Università degli Studi di Catania

Transcription

1 Università degli Studi di Catania Dottorato di Ricerca in Fisica - XXII ciclo FRANCESCO ROMANO Monte Carlo simulations of carbon ions fragmentation in hadrontherapy PHD THESIS Tutor: Chiar.mo Prof. Salvatore Lo Nigro Supervisor: Dr. Giacomo Cuttone PhD coordinator: Chiar.mo Prof. Francesco Riggi

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3 Contents INTRODUCTION RADIOTHERAPY WITH CARBON ION BEAMS FROM THE CONVENTIONAL RADIOTHERAPY TO THE ION BEAM THERAPY RATIONALE OF ION BEAM THERAPY Physical basis of ion beam therapy Biological basis of ion beam therapy TREATMENT PLANNING IN ION THERAPY The INFN TPS project BEAM DELIVERY SYSTEMS AND ACCELERATORS FOR HADRONTHERAPY Beam delivery systems Accelerators in hadrontherapy HADRONTHERAPY FACILITIES IN THE WORLD GEANT4: A MONTE CARLO SIMULATION TOOLKIT THE MONTE CARLO METHOD THE GEANT4 SIMULATION TOOLKIT MAIN FEATURES OF THE GEANT4 TOOLKIT General capabilities and proprieties of the Geant4 toolkit Global structure Development of a Geant4 application...61 II

4 2.4 PHYSICAL PROCESSES IN GEANT Particle transport Electromagnetic processes Hadronic processes Modelling of ion interactions for hadrontherapy applications NEUTRON PRODUCTION FROM CARBON ION BEAMS INTERACTION ON THIN TARGETS THE RIKEN EXPERIMENT THE GEANT4 SIMULATION RESULTS AND COMPARISON WITH THE DATA Double differential cross sections Angular distributions INFLUENCE OF NEUTRON PRODUCTION IN HADRONTHERAPY FRAGMENTATION OF 62 AMEV CARBON BEAMS ON AU TARGETS THE EXPERIMENT AT LNS The detection system E-E identification technique and data acquisition SIMULATION OF THE EXPERIMENT WITH GEANT Simulation of a single E-E telescope detector Simulation of the Hodoscopes RESULTS AND COMPARISONS WITH THE EXPERIMENTAL DATA Data analysis Angular cross sections Double differential cross sections DEPTH DOSE DISTRIBUTION OF 62 AMEV CARBON IONS AND CONTRIBUTION OF FRAGMENTS DEPTH DOSE DISTRIBUTION The experimental setup Simulation with Geant Bragg peak comparison CONTRIBUTIONS OF CHARGED FRAGMENTS Dose and yield of secondaries LET calculations III

5 CONCLUSIONS AND PERSPECTIVES APPENDIX A.1 ELECTROMAGNETIC INTERACTIONS A.2 NUCLEAR INTERACTIONS BIBLIOGRAPHY ACKNOWLEDGEMENTS IV

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8 Introduction Radiotherapy, also called radiation therapy, is the treatment of cancer and other diseases with ionizing radiation. Ionizing radiation deposits energy that injures or destroys cells in the area being treated, the "target tissue, by damaging their genetic material and making it impossible for these cells to continue to grow. The major goal of Radiotherapy is to maximise the dose in the tumour volume and to spare the surrounding normal tissues as far as possible. In other words a compromise between the local control of the tumour and the possible emergence of complications has to be found in order to have an enhanced probability of success in the cure. Hadrontherapy is a radiotherapy technique which uses hadrons, i.e. collimated beams of compound particles made of quarks, for the sterilization of tumour cells. With this collective word, several forms of radiation therapy are usually indicated, such as neutrons, protons, pions, antiprotons, light ions and heavy ions. Heavy ion therapy is a novel technique of high precision external radiotherapy which, in particular, yields a better perspective for tumour cure of radioresistant tumour and it has given indisputably a boost to the use of radiation in the fight against cancer. Indeed, cancer represents today one of the most serious diseases which affect a continuously growing number of people in the world. In the developed country about 30% of people suffer from cancer and every year about oncological patients are treated with high-energy photons. Unfortunately, for the 18% of all cancer patients the local control of the primary tumour without metastases fails. Hence, in this case, innovative and more advanced techniques, such as ion ther-

9 Introduction apy, may increase the probability of cure. The driving force in more than one hundred-year history of conventional radiotherapy was the search of higher precision and greater biological effectiveness of the applied dose. The former could be reached by increasing the energy of the photons yielding a shallow decay of the dose in depth. The higher effectiveness was tested by the additional application of hyperbaric oxygen and drugs as radiation sensitizers. Finally it was the higher physical precision that contributed to the elevated tumour control rate in conventional radiotherapy with x- rays. Nowadays, sophisticated treatments like IMRT (Intensity Modulated Radiation Therapy) try to optimize the dose distribution and confine it within the tumour. In the contest of improving radiotherapy by better targeting and, on the other hand, achieving a larger effectiveness, the transition to hadron beams like neutrons, protons and heavier opens a new age in the radiation therapy field. First trials were done with neutrons, which showed how a greater tumour-control rate could be achieved, especially for radio-resistant tumours. However, because of their poor depth-dose distribution, the damage was too large also for normal tissues outside of the target volume, leading to sever side effects. The clinical use of protons and carbon ions was firstly proposed at the Berkley Cyclotron by Wilson, who measured depth dose profiles with a significant increase in dose at the end of the particle range, giving rise to the so called Bragg peak. First patients were treated with protons in 1954 and some years later first trials with heavier ions started. The motivation of this transition relies on the fact that heavier ions combine the more favourable dose distribution of the protons to the benefits coming by an enhanced Relative Biological Effectiveness (RBE). After several years of study and experimentation, around the 90 s the scientific community involved in hadrontherapy studies was concordant in indicating the carbon ions as the optimal choice. Till now about 3500 patients have been treated with carbon ions worldwide. Energies of about 400 AMeV are necessary to reach deep-seated tumours. The rationale of this choice has to be searched at macroscopic as well as at 2

10 Introduction microscopic level. The advantages of carbon ions, indeed, can be summarized as follows: - Carbon ions deposit most of their energy towards the end of the range, i.e. in the Bragg peak, where they can produce sever damage to the cells while sparing the transverse healthy tissues as well as tissues deeper located. - They penetrate the patient with minor lateral scattering and longitudinal straggling, which are about 3 times lower than protons ones. Moreover, carbon beams can be formed as narrow focused and scanning pencil beams of variable penetration depth, so that any part of a tumour can be accurately covered. - Carbon ion beams undergo nuclear interactions along their path inside the tissue, so that positron emitting isotopes are yielded, allowing the use of innovative techniques such as the on-line PET (Positron Emission Tomography). The location where the dose is deposited can be determined with a high level of precision, therefore critical structures can be preserved. - Beams of carbon ions show a favourable depth profile of the RBE. Indeed, in the entrance channel they mostly produce repairable damages, meanwhile in the last part of their path (where the target tumour is present) they cause more serious effects, showing enhanced RBE values in that region. Hence, for radio-resistant tumours they are more effective. These points clearly indicate the benefits coming from the use of carbon ions respect to the conventional Radiotherapy with photons. Moreover, the last point probably represents the main advantage of carbon ions also respect to protons. To completely take advantage from the use of carbon ions in radiation therapy, a powerful tool, able to handle all the physical and biological parameters involved in this radiation technique, has to be used. This task is fulfilled by the Treatment Planning System (TPS). The TPS is a complex computer tool which permits the computation of the biological effective dose to be delivered to the patients, translating the prescribed dose in a set of information necessary to the radiation treatment design. Some commercial products are already implemented in some ion therapy centres, but substantial improvements could be achieved. For this aim, INFN (Istituto Nazionale di Fisica Nucleare) has recently approved 3

11 Introduction the TPS project, in which the Laboratori Nazionali del Sud (LNS) of Catania are involved. In the last years INFN has showed a great scientific interest on hadrontherapy. Moreover at LNS a long experience in this field has already been matured. Since 2002, indeed, the facility CATANA (Centro di Adroterapia e Applicazioni Nucleari Avanzate) is in operation at LNS for the treatment of eye melanoma. Proton beams are accelerated by a Superconducting Cylocron (CS) and extracted at 62 MeV. With this energy, the depth of 3 cm in eye-tissue can be reached and thickness of some cm can be covered exploiting the passive scattering beam line realized at the CATANA facility. Up to now, about 200 patients have been treated and a local control of the tumour has been obtained in the 96% of cases. The work and the results discussed in this PhD thesis have been carried out at LNS within the research group acting in the hadrontherapy field and developed in the contest of the TPS project. In particular, the issues related to the production of secondary particles by inelastic interaction of the carbon ions with the matter have been faced. Indeed, a part of the highlighted advantages, the main drawback coming from the use of carbon ions relies on the nuclear fragmentation of carbon ion beams, with the consequent production of charged fragments. The reaction products are mostly produced by projectile fragmentation, so that they are characterized by an emission velocity close to that of the primary particles. Final result is the formation of a typical tail of dose just behind the Bragg peak, which is due to the contribution of fragments created along the path of primaries. The higher is the energy of the incident ion, the lower is the peak, because of a great number of inelastic interactions, and the greater is the amount of extra-dose in the tail. Fragmentation reaction may occur already at the stage of acceleration of the beam, especially when passive scattering systems are used as beam delivery systems. Anyhow, the contribution of fragments production coming from the interaction of the beam along the beam transport line can be drastically reduced by means of relatively simple technical expedients. What is instead unavoidable is the nuclear fragmentation inside the patient itself, which is somewhat intrinsic. This aspect has to be taken into account by the TPS calculations, especially 4

12 Introduction when critical structures are close to the distal part of the Bragg peak, where an undesired amount of dose could override the advantages above discussed. However, consequences arising from carbon ion fragmentation have to be undertaken not only beyond the Bragg peak, but also in the whole radiation field, because nuclear reactions occur since the first depths in the patient-tissue. This fact means that, as the primary carbons traverse the matter a continuously growing number of secondary particles is produced travelling together with the primary ions and giving its contribute in terms of dose. In other words, a mixed radiation field is created along the depth, composed mostly by the primary but also by a not negligible number of nucleons and light ions. What complicates the description is the fact that each isotope is characterized by a different RBE. Biological effect, indeed, depends on various quantities, such as the atomic number and the local ionization density of the specific particle considered. The latter is quantitatively described by the Linear Energy Transfer (LET). All these aspects get more importance in the case of using more than one irradiation field, creating even more complexity in mixed radiation fields. A reliable TPS must take into account all these parameters in order to accurately compute the biological dose to be delivered. It is evident that a pure analytical algorithm could not be able to reproduce a so complex picture. At this aim Monte Carlo methods probably represent the most powerful tool for accounting all these different aspects and it should be used as a base of development or at least as a tool of verification for TPS systems design. Monte Carlo methods, indeed, permit the dose computation in a more realistic way than analytical approach and, moreover, can give feedbacks also in particular configurations where experimental verification is difficult or not possible. The aspects just mentioned are wide discussed in the first chapter. In the present work, the Monte Carlo code Geant4 has been used. Geant4 is a simulation toolkit used for particle tracking, able to simulate the interaction of particles with the matter and the production of secondary particles in a wide range of energy. Already used worldwide in the medical domain, in this work it has been exploited to study the fragmentation of carbon ion beams at energy range of interest in hadrontherapy. A general description of its main features to- 5

13 Introduction gether with the physical models of interest, are discussed in the second chapter. The accuracy of a Monte Carlo simulations is related to the reliability of the physical processes implemented, in particular the nuclear ones. Hence, for a realistic estimation of fragmentation products, nucleus-nucleus models inside the code have to be validated versus experimental data. In this work the Geant4 Binary Light Ion cascade model has been used. Not many recent and accurate data are available in literature for carbon ion interactions on thin targets at the energy range of interest in hadrontherapy ( AMeV), especially for nuclear fragments production. The situation is slightly different for neutron yields experiment, for which more accurate data can be found. Hence, a preliminary validation of the nucleus-nucleus Binary Light Ion cascade model has been carried out by comparing neutron double differential cross sections obtained with Geant4 at different angles with experimental data found in literature. In particular data on 12 C ion beams at 135 AMeV impinging on C, Al, Cu and Pb targets have been selected for the benchmark. Results and discussion are shown in the third chapter. Even if some data on fragments production are available at very low energy (up to about 20 AMeV), there is a lack of data for charged fragments production at more intermediate energies. To contribute on covering this gap, an extensive campaign of measurements has been carrying out at LNS. We have already performed a first experiment with 12 C ion beams accelerated at 62 AMeV impinging on a thin Au target. The apparatus used for the reaction products detection was a Hodoscope of two-fold and three-fold telescope detectors, displaced in order to detect most of light projectile fragments which are mainly forward produced. The experimental apparatus has been simulated using the toolkit Geant4. Used methods together with comparisons of angular distributions of charged isotopes and double differential cross sections are shown in the forth chapter. A detailed work has been developed in order to find the best solution in the simulation approach, trying to obtain the final results with acceptable computation time, preserving the level of accuracy. A first realistic simulation approach has been followed by a simpler and more performing one, used for final production of outputs. 6

14 Introduction After this preliminary validation work, which by the way has to be constantly updated in future with new benchmarks and verifications, a characterization of the mixed radiation field due to carbon ions at 62 AMeV has been undertaken in the fifth chapter. Here, a comparison of the total depth dose distribution of a 62 AMeV carbon ion beams transported in air and interacting with a thick water phantom has been also discussed. The experiment has been carried out at LNS, acquiring the depth dose with a ionization chamber positioned at different distances from the phantom window, thanks to an automated system. A brief description of the experiment is done and a comparison with the simulation is shown. Results have been obtained using the Hadrontherapy application, which is an advanced example included in the public release of the Geant4 toolkit. It has been previously developed by our group for the simulation of the CATANA proton beam line, and here customized on purpose in order to reproduce the experimental beam line used for the carbon ion Bragg peak acquisition. Instead, in the second part of the chapter an estimation of the dose and fluence contribution due to the charged fragments produced by thick PMMA targets have been carried out, with the aim of characterizing the mixed radiation field produced further to the primary particles fragmentation. As already pointed out, a detailed description of the radiation quality, i.e. isotopes, fluences, kinetic energy spectra and LET, have to be known for a better correlation with the biological damage on tissue. A Monte Carlo calculation has been developed to take into account all these parameters, computing the depth distribution of avaraged LET weighted on the isotopes fluence, including the contribution of all the secondary particles produced along the carbon ions path. The implemented method could allow a better interpretation of biological responses obtained with human melanoma cells irradiations, whose results are still under analysis. The method, developed inside the Hadrontherapy application, is briefly discussed and predicted results are shown. Finally, the conclusions are reported, where a summary of the main obtained results is presented with future developments and perspectives. 7

15 1. Radiotherapy with carbon ion beams Cancer can broadly be defined as the uncontrolled growth and proliferation of groups of cells. In 1985, deaths were attributed to cancer, in the countries of European Community, which means one out of five cancer patients dies from this disease [1]. In developed countries about 30% of people suffer from cancer, 58% of them at the stage of a localised primary tumour. Unfortunately an increasing trend of the cases is still observed and nowadays about 1 million deaths per year are due to this disease [2]. The growing number of patients suffering from cancer requests for adequate treatments, which may depend on tumour type, stage and diagnosis. In Europe about 45% of all the patients is cured, which means that these patients have a symptom-free survival period exceeding five years. Today, there are various possible methods to heal cancer: surgical removal of the tumour tissue, radiotherapy, chemotherapy and immunotherapy. Surgery and radiotherapy alone are successful in 22% and 12% of cases respectively and, when combined, they account for another 6% of cases. However, for the 18% of all cancer patients the local control of the primary tumour without metastases fails; in these cases, patients could be cured successfully if improved local treatment techniques were available. One of these is the radiotherapy with proton and ion beams, more recently developed, which today is very attractive as regards the maintenance of high quality of life [3].

16 1.1 From the conventional Radiotherapy to the Ion Beam Therapy 1.1 From the conventional Radiotherapy to the Ion Beam Therapy There has rarely been another finding of a physical phenomenon with a similar impact to medicine as Wilhelm Conrad Roentgen s discovery of the X-rays in A carefully reasoned description of his work together with the famous radiograph of his wife s hand was published in a short time: Eine neue Art von Strahlen (i.e.: On a new kind of rays ). While the implication for medical diagnostic had already immediately been foreseen by Roentgen himself, the idea of a benefit from their use in benign and malignant conditions came later. The first successful treatment, which also gave a scientific proof of the therapeutic effectiveness of X-rays was carried out in 1896 by Prof. Leopold Freund in Vienna, who irradiated and removed a hairy mole on the forearm of a patient [4]. But though some negative effects on the skin were soon observed, it took until 1904, when Edison s assistant Clarence Dally died following injuries to his hands and arms [5], that physicians and physicists took the possibly fatal power of the X-rays into account. Up this time and also for some years later radiation therapy remained a more or less empirical science where the major progress originated from clinical application. In the historical development of radiotherapy two general tendencies are visible: the clinical results are improved on one hand by a greater conformity of the applied radiation to the target volume and on the other hand by an increased biological effectiveness of the radiation [6]. Regarding the first one, improvements in radiation therapy are sensibly influenced by the technological progress, which implies the born of different techniques. The earliest radiation sources were provided by gas-filled x-ray tubes. It was W. D. Coolidge who, in 1913, did the groundwork for the present-day x-ray techniques by developing a vacuum tube with a hot tungsten cathode. The earliest tubes of this type, manufactured for general use, operated at a peak voltage of 140 kv with a 5 ma current. Unfortunately, x-rays generated by these tubes were fairly soft and, from the medical point of view, the depth-dose curves were 9

17 1. Radiotherapy with carbon ion beams particularly disadvantageous since the maximum dose would be delivered at the skin surface and then would rapidly fall off with the depth in the tissue (Figure 1.1). Figure 1.1. Depth dose profiles of X-rays, Co-gamma and Roentgen-Bremmstrahlung. It is clearly visible the different dose contribute to the skin. For this reason, during the early experience with vacuum x-ray tubes, first attempts were also made at searching for sources of a more penetrating radiation with attention focused on gamma emitting radionuclides. We are around in the 1950 s, which represents the era of cobalt sources for radiotherapy, containing several kilo curies of 60 Co. The energy of the emitted gamma made it possible to obtain far better depth-dose curves, showing a maximum at about 5 mm below the skin surface and markedly decreasing the dose to the superficial tissues, as visible in. However, the most important developments in radiotherapy arise from the megavoltage therapy. A high voltage accelerator was developed in 1932 by R. Van de Graaff and five years later the first hospital-based accelerator of this type, a 1 MeV air-insulated machine, was installed in Boston. Dramatic increase in the photon radiation energy was made possible by the development of the betatron, by D. W. Kerst, who in 1943 suggested its use as a radiation therapy tool. Anyway, the rilevant weight together with the low beam intensity 10

18 1.1 From the conventional Radiotherapy to the Ion Beam Therapy and the small treatment field area, have contributed to the discontinuation of its production thirty years later. In the mean time, the advances made during the World War II made it possible to use microwave generators for electron acceleration, leading to the born of the first radiofrequency linear accelerators, which were soon to take up a dominant place on the world market of medical accelerators [3]. By that time to today several technological improvements have led up to the development of innovative techniques, with the aim of overcoming limitations of an exponentially decreasing depth dose distribution and achieving an acceptable level of conformation. Nowadays, the most recent Intensity Modulated Radiotherapy (IMRT) uses multiple beams (6-10 entrance ports) pointing towards the geometrical centre of the target in order to conform the dose to the target and spare as more as possible the surrounded healthy tissues. The beams may be non-coplanar and their intensity is varied across the irradiation field by means of variable collimators ( multi-leaf collimators), that are computer controlled. Nowadays in the developed countries, every years about patients out of 10 million inhabitants, are treated with high-energy photons and about 8000 electron linear accelerators (linacs) are used worldwide for cancer treatment [7]. In other techniques, radioactive sources of low range emitters are implanted directly into the tumour to achieve a target-conform exposure. Intra-operative irradiation also tries to spare healthy tissue while maximizing the tumour dose at the same time. As already mentioned, the second tendency which drives the historical development of radiotherapy is the searching of an increased biological effectiveness of the radiation. This interest led to the study of the effects of hypoxic sensitizers to decrease the radio-tolerance of some radioresistant tumours [8]. This method, however, fell short of expectations. Anyway, the greater interest to selectively increase the biological effectiveness opened also investigations on different kind of radiations, attempting to replace the electromagnetic radiation. Indeed, in order to overcome both the physical and the biological limitations of the conventional radiotherapy the use of neutrons, protons and heavier charged particles was proposed, which led to the born of the Hadrontherapy. 11

19 1. Radiotherapy with carbon ion beams Hadrontherapy is a collective word covering all forms of radiation therapies which use beams of particles made up of quarks: neutrons, protons, pions, antiprotons, light ions and heavy ions 1. As in the case of conventional radiotherapy, the use of hadrons for medical applications is sensibly influenced by the technological progress and it is strictly related to the historical development of the accelerators. Hadrontherapy started in 1930 with the invention of the cyclotron by Ernest Lawrence. In 1935 he asked his brother John, who was a medical doctor in Yale, to join him in Berkley and use the new powerful accelerator for medical purposes [10]. The two main applications were the production of radioisotopes and, later, the therapeutic use of fast neutron beams. At the end of 1938, the first patients were treated with neutrons but the technique was primitive and the doses given to healthy tissues too high. Indeed, even if a better tumour control was achieved thanks to the increased biological effectiveness of neutrons respect to photons [11], the poor depth dose profile unfortunately compensated this advantage with severe late effects in normal tissues. For this reason, some years later, in 1948, after the effects evaluated on 226 patients, R. Stone concluded that neutron therapy had not to be continued [12]. Presently, therapy with low-energy neutrons has mostly phased out but high-energy neutrons and boron neutron capture are still subjects of clinical interest [13]. The beginning of hadrontherapy with charged particles is due to Robert Wilson, who was one of Lawrence s students. In 1945 he designed a new 160 MeV cyclotron and, one year later, one year later proposed the use of proton beams in radiation oncology [14]. In fact he had measured depth dose profiles at the Berkley cyclotron with a significant increase in dose at the end of particle range, the so called Bragg peak, which had been observed fifty years before in the tracks of alpha particles by W. Bragg 2 [16]. 1 The term Hadrontherapy was widely accepted and increasingly used since the 1 st International Symposium on Hadrontherapy held in 1993 in Como and organized by U. Amaldi, who first proposed this word in 1992 [9]. 2 Already in 1904 W. Bragg described the peak absorption in the air of alpha particles, which he depicts as [...] a more efficient ionizer towards the extreme end of its course [17]. 12

20 1.1 From the conventional Radiotherapy to the Ion Beam Therapy Figure 1.2. One of the original plots of the Bragg curve, from papers of W. Bragg [17]. Due to Bragg peak, the dose can be concentrated on the tumour target sparing healthy tissues better than what can be done with X-rays. It is interesting to remark that in his paper Wilson mainly discusses protons but he also mentions alpha particles and carbon ions. Two years later, researchers at the Lawrence Berkley Laboratory (LBL) conducted extensive studies on protons and confirmed the predictions made by Wilson. In 1954, the first patient was treated at Berkley with protons, followed by helium treatment in 1957 and neon ions in 1975 [15]. In the first trials at Berkeley, beam application methods used in the treatments were adapted from conventional photon therapy, in which the photon beam is passively shaped by collimators and absorbers. Thus, the energy modulation of charged particle beams was first performed with modified collimator and absorber techniques [19]. This was due to the fact that, at these early times, computer power was too poor for a control system necessary for more modern active beam techniques (see 1.4.1). Although these first methods did not take advantage of the possibilities of charged particles to a maximum, dose distributions could be reached which were superior than those of photon therapy of that time. Especially the increase towards the end of the particle range produced doses that are higher in the target than in the entrance channel (see 1.2.1). In ad- 13

21 1. Radiotherapy with carbon ion beams dition, the fast fall off of dose beyond the Bragg maximum allowed field geometries never possible with electromagnetic radiation. These treatments were performed by mean of particle accelerators that had originally been built for nuclear Physics experiments and were then adapted to tumour therapy. This was the case in Berkeley as well as at Harvard proton cyclotron, which made the largest impact on the development of protontherapy and where, up to now, the highest number of patients have been successfully treated. Some years later, other nuclear physics laboratory in USSR and Japan set up proton beams for therapy and in 1984 the Paul Scherrer Institute (Switzerland) did the same. The clinical proton beam currently used in the facility CATANA for eye melanoma treatments at Laboratori Nazionali del Sud (LNS, Catania, Italy) is also an example of beams produced by a superconducting cyclotron (SC) for nuclear Physics experiments and adapted for the therapy. Even if many times it was felt and said that the hadrontherapy field could not develop without dedicated facilities, this step took almost twenty years. Indeed, the first hospital-based centre was built at the Loma Linda University Center (California), which signed an agreement with Fermilab and treated the first patient in Afterwards, a smooth conversion from a physics laboratory to a hospital facility took place in Japan, where from 1983 to 2000 about 700 patients were treated at the Proton Medical Research Center (PMRC). Finally, USA proton therapy was further expanded during the 1990s. Over patients have worldwide been treated with proton beams by now and other facilities are under construction or in planning stages. [15]. As already mentioned ions heavier than protons, such as helium and argon, first came in use at Berkley in 1957 and 1975 respectively, where about 2800 patients received treatments to the pituitary gland with helium beams. Because of their increased biological effects, the interest toward this kind of beams grew up, so that about twenty years later argon beams were tried at the Bevalac facility (LBL). The main reason for an elevated biological effect is the increase in ionization density in the individual tracks of the heavy particles, where cellular damage becomes clustered and therefore more difficult to repair [20]. This propriety was found useful for treatment of radio-resistant tumours, but problems 14

22 1.2 Rationale of ion beam therapy arose owing to non-tolerable side effects in the normal tissues. Indeed first trials with silicon and neon ions did not give good results and, towards the end of the program, it was found that the charge of these ions was too large: undesired effects were produced in the traversed and downstream healthy tissues [21]. Bevalac stopped operation in After several studies and experimental results, only at the beginning of the 90s carbon ions were chosen as the optimal ion option. In fact in the entrance their effects are similar to X rays and protons ones, while just at the end of their path in the matter, ionization density is definitely larger and more serious damage are there produced to the cellular systems (see 1.2.2). Thus light ions produce a radiation field qualitatively different from ones due either to photons or protons and succeed in controlling radio-resistant tumours, which respond poorly to conventional radiotherapy. The carbon choice was made in Japan by Y. Hirao, who proposed and built HIMAC (Heavy Ion Medical Accelerator in Chiba) in the Chiba Prefecture [22]. In 1994 the facility treated the first patient with a carbon ion beam of energy up to 400 AMeV, corresponding to a maximum range of 27 cm in water. In the same years the pilot project is launched by G. Kraft at GSI (Darmstadt, Germany). Since 1994 other three hospital-based dual facilities (for carbon ions and protons) have either started treating patients or entered the construction phase. 1.2 Rationale of ion beam therapy The main aim of radiotherapy is the loco-regional control of the tumour and, in some situations, of surrounding diffusion paths. In order to reach this scope, a sufficiently high dose must be delivered to the tumour, which may be considered as the target, in physical terms. It could be said that the target tumour is successfully treated only if the applied dose is arbitrarily high enough but in practice it is more complicated. This can be easily understood starting from the definition of dose and looking at the so-called dose-effect curve. By definition, the absorbed dose (D), or simply dose, is the ratio between the energy E D imparted by the irradiation to a small volume of material (tissue) and 15

