles in a certain ratio (in V iba the ratio charged/neutral is taken to be 1.5, according to calculations from Seltzer using the }ABtû code [141]). The assumption is made that the energy transferred to the neutral particles (neutrons, ò s) escapes from the medium. The energy transferred to the charged particles (protons, deuterons, tritons, k-particles, heavy recoil nuclei) is assumed to be deposited locally. For the secondary particles heavier than protons this is valid because their range is very small. To justify this also for the secondary protons, Berger states that the energy imparted at any point to the medium due to the emigration of secondary protons, is compensated by the immigration of protons coming from smaller depths because the rate of production of secondary protons (and their emission spectrum) varies very slowly with depth (except near the Bragg peak where the energy is too low for nuclear reactions anyway). An equivalent formulation: there is a charged particle equilibrium for the secondary protons.

4 56 Monte Carlo tools for proton dosimetry Table 3.1: Some characteristics of the used MC codes V iba }jba 3.21 u,ý"b97 class I IorII IorII geometry semi-infinite sea combinatorial combinatorial medium water user defined user defined source code available yes yes no Yä-P stopping powers yes no yes implementation stand alone assembly of routines stand alone To investigate the validity of the local energy deposition of the nuclear reaction products, the results have to be compared with the results obtained when the secondary protons are transported separately. In a recent Ph.D. thesis [84] an extension of V iba, named Vj ib is described which provides this transport. First results obtained with this code (which also can be found in [84]), indicate that the separate transport indeed has a significant influence on the final dose distribution. To confirm this and to study this effect further we decided that is worthwhile to examine the use of existing, established MC codes which already have been tested extensively for other applications. Two (recently updated) codes claim to be valid in the energy range 250 MeV: }jba 3.21 and u,ý"b97. In this section we will describe these codes, of which the characteristics (together with V iba) are summarized in table 3.1. Because of the larger flexibility, the availability of the source code and the comprehensive documentation of }jba, we decided to focus on this code and later on verify the results with the u,ý"b97 code. Furthermore }jba has the possibility of incorporating different hadronic generators for the treatment of hadronic interactions. One of the possibilities is the use of the hadronic generator from u,ý"b92.wewill show in the next section that as far as the hadronic interactions in the energy region of interest are concerned, the differences in dose distribution are negligible between using the hadronic generator from the old u,ý"b92 version (from now on referred to as u,ý"b) or the up-to-date, stand alone version u,ý"b Description of the GEANT code Overview of physics incorporated in GEANT As can be seen in table 3.1, in }jba the user is responsible to assemble the routines to an executable program (different from u,ý"b97 and V iba which are stand-alone, ready-to-run programs). The technical details of the program we have developed are described in appendix C. In this section we will describe the physics incorporated. The physics routines incorporated in }jba,together with their relationship are shown in figure 3.1. With respect to the electromagnetic interactions, the user has the choice of switching the creation of B-electrons on and off. When the creation of B-electrons is switched

5 3.2 Description of the codes 57 handled by GEANT electromagnetic interactions handled by separate modules hadronic interactions Molière multiple scattering energy loss fluctuations T < 10 kev δ electron production T > 10 kev elastic scattering inelastic scattering nuclear reactions CSDA approximation separate tracking of secondary particles/recoils Figure 3.1: Overview of the }jba code system. The hadronic generators that can be used are: }ûjítûb, u,ý"b, 6íWBV or }WB,Ni. off, }jba functions as a class I algorithm with respect to the electromagnetic interactions. The average energy loss is calculated with the Bethe-Bloch equation (2.1) using ionization potentials and shell corrections based on the old Ziegler and Andersen values [8] (these differ from the íwiý values, as will be shown in the next section). For the energy fluctuations }jba uses the Vavilov/Landau theory (see section 2.1.3) and for the angle deviations the Molière theory (see section 2.1.4). When the creation of B-electrons is switched on (with a lowest possible cutoff of 10 kev) the average energy lost below this cutoff is calculated using the restricted Bethe-Bloch equation (2.14). The fluctuations in energy loss below this cut off are very small, but in case one wants to use very thin layers (for example for microdosimetric calculations) }jba has the possibility of using a sophisticated photo-absorption ionization model to calculate energy loss fluctuations, which is also valid when the energy losses are comparable to the atomic binding energies. }jba also contains electron energy loss and multiple scatter routines to deal with the transport of B-electrons. Also tertiary B-electrons can be produced, with a lower limit of 10 kev, although that probability is small. In figure 3.2A a lateral projection of a small part of a 175 MeV proton track in water is shown, which has been calculated with the interactive version of }jba. It can be seen that the track length of even the most energetic B-electrons is very small (400 kev electrons have a range of 1.3 mm in water, see also table 2.4). With respect to the hadronic interactions }jba makes a separation between elastic and inelastic interactions. During the transport of the primary particle the occurrence of an nuclear interaction is determined by sampling the distribution of free path

