a study on the contribution of nuclear … figure 1.2: the di erent depth-dose distributions of...

Dipartimento di Fisica

Corso di Laurea Triennale in Fisica

A STUDY ON THE CONTRIBUTION OFNUCLEAR FRAGMENTATION IN

PROTON THERAPY

Relatore:

Prof. Alessandro

Lascialfari

Relatore esterno:

Prof. Giuseppe

Battistoni

Correlatore: Dr.

Alessia Embriaco

Codice PACS:

87.10.Rt

Tesi di laurea di:

Marco Barenghi

Matricola:

866475

Anno accademico 2017-2018

Contents

Introduction 3

1 Charged particle therapy 51.1 Introduction to charged particle therapy . . . . . . . . . . . 5

1.1.1 Clinical data and facilities . . . . . . . . . . . . . . . 51.1.2 Physical advantages . . . . . . . . . . . . . . . . . . . 61.1.3 Biological advantages . . . . . . . . . . . . . . . . . . 8

1.2 Physics in charged particle therapy . . . . . . . . . . . . . . 111.2.1 Collisions with atomic electrons . . . . . . . . . . . . 121.2.2 Multiple Coulomb scattering . . . . . . . . . . . . . . 151.2.3 Nuclear interactions . . . . . . . . . . . . . . . . . . 17

1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2 Monte Carlo codes in medical physics 232.1 Monte Carlo methods . . . . . . . . . . . . . . . . . . . . . . 232.2 FLUKA in charged particle therapy . . . . . . . . . . . . . . 24

2.2.1 Introduction to FLUKA . . . . . . . . . . . . . . . . 252.2.2 FLUKA’s application . . . . . . . . . . . . . . . . . . 26

2.3 Simulations with FLUKA . . . . . . . . . . . . . . . . . . . 272.3.1 Generals . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.2 Primary . . . . . . . . . . . . . . . . . . . . . . . . . 272.3.3 Geometry and Media . . . . . . . . . . . . . . . . . . 282.3.4 Scoring . . . . . . . . . . . . . . . . . . . . . . . . . . 282.3.5 Runs and output files . . . . . . . . . . . . . . . . . . 29

2.4 Simulations set ups . . . . . . . . . . . . . . . . . . . . . . . 292.4.1 Fluence simulations . . . . . . . . . . . . . . . . . . . 292.4.2 LET simulations for mono-energetic beams . . . . . . 292.4.3 LET simulations in the case of a SOBP . . . . . . . 30

3 Results and discussion 333.1 Fluence distributions of primary and secondary particles . . 333.2 LET distributions of a single beam . . . . . . . . . . . . . . 363.3 LET distributions in the case of a SOBP . . . . . . . . . . . 393.4 RBE evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 43

Conclusions 45

Bibliography 47

1

2 CONTENTS

Introduction

In this thesis project the effects of nuclear fragmentation in proton therapyhave been analyzed using Monte Carlo simulations. Proton therapy is atechnique that uses proton beams to deliver the dose to the tumor region,while sparing healthy tissues. Its growth in the last decades is due to thephysical and biological advantages that this treatment has, as compared toconventional radiotherapy.

The patient treatment is elaborated using treatment planning systems(TPS) software. For proton therapy, the transport codes in the TPS arebased on the collisions between the protons and the atomic electrons of themedium. However, the contribution of nuclear fragmentation is not takeninto account. Projectile particles can undergo inelastic interactions with thetarget nuclei. From this process nuclear fragments with an atomic numberZ higher than one and low energy are formed. This means that they havea small residual range (of the order of some µm) and a higher LET, whichresults in a higher biological effectiveness than protons.

To account for the effects of nuclear fragmentation in proton therapy aMonte Carlo approach using the software FLUKA has been adopted. In thefirst simulations, the fluence distributions in a water phantom have beenevaluated to study the energies of primary and secondary particles at differ-ent depths. In the second simulations, the LET distributions of primary andsecondary particles have been calculated. Two different cases have been ana-lyzed. In the first one three different mono-energetic beams have considered,while in the second one, in order to study a more realistic case, a SpreadOut Bragg Peak (SOBP) has been simulated. Starting from the resultsof protons LET distributions, the values of RBE at different depths havebeen evaluated using Wedemberg’s analytic model. However, these modelshave been implemented only for protons, without considering nuclear inter-actions. From this study, it emerges that nuclear fragmentation affects witha non-negligible contribution the entrance channel, where healthy tissues areset. Therefore, a future development would be to include the contributionof nuclear fragments in the TPS.

This thesis has been divided into three main chapters. The first oneis intended to be an introduction to proton therapy, in order to presentthe physical and biological advantages and the radiation-matter interaction,focusing on the effects of nuclear fragmentation. In the second chapter theMonte Carlo methods will be briefly exposed. After that, a description ofFLUKA simulations set ups will follow. In the third chapter the results fromthe simulations will be shown. A particular importance will be given to the

3

4 Introduction

results from the simulation of the SOBP to describe the LET distributionsof Helium and protons. The results of the simulations of LET have beenused to evaluate RBE with the Wedemberg model.

Chapter 1

Charged particle therapy

The first chapter intends to present hadrontherapy in a general way. Thischapter will start with some general data, presenting the advantages thatcharged particle therapy has, as compared to radiotherapy. Then, it willfocus the physical aspects of proton therapy: the radiation-matter interac-tion mechanisms will be exposed, focusing on the collisions with the atomicelectrons of the medium, multiple Coulomb scattering and especially on thenuclear fragmentation process.

1.1 Introduction to charged particle therapy

Hadrontherapy is nowadays one the most effective treatments to cure cancer.This field has developed a lot in the past twenty years and still there aresome margins of improvement. Charged particles have a more optimal dose(absorbed energy per mass unit) profile to cover the tumor region thanconventional radiotherapy and have also a higher biological effectiveness.

1.1.1 Clinical data and facilities

Radiotherapy is one the most common ways to treat cancer, and the combi-nation of surgery, chemotherapy and radiotherapy is becoming a standard.Over 80% of these patients receive radiations with X-rays produced by linearelectron accelerators (Linacs). However the total cure rate with the use ofcombined techniques is just around 45%, mostly due to lack of local controlof the tumor and the development of metastases. In the last two decades adifferent treatment has spread, i.e. charged particle therapy (CPT). In 2016only about 0.8% of the radiotherapy patients were treated with charged par-ticle therapy [1], but this number is intended to grow, as figure 1.1 shows.

Proton therapy and charged particle therapy are particularly effective fortumors close to OAR (organs at risk) or deep seated ones. One example arechordomas positioned at the basis of the skull. This kind of tumor is hard tobe surgically removed and radiotherapy needs a high dose (energy absorbedby the target per mass unit), about 60 - 70 Gy. Tumors control reachesbarely 50% at 5 years from treatment. It is possible to reduce the delivereddose and to reach better results with proton therapy and C-ions treatments:

5

6 CHAPTER 1. CHARGED PARTICLE THERAPY

Figure 1.1: Statistics of charged particle therapy facilities in operation andpatients from 1955 to 2014. In the pie chart the geographical distributionof patients and the most common type of charged particles used are shown.The total number of patients treated with hadrontherapy in the entire periodis about 137000, with 15000 patients (11%) treated in 2014 only [2].

the results are respectively 73% and 70% for local control at 5 years fromtreatment [3]. Other good responses come from tumors in the neck and in thehead, prostate and lung cancer. Proton therapy finds also very good resultsfor eyes tumors: the prolific cells are killed and the optic nerve is not effectedfrom the therapy, avoiding possible future complications. Also, chargedparticles is recommended for pediatric patients. In fact, the probability ofthe formation of secondary cancers decreases if charged particles are usedfor the treatment [2].

Studies have shown that:

• about 1% of patients treated with radiotherapy need proton therapy;

• about 12% of patients treated with radiotherapy would significantlybenefit from proton therapy;

• about 3% of patients treated with radiotherapy would significantlybenefit from C-ions treatment.

These statistics justify the charts shown in 1.1 and justify also the big effortthat a lot of countries are doing in building and upgrading their facilities.

1.1.2 Physical advantages

Clear differences emerge from the different dose profiles, as shown in figure1.2. At radiotherapic energies (6 - 25 MeV), photons energy depositionpresents a non-sharp peak at a certain depth, with a decreasing but not nulltrend at higher distances. Instead, heavy ions lose a small portion of theirenergy in the entrance channel and the energy loss is maximum when theyare about to come to rest.

1.1. INTRODUCTION TO CHARGED PARTICLE THERAPY 7

Figure 1.2: The different depth-dose distributions of photon, protons andcarbon ions are compared. At the same range, protons have greater strag-gling than C-ions, but the latter have a prolonged tail beyond the Braggpeak [2].