23 1. Radiotherapy with carbon ion beams the mass of this volume of material. The ratio D = E D /m is measured in gray (Gy). In conventional radiotherapy doses of the order of Gy are deposited in the tumour tissues in amounts of about 2 Gy per session over about 30 days. In the hypothesis of a fairly accurate identification of the target, it is possible to evaluate the probability of obtaining the local control of the tumour through the analysis of the dose-effect curves. They represent for tumour tissues the possibility of obtaining the desired effect as a function of the dose delivered, and for healthy tissues the probability of producing serious or irreversible damage, always as a function of the dose absorbed by the same tissue. In Figure 1.3 the thick line represents, as function of dose absorbed, a hypothetical dose-effect curve for a generic tumour tissue. Figure 1.3. Dose-effect curves for neoplastic (A) and normal (B) tissues; compromise between local control of the tumour and damage to the surrounding healthy tissues is described by the ratio D 2 /D 1. The dashed line represents a dose-effect curve for the healthy tissue involved in the irradiation. As shown, to the absorbed dose necessary to achieve the probability close to 100% of obtaining local control of the tumour corresponds also a very high probability of producing serious complications in the healthy tissue, when this receives the same dose [2]. So, in the daily practice, the compromise between the local control of the tumour and the possible emergence of complica- 16

24 1.2 Rationale of ion beam therapy tions has to be found and it represents the key to have more chance of success in cancer treatment. The possibility to find such a compromise can be expressed quantitatively by the therapeutic ratio. It is defined as the ratio D 2 /D 1 between the dose corresponding to a 50% probability of producing complications D 2 and the dose corresponding to a 50% probability of obtaining the local control of the tumour D 1. On the basis of these considerations, it is clear that the probability of curing the tumour without unwanted side effect increases in line with the ballistic selectivity or conformity of the irradiation delivered [3]. As it will be discuss more in detail in section 1.2.1, the probability of curing tumours can be increased by using charged hadrons beams because the adsorbed dose is more confined in the tumour tissue than electrons and photons one; this allows an enhanced ballistic precision. Moreover hadrons show increased biological effects respect to electromagnetic radiation and also respect to protons (see section 1.2.2). This last feature makes them more successful also in the treatment of radio-resistant tumours. Thus, considering these two different aspects, the therapeutic advantages of ion beams when compared to electron, photon and also proton beams can be found at a macroscopic scale (high level conformation) as well as at the microscopic scale (possibility of varying the radiobiological properties). The latter deals with microscopic distribution of the deposited energy, which changes when different ion beams are considered. For this reason it is often said hadrons as densely ionizing radiation, in contrast to the sparsely ionizing radiation such as X-rays, γ-rays and electrons Physical basis of ion beam therapy The most prominent feature of charged particle beams for their use in radiation therapy is the inversed dose profile, i.e. the increase of energy deposition with penetration depth, which represents the most evident difference respect to electromagnetic radiation. As already mentioned, the increase of ionization density with range was first described for α particles in a publication by Bragg in 1903 [23], and was later confirmed for protons and heavier ions by Wilson[14]. In the interaction of photons with the matter the energy loss is mainly due to three processes: photoelectric effect, Compton scattering and electron-positron 17

25 1. Radiotherapy with carbon ion beams pair production. The relative probability of one of these interactions is a function of the incident photon energy and the atomic number Z of the absorbing material. On the other hand, the energy of the ions is mostly transferred to the target electrons that are emitted as δ-electrons (see section A.1). More than three quarter of the dissipated energy is used for the ionization process and only 10 to 20 % for the target excitation [24]. The interaction strength is directly correlated with the interaction time. At high energies the energy transfer to the target is small but grows when the particles are slowed down. The different physical processes which characterize respectively electromagnetic radiation and charged particles directly affect the different depth dose profiles. In Figure 1.4 (on the left) the depth dose profiles of proton and ion beams is compared to those of protons, neutrons and electromagnetic radiation. Figure 1.4. On the left: comparison of depth dose profiles of photon, neutron, proton and carbon ion beams; the inverse dose profile of protons and ions is clearly visible in contrast to the less favourable profiles of electromagnetic radiation and neutrons. On the rigth: Construction of an extended Bragg-peak by superimposition of single Bragg-peaks of different energy. For low-energy X-rays the stochastic absorption by photoelectric and Compton processes yields an exponential decay of absorbed dose with penetration. For greater photon energies the produced Compton electrons are strongly forwardly scattered and transport some of the transferred energy from the surface to greater depth, yielding an increase in dose in the first few centimetres. For high energy 18

26 1.2 Rationale of ion beam therapy electron Bremsstrahlung, which is mostly used in conventional therapy, this maximum is shifted a few centimetres from the surface of the patient s body sparing the very radio-sensitive skin. In contrast, the energy deposition of charged particles like protons or heavier ions shows a completely different trend. When ions enter an absorbing medium, they are slowed down by losing their kinetic energy. The specific ionization increases with decreasing particle velocity, giving rise to a sharp maximum in ionization near the end of the range. Thus the depth-dose distribution is characterized by a relatively low dose in the entrance region (plateau) near the skin and a sharply elevated dose at the end of the range, which in this case is finite and energy dependent (peak). The ballistic precision of charged particles respect to electromagnetic radiation is evident. Indeed, in the surrounding healthy tissues before or just behind the Bragg peak the dose released is minimized as more as possible respect to the target volume and a better compromise is achieved. However, a monoenergetic beam with a narrow Bragg peak makes possible to irradiate a very small, localized region within the body with an entrance dose lower than that in the peak region [26]. In practical use in therapy, the tumour volume to be treated is normally much larger than the width of the Bragg peak and the lateral spot of the particle beam. In order to fill the target volume with the necessary amount of stopping power particles the peak has to be spread out in the longitudinal direction. This is achieved by superimposing several Bragg peaks of different depths obtained by appropriate selection of distribution of ion energies. The resulting dose depth distribution, known as spread-out Bragg peak (SOBP) shows an extended Bragg peak area which has to accurately overlap the target volume (Figure 1.4, right). The use of ion beams allows tumor conform treatments of enhanced quality respect to those obtained by conventional radiotherapy [25]. In fact, even if the peak-plateau ratio decreases for SOBP respect to a pristine Bragg peak, the final result is still satisfying if we look at depth dose profile for photon beams. The interesting behaviour of charged particles is due to their electromagnetic interactions with the matter. Within the range of therapeutically relevant energies of some hundred AMeV down to rest, the process of energy loss is domi- 19

27 1. Radiotherapy with carbon ion beams nated by inelastic collisions with atomic electron. The average energy loss per unit path length, the so called electronic stopping power, is well described by the Bethe-Block formula [27-29]: de = Z 2 f () v (1.1) dx with Z and v respectively the charge and velocity of the projectile. Details on the dependencies of the formula and relative corrections are given in section A.1.1. In the energy range of interest in hadrontherpy, de/dx is mainly dominated by the term 1/v 2 1/E. Moreover, at very low energies, when the projectile velocity is comparable to the velocity of the orbital electron in the materials, there is a high probability that the projectile picks up these electrons. Thus the charge changes and Z has been replaced by Z eff (the effective projectile charge) which is well described by the Barkas formula [30]: Z eff 2 125βZ 3 = Z(1 e ) (1.2) with β = v/c. Considering these dependences, at not relativistic energies, the energy loss rate grows up as the kinetic energy of the projectile decreases along the penetration depth, in particular in the last few millimetres of the particle path where it shows a much steeper rise. That is the reason why the distribution of the ionizing density produced by the charged particle along the track is characterized by a rather constant plateau, followed by a sharp maximum towards the end, where gives rise at the Bragg peak. Anyway, at the end of the path the stopping power drops because of the rapid reduction of the effective charge Z eff to very low energy values. The relation between electronic stopping power and particles energy is shown in Figure 1.5 for energy range and ions which are of interest in hadrontherapy. As expected, each particle exhibits a de/dx curve which is distinct from the other particle types. Once the stopping power is known, it is possible to calculate the range R of a charged particle. It is the distance it travels before coming to rest. The reciprocal of the stopping power is the distance travelled per unit energy loss. 20

28 1.2 Rationale of ion beam therapy Figure 1.5. Energy loss of different particles as function of the energy; as the atomic number increases the stopping power grows up. Therefore the range R(E) of a charged particle having kinetic energy E is the integral of the reciprocal of the negative stopping power [34]: 0 1 de R ( E) = de (1.3) dx E 0 This equation provides a well defined value but in reality the energy loss processes are affected by statistical fluctuations which are responsible of a dispersion of the path length (range straggling). Moreover, fluctuations in energy loss and multiple scattering processes yield an almost Gaussian energy loss distribution f( E) given by [31][32]: 2 ( E E ) 1 2 2σ f ( E) = e (1.4) 2πσ where σ is the straggling parameter which expresses the half-width at the (1/e)-th height [33]. Hence, statistical fluctuations of energy loss cause a smearing of the range of the stopping particle beam and, consequently, a larger width of the Bragg peak experimentally measured. Range straggling effects for ion beams vary approximately inversely to the square-root of the atomic mass and increase 21

29 1. Radiotherapy with carbon ion beams as the penetration depth grows up. Indeed, at the same penetration depth heavier ions show narrower Bragg peaks and also a steeper distal fall-off, which has a positive effect on the final level of conformation of the radiation to the tumour. A typical fragmentation tail is also well distinguished in case of heavy ions, but this aspect with its implication will be discussed ahead (Figure 1.6). Figure 1.6. Depth dose distribution of photons, protons and carbon ions of 254 AMeV and 300 AMeV; comparison with 135 MeV protons (having the same range value) shows the difference in the Bragg peak width. Moreover, higher incident energies (i.e. higher ranges) correspond to higher peak widths. For greater particle energies and longer penetration the half width of the Bragg maximum becomes larger and the height smaller. Typical values for Carbon ions are given in Table 1.1 [35]. Energy (MeV/u) Range (mm) FWHM (mm) Table 1.1. Typical values for carbon ions Bragg curve. 22 Anyway, more significant for clinical applications are the consequences of

30 1.2 Rationale of ion beam therapy multiple scattering processes in the lateral scattering of the beam, especially when the beam has to pass near critical structures which sometimes can be placed adjacent to the tumour volume. Multiple scattering of an incident ion stems from the small angle deflection due to collisions with nuclei of the traversed material. Numerous small angle deflections in an ion beam lead to lateral spreading of the incident ions away from the central trajectory resulting in larger divergence of the beam. Elastic Coulomb scattering dominates this process with a small strong-interaction scattering correction. The angular distribution of the scattered particles is roughly Gaussian for small deflection angles, and the mean beam deflection is approximately proportional to the penetration depth. The Coulomb scattering of the projectiles is described very precisely in the theory of Molière [36] [37]. Measurements of proton scattering confirmed this theory [38] and a parameterization for small angle scattering having an angular distribution f(α) [39]: 2 α 1 2σα ( α ) = e f (1.5) 2πσ α with 14.1MeV d 1 d σ α = Z + p 1 log 10 (1.6) βpc Lrad 9 Lrad where σ α is the standard deviation, p the momentum, L rad the radiation length and d the thickness of the material. At low energy, multiple coulomb scattering increases and, in general, the effects are more evident for lighter ions. In Figure 1.7 (left) the lateral scattering of a therapy beam is compared for 21 MeV photons, for protons and carbon ions of a range of 14.5 cm in water. The comparison clearly that for protons the lateral scattering exceeds the photon value for penetration depth larger than 7 cm. The lateral deflection of carbon beams is better than 1 mm up to a penetration depth of 20 cm. As expected, carbon ions exhibit a less pronounced lateral deflection, and they are well confined respect to proton beams. This fact represents another advantage of the clinical use of carbon ion 23

31 1. Radiotherapy with carbon ion beams beams and it also contributes to an enhanced ballistic precision. Figure 1.7. On the left: comparison of the lateral scattering of photon, proton and carbon beams as function of the penetration depth [35]. On the right: comparison between lateral distribution of dose deposited by proton and carbon ion beams having approximately the same range; the comparison clearly shows the improved selectivity of carbon ion beams respect to protons [40]. Effects on lateral broadening are much more evident looking at the so called apparent penumbra, which is the sharpness of the lateral dose fall-off [40]. Heavier ion beams exhibit sharper lateral dose fall-offs at the field boundary than lighter ions: in Figure 1.7 the penumbras of proton and carbon beams are compared. The penumbra width increases essentially linearly with the penetration depth of the beam. For low-z ions, such as protons, sharpest dose fall-offs are obtained when the final collimator is at the surface of the patient. For higher- Z ion beams, such as carbon ion beams, active scanning techniques without collimations will produce narrow penumbras. So far, physical effects related to electromagnetic interaction has been discussed. When a heavy ion beam passes through beam line elements and finally traverses the tissues, nuclear reactions take place as well, which cause a small but not negligible amount of nuclear fragments. The result is a reduced number of primary particles reaching the tumour which are replaced by a certain number of secondary particles. An interesting potential for quality control arises from nuclear fragmentation, which by far compensate for the disadvantages, discussed 24

32 1.2 Rationale of ion beam therapy ahead. Concerning the advantages, the stripping of one or two neutrons from the projectile 12 C yields the positron emitting isotopes 11 C and 10 C with half-lifes of 20 min and 19 sec, respectively. The stopping point of these isotopes can be monitored by measuring the coincident emission of the two annihilation gamma quanta following the β + decay. In general, most of the lighter fragments have the same velocity as the primary ions at the collision [42]. The range of these fragments is given by the formula: Z M 2 pr fr R fr = Rpr (1.7) 2 M prz fr with R being the range, Z the atomic number, M fr and M pr the masses of the fragments and the projectiles, respectively [25]. Hence, the range of carbon isotopes is only slightly shorter than that of the primary particle: A R R( C) = R( C) (1.8) 2 Z 12 Thus, from the measured distribution of the annihilation quanta, the range of the stopping particles can be controlled and compared to the calculated range in the treatment planning, providing an in-situ beam monitoring using Positron Emission Tomography (PET). In fact, it is further expected that the spatial distribution of β + -activity induced by heavy-ion beams is strongly correlated with the corresponding dose distribution. Even if carbon beams mostly produce 11 C and 10 C nuclei via projectile fragmentation, also 15 O nuclei are produced from the target [41]. In Figure 1.8 depth-dose distributions of β + -emitters and primary particles in case of proton and carbon ion beams are shown. In contrast to the proton irradiation, the maximum of β + -activity produced by carbon ions is clearly visible. In this second case, the β + -activity shows a prominent maximum, shortly before the peak, formed by 11 C and 10 C fragments of the 12 C projectiles. Hence, all the 12 C isotopes are stopped before the Bragg peak and the sharp fall-off in the β + - activity distribution clearly indicates the position of the Bragg peak [43]. 25

33 1. Radiotherapy with carbon ion beams Figure 1.8. Calculated depth distributions of deposited energy (dash-dotted curves) and β + activity (histograms) for (a) 110 MeV protons and (b) AMeV 12 C nuclei in the PMMA phantom. The distributions of 11 C, 10 C and 15 O nuclei are shown by the longdashed, dashed and dotted histograms, respectively, and their sum is shown by the solidline histogram. The distribution of actual e + -annihilation points is shown by the solidline histogram [43]. All the distributions have been calculated with the Monte Carlo code Geant4. PET monitoring is of great interest in the hadrontherapy field because provides a direct measurement of the beam distribution inside the patient and it represents as well another great advantage coming from the exploitation of carbon ions in radiotherapy. The calculation of projectile range is, in fact, a critical point in treatment planning because human body is composed of a large variety of materials with different densities (bones, muscles, fat, air-filled cavities, etc). Unfortunately, from another point of view beam fragmentation represents also the main disadvantage of using carbon ion beams for tumour treatments. Ion beams suffer nuclear reactions by interacting with the elements present along the 26

34 1.2 Rationale of ion beam therapy beam line as well as inside the tissue itself, as already discussed. The first contribution can be opportunely reduced, and it is strictly dependent on the beam delivery system used; active systems are in this sense advisable (see 1.4.1). The second one is an intrinsic contribution and therefore not eliminable, but it is important to know in details the effects on the delivered dose. According to the equation (1.7) the lighter fragments have a longer range than the primary particles and, thus, are responsible for the undesired dose behind the Bragg peak, usually called tail. In Figure 1.9, the normalized depth-dose distributions in case of SOBP are showed for proton, carbon and neon ion beams having the same range and the tails are clearly visible for ions. Figure 1.9. Comparison of spread out Bragg peak (SOBP) for proton, carbon and neon beams with the same range in water. Tails due to fragmentation are evident for ion beams and more dramatic for neon ion beams [19]. The increasing of the dose just boyind the paek strongly depends on the mass of the ion: in this specific case, it is of the order 15% of the dose in the SOBP for ions like carbon and oxygen, while it can reach 30% in the case of neon ions. This is one of the reasons why, at least from the physical point of view, it is not justified to use ions heavier than oxygen for a really conformal therapy. More- 27

35 1. Radiotherapy with carbon ion beams over, also biological reasons can be address for the exclusion of very heavy ions, as discussed in the next section. Considering also the surface dose in percentage in the plateau region, carbon ions represent a good compromise. However, the effects of fragmentation have to be carefully taken into account in treatment planning also because of the different biological effects characterizing the secondary particles produced, which give rise to a mixed radiation field. The study of fragmentation of carbon ion beam and the calculation of the fragments contribution in terms of dose and ionization density is a key point in hadrontherapy and it represents one of the aims of this work (see chapters 4 and 5). Indeed, for a volume irradiated by a parallel beam of ions, the absorbed dose D can be expressed as function of the ion fluence Φ and the stopping power (- de/dx) by: Φ( x) de D = ρ dx (1.9) where ρ is the density of the stopping material. Because of the fragmentation processes, the particle fluence decreases with the penetration distance according to the relation: ( µ x) Φ ( x) = Φ(0) e (1.10) where Φ(0) is the entrance fluence and µ is the linear attenuation coefficient, proportional to the total microscopic reaction cross-section σ for the ion-tissue interaction [34]. In principle, the dose distribution from each beam could be summed to obtain the total dose distribution. In practical radiotherapy it is not possible, because absorbed dose must be modified by a radiation weighting factor that is energy dependent and changes according to the ion species considered. The composition of this very complex particle field has to be known for dose optimization in heavy ion therapy, in order to take correctly into account the global biological effect in the tissue, due to secondary as well as primary particles. Trying to summarize what as been discussed, these are the main advantages showed by carbon ion beams from the physical point of view [45]: - Carbon ions deposit their maximum energy density in the Bragg peak at the 28

36 1.2 Rationale of ion beam therapy end of their range, where they can produce severe damage to the cells while sparing both the transversely adjacent and deeper located healthy tissues. - They penetrate the patient with minor lateral scattering and longitudinal straggling respect to photon irradiation as well as proton beams. Moreover, being charged, they can easily be formed as narrow focused and scanning single beams of variable penetration depth, so that any part of a tumour can be accurately irradiated with optimal precision. - The location where the dose is deposited by carbon ions can be determined by means of on-line positron emission tomography (PET), which permits exploitation of the millimetre precision of a focused carbon beam, crucial in case of target close to or inside critical structures. Figure Comparison of the planned dose distributions of a carcinoma in the front part of the head. Upper left: IMRT palnning with high energy photons. Lower left: passive proton application. Upper right: active application of protons. Lower right: active application of carbon ions which yields the best dose distribution. (Figures from M Krengly, CNAO, Italy). In conclusion, a comparison of treatment plans of the same clinical case for different modalities is showed in Figure The isodose curves demonstrate as carbon ions yield the best distribution respect to IMRT and also respect to proton 29

37 1. Radiotherapy with carbon ion beams therapy, both for passive and active scanning systems Biological basis of ion beam therapy As already widely discussed, ion therapy shows some interesting physical features respect to conventional radiotherapy and also to more recent proton therapy. However, what makes ion beams really attractive is their enhanced biological effect. In some sense, heavier ions combine more favourable dose distributions of the protons and benefits from the high local ionization and, in particular, carbon ions represent the best compromise between local control of the tumour and negative side effects. Changes in the biological effectiveness are the result of complex interplay between physical parameters, such as ionization density and biological parameters. In order to understand the relation between these different aspects, it is important to discuss at least from a qualitative point of view the modality of energy deposition at a microscopic scale. The deposited dose in a volume is not the only variable to take into account, because it gives indication of the total energy released. For the biological response the track of the particle represents the key information. For an incident charged particle, the ionization occurs along the trajectory of the particle and most of the energy loss is transferred to the liberated electrons, which form a sort of electron cloud around the trajectory of the primary ion, i.e. the ion track. Finally, the action of these electrons determines the biological response together with the primary ionization. It is the higher electron-density, and consequently the ionization-density, that yields a greater biological effectiveness. The formation of a particle track can be regarded more in detail as a twosteps process: first the emission of the electrons by the ion impact and second the transport of these electrons through the material around the particle track, causing secondary ionization and energy deposition. The calculation of track structure is not a trivial task and it has been subject of many publications in which different approaches have been examined. Some groups have developed semi-empirical or analytical models, starting by simplified assumptions [46][47], others have exploited Monte Carlo simulations where each basic interaction is 30

38 1.2 Rationale of ion beam therapy treated individually [48-50]. Although the various models vastly differ in their basic assumptions and their input data, the obtained radial dose distributions are very similar. The diameter of a track depends on the range of the electrons and, consequently, on the velocity of the ion. At higher energy the track is wide and the energy loss is low, therefore the ionization events are well separated. With decreasing energy, the track narrows and the energy loss becomes larger. Consequently, the produced damage has a higher local density resulting in a diminished reparability of the lesion and, therefore, an increased biological effect. Figure Left: schematic view of an undamaged part of DNA (A), two separate single strand breaks (B), a double strand break (C), and a clustered lesion (D); the (*) indicate a base damage. Right: the structure of a proton and a carbon track in nanometre resolution with a schematic representation of a DNA molecule; the higher density of secondary electrons, produced by carbon ions, creates a large amount of clustered DNA damage. The main target of the radiation attack is the DNA (deoxyribonucleic acid) inside cells nuclei. DNA is a very complex system and its integrity is essential for cells survival. Therefore DNA is highly protected by an extremely elaborate repair system so that DNA violations like single strand breaks (SSB) or double strand breaks (DSB) are rapidly restored. But when DNA is exposed to very high local doses, where local refers to the scale of a few nanometres, the DNA lesions become concentrated or clustered and repair system fails to correct the damage, as seen in Figure 1.11 (left) [45]. In general, the energy released by the primary ions is distributed over the volume of the track with a steep gradient of local dose over many orders of 31

39 1. Radiotherapy with carbon ion beams magnitude, from milli-grays at the maximum track radius- to mega-grays in the center [44]. From the previous considerations is evident the difference between densely ionizing radiations, like carbon ions or more in general heavier particles, and sparsely ionizing radiations, like X-rays. The latter, in fact, produces damage in a stochastic manner, where the dose is distributed randomly throughout the cell, giving rise to more reparable lesions. For carbon ions, higher local ionization densities are reached also respect to those obtained with protons. Indeed, for the latter the energy loss is small and the individual ionization events are far from each other, giving rise to more easily reparable DNA damages (Figure 1.11, right). From the quantitative point of view, the energy deposition along the track of the particles in the tissue is represented by the Linear Energy Transfer (LET), measured in kev/µm and defined as the ratio between the energy de deposited by a charged particle in a track element and its length dx, considering only single collisions characterized by energy deposition within a specific value : de L = (1.11) dx It is sometimes called also restricted energy loss or restricted LET. If no limitation in the amount of energy released in any single collision is considered, it is called unrestricted energy loss and it is indicated wit L. Hence, the previous classification of radiations is strictly dependent on LET values: high LET radiations produce more microscopic damages and, thus, they are more biologically effectiveness respect to low LET radiations. Thus, in a high LET track the damage is produced in high density and consequently as clustered lesions that are, to a large amount, irreparable. Moreover, by considering a specific kind of particle, LET is sensibly variable with the penetration depth. Even if there is not a sharp limit between high and low LET values, for many cell systems the biological effectiveness starts being important if LET becomes greater than about 20 kev/µm. LET values of light ions are summarized in table 2 for the range corresponding to 200 MeV protons (262mm of water) 32

40 1.2 Rationale of ion beam therapy [45]. There it is showed that the LET of carbon ions is larger than 20 kev/µm in the last 40mm of their range in water, while in the initial part of an approximately 20 cm range in matter (the so called entrance channel ) LET is smaller that 15 kev/µm. Helium shows high LET values only in the last millimetre. For protons, the range of elevated effectiveness is restricted to a few micrometers at the end of the range, which is too small to have a significant clinical impact. For ions heavier than carbon the residual range of elevated LET starts too early and extends to the normal tissues located before the tumour. After the work done at Berkeley with neon and helium ions, in the beginning of the 1990s, carbon ions were chosen as optimal for the therapy of deep-seated tumours as the increased biological effectiveness, owing to the variation of the LET along the track, could be restricted mainly to the target volume [51]. Charged Particle M N Z E (AMeV) Range = 262 mm LET (kev µm -1 ) at various residual ranges in water (mm) H He Li B C N O Table 1.2. LET values for various charged particles at different residual ranges. The energies of column 2 correspond to a range of 26.2 cm in water [45]. Anyway, LET alone is not the only parameter to be considered for an evaluation of the biological damage. As mentioned before, also parameters strictly related to the biological side have to be taken into account, such as the DNA repair capacity of the cell. The Relative Biological Effectiveness (RBE) combines together both physical and biological aspects and it is the final parameter to be considered in the optimization of the dose to the patient. For ions, 33

41 1. Radiotherapy with carbon ion beams RBE shows complex dependences on absorbed dose, particle energy, atomic number of the radiation, cell line and survival level and it is defined starting from the biological response for sparsely ionizing radiation, such as X-rays. In this case the biological response, which is quantitatively described by the survival S, is a non-linear function of dose and for doses up to a few Gray cell inactivation can be approximated with good accuracy by a linear-quadratic expression in dose D given by: 2 ( α D+βD S = S e ) (1.12) where α and β are specific coefficients characterizing the radiation response. 0 Figure Survival curves for CHO-K1 cells irradiated with X-rays (1) nand carbon ions of different energies: (2) AMeV LET = 13.7 kev/µm, (3) 11.0 AMeV LET = 153 kev/µm, (4) 2.4 AMeV LET = kev/µm [25]. A plot of survival in a semi-logarithmic scale leads to the characteristic shouldered survival curve Figure 1.12, with the size of the shoulder being a measure of the repair capacity of the cell tissue. For particle radiation of increasing LET, the β term becomes smaller and, finally, the radiation response is given by a pure linear dose relationship. The figure clearly shows that the same level 34

42 1.2 Rationale of ion beam therapy of survival is reached by a lower dose in case of carbons, and, in this last case, for lower energies. This fact experimentally demonstrates the different level of damage produced by sparsely and density ionizing radiation, respectively, and its relation with LET. The different action of radiation is quantitatively described in terms of the RBE, which is defined as the ratio of the sparsely ionizing radiation (mostly 220 kev X-rays) and the dose of particles radiation producing the same biological effect [52]: RBE X ray = (1.13) particle Thus, by definition RBE for photons is 1. Because RBE refers to the linear quadratic X-ray dose effect curve, it strongly depends on effect level: it is high for low doses and it decreases with increasing dose. Thus the effect level has to be always given as an index with the RBE level. The ratio of the α-terms as the initial slope of the dose effect curve defines the maximum value RBE α. The value at the 10% survival level RBE 10 is often used in literature for comparisons. In radiotherapy, the topic of the correct RBE has been always crucial for particles therapy and RBE values have to be used that are as close as possible to reality. As mentioned at the beginning of the chapter, the first neutron trials failed because too little was known about RBE, especially, about the behaviour of RBE when the dose is not applied in a single exposure but in many small fractions. For the first period of heavy ion therapy at the Bevalac a large effort was invested for RBE determination in in-vitro experiments. The RBE determination at NIRS mainly uses in-vivo data from corresponding neutron trials and from animal experiments using Carbon beams while the GSI therapy developed a theory for the calculation of RBE that is based on X-ray data taken from the same tumour or healthy tissue that is under particle exposure. For proton therapy a clinical RBE of 1.1 is widespread used. Figure 1.12 also shows that different energies of the carbon ions yield different dose effect curves. Qualitatively, this can be easily explained: for high energies the track is wide and the LET low, thus the ionisation events occur far 35 D D