6 58 Monte Carlo tools for proton dosimetry δ electron production: A elastic hadronic scatter: B recoil proton primary proton electron primary proton Figure 3.2: Lateral projections of 175 MeV proton tracks in water simulated with }jba using u,ý"b as the hadronic module. Grid spacing: 5 mm. lengths. Subsequently it is determined whether an elastic or an inelastic interaction has taken place. After that the energies, angles and multiplicities of the final state particles are sampled from probability distributions. In case of an elastic proton-proton interaction the final state particles will be the recoil and the incident particle (example: figure 3.2B), in case of an inelastic scattering a ò (example: figure 3.3A), the recoil and the incident particle and in case of a reaction the reaction products which can be protons, neutrons, ò s and recoils (example: figure 3.3B). For the hadronic generator one has the following choice in }jba : }ûjítûb u,ý"b 6íWBV [41] is the default hadronic module in }jba. It contains intra-nuclear cascade and evaporation models, of which the validity range generally does not extend to below a few hundred MeV s. In the documentation there is however no mention of the limit below which the use is not valid anymore. [40] is a more sophisticated hadronic module, which contains in addition to the intra-nuclear cascade and evaporation models also a pre-equilibrium model. [70] provides a better treatment of the low energy ( 20 MeV) neutrons. All other interactions are handed over to u,ý"b. }WB,Ni [142] contains the hadronic generator from WB,Ni [45], which is based on the ûj W [10] code. The documentation that accompanies the hadronic generators is not sufficient to determine which code is best suited for our problem. For }ûjítûb and u,ý"b it is

7 3.2 Description of the codes 59 inelastic hadronic scatter: A γ γ hadronic reaction: secondary protons neutron B γ primary proton primary proton γ γ electron Figure 3.3: Lateral projection of 175 MeV proton tracks in water simulated with }jba using u,ý"b as hadronic model. Grid spacing: 5 mm. possible to directly examine the cross sections used, which is done in the next section. In order to investigate how large the effect of using different models is on the dose distribution result, we also have made a direct comparison of the results of the 4 codes, this is discussed in section 3.4. The secondary particles originating from hadronic interactions (which can also be neutral particles) are again put onto the }jba stack and transported further using appropriate routines. For the charged particles the procedure is the same as above, for the neutral particles separate routines exist that deal with the photo-electric effect, Compton scattering, neutron capture, fission etc. (an example of neutron capture can be seen at the right side of figure 3.3, where two tertiary ò s are produced, of which one undergoes a Compton scatter) Cross sections used in GEANT The most basic requirement that a Monte Carlo code has to fulfill is that its cross sections are valid in the energy region of interest, the axiom garbage in-garbage out certainly applies for MC codes. An advantage of}jba that it is easy to extract the energy loss and cross sections tables (unlike u,ý"b97 for example). We can therefore directly compare the }jba stopping powers to the íwiý (see table 2.1, [67]), and thus the V iba values. With respect to the inelastic hadronic cross sections, it is also possible to directly compare }jba with V iba. With respect to the elastic hadronic interactions, however, in V iba no separation between hadronic and Coulomb interactions is made: V iba uses the Molière multiple scattering theory which deals only with the Coulomb interaction. Therefore to verify the elastic hadronic cross sections

8 difference GEANT / ICRU (%) 60 Monte Carlo tools for proton dosimetry stopping power Gd stopping power water stopping power air range air range water range Gd proton energy (MeV) Figure 3.4: The difference between the values used in }jba and the íwiý stopping powers and ranges (as in table 2.1, in % of the íwiý value). in }jba, we have to compare with other, nuclear physics codes. A comparison between the íwiý stopping powers and ranges and the }jba values is shown in figure 3.4. The differences between the }jba and íwiý stopping powers and ranges are material and energy dependent. For water the differences are ä 2% above ä 10 MeV, which means that for proton dosimetry comparisons with u,ý"b97 and V iba (which both contain the íwiý stopping powers), small (ä 1 %) corrections to the initial proton energy have to be applied. In }jba a total hadronic elastic cross section is used. This implies that the electro-magnetic contribution has been separated from the hadronic, since otherwise the total cross section will diverge for scattering angles towards 0 á. Apparently the interference term between the hadronic and electromagnetic contribution has been discarded in }jba, although this is not stated explicitly in the documentation. In order to allow a numerical comparison, we have to separate the electromagnetic from the hadronic interaction in the optical model code/partial wave analysis too. For protons on S O this has been done using the optical model code açýw" (with the parameters for S OfromPerey[105], see also section 2.1.4), by setting the charge of the nucleus to zero and integrating over all angles. For elastic scattering with H it is possible to use the Nijmegen nucleon-nucleon partial wave analysis [122] 1. This provides, however, only the neutron-proton total elastic hadronic cross section. Since in first order the 1 Also on the World Wide Web:

