LET calculations

As already mentioned in different points the effect of ion beams on tumour cells does not depend only on the physical dose, but also biological effects have

5.2 Contributions of charged fragments

137 to be considered, too. The main advantage of the carbon ion beams for radio-therapy is the increased biological effectiveness, RBE, towards the end of the particle range (see section 1.2.2). Besides other parameters like dose, tissue type and biological end-point, RBE depends on the local energy spectrum. The latter is often referred as “radiation quality” and can be characterized by the linear en-ergy transfer. Thus it is reasonable to provide spatial LET distributions in addi-tion to the dose distribuaddi-tions. This might help to localize high LET regions, where the greatest variations in RBE are expected, with the relative conse-quences in cell survival. A Monte Carlo calculation with the toolkit Geant4 of LET for a mixed radiation field has been developed, which afterwards could be compared with the cellular response.

While LET for a monoenergetic beam, not mixed with other secondaries, is easily obtained from tables, the calculation of a realistic mean local LET of a mixed particle beam (and different energies) is a much more complicated task.

There are currently several definitions of LET in use, all of them based on the stopping power. Definitions of restricted and unrestricted linear energy transfer have been already given in section 1.2.2, where it has been discussed as that the latter practically coincides with the electronic stopping power S_el. There-fore, if a quantitative evaluation of the “radiation quality” must be done, we have to be able to give in some way a unique value which takes into account the dif-ferent LET values. Hence, such a kind of “mean LET” has to be calculated. It means we consider only primary particles, to average the stopping powers S(E) depending on the energy spectrum of the primary particles. This mean value can be calculated by averaging the stopping powers of all particles at a certain point in the radiation field, in order to achieve a local mean value of the stopping power S, which by now will be indicated simply with the word LET (which, from now on, stands for mean LET). There are two main common implementa-tions of mean LET: the track averaged LET and the dose averaged LET. The first one is the mean value S weighted by fluence, the second one is instead weighted by its contribution on local dose.

The definitions given regards the mean LET value for a single kind of parti-cle (i.e. only primary partiparti-cles) but when we are in presence of a mixed field, as

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in the case of ion beams, the situation gets more complicated. In this case, at a particular point x, the mean LET for each different particle type has to be sepa-rately calculated. Some analytical models have been proposed in case of mean LET for a single particle type, afterwards validated by Monte Carlo [119]. It is evident that in case of mixed field the use of algorithm becomes a very compli-cated issue.

A Monte Carlo method using the toolkit Geant4 has been proposed here. It is able to take into account the contribution of the mean LET of each isotope pro-duced, and to weight each contribute on the specific isotope, at different penetra-tion depths. In this work, only the fluence averaged LET, i.e. the track LET, has been considered. Indeed, for high LET radiations, as the carbon ions, the energy deposition is more strictly dependent on the track, in the sense that most of the dose is delivered close to the “core” of the particle track, giving rise to a more heterogeneous irradiation (see section 1.2.2). Instead, for low LET radiations, as photons or electrons, the energy deposition for clinical relevant doses is rather uniform over biologically important dimensions (such as cell nucleus), and in this case also the dose averaged LET should be accounted [120].

LET track for a single particle type can be defined using the integral notation reported in ICRU 16 [121] and slightly modifying the formalism, so that we can obtain:

where φE(z) is the particle fluence differential in energy at the position z, and Sel(E) represents the electronic stopping power as function of the energy E.

LET calculations have been implemented inside the Hadrontherapy applica-tion. The simulation has been run exactly with the same geometrical configura-tion, beam parameters and physics used for the secondary dose and fluences dis-tribution, previously discussed (5.2.1). Some modifications have been intro-duced, which allow the calculation of the mean LET at specific pre-defined

5.2 Contributions of charged fragments

139 points along the beam path. At a first stage, we can imagine to have primary par-ticles only of a single specific type (i.e. a fixed isotope), which we call “j”, ne-glecting isotope production at this first stage. The method implemented in the application consists on registering and scoring the kinetic energies of the parti-cles j when they traverse some planes inside the PMMA phantom, placed per-pendicularly to the beam axis at specific positions z. Because of the statistical nature of the interactions, a local spectrum of the j particles is scored at each de-fined depth z, and saved in histograms. For the histograms an energy bin of 250 keV has been chosen together with an energy cut-off of 250 keV, which was tested to have no sensible influence in the final results. Now, if Φi(z) indicates the number of j particles in the energy bin i (i = 1, ..., N) and the stopping power S_i corresponding to the mean of energy bin are taken from tables, thus the LET track can be calculated by:

( ) ( )

which represents the discrete version of the equation (5.1).