43 1. Radiotherapy with carbon ion beams enough to make repair possible, yielding shouldered curves similar to sparsely ionising radiation. When energy decreases the diameter of the track shrinks and the LET increases. This leads to a higher ionisation density where the ionisation events occur closely together with a high possibility of interaction, diminishing the influence of repair and yielding a significantly increased RBE. At very high LET values at the end of the particle range (for carbon ions this is above 200 kev/µm) the local dose density becomes higher than necessary for a lethal damage and RBE decreases again [25]. In other words in these conditions there is an over production of local damage, which gives rise to the so called over-kill effect, resulting also in an effective saturation, while the dominator of equation (1.13) continues to increase linearly. This effect is shown in Figure 1.13 converted into a depth distribution in water. Figure RBE α for carbon irradiated CHO-K1 cells. Data from W. K. Weyrather [53]. According to what has been previously discussed it could be possible to summarize that ions heavier than carbon could achieve the same biological effects or even enhanced. That is in part true but high RBE characterizing a specific radiation is not enough to say that it will be suitable for therapy because, as already stated before, what is crucial is the compromise between sterilization of 36

44 1.2 Rationale of ion beam therapy the tumour and damage to the surrounding healthy tissues. Hence, the impact of the enhanced RBE on tumour killing is higher when the RBE maximum overlaps sufficiently with the Bragg maximum, thus getting together both effects of dose and high RBE but, at the same time, minimizing the biological effects before the peak. For carbon this condition is well satisfied, in fact the strongly elevated RBE region is restricted to the end of the particle range, where RBE has values ranging from 2 to 5, while in the entrance channel it is about 1, which means that reparable DNA damage predominates. In Figure 1.13 that behaviour is already clearly showed but it is demonstrated unambiguously if comparisons with other heavier ions are shown. Figure RBE for C, Ne, Si and Ar ions as function of the penetration depth. For C ions RBE is relatively low at the entrance and becomes higher closeness the peak. Indeed, for ions like Neon, Silicon and Argon the irreparable damage becomes more important in entrance channel, too (Figure 1.14). This is one of the reasons why carbon ions have been considered as the most suitable for hadrontherapy also from the biological point of view and they were chosen in the beginning of 1990 as optimal for treating deep-seated tumours. However, because RBE depends also on the possibility to repair the damage produced in the DNA, the repair capacity of the irradiated tissues becomes relevant. In this contest, a major problem in radiotherapy is the oxygen effect, which 37

45 1. Radiotherapy with carbon ion beams means that irradiation in the presence of oxygen causes higher biological damage than in the absence of oxygen. The ratio of the doses leading to the same effect in aerobic and in hypoxic cells or tissue is called Oxygen Enhancement Ratio (OER). The effect is related to the indirect action of the radiation; indeed the reaction is mainly caused by radiation induced free radicals, which cause biological damage after a chain of events. In the absence of oxygen most of these ionized target molecules can repair themselves and recover. If oxygen is present, it reacts with the free radicals and in this way fixes the radiation lesion. Thus, the often hypoxic tumours may be up to three times more radio-resistant than the well blood-supplied, and therefore aerobic, normal tissues [54]. As the damage from high LET particles does not differ for aerobic or anaerobic cells, the OER is smaller for heavy stopping particles (carbon and heavier). In this case, RBE is higher for the poor oxygenated tissues [55] and, consequently, high LET radiation is more effective. Hence, ion beams are better suited to slow growing and good repairing tumours, which can be radio-resistant to photons or protons. In order to compare the RBE dependence over a large number of experiments with different LET radiation, RBE is frequently given as function of the linear energy transfer. Figure 1.15 summarizes in a qualitative way the RBE- LET relationship for cell inactivation, obtained from a historical series of radiobiological data [56]. Even if it should be considered that the RBE depends on a number of parameters other than LET, the general trend is an increase of RBE with LET, up to a maximum in the region kev/µm, while the RBE decrease observed for greater RBE values is related to the previously discussed over-kill effect. This figure adds some more hints about the rationale for choosing the most suitable particles to be used in hadrontherapy. Protons show radiobiological properties close to those of photons, so that their main advantage relies on the superior dose distribution compared to low-let radiations. Moreover the figure shows that among light ions heavier than protons, carbon ions have the most suitable RBE-LET combination for the energy range of interest in hadrontherapy. Indeed, LET values in a SOBP needed to treat deepseated tumours typically cover the range kev/µm. This range is still before the maximum in the RBE-LET relationship, therefore before the drop due to 38

46 1.2 Rationale of ion beam therapy the over-kill effect. As a consequence, carbon ions give the best combination for RBE at the tumour and RBE at the entrance [57]. Figure 1.15 Summary of LET (here intended as L ) dependence of RBE at 10% survival level. The band includes the results for nine cellular systems exposed to carbon, neon and argon ions at Bevalac (adapted from Blakely et al. [56]). The LET ranges available with protons, carbon and neon ions at the energies of interest in radiobiology and radiotherapy are also shown. Even if what discussed above gives a reasonable and general description of RBE-LET relation, in reality dependence of RBE on LET is different for ions, showing a separate maximum for each atomic number, shifting from 25 kev/µm for protons [58] to higher LET values for heavier ions. In particular maximum measured for carbon ion beams was found at 200 kev/µm [59]. For He ions the RBE maximum is higher than for Carbon and so on. Moreover, the location of the RBE maximum is shifted to greater LET values with increasing atomic number. The finding of separate RBE maxima at different LET values for different atomic numbers indicates that the general energy deposition or LET of a particle does not determine the biological response alone. The experimental finding can be explained with the assumption that the local distribution of ionization density 39

47 1. Radiotherapy with carbon ion beams inside the particle track is as important as the total energy, as already pointed out (Figure 1.11). The LET value gives the energy released in a track element but the radial distribution of the dose depends on the projectile energy. Consequently, the combination of the two parameters, LET and energy, finally determine the RBE and its position in the LET spectrum. These considerations imply that in order to take into account biological effects, beyond the physical dose, the spatial distribution of LET, energy and fluence of all the particles has to be known. This is mandatory in case of mixed radiation fields, as for carbon ion treatments, where LET and energies of primary particles as well as of secondaries have to be considered for an accurate treatment planning. Such a kind of calculation is a very complicated task, which simple analytical algorithms cannot easily carry out. Monte Carlo simulations are useful for these calculations, because they are able to consider the nuclear fragmentation of primary particles (attenuation), the consequently production of secondary fragments and their tracking in the matter, by simulating electromagnetic interactions until the particle comes at rest. These issues and the related calculations are one of the aims of this work (chapter 5). 1.3 Treatment Planning in Ion Therapy In order to fully exploit the attractive potentialities of ions treatment it is necessary to arrange for an algorithm able to make up all the key parameters, physical as well as biological ones, which are necessary to optimize the dose distribution. This delicate aim should be carried out by the Treatment Planning System (TPS). The TPS is a set of tools that allows the translation of the dose prescription into a set of beam energies, positions and intensities needed for the treatment. Innovative contributions in this field are particularly needed in case of ions projectiles. It is evident, from previous considerations, that treatment planning for ion therapy is a multilayered challenge. Initially, it is similar to conventional photon therapy: the tumour is localized based on CT (Computerised X-ray Tomography) and possibly additional MRI (Magnetic Resonance Imaging) and PET (Positron 40

48 1.3 Treatment Planning in Ion Therapy Emission Tomography) images. The target volume to be treated, as well as the critical structures to be spared, are marked by the physician for each slice of the tomogram. The task of the treatment planner is to determine the appropriate particle energies, positions and fluences in order to achieve the prescribed dose in a given target volume. As this is usually done by superposition of many narrow pencil 3 beams, the energy and LET of the primary particle have to be known for each point as well as the number, energy and LET values of the different fragments that contribute to the dose. In the next step the RBE must be included iteratively for each irradiation point [25]. Due to the complex dependences from all factors mentioned in the sections before, this is the most time consuming part of the planning. At GSI these calculations are done using the standard treatment planning software TRiP98 [60][61]. It includes a physical model to describe depth dose profiles and nuclear fragmentation of pencil beams of 12 C ions, and it also includes the radiobiological model LEM (Local Effect Model), which allows the calculation of the biologically effective dose, RBE and cell survival (or other biological endpoints 4 ) for any dose level and radiation field composition, provided the photon sensitivity for the tissue under consideration is known [62]. In Figure 1.16 a comparison of the physical absorbed dose with the biological one necessary to obtain a SOBP is showed. RBE values have been calculated from the measured cells survival. As evident, the physical dose has to decrease at the distal part in order to achieve a homogeneous biological effect over the target region. A project has been carrying out by the INFN (Istituto Nazionale di Fisica Nucleare) with the aim of the design of a new advanced Treatment Planning System for ion therapy with active beam scanning, with a particular attention to carbon ion beams. This project put together different competencies with the scope of giving rise to an innovative tool to be used for ion beams therapy. 3 A pencil beam, narrow beam or pencil of rays is a beam of electromagnetic radiation, typically in the form of a narrow cone or cylinder. 4 A clinical endpoint is an observed or measured outcome in a clinical trial to indicate or reflect the effect of the treatment being tested. 41

49 1. Radiotherapy with carbon ion beams Figure Top: Schematic representation of the variation with depth of RBE, showing the profile of the physical dose (dotted line) required in order to obtain a flat SOBP in terms of biological dose (continuous line). Bottom: RBE calculation as function of the depth The INFN TPS project Hadrontherapy has been of direct INFN involvement for more than fifteen years. Successful examples of cooperation between INFN, radiotherapists and oncologists have been accomplished. In this framework, the development of the CATANA (Centro di Adroterapia e Applicazioni Nucleari Avanzate) facility at the Laboratori Nazionali del Sud (LNS) of INFN in Catania, for the treatment of ocular melanoma by means of a proton beam, has been an example of particular significance: hundreds of patients have already been successfully treated [63]. At present INFN is also deeply involved in the construction of the synchrotron for protons and 12 C ions at CNAO (Centro Nazionale di Adroterapia Oncologica), in Pavia, based on active beam scanning technology [64]. Starting from these experiences several INFN research groups, active in different scientific areas (experimental and theoretical nuclear physics, Monte Carlo calculations and 42

50 1.3 Treatment Planning in Ion Therapy techniques for numerical analysis, radiobiology and hardware/software development for dose monitoring purposes), propose to cooperate in order to develop an improved Treatment Planning System for ion therapy with active scanning [65]. Due to the very large number of degrees of freedom, especially in case of innovative treatment techniques such as the Intensity Modulated Particle Therapy (IMPT), computer-aided inverse planning techniques are mandatory to create reliable IMPT treatment plans. Within these techniques, beam spot positions, energies and particle fluences are determined from the prescribed dose distribution by means of an optimization procedure over an appropriate objective function. The optimization has to be performed in order to have a TPS able to compute the dose delivered to the patient with the highest probability of eradicating all the clonogenic tumour cells (sparing the healthy tissues) and, at the same time, able to evaluate the treatment plan in real time and with a high level of interactivity. Thus, the TPS project, already approved by INFN, consists in the development of the optimization techniques by facing two different aspects. Here research activities are classified into two main streams: to improve the knowledge on the biological effects of radiation by effectively including various physical, biological and other factors into the objective function; on the other hand, to evaluate and minimize the objective function, within the hardware constraints (computation time, memory management) by developing different mathematical methods and algorithms. As widespread discussed in the previous sections, RBE depends non-linearly on several factors and, in case of ion beams, these relationships are more and more complex. Indeed, the biological effectiveness of primary ions as a function of depth in the irradiated tissues has to be evaluated carefully by taking also into account the effects due to all the secondary fragments produced in nuclear reactions within the irradiated volume. In general, the exposure to ion beams will result in a superposition of different radiation fields. Considering the state of art (TRiP98), the INFN TPS projects has been proposed with the aim of achieving substantial improvements by putting forward innovative solutions for different open issues. Some of them are: the multi-field optimization, improvement of physical models for fragmentation (trying to cover 43

51 1. Radiotherapy with carbon ion beams the lack of experimental data for carbon ion fragmentation between 20 e 200 AMeV), simulation tool to verify the implemented TPS (Monte Carlo simulation for independent verifications of the TPS output), techniques for on-line monitoring of the dose delivered (PET on-line techniques), factorization of the RBE computation from the cell type (improvements of the existing radiobiological models), inclusion of 4D treatment for moving target optimization (movements affecting some organs, i.e. breathing movements of lung), water equivalent approximation (different density gradients contribution), optimization of the field definition (search of the best field angle) and finally factorization of the dependence of the TPS on the accelerator and beam control system. Consequently, five main research areas have been identified, where each group of the collaboration will give its own contribution [65]. The work carried out in this PhD thesis has been undertaken in the contest of the INFN TPS project, at LNS-INFN in Catania. At LNS, measurements of projectile fragmentation of carbon ion beams at 62 AMeV on thin targets have been performed [67], and other extensive measurements are scheduled in the future. Monte Carlo simulations have been developed with the toolkit Geant4 [68] in order to reproduce the experimental apparatus used in the experiment and to compare the secondary fragments produced. The results (showed in chapter 4) have permitted first verifications of nucleus-nucleus interaction models implemented in the code; few existing data from literature have been considered, even if they usually deal with neutron production (see chapter 3). Finally, the contribution of secondary fragments in terms of dose and fluences have been calculated with the Geant4 simulations, as well as spatial distributions of LET due to primary and secondary particles (chapter 5). As already pointed out, these parameters are strictly related to the final biological effectiveness and their accurate prediction is crucial in presence of mixed radiation fields. In general, Monte Carlo methods represent a powerful tool because of their capacity of accurately taking into account different variables together, unlike traditional analytical calculations which inevitably lead to shortcomings and approximations, sometimes not negligible. For this reasons, the use of a reliable Monte Carlo could represent one of the major improvements in the development of an innovative TPS. 44

52 1.4 Beam delivery systems and accelerators for hadrontherapy 1.4 Beam delivery systems and accelerators for hadrontherapy In order to fully to take the previously discussed physical and biological advantages of heavy ions, particular requirements with stringent specifications have to be satisfied by the dose delivery systems and, sometimess, complex technologies have to be developed to deliver the optimum radiation dose distributions, obtaining the tumour cells inactivation and at the same time the sparing of surrounding healthy tissues. An hadrontherapy centre should meet all these requirements which can be briefly summarized as follows [3]: - ability to treat most cancers at any depth; - treatment times as short as possible; - high ability to discriminate between the volume to be treated and the surrounding - healthy tissues; - availability of treatment rooms with access for beam fixed and mobile lines; - great accuracy ( 3%) in absolute and relative dosimetry; It is therefore necessary to define a beam of hadrons in terms of path, modulation of the Bragg peak, adjustment of the depth of penetration (range), dose rate, field size, uniformity and symmetry of the field, penumbra side and dose distal fall off. The minimum range required for a line dedicated to the treatment of deep seated tumours is 22 cm in water. The range of a beam a 250 MeV proton beam is approximately 38 cm in water. The equivalent energy demanded to a carbon ion beam to reach acceptable penetration depths is equal to 400 AMeV, which means about 28 cm in water Beam delivery systems Ion beams are accelerated to the required energies by acceleration machines. Then the beam is transported through an energy selection system, which consists of a variable thickness degrader, used to modulate the energy of the beam. It is mandatory in case fixed-extraction energy accelerators are used in the centre. 45

53 1. Radiotherapy with carbon ion beams Beam lines provide the required focussing and achromatic bends to deliver a round beam with the appropriate trajectory for transmission through the gantries 5 or the fixed beam lines. The two available options for beam delivery systems are: passive scattering systems and active scanning systems. Figure Scheme of a passive beam shaping system. Top: lateral beam profile changes according to the elements traversed. Bottom: depth dose distribution before and after the passage through range-modulator and rang-shifter. In the middle: schematic representation of the elements. In general, the passive systems have two tasks: lateral scattering of the beam across the tumour volume and depth modulation. An example of passive system is showed in Figure In the passive scattering system a scatterer is used to expand the beam, followed by a collimator which cuts far-axis particles. A fine degrader adjusts continuously the beam energy according to the tumour depth. A ridge filter is usually used as a range modulator: it expands the energy spread of the beam to the extent which corresponds to the tumour thickness. Thus, it is possible to achieve an appropriate SOBP for each different tumour. A bolus, which is a sort of absorber compensator, conforms the dose distribution in the longitudinal direction and a final colli- 5 Particular devices used in order to provide variable incidence angles between the horizontal plane and the beam; it is often considered as a major component of a facility for its complexity and cost. 46

54 1.4 Beam delivery systems and accelerators for hadrontherapy mator in the lateral direction, following the tumour configuration [69]. In the active scanning system the lateral adjustment to the tumour volume is done by intensity controlled magnetic scanning systems, which allow the irradiated volume to be conformed to the planned target volume within a millimetre precision. An example of active system is represented in Figure Each layer of equal particle energy is covered by a grid of individual picture elements (pixels), for which the individual number of particles has been calculated before, to achieve the desired dose distribution [70]. The beam is moved from one pixel to the next one using a pair of magnets perpendicular to each other and to the beam direction and driven by fast power supplies. The particle fluence for each pixel is measured and the beam is switched to the next pixel when the intensity for one position has been reached. The active shaping system usually is coupled to the direct energy modulation by mean of the accelerator, which can be available, for example, using synchrotrons. Two main approaches are used for magnetic scanning: the spot scanning, in which the intensity of beam is measured for each voxel and the beam is turned off during the change of position, and the raster scanning, in which pixels are much closer and the beam is moved without interruption, almost continuously in case of high beam intensities. Figure Scheme of an active scanning system. The tumour is dissected in slice and each isoenergy slice is covering by a grid of pixels for which the number of particles has been calculated before. During the irradiation the beam is guided by the magnetic system. Passive systems represent the easiest methods when a fast energy variation 47

55 1. Radiotherapy with carbon ion beams from the accelerator is not possible. The disadvantages of this method are: a higher amount of projectile fragments, which contributes to a mixed radiation field and an enhanced lateral scattering, mainly due the protons and other lighter fragments. Moreover the high RBE part cannot be completely restricted to the tumour volume [44]. Active systems are much more precise, so that a tumour adjacent to critical organs like brain stem or optical nerves can be covered with a high dose and the normal tissue is spared to a maximum, but they require a major technological complexity [25]. Till 1997 relatively simple passive spreading systems have been used in overall the centres, most of them based on modified nuclear physics accelerators. Only in 1997 at PSI (Villigen - Switzerland) a novel active spreading system (integrated in a compact gantry) has been implemented [71].In the same years a different active spreading system was installed in the carbon ion beam of GSI (Darmstadt). The required characteristic for the ion beams used in the treatment together with the suitable beam delivery system, have a decisive influence in the choice of accelerator to use. It is a widespread opinion that the particle accelerator is the key element of a hadrontherapy facility because defines unequivocally the overall technological performances, having a great impact on the quality of the radiation treatments produced Accelerators in hadrontherapy The three main options for the accelerator designed specifically to operate in the hospital settings are the cyclotron, the synchrotron and the linear accelerator, and sometimes also a combination of them. Considering the specific requirements which a hadrontherapy centre have to fulfil, linear accelerators do not represent the appropriate choice because of the large space required and their costs of maintenance, therefore, they are not used in hadrontherapy. On the contrary circular machines permit to exploit a unique accelerating structure many times, so that a more and more increased velocity of the accelerated particles is achieved in compact configurations. That is one of the reasons why cyclotrons and synchrotrons have been chosen from the beginning as election machines 48

56 1.4 Beam delivery systems and accelerators for hadrontherapy for external radiotherapy with hadrons. In circular accelerators particles must meet for each acceleration cycle an electric field which has to be concordant with their motion direction (synchronism conditions). Cyclotrons are very compact machines with a constant magnetic field and a fixed frequency of accelerating voltage. The accelerated particles, usually injected in the central part of the machine, move in a nearly spiral trajectory, increasing their energy and orbit radius. All particles travel at the same revolution frequency in the accelerator regardless of their energy and orbit radius (isochronous cyclotron). Thus, acceleration can occur continually and allow continuous extraction of the beam. They are fixed-energy accelerator machines capable of very high current. Moreover its beam structure is so that high intensities can be achieved essentially in a continuous regime. Synchrotrons are pulsed accelerators, with particles moving on a closed, approximately circular trajectory where the magnetic field and frequency of the accelerating voltage vary in time as the energy of particles increases. The energy of the extracted beam can be changed on a pulse-pulse basis, therefore allowing a direct modulation [3]. A great scientific debate is underway for decades on the choice between cyclotrons and synchrotrons, in light of the strengths or weaknesses of each of the two solutions. Cyclotrons are usually appreciated for the simplicity of design and their easy of use. The main advantages are related to the structure of the extracted beam [72]: high stability, which allows greater control in terms of electronic and less staff employed; high currents, which positively influence the treatment time, thanks to the consequent higher dose rate capability; continuous beam, which makes easier the exploitation of modern scanning techniques, such as the breathing active treatment and the repainting. The main disadvantages are the weight of the magnet system, which in case of heavy ion beams can reach hundred of tons, and the fixed extraction energy, which implies the use of passive energy modulation. As already mentioned, passive systems cause the beam fragmentation and the consequently deterioration of the quality field. However, adequate technical solutions permit to partially solve this problem, especially from the radioprotection point of view, for example with the screening of the 49

57 1. Radiotherapy with carbon ion beams produced neutrons. On the other hand, synchrotrons are machine with a wellknown technology, having low weight and power consumption. The main advantage respect to cyclotrons is the possibility to directly modulate the energy of the extracted beam, which avoid the use of passive degradation. Anyway, they require a complex ion injector system, and their dimensions can reach 30 m in diameter, in case of acceleration of heavy ions. They produce pulsed beams with relatively low intensity. As concern the cost, it is a widespread opinion that the synchrotron is a more expensive machine, but it must be taken into account some other elements which can heavily contribute to the total cost of an hadrontherapy facility (for example the gantry), especially in case of heavy ion centres [73]. 1.5 Hadrontherapy facilities in the world Till now only synchrotrons are used to produce carbon ions of about 400 AMeV, because of their high magnetic rigidity. A project for the design of a new superconducting cyclotron has been carried out by the LNS group of INFN in Catania. They have designed the SCENT (Superconductive Ciclotron for Exotic Nuclei and Therapy) superconducting cyclotron, which will be 5 m in diameter and will accelerate protons at the maximum energy of 250 MeV and carbon ions up to a maximum energy of 300 AMeV. In the following table are listed the hadrontherapy centers currently in operation in the world, with the accelerator used, the beam exploited and the number of treated patients. Only a small fraction of these centres, highlighted with the asterisks, uses commercial machines, while the most are located inside research laboratories. Up to now about patients have been treated with proton beams and 3500 with carbon ion beams, 3200 of them in Japan. Moreover, other 4 centres for hadrontherapy with proton and carbon beams are planned or under construction. One of these is the already mentioned CNAO, being built in Pavia, which will be based on a 400 AMeV synchrotron for protons and carbon ions acceleration. 50

58 1.5 Hadrontherapy facilities in the world Center-Country Activity start Accelerator Beam Energy (MeV) Treated patients Harvard, Boston, USA 1961 Cyclotron p (Apr-02) ITEP, Mosca, Russia 1969 Synchrotron p (Dec-06) S. Pietroburgo, Russia 1975 Synchrotron p (oct-06) Chiba, Japan 1979 Synchrotron p (Apr-02) PSI, Switzerland 1984 Cyclotron p (Dec-06) Dubna, Russia 1987 Synchrotron p (Jul-06) Uppsaia, Sweden 1989 Cyclotron p (Dec-06) Clatterbridge, UK 1989 Cyclotron p (Dec-06) Loma Linda, USA 1990 Synchrotron p (Nov-06) Nizza, France 1991 Cyclotron p (Sep-06) Orsay, France 1991 Synchrotron p (Dec-06) i-themba LABS, SA 1993 Cyclotron p (Dec-06) UCSF-CNL, USA 1994 Cyclotron p (Mar-07) HIMAC, Japan 1994 Synchrotron C ion (Aug-06) TRIUMF, Canada 1995 Cyclotron p (Sep-06) PSI, Switzerland 1996 Cyclotron p (Jul-06) GSI, Germany 1997 Synchrotron C ion (Jul-06) HMI, Germany 1998 Cyclotron p (Dec-06) *NCC, Kashiwa, Japan 1998 Cyclotron p (Nov-06) **HIBMC, Hyogo, Japan 2001 Synchrotron p (Sep-06) **HIBMC, Hyogo, Japan 2002 Synchrotron C ion 320 x (Sep-06) ***PMRC, Tsukuba, Japan 2001 Synchrotron p (Jul-06) *NPTC, MGH, USA 2001 Cyclotron p (Oct-06) INFN-LNS, Catania, Italy 2002 Cyclotron p (Oct-06) WPCT, Zibo, China 2004 Cyclotron p (Jul-06) ***WERC, Wakasa, Japan 2002 Synchrotron p (Aug-06) ***WERC, Wakasa, Japan 2002 Synchrotron He, C 200, (Aug-06) **Shizuoka, Japan 2003 Synchrotron p (Nov-06) ****PSI, Switzerland 2007 Cyclotron p (Oct-07) FPTI, Jacksonville, USA 2006 Cyclotron p (Dec-06) MD And. C. center, USA 2006 Synchrotron p (Dec-06) Table 1.3. The hadrontherapy centers currently in operation in the world (from PTGOG). Legenda: *Cyclotron IBA, **Synchtrono Mitsubishi, ***Synchrotron Hitachi, ****Cyclotron Accel. 51

59 2. Geant4: a Monte Carlo simulation toolkit Simulation is the imitation of some real things, states of affairs, or processes. The act of simulating something generally implies representing key characteristics or behaviours of a selected physical or abstract system. Simulation is used in many contexts, including the modelling of natural or human systems in order to gain insight into their functioning. First of all, crucial issues in simulations include the acquisition of valid sources of information about the relevant selection of key characteristics and behaviours, secondly the use of simplifying approximations and finally the validity of the simulation outcomes. Nowadays, one of the most widespread used techniques in this field is represented by the Monte Carlo method, which is actually considered one of the most powerful tools used when it is infeasible or impossible to compute a result with deterministic algorithms. That is the reason why Monte Carlo simulations are well accepted and successfully exploited in different scientific fields, not last in physical sciences. Surprising improvements in computer technology have surely given a boost to the development of this technique, promoting its wide diffusion and giving rise to the born of several Monte Carlo codes, such as the toolkit Geant4. At first developed for the high energy Physics, today it is used in several fields which involve also low energy range of application, like the medical ones. Geant4 simulations of diagnostic devices or radiotherapy techniques have been worldwide carried out in recent years by several research groups involved in medical applications.