9 total elastic cross section (barn) total elastic cross section (barn) 3.2 Description of the codes O elastic FLUKA p FLUKA n DWUCK p 0.4 1H elastic FLUKA p FLUKA n Nijmegen n-p Nijmegen p-p proton energy (MeV) proton energy (MeV) Figure 3.5: Total cross sections of the u,ý"b hadronic generator used in }jba for hadronic elastic scattering together with values calculated with açýw" (see text, for 30, 80 and 180 MeV protons) en Nijmegen partial wave analysis. proton-proton cross section is e of the neutron-proton cross section, due to quantummechanical isospin effects [122], it is still possible to compare the values. For protons on S O we expect that the hadronic elastic cross section for neutrons will be the same as for protons, because the oxygen nucleus contains the same number of protons and neutrons. In figure 3.5 a comparison between the total elastic hadronic cross sections for protons/neutrons on S O and H as reported by the u,ý"b hadronic generator and the values we have calculated using the açýw"/nijmegen partial wave analysis can be seen. The }ûjítûb cross sections (not plotted) agree with the u,ý"b cross section for energies above 50 MeV. For energies below 50 MeV the }ûjítûb cross section shows strange artifacts and becomes unphysical high. From this comparison the following conclusions can be drawn. The total elastic cross section reported by }jba is within the uncertainties in agreement with the values we have calculated by setting the electromagnetic charge to zero in the partial wave analysis code. We can therefore assume that interference term between the electromagnetic and hadronic interaction, which is responsible for the hills and valleys in figure 2.4, is discarded in }jba. To which extent this will affect the validity of the }jba results remains an open question. Also towards the lower energies u,ý"b shows artifacts, since the proton cross section becomes saturated while it is expected to increase until the Coulomb barrier is reached. However, because the final state particles for these energies have a small rangeanyway, the effect on the dose distribution will be small. It can also be seen that the hadronic elastic cross section for scattering on hydrogen is not negligible, which can have a result on the dose distribution because the

10 difference with PTRAN (%) 62 Monte Carlo tools for proton dosimetry 15 16O inelastic FLUKA 10 GHEISHA proton energy (MeV) Figure 3.6: Difference between the inelastic cross sections used in }jba and in V iba (see figure 2.5) for protons on 49 O in % of the ICRU value. energy of the primary particle is split over two particles which will have shorter range than the primary particle would have alone. We therefore decided that it is interesting to tag the protons in our dosimetry program, to determine whether they are primary, originate from elastic hydrogen scatter or are a reaction product. The inelastic cross section of S O (the inelastic cross section of Hisnegligible [71]) can be compared with the V iba cross sections, which are calculated using the }ABtû code (see section 2.1.5). In figure 3.6 the result is shown. u,ý"b has a larger deviations than }ûjítûb for energies above 50 MeV. }ûjítûb has however much larger deviations towards the low energies. With respect to the other possible codes, the only difference between 6íWBV and u,ý"b is the improved treatment of low energy neutrons (from 20 MeV down to thermal energies). Since it is expected that the contribution to the total energy loss from these neutrons will be very small, not much is gained using 6íWBV. A disadvantage is that the computing time becomes considerable longer using 6íWBV. }WB,Ni has the disadvantage that it is not possible to tag the secondary protons, because it does not follow the }jba datastructures/conventions. From the considerations above we decided to start with the use of u,ý"b as hadronic generator in }jba. In section 3.4 we will investigate how the resulting dose distribution is affected when other generators are used. GEANT dosimetry program implementation A summary of the different physics options we have used in our }jba dosimetry program is given below: î B-electron generation with a threshold of 10 kev

11 3.2 Description of the codes 63 î photo absorption ionization model turned off. î hadronic generator u,ý"b î Molière scattering turned on î cut-off energy for hadrons: 1 MeV î cut-off energy for electrons and ò s: 10 kev î automatic calculation of stepsizes switched on. With these options the average time to process 1 primary proton on a Digital Alpha 500/255 workstation was 0.12 s for 180 MeV protons, and 0.08 s for 80 MeV protons. This yields an average computing time of 3.3 h for a run of 10 D protons, which is the number of protons necessary to reach a statistical uncertainty below the percent level. The quantities recorded in our }jba dosimetry program are the dose (ì energy loss) and flux per incident proton as a function of depth. The dose is expressed in MeV cm 2 /g. The flux is per unit area, in the 1 dimensional simulation this means the complete area perpendicular to the beam. In addition to the total dose also the contributions of different particle types are separately recorded: î the dose originating from primary protons (i.e. from energy losses below 10 kev) î secondary protons from nuclear reactions (including tertiary protons created by secondary neutrons) î secondary recoil protons (from elastic hydrogen scattering) î B-electrons above 10 kev. The contribution of heavier recoils is then given by the difference between the total dose and the sum of primary, secondary protons and B-electron contributions. We also have recorded the energy loss spectrum (or Linear Energy Transfer (LET) spectrum) for the different contributions as a function of depth. The other quantity that is recorded is the particle flux per incident proton for the different contributions as a function of particle energy and depth. The flux in a certain region is calculated using the sum of particle track lengths in the region divided by the volume of the region [31].

