Absorbed dose and biological effect in light ion therapy198973/FULLTEXT01.pdf · 2013-08-12 · with a simple track-segment model of ion beam transport. Although the stud-ies show

Absorbed dose and biological effect in light ion therapy

Malin Hollmark

Stockholm University

Medical Radiation Physics Department of Oncology-Pathology

Stockholm University & Karolinska Institutet Stockholm 2008

©Malin Hollmark, Stockholm 2008 ISBN 978-91-7155-671-4 Printed in Sweden by Universitetsservice US-AB, Stockholm 2008 Distributor: Medical Radiation Physics, Stockholm University & Karolinska Institutet

On dedication page: “Dad and Freya” by Freya Hollmark 3 years old

To my family

Abstract

Radiation therapy with light ions improves treatment outcome for a number of tumor types. The advantageous dose distributions of light ion beams en-able exceptional target conformity, which assures high dose delivery to the tumor while minimizing the dose to surrounding normal tissues. The demand of high target conformity necessitates development of accurate methods to calculate absorbed dose distributions. This is especially important for heavy charged particle irradiation, where the patient is exposed to a complex radia-tion field of primary and secondary ions. The presented approach combines accurate Monte Carlo calculations using the SHIELD-HIT07 code with a fast analytical pencil beam model, to pro-vide dose distributions of light ions. The developed model allows for ana-lytical descriptions of multiple scattering and energy loss straggling proc-esses of both primary ions and fragments, transported in tissue equivalent media. By applied parameterization of the radial spread of fragments, im-proved description of radial dose distributions at every depth is obtained. The model provides a fast and accurate tool of practical value in clinical work. Compared to conventional radiation modalities, an enhanced tissue response is seen after light ion irradiation and biological optimization calls for accu-rate model description and prediction of the biological effects of ion expo-sure. In a joint study, the performance of some radiobiological models is compared for facilitating the development towards more robust and precise models. Specifically, cell survival after exposure to various ion species is modeled by a fast analytical cellular track structure approach in conjunction with a simple track-segment model of ion beam transport. Although the stud-ies show that descriptions of complex biological effects of ion beams, as given by simple radiobiological models, are approximate, the models may yet be useful in analyzing clinical results and designing new strategies for ion therapy. Keywords: Light ion therapy, absorbed dose calculation, analytical pencil beam model, Monte Carlo, radiobiological modeling

Aim of the thesis

The principal aim of this thesis is to develop a fast and accurate analytical method for the calculation of dose distributions of light ions in tissue equivalent media. The scope of the thesis also covers studies of methods for evaluation of biological effects on cells from exposure to light ions. To complete these tasks, the following steps have been taken:

1. Development of an analytical pencil beam model, in which an analyti-cal approach is combined with the Monte Carlo code SHIELD-HIT07 to derive physical dose distributions of light ions transported in tissue equivalent media (Papers I and II)

2. Experimental verification of lateral dose distributions of protons as

derived by the analytical pencil beam model for ion beam transport in heterogeneous media (Paper III)

3. Investigation and comparison of some of the existing radiobiological

models that describe cell survival after light ion irradiation (Paper IV) 4. Prediction of the in vitro cell survival from light ion irradiation by us-

ing a track structure approach (Papers V-VII).

List of papers

The thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I M Hollmark, J Uhrdin, Dž Belkić, I Gudowska and A Brahme, 2004, Influence of multiple scattering and energy loss straggling on the absorbed dose distributions of therapeutic light ion beams: I. Analytical pencil beam model, Physics in Medicine and Biol-ogy 49, 3247-3265

II M Hollmark, I Gudowska, Dž Belkić, A Brahme and N Sobo-levski, Analytical model for light ion pencil beam dose distribu-tions: multiple scattering of primary and secondary ions, ac-cepted for publication in Physics in Medicine and Biology

III M Hollmark, I Gudowska, P Kimstrand, E Traneus and N Tilly, Experimental verification of transverse ion beam profiles calcu-lated by an analytical pencil beam algorithm and the Monte Carlo code SHIELD-HIT07, manuscript in preparation

IV M Hollmark, M R Edgren, P Kundrát, B K Lind, A E Meijer, M P R Waligórski and M Wedenberg, A comparison of radiobio-logical models for light ion therapy, manuscript in preparation

V M P R Waligórski, M Hollmark, I Gudowska and J Lesiak, 2004, Cellular parameters and RBE-LET dependences for modelling heavy-ion radiotherapy, Radiotherapy and Oncology. 73, S173-175

VI M P R Waligórski, M Hollmark and J Lesiak, 2006, A simple track structure model of ion beam radiotherapy, Radiation Pro-tection Dosimetry 122, 471-474

VII M P R Waligórski, M Hollmark, M Korcyl, J Lesiak and I Gu-dowska, A simple cellular track structure approach for modeling ion beam radiotherapy, manuscript in preparation

Reprints were made with permission from the publishers.

List of Abbreviations

CSDA Continuous slowing down approximation

GFE Generalized Fermi-Eyges theory

DNA Deoxyribonucleic acid

FE Fermi-Eyges theory

LET Linear energy transfer

LQ Linear-quadratic model

MC Monte Carlo

OER Oxygen enhancement ratio

PMMA Polymethyl methacrylate (acrylic glas) RBE Relative biological effectiveness

RCR Repairable-conditionally repairable damage model

P2S Probabilistic two-stage model

SOBP Spread out Bragg peak

TST Track structure theory

TPS Treatment planning system

Contents

1. Introduction 1

1.1 Light ion beams – physical aspects 2

1.2 Light ion beams – biological aspects 4

1.3 Light ion beams – clinical aspects 6

2. Absorbed dose calculations for light ion therapy 8

2.1 Physical processes – background 9

2.2 Dose calculation methods in the present study 12

2.2.1 The Monte Carlo code SHIELD-HIT07 12

2.2.2 The analytical pencil beam model 15

3. Modeling of radiobiological effects of light ions beams 36

4. Conclusions and outlook 41

Summary of papers 43

Acknowledgements 46

Bibliography 48

1. Introduction

Cancer is the name of a group of diseases that are all constituted of cells with genomic changes that result in malignant tumor cells. There are more than 100 different types of cancer and the disease can affect people at all ages, although there is an increased risk with age for the most common cancer types. The cancer incidence and death rates in Europe 2004 was found to be about 2.9 million new cases of cancer diagnosed and 1.7 million cancer deaths (Boyle and Ferlay 2005). In developing countries, the number of can-cer cases is predicted to reach 10 million in 2015. It is estimated that the probability of dying from cancer is 20-25% in the industrialized countries (IAEA 2008). This makes cancer a major health problem worldwide and the ageing population of the industrialized countries will cause these numbers to increase even further. Most cancers can be treated and well over 50% can be cured. The outcome of the treatment is dependent on the specific cancer type, location and stage, including the presence of metastases. The most common treatments for solid malignancies are surgery, radiation therapy and chemotherapy. Radiation therapy, which is the use of ionizing radiation for curative or palliative tumor treatment, may be used as the primary therapy, although a combination of radiation therapy with the other treatment modalities is also customary. Approximately 60% of patients with solid cancers receive radiation therapy and it is estimated that approximately half of the patients who are cured have benefited from the treatment. Although more sophisticated treatment modalities may be developed, such as gene therapy, hormone therapy, tar-geted therapy and immunotherapy, radiation therapy will likely continue to be a very important, effective and cost effective modality for all types of solid tumors (IAEA 2008). The first medical use of X-rays was reported in January in 1896, only a couple of months after the German physicist Wilhelm Conrad Röntgen’s first found that photographic film in light-sealed containers could be blackened by a “new kind of ray”. Only three years after his discovery, Pierre and Marie Curie isolated the radioactive elements Radium and Polonium. The spread of this new technology was rapid and lead to the births of the fields of diagnostic and therapeutical radiology in the late 19th century.