60 2.1 The Monte Carlo method In this thesis work, some issues related to carbon radiotherapy have been faced using the Monte Carlo simulation toolkit Geant4. The achieved results will be discussed in the next chapters. 2.1 The Monte Carlo method Monte Carlo methods are a class of computational algorithms that rely on repeated random sampling to compute results. They provide approximate solutions to a variety of mathematical problems by performing statistical sampling experiments with random numbers and probability statistics. To give a unique and exhaustive definition of Monte Carlo methods is a very hard task; several definitions can be found in books and papers concerning this subject. Monte Carlo methods are often used to simulate physical and mathematical systems and they are especially useful to study systems with a large number of coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cells. More broadly, Monte Carlo methods are useful for modelling phenomena with significant uncertainty in inputs. Among all numerical methods, which rely on n-point evaluations in m-dimensional space to produce an approximate solution, the Monte Carlo method has absolute error of estimate that decrease as n -1/2 whereas, in the absence of exploitable special structure all others have errors that decrease as n -1/m at best. This property gives the Monte Carlo method a considerable edge in computational efficiency as m, which represents the size of the problem, increases [74]. In the same way as its definition, history of Monte Carlo method is somewhat complex and not so clear. Even if some isolated and undeveloped instances were traced also during the nineteenth century, the born and the developing of this technique, as a systematic tool, dates from the 1940s, when the term "Monte Carlo method" was coined by physicists working on nuclear weapon projects at the Los Alamos National Laboratory [75]. Physicists at Los Alamos Scientific Laboratory were investigating radiation shielding and the distance that neutrons would likely travel through various materials. Despite having most of the necessary data, such as the average distance a neutron would travel in a substance be- 53

61 2. Geant4: a Monte Carlo simulation toolkit fore it collided with an atomic nucleus, the problem could not be solved with theoretical calculations. John von Neumann and Stanislaw Ulam suggested that the problem could be solved by modelling the experiment on a computer using chance. Being secret, their work required a code name. Von Neumann chose the name "Monte Carlo". The name is a reference to the Monte Carlo Casino in Monaco where Ulam's uncle would borrow money to gamble and, more in general, it relates to the random character of the games of chance, such as the roulette. It was only the use of first electronic computers that Monte Carlo methods began to be studied in depth. In the 1950s they were used at Los Alamos for early work relating to the development of the hydrogen bomb, and became popularized in the fields of Physics, physical chemistry, and operations research. Monte Carlo methods usually follow different approaches, depending on the particular field of application. However, these approaches tend to follow a particular scheme: define a domain of possible inputs, generate inputs randomly from the domain using a certain specified probability distribution, perform a deterministic computation using the inputs and aggregate the results of the individual computations into the final result. Thus, the reliance computation is sensitively dependent on the goodness of the random numbers, which are required in large amounts to achieve good statistics. This requirement led to the necessity of using methods which in some a way are able to create long chain of random numbers, giving rise to the development of the so called pseudorandom numbers generators. A pseudorandom number generator is an algorithm for generating a sequence of numbers that approximates the properties of random numbers. The sequence is not truly random in the sense that it is completely determined by an initial state, usually named seed. When Monte Carlo simulations are exploited to produce a unique result by splitting the process in several tasks, the use of different seeds is mandatory. 2.2 The Geant4 simulation toolkit As already mentioned, several Monte Carlo codes have been recently devel- 54

62 2.2 The Geant4 simulation toolkit oped, which have found applications in various fields. From the historical point of view the development of Monte Carlo codes in Physics sciences is due in particular to the nuclear and high energy physics, because of the ever-increasing demand for large-scale and accurate simulations of complex particle detectors. However, similar requirements arise also in other fields, such as radiation physics, space science and nuclear medicine, where Monte Carlo codes have burst into for several years. The Geant4 toolkit was developed about forty years ago in response to this, and now it is world wide used in different research areas. Geant4 is a body of C++ code which models and simulates the interaction of particles with matter. The code is freely distributed under an open software license [76]. Documentation, source code, databases and, for some computer platforms, binary libraries can be downloaded from the Geant4 website [77]. GE- ANT is an acronym for GEometry ANd Tracking and is usually pronounced like the French géant, meaning giant. Its origins can be traced back to the 1970s. The original designs were in Fortran, culminating in GEANT3 [80] in 1982, which is still used but no longer developed. In 1993, two independent studies, one at CERN (The European Organization of Nuclear Research) in Geneva, and one at KEK (The High Energy Accelerator Research Organization) in Tsukuba, considered the application of modern computing techniques to this sort of simulation, focalizing the attention on object oriented programming. This led to a joint development project under the CERN Research and Development Programme in The merging language C++ was chosen and the first version of Geant4 was released in It now runs to almost a million lines of code [82]. In beginning design phase of the project, it was recognised that while many users would incorporate the Geant4 tools within their own computational framework, others would want the capability of easily constructing stand-alone applications which carry them from the initial problem definition right through to the production of results and graphics for publication. For this aim, the toolkit includes built-in steering routines and command interpreters which allow all parts of the toolkit to work in concert. Geant4 was designed and developed by an international collaboration, formed by individuals from a number of cooperating institutes and it builds on the accumulated experience of many contributors to 55

63 2. Geant4: a Monte Carlo simulation toolkit the field of Monte Carlo simulation of physics detectors and physical processes. The Geant4 Collaboration represents, in terms of the size and scope of the code and the number of contributors, one of the largest and most ambitious projects of this kind [82]. Recently, our physics research group at LNS-INFN in Catania, has taken part the to development of the Geant4 toolkit, enjoying the Geant4 collaboration and giving some contributions as concern the low energy electromagnetic and hadronic models validation and the developing of medical physics applications. 2.3 Main features of the Geant4 toolkit The Geant4 toolkit has been built as the basis for specific simulation components. Thus it was required: - to have well-defined interface to other components and - to provide parts to be used by other components. Other design requirements are modularity and flexibility and implementation of physics has to be transparent and open to user validation. It should allow the user to understand, customize and extend it in all domains. The high level design was based on an analysis of initial user requirements. This, finally, led to a modular and hierarchical structure for the toolkit where sub-domains are linked by an uni-directional flow of dependencies. The key domains of the simulation of the passage of particles through matter are: - geometry and materials - particle interaction in matter - tracking management - digitalization and hit management - event and track management - visualization and visualization framework - user interface Naturally these domains led to the creation of class categories with coherent interfaces and, for each category, a corresponding working group with a well de- 56

64 2.3 Main features of the Geant4 toolkit fined responsibility. GEANT4 exploits advanced software engineering techniques to deliver these requirements of functionality, modularity and extensibility General capabilities and proprieties of the Geant4 toolkit Geant4 offers users the ability to create a geometrical model, with a large number of components of different shapes and materials, and to define sensitive elements which record information (hits) needed to simulate detector responses (digitization). Primary particles of events can be derived from internal and external sources. Geant4 provides a comprehensive set of physical processes to model the behaviour of particles. The user is able to choose from different approaches and implementations, and to modify or add to the set provided. In addition the user can interact with the toolkit through a choice of (graphical) user interface and visualize the geometry and tracks with a variety of graphics system through a well-defined interface. In general, the classes in the toolkit are designed in a highly reusable and compact way so that the user can extend or modify their services for specific applications, following the discipline of object-oriented technology. The result is a highly granular implementation, which allows the detailed inspection of each component of the code, at source level. This is a very important feature especially in the medical physics field, where the constraints related to the required high level of accuracy often implies ad-hoc customizations Global structure The design of GEANT4 has evolved during its development. It currently includes 17 main categories. Figure 2.1 shows the top-level categories and illustrates how each category depends on the others. There is an unidirectional flow of dependencies, as required. Categories at the bottom of the diagram are used virtually by all higher categories and provide the foundation of the toolkit. These 57

65 2. Geant4: a Monte Carlo simulation toolkit include: - the category global covering the system of units, numeric constants and random number handling; - materials, particles, graphical representations, geometry including the volumes for detector description and navigation in the geometry model; - intercoms which provides both a mean of interaction with GEANT4 through the user interface and a way to communicate between modules. Figure 2.1. The Top Level Category diagram of the Geant4 toolkit. The open circle on the joining lines represents a using relationship: the category at the circle end uses the adjoined category. Moreover, we can find reside categories required to describe the tracking of particles and the physical processes they undergo. The track category contains 58

66 2.3 Main features of the Geant4 toolkit classes for tracks and steps, used by processes which contain implementations of models of physical interactions. Additionally one process, transportation, handles the transport of particles in the geometry model and, optionally, allows the triggering of processes parameterizations. All these processes may be invoked by the tracking category, which manages their contribution to the evolution of a track s state and undertakes to provide information in sensitive volumes for hits and digitization. Over these the event category manages events in terms of their tracks and run category manages collections of events sharing a common beam and detector implementation. Finally capabilities which use all of the above and connect to facilities outside the toolkit (through abstract interfaces) are provided by the visualization, persistency and user interface categories [78]. In order to better understand the structure of the code and its specific functionality from a more practical point of view, some of these categories are discussed more in details, as follows: Event - The event category provides an abstract interface to external physics event generators for the generation of the primary particles which defines a physical event. The class G4Event represents an event, which is the main unit of simulation. An event starts with the generation of the primary particle and stops when both primary and secondary particles end their path or exit from the simulated volumes. Run - A run is defined as a collection of events whose setup characteristics remain unchanged. The class G4Run represents a run, during which the user cannot modify the geometry, the characteristic of the detector or the physics processes. However, geometry configuration, primary particles and other features can be modified between a run and another one. Tracking - The tracking category steers the invocation of processes and it does not depend on the particle type or specific process. In Geant4 particles are transported, instead of being considered self moving. All physical processes associated with the particle propose a step, which is managed by the class G4Step. Each particle is moved step by step with a tolerance that permits significant optimizing of execution performance but which preserves the required 59

67 2. Geant4: a Monte Carlo simulation toolkit tracking precision. The smallest step length is chosen between the following: the maximum allowed step (stipulated by the user) and the steps proposed by the actions of all attached processes, including that imposed by the geometrical limits. Depending on its nature, a physics process has one or more characteristics represented by the following actions handled by the tracking: at rest, for particles at rest (e.g., decay at rest); along step, such as energy loss or secondary particle production that happens continuously along a step (e.g., Cherenkov radiation); post step, which is invoked at the end of the step (e.g., secondary particle production by a decay or interaction). Along step actions take place cumulatively, while the others are exclusive. The tracking handles each type of action in turn. For these three actions, each physics process has a GetPhysicalInteractionLength, which proposes a step, and a DoIt method that carries out the action. Finally the tracking scans all physics processes and actions for a given particle, and decides for the one which returns the smallest distance. Particles and Materials - These two categories implement facilities necessary to describe the physical properties of particles and materials for the simulation of particle-matter interactions. Particles are based on the G4ParticleDefinition class, which describes the basic properties of particles, like mass, charge, etc., and also allows the particle to carry the list of processes to which it is sensitive. The design of the materials category reflects what exists in nature: materials are made up of a single element or a mixture of elements, which in turn are made of a single isotope or a mixture of isotopes. Characteristics like radiation and interaction length, excitation energy loss, coefficients in the Bethe-Bloch formula, shell correction factors, etc., are computed from the element. Geometry - The geometry category offers the ability to describe a geometrical structure and propagate particle efficiently through it. Detector geometry is made of different volumes. The largest volume which has to contain each else defined volume is the World volume. Each volume is completely defined in Geant4 if its shape, properties and positions are defined. Thus, these different pieces of information are divided into three distinct concepts: solid volume (shape, dimensions, kind of sub-category), logical volume (material, sensitive- 60

68 2.3 Main features of the Geant4 toolkit ness, magnetic field) and physical volume (position, rotations, replicas, parameterizations). Physics - It manages all the physics processes and models implemented in Geant4. There is a wide set of physics models to handle the interactions of particles with matter across a very wide energy range. The Geant4 toolkit contains a large variety of complementary and sometimes alternative physics models covering the physics of photons, electrons, muons, hadrons and ions from 250 ev up to several PeV. There are seven major sub-categories: electromagnetic, hadronic, transportation, decay, optical, photolepton_hadron, and parameterisation. The first two, electromagnetic and hadronic, are further sub-divided. For a particle interaction it is important to stress the difference between the process, i.e., a particular initial and final state, which therefore has a well defined crosssection or mean-life, and the model that implements the production of secondary particles. Geant4 allows the possibility of offering multiple models for the same process. Hits and digitization - The hit, managed by the G4Hit class, represents a snapshot of a physical interaction of a track in a sensitive part of the detector. If a volume is defined sensitive, it is possible to retrieve several information related to the interaction occurred, by using a collection of hits. A digit represents a detector output. Visualization - It is a high level category which uses interfaces for the visualization of volumes implemented, particle trajectories, tracking steps, hits; it helps the user to prepare and execute detector simulations. Drivers for several graphics systems are offered and can be instantiated in parallel Development of a Geant4 application A typical Geant4 application is a program composed by several files all managed by the main file. The first thing which a main file has in charge to do is to create an instance of the G4RunManager class. This is the only manager class in the Geant4 kernel which should be explicitly constructed in the user's main. It controls the flow of the program and manages the event loop within a run. When G4RunManager is created, the other major manager classes are also 61

69 2. Geant4: a Monte Carlo simulation toolkit created. One of these is the user interface manager, G4UImanager. Beyond these, there are three user classes which have to be mandatory implemented [83]: - G4VUserDetectorConstruction to define the material and geometrical setup of the detector. Several properties, such as detector sensitivities and visualisation attributes, are also defined in this class. - G4VUserPhysicsList to define all the particles, physics processes and cut-off parameters. - G4VUserPrimaryGeneratorAction to generate the primary vertices and particles. For these three user classes, Geant4 provides no default behaviour. Moreover there are some optional classes which allow the user to modify the default behaviour of Geant4: - G4UserRunAction for actions at the beginning and at the end of each run. - G4UserEventAction for actions at the beginning and at the end of each event. - G4UserStackingAction to customise access to the track stacks. - G4UserTrackingAction for actions at the creation and at the completion of every track. - G4UserSteppingAction to customise behaviour at every step. 2.4 Physical processes in Geant4 A particle in flight is subjected to many competing processes. As already mentioned, in simulation the particle proceeds in steps and each process proposes a step, according to its physical interaction length. Moreover, different models are available for the same process, so that a criterion of choice has to be followed by the user, according to his requirements and physical constraints. As stressed at the beginning of the chapter, opening of physics implementation is a distinct feature of the Geant4 code, not always present in other similar codes. This specific characteristic can be a powerful tool for expert users but, at the same time, it could be not always safe for beginner users, who may not be 62

70 2.4 Physical processes in Geant4 deeply conscious of the choices done and would probably prefer somewhat wide tested and ready to use. In order to keep both the possibilities open, Geant4 has included different level of implementation of the physical processes in the applications, so that also pre-compiled physics package are provided to the user. The User can choose to explicitly call, for each particle and in a given energy range, the appropriate process instantiating the corresponding models. At this level the user has the maximum control of the physics but he is exposed to a bigger probability to make errors. In the second and third levels, on the other hand, the user can directly call one or more of the pre-compiled and ready-to-use set of physics models. These sets, also commonly called, in the Geant4 slang, Physics Lists and Reference Physics Lists, are contained in the installation directory of the code. The Physics Lists permits separately, the activation of models for group of processes, where the energy limits of application are fixed and well-packed. The Reference Physics Lists permit the third and even the simpler and high-level methodology for the physics implementation in a Geant4 application. They are in fact, built-in package containing, at the same time, a complete set of physics models useful for a given application. A Reference Physics Lists, whose activation can be done simply calling its name in the physics file of the application, can be considered as a set of specifically grouped Physics Lists. These functionalities are provided for customization requirements and for simulation time optimization. Compromise between accuracy and performance is always a key point in Monte Carlo simulation and usually it represents a crucial point in the developing of user applications. In this concern, a very useful tool is also the definition of production thresholds by using the SetCut method. If implemented, a process may choose to suppress secondary particles (below a define threshold) which would have been produced because of an eventual interaction. Unlike other codes, which usually provide energy thresholds, Geant4 offers the possibility to set a cut-in-range, which means the suppression of particles which, if produced, would have a range smaller than a user-defined value, named range cut. The idea of a "unique cut value in range" is one of the most important features of Geant4 and is used to handle cut values in a coherent manner, so that the code itself internally converts the value into energy for individual 63

71 2. Geant4: a Monte Carlo simulation toolkit materials. This feature allows the user to easy choose the best threshold value, decoupling that from the several implemented materials. Moreover for improving performance, cut is applicable to some defined regions and not necessarily to the whole geometry. What is important to stress is that cut-in-range is in Geant4 a production cut and not a track threshold, meaning that if a secondary is produced it will be tracked up to zero energy. A typical case of application is the δ- rays and bremstrahlung production, where the use of a cut is a necessity in order to suppress the generation of large numbers of soft electrons and gammas. Among the several physics models available in Geant4, in the next sections the electromagnetic and the hadronic models will be quickly discussed, preceded by a short description of the transport model implemented in the code. In particular, hadronic processes are of great interest in the contest of hadrontherapy, because responsible of the ion beam fragmentation Particle transport The interaction of a generic particle is simulated and transported in Geant4 using the so called differential approach. A short description is given as follows. The interaction Length or Mean Free Path Let us consider a simulated particle traversing a material. Two cases have to be distinguished. In the case of simple material the number of atoms per volumes is: Nρ n = (2.1) A where: N is the Avogadro s number, ρ the density of the medium and A is the mass of a mole. In a compound material the number of atoms per volume of the element is: n i NρW A i = (2.2) i where w i and A i are respectively the proportion by mass and the mass of a mole 64

72 2.4 Physical processes in Geant4 of the i th element. The mean free path of a process λ, also called interaction length, can be given in terms of its total cross section: 1 λ ( E) = [ niσ ( Z i, E) ] (2.3) i where σ(z, E) is the total cross section per atom of the process and the sum runs over all elements composing the material. The quantity in brackets is also called macroscopic cross section, so that the mean free path is the inverse of this quantity. Inside GEANT4 simulation cross sections per atoms and mean free path values are tabulated during initialization. Determination of the interaction point The mean free path of a particle for a given process depends on the medium and cannot be used directly to sample the probability of an interaction in an heterogeneous detector. The number of mean free paths which a particle travels is: 2 x dx nλ = (2.4) λ x 1 and it is independent on the material traversed. If n r is a random variable denoting the number of mean free paths from a given point of interaction, it can be shown that n r has the distribution function: P ( x) nλ ( n n ) = e r < 1 (2.5) λ The total number of mean free paths the particle covers before the interaction point, n λ, is sampled at the beginning of the trajectory as: n λ = logη (2.6) with η a random number uniformly distributed in the range (0,1). The value n λ is updated after each step x according to the formula: 65

73 2. Geant4: a Monte Carlo simulation toolkit n λ x = nlambda (2.7) λ until the step originating from s(x) = n λ λ(x) is the shortest and this event triggers the specific processes. The short description given above is the differential approach to the particle transport, which is used in most of the simulation codes. In this approach besides the other (discrete) processes, the continuous energy loss imposes a limit on the step size too, because the cross sections depend on the energy of the particle. Then it is assumed that the step is small enough so that the cross sections of the particles remain approximately constant during the step. In principle one has to use very small steps in order to have an accurate simulation but the computing time increase if the step size decreases. ( x) Electromagnetic processes Geant4 electromagnetic physics manages the electromagnetic interactions of leptons, photons, hadrons and ions. The electromagnetic package is organised as a set of class categories: - standard: handling basic processes for electron, positron, photon and hadron interactions; - low energy: providing alternative models extended down to lower energies than the standard category; - muons: handling muon interactions; - X-rays: providing specific code for X-ray physics; - optical: providing specific code for optical - utils: collecting utility classes used by the other categories. Public evaluated databases are used in electromagnetic processes without introducing any external dependence, while keeping the physics open to future evolutions of available data sets. This feature also contributes to the reliability and the openness of the physics implementation. The results showed in this work have been carried out by using the standard electromagnetic processes, which have been widely validated and, moreover, 66

74 2.4 Physical processes in Geant4 cover the range energy of interest for hadrontherapy applications. Moreover a huge number of improvements and developments have been recently done in this package [84]. In the standard electromagnetic processes category, there is a variety of implementations of electron, positron, photon and charged hadron interactions. The class G4eIonisation calculates for electrons and positrons the energy loss contribution due to ionisation and it simulates the discrete part of the ionisation. The class G4eBremsstrahlung computes the energy loss contribution due to soft bremsstrahlung and simulates the discrete or hard bremsstrahlung. For the electromagnetic processes of hadrons, the G4hIonisation class computes the continuous energy loss and simulates d-ray production. In this case, energy loss, range and inverse range tables are constructed only for proton and anti-proton; the energy loss for other charged hadrons are computed from these tables at the scaled kinetic energy. Photon processes include Compton scattering, γ-conversion into electron and muon pairs and photo-electric effect. The G4MultipleScattering class simulates the multiple scattering of charged particles in material after a given step, then it computes the mean free path length correction and mean lateral displacement [85] Hadronic processes The basic requirements for the physics modelling of hadronic interactions in a simulation toolkit span more than 15 orders of magnitude in energy. The energy ranges from thermal for neutron cross sections and interactions, through 7 TeV (in the laboratory) for high energy experiments, to even higher for cosmic ray physics. In addition, depending on the setup being simulated, the full range or only a small part might be needed in a single application. Of this huge domain of energy, incident particle, target isotope and level of precision, much is already available in the standard distribution of the Geant4 toolkit. The hadronic process category includes several families of classes: processes, which define each possible process and provide connections to the underlying cross-sections and models which implement the process; management, containing classes which provide steering mechanisms for the application of ap- 67

75 2. Geant4: a Monte Carlo simulation toolkit propriate interaction models; cross-sections, which encapsulate all cross-section data and associated calculation methods for computing the occurrence of processes; stopping processes, a distinct category for particles stopping or at rest; models, each of them implements the final state generation of a particular process or a set of processes for a particle or class of particles within a specified energy range; and utility classes, which provide various standardised computational methods for use by models. Within the models category are a number of subsystems: low energy, high energy, generator, neutron hp, radioactive decay, and so on, organised according to energy range, methodology and reaction type, and so on. Among this large array of models, there are some models largely based on evaluated or measured data, some models predominantly based on parameterisations and extrapolation of experimental data under some theoretical assumptions, and models which are predominantly based on theory. According to this distinction, three main classes of models can be distinguished: Data driven models. They are based on evaluated or measured data which, in some cases, are able to cover wider ranges of energy. The main data driven models in Geant4 deal with neutron and proton induced isotope production, and with the detailed transport of neutrons at low energies. Moreover, this kind of approach is also used to simulate photon evaporation at moderate and low excitation energies, and for simulating radioactive decay. Parameterised models. Parameterizations and extrapolations of crosssections and interactions are widely used in the full range of hadronic energies, and for all kinds of reactions. Parameterised models mainly include fission, capture and elastic scattering, as well as inelastic final state production. Theory based models. Theory based modelling is the basic approach in many models that are already provided or are under development. It includes a set of different theoretical approaches to describe hadronic interactions, depending on the addressed energy range and computing performance needs. Data driven modelling is known to provide the best approach to low energy neutron transport for radiation studies in large detectors. Parameterisation driven modelling has proven to allow for tuning of the hadronic shower and is widely 68

76 2.4 Physical processes in Geant4 used for calorimeter simulation. Theory driven modelling is the approach which promises safe extrapolation and allows for maximal extendibility and customisability of the underlying physics. Anyway, models belonging to different classes can be combined in order to achieve the required level of accuracy for the specific simulation. In the following sections, a brief overview of the Geant4 physical models used in this work for the study of carbon ions beams fragmentation is presented Modelling of ion interactions for hadrontherapy applications In order to simulate carbon ion beams interaction with the matter for hadrontherapy purposes and study how the primary particle fragmentation can influence the radiation field, the suitable hadronic models have to be accurately chosen, looking at their features and energy range of applicability. In general, the energy range of interest in hadrontherapy with carbon ion beams is about AMeV which, as already mentioned allows to reach depths in water up to 28 cm. In particular, in this work studies of carbon fragmentation and comparisons with experimental data for different targets at 135 AMeV (chapter 4) and 62 AMeV (chapters 5, 6) have been carried out, respectively for neutron production and charged fragments production. Total reaction cross section has to be implemented, as well as models describing the elastic and inelastic scattering; for the latter, also isotope production and following de-excitations and decays have to be simulated with models. Total reaction cross section in nucleus-nucleus reactions As already mentioned before, an important input parameter for simulation of particles transport in the matter is the total reaction cross section σ R, which is defined as the total σ T minus the elastic σ el cross section fro nucleus-nucleus reactions: σ = σ σ (2.8) R T el 69

77 2. Geant4: a Monte Carlo simulation toolkit The total reaction cross section has been studied both theoretically and experimentally and several empirical parameterizations of it have been developed. In Geant4 the total reaction cross sections are calculated using four parameterizations: the Sihver [86], Kox [87], Shen [88] and Tripathi [89] formulae. In the present work the Tripathi formula has been used, because it works in a wide energy range (from 10 AMeV up to 1 GeV/n) with uniform accuracy for any combination of interacting ions. The Tripathi formula incorporates different important effects, such as Coulomb interactions at lower energies and strong absorption model, its form is: 2 ( ) B AP + AT + δ E E 1 1 R = σ πr0 1 (2.9) CM where r 0 = 1.1 fm, and E CM is the colliding system centre of mass energy in MeV. The last term in equation (2.9) is the Coulomb interaction term, which modifies the cross section at lower energies and becomes less important as the energy increases (typically after several tens of AMeV). In equation (2.9), B is the energy-dependent Coulomb interaction barrier which is given by: 1.44Z PZT B = (2.10) R where Z P and Z T are the atomic numbers of the projectile and target, respectively. R is the distance for evaluating the Coulomb barrier height, given by: ( AP + AT ) 1.2 R = rp + rt + 1 (2.11) 3 E where r i is equivalent sphere radius and is related to the r rms,i radius by: with (i = P, T). The energy dependence in the reaction cross section at intermediate energies results mainly from two effects: trasparency and Pauli blocking. This energy dependence is taken into account in δ E, which is given by: 70 CM r i 1.29r rms, i = (2.12)

78 2.4 Physical processes in Geant4 ( A 2Z ) 0.16S 0.91 T T Z P δ E = 1.85S + CE + (2.13) 1 3 ECM AT AP where S is the mass asymmetry term given by: AP AT S = (2.14) A + A 1 3 P 1 3 T and is related to the volume overlap of the collision system. The last term on the right side of the equation (2.13) accounts for the isotope dependence of the reaction cross section. The term C E is related to both the transparency and Pauli blocking and is given by: C E E E ( 1 e ) 0.292e cos( 0.229E ) = D (2.15) where the collision kinetic energy E is in units of AMeV. Here, D is related to the density dependence of the colliding system, scaled with respect to the density of the 12 C + 12 C colliding system: D ρ A + ρ p at = (2.16) ρ + ρ AC AC The density of a nucleus is calculated in the hard-sphere model. In effect, D simulates the modifications of the reaction cross sections caused by Pauli blocking. The Pauli blocking effect, which has not been taken into account in other empirical calculations, has been introduced here for the first time. At lower energies (below several tens of AMeV) where the overlap of interacting nuclei is small (and where the Coulomb interaction modifies the reaction cross sections significantly), the modifications of the cross sections caused by Pauli blocking are small and gradually play an increasing role as the energy increases, which leads to higher densities where Pauli blocking becomes increasingly important. The Tripathi formula is implemented inside the class G4TripathiCrossSection but it is still not fully implemented, so that it is valid for projectile energies less than 1 AGeV. 71