12 64 Monte Carlo tools for proton dosimetry 3.3 Comparison with experiments Introduction The proton therapy beam that impinges on the phantom (or patient) can be completely described by its phasespace. In principle it is possible to determine this phasespace by following the beam as it leaves the cyclotron, travels through absorbers and scatter foils, collimators, and finally reaches the phantom. However, when one wants to compare the results of MC simulations with experimental results it is better to use a simpler approach, because this reduces the number of parameters. A basic test is the comparison between the calculated and measured depth-dose distributions, which is essentially a one-dimensional problem. When the beam area is large compared to a few times the 80 % - 20 % lateral fall-off distance, say è 3cm, and the radiation detector is small compared to the beam area, the experimental results can be compared with simulations of a pencil beam from which the energy loss only as a function of depth is scored [35]. This is illustrated in figure 3.7. This approach is valid under assumption that at each point the same number of protons scatter in and out of the detector. With respect to multiple scattering this is clear since the scattering angles are small. With respect to the hadronic interactions, however, this will have to be verified since the angle of the secondary particles with respect to the beam direction can be large. For our experiments we have used passively scattered beams of the Uppsala and Louvain-la-Neuve facilities and the active spot scanning beam of PSI. A complication to comparisons with passively scattered proton beams is that they are not parallel. It is therefore necessary to correct for the geometric attenuation of the beam, which is proportional to *- 2, where - is the distance between source detector. For a typical distance between the first scatter foil and the isocentre in the order of 2.5 m it is possible to use the position of the first scatter foil as a virtual focus [54]. To compensate for this effect we have measured the depth dose distribution in two different ways: in a water medium with a detector that moves in depth and with the detector fixed at the isocentre broad beam pencil beam detector detector Figure 3.7: The equivalence of a small detector in a broad beam and a broad detector in a pencil beam.

13 3.3 Comparison with experiments 65 position, and a variable number of polystyrene slabs before the detector. The difference between the two measurements (apart from differences in the physical properties of the medium) is the geometric attenuation. This correction is not necessary for comparison with the measurements in the spot scanning beam of PSI, which has a parallel beam because of its ion optics. Furthermore in PSI we could make use of a water bellows system, which makes it possible to do a fixed position measurement in a water medium. Another item of interest in comparing data with calculations is the influence of the detector on the measured results. In section 2.2 it has been shown that the influence of a small air-filled chamber on the dose profile is small with respect to the differences in stopping power and integration effects. We have therefore used a plane parallel ABWV-02 chamber which has a 2 mm gap perpendicular to the beam direction and a diameter of 10 mm. In first approximation we assume that the detector signal of this chamber is proportional to the dose in water. Smaller effects concerning the effects of B-electrons will be examined later. At PSI we have used a different reference dosimeter: a largearea(b=80 mm) ionization chamber, which covered the complete field Method The applied procedure is: î For the data taken at the Louvain-la-Neuve and Uppsala facilities: divide the ABWV-02 signal in water by E5ú#*E5ú# n5ää 2 where 5ú# is the distance from the first scatter foil to the isocentre and 5 the depth in water (5 'fcoincides with the isocentre) in order to correct the experimental data for the power law effect. î determine the experimental range - f defined as the point where the dose has dropped to 80 % of the Bragg peak. î determine the experimental steepness of the tail of the Bragg peak by determining the distance at which the dose drops from 80 % to 20 % î vary the initial proton energy in the various models/codes until the experimentally determined range is obtained in a water medium. We have set the contribution to the tail of low energy protons (0 in equation (2.28) to zero. î vary the initial proton energy spread until the experimentally determined steepness is obtained in a water medium. î with these beam parameters perform a simulation in a polystyrene medium. î this result can now directly be compared with the experimental polystyrene data.