What discussed so far deals with calculation of LET track at different depth z for only a single kind of particle j. This could be valid, in a first approximation, for proton beams, where the contribution due to secondary particles could be ne-glected. However in carbon ions radiotherapy the situation is largely different.

The method above discussed has been extended at all the isotopes produced and the equation (5.2) has been calculated for each charged isotope j produced be-cause of the fragmentation of the primary carbon ions. A criterion of choice has been established in order to avoid too many calculations iterated also in case of isotopes rarely produced. This has been done by performing a first long simula-tion run where all the particle fluences per isotope have been stored (with same methods discussed in section 5.2.1) and statistics of isotope production has been obtained. Secondary charged particles which are produced below a specific threshold have been neglected for production of final results.

Hence, LET track of the charged fragments produced along the beam path as

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well as of the primary ions, have been calculated for some fixed depths (Figures 5.7-5.10).

Moreover, the calculation of a “total” LET track has been also implemented in order to have a unique mean LET value for each depth position z, which takes into account also the LET contribution of secondary particles. A mean of LET value previously found for each isotopes has been calculated by averaging on the isotope fluence. In formula:

where k is the total number of isotopes species produced and considered in the calculation. This quantity represents the “total” mean value at a specific PMMA depth, including primary as well as secondary particle contributions.

LET calculations have been performed at the following depths of PMMA:

Depth [mm] Percentage dose

3.23 14,40%

Table 5.2. Depth values in PMMA where the LET calculations have been performed.

The corresponding percentage values in the normalized Bragg curve are also shown.

In Figures 5.7-5.9, LET track plots are showed, selecting some of the

5.2 Contributions of charged fragments

141 charged fragments produced. In each plot, the Bragg peak in arbitrary unit ob-tained in the simulation has been superimposed in order to have a clear reference on peak position, entrance channel and tail. Hence it gives just a graphical guide to better individuate depth dose positions.

For protons and alpha particles (as well as for their isotopes) a common trend has been found, as expected, even if different absolute values are obvi-ously achieved. They have a rather constant value of LET track in overall the position depth considered, due to the fact that energy spectra are quite similar in the different positions: this is a direct consequence of the fact that in the depth positions considered their velocity is fairly constant, also behind the peak where they are no more produced. Longer distances are necessary to see variations on LET. Found values range on 3-4 keV/µm for protons and 10-15 keV/µm for al-pha particles.

Different behaviours can be observed from Li fragments up to heavier iso-topes. In this case, indeed, a different trend in the two regions, before and after the peak, starts being evident. In the entrance channel they are continuously pro-duced with similar velocities, giving rise to overall constant LET values. After the peak, LET track increases because the fragments previously produced slow down, decreasing their velocity much more rapidly than protons and alpha. So an increasing in LET is found, as expected. It is much steeper as the atomic mass of the considered fragment increases. This is clearly visible for example in case of ⁷Li LET track plot. Values found range on 15-30 keV/µm for Li fragments, 30-90 for Be and 50-250 for B.

In Figure 5.9 (right) LET calculations for the primary ¹²C ions is shown. It is the steepest one, as expected, reaching values up to 800 keV/µm in proximity of the distal fall-off of the Bragg peak. In this region, indeed, carbon ions are at very low energy, the energy released is consequently highly localized and their track is narrow.