79 2. Geant4: a Monte Carlo simulation toolkit Elastic scattering The model of nucleon-nucleon elastic scattering is based on parameterization of experimental data in the energy range of MeV. The simulation of elastic scattering is quite simple, since the final states are identical with the initial ones. At higher energies the hadron-nucleus elastic scattering can be simulated using the Glauber model and a two Gaussian form for the nuclear density. This expression for the density allows to write the amplitudes in analytic form. This assumption works only since the nucleus absorbs hadrons very strongly at small impact parameters, and the model describes well nuclear boundaries. The G4LElastic class has been used in the simulations, which is valid for all incident particles and, included ions, and for both low and high energies. Nucleus-nucleus models The state of the art of nucleus-nucleus models in Geant4 at the energy range of interest in hadrontherapy is characterized by the presence of three models: - Abrasion-ablation model [90]. - Binary Light Ion Cascade model [91]. - Quantum Molecular Dynamic model [92]. Recently an alternative model, also belonging to the family of the intranuclear cascade models, has been introduced, the INCL-ABLA model [93], but it is applicable only to lighter ions (up 4 He incident particles). The results carried out in this work has been obtained using the Binary Light Ion Cascade model which, according to some inter-comparisons previously performed between the different models, represents a good compromise between an acceptable level of accuracy in isotope production predictions and computing performance. The Binary Light Ion Cascade model The Binary Light Ion Cascade model is an improved version of the Binary Cascade model (applicable only to pions, protons and neutrons). It is an intranuclear cascade model used to simulate the inelastic scattering of light ion nuclei 72

80 2.4 Physical processes in Geant4 and it is applicable for projectile energy ranging on about 80 MeV to 10 AGeV. In simulating light ion reactions, the initial state of the cascade is prepared in the form of two nuclei, as described in the above section on the nuclear model. The lighter of the collision partners is selected to be the projectile. The scattering is modelled by propagating the nucleons of the ion as free particles through the nucleus [94]. In the model, primary nucleons are propagated within a nucleus (3- dimensional detailed model of the nucleus) where it suffers binary scattering with an individual nucleon of the nucleus; secondary particles eventually produced in the interaction are propagated and can, in turn, re-scatter with nucleons, creating the cascade. The nucleus is modelled by explicitly positioning nucleons in space, and assigning momenta to these nucleons. This is done in a way consistent with the nuclear density distributions, Pauli s exclusion principle and the total nuclear mass. Free hadron-hadron elastic and reaction cross-section are used to define collision locations within the nuclear frame. Where available, experimental cross-sections are used directly or as a basis for parameterizations used in the model. The propagation of particles in the nuclear field is done by numerically solving the equations of motion, using time-independent fields derived from optical potentials. The cascade begins with a projectile and the nuclear description, and it stops when the average energy of all participants within the nuclear boundaries are below a given threshold. The remaining pre-fragment will be treated by pre-equilibrium decay and de-excitation models. The initial condition of the transport algorithm is a 3-dimensional model of a nucleus and the primary particle's type and energy. As regards the first one, a 3- dimensional model of the nucleus is constructed from A nucleons and Z protons with coordinates r i and momenta p i, with i = 1, 2,..., A. Nucleon radii r i are selected randomly in the nucleus rest frame according to the nuclear density ρ(r i ). For nuclei with A > 16 a Woods-Saxon form of the nucleon density is used [95]: where ρ 0 is approximated as ρ 0 ρ ( ri ) = (2.17) 1+ exp [( r R) / a] i 73

81 2. Geant4: a Monte Carlo simulation toolkit a π ρ = πR R (2.18) with a = fm, and R = r 0 A 1/3 fm, and the correction r 0 = 1.16(1 1.16A -2/3 ) fm. For light nuclei the harmonic-oscillator shell model is used for the nuclear density [96]: ( r ) = ( πr ) 3 / exp( r / R ) ρ (2.19) i where R 2 = (2/3)<r 2 > = A 2/3 fm 2. To take into account the repulsive core of the nucleon-nucleon potential it is assumed inter-nucleon distance of 0.8 fm. The nucleus is assumed to be spherical and isotropic, i.e. each nucleon is placed using a random direction and the previously determined radius r i. The momenta p i of the nucleons are chosen randomly between 0 and the Fermi momentum p max F (r i ). The Fermi momentum, in the local Thomas-Fermi approximation as a function of the nuclear density ρ is: max p F 2 ( r) hc 3π ρ( r) i ( ) 1/ 3 = (2.20) The total vector sum of the nucleon momenta has to be be zero, i.e. the nucleus must be constructed at rest. The effective of collective nuclear interaction upon participants is approximated by a time-invariant scalar optical potential, based on the properties of target nucleus. For protons and neutrons the potential used is determined by the local Fermi momentum p F (r) as: V () r ( r) 2 pf = (2.21) 2m where m is the mass of the neutron or the mass of the proton, respectively. As concern the primary particle, an impact parameter is chosen randomly on a disk outside the nucleus, perpendicular to a vector passing through the center of the nucleus. The initial direction of the primary is perpendicular to this disk. Using straight-line transport, the distance of closest approach d min i to each nu- 74

82 2.4 Physical processes in Geant4 cleon i in the target nucleus, and the corresponding time-of-flight t d i is calculated. The interaction cross-section σ i with target nucleons is calculated based on the momenta of the nucleons in the nucleus, and the projectile momentum. Target nucleons for which the distance of closest approach d min min i is smaller than d i < (σ i /π) 1/2 are candidate collision partners for the primary. All candidate collisions are ordered by increasing t d i. In case no collision is found, a new impact parameter is chosen. The primary particle is then transported in the nuclear field by the time step given by the time to closest approach for the earliest collision candidate. Outside the nucleus, particles travel along straight-line trajectories. Inside the nucleus particles are propagated in the nuclear field. At the end of each step, the interaction of the collision partners is simulated using the scattering term, resulting in a set of candidate particles for further transport. The secondaries from a binary collision are accepted subject to Pauli's exclusion principle. In the allowed cases, the tracking of the primary ends, and the secondaries are treated like the primary. All secondaries are tracked until they react, decay or leave the nucleus, or until the cascade stops due to the fact that the mean energy of all propagating particles in the system is below a threshold ( 15 MeV). At this stage the state of affairs has to be treated by means pre-equilibrium decay and de-excitation models. Hence, the residual participants (pre-fragment), and the nucleus in its current state are then used to define the initial state for preequilibrium decay. Pre-equilibrium decay and de-excitation models At the end of the cascade, a fragment is formed for further treatment in Precompound and nuclear De-excitation models. These models need some information about the nuclear fragment created by the cascade. The fragment formed is characterized by several parameters, such as the number of nucleons in the fragment, the charge, the four momentum of the fragment and so on, which are used as input parameter by the pre-equilibrium and de-excitations models. The GEANT4 Precompound model is considered as an extension of the hadron kinetic model. It gives a possibility to extend the low energy range of the hadron kinetic model for nucleon-nucleus inelastic collision and it provides a 75

83 2. Geant4: a Monte Carlo simulation toolkit smooth transition from kinetic stage of reaction described by the hadron kinetic model to the equilibrium stage of reaction described by the equilibrium deexcitation models [97]. The Precompound model is applicable in the energy range MeV. Only emission of neutrons, protons, deutrons, thritium and helium nuclei are taken into account. The precompound stage of nuclear reaction is considered until nuclear system is not an equilibrium state. Further emission of nuclear fragments or photons from excited nucleus is simulated using equilibrium models implemented in Geant4, such as the Evaporation, the Fermi breakup and the Statistical Multifragmentation models. At the end of the pre-equilibrium stage, the residual nucleus is supposed to be left in an equilibrium state, in which the excitation energy E* is shared by a large number of nucleons. Such an equilibrated compound nucleus is characterized by its mass, charge and excitation energy with no further memory of the steps which led to its formation. If the excitation energy is higher than the separation energy, it can still ejects nucleons and light fragments (d, t, 3 He, α), whose description is treated by the Evaporation model (for excited nuclei at relatively low excitation energies). These constitute the low energy and most abundant part of the emitted particles in the rest system of the residual nucleus. The emission of particles by the Evaporation model is based on Weisskopf and Ewing model [98]. Fission (for nuclei with A > 65) and Photon Evaporation can be treated as competitive channels in the evaporation model. The Statistical Multifragmentation and Fermi break-up models are used at excitation energies E * above 3 AMeV. The Multifragmentation model is capable to predict final states as a result of a highly excited nucleus statistical break-up. The initial information for calculation of multifragmentation stage consists in the atomic mass number A, charge Z of excited nucleus and its excitation energy E *. At higher excitation energies (E * > 3 AMeV) the multifragmentation mechanism, when nuclear system can eventually breaks down into fragments, becomes the dominant. Later on the excited primary fragments propagate independently in the mutual Coulomb field and undergo de-excitation [99]. 76

84 2.4 Physical processes in Geant4 The Fermi break-up model is capable to predict final states of excited nuclei with atomic number Z < 9 and A < 17. For light nuclei the values of excitation energy per nucleon are often comparable with nucleon binding energy. Thus a light excited nucleus breaks into two or more fragments with branching given by available phase space. This statistical approach was first used by Fermi to describe the multiple production in high energy nucleon collision [100]. 77

85 3. Neutron production from carbon ion beams interaction on thin targets Open issues in hadrontherapy with carbon ion beams have led to the use of Monte Carlo simulations for a detailed particle transport in the matter. As previously discussed (chapter 2), Monte Carlo codes are able to simulate all the possible interactions of primary particles with the matter and track the secondary particles produced, which are finally responsible of the mixed field. A detailed estimation of the composition and characteristics of the secondary particle field is of crucial importance in ion therapy (section 1.3). Hence, Monte Carlo codes, widely used for this aim, need to be validated against experimental data. The accuracy of their predictions ultimately relies on their ability to correctly model the relevant production cross sections. To that end, a stringent test of the physics input to ion transport model calculation is surely represented by comparisons against thin target experiments for charged fragments production. Double differential cross sections d 2 σ/dedω and angular distributions dσ/dω are probably the most suitable wished quantities. Unfortunately, there is a lack of data in the energy range of interest for hadrontherapy ( AMeV), especially for carbon

86 3.1 The RIKEN experiment ion incident particles. There are some more chances in case of thick target experiments but they are often considered not the best choice in case of nucleusnucleus models validation, even if widely exploited for shielding design, where they are useful for radiation background estimation. For thick target experiments, indeed, primary particles go through the target decreasing their initial energy down very low values and, in some cases, stopping into that. Therefore, different processes at different energy values can occur, giving rise to a mixed field composed also by secondary particles which, in turn, can interact with other target nuclei producing other secondaries. It is evident that a superimposition of different processes provides the final spectrum obtained, so that definitive and detailed conclusions on the predictions of the interaction model studied cannot be drawn. Some recent data are instead available in literature for thin target experiments concerning the neutron production. Even if they do not give direct information about light ions produced, however they can be considered a good tool for preliminary comparisons of nucleus-nucleus interaction models implemented in Monte Carlo codes. A set of neutron production experimental data, which have been selected between the most accurate and recent data present in literature, has been used for a first validation of the Binary Light Ion Cascade model of the Geant4 toolkit. 3.1 The RIKEN experiment Up to about ten years ago, for thin target neutron yields, there existed only a few published data on the double differential cross section for 337 AMeV Ne ions on C, Al, Cu and U target [101] and 790 AMeV Ne ion on Pb target [102]. Under these circumstances, a series of systematic studies have been performed since 2000 at The Institute of Physical and Chemical Research, RIKEN, in Japan, with the aim of measuring the neutron production cross sections by heavy ions with energies from about 100 to 800 AMeV. In particular, for the aim of this thesis, measured data from thin targets of C, Al, Cu and Pb bombarded by 135 AMeV C ions and published in 2001 by Sato et al. [103] have been used for 79

87 3. Neutron production from carbon ion beams interaction on thin targets comparisons with Geant4. The measurements were carried out by a research group working at the RIKEN Ring Cyclotron [103]. A beam of 135 AMeV carbon ions has been transported through a beam swinger system into a scattering chamber that held the target. A diagram of the experimental setup may be seen in Figure 3.1. A single NE-213 neutron detector was placed 847 cm far from the target position. After passing through the target, beam particles were bent by a dipole magnet towards a beam dump. Neutrons moving in the forward direction from the target position continued 7 m through a beam pipe, and exited through a 3-cm thick acrylic vacuum window, into air. Figure 3.1. The experimental setup used in the RIKEN experiment for the neutron detection [103]. 80 Before entering the neutron detector, the neutrons go through a square iron

88 3.1 The RIKEN experiment collimator. Concrete blocks surround the collimator in order to shield the detector from background neutron that come from the beam dump. Moreover, a 0.5 cm thick NE-102A plastic scintillator is placed directly in front of the neutron detector in order to detect any charged particles that were incident upon the neutron detector. The number of beam particles incident upon the target were counted with a Faraday cup coupled to the beam dump. The beam swinger allowed measurements of neutron spectra at 0, 15, 30, 50, 80 and 110 degrees relative to the beam direction into the target. Measurement of the background was not possible for geometrical reason but, however, they estimated it to be low, due to the extensive shielding around the collimator and beam dump. The targets used were C (0.216 g/cm 2 ), Al (0.162 g/cm 2 ), Cu (0.268 g/cm 2 ) and Pb (0.340 g/cm 2 ). Data were acquired on an event-by-event basis. Charged particles were eliminated by using E-E plots obtained with the two scintillators and gamma-ray events were eliminated using pulse-shape discrimination properties of NE-213 liquid scintillator. Neutron energies were calculated with the time-of-flight measurements. The authors of the paper give some more details about the experimental uncertainties. The energy resolution for a 100 MeV neutron was about 3%. The statistical uncertainties varied from 2 to 5 percent for low to medium energy neutrons (10 to 100 MeV), and around 30% for the highest energies. The background component not directly measured but calculated by measurements with and without target, was found to be less than 10%. The systematic uncertainty in the number of incident beam particles was estimated to be less than 5%. The uncertainty in the angular acceptance was less than 2%. The uncertainty in the efficiency calculation was estimated to be about 16%. The overall normalization uncertainty, then, was within 17%. Details on experimental data, as well as the numerical values of double differential neutron cross sections, have been taken from the publication [103] and from the Experimental Nuclear Reaction Data (EXFOR) database, which collects a compilation of nuclear reaction data [104]. 81

89 3. Neutron production from carbon ion beams interaction on thin targets 3.2 The Geant4 simulation For comparisons with the experimental data, a low-level approach has been used in the simulation. A model-level test for hadronic model validation, called test30, developed by the Geant4 collaboration, has been used and modified to obtain the neutron production spectra in this case of study [105]. Test30 is a generic verification suite based on an abstract interface to a final state hadronic interaction generator. Single events with a known incident particle energy are shot and an explicitly defined hadronic final state generator is chosen. The kinematics of secondaries produced in the interaction are analyzed and the resulting angular, momentum and energy spectra are stored in histograms. The test has been customized in order to implement the nucleus-nucleus model of interest, as well as the de-excitation models, and parameters of the experimental configuration has been used as input in the simulation. The nucleusnucleus model used for direct interaction was the Binary Light Ion Cascade [91] and the Tripathi total reaction cross section has been implemented [89]. The excited nuclear remnants created after the first cascade stage of interaction are passed to the de-excitation models, for nucleon evaporation end emission of light fragments. The Evaporation model has been used for the description of nucleon evaporation from excited nuclei at relatively low excitation energy. Fermi break-up channel is activated when Z 9 and A 17 and Statistical Multifragmentation model for excitation energies E * 3 MeV. Some more theoretical and technical details about the used models can be found in section The results, shown in the following sections, have been obtained using the Geant4.9.2.patch02 version of the Monte Carlo toolkit. Beam particle hits the nucleus target and, according to the impact parameter and the cross sections predicted by the model, secondary particles are produced at different energies and angles. Neutrons produced in the nuclear reactions are selected according to their direction of emission (laboratory angles 0, 15, 30 50, 80 and 110 degrees) and stored in ROOT histograms for further analysis. The angles have been calculated in order to cover the detection angle θ with a certain width θ given by the surface of the detector and its distance from the target 82

90 3.2 The Geant4 simulation (figure...), which has been calculated to be θ = In this way, all the neutrons emitted at the angle θ ± θ are collected in the simulation, with both the angles θ 1 = θ + θ and θ 2 = θ θ making a solid angle given by: ( cosθ ) Ω = 2π 2 cosθ1 (3.1) Figure 3.2. Schematic view of θ in the experiment. The cross sections contributing to the i-th bin of the histogram are given by aσ N R i σ i = (3.2) ninc where σ R is the total reaction cross section, a is the conversion factor from cm 2 to barn, n inc is the number of incident primary particles and N i is the number of times bin i is incremented during the run. Each bin represents a small region of phase space Ω E (with E the width of the bin in energy) into which the secondary particles go after being produced in the interaction [105]. Finally, the double differential cross section is estimated as: 2 d σ dωde σ i = E Ω (3.3) In some words, the method implemented in test30 application forces each primary particle shot in the simulation to undergo nuclear interactions, because the total reaction cross section σ R is directly picked up from the code and used for each event (according to the energy of the projectile and the atomic and mass numbers involved). In this way, all the useless events consisting on passing of 83

91 3. Neutron production from carbon ion beams interaction on thin targets primary particles through the target (i.e., only electromagnetic interactions) are skipped. The employment of this level-model test has noticeably reduced the CPU time without affecting the accuracy of the final results. Comparisons with the traditional approach, where the complete geometry of the experimental setup is simulated, have shown that same results can be carried out with calculation times decreased up to one order of magnitude. However, more in general, differences in time performances are sensibly dependant on particle type and energy as well as the target atomic mass. The simulation was run, shooting a total number of primaries for each target, which gives statistical uncertainties within 1%. An energy threshold of 10 MeV has been fixed in the simulation, which corresponds to the minimum neutron energy measured at each angle in the experiment. 3.3 Results and comparison with the data Double differential cross sections Comparisons between double differential cross sections d 2 σ/dωde obtained with the Geant4 simulation and ones obtained in the experiment are showed in Figures 3.3 to 3.6. In general, the spectra in the forward direction (0 ) are characterized by a prominent peak centred near the beam energy per nucleon, which is predicted also by the Geant4 simulations but under-estimated in the absolute value. As the target mass becomes heavier the high energy peak becomes less significant but, at the same time, the discrepancy between Geant4 and the data grow up. The dependence on target mass can be clearly seen comparing, as an example, the C + C system with the C + Pb system (Figures 3.3 and 3.6). Moreover as the angle increases, the peak becomes less prominent, so that at about 20 the peak is not more visible, both in the two spectra. This is a common 84

92 3.3 Results and comparison with the data trend for each target and, thus, clearly visible for each double differential cross section plot. The high energy neutrons in the region of this forward peak come mainly from the break-up of the projectile, along with the direct knock-on neutrons from the target. In these two kinds of processes, the momentum transfers from projectile to target nuclei adds a momentum kick during the collision, especially for light nuclei, so that neutrons with up to 2 or 3 times the beam energy per nucleon are detected in this region ( 400 MeV). A different behaviour can be seen in the low energy component of the neutron spectra. Indeed, at energies below 20 MeV, the spectra are dominated at all angles by the decay of the target remnant. Because velocities of targets in the laboratory frame are low, that source of neutrons is essentially isotropic and, as such, target-like neutrons can be seen at all angles. The exponential behaviour of the cross section with energy in this region suggests the target remnant decays by equilibrium processes, which seem to be over-estimated in the simulation especially for larger angles and heavier targets while better predicted, for example, for the C + C system at 0 and 15. More in general it can be seen that as the target mass increases the relative contribution to the overall spectra from target break-up increases, as evident comparing C and Pb targets. At intermediate energies (from about 20 MeV to the beam energy per nucleon), the spectra are dominated by the decay of the overlap region, where a sizable number of projectile nucleons and target nucleons mix and can undergo several nucleon-nucleon collisions. As the impact parameter of the projectiletarget interaction decreases, likelihood of a surviving projectile source and target source also decreases, and the contribution of the pre-equilibrium decay of the overlap region becomes the dominant source. In this region there is a component that becomes less pronounced as the angle increases. Trying to summarize, at forward angles and high energies there is an overall good agreement between experimental data and Geant4 simulation in case of light targets, but as the target mass increases the prominent peak centred near the beam energy per nucleon is not well reproduced. An under-estimation of neutrons at 0 is evident in the case of Pb target, which, for these systems, could be due to an un-correct prediction of the projectile break-up with its subsequent de- 85

93 3. Neutron production from carbon ion beams interaction on thin targets excitation. At lower energies, a general overestimation of neutron production is found, which becomes more evident at larger angles and for heavier targets. However, the target remnant decays due to the equilibrium processes are well reproduced for smaller angles and lighter targets. Finally, in the intermediate energy region and at intermediate angles the agreement gets better as the target mass increases. For lighter targets, the neutrons spectra show a different trend, as clearly visible in the C + C system at 30 and 50. For each plot semi-logarithmic scale has been adopted, both for the experimental data and the simulation result, in order to better visualize the results and distinguish the two components at low and high energies of the neutron spectra. 86

94 3.3 Results and comparison with the data Figure 3.3. Neutron production cross sections for 135 AMeV C + C collisions at

95 3. Neutron production from carbon ion beams interaction on thin targets Figure 3.4. Neutron production cross sections for 135 AMeV C + Al collisions at

96 3.3 Results and comparison with the data Figure 3.5. Neutron production cross sections for 135 AMeV C + Cu collisions at

97 3. Neutron production from carbon ion beams interaction on thin targets Figure 3.6. Neutron production cross sections for 135 AMeV C + Pb collisions at Angular distributions Angular distributions of emitted neutrons from all systems are shown in Figures 3.7 and 3.8, where, as in the previous case, the results obtained with the Geant4 Binary Light Ion Cascade model are compared with the experimental data. 90

98 3.3 Results and comparison with the data The distributions were generated by integrating the spectra over neutron energies above 20 MeV, both for the data and the simulations. Error bars, both for the angles and the cross sections, are within the points width and not well distinguished. The following plots illustrate some of the points made above in the discussion of the double differential cross sections. The two different distributions (Geant4 and data) show similar trends, even if some relative difference is present in some cases. In general, as the target mass decreases, the distribution becomes more forward focussed and the cross section decreases. That behaviour is quite well reproduced by the Geant4 simulation. At forward angles, the yield decreases rapidly with increasing angle. There the yield is primarily due to direct knock-on processes and break-up of the projectile, which mainly contribute for small angles. Looking at this kind of plots it appears more evident the neutron production under-estimation at 0 of the simulation results, especially for heavier mass targets. At large angle the neutron yield rapidly increases as target mass increases. This may be due to the decrease in the number of target-like neutrons that contribute to the bulk of the yield. For intermediate angles an overall good agreement is found for each target. However, considering that angular distributions are obtaining by integrating over all energies (starting from 20 MeV), the contribution from target-like neutrons is dominated from the contributions coming from the other sources. In conclusion, it can be generally said that the overall trend of the neutron angular distributions is better reproduced by the Geant4 simulation for lighter targets, while for heavier ones it gets worse, as it can be seen in case of Pb targets where the two distributions show different slopes. 91

99 3. Neutron production from carbon ion beams interaction on thin targets 10 5 C + C -> n + X 10 4 Experimental data Geant4 dσ/dω [mb sr -1 ] Angle [ deg ] 10 5 C + Al -> n + X 10 4 Experimental data Geant4 dσ/dω [mb sr -1 ] Angle [ deg ] Figure 3.7. Angular distributions of neutron production cross sections integrated above 20 MeV, for C (top) and Al (bottom) targets. 92

100 3.3 Results and comparison with the data 10 5 C + Cu -> n + X 10 4 Experimental data Geant4 dσ/dω [mb sr -1 ] Angle [ deg ] 10 5 C + Pb -> n + X 10 4 Experimental data Geant4 dσ/dω [mb sr -1 ] Angle [ deg ] Figure 3.8. Angular distributions of neutron production cross sections integrated above 20 MeV, for Cu (top) and Pb (bottom) targets. 93

101 3. Neutron production from carbon ion beams interaction on thin targets 3.4 Influence of neutron production in hadrontherapy The discussed comparisons show that the Geant4 Monte Carlo toolkit is capable to predict the neutron production with an acceptable level of accuracy, even if some more efforts have to be done in order to improve the results, especially in the forward direction and for heavier targets. However the validation work up to now discussed will be undertaken also for other incident energies and eventually taking into account also the other nucleus-nucleus models present in Geant4. Anyway, as already previously mentioned, neutron production cross section do not provide a definitive benchmark of ion interaction models, due to a lack of information which only the detection of all the charged fragments produced can completely give. Considering the very few data in the energy range of interest for medical applications of hadrons, they however can be considered for preliminary comparisons. On the other hand, they are of great interest in shielding design and for space applications. Concerning the influence on hadrontherapy of neutron production from carbon ion beams fragmentation, various works have been carried out by different research groups, both from the experimental point of view [107] and also using Monte Carlo simulations [108]. Even if the study of neutron yield effects on tissue-like media is out of the scope of this thesis, it could be interesting just to mention a study carried out by I. Pshenichnov and his collaborators, who have estimated the energy distribution due to secondary neutrons produced by 12 C beams in water phantoms, using the toolkit Geant4 [109]. They simulated the interaction of 135 AMeV and 330 AMeV carbon beams, calculating the dose due to the neutron produced in the nucleus-nucleus interactions. Neutrons, as neutral particles, do not lose their energy directly via ionization, but rather via secondary reactions involving charged hadrons, i.e. secondary protons or nuclear fragments which thus produce ionization. Elastic scattering on target protons and nuclei are the main processes responsible for the energy loss by low-energy neutrons in water, hence, for those neutrons produced in de-excitations of remnants 94

102 3.4 Influence of neutron production in hadrontherapy target nuclei. For neutrons with energies close to the beam energy per nucleon, mostly originating from projectile fragmentation, other kind of reactions occurs, creating a kind of particle shower. However, for both the regime, the final result consists mainly in production of protons which are responsible for the 80-90% of the neutron energy deposition, according to their calculation. What is important at the end is that the neutron-induced ionization processes, due to secondary protons and nuclear fragments, contribute not more than 1-2% of the total energy deposited in a water phantom. Comparing the results also with those obtained for proton beam irradiation they conclude that additional neutrons produced by ion beams compared to proton beams do not appear to be a serious hazard to the human body. The situation is dramatically different in case of charged fragments; whose contribute in the total secondary dose is up to two orders of magnitude higher than the neutron contribution, especially in the core region of the beam [108]. That is the reason why the validation of nucleus-nucleus models against secondary charged particles production and the estimation of contribute in dose is of crucial importance in radiotherapy with carbon ion beams. These two aspects have been treated in this thesis and results have been shown respectively in chapter 4 and chapter 5. 95

103 4. Fragmentation of 62 AMeV carbon beams on Au targets As already pointed out, the production of projectile charged fragments is one of the most important problems to be considered when planning for the utilization of heavy charged particles in radiotherapy. Monte Carlo codes are powerful tools for this purpose but they need wide validation versus data, in order to trust in their predictions of secondary produced. Lack of accurate data on thin target experiments for charged fragments production in the energy range AMeV has led to initially compare nucleus-nucleus models versus neutron production data (chapter 3). As discussed in the previous chapter, these can be used as preliminary comparison but they are not suitable to give complete information on fragmentation reactions. Some data about fragments production from thin targets are available at very low energy (up to about AMeV) [110] but they cover just a little part of the energy range of interest for ion therapy. To cover this gap at intermediate energy, an extensive campaign of measurements has been recently started at Laboratori Nazionali del Sud (INFN) in Catania thanks to the collaboration between our research group, more strictly involved in hadrontherapy issues, and a research group involved in nuclear Physics. A first experiment has been already carried out: a 12 C beam, delivered by the Supercinducting Cyclotron (CS) of LNS at 62 AMeV impinging on a 197 Au target. Reac-