The treatment has to be planned in such a way that the surroundingtissues or OAR close to the tumor receive the least possible dose, whilemaximizing the energy released in the tumor volume. In radiotherapy, dueto the dose profile, surrounding tissues are unavoidably hit. This fact limitsthe total delivered dose, due to possible complications in normal tissues. Inorder to reduce this possibility, different entrance channels are used. One ex-ample of such techniques is given by the Intensity Modulated Radio Therapy(IMRT) [4], which is considered to one of the most effective radiotherapictreatments. It uses up to 10 different X-rays beams and multi-leaf collima-tors in order to precisely cover the shape of the tumor. Another sophisticatedradiotherapy technique is image-guided radiotherapy, which consists of theacquisition of medical, usually using CT scanning systems, in order to de-tect the movements of critical organs, such as the prostate, and correct thetreatment [5].

Charged particles have a dose profile that allows more safety for normaltissues and a higher energy deposition in the tumor region. In figure 1.2 itis possible to observe the typical Bragg peak profile, which consists of a flatpart in the entrance channel and a steep slope in the final part. Chargedparticles deposit a small fraction of their energy at the body’s surface (from10% to 20%) when their velocity is high, and deposit most of it just beforethey come to rest.

In proton therapy the energy range to cover tumor regions is between60 - 250 MeV, while for carbon ions the energy ranges between 100 - 400MeV/u [6]. Obviously, a single beam would not be able to irradiate theentire tumor region. To cover homogeneously the entire volume, overlappingbeams are used, as shown in figure 1.3. This technique is called Spread OutBragg peak (SOBP) [7]. Once the structure of the tumor is known, thanksto imaging techniques such as MRI and CT, treatment planning systems


Figure 1.3: To cover the entire volume of the tumor in charged particletherapy different beams of different energies are used for the Spread OutBragg Peak technique [2].

(TPS) are used to plan the treatment and therefore to set the energies andthe fluence of the particles. There are two methods to irradiate the entiretumor region. The first one is covering the volume by changing the energiesand the position of the delivered beam and is known as active modulation.Instead, for passive modulation, filters in between the nozzle and the patientsare applied to modify the shape of the radiation field and its energy.

1.1.3 Biological advantages

One of the advantages that charged particle therapy has, as compared toradiotherapy, is a higher biological effectiveness when the same dose is con-sidered for the two different treatments. In fact, in order to obtain the samebiological effect (as 10% of surviving cells after irradiation) charged particletherapy require less dose. This concept is expressed by a parameter knownas Relative Biological Effectiveness (RBE), defined as

RBE =Dγ

Dparticle

(1.1)

where the ratio is calculated for the same biological effect and Dγ is thedelivered dose from the reference radiation (X-rays). RBE is not easy tocalculate, because of its biological nature. It depends both on physical andbiological quantities. From a biological point of view, it depends mainly onthe cell lines used for the experiment and if it is performed in vivo or in vitro.Instead, from a physical point of view it depends on the dose, particles LET(Linear Energy Transfer, defined as the absorbed energy per length unit),the fraction scheme and the oxygen concentration. The dependence on LETis shown in figure 1.4.

The most common one is RBE10%, the ratio of the doses for which thesurvival probability of the cells is 10%. Other percentages, or other effects,

1.1. INTRODUCTION TO CHARGED PARTICLE THERAPY 9

Figure 1.4: RBE10% values are studied as a function of LET for two differentcell lines [8].

can be used to calculate and define different RBE. For clinical purposes, forprotons it is considered to be sufficient to take an average value for RBEof 1.1. Instead, for C-ions it is important to consider the variation of RBEalong the range of the particles: values up to 3 or more have to be considered.

LET is one the most important physical parameters that characterizethe quality of different radiations. As shown in figure 1.4, RBE reaches amaximum around 30 keV/µm, 100 - 120 keV/µm and 180 - 220 keV/µm forprotons, Helium and Carbon ions respectively [4]. If LET ranges in theseintervals (which are usually referred to as optimal LET regions), the bio-logical effectiveness of the particles reaches its highest value. In order tokill a cell or to inhibit its prolific ability, a double strand break of the DNAchain (DSB) is required. This can happen in two ways: two single strandbreaks (SSB) next to each other (spatially and temporally) or a single eventthat breaks the DNA chain in two close points [5]. The higher the LET, themore likely DSB will occur. In fact, LET is linked to the mean free path λof the radiation. So, if λ is close to the distance between the two chains ofDNA, which is about 2 nm, the possibility of DSB is higher. This justifiesthe trend of the charts in figure 1.4. If LET is higher the optimal values, therelative biological effectiveness drops, because the relationship between doseand LET is D ∝ Let × Φ, where Φ stands for the fluence of the particles.If LET is too high, the fluence drops and this causes a consequent drop ofRBE too.

Usually, for charged particles the cell survival probability is parameter-ized as a function of the delivered dose

S(D) = e-αD-βD2

(1.2)

As shown in figure 1.5, the surviving curve for protons is lower than thephotons once, meaning that protons have a higher RBE than photons. Also,since the curve are not perfectly parallel, it is possible to observe the depen-dence on the delivered dose. The ratio α/β has a direct biological meaning:


an high ratio (∼ 10 Gy, as in the case of the skin) is usually associatedto those tissues with relatively high sensibility to radiotherapy and usuallythey do not present hypoxic regions. Instead if the ratio is low (∼ 2 - 6Gy, as in the case of the kidneys) the tissues are usually more resistant toradiotherapy and can present hypoxic regions. For the latter, the chart infigure 1.4 shows that RBE reaches higher values, which means that chargedparticle therapy is more effective for those tumors that are more resistant toradiotherapy.

Radiations are usually divided into densely and sparsely ionizing ones,depending on their LET. Protons in the range of therapeutic energies reachonly a few keV/µm, so their biological power is not very different from pho-tons. Their use is due to the more advantageous dose-depth profile. Thedifference between proton therapy and radiotherapy hides in the radiation-matter interaction. DBS in radiotherapy are mostly caused by reactive oxy-gen products (ROS) which break the DNA chain through highly reactivemolecules (such as HO• radicals, produced during hydrolysis) that reactwith the sugars of the nucleotides. The presence of molecular oxygen is nec-essary to fix the damage. This is the main reason why hypoxic tumors areresistant to radiotherapy. Protons and ions tend to interact through colli-sions with atomic electron, ionizing the atoms of the medium. The damageof the DNA chain is due to a direct energy deposition. The difference be-

Figure 1.5: Experimental data of cell inactivation of CHO cells (Chinesehamster ovary cells) after irradiation with 60Co γ-rays and with a 160 MeVproton beam, respectively reported with circles and triangles. The dataare shown with the respective modeling analysis, represented by the dashedand solid lines. For the proton model nuclear interactions are not take intoaccount [1].

1.2. PHYSICS IN CHARGED PARTICLE THERAPY 11

tween proton and heavier ions is their LET: protons are considered as asparsely ionizing radiation, so the DSB will most likely to be produced bythe interaction of two SSB caused by two different particles, while heavierions mostly cause DSB by direct energy depositions on both sides of thechain.

1.2 Physics in charged particle therapy

The physics that stands behind proton therapy and charged particle therapycan be summarized by describing the interaction of heavy ions with matter.The main two features of the passage of charged particles through a mediumare the loss of energy and the deflection from their incident direction. Thereare different ways through which incident radiation interacts with matter,and each one of them has a certainly probability, expressed in terms of crosssection. It can be useful to remind that the total cross section σtot and themean free path λ are linked trough the equation

λ =A

NAρσtot(1.3)

where A is the atomic mass number of the particle, NA the Avogadro con-stant and ρ is the density of the medium.

It also has to be remembered that the radiation-matter interaction is adiscrete and stochastic phenomenon: this will lead to make certain consid-erations about the limited precision of charged particle treatments.

The processes through which charged particles can interact with matterare the following:

• inelastic collisions with the atomic electrons of the material;

• scattering from nuclei;

• emission of Cherenkov radiation;

• bremsstrahlung;

• nuclear reactions.

Of the two electromagnetic interactions, i.e. inelastic collisions with theatomic electrons of the material and elastic scattering from nuclei, the firstone has generally a higher cross section and this leads to the fact that theenergy loss of the incident charged particles is mainly due to collisions withatomic electrons. The amount of transferred kinetic energy is small, but incondensed matter λ is so large that a particle loses all its initial energy in asmall length. For example, a 200 MeV proton loses all of its energy in about25 cm of water. These atomic collisions are divided in two groups: softcollisions cause and excitation of the hit atom, while hard collisions causeionization. It is possible that a certain amount of energy is transferred tothe knocked out electrons so that the latter may be able to cause secondaryionizations. These high-energy electrons are referred as δ rays.


Scattering from nuclei has usually a smaller cross section. In general theenergy transfer is smaller, since the masses of the projectile and the mediumhave the same order of magnitude. The major part of the energy loss is dueto atomic electron collisions.