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The major therapeutical radiology, or radiation therapy, today is external radiation therapy, but for certain tumor types, brachytherapy, i.e. internal treatment with sealed radioactive sources, is considered effective. Since this thesis deals only with external beam radiation therapy, the term “radiation therapy” will hereafter indicate external radiation therapy. Currently, the main treatment modalities are electron and photon beams, but there are other modalities available, such as protons, neutrons, pions, antiprotons, and heavy charged particle beams. In radiation therapy, the treatment is usually delivered with a margin, which means that the dose (i.e. the energy deposited per unit mass; 1 Gy = 1 J/kg) is delivered to the tumor but also to adjacent healthy tissues, in which a risk of malignant spread is likely. This margin of normal tissue around the tumor, which also allows for uncertainties in daily patient set-up and internal tumor and normal tissue motions, can result in unnecessary damage to otherwise healthy tissues and might even cause radiation-induced secondary cancers. One of the goals of radiation therapy is thus to deliver a maximum dose to the target volume and a minimum dose to the surrounding healthy tissue. In the conventional radiation modalities, i.e. using electron and photon beams, numerous techniques have been developed in attempts to spare the normal tissues, such as arc therapy, 3-dimensional conformal treatments and intensity modulated radiation therapy (IMRT). Fast neutron therapy, which offers an increased relative biological effectiveness (RBE), has proved very efficient, particularly for hypoxic tumors (Maughan et al 1999). However, although high energy neutrons are very effective in killing tumor cells, they can be equally lethal to normal tissues and the side effects have thus been severe for this modality (Wambersie 1992). With light ion therapy, in addi-tion to generating higher biological effects compared to the conventional modalities, improved conformity to the tumor volume and diminished severe normal tissue toxicity can be achieved thanks to the physical characteristics of these charged particle beams.

1.1 Light ion beams - physical aspects W H Bragg was the first to observe the high energy deposition where α-particles stopped at the end of their range (Bragg and Kleemann 1905). In 1946, Robert Wilson investigated the depth dose profile of protons at the Berkeley cyclotron and also observed this so-called Bragg peak, i.e. a steep increase of energy deposition at the end of the particle range (Wilson 1946). He then pointed out the potential to use protons for tumor therapy. In 1954, at the Lawrence Berkeley Laboratory (LBL) at the University of California, USA, the first treatment of a human pituitary gland was performed with a

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340 MeV proton beam where the plateau of the dose distribution was util-ized (Tobias et al 1955, Lawrence et al 1958). Later, the therapeutical ad-vantages of the Bragg peak was used instead by applying beams of lower energies and placing the Bragg peak in the target. In 1957 in Uppsala, Swe-den, the Gustav Werner cyclotron was the first to use a modulated proton spread out Bragg peak (SOBP) for treatment of human cancer (Larsson 1967). The first clinical experience with ions heavier than protons, such as helium, carbon and neon ions took place 1977 at the LBL. Figure 1 is a schematic illustration of the dose deposition of high energy electrons, 50 MV X-rays and 150 MeV protons in water. The figure illus-trates the physical advantage of light ion therapy, where the energy deposi-tion of protons clearly indicates a very low dose delivery behind the Bragg peak and also, at least when compared to electron beams, at the entrance. The rapid increase and the fast fall-off of the dose at the end of the ion range, resulting in the Bragg peak, allows for a conformal tumor irradiation with simultaneous sparing of the healthy tissue, not possible with the conven-tional modalities. In this thesis, the distinction between light and heavy ions is made as recommended by Chu (1999) and the International Atomic En-ergy Agency (IAEA 2008). This recommendation is that light ions are those nuclei with atomic number . 10Z ≤

50 MeV e-

150 MeV protons

Dos

e no

rmal

ized

to m

axim

um

Depth (cm)

50 MV X-rays

SOBP

Figure 1. Schematic illustration of the dose deposition of high energy electrons, 50 MV X-rays and 150 MeV protons in water.

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A disadvantage in ion therapy is the production of beam fragments which may decrease the dose conformity to the target. However, when radioactive fragments, such as 11C6+ and 10C6+ ions, are produced, this drawback can be exploited by enabling in-situ observation through the use of positron emis-sion tomography (PET), which, in combination with computer tomography (CT), facilitates monitoring of the dose deposition in a patient during a treatment (Enghardt et al 1999, Brahme 2003, Enghardt et al 2004).

1.2 Light ion beams - biological aspects When eradicating tumors by ionizing radiation, the DNA in the cell nucleus is considered to be the main target, although lesions to other cellular compo-nents, such as the plasma membrane and other organelles, can be induced (Hall and Giaccia 2006). The amount and type of damage is strongly de-pendent on the genetic characteristics of the irradiated cell as well as on the radiation quality, the dose delivered and the exposure time. The radiation quality is specified by the type of incident particle used, its charge state and energy, all of which determine the linear energy transfer (LET) of the particle, i.e. the amount of energy transferred to a medium per unit length of the particle track. A distinction is usually made between low- and high-LET radiation. Low-LET radiation, such as electrons and photons, produces randomly scattered ionizations separated by fairly large distances and delivers an ionization density of about 0.2 – 10 keV μm , whereas the corresponding values for high-LET radiation for ions are characterized by dense ionizations and LET values of about 20 – 200 keV μm . Low-LET radiation produces mostly base damages and single-strand breaks (SSB) for which the cell has a very efficient repair system. Also double-strand breaks (DSB) produced by low-LET radiation can be repaired with high fidelity, even if DSBs are considered to be more difficult to repair. The cell survival after high-LET exposure is lower than that after low-LET irradiation since high-LET irradiation can produce so-called multiply damage sites consisting of more complex DNA damages that are difficult for the cell to repair (Ward 1988, Goodhead 1994, Stenerlöw et al 1996). The biological advantage with light ion therapy compared to the low-LET conventional modalities is thus the enhanced response of the biological systems to the high-LET exposure. This is reflected in the increased RBE, the decreased oxygen enhancement ratio (OER), and the decreased cell-cycle dependence of the response.

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Relative biological effectiveness (RBE) The biological effect of different types of radiation can be compared by us-ing the concept of RBE, which is defined as the ratio of the dose from a ref-erence radiation quality, , to the dose from the test radiation, XD D , that produce equal biological effect

XDRBED

= . (1)

The reference radiation is usually 60Co γ-rays or 220 – 250 kV X-rays, al-though differences in the RBE between these two modalities have been found (Hill 2004). The cell survival, and hence the RBE, is highly dependent on the biological system, type and level of effect (i.e. the endpoint) and, as mentioned, on the radiation quality. The clonogenic cell survival RBE in-creases with increasing LET to a peak value at about 100 keV μm of the test radiation which means that there is an increase in the efficiency of tumor cell eradication with increasing LET. At higher LET values, the RBE decreases again due to the effect over “over-kill” where the ionization density exceeds what is needed to kill a cell (Kraft 2000). Oxygen enhancement ratio (OER) Many tumors are hypoxic, i.e. they have a low oxygen tension. It has been found that the dose required from conventional radiation therapy under hy-poxic conditions can be up to about three times greater than the dose needed under fully aerated conditions. The OER is the ratio of the dose required to produce a given effect in the absence of oxygen to the dose required for the same effect in the presence of oxygen. For sparsely ionizing radiation, such as X- and γ-rays, the OER at high doses has a value between 2.5 and 3. For densely ionizing radiation, such as neutron and ion radiation, it has been shown that there is a considerably reduced oxygen effect, i.e. the dose re-quired for cell kill is almost the same whether the irradiation is conducted under hypoxic or aerated conditions. Cell cycle sensitivity Cells show differences in radiosensitivity when exposed to sparsely ionizing radiation in different phases of the cell cycle. Generally, cells in the late S-phase, where DNA is synthesized, and in the early G2-phase, have shown to be more radioresistant when compared to cells in the mitotic (M) and G1-phase. In the high-LET range, these differences are diminished and cells in different cell cycle stages are more equally radiosensitive (Hall and Giaccia 2006).