104 4.1 The experiment at LNS tion products are detected by a multi-detector array. The experimental setup has been simulated with the Geant4 Monte Carlo toolkit and its predictions have been compared with the experimental data, as shown in the next sections. Another experiment has already been performed and data analysis is still in progress. Other experiments at higher energies (up to 400 AMeV) are planned for the next year in other laboratories, so that further benchmarks of the Geant4 models will be able in the future. 4.1 The experiment at LNS Experimental data, reported in this work, have been collected at Laboratori Nazionali del Sud (LNS, Catania, Italy) in the framework of TPS project (see section 1.3.1). The aim of the experiment is to measure the angular distribution of the emitted fragments and absolute differential cross section. Figure 4.1. Schematic map of beam lines in the experimental rooms of LNS. Inside the scattering chamber a three positions target holder (empty frame 97

105 4. Fragmentation of 62 AMeV carbon beams on Au targets and µm 197 Au and 1 mm 12 C) is set up. Empty frame was used for background events measurements (due, for example, to particles interaction with frame edges) and 197 Au frame for the experiment. After fragmentation of the primary beam on the target, reaction products are detected by a complex system of telescopes and resolved with the E-E technique. In Figure 4.2 a picture and a scheme of the scattering chamber have been shown. Figure 4.2. Left: picture of the experimental set up. Right: schematic view of the experimental set up in the scattering chamber The detection system The interaction of ion beams at this energy is mainly dominated by projectile fragmentation. Reaction products have velocities similar to the primary beam and are emitted mainly in the forward direction. Hence, different isotopes have to be detected with atomic mass ranging from that of protons up to mass equal to the primary particle one. To investigate the beam fragmentation influence on dose, in hadrontherapy, the measurements of the yield of the different isotopes is necessary. Therefore, a detection system used for this aim has to be able to distinguish particles in term of atomic Z and mass number A, with a good resolution in energy and high efficiency, covering a solid angle relatively small around the incident beam direction. Hence, in our experiment two Si-CsI hodoscopes of two-fold and three-fold telescope detectors with different granularity have been used, called Hodo-small 98

106 4.1 The experiment at LNS and Hodo-big, respectively placed at a distance of 0.8 m and 0.6 m from the target [111]. Each element of the Hodo-small consists of 300 µm thick silicon E ( E) detector 1x1 cm 2 followed by a CsI(Tl) crystal E (E) detector with the same area and 10 cm long, with photodiode read-out. The detectors of the Hodo-small are arranged in a cube of 81 telescopes, covering a total angular range of θ lab = ± 4.5 (Figure 4.3, right). In the final version of the Hodo-big each element of the multi-detector array is a three-fold telescope consisting of a 50 µm thick Silicon E ( E 1 ) detector followed by a 300 µm thick Silicon E ( E 2 ) detector and a 6 cm long CsI(Tl) crystal E (E) detector with photodiode read-out. Each element has an active area of 3x3 cm 2. The implementation of the 50 µm E 1 detector in the Hodo-big allows a lower energy threshold of 3 AMeV. The 88 modules of the Hodo-big are arranged on a spherical surface of radius r = 60 cm centered on the target position. They cover a total solid angle of 0.23 sr with a geometrical efficiency of 90%. The Hodo-big covers the angular range between ± 6.09 θ lab ± (Figure 4.3, left). Figure 4.3. Left: the 88 modules of the Hodo-big arranged on a spherical surface of radius r = 60 cm centred on the target position. Right: the detectors of the Hodo-small are arranged in a cube of 81 telescopes at 80 cm far from the target. The two hodoscopes cover a solid angle of 0.34 sr with a geometrical effi- 99

107 4. Fragmentation of 62 AMeV carbon beams on Au targets ciency of 72% [112]. In Figure 4.4 one of the Hodo-big telescopes is shown. In order to maximize packing there is no housing around the scintillators, and the E detectors are self-supporting silicon chips. Figure 4.4. Schematic representation (left) and image (right) of a telescope of Hodo-big. The CsI(Tl) scintillator has been chosen because of the advantages that it presents with respect to other possible candidates. It is only slightly hygroscopic so that hermetic sealing is not needed. Moreover it has good mechanical properties so that it can hold mechanical shocks fairly well and can be easily machined. Its high density (4.51g/cm 3 ) makes the scintillator suitable to stop ions, in the energy range of interest, in few centimeters. Furthermore the light emission spectrum fits well with the response curve of silicon photodiodes: the quantum efficiency is around 70% at 550 nm. A disadvantage is represented by the dependence on the charge, mass and energy of the particle and the scintillation quenching, which requires specific calibration procedures for each ion. The CsI crystals of the Hodo-big have been accurately cut in a pyramid-trunk shape (31x31 mm 2 and 34x34 mm 2 ) in order to be centered at the same distance (60 cm) on the target location, and to make possible each of them subtends an equal solid angle of 2.8 msr. The CsI scintillators of the Hodo-small do not require to be shaped accordingly since they are simply packed one over the other because the small angular aperture with respect to the distance of the target ensures that 100

108 4.1 The experiment at LNS they are all placed at around the same distance within mechanical errors. Photodiodes have been chosen because they require small space and have a good quantum efficiency for the CsI(Tl) light emission spectrum E-E identification technique and data acquisition Working principle of a generic radiation detector depends essentially on the interaction modalities between the incident radiations and the absorbing material. In case of charged particle beams at intermediate energies, the main mechanism for ion energy loss consists in the Coulomb interaction with orbital electrons of the traversed medium. In particular, traversing a thickness of material, charged particles collide with electrons close to their trajectory and delivery, in every collision, a small fraction of their initial kinetic energy producing excitation or ionization of the medium atoms. The stopping power S, described by the Bethe-Bloch formula (see section A.1.1), is proportional to S (mz 2 /v 2 ) where m, z and v are respectively mass, charge and velocity of the incident particle. The E-E technique is based on this relation. Indeed, telescopes generally consist of a thin detector E, where the projectile loses only a small fraction of its total energy, positioned before a second detector E, thicker than the first one, where the incident ion is completely stopped. In this way, considering only events in coincidence in the detectors, the energy loss ( E signal) and the total energy (E tot = E+E) of the projectile are measured (the last as sum of E and E signals). Thus, it is possible to identify the particle which has traversed the telescope and to resolve also for the different isotopes [113]. Figure 4.5 shows a typical E-E calibrated matrix relative to one of the Hodo-big telescopes used during the experiment, where it is possible to resolve the atomic and mass number of the isotope produced. It is also clearly visible the elastic peak of the incident 12 C particles. Signals coming from the detectors are handled by means of standard electronics chains for the pulse height analysis, based on charge preamplifier, shaping amplifier, stretcher and analog-to-digital converter. The signals are then digi- 101

109 4. Fragmentation of 62 AMeV carbon beams on Au targets tized by 96-channel Fastbus LeCroy QDC (charge-to-digital converter) with a common gate derived from the data acquisition trigger. Figure 4.5. A E 2 -E CsI matrix of one of the Hodo-big telescopes (θ = 8.62 ). It is clear that the resolution in terms of mass A is good up to Z 4; i.e. for H, He, Li and Be isotopes. Instead, Boron and Carbon isotopes are not well distinguishable by means of a simple graphic analysis. 4.2 Simulation of the experiment with Geant4 In spite of the neutron production case, discussed in the previous chapter, for the simulation of this experiment a more realistic approach has been followed, where in a first stage the complete geometry of the experiment has been reproduced. This choice is mainly motivated by the fact that the particles detected by the experimental apparatus are charged isotopes; considering the thin target used (113 µm), energy spectra of charged fragments (especially the heavier ones) could be slightly influenced in energy values, depending on the interaction point 102

110 4.2 Simulation of the experiment with Geant4 of the primary particles in the target, because of their electromagnetic interactions (and consequently energy losses) with the target itself. Hence, in order to have a more realistic comparison with the data and considering that the experiment has been carried out by our research group (therefore, each kind of details was in our knowledge), the full simulation approach has been initially adopted. At first a single element of the Hodo-big detector has been reproduced. Hence, a single E-E telescope detector has been simulated, considering the energy released in the different fold of the telescope and collecting the output in a way that a typical E-E matrix was reproduced. Results have been afterwards compared with those obtained with a more performing method, where kinetic energies of the isotopes are stored just when they hit the surface of the E detector, showing that no relevant differences are present in the results. Finally, the second method, chosen for the final configuration of the simulation, has been extended to larger solid angles in order to achieve a higher statistics. Details on the simulation approach are discussed in the next section. Concerning the physical processes implemented, some hadronic processes used for the neutron production simulation (see section 3.2) has been implemented here. Thus, also in this case the Binary Light Ion Cascade nucleusnucleus models have been used, with the Tripathi total reaction cross section and the G4Lelastic model for the elastic scattering. In order to treat the equilibrium stage, the following de-excitation models have been implemented: Evaporation (for nucleon evaporation at low excitation energy), Fermi break-up (for Z 9 and A 17) and Statistical Multifragmentation (for excitation energies E * 3 MeV). For more technical details on models see section The Standard Electromagnetic Physics package was chosen for the description of the energy losses of primary and secondary charged particles (section for details). All the classes handling the most important processes have been implemented, considering the models for ionization processes and multiple scattering. Outputs of the simulations have been stored at the end of each Run in ROOT histograms and ASCII files. Simulations and final results have been achieved with the Geant4.9.2.patch02 version of the Monte Carlo toolkit. 103

111 4. Fragmentation of 62 AMeV carbon beams on Au targets Simulation of a single E-E telescope detector As mentioned, in a first stage, a realistic reproduction of the E-E telescope detectors used in the LNS experiment has been developed in the simulation. This more realistic method, although very time consuming, has allowed testing the reliability of the more performing simulation approach finally used for comparison with the experimental data. The characteristic of the beam have been defined in the PrimaryGeneratorAction class (see 2.3.3). The 12 C ion beam, has a Gaussian distribution both in energy and in the traversal plane position. The total nominal energy was 744 MeV, with a estimated energy spread of about 0.1% and a spot characterized by a spatial distribution of σ y = σ z = 0.5 mm. The geometry configuration has been defined in the DetectorConstruction class, where a World volume contain all the daughter volumes. Along the beam direction the target has been placed: it is a 197 Au square shaped foil of 7 cm side and 113 µm thickness. At the distance of 60 cm and at the angular position of 8.62 has been placed one element of the Hodo-big detector. For this first test it was decided to not consider the first 50 µm Si detector ( E 1 ), but only the E 2 and E stages. Thus, a 300 µm thickness silicon E detector of 3 x 3 cm 2 of area was placed, followed by a CsI crystal pyramid-trunk shaped (31x31 mm 2 and 34x34 mm 2 ) of 60 mm of thickness. The simulation was run with a total of primaries in order to achieve an acceptable statistics. Considering the very long simulation time the simulation has been divided in 20 runs of events with different random number sequences 6, exploiting a CPU cluster present at LNS. The energy released by the secondary charged particles produced during the interaction of carbon ion beams with the target, as well as carbon ions them- 6 When simulations are divided in different jobs for finally collect the results, it is mandatory to run each job with different random number sequences in order to achieve different numerical results. This is possible by submitting each job with a different initial seed (see 2.1). 104

112 4.2 Simulation of the experiment with Geant4 selves deflected for elastic scattering, has been computed, tracking each particle inside the two folds of the telescope and simulating the eventual secondaries which in turn could be produced into the detector itself. For each event, which includes the tracking of the primary carbon and of all the secondary particles, energy values are stored together with the number of event and the atomic and mass number (Z and A) of the isotope. In Figure 4.6 a snapshot of the simulation is shown where primaries hitting the target are visible, together with the E-E telescope. This configuration has been called setup1. The following figure has been obtained using the visualizations tools present in Geant4. Figure 4.6: Snapshot of the Geant4 simulation. Target and telescope are visible. Blue tracks are positive charged particles and green are neutral particles. Results from the simulation are shown in Figure 4.7, which reasonable reproduces a typical E-E matrix plot, also obtained with the analysis of the experimental data and previously shown (Figure 4.5). Different atomic mass species are clearly visibly up to 12 C. In this case the elastic peak seems also well reproduced by the simulation. As expected, summing the E and the E energy value related to the 12 C elastic peak, it is obtained a value (~ 704 MeV), which is about equal to the nominal beam energy subtracted by the energy loss in the target. Energy spectra for each isotope produced has been calculated. 105

113 4. Fragmentation of 62 AMeV carbon beams on Au targets Figure 4.7. E-E matrix plot obtained with the Geant4 simulation. Also in this case, as in the experimental one, different isotopes can be resolved. Another simulation in the same conditions (physical as well as geometrical) has been repeated but with only a difference: in this case the whole telescope was not reproduced but just the first face of the E detector (which is considered as a detection plane ): kinetic energy, Z and A of the particles (both secondaries and primaries) hitting the face are calculated in the SteppingAction class and stored; after that, the particles are killed. This method avoids the tracking inside the telescope up to zero energy, allowing higher computing performance (about five times less time-consuming). This configuration has been called setup2. However, results obtained with the second method (setup2) have been compared with the first one (setup1) in order to check the reliability of the second method, slightly less realistic but less time consuming. Energy spectra of isotopes produced at that angle has been compared, showing that coherent results have been found in the two different cases. In Figure 4.8 is shown an example of 106

114 4.2 Simulation of the experiment with Geant4 inter-comparison for alpha isotope production. A slightly over-estimation of secondary in the high energy region has been found in the setup2 configuration, probably due to the probability of interaction of particles inside the telescope, especially in the CsI scintillator where the particles stop. Moreover, as expected differences at low energy are found, due to the threshold of the E Si detector. However, taking into account the thresholds (the same of the data) results obtained with setup2 well reproduce the results obtained with setup1, therefore setup2 was finally used for computing results at the other angles, considering it is less time-consuming and much more easier to implement in the case of complex multi-detectors arrays as the LNS Hodoscope Comparison of energy spectra for the two different setup setup1 setup Counts Energy [MeV] Figure 4.8. Comparison of alpha energy spectra obtained with the two different setups, at θ = Simulation of the Hodoscopes Once the setup2 method has been chosen, the DetectorConstruction class of the Geant4 application developed, has been modified in order to implement and simulate all the telescopes composing the hodoscopes (just considering 107

115 4. Fragmentation of 62 AMeV carbon beams on Au targets the first face, as previously discussed). Hence, at a distance of 60 cm from the target have been placed and reproduced the 88 telescopes of the Hodo Big, and 80 cm far from the target the 81 telescopes of the Hodo small. As in the experimental apparatus used for the measures, the Hodo Big detectors were arranged in a hemi-spherical shape centered in the center of the target, and the Hodo small detectors were placed in a square plane, reproducing the dead-space between the different elements. A scheme of the whole apparatus reproduced is showed in Figure 4.9. Figure 4.9. The scheme of the experimental apparatus reproduced in the simulation. Carbon beam, coming from the left hit the target, produce secondaries which are then detected by Hodo Big (60 cm far) and Hodo Small (80 cm far). For comparison with the experimental data six angles θ has been chosen, correspondent at six different detectors which were the same chosen for the experimental analysis [114]: θ = 18.97, 15.17, 11.38, 8.62 for the Hodo Big, and θ = 3.71, 3.34 for the Hodo small. A last modification to the geometrical configuration has been implemented in order to increase statistics with reasonable calculation time. Indeed, considering the displacement of each detector, it is possible to exploit the symmetry showed by the experimental apparatus and collect much more counts. 108

116 4.2 Simulation of the experiment with Geant4 Figure Left: front view of the scheme of the apparatus; red lines indicate the spherical crown containing a specific detector (15.17 ), which has to be considered in the construction of the final ring-type detector. Right: An example of a ring-type detector arranged in a spherical surface, where particles coming from the target are registered (the ring dimension is not that of the angles finally chosen) For example, fixing the distance r, the detector placed at a specific inclination angle θ = and azimuth angle φ = 0 will be symmetric to another detector placed at the same inclination angle θ and, for example, at the azimuth angle φ = 180 (Figure 4.10, left). These two detectors will detect the same number of reaction products and, consequently, they will give approximately the same counts, according to the statistical fluctuations (as confirmed by the simulations). Therefore, large ring-type detector on a spherical surface can be used with an angle equivalent to the experimental one (Figure 4.10, right). In such a way, the solid angle covered by a single detector is included in the much larger solid angle covered by the ring-type detector, achieving enhanced statistics in less computation time, without affecting the final results. This final setup (setup3) has been definitively exploited for the production of output to be compared with the experimental data. In this configuration, particles (secondaries as well as primaries) coming from the target interaction point are registered in the SteppingAction class when they hit the ring-type detector and information stored (kinetic energy, Z and A) before killing them. A total 109

117 4. Fragmentation of 62 AMeV carbon beams on Au targets number of of primary particles have been used, achieving statistical uncertainties within 2% 4.3 Results and comparisons with the experimental data Data analysis As mentioned in section A.2.3, first experiments on fragmentation processes at intermediate energies (20 AMeV E/A 200 AMeV) have shown that already in the energy range projectile-like fragments exhibit many features observed above 100 AMeV, such as broad mass distributions and nearly Gaussian energy spectra of expected width. These kinds of fragments are produced with a velocity slightly lower than the incident particle velocity and are emitted in a relatively small angular cone around zero degree. Anyway, some typical low energy phenomena, such as direct surface transfer reactions and relatively large energy dissipation seem to survive and, thus, they indicate that we are not completely in the high energy regime. This causes deviations from specific projectile fragmentation processes. These deviations mainly consist on mean velocity slightly lower than the beam velocity and low energy tails in the energy spectra. Starting from these consideration, the output of the simulations have been analyzed fitting with a Gaussian curve the energy spectra related to each isotope (for each considered angle), following exactly the same method used for the experimental data analysis. In such a way, comparisons between experimental data and simulation results are coherent. The β for each isotope (with β = v/c) has been calculated so that the projectile fragmentation peak has been selected and Gaussian fits superimposed. Carbon ions hitting the target at their nominal energy (744 MeV) emerge from the target with an energy of about 703 MeV, because of the electromagnetic interactions with the target electrons. This energy value corresponds to β = The energy loss value of about 40 MeV has been confirmed both by the simulation results and the experimental data. In Figure 4.11 it has been shown, as an example, the spectrum of alpha parti- 110

118 4.3 Results and comparisons with the experimental data cles at θ = obtained with the Geant4 simulations and the related Gaussian fit. In this case it has been relatively simple to fit the energy distribution. However, it clearly exhibits a small tail in the lower energy side of the Gaussian distribution due to the energy dissipation in the fragmentation processes, found in the experimental spectra as well. Similar results have been found also for the other isotopes, even if in some cases it was not so easy to well distinguish the projectile fragmentation peak (especially for lighter isotopes). Figure Energy spectrum of alpha particles at θ = 11.38, fitted with a Gaussian curve. The peak related to the projectile fragmentation is clearly visible together with a small tail at lower energies, due to dissipative processes. A first qualitative calculation has been carried out in order to evaluate the angular distribution of the fragments produced and compare their behaviour as a function of the atomic mass Z. For each isotope and angle the energy spectra have been integrated in order to obtain the number of particles detected at the specific angle. Results have been summed for particles with same Z and then normalized on the solid angle covered at the specific angular position. The values obtained have been plotted for each atomic number normalizing respect to the smaller angle (3.34 ). Results have been showed in Figure 4.12, where counts per solid angle are compared in a semi-logarithmic scale for each atomic number Z at the different considered angles. Errors have been calculated with the error propagation laws. For the angular uncertainty, half value of the inclination 111

119 4. Fragmentation of 62 AMeV carbon beams on Au targets angle θ related to each coronal crown has been considered. The angular distributions have been fitted with a straight line, which well approximate the distributions. This demonstrates that the angular distribution of the reaction products exponentially decreases with the angle, confirming that the projectile fragments are mainly produced in the forward direction. This general behaviour is followed for each Z; moreover as the Z increases the angular distribution is steeper, showing a higher gradient. Thus, heavier particles are more forwarded emitted than lighter. The trend showed is also confirmed by experimental data [114]. Counts per solid angle normalized [A. U.] ,1 0,01 0,001 Geant4 H He Li Be B C 0,0001 0,00 5,00 10,00 15,00 20,00 25,00 Angle [degree] Figure Counts per solid angle for each atomic number Z as function of the angle, normalized to the smaller angle (3.34 ). The figure shows that the projectile fragments are mainly emitted in the forward direction; moreover, the angular distribution, which exponentially decreases, is steeper for heavier fragments. This comparison gives an interesting, but still qualitative, description of the fragmentation processes occurred and the consequent reaction products, confirming some expected more general trends. However, to validate the Geant4 nucleus-nucleus model used in the simulations and evaluate its accuracy in the 112

120 4.3 Results and comparisons with the experimental data prediction of fragments produced, comparisons of the absolute angular cross sections are necessary Angular cross sections The calculation of the angular cross section has been done integrating the energy spectra (fitted with a Gaussian curve) obtained with the Geant4 simulation for each isotope at each angle θ, which gives the total counts N D at the specific angle, and normalizing respect to the solid angle Ω and the total number of incident particles n inc (primaries). In formula: dσ = dω n inc N t arg et ( cosθ cos ) ρδx2π θ where A target is the atomic mass of the target expressed in g ( g), ρ the density (19.32 g/cm 3 ), δx the target thickness ( cm) and Ω = 2π(cosθ 2 -cos θ 1 ) is the solid angle covered by the spherical crown which includes the detector. Errors on angular cross sections have been calculated with the error propagation of counts, angle and target thickness. The angular cross sections obtained with the Geant4 simulations have been then compared with the experimental ones. Unfortunately, the experimental absolute cross section was not directly calculated because the number of incident particles was not directly measured, due to technical problems during the experiment. Hence, an indirect method has been implemented for the normalization of the experimental counts, which arises large errors on the estimation of the experimental cross section ( 75%). This method consists on calculating carbon ions number per solid angle coming from elastic scattering and detected at an angle θ smaller than the grazing angle ( 4.9 in this specific case) and comparing this value with the Rutherford cross section per solid angle at the same angular position. From this comparison a calibration factor can be obtained to convert counts per solid angle (sr -1 ) into cross section per solid angle (barn/sr). Unfortunately only two detectors are placed within the grazing angle, hence just two points can be considered in the calculation. Thus, in the experimental analysis stage, it has been decided to use the mean angle (in the centre of mass sys- 113 D A 2 1 (4.1)

121 4. Fragmentation of 62 AMeV carbon beams on Au targets tem, as required by the Rutherford formula) between them and the mean value N of the corresponding counts N 1 and N 2 of detected carbon ions per solid angle. Unfortunately this method implies a very large error, above all in the N value, because the most probable value of the mean counts N is given by any number within N 1 + δn 1 and N 2 - δn 2 (where δn 1 and δn 2 are the errors of N 1 and N 2, respectively and N 1 > N 2 ). In the Figures 4.13 to 4.20 some comparisons between experimental angular differential cross sections and results obtained with the Geant4 simulations are shown in semi-logarithmic scale for the following isotopes: proton, deuteron, triton, alpha, 6 Li, 7 Li, 9 Be and 10 B. Linear fit of both the experimental data and the simulation results have been done. The general trend already discussed is here confirmed also by data. Indeed, both the angular distributions are forward peaked, especially for heavier fragments, which exhibit a steeper gradient. Indeed, the considerations made about the exponential decreasing of projectile fragments with the angle increasing are confirmed by experimental data. However, even if for almost all isotopes the trend showed is well reproduced (only in case of 10 B the slope is not well reproduced), the absolute value of the angular cross sections related to simulations differ from data up to one order of magnitude, in some cases. For lighter secondaries, like protons, deuterons and tritons, the agreement of the absolute values is slightly better (within the errors) but for heavier fragments the differences increase. The discrepancies would seem to show that the nucleus-nucleus interaction model used in the Geant4 simulations does not satisfactorily reproduce the projectile fragmentation, producing a not adequate number of secondary particles. But, a consideration has to be done, looking at the comparisons. Assuming that the number of inelastic interactions per incident particle is correctly implemented (i.e. the total reaction cross section is correct), it is reasonable to expect a certain asymmetry in the discrepancies. In other words, one does not expect to have an underestimation or an over estimation of each secondary particle produced, which is exactly what have been found in the comparison. Hence, the fact that isotopes obtained with Geant4 are underestimated may be due to the reli- 114

122 4.3 Results and comparisons with the experimental data ability of total reaction cross sections or to normalization issues. The latter, as already pointed out, was one of the main problems faced during the experiment. Indeed, the indirect normalization method used in the data analysis implies big systematic errors in the estimation of the conversion factor. Moreover, in the data analysis some difficulties have been found for some cases in the discrimination of the projectile fragmentation peak (due to electronic noise), which may have caused imprecision in the Gaussian fit and subsequent integration for the calculus of the angular cross section. Of course, the fact that the discrepancy is not constant for each isotope and angle indicates that also the nucleus-nucleus model used in the simulation, in some case, may not correctly work. Cross Section per Solid Angle (barn/sr) 1,0E+03 1,0E+02 1,0E+01 1,0E+00 1,0E 01 1,0E 02 p Experimental data Geant4 1,0E 03 0,00 5,00 10,00 15,00 20,00 25,00 Angle (degrees) Figure 4.13: Angular distributions of proton production cross sections in semilogarithmic scale. 115

123 4. Fragmentation of 62 AMeV carbon beams on Au targets 1,0E+03 d Experimental data Geant4 Cross Section per Solid Angle (barn/sr) 1,0E+02 1,0E+01 1,0E+00 1,0E 01 1,0E 02 1,0E 03 0,00 5,00 10,00 15,00 20,00 25,00 Angle (degrees) Figure Angular distributions of deuteron production cross sections in semilogarithmic scale. Cross Section per Solid Angle (barn/sr) 1,0E+03 1,0E+02 1,0E+01 1,0E+00 1,0E 01 1,0E 02 t Experimental data Geant4 1,0E 03 0,00 5,00 10,00 15,00 20,00 25,00 Angle (degrees) Figure Angular distributions of triton production cross sections in semi-logarithmic scale. 116

124 4.3 Results and comparisons with the experimental data 1,0E+03 α Experimental data Geant4 Cross Section per Solid Angle (barn/sr) 1,0E+02 1,0E+01 1,0E+00 1,0E 01 1,0E 02 1,0E 03 0,00 5,00 10,00 15,00 20,00 25,00 Angle (degrees) Figure Angular distributions of alpha production cross sections in semi-logarithmic scale. 6 Li Cross Section per Solid Angle (barn/sr) 1,0E+03 1,0E+02 1,0E+01 1,0E+00 1,0E 01 1,0E 02 Experimental data Geant4 1,0E 03 0,00 5,00 10,00 15,00 20,00 25,00 Angle (degrees) Figure Angular distributions of 6 Li production cross sections in semi-logarithmic scale. 117

125 4. Fragmentation of 62 AMeV carbon beams on Au targets 7 Li Cross Section per Solid Angle (barn/sr) 1,0E+03 1,0E+02 1,0E+01 1,0E+00 1,0E 01 1,0E 02 Experimental data Geant4 1,0E 03 0,00 5,00 10,00 15,00 20,00 25,00 Angle (degrees) Figure Angular distributions of 7 Li production cross sections in semi-logarithmic scale. 1,0E+03 9 Be Experimental data Geant4 Cross Section per Solid Angle (barn/sr) 1,0E+02 1,0E+01 1,0E+00 1,0E 01 1,0E 02 1,0E 03 0,00 5,00 10,00 15,00 20,00 25,00 Angle (degrees) Figure Angular distributions of 9 Be production cross sections in semi-logarithmic scale. 118