14 66 Monte Carlo tools for proton dosimetry Table 3.2: Overview of the applied measurement methods Louvain Uppsala PSI NACP, moving in depth, water medium î î NACP,fixedindepth, polystyrene medium î î 80 mm chamber, fixed in depth, water medium î We have compared the experimental data with }jba (using the u,ý"b hadronic generator), u,ý"b97, V iba calculations and with the analytical model (2.28) from section For the calculations it is possible to directly compare the absolute dose normalized to the incident proton flux. For comparison with experimental data this is not possible, therefore we have normalized all models to the entrance dose. A summary of the different measurements methods can be found in table 3.2. With V iba in its present form it is not possible to handle media other than water. With the analytical model (2.28) it is possible to calculate the dose distribution in polystyrene. In that case the parameters are different from the ones given in 2.3 which hold for water. The parameters D and R can be determined with a fit to the íwiý data, the value for E is calculated from the nuclear survival probability given in [69]. The numerical values are listed in table Results Figures 3.8, 3.9 and 3.10 show the data and calculations for the Louvain-la-Neuve, Uppsala and PSI beams, respectively. In table 3.4 the parameters we have used in the model calculations are listed. The analytical and V iba curves overlap within the statistical uncertainty,asdo}jba and u,ý"b97 curves. The experimental data points lie in between the curves. Several observations can be made: 1. The V iba code agrees with the analytical model within one percent, with some small deviations in the energy spread parameter. u,ý"b97 and }jba also agree with each other within one percent, again with some deviations in the energy and energy spread caused by the different stopping powers. This implies that the way nuclear secondaries are treated is the predominant factor accounting for the difference in the dose distribution, because in this respect both V iba and Table 3.3: analytical model (2.28) parameters for polystyrene parameter value unit D ü10 ý6 cm MeV ýs s E cm ý g/cm ô

15 relative dose (%) relative dose (%) 3.3 Comparison with experiments Louvain H 2 O Louvain polystyrene measurement GEANT analytical measurement GEANT analytical depth (mm) depth (mm) Figure 3.8: Comparison of depth dose curves measured with a ABWV chamber in Louvain-la-Neuve, calculated with }jba and the analytical formula (2.28). The parameters that have been used in this calculation are listed in table 3.4. The u,ý"b97 curve overlaps with the }jba curve, the V iba curve overlaps with the analytical model. 100 Uppsala H 2 O Uppsala polystyrene measurement GEANT analytical measurement GEANT analytical depth (mm) depth (mm) Figure 3.9: Comparison of depth dose curves measured with a ABWV chamber in Uppsala, calculated with }jba and the analytical formula (2.28). The parameters that have been used in this calculation are listed in table 3.4. The u,ý"b97 curve overlaps with the }jba curve, the V iba curve overlaps with the analytical model.

16 relative dose (%) 68 Monte Carlo tools for proton dosimetry 100 PSI H 2 O NACP measurement GEANT analytical depth (mm) Figure 3.10: Comparison of depth dose curves measured with a ABWV chamber in Uppsala, calculated with }jba and the analytical formula (2.28). The parameters that have been used in this calculation are listed in table 3.4. The u,ý"b97 curve overlaps with the }jba curve, the V iba curve overlaps with the analytical model. Table 3.4: MC calculation parameters for the different beams parameter Louvain Uppsala PSI source-axis distance 2.47 m 2.25 m 4 }jba beam energy (MeV) }jba energy spread j (MeV) u,ý"b97 beam energy (MeV) u,ý"b97 energy spread j (MeV) V iba beam energy (MeV) V iba energy spread j (MeV) analytical beam energy (MeV) analytical energy spread j (MeV)

17 3.3 Comparison with experiments 69 the analytical model differ from }jba and u,ý"b97. This also means we can limit ourselves from now on to the analytical model and }jba. 2. All models are able to predict the change inrange caused by the change in medium from water to polystyrene (5.2 % for the Uppsala beam, 4.7 % for the Louvain-la-Neuve beam). This is not a trivial scaling with 4 ø (see equation ~ (2.1)): in that case we would expect a 7 % change inrange for both cases. The difference is due to the different excitation potentials U: U ê@ ih =75eV,U TL*)t )hi?i =69eV. 3. The change in the ratio dose in the Bragg peak to entrance dose (the peak-toentrance ratio ) from water to the polystyrene (5.9 % for Uppsala, 1.7 % for the Louvain-la-Neuve) is the same for the various calculations. This indicates again that the differences between the models and the experimental data are caused by the treatment of nuclear secondaries, because both at the entrance (because of the non-local deposition) and in the Bragg peak (because of the low cross section) their contribution is negligible. 4. The agreement between experimental and model peak-to-entrance ratios is quite good. This means that the way the energy spread was chosen (fitting to the tail 80 % - 20 % distance) is consistent with the measured peak-to-entrance ratio. For the Louvain-la-Neuve beam there might be a small deviation with the experimental data in polystyrene. However this can be explained by the experimental difficulty of finding the exact height of the Bragg peak in case of low energy proton beams. The smallest slab we have used was 0.5 mm polystyrene, which means that the Bragg peak can be missed by the measurement because it is positioned inside a slab. For highenergyproton beams this problem does not occur, since the Bragg peak is there much broader. 5. The largest deviation between the model predictions and experimental data occurs in the region 2 to ô of the range: up to 13 %. e The deviation between the model predictions and experimental data maybe due to two effects: the first one is that the contribution of secondaries as discussed in section 3.2 is not correct. If so, the local deposition assumption of the analytical model underestimates the contribution of secondaries for the Uppsala and PSI beams (with relative high initial beam energies), while the agreement for the Louvain-la-Neuve beam (with a low initial beam energy) is reasonable. }jba appears to overestimate the contribution from secondaries in all cases. It is therefore useful to examine the contribution from the nuclear secondaries in more detail, which is done in section 3.4. The other possibility could be that the assumption that the one dimensional calculation corresponds to a 3 dimensional reality, in case the detector is small compared to the beam area, is too simple. We have therefore also examined the effect of extending the simulation to more dimensions in section