Finally, in Figure 5.10, the total LET track, averaged on the different contri-butions due to the charged fragments is shown, compared with the LET track of primary ions. The LET track in the entrance surface region is mainly due to the primary ¹²C ions, as expected. However, the contribution of other low LET

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ticles, which composed the mixed radiation field in that region, tend to decrease the total average LET, so that in the plateau region is about 15-20 % less than the single ¹²C LET. The discrepancies between only primary LET and total dras-tically increase at the peak region, where the primary LET reaches values up to two times higher than the total track LET. Enhanced biological effects are ex-pected in this region. Finally, in the distal part, the contribution of primary drop to zero, and the total LET is essentially dominated by charged fragments contri-bution, reaching an almost constant value of about 8-10 keV/µm.

The shown results clearly indicate that in case of carbon ion therapy, taking into account only the track value of primary carbon ions, we would introduce er-rors in the LET calculation. A more realistic prediction is achieved if we con-sider also the contribution of all the charged fragments produced, averaged on the fluence. In this case, the total averaged LET track can be actually considered as a quantitative estimation of the “radiation quality” and it can further related to biological effects and cell survivals responses. This will be done in the early fu-ture by trying to relate the outcomes found from the analysis of results obtained after cells irradiation.

However, the LET calculations, as well as yield and dose distributions of secondaries, will be repeated at higher energies in order to verify the method adopted here at energy range used in radiotherapy for deep seated tumours. Next step should be also the extension of the Monte Carlo method at three-dimensional geometries, in order to have more realistic predictions of spatial LET distribution. Different tissue compositions and inhomogeneities could be also taken into account.

The implemented functionality in the Hadrontherapy application will be in-cluded in the official version of the example and, thus, they will be found in the next official public release of the Geant4 toolkit. Indeed, the methods so far pre-sented can be applied to study the carbon fragmentation at the energy range typi-cally used in hadrontherapy.

5.2 Contributions of charged fragments

Figure 5.7. LET calculation for protons (left) and alphas (right) for 62 AMeV ¹²C ion beams on PMMA. A Bragg curve (in arbitrary unit) obtained by simulations in the same setup has been superimposed here and in the following plots, in order to have clear refer-ences on the different regions of interest (plateau, peak, tail).

0 2 4 6 8 10 12 14

Figure 5.9. LET calculation for ¹⁰B (left) and primary particles ¹²C (right).

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0 2 4 6 8 10 12 14

0 100 200 300 400 500 600 700 800

PMMA depth (mm)

LET (keV/um)

LET TRACK

Relative dose distribution LET only primary LET primary + secondary

Figure 5.10. Comparison between LET of primary ¹²C (red triangles) and total LET (green dots) for 62 AMeV ¹²C ion beams on PMMA.

Conclusions and perspectives

Monte Carlo methods represent one of the most effective tools for verifica-tion of dose computaverifica-tion in radiotherapy with carbon ion beams. The physical processes involved in this novel radiation therapy technique, as well as the in-volved biological aspects give to the description a level of complexity which analytical methods and experiment do not always cover at all. In this picture, Monte Carlo simulations are able to achieve a more realistic description of the physical processes, taking into account the effects due to the primary particles as well as to the secondary ones produced along the path in the matter. The tracking and the consequently interactions of each particle produced are possible for all the energy range of interest, and realistic geometrical configurations can be im-plemented in detail, taking into account the presence of different material com-positions. If reliable results have to be expected, the physical models imple-mented inside Monte Carlo codes have to be validated versus experimental data.

In particular, the accuracy of nucleus-nucleus interaction models is crucial in carbon ion therapy, in order to have a reliable prediction of the produced nuclear fragments.

In this work, the Monte Carlo toolkit Geant4 has been used. An extended validation of the Geant4 nucleus-nucleus models at the energy range of interest in hadrontherapy has not been fully achieved yet, especially for nuclear frag-ments production predictions on thin targets. The work discussed in this thesis represents a contribution in this direction. Indeed, the results carried out and

dis-146

cussed have allowed, at first, to compare the Binary Light Ion Cascade model implemented in the Geant4 toolkit with experimental data found in literature and, then, also with our data. Afterwards, the code has been used in order to es-timate the “radiation quality” of carbon ion beams used in radiotherapy, by com-puting the dose due to the charged fragments produced and calculating the depth dose distribution of Linear Energy Transfer (LET) characterizing both primaries and the different isotopes produced.