126 4.3 Results and comparisons with the experimental data 1,0E B Experimental data Geant4 Cross Section per Solid Angle (barn/sr) 1,0E+02 1,0E+01 1,0E+00 1,0E 01 1,0E 02 1,0E 03 0,00 5,00 10,00 15,00 20,00 25,00 Angle (degrees) Figure Angular distributions of 10 B production cross sections in semi-logarithmic scale Double differential cross sections The explanation proposed so far, which in few words considers the big discrepancies found mainly dependant on normalization errors in the experimental data, seems also to be in part confirmed looking at the comparisons of double differential cross sections, which give more detailed feedbacks. Comparisons for deuteron, alpha and 7 Li isotopes at the angles θ = 8.62, 11,38 are shown as an example (Figures ). Of course, discrepancies in the absolute values are even more evident than in the angular distributions but what is interesting to note is that the shapes of the spectra are quite well reproduced, especially in case of lighter fragments, such as deuterons and alpha particles. For example, if we look at the alpha spectra comparisons, the trend is well reproduced at low and as well as at higher energy ranges. In particular for θ = 8.62 a prominent peak centred at about 230 MeV is visible in the experimental data as well as in the simulation results. It corresponds to the beam energy per nucleon, indicating that fragmentation processes of the projectiles are reproduced also by the simulations. Similar comparisons 119

127 4. Fragmentation of 62 AMeV carbon beams on Au targets have been found for other light fragments. Slightly different considerations have to be done in case of heavier fragments which, beyond discrepancies in the absolute values, show also a quite different spectrum, compared to the experimental data, especially at low energy. One example is the 7 Li fragment spectrum, where the peak at the energy beam per nucleon is present as well, but discrepancies in the shapes with the experimental data are evident at lower energies. This fact may be explained by considering that the de-excitation models used in the simulation could not correctly reproduce heavier fragments emission from the target remnant. 7 Li isotopes can be produced as residual in the Evaporation model and as fragment from compound nucleus explosion in the Fermi break-up and Multifragmentation models, provided that in this last case the excitation energy is E * = 3 MeV (see section 2.4.4). Instead light particles from protons up to alphas can also be produced as ejectiles in the Evaporation model. Anyway, a new generalized de-excitation model has been releasing inside the next Geant4 official version. New comparisons with this improved version will be carried out. Finally, comparing the angular cross sections of all the considered isotopes, it can be said that, in general, light particles are mainly produced in the inelastic interactions of primary carbon ions with the target, in particular alpha fragments. This fact has been confirmed also in case of thick target simulations (see chapter 5), where light fragments, namely for Z = 1, 2, are mainly produced in the inelastic interactions with thick phantom. In particular alpha particles are very stable nuclei respect to other light ones because of their greater binding energy ( 7 AMeV). This aspect gives rise to clustering phenomena, so that in the specific case carbon ions can be considered as alpha cluster structures and their break up into three alpha particles is extremely probable. Production of alpha particles from projectile fragmentation has to be taken into account especially when passive energy beam degrader are used in carbon ion treatment, as in case of fixed energy beam machine, such as the cyclotron (see section 1.4). In this case, indeed, alpha particles cannot be easily eliminated by the action of a magnetic system because they show the same velocity and charge to mass ratio of the carbon beam. However, further studies have demonstrated that by using specific techni- 120

128 4.3 Results and comparisons with the experimental data cal solutions, their contribution to the dose patient is negligible respect to the contribute coming from fragmentation inside the patient himself [115]. Considering the uncertainties of the experimental data, the measurements have been recently repeated at the same energy and target, reproducing the same conditions of the previous experiment. In such a way it will be possible to eventually correct the absolute values previously found, thanks to direct measurement of the incident carbon ions in the target. During this run, indeed, a Faraday cup has been used, which has allowed to continuously measure the beam current for the whole experiment. Moreover also a run of 12 C beams on 12 C target at 62 AMeV has been performed in order to study fragmentation processes for targets, which are more interesting from the biological point of view. An extended campaign of measurements is starting for covering the lack of data in the energy range of interest in ion therapy. An experiment, indeed, is scheduled on the GSI (Darmstadt, Germany) laboratory where fragments produced in the interactions of 12 C ion beams at 200 AMeV and 400 AMeV on 12 C and 197 Au targets will be detected with the ALADIN spectrometer. All these measurements will also contribute to continue the validation of nucleus-nucleus interaction models of the Geant4 toolkit, necessary for a correct estimation of the secondary particles produced in the fragmentation of carbon ion beams inside the patient. 100 d Experimental data GEANT4 100 d Experimental data GEANT4 d 2 σ/dedω (barn/sr/mev) , d 2 σ/dedω (barn/sr/mev) , , E (MeV) 0, E (MeV) Figure Deuteron double differential cross sections at 8.62 (left) and (right). 121

129 4. Fragmentation of 62 AMeV carbon beams on Au targets 1000 α Experimental data 1000 α Experimental data d 2 σ/dedω (barn/sr/mev) ,1 GEANT d 2 σ/dedω (barn/sr/mev) ,1 GEANT , E (MeV) 0, E (MeV) Figure Alpha double differential cross sections at 8.62 (left) and (right). d 2 σ/dedω (barn/sr/mev) ,1 7 Li Experimental data GEANT d 2 σ/dedω (barn/sr/mev) ,1 0,01 7 Li Experimental data GEANT , E (MeV) 0, E (MeV) Figure Li double differential cross sections at 8.62 (left) and (right). 122

130 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments Advantages of using carbon ion beams for tumour treatments have been widespread discussed in chapter 1, where the main physical characteristics and also the biological effects related with this kind of irradiation have been highlighted. It has also been illustrated how, on the other hand, the fragmentation processes occurring along the primary particles path, produce different nucleons and light charged fragments which pass through the tissue together with the carbon ions. As showed in the previous chapters, many of these charged fragments have been shown to have velocity close to that of the incident particles, reaching a region beyond the range of primary particles and, consequently, causing an undesired dose amount, called fragment tail. This tail usually characterizes the distal part of the Bragg peaks related to heavy ions and it must be carefully considered, especially when critical healthy structure are close to the tumour (section 1.2.1). However, the contribution of fragments has to be taken into account not only in the tail but also in the whole radiation field because the biological effectiveness on tissue changes in presence of mixed radiation field (section

131 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments 1.2.2). The dose, yield, energy and linear energy transfer (LET) distribution of each particle species are all quantities to be taken into account for the characterization of the quality of a radiation beam. These quantities as a function of the body depth are therefore indispensable for a precise estimation of the clinical effect of any therapeutic beam, and for a final determination of the most suitable kind of beam for radiotherapy. Retrieving such a kind of information from measures in each possible configuration would be a very hard issue. Hence, Monte Carlo calculations are widespread used to take into account the contribution of the different parameters involved and to give more accurate and detailed basis for treatment planning calculations (section 1.3). Of course, nucleusnucleus interaction models implemented in Monte Carlo codes have to be largely validated against data. The obtained results at this regard have been already discussed in chapters 3 and 4, although further benchmarks at different energies and with different targets will be carried out in the future. In this chapter, total depth dose distribution of a 62 AMeV carbon ion beam calculated with the Geant4 simulation is compared with the experimental distributions, acquired during an experiment performed at Laboratori Nazionali del Sud (LNS) in Catania. This comparison allows to understand and verify if the interactions occurred along the primary particles path, and the consequent energy losses, are well reproduced by the code. Moreover, the mixed radiation field created because of the fragmentation processes has been studied, calculating with the Monte Carlo the dose contributions due to the charged fragments as well as their yield. Finally, an estimation of the distribution of LET in the depth has been carried out, considering the contribution related to each isotope produced and trying to obtain a unique value of LET calculated separately for each isotope. This represents a kind of total averaged LET, which should give a quantitative description of the radiation quality as a function of the depth. 124

132 5.1 Depth dose distribution 5.1 Depth dose distribution The experimental setup The experiment has been carried out at Laboratori Nazionali del Sud in Catania. Carbon ion beams at 62 AMeV are extracted by the LNS Superconducting Cyclotron and transported toward the 0 degree cave (Figure 4.1). The beam is transported in vacuum by various magnetic systems up to the final part of the beam line, where a kapton window of 0.05 mm of thickness is placed. After passing through the window, the carbon ion beam travels in air and reaches a monitor chamber (Figure 5.1, left). Figure 5.1. Left: a picture of the monitor chamber used for the beam monitoring during the experiment. Right: the water phantom equipped with the Markus chamber; the circular thin PMMA window is also visible This is a transmission ionization chamber installed along the beam line with the aim of monitoring the delivered dose. Monitor chambers are also used to do an on-line check of the beam. Indeed, they are connected to a feedback circuit which interrupts the beam irradiation when a given amount of dose (i.e. of MUs) is already released. After its calibration versus an absolute dosimeter it is also able to give a response in terms of Grays per Monitor Units (MU). The monitor chamber is made up of two kapton foils 25 µm of thickness and, adjacent to these, two copper foils of µm which contain an air thickness of 9 mm. 125

133 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments At a distance of 50 cm from the monitor chamber, a water phantom has been placed, where the dose distribution is measured. The phantom is a PMMA (polimethilmethachrilate) cubic tank, 40 cm of side, with a window of 1.5 mm of thickness placed on the beam direction side (Figure 5.1, right). It is filled of water and ionization chamber for the acquisition of the ionization current is placed inside. For this measurement, the parallel plate Markus type ionization chamber has been used. It has been polarized with an electric field of 300 V and coupled to an electrometer PTW-UNIDOS. The chamber is composed of two electrodes containing 2 mm of air and an active volume of about cm 3. Thanks to its very small active volume and the presence of a guard ring to minimize the perturbation effects, it is considered one of the most precise devices for this kind of measurements. For the experimental acquisition of Bragg curves the Markus chamber is moved inside the water box by means of high precision computer controlled stepping motors. Ionization current produced as function of the depth in water is acquired with a spatial resolution of the order of 50 µm. Acquisitions with the ionization chamber have been repeated 5 times for each penetration depth, obtaining an uncertainty within 2% Simulation with Geant4 Carbon ions depth dose distributions inside the water phantom have been reproduced by simulating the LNS beam line used for the experiment. The Hadrontherapy Geant4 advanced example has been used at this aim. It is a fullyopen source application initially developed by our group with the scope of simulating the passive proton beam line of the CATANA facility, at LNS, for the proton treatment of ocular tumours [116]. Proton beams at 62 MeV are transported in the CATANA room (Figure 4.1), where a passive beam line for delivering of the dose to the patient has been setup. The beam line is composed by a set of scattering systems and beam delivery monitors which have been completely simulated with Geant4 inside the Hadrontherapy application. Since 2004 the application has been included among the public examples of the Geant4 toolkit and regularly updated. Thus, users can freely download it from the website [77] and customize the geometry for the specific beam lines to be simulated [117]. 126

134 5.1 Depth dose distribution The Hadrontherapy application has been modified in order to reproduce the experimental setup used at the 0 degree beam line, previously described. In the DetectorConstruction class the vacuum area with the kapton window has been implemented together with the monitor chamber and the water phantom, placed at the same distances as in the experiment. The ionization chamber has been reproduced placing inside the simulated phantom a cubic sensitive detector 40 mm of side, divided in 4000 slabs of 10 µm of thickness orthogonal to the beam axis. 12 C beam PMMA window slabs Sensitive detector Figure 5.2. Scheme of the sensitive detector implemented in the Geant4 simulation. Dose released by the carbon in beam is stored in the slabs composing the phantom. This thickness has been chosen in order to have a good spatial resolution in the simulated results. From each slab it is possible to retrieve the total energy deposited by carbon ion beams as well as by the produced secondaries and to store it in ASCII or ROOT files. Each slab simulates the active volume of the ionization chamber. Hence, in a unique simulation run the total energy deposited along the penetration depth is calculated and the Bragg curve reproduced for further comparison with the data. In Figure 5.2 a representation of the sensitive de- 127

135 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments tector divided in slabs is showed. Physical models have been activated by simply recalling the appropriate Physics Lists. Hadrontherapy application, indeed, has been developed in order to allow the user to implement physical processes with all the possible methods available in Geant4: by hand, using the Physics Lists or using the packed Reference Physics Lists (see section 2.4). In particular the G4EmStandard- Physics_option3 physics list has been activated for the electromagnetic processes, which shows an enhanced level of accuracy on energy loss (this option3 version has been expressly developed by Geant4 collaboration to meet requirements of medical applications, where high level of accuracy is required). The nucleus-nucleus inelastic interactions have been simulated implementing also in this case the previously used Binary Light Ion Model, together with the deexcitation models for the evaporation of nucleons and light fragments in the equilibrated stage (see section for details on models). De-excitation stage has been activated implementing the same limits used for the neutron production simulation (section 3.2) and charged fragments simulation (section 4.2). Once the geometry set-up has been defined and the physical processes activated the final information Hadrontherapy application requires is the primary beams characteristics. A nominal energy of 744 MeV with a spread of MeV (i.e., 0.1% of the nominal energy) has been considered in the simulation. This value of energy spread has been chosen after comparing the results of simulations run with different energy spread values with the experimental Bragg peak. The nominal spread value (i.e MeV) showed a better agreement and, thus, it was finally adopted for the production of the results. A production cut of 0.1 mm has been adopted, after having been verified not to affect final outcomes and as well as to reduce calculation time. A total of primary events have been shot, which give statistical uncertainties within 1%. The Geant4 version 9.2.p02 has been used. In Figure 5.3 a snapshot from the Geant4 simulation is shown, where is clearly visible the carbon ion beam (blue tracks) impinging the water phantom (red cube) and producing secondary particles (green and blue tracks) and electrons (red tracks). 128

136 5.1 Depth dose distribution Figure 5.3. Snapshot of the Geant simulation. 12 C particles (blue tracks) coming from the left side interacts with the water phantom (red), producing secondary particles. Red tracks represent negative charged particles and green ones neutral particles Bragg peak comparison The Bragg curve obtained with the Geant4 Monte Carlo simulation has been compared with that obtained with experimental data. Total dose distributions are, of course, the final result of different effects and processes which occur at each depth position, from the incident particle energy up to the zero value. Along the path of primary carbon ions, ionization and excitation processes are produced and secondary particles are emitted for nuclear interaction. Secondaries release their energy ionizing the material traversed and can, in turn, create other else particles. Hence, different particles concur to the total energy deposit which, therefore, is the integral effect of various physical processes suffered by primary particles and its products. However, depth dose distributions are of great importance in hadrontherapy because they give crucial information about some dosimetric quantities used also as input parameter for treatment planning. So a comparison between Bragg curves, even if it is somewhat of integral, indicates if the predicted distribution is accurate or not. In the specific case of study, it indi- 129

137 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments cates how the Monte Carlo simulations well reproduce the energy loss processes suffered by the incident particles and secondary ones too. From a quantitative point of view, the agreement can be also evaluated by calculating some of the mentioned dosimetric quantities, such as: - Peak-plateau ratio, defined as the ratio of the deposited energy at the peak position and at the entrance. - Practical range, defined as the distance from the entrance surface of the beam and to the distal point where the 10% of the maximum dose is measured. - Full width at half maximum (FWHM), defined as the width of the Bragg peak at the points correspondent to the 50% of the maximum dose value. - Distal dose fall-off, defined as the distance between the points of 80% and 20% of absorbed dose along the beam axis beyond the Bragg peak Arbitrary units Depth [mm] Figure 5.4. Bragg curve of a 62 AMeV 12 C beam on water. The points give the experimental data acquired at LNS while the black line represents the Geant results. 130 Comparison between experimental Bragg peak and that obtained with the

138 5.1 Depth dose distribution Geant4 Hadrontherapy application is shown in Figure 5.4, where the Bragg curves have been normalized to the area. The depth dose distributions are in good agreement, as visible by the superimposition of the curves, both in the entrance channel (plateau) and at the peak, where the simulation slightly overestimates the released dose. In Table 5.1 are summarized the correspondent values of the quantities above defined both for the data and the simulated peak. Practical range, FWHM and distal fall-off show a very good agreement, considering that the highest discrepancy is of 30 µm (for the FWHM), which is within the position uncertainty of 50 µm. Parameters Exp. data Geant4 Peak-plateau ratio Practical range [mm] FWHM [mm] Distal fall-off [mm] Table 5.1. Comparison between dosimetric parameters obtained with the experimental data and with Geant4. Anyway, a discrepancy in the peak-plateau ratio of about 9% has been found. This value is larger than the uncertainties of the measured and simulated deposited energy which are respectively of 2% and 1%. The discrepancy is mainly due to the position uncertainty which affects the localization of the measuring point of the Markus chamber in the phantom. Indeed, if we look at the Bragg peak and the gradient of dose just before and after that, it is evident that very small uncertainties in position can cause large difference in the Bragg curve values. This is a typical characteristic of carbon ions Bragg curve, especially for low energy incident particles, as in this case. In this configuration the peak is very steep, the gradient in the region of the peak is high and it can be difficult, from the experimental point of view, to put the detector at the peak posi- 131

139 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments tion. This problem is not found when the incident energy of carbon ion beams is higher ( AMeV) or, as an example, in case of proton beams (see Figure 1.6, chapter 1). Proton beams accelerated at the same energies, indeed, exhibit distal fall-offs roughly one order of magnitude higher and peak-plateau ratios slight larger than the half of carbon ion case [116]. Moreover, at these low energies, the fragment tail is not clearly visible because few fragmentation events occur in this case. 5.2 Contributions of charged fragments As already mentioned at the beginning of the chapter, the knowledge of the beam quality of a carbon ion beam is of crucial importance in treatment planning. An ion beam is surrounded by a halo of secondary fragments when traversing a specific material, because of nuclear processes. Secondary particles are responsible of the mixed radiation field and of the presence of the fragments tail. As widely discussed in section 1.2.2, it is important to know the spatial distribution of the produced fragments, their energy distribution and dose contribution. These and other else factors, indeed, are necessary to the calculation of the Relative Biological Effectiveness (RBE) which finally allows to know the biological dose to be delivered to the tumour target. In order to study how the mixed radiation field can influence the final biological response on tumour tissue, an experiment has been performed in the same room and with the same experimental configuration as the previous one. HTB140 human melanoma cells have been irradiated with 62 AMeV 12 C beams at different depth positions along the Bragg curve, in order to evaluate the cell survival and the quality of the cellular damage. Cells have been positioned behind various PMMA thin slabs of increasing thickness. Thus, what is different from the previous setup is the use of PMMA thickness. It is easier to handle PMMA slabs than water phantoms, when cells have to be positioned at different penetration depths. However this experimental method has been tested in the past and used with good results for proton beam irradiations [118]. In the next sections, results obtained from Monte Carlo calculation will be 132

140 5.2 Contributions of charged fragments shown. Depth dose distribution and fluence of secondary calculated with the simulation will be discussed. Finally, a method of calculation of the LET distribution of primary and secondary particles will be showed, together with results achieved for some particular points of the Bragg curve where, during the experiment the cells have been placed. However, only the results obtained by Monte Carlo calculations will be here discussed, because outcomes from the biological response are still under study and a deeper interpretation is needed, according to the Monte Carlo outputs Dose and yield of secondaries To obtain the dose and yield contribution due to the secondary particles produced, the Hadrontherapy application has been modified retrieving single dose distributions curves related to each charged fragment produced along the carbon ions path. A PMMA cubic phantom, similar to that used for cells irradiation, has been simulated. The same methods of retrieving energy deposited in the sensitive detector and the same physical processes and parameters previously implemented (see section 5.1.2) have been used. In order to have more readable plots, contributions of the isotopes have been summed, so that depth dose distributions for H, He, Li, Be and B fragments are shown in Figure 5.5, without distinguishing the mass number A. The dose contribution relative to carbon ions, which have undergone hadronic interaction along the path ( 10 C, 11 C), have been also taken into account, separating this contribute from the depth dose distribution of the primary 12 C ions. Moreover, total depth dose distribution (Bragg peak) and total dose due to the fragments are shown in the same plot. Semi-logarithmic scale has been used, otherwise dose contributions due to the secondaries and the fragment tail behind the peak would be not clearly visible 7 (Figure 5.5). A total number of primary events ( 12 C) have been shot in the simula- 7 At the this low energy ( 12 C at 62 AMeV), effects of secondaries in dose and fluence are, indeed, much less visible than for higher energies (up to 400 AMeV). 133

141 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments tion in order to have statistical fluctuations on secondary particles production almost overall within 5%. Indeed, a huge number of primary particles is necessary in order to have relatively smooth depth dose curves also for the secondaries. A practical range of 9 mm has been found for primary particles. In the overall plateau region, dose of secondary particles contribute only to about 1-3% on the total deposited dose registered at the same position and to the 0.5% on the peak. Thus, total dose (black line) and primary 12 C dose (red line) practically coincide up to the peak position. Of course, just behind the Bragg peak total dose is only due the secondary particle (100%), indeed 12 C dose drastically drop to zero. Concerning the contribution of each particle on the total secondary dose, the percentage dose respect to the total dose due to the fragments at 1 mm behind the Bragg peak has been evaluated. As expected, the main part is due to the He fragments, which release the 56% of dose, followed by B fragments with 18% and H with 15%. Contributions of other fragments are below 5%. Going more in depth the contribution is practically due only to H and He fragments which give a total dose which at 30 mm from the peak is still not negligible (~ 7% of the secondary dose obtained 1 mm behind the peak). This fact confirms what previously found in chapter 4 and discussed for isotopes production in thin target experiments. Many alpha particles are, indeed, produced (together with H fragments) because of the specific alpha cluster structure characterizing carbon ions. Moreover, the presence of light particles relatively far from the range of the primary also confirms that projectile fragments are produced at velocities close to the primary particle ones, as expected according to the thin target results (see section 4.3). It is also interesting to highlight that 1 mm before the Bragg peak the main contribution is given again by He fragments (46%), but in this case also C isotopes give a not negligible contribution (25%). An ambiguous step of He fragments in the entrance remains to be better understood. It may be due to some nuclear processes which could be not activated for this specific energy range, but deeper investigations are needed in order to understand its cause. 134

142 5.2 Contributions of charged fragments However, it must be stressed that the following plot represents just a calculation of the physical dose, not giving detailed information on where it is expected the major biological damage. To do that, the beam quality have to be globally considered and LET calculation can give, in this direction, more realistic responses, as it will be discussed in the next section. Arbitrary units H He Li Be B C C12 Total Dose Dose of secondaries Depth [mm] Figure 5.5. Total depth dose distribution (Bragg curve, in black) and contribution of charged fragments (coloured lines) in PMMA. Dotted line represents the total contribution from secondary particles. More or less the same general comments are valid for the fluence distribution (Figure 5.6). H and He ions are produced for overall the depth roughly with the same amount. This is, obviously, coherent with what previously found for the dose: He fragments, indeed, release about four time the dose released by H fragments, because of the Z 2 dependence of the stopping power (see sections and A.1.1). Thus it has been found a released dose about 4 times higher. Concerning the primary beam, about 82% of the incident carbon ions 135

143 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments reaches the peak region, because stripping of one or two neutrons produce respectively 11 C and 10 C isotopes (mostly 11 C, which are about the 90% of secondary C fragments yield). As already pointed out, these isotopes are of great importance in hadrontherapy because they give the possibility to use the modern in-situ PET (Positron Emission Tomography) techniques for the monitoring of the delivered dose to the patient (see section 1.2.1). A detailed calculation of the secondary particles yeild is crucial when calculation of LET and its spatial distribution have to be evaluated for trying to relate the cellular damage to the radiation quality H He Li Be B C C12 Fluence Depth [mm] Figure 5.6. Fluence depth distribution of primary and secondary particles in PMMA 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 136

144 5.2 Contributions of charged fragments to be considered, too. The main advantage of the carbon ion beams for radiotherapy 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 energy transfer. Thus it is reasonable to provide spatial LET distributions in addition to the dose distributions. This might help to localize high LET regions, where the greatest variations in RBE are expected, with the relative consequences 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. Therefore, 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 different 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 implementations 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 particle (i.e. only primary particles) but when we are in presence of a mixed field, as 137

145 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments 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 separately 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 complicated 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 produced, and to weight each contribute on the specific isotope, at different penetration 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: L t ( z) = 0 S el 0 ( E) ϕ ( z) ϕ E E ( z) de de (5.1) where φ E (z) is the particle fluence differential in energy at the position z, and S el (E) represents the electronic stopping power as function of the energy E. LET calculations have been implemented inside the Hadrontherapy application. The simulation has been run exactly with the same geometrical configuration, beam parameters and physics used for the secondary dose and fluences distribution, previously discussed (5.2.1). Some modifications have been introduced, which allow the calculation of the mean LET at specific pre-defined 138

146 5.2 Contributions of charged fragments points along the beam path. At a first stage, we can imagine to have primary particles only of a single specific type (i.e. a fixed isotope), which we call j, neglecting isotope production at this first stage. The method implemented in the application consists on registering and scoring the kinetic energies of the particles j when they traverse some planes inside the PMMA phantom, placed perpendicularly 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 defined 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 Φ i (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: L j t ( z) N i= 1 = N i= 1 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 neglected. 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 because 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 simulation 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 139 Φ i, j Φ ( z) i, j S, ji ( z) (5.2)

147 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments well as of the primary ions, have been calculated for some fixed depths (Figures ). 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: L t ( z) k j= 1 N j= 1 i= 1 = k N Φ i= 1 i, j Φ ( z) i, j S, ji ( z) (5.3) 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 ,40% ,97% ,95% ,99% ,00% ,16% ,00% ,00% ,00% ,00% ,03% ,40% ,30% ,20% 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. 140 In Figures , LET track plots are showed, selecting some of the

148 5.2 Contributions of charged fragments charged fragments produced. In each plot, the Bragg peak in arbitrary unit obtained 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 obviously 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 kev/µm for alpha particles. Different behaviours can be observed from Li fragments up to heavier isotopes. 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 produced 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 7 Li LET track plot. Values found range on kev/µm for Li fragments, for Be and for B. In Figure 5.9 (right) LET calculations for the primary 12 C 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 contributions 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 12 C ions, as expected. However, the contribution of other low LET par- 141

149 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments 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 % less than the single 12 C LET. The discrepancies between only primary LET and total drastically 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 expected 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 contribution, 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 errors in the LET calculation. A more realistic prediction is achieved if we consider 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 future 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 threedimensional 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 included 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 presented can be applied to study the carbon fragmentation at the energy range typically used in hadrontherapy. 142

150 5.2 Contributions of charged fragments 10 proton LET contribution 50 alpha LET contribution 9 energy deposited (a.u.) LET Track 45 energy deposited (a.u.) LET Track LET (kev/um) LET (kev/um) PMMA depth (mm) PMMA depth (mm) Figure 5.7. LET calculation for protons (left) and alphas (right) for 62 AMeV 12 C 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 references on the different regions of interest (plateau, peak, tail). 50 Li7 LET contribution 200 Be9 LET contribution 45 energy deposited (a.u.) LET Track 180 energy deposited (a.u.) LET Track LET (kev/um) LET (kev/um) PMMA depth (mm) PMMA depth (mm) Figure 5.8. LET calculation for 7 Li (left) and 9 Be (right). 400 B10 LET contribution 800 Primary C12 LET contribution energy deposited (a.u.) Relative dose distribution 350 LET Track 700 LET Track LET (kev/um) LET (kev/um) PMMA depth (mm) PMMA depth (mm) Figure 5.9. LET calculation for 10 B (left) and primary particles 12 C (right). 143

151 5. Depth dose distribution of 62 AMeV carbon ions and contribution of fragments Relative dose distribution LET only primary LET primary + secondary LET TRACK 600 LET (kev/um) PMMA depth (mm) Figure Comparison between LET of primary 12 C (red triangles) and total LET (green dots) for 62 AMeV 12 C ion beams on PMMA. 144

152 Conclusions and perspectives Monte Carlo methods represent one of the most effective tools for verification of dose computation in radiotherapy with carbon ion beams. The physical processes involved in this novel radiation therapy technique, as well as the involved 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 implemented in detail, taking into account the presence of different material compositions. If reliable results have to be expected, the physical models implemented 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 fragments production predictions on thin targets. The work discussed in this thesis represents a contribution in this direction. Indeed, the results carried out and dis-