The other mechanisms of interactions have a smaller cross section inthe therapeutic energy range, or their threshold is higher than hundredsof MeV/u. The Cherenkov radiation is a phenomenon that occurs whenthe velocity of the charged particle exceeds the speed of light in a medium.Bremsstrahlung is electromagnetic radiation produced by the deceleration ofcharged particle by the electric field of the medium. It finds an applicationin medicine in the X-ray tube. It is an important effect for electrons, but itbecomes important at energies higher than the therapeutic ones for heavyions. Nuclear reactions instead are present in proton therapy and chargedparticle therapy, and this kind of interaction will be analyzed.

1.2.1 Collisions with atomic electrons

Inelastic collisions with the atomic electrons is the main cause of energyloss of an incident charged particle. Although every sort of interaction hasa stochastic and discrete nature, they can be modeled by a function: theBethe-Block formula [9] describes the energy loss per length unit, knownas stopping power, due to the inelastic collisions of heavy ions with atomicelectrons.

−dEdx

= 2πNAr2emec

2ρZ

A

z2

β2

[ln

2meγ2v2Wmax

I2− 2β2 − δ − 2

C

Z

](1.4)

where the constants re, me and NA are respectively the classical electronradius, the electron mass and Avogadro’s number; the absorbing materialconstant I, Z, A, ρ are respectively the mean excitation potential, the atomicnumber, the atomic weight, the density; the incident particle characteristicsz, β, γ, Wmax are respectively the charge in units of e, v/c and 1/

√1− β2

and the maximum energy transfer in a single collision and the corrections δand C are respectively the density and shell corrections.

It is now to possible to explain the Bragg peak (figure 1.2): chargedparticles have a stopping power ∝ 1/β2, so they tend to lose the biggerfraction of their energy at the end of their path. This fact is also shown byfigure 1.6.

At non-relativistic energies, the trend of the stopping power is dominatedby 1/β2. It decreases to a minimum point, at about v ≈ 0.96c, known asminimum ionization point. Then the dE/dx rises again due to the logarith-mic dependence of Eq.1.4.

At high and low energies the original formula of Bethe-Block does notfit well experimental data, so two corrections were added. The density ef-fect arises from the interaction of the incident particle electric field and themedium. The field tends to polarize the atoms along the particles path. Theenergy loss through collisions with atomic electrons will therefore contributeless to the total one, predicted from the formula. The shell correction in-stead becomes important ad low energies and it arises when the velocity


Figure 1.6: Stopping power of different particles as a function of the energy.The low energy region is not shown. It is possible to see that α particleshave a dE/dx about four times greater and protons given a certain kineticenergy per nucleons, due to the dependence from z [9].

of the incident particle is comparable to the orbital velocity of the boundelectrons. In this condition, the assumption that the electrons can be seenas stationary in no longer valid and the formula breaks down. Other cor-rections that take into account other effects can be added. This formula isvalid in the region 0.05 < βγ < 1000 [9]. Even though there are plenty ofcorrections, the Bethe-Block breaks down at higher or lower energy, wherethe cross section of other phenomenons become comparable. For example,at lower energies the main effect is the capture of electrons of the atoms ofthe medium.

From the Bethe-Block formula, it is possible to estimate the range ofan incident particle. The range can be expressed as function of the initialkinetic energy as

R(T0) = R0(Tmin) +

∫ T0

Tmin

(dE ′

dx)-1 dE (1.5)

where Tmin is the minimum energy at which the Bethe-Block formulais valid and R0(Tmin) is an empirical parameter to account the low energybehavior.

Experimentally, the range can be measured by forcing a beam to passthough different thicknesses of the material and measuring the fraction oftransferred incident particles. In figure 1.7 a typical graph of transmittedrelative intensity as function of the depth is shown.

Since the range can be calculated from the Bethe-Block formula, it can bethought as a precise quantity, once the initial kinetic energy of the particle isknown. However, radiations interact with matter in a stochastic way, so the


Figure 1.7: The number-distance curve shows the fraction of transmittedparticles as a function of the depth. Also, the range distribution is shown,with a shape that can be approximated to a Gaussian [9].

range is a stochastic quantity as well. The Bethe-Block is just a model forinelastic collisions with the electrons of the medium, so the particles rangeis subjected to fluctuations. This phenomenon is known as range straggling.In first approximation, the distribution is a Gaussian: its mean value isknow as mean range and it corresponds to the midpoint of the descendingslope. It corresponds to the thickness at which half of the incident particleare stopped. The practical range has also been defined: it corresponds tothe thickness obtained by extending the tangent line at the midpoint to thezero-level [9].

The Bragg peak of a beam is broadened and this can be described by theasymmetrical Valinov distribution that can be approximated by a Gaussianfunction, in the limit on many collisions, as shown in figure 1.7. Of course,different ions have a different level of range straggling. The relative rangestraggling of a particle is given by

σRR

= (M)-1/2ϕ( E

Mc2

)(1.6)

where E is the energy of the particle, M is its mass and ϕ( EMc2

) is afunction that varies slowly [2]. At the same range, a useful scaling law canbe found to compare the straggling of two different particles

σR1

σR2

=

√M2

M1(1.7)

According to this equation, the relative straggling for C-ions is about3.5 times smaller than for protons. Range straggling does not represent aproblem in charged particle therapy. In treatment planning system (TPS)programs the longitudinal range distribution, that is normally close to aGaussian form, is taken from databases. This means that the TPS are able


Figure 1.8: Mean ranges for different particles traversing water. Given acertain initial energy, light particles have a longer range than heavier ones [6].

to account for this effect and to modulate the radiation field to obtain theSOBP.

There are not quick and handy formulas to determine the range of a par-ticle, but useful scaling laws can be calculated. Given two charged particlesin the same material and at the same velocity this scaling formula can beused:

R2

R1

=M2

M1

(z1z2

)2 (1.8)

For instance, protons and 4He-ions have the same range, while protonshave a range 3 times greater the 12C-ions, as shown in figure 1.8.

For the same particle in different materials the scaling rule is

R1

R2

=ρ2ρ1

√z1√z2

(1.9)

1.2.2 Multiple Coulomb scattering

Charged particles passing through matter suffer repeated elastic Coulombscattering from nuclei. This phenomenon affects the lateral spread of thebeam. When an ion hits an atomic electron, ionizing the atom, its directiondoes not change, due to the small mass of the electron. Instead, if an elasticscattering occurs between two nuclei, a small deviation from its originaldirection is expected, as suggested by the differential Rutherford formula ofthe cross section

dσ

dΩ∝ z1

2z22mc

βp

1

4sin4(θ/2)(1.10)


Figure 1.9: FWHM of protons and C-ions beams at different energies travers-ing 1 meter of air and entering in a water phantom [6].

where z12 and z2

2 are the projectile and target charges, the terms m, βand p refer to the projectile and θ is the deflection angle. The cross sectionshows a maximum when θ is small. If the energy of the ions is high enoughthat their range is of the order some centimeters, the effect of the elasticscattering with the nuclei of the medium becomes cumulative, spreading thebeam laterally.

Usually the condition N > 20 (where N is the number of the collisions)is a good assumption: it is possible to treat this phenomenon statisticallyand to introduce the Multiple Coulomb scattering: the lateral profile hastherefore a Gaussian shape. The following formula [6] emerges

√< θ2 > = ZP

13.6MeV

pcβ

√x

X0

[1 + 0.038ln

( x

X0

)](1.11)

where ZP is the projectile charge expressed in unit of e, x is the thicknessof the material, and X0 radiation length of the material. In general thiseffect decreases for materials with heavy elements, thus we can expect alarger spread of the beam in the bones instead of biological tissues, whichcan be approximated to water.

Instead, by comparing different particle at the same range, we note that[6] ( Zp

pcβ

)protons

∝ 3( Zppcβ

)12C−ions

(1.12)

At the same range, lighter particles show a larger spread of the beam,which increases with the depth, as suggested by Eq.1.12

Again, the TPSs are able to take this effect into account. For everydepth, the beam is moved considering its FWHM is, so that every layer ofthe tumor receives the right amount of dose.


1.2.3 Nuclear interactions

It is important to understand the nuclear processes that occur in protontherapy and charged particle therapy in order to predict their effects. Aprojectile nucleus can undergo elastic or inelastic scattering with the nucleiof the medium and fragments of different nature and energy can be produced.All nucleus-nucleus interaction models used in therapy transport codes arebased on the two-steps abrasion-ablation model [10]. The time-scale of thisprocess corresponds to the one of strong interactions: 10-22 - 10-23 s.

Figure 1.10: Scheme of the abrasion-ablation model for the collision betweentwo nuclei [11].