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1.3 Light ion beams - clinical aspects From the 70’s through the 80’s, there were research-based proton facilities located in the former Soviet Union, Sweden, Switzerland and the USA. Loma Linda University Medical Center in the USA opened the world's first hospital-based proton treatment facility in 1990 and has been followed by several similar centers in Japan, Germany, France, Canada, China, Italy and South Africa. By the end of 2006 more than 53,000 patients had received proton irradiation. Today, there are more than 25 proton therapy centers around the world and more are under construction or in the planning phase (Schulz-Ertner et al 2006). A proton treatment facility in Uppsala, Sweden is planned to start treatments in 2012. In different periods between 1977 and 1992, about 2,500 patients in total were treated with particles heavier than protons, such as helium, carbon and neon ions, at the LBL. In 1994, the first medically-dedicated carbon ion therapy center opened at HIMAC in Chiba, Japan. Today there are two car-bon ion centers in Japan and more than 4,000 patients had been treated as of December 2006 (cf. Tsujii et al 2007). In 1997, patient treatments started in a carbon ion therapy project at Gesellschaft für Schwerionenforschung (GSI) in Darmstadt, Germany (Debus et al 2000) and following pilot project, the hospital-based Heidelberg Ion-Beam Therapy (HIT) Center is planned to start treatments end of 2008 (Haberer et al 2004, Amaldi and Kraft 2007). In Germany, two additional centers are planned: PTC in Marburg and PTZ in Kiel. Further examples of present European proton and light ion therapy projects are ETOILE in France (Bajard et al 2004, Amaldi and Kraft 2007), Med-Austron in Austria, CNAO in Italy (Amaldi and Kraft 2007) and a pro-ject at the Karolinska University Hospital in Stockholm, Sweden. In this latter project, the use of light ions is being considered with an expected beam towards 2012 (Brahme et al 2001, Svensson et al 2004). Treatment planning systems Regardless of which treatment regime is used, a computer aided treatment and dose delivery program for an individual treatment planning is needed. This treatment planning system (TPS) should be able to prescribe a treatment that lethally damages the tumor while minimizing normal tissue damage. A typical treatment plan depends on tumor (e.g. tumor location, type and ex-tent) and patient characteristics (e.g. patient age, health and weight) as well as on the available treatment alternatives. The process involves acquiring the optimal beam type, energy and incidence angle as well as outlining key ana-tomical features, such as the tumor and organs at risk. Aided by three-dimensional medical imaging, using mainly CT, but also PET, magnetic resonance imaging (MRI) and magnetic resonance spectroscopic imaging

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(MRSI), the treatment planning system performs beam transport simulations and optimizes the absorbed dose in three dimensions. In addition to the physical dose optimization in a TPS, an optimization of the biological effect is also required. This optimization process aims at improv-ing the treatment outcome by taking into account the differences in biologi-cal response of different tumor cells and normal tissues. Today there are only a few TPSs dedicated to proton therapy, e.g. Helax-TMS (Dahlin et al 1983, Jung et al 1997), EYEPLAN (Goitein and Miller 1983), OCTOPUS (Pfeiffer and Bendl 2001), KonRad Pro (Oelfke and Bort-feld 2001), Eclipse (Varian 2005) and FOCUS (Herault et al 2007). The majority of ocular proton therapy patients receive treatment planning using EYEPLAN. In clinical applications of carbon ion therapy, a treatment planning platform Voxelplan combined with the dose optimization program TRiP (Krämer et al 2000, Krämer and Scholz 2000, Jäkel et al 2001), based on the semi-analytical transport model YIELD, and the graphical user interface VIR-TUOS, is used at the GSI facility in Darmstadt, Germany. Another software, HIPLAN, is used at the HIMAC facility in Chiba, Japan (Endo et al 1996). A TPS with biological optimization for light ion therapy, based on PET-CT predictive assay of tumor responsiveness, is under development in collabora-tion between Karolinska Institute and Raysearch Laboratories (Brahme 2003, Brahme 2004). To be useful in a TPS, an analytical algorithm for the calculation of absorbed dose distributions must be both accurate and fast. The accuracy is necessary to ensure that the correct dose is delivered to the tumor and to obtain the proper correlation between the dose delivered and the clinical effect. The algorithm should also be fast enough to facilitate recalculation for optimiza-tion of the physical dose and biological effect of the treatment outcome. In the following chapter, a developed analytical model for the calculation of physical pencil beam dose kernels of light ions in matter is presented together with a brief background considering the most important charged particle interaction processes of relevance to this model. Also, a short introduction of the applied Monte Carlo code SHIELD-HIT07 is given. In chapter 3, some radiobiological aspects of light ions are presented. A few radiobiological models, applicable in light ion therapy and used in the undertaken collaborative study, are described and discussed.

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2. Absorbed dose calculations for light ion therapy

In this thesis, by physical dose is meant the dose absorbed in a medium and a dose kernel is the dose deposited in a semi-infinite medium from a point-monodirectional beam. Besides actual measuring, there are two main ap-proaches to generate the physical dose kernels needed as input into a TPS (Goitein 1992): analytical models, such as the pencil beam model, and Monte Carlo (MC) techniques. Analytical pencil beam models have been in extensive use for the dose cal-culation in external radiation therapy treatment planning, see e.g. (Hogstrom et al 1981, Brahme et al 1981). The general theory behind the pencil beam approach is presented in section 2.2.2.1. There are several existing analytical pencil beam algorithms for proton or light ion therapy that are either based on calculated or measured dose kernels or rely on parameterizations of ex-perimental data or MC-calculated doses (Lee et al 1993, Bortfeld 1997, Deasy 1998, Schaffner et al 1999, Krämer et al 2000, Szymanowski et al 2001, Pedroni et al 2005, Kanematsu et al 2006, Kempe and Brahme 2008, Kempe private communication). The use of Monte Carlo techniques for hadron transport has been successful in wide range of applications, such as radiation shielding in space, accelera-tor and medical physics. In radiation therapy, MC techniques have as of yet been too time-consuming and have thus not been applied in individualized treatment planning. The MC approach is on the other hand an exceptional technique for generic dose calculations, especially when dealing with tissue heterogeneities. Moreover, as computer storage and speed continue to in-crease, the MC techniques might potentially be used in the future for indi-vidual treatment plans. Several MC codes exist that are capable of computing dose distributions for light ion radiation therapy. For clinical applications of proton beams, the MC codes MCNPX (Newhauser et al 2005, Herault et al 2007, Newhauser et al 2007), FLUKA (Biaggi et al 1999, Parodi et al 2007), GEANT4 (Paganetti 2004, Paganetti et al 2005), VMCPro (Fippel and Soukup 2004) and an in-house made MC code (Tourovsky et al 2005) have been used. For ion ther-

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apy purposes, mainly with carbon ions, FLUKA (Sommerer et al 2006), PHITS (Niita et al 2006), GEANT4 (Pshenichnov et al 2005, Kase et al 2006) and SHIELD-HIT (Gudowska et al 2004, Geithner et al 2006) have been tested. Prior to introducing the present approach for absorbed dose calculation, a short background on the charged particle interaction processes that are appli-cable to this thesis is presented.

2.1 Physical processes - background A proper description of light ions penetrating matter should account for the following processes of the ions: (i) energy losses, (ii) nuclear reactions, (iii) multiple scattering of both primary ion and fragments and (iv) energy strag-gling. (i) Energy losses The stopping power of a medium reflects the slowing down process of a penetrating heavy charged particle and is defined as the average energy loss of a particle per unit path length. The ionization density increases along the path of an ion and reaches its maximum in the Bragg peak. The main contribution to the stopping power is due to the electronic stopping caused by inelastic collisions between the incident particle and the target electrons. These collisions lead to ionizations and excitations, where the major part of the energy loss (60-80%) ends up in the ionization of the target atoms. In the ionization process, the target atoms emit high- and low-energy electrons, called -rays, which form the track core around the particle trajectory. δ Nuclear stopping power is caused by elastic collisions between the incident particle and the target nuclei and increases with increasing particle mass. For heavy charged particles with energies in the range of interest for therapeutic applications (up to 500 MeV/u), the effect of these elastic collisions, leading to recoil of the target atoms, is relevant only at low particle energies (∼10 keV/u) and results mainly in small deflections of the particle. (ii) Nuclear reactions Elastic nuclear collisions between a particle penetrating a medium and target nuclei are mainly responsible for the changes of the particle trajectory (scat-tering processes). In a nuclear inelastic collision between a primary particle penetrating a me-dium and the target nuclei, fragmentation processes of the particle itself

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(beam-produced fragments) (Schardt et al 1996, Gunzert-Marx et al 2004) or of the target nuclei (target fragments) can occur. In ion therapy, beam frag-mentation processes of high Z primary particles result in an undesirable dose contribution beyond the Bragg peak, since the beam-produced fragments generally have longer ranges than the primary particle. Simultaneously, the well-defined depth dose distribution is deteriorated due to the broadening of the Bragg peak. The target fragments are generally characterized by low energies and a local energy deposition close to the ion track.