18 primary proton flux per incident proton 70 Monte Carlo tools for proton dosimetry Louvain water Louvain polystyrene Uppsala water Uppsala polystyrene depth (cm) Figure 3.11: The flux reduction of primary protons calculated with }jba for the Louvain-la-Neuve (80 MeV) and Uppsala (175 MeV) case. 3.4 Influence of nuclear interactions Since we are able to tag the different particles in }jba, the separation of contributions from different types of secondary particles enables the study of the influence of nuclear interactions Primary proton flux reduction We start by examining the flux reduction of the primary protons for the Louvain-la- Neuve and Uppsala beam (the PSI beam only differs from the Uppsala beam in the initial energy spread, which is not of interest for the study of nuclear interactions), which is shown in figure The reduction is approximately linear in both cases up to the Bragg peak region. The results of a linear fit are shown in table 3.5. Table 3.5: The flux reduction per cm medium of the primary protons compared for PTRAN/analytical model (see section 2.1.5) and GEANT/FLUKA flux reduction (cm 3 ) GEANT PTRAN Louvain water Louvain polystyrene Uppsala water Uppsala polystyrene

19 relative contribution to dose (%) 3.4 Influence of nuclear interactions Louvain Bragg peak primary protons Uppsala Bragg peak primary protons 10 δ electrons δ electrons reaction protons 1 reaction protons recoil protons recoil protons heavy recoils depth (mm) heavy recoils depth (mm) Figure 3.12: Contribution of different particle types to the total dose (plotted in figure 3.9 and 3.8) in the Louvain (range U 3 : 48.4 mm) and Uppsala case (range U 3 : mm) The }jba /u,ý"b values for water are in agreement with the V iba values (see figure 2.6 in section 2.1.5), which is expected because the differences of the inelastic cross sections are small. Also it is clear that neglecting the dependence on the initial beam energy (as has been done in the analytical model to determine E) might not be justified. However since the analytical model calculation overlaps with the V iba calculation (in which the energy dependence is taken into account), the effect must be negligible small Relative contribution of different secondaries In figure 3.12 the contribution to the dose of nuclear secondaries as calculated with }jba (using the u,ý"b hadronic generator) is plotted as a function of depth for the Louvain-la-Neuve and Uppsala beams. For reference also the electromagnetic contributions are shown: primary protons (with an energy loss 10 kev) and B-electrons (: 10 kev). The contribution of B-electrons : 10 kev is in agreement with the theory (see figure 2.3 in section 2.1.3). The dose due to heavy recoils can be neglected. The recoil protons (for example shown in figure 3.2B) are produced by proton-hydrogen or neutron-hydrogen collisions. These collisions do not contribute to the primary flux reduction, since the primary proton survives after such a collision. Their contribution to the dose is small, which corresponds with the small elastic hadronic cross section (see figure 3.5). The reaction protons (for example shown in figure 3.3B) are the protons originating from a proton-oxygen or neutron-oxygen reactions. The primary particle

20 difference with FLUKA (%) difference with FLUKA (%) 72 Monte Carlo tools for proton dosimetry depth (mm) depth (mm) total dose reaction protons GHEISHA MICAP GCALOR primary protons recoil protons depth (mm) depth (mm) difference with FLUKA (%) difference with FLUKA (%) Figure 3.13: Comparison of the different hadronic modules in }jba for the Uppsala beam (175 MeV): the difference between the dose calculated with u,ý"b and }ûjítûb (solid line), 6íWBV (dashed line) and }WB,Ni (dotted line) in % of the FLUKA value as a function of depth for the total dose, primary protons, reaction protons and recoil protons. will disappear in that case. The dose due to reaction protons can not be neglected: it can be as large as 10 % for the Uppsala beam. The inelastic proton-oxygen scattering has not a large effect on the energy deposition, since the primary proton survives (albeit with a smaller energy), while the only secondary particle is a ò Hadronic generator comparison To investigate the influence of the hadronic generator on the resulting dose distribution we have compared the 4 generators of section 3.2 with respect to the dose distribution for the various secondary particles. In figure 3.13 the difference with u,ý"b is plotted