As shown in figure 1.10, in the overlapping zone of two nuclei during anion-ion interaction, there is the formation of a fireball. In the following phasethe latter and the other residual nuclei (called pre-fragments) de-excite bythe emission of light particles and γ-rays. Depending on geometric condi-tions, there can be a small mass removal (in case of peripheral collisions)or the interaction can lead to the total destruction of the two nuclei. Inproton therapy, only target fragmentation occurs, because nucleons breakdown in quarks at higher energies, while in charged particle therapy also theprojectile nuclei can break down into fragments. The main difference is thatthe fragments from the target have a low energy and a short range, whilethe projectile fragments have a similar velocity to the incident particles, re-sulting in a longer range.

One of the most used parameterization of the fragmentation process is thegeometric approximation, where the nuclei are assumed to be black sphereswith radius a. The nucleus-nucleus reaction cross section [2] is given by

σR = σT − σel = π(aP + aT )2 (1.13)

This cross section can be expressed as the difference between total andelastic cross section and it gives the result of π(aP+aT )2, where aP and aT arerespectively the projectile and the target nuclei radius. There are differentparameterizations of the nuclear radius, such as a = r0Aπ(aP + aT )1/3 − b,where r0 is the nucleon radius and b is a correction factor. Substituting aPand aT , σR can be written as

σR = πr02(aP

1/3 + aT1/3 − b)2 (1.14)


Figure 1.11: Expected energy, LET and range of the fragments produced inwater by a 180 MeV proton. The expected energies are calculated accordingto the Goldhaber formula. It is noteworthy that these fragments have veryhigh LET values [1].

know as the Bradt-Peters formula. However, this formula is not reallyaccurate for therapeutic energies, where the reaction cross section showssome energy dependence. Correction for this energy range are available. Inparticular, a common parameterization of the Bradt-Peters formula is

σR(EP ) = σ0f(EP , ZT ) (1.15)

where σ0 is the following geometrical factor

σR = πr02(1 + aT

1/3 − b0(1 + aT1/3))2 (1.16)

and the parameter b0 is a polynomial expansion of the mass target num-ber

b0 = 2.247− 0.915(1 + aT1/3) (1.17)

The function f(EP , ZT ) depends on the energy of the projectile particleand the atomic number of the medium atoms. For a proton with energy EP≈ 200MeV the reaction cross section can be, and it often is, approximated [2]as

σR ≈ 53aT2/3mb (1.18)

In water, using equation 1.3 to evaluate the mean free path, a protonwith EP ≈ 200 - 300 MeV has a λ = 82 cm: this means that approximately20% of protons will undergo inelastic reaction in a typical treatment planfor a deep seated tumor [1]. From this study on the target fragmentation,the table in figure 1.11 reports the expected energy, LET and range of thefragments produced in water by a 180 MeV proton beam, calculated withan analytic formula. These fragments have a certain spectrum of energy,but it will remain in the region of a few MeV. Protons in water are able toproduce different types of fragments, as shown in figure 1.11, with very highLET. Therefore, they are most likely to have a clinical relevance. Nuclear


Figure 1.12: Attenuation of different primary beams as function of the depthin water [2].

fragmentation affects in two ways the distribution of the dose in the patients.On one hand, not all the particles of the beam reach the tumor region. Aproton beam with EP ≈ 200 MeV approximately loses about 20% of theinitial particles due to nuclear fragmentation. For heavier ions, such ascarbon, the loss is higher: about 50% of primary particles are lost due tothis process for a deep seated tumor [8].

Figure 1.12 shows that the transmitted fraction of particles decreases asthe atomic number increases, at a fixed depth. About 1% of protons are lostin one centimeter of water due to nuclear interactions, while about 4% ofC-ions are lost.

On the other hand, particles that do not reach the tumor deposit theirenergy in normal and healthy tissues. The effects are not negligible: thisfragments have high values of LET, which means that their biological ef-fectiveness is higher than the primaries one. Protons are seen as a sparselyionizing radiation, especially in the entrance channel. Fragments with highRBE do affect the response of normal tissue. Instead, the Bragg peak regionis not affected in a very significant way, because in this region the fractionof nuclear interaction is small and the biological effectiveness of the primaryparticle is higher. This topic is well represented by figure 1.13.

Even though the inelastic nuclear reaction cross section grows with thedepth, it is possible to observe that the contribution of the fragments ishigher in the entrance channel, while it is significantly lower in the SOBPregion. As in figure 1.13, about 8% of the cells in the entrance channel arekilled from the deposition of energy of the target fragments.

In the work of [12] a simulatory study performed with FLUKA has beenperformed, with a 220 MeV proton beam irradiating a polyelthylene tar-get. The deposited dose from secondary particles has been evaluated andit contributes for 12-13% of the total dose in the entrance channel. TPSfor proton therapy do not take into account the effects of nuclear fragmen-tation. It is possible with software based on Monte Carlo codes, such asFLUKA, to simulate at any depth the fragments produced by protons inwater, to increase the accuracy of treatments. In therapies with ions heavier


Figure 1.13: Impact of ionization and target fragmentation in sections of 1mm2 in the entrance channel end in the Bragg peak region. The fraction ofkilled cells in the two different regions shows that nuclear fragmentation doaffect the killing rate in the entrance channel. In particular, about 1/12 ofthe cells are killed due to highly ionizing fragments in the entrance channel,it happens for 1/40 in the region of the Bragg peak [1].

than protons, the projectile can undergo fragmentation too. The projectilefragments keep approximately the same velocity, thus their range is biggerthan the one of the projectiles, as suggested by figure 1.8. This causes a tailbeyond the Bragg peak, as shown in figure 1.2.

The effects of the target fragmentation can be observed also in the lat-eral dose profile. The spreading from the trajectory of the primary beamis mostly due to the multiple Coulomb scattering, but nuclear interactioncontribute to the widening of the beam too. In fact, if the dose delivered bythe fragments is taken into account, the lateral dose profile is no more welldescribed by a Gaussian: other parameterizations of this phenomenon areconsidered [13], [14]. These are in general given by the sum of Guassians, toaccount for the scattering of primary and secondary particles, or the sum ofa Gaussian for the primary beam and another function to account for singlescattering events due to nuclear fragmentation at wider angles.

In order to properly account for the contribution of nuclear fragments tothe physical and biological dose, a characterization of secondary particles isrequired. In literature not a lot of data can be found for therapeutic energies,even though some experiments are being planned, such as the FOOT exper-iment [15]: it aims to measure the heavy fragments cross sections and theirenergy spectra with a maximum uncertainty of 5% and an energy resolutionof the order of 1 - 2 MeV/u, in order to contribute to a better radiobiolog-ical characterization of protons. Another project is MoVe-IT: its purposeis to introduce new biological models in order to optimize the treatmentplannings with ion beams. A particular attention is given to the impact oftarget fragmentation and the evaluation of RBE for protons.

Due to the lack of experimental data and since these project have not

1.3. CONCLUSIONS 21

yet given a contribution to the study on the impact of nuclear fragmenta-tion, nowadays Monte Carlo simulations are a useful tool to provide a firstcharacterization of the nuclear fragments.

1.3 Conclusions

Proton therapy can be a valid alternative to radiotherapy for a certain kindof tumors. The physical advantage is the delivered dose profile characterizedby the typical Bragg peak, which is due to the collisions with the atomicelectrons. Another important phenomenon that should be taken into ac-count by the TPS is the fragmentation of the target nuclei. Approximatingthe biological tissues to water, an assumption that is usually made in exper-iments and also in the treatment plannings, nuclear fragments with Z thatvaries from one (Hydrogen) to eight (Oxygen) can be produced. They havea low energy, so LET is high and can reach 100 - 1000 keV/µm for some sec-ondary particles. Although the cross section of nuclear fragmentation growswith the depth, it gives a significant contribution to the dose in the entrancechannel, where healthy tissues are. The purpose of this thesis is to accountfor the effects of target fragmentation in the entrance channel, by using theFLUKA code to calculate the fluences of the secondary particles at variousdepths and to characterize their impact studying their LET distributions.

Chapter 2

Monte Carlo codes in medicalphysics

The second chapter focuses on the application of Monte Carlo simulations inmedical physics. After a short presentation of the Monte Carlo methods, thefeatures of FLUKA will be presented. FLUKA is a Monte Carlo code whichsimulates transport and interaction of particles and nuclei with matter. Itcan deal with complex geometries and allows to simulate the measurementsof different physical quantities [16]. Its fields of application will be brieflypresented, focusing on medical physics. After this introduction, the set upsof the simulations that have been run for this study will be presented.

2.1 Monte Carlo methods

In physics, Monte Carlo simulations become interesting and useful whenstochastic phenomena are considered. These codes are able to follow theevolution of a model which does not evolve in terms of a deterministic way.In this case, the results are not given by the solutions of rigorous equationsthat rule the physics involved: the results of the investigated problem aregiven by various simulations which depend on a sequence of random num-bers generated or sampled during the simulation. This method providesapproximate solutions to mathematical and physical problems, like the so-lution of a non-standard integral or the simulation of experiments that arenot approachable by hands or, in the second case, would take too long torun, or simply are not practical.