(i.e. 1Z > )

For the energy deposition distal of the Bragg peak, the dose distribution is thus dominated by the energy deposition of the charged fragments. Conse-quently, the fragments modify the physical dose distribution as well as the biological effect and it is essential to estimate both the primary and the sec-ondary particle contribution to the ion radiation field. This presents a prob-lem in TPSs for which the RBE of fragments with 1Z > must be considered in the area around the Bragg peak (Matsufuji et al 2003, Kempe et al 2007). Neglecting this in treatment planning might lead to detrimental effects in and around the target volume. (iii) Multiple scattering Multiple scattering occurs whenever the direction of motion of a particle penetrating a target is changed through collisions with the target atoms. For charged particles, the multiple scattering processes are dominated by elastic collisions in the Coulomb field of the target nuclei (Scott 1963), but inelastic collisions with atomic electrons should also be taken into account (Striganov 2005). The angular deflection of a particle colliding with atoms of a medium can be described by the mass scattering power. It has been shown in the first ap-proximation that the mean square angle of scattering increases linearly with the thickness of the absorber (Rossi 1952). The mass scattering power, T ρ , can thus be defined as the increase in the mean square angle of scattering,

2dθ , per unit mass thickness traversed, dzρ (Brahme 1972, ICRU 1984). The following equation for the mass scattering power, T ρ , is given by ICRU (1984) for electrons and somewhat differently by Rossi (1952) for heavy charged particles

( )

2 22

2 2, 2

121 1

4 lnd 1 1 1d 1A e P eff

A

m mMTN r Z

Zz M μ μ

θ θθ θβ

θ πρ ρ τ

−

= =⎧ ⎫⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎪ ⎪⎢ ⎥ ⎢ ⎥+ − + +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎨ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥+⎝ ⎠ ⎪ ⎪⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦⎩ ⎭

(2)

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where AN is Avogadro’s number, is the classical electron radius, er MZ and

AM are the atomic number and the molar mass of the medium, respectively, is the ratio of the kinetic to the rest energy of the incident particle, τ β is

the velocity of the particle normalized to the speed of light, is the cut-off angle due to the finite size of the nucleus and

mθμθ is the screening angle due

to the screening of the nucleus by orbital electrons. The effective charge of the incident particle, ,P effZ , is approximated by the Barkas formula

( )( )2/3, 1 exp 125P eff P PZ Z Zβ −= − − , (3)

where PZ is the charge of the particle. In the same way as the incident particle, the beam-produced fragments will undergo elastic scattering with the target nuclei. Heavy beam fragments with atomic number 2Z > , generally scatter through small angles, whereas the scattering of lighter beam fragments of 2Z ≤ results in a larger angle scat-tering which broadens the beam. The larger angle scattering, due to atomic processes during beam penetration of light fragments, should be distin-guished from the large-angle intra-nuclear scattering due to fragmentation reactions of the incident particle. Since the intra-nuclear Fermi motion is taken into account for both nucleus-projectile and nucleus-target interac-tions, and also because nucleon-nucleon scattering possesses its own angular distribution, all the products of these interactions exhibit certain angular deviations relative to the incident particle direction. Fast beam-produced secondaries are collimated mainly in the forward direction, but can also have a noticeable angular spread. Target-produced secondaries on the other hand, have a much wider distribution, but as target fragments in general have low energies and deposit their energy close to the ion track, they will conse-quently not influence the beam width appreciably compared to the beam-produced fragments. (iv) Energy straggling The slowing down processes of a charged particle penetrating a medium involve individual interactions which are characterized by a distribution of probable energy transfers. The energy loss of a charged particle is conse-quently a stochastic process and the local value will fluctuate around the expected mean value along the track of the particle. This statistical fluctua-tion in the energy-loss process is known as energy loss straggling, which describes the spread of the energy distribution and varies with the energy of

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the incident particle. Landau (1944) solved the transport equation for energy loss straggling and derived the Bothe-Landau formula for the energy-loss spectrum for the straggling function and the most probable energy loss. This formula was later extended by Vavilov (1957), Maccabee et al (1968), Me-jaddem et al (2001) and others. For thin targets, the straggling distribution is usually described by a skewed Gaussian according to the Vavilov distribu-tion (Vavilov 1957). The straggling distribution changes gradually from a skewed distribution to an almost symmetrical Gaussian as the path length increases. This is due to the interaction of many random processes according to the central limit theorem (which roughly states that data which are influ-enced by many small and unrelated random effects are approximately nor-mally distributed, i.e. following a Gaussian distribution). Thus, for thick targets, the energy-loss spectrum is Gaussian-shaped and centered on the initial beam direction. The energy loss straggling process results in a range distribution of the parti-cles that is different from the range predicted by the continuous slowing down approximation (CSDA). The range straggling for particles of the same initial energy is defined as the fluctuation in the true path length. This fluc-tuation, which is especially significant for lighter particles of 2Z ≤ , will reduce the height and increase the longitudinal width of the Bragg peak and this is of prime importance when treating a tumor close to an organ at risk in front of or in the distal part of a tumor. Further details on energy loss and range straggling can be found in Paper I.

2.2 Dose calculation methods in the present study For many clinical treatment planning purposes, the combination of fast ana-lytical calculations and accurate MC codes has shown to be a very practical approach for absorbed dose calculations and has been explored by many authors for proton therapy (Petti 1992, Hong et al 1996, Carlsson et al 1997, Sandison et al 1997, Ciangaru et al 2005, Soukup et al 2005, Kimstrand et al 2007). The following sections present the analytical pencil beam model and the MC code SHIELD-HIT07 that are combined in the developed analytical pencil beam approach for dose kernel calculations of light ions.

2.2.1 The Monte Carlo code SHIELD-HIT07 The MC code SHIELD-HIT07 (Gudowska et al 2004, Geithner et al 2006) simulates the interactions of hadrons and atomic nuclei of arbitrary charge and mass number with complex extended targets in a wide energy range, from 2 GeV/u down to 25 keV/u, and to thermal energies in the case of neu-trons. The SHIELD-HIT07 code extends the original hadron transport code

12

SHIELD (Dementyev and Sobolevsky 1997). SHIELD-HIT07 describes inelastic nuclear reactions using the Russian nuclear models, which give a detailed description of all the stages of hadron–nucleus and nucleus–nucleus inelastic interaction, such as fast intra-nuclear cascade, pre-equilibrium de-cay and, finally, equilibrated de-excitation of residual nuclei. The Russian models are described elsewhere (Botvina et al 1997, Dementyev and Sobo-levsky 1997, Sobolevsky 2001, Gudowska et al 2004, and references therein). To simulate transport of hadrons and nuclear fragments in matter, the nu-clear total cross sections and inelastic cross sections (i.e. the reaction cross sections) of hadron–nucleus and nucleus–nucleus interactions are required. The SHIELD/SHIELD-HIT07 codes use the well established parameteriza-tions of these cross sections based upon fitting the optical model of nuclear reactions (Barashenkov 1993, Barashenkov and Polanski 1994) to the repre-sentative set of experimental data. For all generations of secondary frag-ments and for both types of interaction (inelastic and elastic), the SHIELD/SHIELD-HIT07 calculations allow for the initial angular distribu-tion after intra-nuclear reactions. The code provides spatial distributions of the energy deposition of primary particles, nuclear fragments, recoil nuclei and charged secondary particles from neutron interactions. Also, the fluence differential in energy, both of the primary ions and fragments, including higher order generations of all particles is defined at selected zones in the simulated geometry. In the present work, the ionization energy loss of heavy charged particles in tissue equivalent media is calculated using the data from ICRU (1993) and ICRU (2005) and fluctuations of energy loss are assumed to be Gaussian-distributed. The Landau as well as the Vavilov models of energy loss fluc-tuations are also included in SHIELD-HIT07, but are not applied in the pre-sent studies. For multiple Coulomb scattering processes, two models are implemented in SHIELD-HIT07: Gaussian (the so-called Fermi distribution) and Molière. The Fermi distribution results in the Gaussian approximation for small scattering angles, whereas the Molière theory is more general. For heavy charged particles the Molière distribution coincides with Gaussian distribution for small angles, but shows wider tails for larger angles. SHIELD-HIT07 includes the codes for simulation of Molière scattering from the GEANT 321 package, which can be freely downloaded from the CERN program library (CERNwebpage). Detailed description of the Molière codes may be found in the document “CERNwriteup 1994” from the mentioned library. For the present purposes, we run SHIELD-HIT with the Molière multiple scattering model.