21 3.5 Simulation of dosimeter response 73 for the Uppsala beam, for which these contributions are the largest. The differences between the different hadronic generators are large enough to have an effect on the resulting total dose distribution. However, none of the hadronic generators is able to simulate the data in the region of 2 to ô of the range, since that would require smaller e dose values (é 13 %) than the u,ý"b calculations (which are shown in figures ) for that ranges. When we compare the 6íWBV with the u,ý"b values, it can be seen that the improved treatment of the low energy neutrons has only significant consequences for the recoil protons, but as their contribution is small, the total dose is not influenced by much. When we compare the }ûjítûb with the u,ý"b values, it can be seen that the difference between their inelastic cross sections (as shown in figure 3.6) also influences the primary proton flux reduction. A larger difference can be seen with respect to the reaction protons: up to ä 50 %. Since their contribution to the total dose is ä 10 %, it can be expected that the total dose deviates up to 5 %, which is the case. The differences between the }WB,Ni and u,ý"b values are smaller than the other hadronic generators, and also }WB,Ni does not simulate the experimental data with sufficient accuracy. 3.5 Simulation of dosimeter response Introduction Another possible explanation for the deviation between the }jba simulation (including the u,ý"b hadronic generator) and the experimental data shown in figures 3.8, 3.9 and 3.10 is an effect of the dosimeter on the measured dose distribution (see section 2.2). Since the secondary protons in general will have a larger angle compared to the primary protons, the condition of lateral equilibrium of the secondary particles (see section 3.3) may not be fulfilled, which means that the contribution of dose due to secondary protons would be less in a dosimeter with a small size compared to the beam profile. It is interesting to note that the deviation in the Vtí beam is smaller, which may have to do with the use of a large (B=80 mm) ionization chamber (which covers the total beam area) instead of an small ABWV (B=10 mm) chamber. To investigate the proportionality of the dose in a dosimeter to the dose in a medium, we have extended the simple 1D slab geometry used in previous sections to a 2D cylinder, which contains 3 layersasshowninfigure We have studied the influence of the dosimeter response: î to examine whether the difference between the }jba /u,ý"b simulations and the measurements can be explained by the angular distribution of the nuclear secondaries (section 3.4). Here only the geometry is of interest, and the detector physical properties are kept equal to those of the medium. î to examine to which extent the effect of B-electrons on the dose measurement with a thin, air filled ionization chamber can be simulated (see section 2.2.4). For

22 74 Monte Carlo tools for proton dosimetry incoming pencil beam with energy E and energy spread σ medium layer 1 detector layer radial distance depth medium layer 2 Figure 3.14: Extension of the 1D geometry to simulate the influence of dosimeter on the final dose distribution this it has to be verified if very small layers still yield valid results, since a typical ionization chamber thickness of 2 mm air has a water equivalent thickness of 2.4 >m, which is in the same order of magnitude as the rangeof10kevb-electrons. î to examine whether we can simulate the influence of a thick dosimeter such as a scintillator screen on the observed dose distribution. This is interesting in connection with the experimentally observed decrease of the signal in the Bragg peak (section 4.4.3). For this both the geometry of the detector and the physical properties are of interest Effect of nuclear secondaries on the dose measurement To investigate the effect of nuclear secondaries we first want to verify the assumption that there is a lateral equilibrium for all particles, which is necessary in order to be able to compare a small detector in a broad beam with a broad detector in a small pencil beam (see figure 3.7). We used two approaches for this: î in u,ý"b97 it is relatively straightforward to use broad beams, we have therefore used a geometry ofasmall(b= 1cm) detector cylinder which is contained in a broad medium cylinder. Both the detector and medium material are water. î }jba is not optimized for using broad beams, we therefore have used a pencil beam and recorded the dose from the different particles as a function of depth and