Monte Carlo simulations apply to a big variety of fields: the first appli-cation goes back to the Second World War [17]. The scientists involved inthe Manhattan Project to develop the first atomic bomb were facing difficultequations, almost impossible to solve by hand, to calculate the probability ofthe second fission of Uranium caused by a previously produced neutron. Thedifficulties were hidden in the complicated geometry of the bomb, which wasnecessary in order to obtain the right solution. They used to solve the equa-tions for single particles, implementing the calculations with a mechanical

23

24 CHAPTER 2. MONTE CARLO CODES IN MEDICAL PHYSICS

calculator. The trajectories of the particles were chosen by picking randomnumbers, since these trajectories would have the same statistical propertiesof the real neutrons trajectories. The key in solving the problem was tosimulate enough trajectories to make the results converge to the true natureof the phenomenon. Nowadays, computers are used to simulate every sort ofphysical process, which has to be implemented in models that well representthe nature of the physics involved.

Monte Carlo simulations allow to simulate statistical processes, such asthe interaction of particles with matter, and to simulate the measurementsof different physical quantities [7]. Since the radiation-matter interaction isa stochastic process and does not follow deterministic laws, software basedon Monte Carlo simulations are the right tool to solve various tasks. FLUKAobtains results by simulating the interaction of single particles and recordingonly some requested information.

However, one of the possible disadvantages of the simulations is that,given a number of histories N , the error of the simulated quantity is pro-portional to 1/

√N . This fact to make the result converge, a big number of

simulations have to be run, making the time of computation grow [18].

2.2 FLUKA in charged particle therapy

Software based on Monte Carlo codes are increasingly becoming an irre-placeable tool in the field of charged particle therapy. Different codes haveborn through the years, such as FLUKA, Geant4, PHITS and MCNP. Eventhough they are not specifically designed for medical physics, they allow tosimulate the passage of ions (such as protons, Helium, Carbon and Oxygen)through matter with good agreements with experimental data. An exampleis reported in figure 2.1.

Figure 2.1: Comparison between experimental data and simulation usingFLUKA for the energy deposition in water for a 12C beam of energy 338MeV/u. The experimental measures have been performed at CNAO [8].

2.2. FLUKA IN CHARGED PARTICLE THERAPY 25

2.2.1 Introduction to FLUKA

The history of FLUKA goes back to 1962 at CERN , when a work on hadroncascades was being performed [16]. FLUKA is a tool for the calculations ofparticle transport and interaction with matter, covering an extended rangeof energy and particles. It can simulate with accuracy the propagation andphysical quantities such as delivered dose, deposited energy and fluence ofabout 60 different particles (keeping out from this count all positive nuclei)from hundreds of eV up to thousands of TeV. The physics of the simula-tions is implemented by various models, which are applied depending on theenergy and the particles involved [19]. The program can also transport pho-tons and it is able to perform the time evolution and tracking of the emittedradiation from unstable residual nuclei. It is also possible to implement amagnetic field in a desired region.

At the energy range of interest in hadrontherapy the model PEANUTis used for hadron-nucleus collisions. The package PEANUT includes adetailed Generalized Intra-Nuclear Cascade (GINC) and a pre-equilibriumstate. After the latter, models for evaporation and gamma de-excitation ofthe residual nuclei are used. Light residual nuclei might instead be frag-mented into different bodies according to the Fermi break-up model [6].

When the energies involved are lower than 5 GeV/u, hadron-nucleusinteraction is described as a cascade of two-body collisions, where the targetnucleus is considered as a cold Fermi gas of nucleons in a potential well.Quantum effects have been included in the model to better describe theinteractions. Secondary particles produced in the cascade of collisions aretreated as primaries.

The intra-nuclear cascade modeled by GINC is taken on until all nucleonsare below an energy cut-off around 50 MeV. The pre-equilibrium particleemission is then described by different statistical models. This stage stopswhen a thermal equilibrium is reached.

The equilibrium stage is the last one involved in hadron-nucleus inter-actions. After the middle stage, the residual nucleus is assumed to have acertain excitation energy shared among its nucleons. If the excitation energyis higher than the separation energy, nucleons and light fragments can beemitted. Evaporation is a process in competition with fission, but it is par-ticularly important only for heavy targets (with Z > 70). Instead, the Fermibreak-up is important for light targets and can occur when the excitationenergy is higher then the binding energy: in this case light nuclei can breakup into two or more fragments. If an amount of excitation energy is leftafter these processes, de-excitation occurs by emission of γ-rays.

When nucleus-nucleus interactions (as in case of a carbon beam in water)are simulated, essentially the same models for hadron-nucleus interactionsare used, even though they are slightly more complicated. The relativis-tic Quantum Molecular Dynamics (rQMD) is used until a threshold of 125MeV/u. At lower energies, the Boltzmann Master Equation (BME) is con-sidered.


2.2.2 FLUKA’s application

FLUKA is extensively used for radioprotection calculations and to designthe facilities for forthcoming projects. The use of FLUKA is required forsecondary beam design, energy deposition, radiation damage, shielding ac-tivation and waste disposal. FLUKA has also been used for various ex-periments of high energy physics, concerning electron, muons, hadron andneutrino physics. [20].

FLUKA and the other software based on MC codes have become anessential tool in medical physics. The treatment plannings are based oncalculation based on Monte Carlo simulations. The machinery is set up byirradiating a phantom (typically water) and the measures have to be wellreproduced by the calculations of the TPS. The energy deposition is a keyfactor in planning the treatment: once the geometry of the target has beenwell defined by medical images, the TPS has to find the best combinationof beam to maximize the dose in the tumor region and minimize the one tothe healthy tissues. A simulated energy deposition and the correspondingBragg peak is shown in figure 2.2.

Figure 2.2: Example of a simulation of the energy deposition for a protonbeam (N=108) with an initial energy of 150 MeV, in a water phantom. Thenozzle position is in (0, 0, 0.1cm) and the direction is along the z axis. Thefirst plot shows the energy deposition on the YZ plane, while the second oneshows the Bragg profile, through a projection of the latter on the z axis.

2.3. SIMULATIONS WITH FLUKA 27

Another application in which FLUKA has been being recently used isthe Positron Emission Tomography (PET) [21] that can be performed afterthe irradiation as an in vivo verification of the treatment delivery [21]. Inthis study, the positron emitters distributions are calculated by combiningproton fluence with experimental data for some β+ emitter isotopes, as 11C,13N, 14O, 15O, 30P and 38K.

FLUKA also allows to simulate physical phenomena that occur in thebiological tissues, like the energy deposition and other useful quantities asLET or the range the particles involved. This gives the opportunity tostudy possible effects that might be difficult or too expensive to study in anexperiments.

2.3 Simulations with FLUKA

For this study the program FLUKA has been chosen to run the simulations.The main reason is that the physics models of FLUKA are particularly

accurate. They are continuously updated and benchmarked with new ex-perimental data as soon they are available. This software is used in manyhadrotherapic centers and it has been validated on various experimentaldata, as shown in figure 2.1. Also, FLUKA comes with the graphic inter-face FLAIR: this makes the simulation set up quicker and easier; with thegeometry package it is possible to visually check the geometry built for thesimulation. Before presenting the study and the data, it might be instruc-tive to briefly describe how a simulation is set up using the FLAIR interface.

A flair FLUKA project contains all the information to run the simulation.A file .inp is necessary as an input file. The input can be divided into differentsections. A project is made of the composition of different cards that arerequired for the simulation.

2.3.1 Generals

In this section the default physics of the simulation has to be declared.The card DEFAULTS with the option PRECISIO allows the user to usegeneric models for the passage of different particles in a medium. Anotherimport card is RANDOMIZ: this is the one that generates random numbersto simulate every stochastic event. A card STOP is required to stop theprogram once the simulation is over.

2.3.2 Primary

The section ”Primary” allows the user to choose the particle beam with thecard BEAM. The type of particle has to be chosen and the energy, or themomentum, of the beam has to be set. It is possible to characterize theshape and the uncertainties of the energy or of the geometry of the beam.Using advanced user routines it is also possible to read an external file with


a certain energy or momentum distribution and use it as a primary beam.This is useful to study more complicated case, as an homogeneous externalfield (which can be applied for a radioprotection simulation in space) or aradiation field in order to create a SOBP. The position of the beam has tobe set using the card BEAMPOS. The default direction of the beam is alongthe z axis, but it can be modified by the user. A START card is needed tostart the simulation and to set the number of primary particles.