13

SHIELD/SHIELD-HIT07 are universal MC codes open to applications in different fields, ranging from space research via radiation shielding for ac-celerators to simulation of the passage of ions through tissue. For the latter applications, the most important is to emphasize that SHIELD-HIT07 accu-rately simulates the ionization/excitation loss of energy, energy loss strag-gling and multiple scattering for the primary and secondary particles. The SHIELD/SHIELD-HIT07 codes have shown good agreement with available experimental data (Dementyev and Sobolevsky 1999, Gudowska et al 2004, Gudowska and Sobolevsky 2006, Geithner et al 2006, Heilbronn et al 2006). Figure 2(a) shows SHIELD-HIT calculations of the relative depth dose dis-tribution and figure 2(b) the normalized fluence of the primary particles and fragments with 1Z = , and for a 279 MeV/u 12C6+ ion beam transported in a PMMA phantom (Gudowska and Sobolevsky 2006). The SHIELD-HIT07 results show good agreement with experimental data from Matsufuji et al (2003). For the production of heavier secondaries ( ) , benchmarking with experimental data shows larger discrepancies of 10–30% between calculations and measured fragment yield (Gudowska et al 2004, Geithner et al 2006). The statistical uncertainties in SHIELD-HIT07 calcula-tions of primary and secondary particle doses are typically less than 1%. Systematic uncertainties in these calculations contain uncertainties mainly in the cross section data for inelastic nuclear interactions in the order of 10 – 40 % (Gudowska et al 2004).

2Z = 6Z =

2Z >

Figure 2. Experimental data and SHIELD-HIT calculations of 279 MeV/u 12C6+ ions in PMMA (Gudowska and Sobolevsky 2006); (a): the relative depth dose distribu-tion calculated by SHIELD-HIT07 (solid line) compared with experimental data (full squares) and (b): the normalized fluence of the primary particles and fragments with , and calculated by SHIELD-HIT07 (lines with open symbols) compared with experiments (full symbols).

1Z = 2Z = 6Z =

14

The rationale for using SHIELD-HIT07 data in this thesis is threefold: • First, the SHIELD-HIT07 calculated integral dose distributions, without

energy loss straggling and multiple scattering processes, but with nuclear processes included, are used as input into the pencil beam model. Also, SHIELD-HIT07 provides the contribution to the dose from all the pro-duced secondaries.

• Second, full SHIELD-HIT07 runs with all relevant physical processes included were performed in order to obtain the transverse and longitudi-nal dose distributions of all produced secondaries needed for the extrac-tion of the lateral deflection parameters in the analytical pencil beam model.

• The third motivation for using SHIELD-HIT07 is to compare the results of the analytically calculated dose distributions with the corresponding distributions obtained with SHIELD-HIT07.

2.2.2 The analytical pencil beam model Most of the current analytical models for ion dose calculation have been developed for specific ion species and/or a specific beam line and generic algorithms are scarce. The goal of the present study is to develop a fast and accurate “all-round” analytical technique to produce physical pencil beam dose distributions of light ions, of variable energies, transported in tissue equivalent materials. The following section will briefly describe the analyti-cal method used in this thesis and additional details can be found in Papers I and II. The presented analytical method uses a pencil beam algorithm with multiple scattering and energy loss straggling processes accounted for analytically and the planar integral depth dose distributions (i.e. the depth dose in an infinite medium from energy depositions in a lateral plane) derived from the MC code SHIELD-HIT07. The contribution to the dose from fragmentation processes is not presently included in the analytical algorithm. Instead, the effect of nuclear reactions (both elastic and inelastic) is included in the SHIELD-HIT07 calculations of the planar integral dose distributions. The calculations of dose distributions are in this work based on energy deposition in water, which is assumed to be a suitable tissue equivalent medium. Stud-ies of the dose deposition in other media, such as air and PMMA, have also been performed (see Paper III). The stopping power for protons and 4He2+ ions are taken from ICRU (1993) and for heavier ions from ICRU (2005).

15

2.2.2.1 Definition of a pencil beam

A broad beam emerging from an accelerator or arbitrary radiation source can be considered as composed of a superposition of identical narrow pencil beams (Brahme 1975, Brahme et al 1981, Hogstrom et al 1981). A pencil beam is a mono-directional (i.e. moving in one direction) beam of mono-energetic particles, originating from a point source, passing through an in-finitesimal area x yδ δ , as illustrated in figure 3(a). The particles in this nar-row circular symmetric beam have an average angular divergence . The angles

θxθ and are the projections of the angle on the x- and y-axis,

respectively, and yθ θ

2 2 2x yθ θ θ+= . The radial divergence of the beam is

, the mean square radial spread 2 2r x y2= + 2 2 2y= +r x , and the mean co-variance of the radial and angular spreads is x yr x yθ θ θ= + . A cylindrical coordinate system, used to describe the lateral and angular distribution of particles, with the beam incident along the z-direction, is given in figure 3(b).

θ x

� �δ δ

����������

����� �������

� �

(a) (b)

θx

z

θy

θ θ

r

u r

y

x Figure 3. (a) Schematic representation in the ( )x z−

θ

-plane of a particle beam inci-dent on a patient (modified from Hogstrom et al (1981) ). (b) The coordinates of the particle are and the direction of motion is . The variable u is an integra-tion variable along the z-axis.

( , ,r x y z= )

))

tion

Equation (4) shows how the dose distribution, , of a point-

monodirectional , mono-energetic pencil beam can, for a homogene-ous medium, be factored out as a product of two parts, one of which is a planar integral dose distribu , ( )

(pm ,D r z

(pm

D z , and the other is a depth dependent

16

radial pencil beam profile, a spread function, , defined in the next section

( ),S r z

( ) ( ) ( )pm , ,D r z D z S r z= . (4)

The planar integral dose distribution, ( )D z , is sometimes also called the broad beam central-axis depth dose and is the distribution of absorbed dose in an infinitely broad plane-parallel beam, in emi-

nar integral dose is recovered according to ( ) ( )pm0

, 2 d

cident normally on a sinfinite medium. By integrating the pe dose over all radii, the pla-ncil beam

D z D r z r rπ∞

= ∫ . In this

ork, the MC calculated planar integralwto

depth ibutions are required account for the contribution to the dose fro nuclear reactions as well as

eous slab of semi-infinite extension, provided the particles are aussian distributed in both coordi mi 1940,

dose distrm

nate and direction space (Fer

for the secondary ion transport.

2.2.2.2 Scattering of primary and secondary ions

The well-known Boltzmann equation describes the transport of charged par-ticles in matter and numerical solutions to this equation with various ap-proximations exist (Bethe et al 1938). Under the assumption that only a small portion of energy is transferred from the incident charged particle to the target nuclei or target electrons, the CSDA can be used. Assuming also that the incident particle will scatter through small-angles, the Fokker-Planck solution to the Boltzmann equation (BFP) is obtained. Enrique Fermi derived the first analytical solution to the BFP (Fermi 1940) and Leonard Eyges later accounted for the energy losses within the CSDA approximation (Eyges 1948). Also, Chen Ning Yang added the equation for the path length distri-bution (Yang 1951). This resulted in the Fermi-Eyges (FE) approximation, which is valid for a particle beam of any radial and angular spread incident on a homogenGEyges 1948). The multiple scattering is in this work based on a generalization of the Fermi-Eyges (GFE) theory for small-angle multiple scattering ( )sinθ θ≈ of charged particles (see Paper I). Application of the GFE theory for calculating the dose distributions of electrons was initiated in the seventies (Brahme 1975) and the early eighties (Hogstrom et al 1981, Brahme et al 1981, Lax et al 1983, ICRU 1984, Lax and Brahme 1985, Ahnesjö et al 1992). The colli-sions of incident electrons result in large angle deflections, whereas heavier charged particles will, due to their large mass (relative to the electron mass),

17

be deflected through very small angles in a narrow forward cone ( )0θ ≈ (Bichsel 1968). This indicates that the small-angle multiple scattering ap-proximation, with Gaussian distributions in the lateral coordinates, can even more accurately describe the radial distribution of light ions than that of electrons due to the low multiple scattering of the former. It has been shown by several other authors that the GFE theory allows for an accurate evalua-tion of spatial dose distributions for protons (Petti 1992, Russell et

ong et al 1996, Sandison et al 1997, Carlsson et al 1997, Sandatsu et al 2008).

al 1995, ison and H

Chvetsov 2000, Kanematsu et al 2006, Kanem

In the present approach, a spread function, ( ) ( )2 2, ,S r z S x y z≡ + , based

e GFE theory and applied for light ions, is introduced and a pure mono-chromat am is conceived as entering the wat