23 3.5 Simulation of dosimeter response 75 radial distance. This enables the study of the dose distribution in radial direction for the different particles. We have used the Uppsala beam parameters from section 3.3 (listed in table 3.4). In figure3.15a2dimensional intensity plot of the dose as a function of depth and radial distance is plotted for the different contributions. To quantify the difference a radial profile at 10 cm depth is plotted in figure It can be seen that at that depth the dose due to nuclear secondaries becomes larger than the dose due to primary protons for radial distances greater than 2.3 cm. Since the dosimeter for the measurements in figure 3.9 was positioned at 3 cm from the edge of the field, this might explain the difference between the 1D simulation and the measurement, although the contribution to the total dose is already very small at that distance. In order to verify this directly we have also performed simulations with the u,ý"b97 code. In these simulations we have used a circular beam with two different diameters, B=5 cm and B=7 cm. The detector was a water filled cylinder with B=1 cm. For the beam energy and energy spread we have used the Uppsala values (table 3.4). We have simulated the tracks of 10 D protons, taking a computing time of 14 hour on a Digital Alpha 500/255 workstation. Within the statistical uncertainty of 1 %, there was for both beam diameters no difference with the 1 dimensional results shown in figure 3.9. Hence the angular distribution of nuclear secondaries does not seem to be the explanation for the deviation between the }jba /u,ý"b Monte Carlo simulations and measured depth-dose data Effect of delta-electrons on the dose measurement The motivation to study the influence of B-electrons on the dose measurement with an air-filled chamber originated from the experimental finding that the dose of 180 MeV protons measured with a 0.6 cm ô Farmer type chamber in air, decreased by 6%when the plastic protection cap (thickness 1 mm V66B) was removed. Since the plastic cap is very thin, it does not have a detectable influence on the proton energy or primary proton flux. The effect might be caused by a change intheb-electron flux, so we decided to see if we could simulate it with }jba. Before we are able to do so, we first have to investigate if }jba still yields valid results in case of very thin layers. We have therefore simulated the energy and energy loss (i.e. LET) spectra in the detector for the different particles. For the detector we used a thin, 0.1 mm, water filled detector (which is still ä 50 times thicker than the water equivalent thickness of a Farmer chamber: 2.4 >m). We have performed calculations for 2 beams: the 80 MeV Louvain-la-Neuve beam and the 175 MeV Uppsala beam (see table 3.4). In figure 3.17 the energy flux and LET spectra are shown at a depth of 1 cm and at the position of the Bragg peak (that is the position of the maximum dose in figures 3.8 and 3.9, see also section 2.1.6). We only show the primary and B-electron spectra, since the contribution of secondary protons is negligible both at the entrance

24 relative dose 76 Monte Carlo tools for proton dosimetry 60 primary protons radial distance (mm) radial distance (mm) depth (mm) nuclear secondaries depth (mm) Figure 3.15: Contour plot of the dose distribution of the Uppsala beam in water. The contour lines have a value of 0.1, 0.25, 0.5, 0.75, 1, 2.5 and 5 % of the maximum dose, both for the primary protons and nuclear secondaries primary protons depth = 10 cm nuclear secondaries radial distance (mm) Figure 3.16: Profile of the radial dose distribution at a depth of 10 cm. The dose has been normalized to the maximum dose of the primary protons, that occurs at r=0.

25 3.5 Simulation of dosimeter response 77 Table 3.6: Comparison of the energy loss in a 0.1 mm thin water slab, calculated from the energy fluence bins and from the energy loss as plotted in figure Q õ is the content of the flux bin (width ü õ ), Q OHW is the content of energy loss bin (width ü OHW ) Units are MeV cm 5 /g Q õ ü V ü ü x Q OHW ü ü OHW difference (%) Louvain entrance Louvain Bragg Louvain entrance Louvain Bragg Uppsala entrance Uppsala Bragg Uppsala entrance Uppsala Bragg and in the Bragg peak. In appendix C we show the two dimensional intensity plots of these spectra for all contributions at all depths in figures C.2-C.4. The mean dose in a 0.1 mm thin slab at a certain depth due to continuous energy loss can be calculated in two ways: î by multiplying each bin of the energy flux spectrum with the appropriate stopping power: the restricted stopping power for the primary protons (see equation (2.14) in section 2.1.3) and the total mass collision stopping power for the electrons (see table 2.4 in section 2.2.4). î The total dose in the slab is given also given by the sum of LET bins. The results of these two calculations are shown in table 3.6. For the protons the two methods yield essentially the same numbers. Only in case of the Louvain Bragg peak are the differences statistically significant. This might have to do with relative size of the detector compared to the width of the Bragg peak. A number of observations can be made: in the entrance the spread in energy loss is small, and the total loss is equal to the (restricted) proton stopping power since there are almost no protons lost due to nuclear reactions. The numerical value differs slightly from the íwiý values, which is expected. In the Bragg peak the energy spread has become large, which is also according to expectation. The mean proton energy in the Bragg peak is higher, and thus the proton LET lower, for the Uppsala beam than for the Louvain beam. For the electrons the results show large differences. From a comparison of the electron energy loss contributions calculated by taking the sum of LET bins (i.e. the total dose due to B-electrons) and the theory of section it can be concluded that the electron energy loss is in agreement with theory. This means that the electron flux spectra are not correct. Since the default stepsize in }jba (0.07 mm for a water medium) is