2.3.3 Geometry and Media

These two sections allow the user to set the geometry of the simulation andto characterize it with different materials. To set it up, the first part con-sists of the creation of different bodies. FLUKA offers the opportunity touse different bodies which are implemented in the code and to set their di-mensions. Boxes, spheres and other more complex bodies can be created, aswell as infinite planes. After all the necessary bodies have been created, theuser has to create the regions, using the operations of union, subtraction andintersection on the different bodies. The final step is to assign the desiredmaterial to the different regions. It has to be highlighted that to the mostexternal region the material BLCKHOLE has to be assigned. This virtualmaterial has the property to absorb every particle that hits it, not allowingany of them to exit the region of the simulation, in order to avoid followingthe particles to infinity.

There are different options that can be added to the materials. One usefulexample is given by the card MAT-PROP, that allows the user to changethe default properties of the materials. The mean ionization potential canbe modified: this might be useful for water. This is one of the most criticalparameters that affects the Bethe Block formula. Usually hadrotherapiccenters and research bodies use the value of the mean energy potential thatbetter fits experimental data. It’s value fluctuates between 75 and 78 eVand it affects the position of the Bragg peak, as eq. 1.4 suggests.

2.3.4 Scoring

In the scoring section the user can select the cards in order to declare whattype of measurements have to be simulated. The regions of interest have tobe declared as well. In general, every card allows the user to choose differentkinds of simulation. The USRBIN card allows the user to use evaluate theenergy deposition in a declared regions. The USRTRACK instead allowsthe user to calculate the fluence of precise particles in a volume or though asurface. The USRYIELD allows different options of scoring, for example tocalculate the LET of the particles passing through a surface. A very usefulcard that has a filtering functions is AUXSCORE. Associated to anotherscoring card, it allows the user to select only one type of particle and tostudy only its properties.

2.4. SIMULATIONS SET UPS 29

2.3.5 Runs and output files

Once the simulation is set up, the runs are ready to start. The default num-ber of runs is one, but it can be changed to use different seeds of the randomgenerator, in order to avoid statistic recurrences. A logic unit of output isassociated to every scoring card to generate the output files. FLAIR is alsoable to plot basic graphs of different types of simulated measures.

2.4 Simulations set ups

Three different kinds of simulations have been run. The first one intendscharacterize the fluence and the energy spectra of both primary and sec-ondary particles at different depths. The second one intends to study thecontribution to the total LET of primaries and secondaries in the case ofmono-energetic beams. The last one maintains this aim, but LET is evalu-ated in the case of a SOBP, in order to study a more realistic case.

2.4.1 Fluence simulations

The first simulation was done using a single and point-like proton beam withinitial energy of 150 MeV. The position of the beam has been set at (0, 0,-1 cm) with a positive direction along the z axis. The geometry of the firstsimulation was the following: in the vacuum a box of water has been created,it is 30 cm long (0 < z < 30 cm) with squared basis of 1600 cm2 centered in(0,0) on the XY plane. It has to be observed that the Bragg peak position ofthis beam in water is at about 15.8 cm, so the box is big enough to containthe entire deposition of dose. The box was then divided intersecting infinityXY planes in order to create regions with a ∆z of 1 millimeter at differentdepths. Every simulation has been done with 107 primary particles for tendifferent runs, summing up to 108 protons. This regions were set at: 0.5,1, 2, 3, 4, 5, 7.5, 10, 12.5, 15 cm. It has to be highlighted that for everysimulation the mean ionization potential has been set at 77 eV: such valueis used at CNAO to run experiments and simulations.

The fluence of the primary and secondary particles has been simulated us-ing the card USRTRACK combined with the card AUXSCORE, to measurethe fluence of every kind of particle. The fluence of the following isotopeshave been studied: 1H, 2H, 3H; 3He, 4He, 6He; 6Li, 7Li; 7Be, 9Be; 10B, 11B;10C, 11C, 12C; 13F, 14F; 15O. The energy ranges of the studied isotopes are0 - 150 MeV for primary particles with 300 bins and 0 - 20 MeV for thesecondaries with 200 bins.

2.4.2 LET simulations for mono-energetic beams

In order to study the LET of every particle, of primary protons and of 4He,different simulations with proton beams of 100, 150 and 200 MeV initial ki-netic energy have been run. Infinite XY planes have been created to identifydifferent depths in the box of water. For each case, different XY planes have


been positioned in the regions of interest, with a high density of planes inthe Bragg peak region of each simulation. For instance, for the 150 MeVproton beam, the planes have been set at 1, 2, 3, 4, 5, 8, 10, 14, 15, 15.5,15.6, 15.7, 15.8, 15.9, 16.0 cm. The simulations have been performed usingthe card USRYIELD. The study was done on a LET range between 0 and250 keV/µm with 500 bins.

2.4.3 LET simulations in the case of a SOBP

This simulation has the same scoring options of the LET simulations formono-energetic beams, but a lot changes in the set up, in order to obtain aSpread Out Bragg Peak (SOBP).

Tumors can be seen as 3D deep seated objects, so to irradiate it entirelythe SOBP has to be extended in three dimensions. In order to simulatea clinical case, the purpose of this last simulation is to irradiate a 5x5x5cm3 water cube, with 10 < z < 15 cm, as figure shows 2.3. To create aflat plateau, as shown in figure 1.3, it is necessary to overlap different singlebeam with different energies. The weights and the energies of these beamshave to produce a flat plateau in at a desired depth. In literature manyalgorithms for the calculations of the SOBP can be found. For this studythe algorithms from the work of [22,23] have been chosen. They are based on

Figure 2.3: Energy deposition as a function of the depth for a SOBP. Theparameters from table 2.1 have been used. The first plot shows the energydeposition on the YZ plane, while the second one shows projection on the zaxis. The purpose to cover a 5x5x5 cm3 water cube has been accomplished.

2.4. SIMULATIONS SET UPS 31

ek (MeV) weights115.52 0.003839116.19 0.011591116-86 0.019446117.52 0.027408118.18 0.035479118.84 0.043665119.49 0.051969120.15 0.060397120.80 0.068952121.44 0.077640122.08 0.086467122.73 0.095438123.37 0.104560124.00 0.113840124.64 0.123285125.27 0.132902125.90 0.142700126.53 0.152689127.15 0.162878127.77 0.173278128.39 0.183900129.01 0.194758129.63 0.205866130.24 0.217239130.85 0.228895

ek (MeV) weights131.46 0.240851132.07 0.253130132.67 0.265755133.26 0.278753133.88 0.292153134.48 0.305990135.07 0.320302135.67 0.335136136.26 0.350542136.85 0.366583137.44 0.383332138.03 0.400875138.61 0.419318139.19 0.438794139.77 0.459466140.35 0.481548140.93 0.505320141.51 0.531166142.08 0.559635142.65 0.591557143.22 0.628292143.79 0.672355144.36 0.729473144.92 0.820820145.49 1

Table 2.1: The energies ek of the single beams are reported with the cor-responding weights. The first weight, not included in this table, is zero, inorder to create 50 intervals [ek; ek+1]

a simple formula that links the energy of a single beam, in an homogeneousmedium, with its mean range. It is referred to as the Kleeman-Bragg formula

R = αEp0 (2.1)

where α is typically 0.022 cm/MeV, E is the energy of the beam and pis a number that varies between 1.5 and 2. In the work of [22] it is possibleto find a table for the best parameter p0, which depends on the highestenergy of the beams E0 and the percentage extension of the SOBP χ. Ithas to be decided how many single beams compose the SOBP. By doing so,the k-ranges of the beam are calculated. To each of them is associated anenergy

ek =(rkα

)1/p0 (2.2)

In order to create a flat plateau the beams must have different fluence.This fact is implemented by assigning different weights from 0 to 1 calculated


with a given algorithm. For this process the rejection method is used [18]. Itis based on the fact that it is possible to sample a random variable performinga uniform sampling of the cartesian plane, keeping the samples in the regionunder (or over) the graph of a given density function. In this case, a randomnumber r from 0 to 1 in generated. Each time, given an energy ek+1, it isselected if rε[wk;wk+1].

For this study 50 beams have been used in order to create the SOBP.The energies with the corresponding weights are reported in table 2.1.

To create a SOBP in FLUKA the user has to add a card SOURCEin the Primary section and modify the subroutine source.f. In the sourceroutine, the spatial standard deviations and the energy standard deviationshave been set different from zero. In particular the standard deviationsare σxy = 0.94 cm σE = 0.8MeV. Again, in order to study the total LETand the contribution of primary particles and Helium fragments, the cardUSRYIELD has been used for different depths: 1, 2, 3, 4, 5, 8, 10, 11, 12,13, 14, 15 cm.