0z = . T n of a rotationa ly symmetric beam is norm

on thic primary ion be er phhe spread functio l

at each

antom at alized

so that ( )0

, 2 d 1S r z r rπ∞

=∫ , depth z , and is defined as

( )( ) ( )

2

2 2

1, exp rS r zr z r zπ

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠with

, (5)

( ) ( ) ( ) (2 0z r= + ) ( ) ( )22 2 2

00 2 0 -

zr r z z z u T u duθ θ+ + ∫ , (6)

where ( )2 0r is the initial ( )0z = mean square radial spread of the beam,

( )2 0θ is the initial mean square angular spread and ( )0rθ is the initial covariance of the radial and angular distributions (the bar indicates the stan-dard average over collisions). The relationship between the root mean square (rms) radius, also called the 1/e radius, and the standard deviati

radial distribution is

on of the 2

1/e rr r σ= = . The mean square angle and the covari-ance in equation (6) are respectively given by

( ) ( ) ( )2 2

00

zz Tθ θ= + ∫ u du , (7)

and

( ) ( ) ( ) ) ( )2

00 0r z r z u T u duθ θ θ= + + . (8) ( -

zz∫

The linear scattering power, ( )T u , is calculated using equation (2) and u is an integration variable along the z -axis. During the development of the presented algorithm, three different subse-quent approaches have been applied in an attempt to obtain improved lateral

18

dose profiles: (i) the total dose contribution from primary and secondary ions is approximated by the multiple scattering of the primary ions (see Paper I), (ii) the dose contributions from primary and secondary ions are separated and the multiple scattering processes for each ion are calculated separately (see Paper II) and (iii) parameterization of the beam spread due to secondary articles, i.e. the wider initial spread of the secondary particles is accounted

). pfor by adjusting the deflection parameters in the GFE theory (see Paper II (i) Total dose contribution from primary and secondary ions The scattering processes of both the primary ions and the beam-produced fragments are described by the GFE theory, which is valid for small angle scattering. The scattering of the primary ions is well described by this method (see Paper I), but since the lighter secondary particles with 1Z = will suffer from larger angle scattering (Matsufuji et al 2005), the GFE the-ory is not fully sufficien r a proper description of the lateral scattering of the fragments. This is evident when, in this first approximation, the total mass scattering power,

t fo

T ρ , for all transported particles (primary and sec-ondary) is approximated by the mass scattering power for the primaries only. This approach will underestimate the total mass scattering power. The ana-lytical model results of absorbed dose distributions of ions will consequently how an overestimation of dose on the central-axis compared to the corre-

a modi

are calculated separately tion modifies thesion (equation (4)) as

,

ssponding results by the Monte Carlo code SHIELD-HIT07 (cf. figure 4). (ii) Separation of dose contributions from primary and secon-dary ions One approach to solve the above mentioned problem is to evaluate the small-angle scattering processes for primary and heavy secondary ions separately. A more correct estimation of the scattering processes is obtained in -fied analytical model (described in Paper II), in which the spread functions of the primary particles, ( ) , and of the ith secondary ions, ( )P ,S r z

. This separaS, ,iS r z ,

pencil beam expres-

( ) ( ) ( ) ( ) ( )pm P P S, S,, , i ii

D r z D z S r z D z S r z= +∑ , (9)

with

( )( ) ( )

2

P 2 2P P

1, exp rS r zr z r zπ

⎛ ⎞= −⎜ ⎟⎜ ⎟

⎝ ⎠, (10)

and

( )( ) ( )

2

S, 2 2S, S,

1, expii i

rS r zr z r zπ

⎛ ⎞= ⎜− ⎟

⎜ ⎟⎝ ⎠

. (11)

19

The quantities ( )PD z and ( )S,iD z are the planar integral energy deposition distributions, of the primary and the ith secon ions, respectively. The

square radial spread of the primaries, ( )Pr z , and the seco ,

0 0i

this es on, the linear scattering powers (equation (2)) for the primary ions, ( )PT u , as well as for each of the secondaries, ( )S,iT u , are calculated separately. The mean energy at every depth of each of the i secondaries is then needed as input and is in this

dary mean ndaries

tion (6). The radial distributions,

work evaluated by the fluence weighted ean energy according to

2

dr r∞ ∞

∫ ∫

( )2S,ir z

and S,i

t

, are calculated using eq

, are normaliz

ua

ed as

( )P ,S r z

d 1r r= = . In ( ),S r z

imati

( ) ( )P S,, 2 , 2S r z S r zπ π

m

( )( )

d

dE

E

E z EE

z E

Φ=

Φ∫∫

, (12)

HIELD-HIT07 track length fluence data for every type of secondary ion.

As can be seen in figure 4, for this beam

for which the fluence differential in energy, ( )E zΦ , is extracted from S Figure 4 displays the total energy deposition on the central-axis of 391 MeV/u 12C6+ ions in water, calculated analytically and with SHIELD-HIT07. In this thesis, the SHIELD-HIT07 results are considered a benchmark. The analytical results from method (i), described above, without separation of the spread functions, are compared with the modified analytical model results, where the spread functions of the primary and secondary particles are sepa-rated (method (ii)), as well as with results from method (iii) described below.

with divergence parameters

( )2P 0 0.0625r = cm2, ( )2

P 0 0θ = and ( )P 0 0rθ = , method (i) gives a too high dose on the central-axis when compared to the SHIELD-HIT07 results. The improvement by accounting for the multiple scattering of secondaries can be een in the plateau region and beyond the Bragg peak.

s

20

0 5 10 15 20 25 300

200

400

600

800

SHIELD-HIT method (i) method (ii) method (iii)

Ene

rgy

depo

sitio

n (M

eV/g

)

Depth (cm) Figure 4. Analytical and SHIELD-HIT07 calculations of the total energy deposition on the central-axis of 391 MeV/u 12C6+ ions in water. Solid histograms are SHIELD-HIT07 results, long dashed curve are results of method (i), i.e. without separation of the spread functions for primary and secondary particles, the dash-dotted curve dis-plays the results from method (ii), where the dose contributions from primary ions and fragments are separated, and the dash-dot-dotted curve shows results from method (iii), where the dose contributions are separated and the divergence parame-ters have been adjusted. (iii) Parameterization of the beam spread due to secondary par-ticles The improvement obtained from the method described above is however not enough since the single-Gaussian expression in the GFE theory is not able to correctly estimate the tails of the radial profiles of the secondaries, i.e. the larger angle scattering of light secondary particles will be underestimated. In Paper II, it was shown that the use of a double-Gaussian distribution, as an alternative to the GFE theory, to take this larger angle scattering into account improves the results, especially for fragments with 1Z = . However, it was concluded in Paper II that a detailed description of the radial dose tails would not lead to a significant modification to the final outcome since the contribution from fragments with 1Z = to the total dose in water is of the order of 10%. As an alternative to the double-Gaussian distribution, the single-Gaussian of the GFE theory is proposed to be used with the larger angle scattering of the fragments approximated by estimating the divergence parameters, i.e. the initial radial ( ( )2

S, 0ir ) and angular ( ( )2S, 0iθ ) spreads and their covariance

21

( ( )S, 0irθ ), for each of the secondary ions (see Paper II). This is done by adjusting the parameters through a fitting procedure. The fitting is done through the Levenberg-Marquardt least square minimization of the variance (Press et al 1986) between the analytical dose distribution defined by equa-tion (4) and the results from a full (i.e. with all relevant physical processes included) SHIELD-HIT07 calculation of the dose distribution of each of the secondary ions. Details on the fitting procedure can be found in Paper II. The derived parameters allow for a considerably improved description of the radial distributions of secondary ions at every depth. This approximate cal-culation of the radial spread of secondaries is not aimed at providing a proper description of the scattering processes of fragments, but rather to provide an estimated description of the shape of the radial profiles of the dose contribution due to fragments. An illustration of the improvement in the radial dose profile of secondaries that is achievable with the use of method (iii) is displayed in figure 5, in which the analytical and SHIELD-HIT07 results from a dose distribution calculation of proton fragments produced in a 329 MeV/u 11B5+ ion beam in water are shown. Figure 5(a) shows the analytical results when method (ii) described above is applied. In this case, the deflection parameters of the fragments are the same as for the primary ions, i.e. ( )2

P 0 0.0625r = cm2,

( )2P 0 0θ = and ( )P 0 0rθ = . Figure 5(b) shows the corresponding results from

a full run with SHIELD-HIT07 for a beam of the same size as in the analyti-cal calculations. Here, the wide distribution of the proton fragments close to the phantom surface can be seen. Figure 5(c) shows the improvement that is obtained by applying method (iii) where the deflection parameters have been adjusted through fitting procedures to the SHIELD-HIT07 data, in this case to ( )1

2S, H 0 0.1r + = 5 cm2, ( )1

2S, H 0 0.03θ + = rad2 and ( )1

5S, H 0 1.28 10rθ +

−= ⋅ cm

rad. Figures 6 and 7 show the corresponding results for 4He2+ and 7Li3+ fragments produced in a 329 MeV/u 11B5+ ion beam in water with noticeable improvements obtained by using method (iii). The same can be seen in fig-ures 8 and 9 where the secondary protons and heavier 7Li3+ fragments, pro-duced in a 468 MeV/u 16O8+ primary ion beam, are shown.