In the third chapter a preliminary comparison of the nucleus-nucleus Geant4 model with data of literature on neutron production cross sections is presented.

Interaction of 135 AMeV ¹²C beams on C, Al, Cu and Pb targets has been simu-lated. Comparisons between the double differential cross sections and angular distributions show a discrete agreement in the intermediate energy region of the neutron spectra, for intermediate angles. The prominent peaks present in the for-ward direction spectra are reproduced also by simulations, even if underesti-mated in the absolute value. The discrepancies grow up as the target mass in-creases. On the contrary an overestimation of neutron production at low energies is found, which increases as large angles where the de-excitation processes of target remnants mainly dominate. Angular distributions are quite well repro-duced, except at forward angles and for heavier mass targets.

Even if neutron production is widespread used for interaction models bench-marks, a more stringent verification for hadrontherapy purposes is achieved comparing the charged reaction products. This is the item of the forth chapter, where comparison of charged isotopes angular distributions are presented and discussed. Due to lack of data up to 400 AMeV regarding isotopes productions after carbon ion interactions, we have carried out an experiment at Laboratori Nazionali del Sud (LNS) - INFN of Catania. ¹²C ion beams, accelerated by a Su-perconducting Cyclotron (SC) at the energy of 62 AMeV impinge on a ¹⁹⁷Au thin target. Charged particles produced because of fragmentation of primary beams are detected by on Hodoscope detector, composed of 81 two-fold and 88 three-fold telescopes detectors. The apparatus has been simulated with Geant4, following a consequential approach: at first, a single telescope of the Hodoscope has been reproduced and output were inter-compared with a simpler, slightly

Conclusions and perspectives

147 less realistic but less time consuming method. Afterwards, once the reliability of the simplified method has been verified, the simulation of the whole apparatus has been provided by using this simplified geometrical model. The results show that, as expected, the angular distributions are forward peaked, especially for heavier fragments, confirming that projectile fragments are mainly produced in forward direction. The trend of the angular cross sections is quite well repro-duced by the Geant4 simulations for each isotope. Anyway, discrepancies in the absolute value up to one order of magnitude have been found. The fact that all the simulated isotopes spectra are underestimated suggest two possible explana-tions for the observed discrepancies. On one hand, they could be due to reliabil-ity of the total reaction cross section implemented in the code, which at this en-ergy range has been not completely validated. On the other hand, discrepancies may come from normalization problems. The latter is actually one of the main difficulties faced in the data analysis, because a direct measurement of the inci-dent particles number was not achieved. Indirect methods have been afterwards used for conversions of counts per solid angle into cross section per solid angle, which may have introduced systematic errors. This hypothesis seems to be in part confirmed by looking at the double differential spectra, which exhibit a similar general trend. However, measurements have been recently repeated and analysis is still in progress.

Finally, in the fifth chapter, aspects related to the effects of the mixed radia-tion field produced by the projectile fragmentaradia-tion have been treated. The cumu-lative effect of nuclear as well as electromagnetic interaction have been verified by comparing the Bragg curve of a 62 AMeV ¹²C beam interacting with water.

The experiment has been performed at LNS-INFN and the beam line has been simulated using and ad-hoc customizing the Geant4 Hadrontherapy application, developed some years ago for the simulation of the CATANA proton treatment facility. The results show a nice agreement between the two depth dose distribu-tions, confirmed by the comparison of the main dosimetric parameters used in clinical practice. A slight disagreement has been found in the peak-plateau ratio, which however can be explained considering the very steep distal fall-off of

The experiment has been performed at LNS-INFN and the beam line has been simulated using and ad-hoc customizing the Geant4 Hadrontherapy application, developed some years ago for the simulation of the CATANA proton treatment facility. The results show a nice agreement between the two depth dose distribu-tions, confirmed by the comparison of the main dosimetric parameters used in clinical practice. A slight disagreement has been found in the peak-plateau ratio, which however can be explained considering the very steep distal fall-off of