Chapter 3

Results and discussion

In the third chapter the results obtained with FLUKA simulations will bepresented. At first, the fluence distributions of primary and secondary par-ticles at different depths, for a mono-energetic beam of 150 MeV, will beshown. After the characterization of the energies of primaries and secon-daries, a study on the LET distributions have been performed concerningthree different mono-energetic beams of 100, 150 and 200 MeV. Besides, theanalysis on LET has been extended by implementing a simulation with aSOBP. This gives the opportunity to study the LET distributions of primaryprotons and of the fragments in a more clinical case. From these results, theanalytic model of Wedemberg has been considered in order to estimate pro-tons RBE. The LET distributions of 4He have been strongly investigated,due to the important biological impact of these fragments.

3.1 Fluence distributions of primary and sec-

ondary particles

In this section, the fluence of primary and secondary particles at differentdepths has been simulated for a mono-energetic beam of 150 MeV, usingthe simulation set up reported in section 2.4.1. At every depth the fluenceof the most abundant isotopes will be shown: every spectrum is followed bya zoom image on the low energy region of 0 - 20 MeV, in order to bettervisualize the distributions of the secondary particles with Z > 1 .

The fluence concerns the depths at 3, 10 and 15 cm, shown in figure 3.1,3.2 and 3.3.

Figure 3.1.a shows that after 3 centimeters protons have lost only a smallfraction of their initial energy: their spectrum is peaked at about 130 MeV.At lower energies it is possible to observe the contribution of secondaryprotons. The energy spectra of secondary particles is restricted to lowerenergies. In figure 3.1.b, 4He is the most energetic particle, with an energyend-point of about 13 MeV/u. The other secondary particles have a lowerenergy end-point, as 7Li and 9Be, that barely reach 3 MeV/u.

At a depth of 10 cm (figures 3.2.a and 3.2.b) protons have lost a signif-icant fraction of their initial energy. The peak is more broaden due to thestochastic nature of radiation-matter interaction. Also, a small portion of

33

34 CHAPTER 3. RESULTS AND DISCUSSION

(a)

(b)

Figure 3.1: Energy spectra of the most abundant isotopes at the depth of 3cm. Panel (b) is the zoom image in the region 0 - 20 MeV.

(a)

(b)

Figure 3.2: Energy spectra of the most abundant isotopes at the depth of10 cm.

3.1. FLUENCE DISTRIBUTIONS OF PRIMARYAND SECONDARY PARTICLES35

(a)

(b)

Figure 3.3: Energy spectra of the most abundant isotopes at the depth of15 cm.

the protons population appears at higher energies than the peak ones. Thisis due to the protons scattered by neutrons, which have been previouslyproduced from nuclear interactions. Since primaries have a smaller energiescompared the ones of the entrance channel, the energy end-points of thesecondaries decrease.

Close to the region of the Bragg peak (figures 3.3.a and 3.3.b) protonshave lost a big fraction of their initial energy, the peak is now around 25-30MeV. The contribution of secondary protons is less present at lower energiesthan the entrance channel, while at higher energy the population of scat-tered protons from neutrons has grown. Since the energy of protons is low,it reflects on the spectrum of secondary particles: their energy end-pointshave become about 6 MeV/u for 4 He and they do not exceed 1 MeV/u forthe heavier nuclei.

From the plots in the figures 3.1.b, 3.2.b and 3.3.b it is possible to ob-serve that 4 He is the most abundant isotope among secondary particles. Itcan be interesting to evaluate its relative contribution respect to the othersecondaries heavier than protons: in figure 3.4 the relative contribution of4He to the total secondaries has been studied at different depths. The ratioshave been obtained by integrating the energy spectra of every secondaryand dividing the value for 4He by the sum of the integrals of the other par-ticles. The relative contribution of Helium is high: it starts from 80% in


Figure 3.4: Relative contribution of 4He to the total secondary particles withZ > 1.

the entrance channel and reaches about 98% on the Bragg peak. This resulthints that the secondary particle which contributes the most to the biologicaldamage is 4He. For this reason it has been decided to highlight the behaviorof this particle in the next simulations.

3.2 LET distributions of a single beam

After the study on the fluence and the energies of produced fragments atdifferent depths, it emerges that secondaries tend to have a low energy, atmaximum a few MeV. Due to their small kinetic energy and a higher atomicnumber than protons, they are expected to have high LET. For this reason,it has been decided to study the LET distributions of primary and secondaryparticles using three different mono-energetic beams. The set up of sections2.4.2 has been considered. The plots, reported in figure 3.5, correspond todifferent depths for the 100 MeV beam: this positions can respectively rep-resent the distributions of the entrance channel, the intermediate region, adepth close the Bragg peak and the Bragg peak itself. For the beams withinitial energy of 150 and 200 MeV, two plots are shown, respectively in figure3.6 and 3.7.

In the entrance channel (figure 3.5.a) primary protons have lost a smallfraction of their energy, so their LET is low, not exceeding 2 keV/µm. Asit was expected, 4He reaches high values of LET, especially as comparedto the protons ones. It ranges from 15 to 225 keV/µm and has an almostconstant trend. It is possible to observe that in the region between a fewkeV/µm to about 30 keV/µm, primary particles and 4He do not contributeto the total distribution: this range is dominated by secondary protons,that have a wider and lower energy spectrum in the entrance channel ascompared to primary protons (as shown in fig 3.1.a). The 4He contributionis important in the high LET region, in fact the total distribution of LET isalmost indistinguishable from the 4He one above 80 keV/µm.

At an intermediate depth (figures 3.5.b, 3.6.a and 3.7.a) protons havelost a considerable amount of their initial energy, and this fact is reflected in

3.2. LET DISTRIBUTIONS OF A SINGLE BEAM 37

(a)

(b)

(c)

(d)

Figure 3.5: LET distribution of primary particles (PRIMARY), 4He (4He)and of every particle (TOTAL) at 1 cm (a), 3 cm (b), 7 cm (c) and 7.7 (d)for a 100 MeV beam.


(a)

(b)

Figure 3.6: LET distribution of primary particles, 4He and of every particleat 10 cm (a) and 15.8 cm (b) for a 150 MeV beam.

(a)

(b)

Figure 3.7: LET distribution of primary particles, 4He and of every particleat 15 cm (a) and 25.9 cm (b) for a 200 MeV beam.

3.3. LET DISTRIBUTIONS IN THE CASE OF A SOBP 39

an enhancement of their LET. The distribution of 4He remains similar forall three plots.

In the region close to the Bragg peak (figure 3.5.c) the contribution ofprimary particles is approaching the total distribution, hinting that the pro-tons contribution is starting to dominate among the secondaries. However,the higher LET region is still dominated by 4He. The shape of its distri-bution has visibly changed and the peak has moved to higher values, ascompared to the first centimeters.

On the Bragg peak (figures 3.5.d, 3.6.b and 3.7.b) the total distributionis almost identical to the protons one and 4He contribution has drasticallydecreased, due to the small kinetic energy of primary particles.

From the study on the LET distributions of the mono-energetic beamsit is possible to identify different aspects, depending on the depth that isconsidered. In the entrance channel protons have a very low LET, hence asmall RBE is expected. However, in this region 4He has a wide distributionthat reaches high values of LET and as a consequence it will contribute witha non-negligible biological damage. It has to be highlighted that 4He reachesvalues of LET that includes the region of the so called optimal LET, whichcorresponds to the higher possible biological effectiveness. Since protonstend to acquire higher LET as the depth increases, they reach higher RBEvalues and the damage produced by 4He will have a smaller relative impact.

3.3 LET distributions in the case of a SOBP

From the study on the last simulations it emerges that 4He could be respon-sible for a significant contribution to the biological damage in the entrancechannel, while protons dominate the entire contribution on the Bragg peak.

After these results it becomes interesting studying the LET distributionson a more clinical case, as the case of a SOBP (Spread Out Bragg Peak).An analysis on protons LET and 4He has been performed. Their mean LETas a function of the depth has been investigated. By the data from protonsit is be possible to evaluate their RBE along the entire SOBP. From thedistributions, it is also possible to account for the relative contribution by

Figure 3.8: LET distribution of all particles, primary protons and 4He inthe entrance channel at 1 cm


(a)

(b)

(c)

(d)

Figure 3.9: LET distribution of all particles, primary protons and 4He onthe SOBP at 5 (a), 10 (b), 13 (c) and 15 cm (d).

3.3. LET DISTRIBUTIONS IN THE CASE OF A SOBP 41

protons and 4He to the total LET, in order to compare the relative effec-tiveness of these particles.

In order to implement a SOBP using FLUKA the source has to be com-posed of different mono-energetic beams (reported in table 2.1) with appro-priate weights in order to create a SOBP from 10 to 15 centimeters, as itwas exposed in section 2.4.3.

In the first centimeter (figure 3.8) protons have a low LET of a fewkeV/µm. Helium too maintains the same behavior, with a wide distribution.The intermediate LET region is not dominated nor by primary protons or4He: it is still dominated by secondary protons.

As the depth grows, at 5 centimeters (figure 3.9.a) the distribution ofprotons has broaden toward higher LET values, due to the energy loss,while the 4He one remains similar.