22

Figure 5. Energy deposition distribution of secondary protons produced in a 329 MeV/u 11B5+ ion beam in water; (a): analytical results without fitting of the deflec-tion parameters (i.e. method (ii)), (b): SHIELD-HIT07 results and (c): analytical results with adjusted deflection parameters (i.e. method (iii)).

Figure 6. Energy deposition distribution of 4He2+ fragments produced in a 329 MeV/u 11B5+ ion beam in water; (a): analytical results without fitting of the deflec-tion parameters (i.e. method (ii)), (b): SHIELD-HIT07 results and (c): analytical results with adjusted deflection parameters (i.e. method (iii)).

23

Figure 7. Energy deposition distribution of 7Li3+ fragments produced in a 329 MeV/u 11B5+ ion beam in water; (a): analytical results without fitting of the deflec-tion parameters (i.e. method (ii)), (b): SHIELD-HIT07 results and (c): analytical results with adjusted deflection parameters (i.e. method (iii)).

Figure 8. Energy deposition distribution of proton fragments produced in a 468 MeV/u 16O8+ ion beam in water; (a): analytical results without fitting of the deflec-tion parameters (i.e. method (ii)), (b): SHIELD-HIT07 results and (c): analytical results with adjusted deflection parameters (i.e. method (iii)).

24

Figure 9. Energy deposition distribution of 7Li3+fragments produced in a 468 MeV/u 16O8+ ion beam in water; (a): analytical results without fitting of the deflection pa-rameters (i.e. method (ii)), (b): SHIELD-HIT07 results and (c): analytical results with adjusted deflection parameters (i.e. method (iii)).

An example of the SHIELD-HIT07-calculated primary and secondary pencil beam dose distributions, with multiple scattering and energy loss straggling processes included, used in the fitting procedures, are shown in figure 10. In this case, a full run with SHIELD-HIT07 has been performed for a 391 MeV/u 12C6+ ion beam in water, with initial radial size = 2.5 mm. The pencil beam expression in equation (4) is fitted to the shown distributions in order to extract the divergence parameters,

( )0rσ

( )2S, 0ir , ( )0i

2S,θ and ( )S, 0irθ , for

each secondary particle. As can be seen from figure 10, the contribution to the total dose from fragments of 1Z = is very low and, also, these fragments show a wide lateral spread compared to the heavier fragments. This latter characteristic of the lightest fragments presents a difficulty in the fitting pro-cedure of the GFE theory of multiple scattering, since only the small-angle scattering is accounted for in this theory and the fragments under discussion clearly display a larger angle scattering. Nevertheless, the divergence pa-rameters are adjusted for all fragments to approximate the larger-angle scat-tering. The results of the fitting procedure for secondaries, produced by dif-ferent primary ions with a range of 26 cm in water, are shown in figures 11-13, where the parameters extracted from the fitting are plotted against the atomic mass number of the fragments.

25

Figure 10. SHIELD-HIT07 calculations of the dose distributions of primary 12C6+ ions and each secondary particle species produced in a 391 MeV/u 12C6+ ion beam in water.

0 2 4 6 8 10 12 14 16 18 200

0.05

0.10

0.15

0.20

0.25 Primary 1H+

Primary 4He2+

Primary 7Li3+

Primary 11B5+

Primary 12C6+

Primary 16O8+

r S,i2 (0

) (cm

2 )

Atomic mass of the fragment Figure 11. Derived initial mean square radial spreads, ( )2

S, 0ir , of fragments produced from different primary particles with ranges of about 26 cm in water.

26

0 2 4 6 8 10 12 14 16 18 200.0

5.0x10-4

1.0x10-3

1.5x10-3 Primary 1H+

Primary 4He2+

Primary 7Li3+

Primary 11B5+

Primary 12C6+

Primary 16O8+

θ S,i2 (0

) (ra

d2 )

Atomic mass of the fragment

Figure 12. Derived initial mean square angles, ( )2S, 0iθ , of fragments produced from

different primary particles with ranges of about 26 cm in water.

0 2 4 6 8 10 12 14 16 18 20

0.0

5.0x10-3

1.0x10-2

1.5x10-2

2.0x10-2

2.5x10-2

3.0x10-2

3.5x10-2

Primary 1H+

Primary 4He2+

Primary 7Li3+

Primary 9B5+

Primary 12C6+

Primary 16O8+

rθS,

i(0) (

cm ra

d)

Atomic mass of the fragment

Figure 13. Derived initial covariance parameters, ( )S, 0iθr , of fragments produced from different primary particles with ranges of about 26 cm in water.

An advantage of using the GFE theory, instead of for instance a double-Gaussian function, for the description of radial dose distributions, is the in-

27

formation that is acquired concerning the beam transport characteristics by applying the transport equations (6)-(8). By means of the parameters in the GFE theory, together with the mass scattering power, T ρ , transport equa-tions for describing the distribution in angle and position of a charged parti-cle beam penetrating matter are provided. Further improvement of the model would be achieved if universal divergence parameters for the fragments could be obtained. Such improvements would reduce the reliance upon time-consuming MC calculations and, thus, create a fast and accurate technique for calculating physical dose distributions. In figures 11-13, the curves roughly agree and this could indicate that the parameters of the secondaries are rather similar regardless of primary ion. In figure 14, a comparison of the adjusted fitting parameter ( )2

S, 0ir of fragments, produced in primary beams of different energies, is shown. The results for primary protons, 7Li3+ and 12C6+ ions with ranges 10, 20 and 26 cm in water illustrates that a selection of universal divergence parameters for the fragments could be accomplished for primary beams of different energies. The corresponding results from the fitting procedures for the parameters ( )2

S, 0iθ and ( )S, 0irθ are presented in figures 15 and 16. In these figures, similar tendencies of the parameters can be seen. The initial mean square angular spread, ( )2

S, 0iθ , of the fragments, in figure 15, decreases with increasing atomic number of the fragments and, also, as expected, the parameter for a specific fragment increases with de-creasing incident energy (i.e. with shorter range) of the primary particle is seen. The largest spread of the parameters is seen for the lighter fragments

and this is most likely due to the inability of the GFE theory to properly describe the larger widths of the radial spreads of fragments. As mentioned in the previous section, this will however not influence the accu-rate description of the total dose distribution to a large extent. Figure 16 also shows that the largest differences between the extracted parameters are seen for the lightest fragments. The spread of the divergence parameters

( 4Z < )

( )2S, 0iθ

and ( )S,irθ 0 of the lightest fragments could be due to insufficient statistics in the calculation of the dose contribution from these fragments in the beams; hence the difficulty of fitting.

28

0 2 4 6 8 10 120.0

0.1

0.2

0.3

Range 26 cm: Primary 1H+

Primary 7Li3+

Primary 12C6+

Range 20 cm : Primary 1H+

Primary 7Li3+

Primary 12C6+

Range 10 cm : Primary 1H+

Primary 7Li3+

Primary 12C6+

r S,i2 (0

) (cm

2 )

Atomic mass of the fragment Figure 14. Derived initial mean square radial spreads, ( )2

S, 0ir , of fragments produced from primary protons, 7Li3+ and 12C6+ ions with ranges 10, 20 and 26 cm in water.