153 Conclusions and perspectives 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 estimate the radiation quality of carbon ion beams used in radiotherapy, by computing 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 12 C beams on C, Al, Cu and Pb targets has been simulated. 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 forward direction spectra are reproduced also by simulations, even if underestimated in the absolute value. The discrepancies grow up as the target mass increases. 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 reproduced, except at forward angles and for heavier mass targets. Even if neutron production is widespread used for interaction models benchmarks, 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. 12 C ion beams, accelerated by a Superconducting Cyclotron (SC) at the energy of 62 AMeV impinge on a 197 Au 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 146

154 Conclusions and perspectives 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 reproduced 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 explanations for the observed discrepancies. On one hand, they could be due to reliability of the total reaction cross section implemented in the code, which at this energy 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 incident 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 radiation field produced by the projectile fragmentation have been treated. The cumulative effect of nuclear as well as electromagnetic interaction have been verified by comparing the Bragg curve of a 62 AMeV 12 C 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 distributions, 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 carbon ions at low energies, as those used in the experiment. As widely discussed in 147

155 Conclusions and perspectives the first chapter, range straggling effects are less evident at low incident energies, so that narrower Bragg peaks are achieved. If the position uncertainties are comparable with distal fall-off, as in this case, hence it may be difficult to place the detector actually at the peak. Anyway, although total depth dose depositions are, in the case of ion beams, the global results of different physical processes, the good agreement showed in the results give a general confirm of the reliability of the Geant4 simulation developed. Then, we pay our attention on secondary particles contributions on dose and fluence. The results show, as expected, that the unwanted dose in the fragment tail is mainly due to protons and alpha particles, which travels more in depth than light ions. Moreover, what has been found is coherent with results presented in the forth chapter, where mainly alpha particles are produced at the different angles. However, at this incident energy, deposited dose due to charged fragments represent a small contribution, unlike high energy primaries. A Monte Carlo calculation of the LET dose distributions has been also performed, calculating the so called LET track for each isotope at some specific depths, and finally averaging the specific contributions of each secondary particle. As widespread discussed this parameter is of great interest when biological effects have to be taken into account. Mean LET track can give, indeed, a quantitative estimation of the radiation quality, which is a crucial information for carbon ions beam irradiation. The results show a common trend for protons and alpha particles, which exhibit a rather constant LET, while a different behaviour was found for light ions. They indeed show two different trends before and after the peak, presenting respectively roughly constant (except for B fragments) and increasing LET values. These results are somehow coherent with the fluences of secondaries. Moreover, comparing the mean LET due to the overall contribute of the mixed field and LET of primary carbon ions, it is evident the LET in the entrance channel is mainly dominated by primaries, although slightly decreased by the low LET components, while in the tail behind the Bragg peak it is essentially due to the fragments. In the early future these results will be related to the biological response of melanoma cells which were irradiated at LNS with 62 AMeV 12 C beams in some of the depth positions chosen for the LET calculation. Simu- 148

156 Conclusions and perspectives lation results may help to better interpret the survival results found at the different depths. In conclusion this work has provided a verification of one of the nucluesnucleus models implemented in the Geant4 code, in order to study the reliability of the nuclear processes at the energy range of interest for hadrontherapy applications. It has also given a more general contribution to the validation plan which the Geant4 Collaboration has been carrying forward and which has to be constantly updated according to the improvements introduced in the code. Anyway, results demonstrate that many efforts should still be done in order to better tuning the interaction processes which are involved in carbon ion fragmentation up to 400 AMeV. Other else comparisons are planned in the future and, moreover, an extended campaign of measurements has been scheduled. An experiment on 12 C ion beams impinging on C target has been already carried out at LNS-INFN, and data analysis is still in progress. Moreover, measurements at higher energies will be performed in 2011 at GSI laboratory, in Darmstadt. All these experiments will allow to validate and eventually improve the nuclear interaction models implemented in Geant4, as well as in other Monte Carlo codes. Moreover, Monte Carlo studies on contribution of secondary products will be repeated at higher energies in order to study LET distributions also at larger depths, where a characterization of the radiation quality could be related to other biological measurements, for deep seated tumours cases. In this concern, we have in program to implement inside the Monte Carlo calculations the Local Effect Model (LEM), for a complete prediction of biological responses. In general, the presented work represents a possible example of how Monte Carlo simulations can improve the knowledge of the physical processes related to the hadrontherapy, even contributing to the clinical efficiency. 149

157 Appendix A.1 Electromagnetic interactions In general, two main features characterize the passage of charged particles through matter: a loss of energy by the particle and a deflection of the particle from its incident direction. These effects are primarily the results of two processes, respectively: inelastic collisions with the atomic electrons of the material and elastic scattering from nuclei. These reactions occur many times per unit path length and it is their cumulative result which accounts for the two principal effects observed. Of the two electromagnetic processes, the inelastic collisions are almost solely responsible for the energy loss of heavy particles in the matter. In these collisions energy is transferred from the particle to the atom causing a ionization or excitation of the latter. The amount transferred in each collision is generally a very small fraction of the kinetic energy of the particle; however, in normally dense matter, the number of collisions per unit path length is so large, that a substantial cumulative energy loss is observed even in relatively thin layers of material. These atomic collisions are customarily divided into two groups: soft collisions in which only an excitation results, and hard collisions in which the energy transferred is sufficient to cause ionization. In some of the hard reactions, enough energy is, in fact, transferred such that the electron itself causes substantial secondary ionization. These high-energy recoil electrons are sometimes referred to as δ-rays or knock-on electrons [122]. As discussed in the previous

158 A.1.1 The Bethe-Block formulation chapter, the energy and spatial distribution of the electrons produced are finally responsible of the quality of the radiation in a particular point of the field, because they characterize the structure of the primary particle track. Elastic scattering from nuclei also occurs frequently although not as often as electron collisions. In general very little energy is transferred in these collisions since the masses of the nuclei of most materials are usually large, compared to the incident particle. The inelastic collisions are, of course, statistical in nature, occurring with a certain quantum mechanical probability. However, because their number per macroscopic path length is generally large, the fluctuations in totally energy loss are small and it is possible to work with the average energy loss per unit path length. This quantity, called stopping power de/dx, was first calculated by Bohr in 1913 using classical arguments [123], and later by Bethe [27], Block[29] and others[30] using quantum mechanics. A.1.1 The Bethe-Block formulation The correct quantum-mechanical calculation was first performed by Bethe, Bloch and other authors. In the calculation the energy transfer is parameterized in terms of momentum transfer rather than the impact parameter. The formula obtained is: de 2 2 Z z 2m ec β γ Wmax 2 C = 2π N r m c A e e ρ ln 2β δ 2 2 (A.1) 2 dx A β I Z commonly known as Bethe-Block formula, valid for heavy charged particles for energies ranging from MeV up to GeV. In Table A.1 all the variables used in the equation are summarized. The last two terms in the formula, δ and C, are respectively the density effect correction and the shell correction. The density effect correction is important at high energy. It arises from the fact that the electric field of the particle also tends to polarize the atoms along its path. Because of this polarization, electrons far from the path of the particle will be shielded from the full electric field intensity. 151

159 A.1 Electromagnetic interactions Symbol Definition Units or value N A Avogadro s Number mol -1 2 ( 4πε m c ) r m e c 2 Electron energy at rest MeV ρ Density of absorbing material g cm -3 Z 2 e = e 0 e Classical electron radius cm A z Atomic number of absorbing material Atomic weight of absorbing material Charge of incident particle in unit of e g mol -1 β = v c Relativistic term v Velocity of incident particle cm/s γ 2 = 1 1 β Relativistic term W max Maximum energy transfer in one collision Coulomb -1 ev I Mean excitation potential ev δ Density effect correction C Z Shell effect correction Table A.1: Variables use in the equation (A.1) Collisions with these outer lying electrons will therefore contribute less to the total energy loss than predicted by Bethe-Bloch formula. This effect becomes more important as the particle energy increases. Moreover, it is clear that this effect depends on the density of the material (hence the term density effect), since the induced polarization will be greater in condensed materials than in lighter substances such as gas.the density effect strongly depends on dielectric properties of the traversed medium and, moreover, it decreases when the atomic number of the medium increases, because in this case electrons result closer and less contribute to the polarization effect. Sternheimer derived the value of δ for a series of materials and obtained results that agree very good to the experimental data except in the low energy region where, however, density 152

160 A.1.1 The Bethe-Block formulation effects are negligible [124]. The shell effect is important at low energies. It accounts for effects which arise when the velocity of the incident particle is comparable or smaller than the orbital velocity of the bound electrons. At such energies, the assumption that electron is stationary with respect to the incident particle is no longer valid and the Bethe-Block formula breaks down. The correction is generally small and its description is usually done by using empirical formulae [125]. In addition to the shell and density effects, the validity and accuracy of the Bethe-Bloch formula may be extended by including a number of other corrections pertaining to radiation effects at ultra-relativistic velocities, kinematical effects due to the assumption of an infinite mass of the projectile, higher-order terms in the scattering cross section, correction for the internal structure of the particle and electron capture at very low energy [126][127]. However, with the exception of the latter, which could be not negligible with very heavy ions, the others are usually of the order of 1%. The maximum energy transfer W max is that produced by knock-on collisions. Starting from kinematics considerations and considering incident particles with mass M» m e, it is given by: Wmax 2 γ mec v (A.2) This quantity, which is responsible of the kinetic energy of the electron produced by primary particles, is also strictly connected to the track diameter. Indeed, it clearly shows, from a quantitative point of view, that electrons produced by particles travelling with high velocity v are in turn emitted with high kinetic energy, giving rise to a more wide diameter track (section 1.2.1). An important parameter in the Bethe-Block formula is the mean excitation potential I, which is very difficult to theoretically calculate. Indeed values of I for several materials have been deduced from actual measurements of de/dx and semi-empirical formulae [128]. The ionization potential varies with the atomic number of the absorbing medium Z in a complicated manner, because of local irregularities due to the closing of certain atomic shells. This parameter is crucial in hadrontherapy because Bragg peak position is sensibly influenced by consid- 153

161 A.2 Nuclear interactions ering its slight variations. The standard value for the ionization potential of water widespread used is 75 ev, according to the ICRU (International Commission on Radiation Units) recommendations [129], but values up to 80 ev can be found elsewhere [130]. When Monte Carlo codes are used for the transport of therapeutic ion beams in the matter, the accurate definition of I actually represents an open issue and usually it has to be slightly modified in order to tune the simulated ranges of the primary particles (i.e. the Bragg peak position) with the experimental data [131]. At non relativistic energies, de/dx is dominated by the overall 1/β 2 factor and decreases with increasing velocity until about v 0.96c, where a maximum is reached. Particles at this point are known as minimum ionizing. The minimum value of de/dx is almost the same for all particles of the same range. As the energy increases beyond this point, the term 1/β 2 becomes almost constant and de/dx rises again due to the logarithmic dependence. However the relativistic rise is partially compensated by density correction. For energy below the minimum ionizing value, each particle exhibits a de/dx curve which, in most cases, is distinct from the other particle types. At very low energy, the stopping power reaches a maximum and then drops rapidly, due to the tendency of particles to pick up electrons. A.2 Nuclear interactions A.2.1 Overview The study of the interaction of two ions is a subject of growing interest in nuclear physics. The complex nature of the projectile makes it possible that a number of new reactions occur, and also when the projectile fuses with the target nucleus creating a compound nucleus one has consider the special features of the heavy-ion reaction due to the large angular momentum carried in by projectile. At low energies two ions interact only through their Coulomb fields, and can scatter elastically or inelastically. Nuclear interactions can only take place if the 154

162 A.2.1 Overview two-ion energy E cm in their centre of mass is high enough to overcome the Coulomb barrier, and then the associated wave length ( λ = h / 2MEcm ) is much less than the nuclear dimensions. In such circumstances the interaction shows semiclassical features, and in particular it is appropriate to consider the ions moving along their classical orbits. This semiclassical nature of the heavy-ion interactions makes it possible to give an overall description in terms of the minimal distance between the two interacting ions r min, which is simply related to the impact parameter b by [132]: r min b = (A.3) V ( rmin ) 1 E cm where V(r min ) is the nuclear potential acting between the two ions. Even though such a description is only qualitative and a full treatment must take account of the quantum mechanical nature of the process, it is possible to distinguish four regions where the different reaction mechanisms predominate as the minimal distance between the two ions increase: - the fusion region, with 0 r min R F ; - the deep inelastic and the incomplete fusion region with R F r min R DIC - the peripheral region, with R DIC r min R N ; - the Coulomb region, with r min > R N, where R N is the distance above which nuclear interactions are negligible. The ion orbits corresponding to these regions are schematically shown in Figure A.2.1. In the fusion region the two-ion interaction leads predominantly to the formation of a composite nucleus which initially is quite far from statistical equilibrium since a large part of its excitation energy is in the form of an orderly collective translational motion of the nucleons of the projectile and the target. This orderly motion is transformed into chaotic thermal motion by means of a cascade of nucleon-nucleon interactions. 155

163 A.2 Nuclear interactions Figure A.2.1: Classical picture of heavy ion interactions showing the trajectories corresponding to close, grazing, peripheral and distant collisions. This thermalization eventually leads to a compound nucleus in statistical equilibrium, but, before reaching this, a considerable fraction of the excitation energy may be carried off by fast particles or clusters of particles with energy considerably greater than the average thermal energy. The importance of these pre-equilibrium emissions increases with the two-ion relative energy, but cannot be completely neglected even at energies only slightly greater than that of the Coulomb barrier acting between the two interacting ions. The amount of preequilibrium emissions and the energy distribution of the emitted particles depend on the mean field acting between the two ions, the type of interacting ions and on the competition between the emission of particles into the continuum and nucleon-nucleon interactions which redistribute the energy of the two interacting nucleons. Once the compound nucleus is formed, its angular momentum may be so high that it is unstable toward fission into two fragments, and, even when this does not occur, the high angular momentum it possesses makes fission much more likely than in the case of light projectile reactions. In the deep inelastic and incomplete fusion regions the overlap of the ions is much less than in the case of fusion, but it is sufficient to allow a strong interaction between the two ions which transforms a sizeable fraction of the kinetic en- 156

164 A.2.1 Overview ergy into internal excitation energy. This may occur by a process called deep inelastic collision in which the two ions form a dinuclear system which lasts for some time. The two ions are connected by a neck through which a substantial energy transfer occurs, but only a few nucleons are transferred from one to the other. When the dinuclear system breaks, two fragments fly apart which are similar to the projectile and the target and are called the projectile-like and the target-like fragments. In this impact parameter window another process may occur in the case of light projectiles: the break-up fusion or incomplete fusion reaction in which the projectile breaks-up into two fragments one of which fuses with the target, while the other flies away almost undisturbed. In the peripheral region the ions brush past each other. The processes which may occur are, with increasing impact parameter, the transfer of one or a few nucleons from one ion to the other, and elastic and inelastic scattering. Finally in the Coulomb region the nuclear interaction between the two ions is negligible, but they may still interact via the Coulomb excitation. The impact parameter b can be related to the corresponding orbital angular momentum L by the classical relation: Lh = µ vb = pb = hkp (A.4) where µ is the two-ion reduced mass, and v and p the two-ion relative velocity and momentum. Previous considerations make it possible to assume, to a first approximation, that the different processes dominate in different L-windows and give approximate expressions for the reaction cross-sections in the different regions as: L= Li + 1 π σ = (2L + 1) (A.5) k i T 2 L L= Li where, to a good approximation, the transmission coefficient T L 1 for L L max, and i = 1, 2, 3, 4 corresponds, respectively, to fusion, to deep inelastic or breakup fusion, to peripheral and to Coulomb reactions and L = L 1, L 2, L 3, L 4, L 5 equals, respectively 0, kr F, kr DIC, kr N, L max, with 157

165 A.2 Nuclear interactions L max = σ Rk π 2 1 (A.6) where σ R is the total cross-section including Coulomb excitation. The total reaction cross-section to a good approximation is given by the semiclassical expression: ( R ) = 2 V R int σ πrint 1 (A.7) E CM where the interaction radius R int and the interaction barrier V(R int ) may be obtained by plotting the product σ R E CM against the two-ion centre-of-mass energy. The decomposition of the total cross-section into the contributions discussed above is schematically shown in Fig... Figure A.2.2: Schematic picture of the contribution of different partial waves to the reaction cross-section for a collision between two heavy ions. The relative proportions of the various non-elastic events vary with energy and masses of the ions. A.2.2 Fusion reactions Heavy ions reactions have been extensively studied because they offer the possibility of producing nuclei with high excitation energy and high spin thus 158

166 A.2.2 Fusion reactions allowing the study of nuclear matter in conditions which are not easily formed in other reactions. Fusion reactions have also been used to try to produce superheavy nuclei and to produce proton rich nuclei very far from the stability line. We now consider a few important features of fusion reactions: - for light projectiles and low incident energies the fusion cross-section may be a considerable fraction of the reaction cross-section; - with increasing charge of the interacting ions the fusion probability falls abruptly, as show in Fig. - the fusion cross-section at first increases linearly with 1/E CM, reaches a maximum thereafter decreases linearly with 1/E CM, as show in Fig... for the interaction of 16 O with 27 Al; - the detailed energy dependence of the fusion cross-section may differ quite considerably for different interacting ions forming the same composite nucleus, as shown in Fig. for the interaction of 16 O with 16 B and 14 N with 12 C; - in general the fusion cross-section varies quite smoothly with the energy, but in the case of light systems it may show quite large oscillations. Figure A.2.3: Some features of the fusion cross section: ratio of the fusion and reaction cross-sections as a function of the projectile and target charges (left); fusion cross section of 16 O wit 27 Al. These features may be explained, at least qualitatively, by considering (a) the 159

167 A.2 Nuclear interactions dependence of the two-ion potential on the distance and the relative angular momentum, and (b) the limitation to the formation of the composite nucleus due to the absence of states below the yrast 8 line which gives the minimum excitation energy a nucleus may have as a function of the total angular momentum [133]. Figure A.2.4: Sum of the nuclear, Coulomb, and centrifugal potentials for 18 O Sn as a function of radial distance for various values of orbital angular momentum L. The nuclear potential has the Saxon-Woods form with V = 40 MeV, r 0 = 1.31, a = The horizontal line at E CM = 87 MeV corresponds to an incident energy of 100 MeV. The turning points for various values of L are marked by dots The effective potential acting between the two ions is assumed to be a complex one-body potential, the real part describing the refraction of the incident particle by the nucleus and the imaginary part the absorption by all non-elastic interactions. For the purpose of the following discussion we have to consider only the real part of this potential. The Coulomb potential may be approximated 8 In a plot of excited-state energy as a function of the orbital angular momentum, the yrast line is the line under which no more states are present. 160

168 A.2.2 Fusion reactions with sufficient accuracy by that between two uniformly charged spheres, perhaps with some allowance for surface diffuseness. The strongly attractive nuclear potential represents all the complicated interactions between the two ions and for simplicity we use a Saxon-woods form: 3 3 where R r ( A ) 1/ + 1/ 0 1 A2 V V 0 N () r = ( r R ) a 1+ e (A.8) =, with A 1 and A 2 mass number of interacting ions. Finally the centrifugal term V N (r) is (ħ 2 /2µ) L(L+1)/r 2. The relative contributions of these three potentials depend on the energy and on the masses and charges of the interacting ions. As an example, the predicted total potential for the interaction of 18 O Sn is shown for representative values of L in Figure A.2.4. This Figure shows that fusion may occur only up to a critical angular momentum L crit. The maximum value of the fusion reaction then occurs at the energy corresponding to the relative momentum: ( R ) ( R ) V Lcrit = 1 (A.9) R F k F k = ECM If L is less than L crit fusion is possible. This justifies why for light ions the fusion cross section is an important fraction of the total one and why it decreases as the interacting ions charge increases. Indeed, the decrease is due to the larger influence of the repulsive Coulomb potential on high Z ions also at higher energies, i.e. at lower distances; obviously if ions do not approach enough they cannot fuse in a compound nucleus. In terms of orbital angular momentum, the Coulomb potential reduces L, causing a decrement of fusion probability. Moreover, also for high relative angular momentum, two ions could be unable to make fusion; this happens when they are not able, together, to give to the compound nucleus the necessary energy for yrast line overcoming for the gained total angular moment J. When the two ions fuse, the composite nucleus is formed with the energy: E F = E Q (A.10) CN CM + 161

169 A.2 Nuclear interactions where E CM is the relative energy of the two ions in the centre of mass system and Q is the fusion Q-value, the energy which is gained or lost when the two ions fuse and is given by ( M + M M ) 2 Q = (A.11) 1 2 CN c where M 1, M 2, and M CS are the ground state masses of the fusing ions and the composite nucleus. Thermalization of the composite nucleus When two heavy ions fuse they form a composite nucleus which is far from statistical equilibrium since a large fraction of its energy is in form of an orderly collective translational motion of the nucleons of the projectile and the target. This orderly motion transforms into chaotic thermal motion through a cascade of nucleon-nucleon interactions during the thermalization of the composite nucleus. This takes some time and before reaching thermal equilibrium nucleons or clusters which still have an energy considerably higher than their equilibrium thermal energy may be emitted into the continuum. These pre-equilibrium emissions must be taken into account to reproduce the multiplicity and the spectra of the ejectiles measured in heavy ion fusion reaction. The ejectiles emitted in a fusion reaction at rather low incident energies are detected in coincidence with the residue, of mass near to the composite nucleus mass, emitted at a very forward angle with a velocity about equal to that expected for a complete momentum transfer. This velocity, indicating by M p and M T the projectile and the target mass and by v p the projectile velocity in the laboratory system is: v CN M p = v p (A.12) M p + M t This is the velocity of the composite nucleus in a fusion reaction and the previous relation simply expresses the conservation of linear momentum and is valid on the assumption that the velocity of a heavy residue (often called the evaporation residue, ER) remains equal to that of the composite nucleus even 162

170 A.2.2 Fusion reactions after its deexcitation by particle emission. This is a valid assumption when the ejectiles are emitted isotropically or symmetrically about 90 in the centre-ofmass system as is expected to occur for particles evaporated from the compound nucleus. It is no longer valid if pre-equilibrium particles are also emitted because they are preferentially emitted in the forward direction, thus reducing the velocity of the residue. However, it is approximately valid if the ejectiles have mass and energy considerably smaller than those of the projectile, as occurs at not too high incident energies (E inc 25 AMeVucleon). The velocity of the evaporation residue is estimated by its time of flight from the target to the detector and its mass by measuring its energy. At higher incident energies this procedure is no longer valid since the excitation energy of the composite nucleus is very high ant it emits, before reaching statistical equilibrium, so many particles that the velocity of the residue is considerably less than that given by the equation... In conclusion, the thermalization of the composite nucleus is due to a cascade of nucleon-nucleon interactions which starts as soon as the two ions overlap. Some calculations suggest that with increasing bombarding energy an increasing fraction of the excitation energy is dissipated by emitting preequilibrium particles (about 50% at an energy of about 30 AMeV), greatly reducing the probability of forming very excited equilibrated nuclei [134]. Evaporation from equilibrated nuclei If the compound nucleus has both high excitation energy and high angular momentum, fission in two pieces may happen immediately after its formation. Indeed, a large angular momentum causes, during the rotation, a deformation along the axis perpendicular to the nucleus rotation one; in this conditions, the equilibrium shapes are ellipsoids since these minimize the rotational energy. As the angular momentum J increases, these ellipsoids become more and more elongated until the nucleus becomes unstable against division into two parts. But in the decay process it is not possible to populate states of the residual nucleus below the yrast line. Hence, the compound nucleus mainly emits nucleons; then 163

171 A.2 Nuclear interactions fission happens if energy and angular momentum are still high enough. These emissions are called evaporation because the nucleus comes off similarly to molecules of a liquid evaporation. Also alpha particles are emitted at the end of the evaporation chain, together with photon emission. Evaporation particles are less energetic than pre-equilibrium ones and are emitted in an isotropic way. Moreover, these two kinds of emission happen in different moments: the first one persists up to sec after the impingement between projectile and target nuclei, while second one stops after sec. Finally, evaporation emission occurs as soon as Coulomb barrier is overcome, whereas pre-equilibrium emission appears if the relative energy between two ions increases. A.2.3 Projectile fragmentation In the projectile fragmentation process, nuclei accelerated to energies several times above the coulomb barrier break-up on a fixed target. The produced fragments have a velocity slightly lower than that of the incident beam and they are emitted in a rather small angular cone around zero degree. Experimental and theoretical studies have been performed in order to describe the reaction mechanism and to evaluate the production cross section for any projectile-target combination so that predictions could be made for future experiments. One of the proposed models for the projectile fragmentation is due to Serber, in 1947 [135]. According to this model, the peripheral highly energetic heavy ion reaction can be described as a two-step process in which each step occurs in clearly separated time intervals. The initial, and faster (time order of seconds), process of interaction consists on the overlapping between projectile and target nuclei. The spectator nucleons in the projectile are assumed to undergo little change in momentum, and likewise for the spectators in the target nucleus. This step can lead to highly excited objects (pre-fragments) which are usually very different from the final observed fragments. Before being detected, the pre-fragment, which continues in the original beam direction with practically unaltered beam velocity, looses its excitation energy through the emission of particles (neutrons, protons and small clusters) and γ-rays and re-arranges itself 164

172 A.2.3 Projectile fragmentation corresponding to the remaining numbers of protons and neutrons. This second step (de-excitation) occurs slowly relative to the first step and typically occurs at time scales on the order of to seconds (depending on the excitation energy of the pre-fragment). A simple portrayal of this process is shown in figure 3.5. Figure A.2.5.: A simplistic picture of the projectile fragmentation. The overlapping region of the target and the projectile is shared off, leaving an excited pre-fragment. The pre-fragment de-excites through statistical emission and becomes the final observed fragment. The presented scenario allowed to Goldhaber the developing of a statistical model of the fragmentation process as a rapid process in which the momentum distribution of the products is related to the momentum of the nucleons inside the projectile at the time of the reaction [136]. According to this model: - fragments are produced with a velocity slightly lower than that of the beam; - fragments are emitted in a rather small cone around the direction of the incident beam; - the momentum distribution has a Gaussian shape and it is isotropic in the centre-of-mass system. The width of the distribution is given by: ( A A ) A 2 2 f p f σ = σ 0 (A.13) A 1 p where A f and A p are the fragment and projectile masses respectively, and σ 0 is the reduced width which can be estimated from the relation σ 0 = p 2 f /5, where p f is the Fermi momentum of the nucleons within the projectile. 165

173 A.2 Nuclear interactions On the other hand the probability of formation of a given fragment is expected to be exponential with a maximum in the projectile value [137]. This relation is very well reproduced by experimental data as reported in figure 3.6 for the Xe projectile fragmentation where the fragment rate decreases by approximately one order of magnitude for each missing neutron. Figure A.2.6. Isotope yields for projectile fragmentation of 124,129 Xe (red and blue point) and fission of 238 U (green points) [CDR01]. The incident energies used for projectile fragmentation experiments have changed with technological advances. The processes which occur at various energies differ sensibly. The mechanism of production is intermediate between the transfer reactions observed at low energy and a pure fragmentation processes observed at higher energies up to several GeV per nucleon. Low energy fragmentation was available for many years, and a large amount of data has been accumulated [138]. At incident energies E/A 20 AMeV, several different reaction mechanisms contribute to the process. Reactions in this energy regime are generally not considered true fragmentation and cannot be described by Serbers simple two-step process [135]. The time of interaction is long, due to the slow relative velocity between the target and the projectile, and 166