At the beginning of the SOBP (figure 3.9.b), it is possible to observea substantial difference from the simulations with mono-energetic beams:although the highest values of energy deposition are reached at this depth,the 4He contribution is still present and still dominates in the region ofhigher values of LET. This behavior was not observed on the Bragg peakfor the single beams, because in the case of the SOBP only a fraction ofprimary particles stops at this depth and the remaining population still hasthe energy to cause nuclear fragmentation.

In fact at 13 centimeters (figure 3.9.c), the region up to about 80 keV/µmis entirely dominated by the contribution of protons, but there still is a non-negligible contribution of 4He for high values of LET.

At the end of the SOBP (figure 3.9.d), as expected, the contribution of4He drops, due to the fact the the last primary protons are about to stop,and their kinetic energy is not high enough to produce nuclear fragments.

After these considerations, the trend of the LET distributions has beenseparately investigated for protons and 4He.

Protons contribution

From the previous simulations, by calculating the means of the LET distri-bution for each depth, it is possible to observe in figure 3.10 that primaryprotons have an almost constant mean LET in the first 8 centimeters, withvalues a little above 0.80 keV/µm. At a depth of 10 centimeter, the meanLET starts to increase, due to the fact that a fraction of the protons popu-lation is about to stop and as a consequence their LET reaches high values.As the depth grows, the mean LET increases up to 5.5 keV/µm.

Helium contribution

The mean LET of 4He is set around higher values: it ranges from about 75to 120 keV/µm, as shown in figure 3.11. It has to be underlined that thesevalues include the region of the optimal LET, that lays at around 100 keV/µfor Helium, as figure 1.4 shows.


Figure 3.10: Means of the LET distributions of primary protons in the caseof a SOBP at different depths.

Figure 3.11: Means of the LET distributions of 4He at different depths inthe case of a SOBP.

Figure 3.12: Relative contribution to the total LET of 4He, considering allpositive particles.

Since 4He is expected to have a high biological effectiveness, it is inter-esting to study the relative contribution to the total LET, to quantify itsabundance among all the other ions. The plot in figure 3.12 has been ob-tained by integrating the LET distributions of all positive ions (obtainedthrough another simulation, which has not been shown) and the 4He one,to calculate the ratios at different depths. Figure 3.12 shows that 4He con-

3.4. RBE EVALUATION 43

tributes in the amount of some per thousands in the entrance channel, andits contribution decreases with the depth. As expected, it is possible tostate that the highest impact of 4He, and in general of all the secondaries,is reached in the entrance channel, due to the fact that primary protonshave a low LET and a small biological effectiveness. On the SOBP, eventhough 4He is still present, its contribution is lower due to protons higherRBE values.

3.4 RBE evaluation

From the results of the LET simulations in the case of a SOBP, it emergeshow protons LET assumes values which grow with the depth, therefore ahigher contribution of primary protons to the biological damage is expected.

In literature it is possible to find various analytic models to calculateRBE of protons, depending on LET, the delivered dose and the cell line.The most common models are Wedemberg [24], Wilkens [25], McNamara [26]and Carabe [27]. All these models are based on experimental data: the pa-rameterization of RBE is obtained by the comparison between the survivingfraction of cells after the irradiation with photons and mono-energetic pro-ton beams, to which correspond well defined LET values [22, 28]. Someassumptions are common for every model: a linear dependence from of theradiobiological parameter α is assumed, while β is supposed to be constant.Also, they can be applied on a limited LET range, up to 30 keV/µm, becausethey do not take into account the decrease of RBE after this point. It has tobe highlighted that most of these models are used to evaluate protons RBEusing LETD, which is the dose-averaged LET [28]. This quantity intendsto average the LET contributions of different particles, weighted for theirdeposited dose.

In this work, it has been decided to consider the Wedemberg modeland to calculate RBE considering only protons LET as an approximation.This assumption is justify by the fact that from the previous simulation, ithas been calculated that protons contribute to the total LET from 98.5%to almost 100% on the SOBP, hinting that LET values are similar to theLETD ones.

In the Wedemberg model [24], RBE is given by the formula

RBE(LET,D, α/β) =

− 1

2D

(αβ

)ph

+1

D

√1

4

(αβ

)2ph +

(qLET +

(αβ

)ph

)D +D2

(3.1)

whereD is the dose. The calculations have been implemented consideringa dose of 2 Gy, which is a standard value for a scheme fractions. For thisstudy the cell line V79 (Chinese Hamster fibroblasts) has been considered,because this is one of the most used cell lines for radiobiological experiments.This model is based on the assumption that α grows linearly with LET withthe formula

αphα

= 1 +q

(α/β)phLET (3.2)


Figure 3.13: RBE values obtained with the Wedemberg model for primaryprotons and both primary and secondary once.

where q is a parameter obtained by experimental data, different for eachcell line. For V79 cells, the value of q

(α/β)phis 0.1085 (keV/µm)-1 .

RBE has been evaluated in two different ways. In the first case the meanLET of primary protons in the case of the previous SOBP has been con-sidered, while in the second case both primary and secondary protons havebeen considered (LET values have been obtained through another simula-tion, which has not been shown). The choice of considering both primaryand secondary protons is in the order to analyze a more realistic trend ofRBE and to observe if secondary protons cause a significant enhancementof this parameter, since this has been reported in previous works [12]. Also,this choice has been suggested in the work of Mairani [29]. The plots areshown in figure 3.13.

Primary protons, as shown in figure 3.10, have constant LET in theentrance channel, therefore a constant RBE value of 1.05 is found in the en-trance channel. Due to the enhancement in the last centimeters, RBE growson the SOBP, reaching 1.3. However, if secondary protons are considered, itis possible to observe higher RBE values in the entrance channel. As shownsecondary protons have lower energies than the primary ones, resulting onan increase of LET if they are considered. The value at the first centimetersis 1.1: secondary protons contribute to an enhancement of RBE of about5%, while at higher depths their contribution becomes negligible. Therefore,secondary protons contribute to an enhancement of the biological damagein the entrance channel, to which it should also be accounted the one causedby heavier fragments.

Conclusions

In this thesis the impact of nuclear fragmentation has been investigated bymeans of Monte Carlo simulations. From the fluence simulations it has beenobserved that secondary fragments have a more restricted energy spectrathan protons, and only 4He, which is the most abundant isotope among allsecondary particles, exceeds 10 Mev/u in the entrance channel. From theseresults the LET distributions of primary and secondary particles have beenanalyzed, highlighting the behavior of Helium. From the study concerningmono-energetic beams it emerged that 4He has wide LET distributions atany depth. This fact has been remarked also in the simulation of a SOBP,which has been studied to consider a more clinical case. From the latter, themeans of the LET distributions of primary particles and Helium have beenobtained.

These results have been used in order to evaluate protons RBE throughthe Wedemberg model. It is possible to observe that it does not have aconstant value. Also, fixing it at 1.1 might result in an underestimation ofthe biological damage on the SOBP, due to the increasing LET of primaryprotons on the distal fall-off. In this region the dose is supposed to decrease,as the entire depth of the tumor has been already irradiated. In the distalfall-off, it has been estimated that RBE reaches values around 1.3, meaningthat 20% of non-calculated biological dose is delivered: due to higher LET inthe region of the SOBP, there is an increase of RBE that can cause an over-killing and might be dangerous especially if a critical organ risk is seatedbehind the tumor region. Also, it is possible to observe the contribute ofsecondary protons to RBE. If these particles are considered too, RBE reachesvalues of 1.1 in the first centimeters, hinting that these secondaries contributefor about 5% to the total biological damage.

Besides, secondary particles with Z > 1 should be considered too, in or-der to calculate the right amount of both physical and biological dose. Inthis work, it has been shown that these fragments have a low energy. Dueto their energy spectra, the fragments have a residual range in the order ofsome µm and their LET assumes high values. The highest impact is there-fore situated in the entrance channel, where protons have low LET and asmall biological effectiveness. From the results it has emerged that 4He isthe most abundant isotope at any depth and its mean LET values are seatedin the region of the so called optimal LET, which means that the biologicaleffectiveness of this particle reaches its maximum. Even though the contri-bution to the total LET is of the order of some per thousands, its impactcan’t be neglected, due to the high LET and therefore high RBE. However,for the calculations of RBE for particles heavier than protons, there is still

45

46 Conclusions

a lack of manageable models that might have clinical applications.

From this thesis project, it emerges that the effects of nuclear fragmen-tation should be taken into account in the TPS for proton therapy. Thedevelopment of the software should be supported with the implementationof RBE models for particles with an atomic number higher than one, in or-der to properly evaluate the biological dose that is absorbed by the tissues,both healthy and tumor ones.

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a study on the contribution of nuclear … figure 1.2: the di erent depth-dose distributions of...

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