0 2 4 6 8 10 12

0.000

0.002

0.004

0.006Range 26 cm:

Primary 1H+

Primary 7Li3+

Primary 12C6+

Range 20 cm : Primary 1H+

Primary 7Li3+

Primary 12C6+

Range 10 cm : Primary 1H+

Primary 7Li3+

Primary 12C6+

θ S,i2 (0

) (ra

d2 )

Atomic mass of the fragment Figure 15. Derived initial mean square radial angles, ( )0θ 2

S,i , of fragments produced from primary protons, 7Li3+ and 12C6+ ions with ranges 10, 20 and 26 cm in water.

29

0 2 4 6 8 10 12-0.005

0.000

0.005

0.010

0.015

0.020

0.025

Range 26 cm: Primary 1H+

Primary 7Li3+

Primary 12C6+

Range 20 cm : Primary 1H+

Primary 7Li3+

Primary 12C6+

Range 10 cm : Primary 1H+

Primary 7Li3+

Primary 12C6+

rθS

,i(0) (

cm ra

d)

Atomic mass of the fragment

Figure 16. Derived initial covariance, ( )S, 0irθ , of fragments produced from primary protons, 7Li3+ and 12C6+ ions with ranges 10, 20 and 26 cm in water.

The derived modified analytical model contains several simplifying ap-proximations in order to supply practical descriptions of primary and secon-dary ion dose distributions. For instance, at zero depth , mainly pri-mary ions and only a few back-scattered fragments are present. The values of the GFE parameters

( 0z = )

( )2S,ir z , ( )2

S,i zθ and ( )S,ir zθ should thus be zero at for the produced fragments. However, the reason for assuming non-

zero values of these parameters for fragments, introduced at , is merely to approximate the initial size of the subsequent larger angle scattering (cf. figure 5) and not to describe the physics adequately.

0z =0z =

The presented analytical method is reliant upon the accurate, but fairly time-consuming MC code SHIELD-HIT07 for the calculation of light ion planar integral dose distributions. Improvements concerning the speed of the ana-lytical method could be achieved by establishing a database of planar inte-gral dose distributions of the desired ion species with varying initial ener-gies, traversing different geometries. Such improvements would reduce the reliance upon SHIELD-HIT07 and, thus, create a more flexible technique for calculating physical dose distributions. Currently, the overall time saved by using the analytical model presented in this work, instead of a full SHIELD-

30

HIT07 run, for the calculation of e.g. a 468 MeV/u 16O8+ ion pencil beam dose distribution in a water phantom is of the order of 60%.

2.2.2.3 Energy loss straggling calculations

In the present model, the planar integral energy deposition, where both the energy loss straggling and multiple scattering processes are turned off,

( )0D z

( ,

, is simulated by SHIELD-HIT07. The missing effect of the energy loss straggling is subsequently incorporated by an analytical function,

)F zΔ

( ), in order to derive the straggled planar integral energy deposition,

D z , through the convolution ( ) ( ) ( )0 ,D z D z F z= ⊗ Δz

. Here, Δ is the energy loss affecting a particle traversing a path length . The energy loss and range straggling processes are assumed to be Gaussian-shaped, with variances and , respectively, calculated using the expressions from Paper I and ICRU (1993).

2Eσ 2

zσ

In a first assessment, the range straggling for all particles, i.e. for both pri-mary ions and for beam-produced fragments, were taken into account in the absorbed dose calculations. Tests using the SHIELD-HIT07 code showed that the straggling of the secondaries was negligible, therefore only the straggling of primary ions is accounted for in the analytical model. Figure 17 shows the range straggling parameter, , for various primary ions with different energies (chosen to attain a range of 26 cm in water). For primary protons, 7Li3+ and 12C6+ ions, results for ranges of 10 and 20 cm are included as well.

zσ

As mentioned in section 2.1, the total energy loss in a track segment is a stochastic quantity whose distribution is described as a straggling function. This function is not symmetrical around the mean energy loss, which indi-cates that the most probable energy loss, at which the straggling function has its highest value, is not equal to the mean energy loss. The distribution of energy losses is in its most general form described by binomial statistics depending on the exact number of single collisions that each ion has under-gone, but it is also well described by the Vavilov distribution (Vavilov 1957). This distribution varies with the energy of the particle and is usually described by a skewed Gaussian. When collisions with energy transfers close to the maximum possible value are very frequent the distribution coincides with a pure Gaussian distribution (Mejaddem et al 2001). Also, the strag-gling distribution has been seen to gradually change from a skewed distribu-tion to an almost symmetrical Gaussian as the path length increases. This is due to the interaction of many random processes according to the central

31

limit theorem. The Gaussian probability that a particle traversing a medium will be affected by a certain energy loss is given by e.g. Rossi (1952) and ICRU (1993)

0 2 4 6 8 10 12 14 16 180.00

0.05

0.10

0.15

0.20

0.25

0.30

Range 26.2 cm Range 20 cm Range 10 cm

16O8+12C6+11B5+9Be4+

7Li3+

4He2+

1H+

Ran

ge s

tragg

ling,

σz, (

cm)

Atomic mass of primary ion

Figure 17. Analytically calculated range straggling parameter of primary ions with ranges 10, 20 and 26 cm in water.

2.2.2.4 Total dose distributions

In Paper II, the results of the analytical calculations of pencil beam dose distributions are compared with results of SHIELD-HIT07 calculations for pencil beams of the same radial size and energy. The initial radial size of the incident beam is arbitrarily chosen to be = 2.5 mm in both the SHIELD-HIT07 and the analytical calculations. Comparisons of the analyti-cally calculated pencil beam energy deposition distributions on the central-axis, of 202 MeV protons, 200 MeV/u 4He2+, 234 MeV/u 7Li3+, 329 MeV/u 11B5+, 391 MeV/u 12C6+ and 468 MeV/u 16O8+ ions, with the corresponding SHIELD-HIT07 calculations are shown in figure 18. The results for primary and secondary ions as well as for the total energy deposition distributions of the analytical method and SHIELD-HIT07 are consistent with each other for all ions. Moreover, the analytical results compare favorably to experimental data for proton beams of 155 MeV and 180 MeV, as shown in Paper III.

( )0rσ

32

0 5 10 15 20 25 300

5

10

15

20

25

Ener

gy d

epos

ition

(MeV

/g)

0 5 10 15 20 25 300

20

40

60

80

100

0 5 10 15 20 25 300

50

100

150

200

250

Ener

gy d

epos

ition

(MeV

/g)

0 5 10 15 20 25 300

100

200

300

400

500

600

234 MeV/u 7Li3+

391 MeV/u 12C6+ 468 MeV/u 16O8+

200 MeV/u 4He2+202 MeV 1H+

329 MeV/u 11B5+

0 5 10 15 20 25 300

200

400

600

800

Ener

gy d

epos

ition

(MeV

/g)

Depth (cm)0 5 10 15 20 25 30

0

200

400

600

800

1000

1200

1400

Secondary ion dose

Secondary ion doseSecondary ion dose

Secondary ion dose Secondary ion dose

Total dose

Total doseTotal dose

Total dose

Total dose

Primary ion dose

Primary ion dosePrimary ion dose

Primary ion dosePrimary ion dose

Secondary ion dose

Primary ion dose

Total dose

Depth (cm)

Figure 18. The central-axis distributions of the total energy deposition as well as the energy deposition of primary ions and fragments. The solid histograms show the SHIELD-HIT07 results and dashed lines are the analytical results.

33

2.2.2.5 Model extensions

Broad beams The definition in section 2.2.2.1 of a pencil beam states that the spatial dis-tribution of a beam of any given finite initial radius can be obtained by the superposition of elementary pencil beam dose distributions from point-monodirectional beams. The following formula can be used to generate a broad beam depth dose distribution, ( , ,bb )D x y z , for a rectangular beam with sides 2a and 2b (shown in figure 19)

( ) ( )2 2 2

, ,4bb

D z x a x a y b y bD x y z erf erf erf erfr r r r2

⎡ ⎤⎛ ⎞ ⎛⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎞⎢ ⎥⎜ ⎟ ⎜⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟= ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝

⎟⎠⎣ ⎦

− + − ++ × + , (13)

where erf denotes the error function. Figure 20 present a broad beam repre-sentation of 234 MeV/u 7Li3+ ions in water, with a = b = 2.5 cm, a calculated using the expression for the mean square radius, ( )2r z , of a pencil beam (equation (6)) and the planar integral dose ( )D z calculated with SHIELD-HIT07.

x

y

2b

2a Figure 19. A schematic illustration of the rectangular beam calculated with equation (13).

34

Figure 20. Analytically calculated broad beam energy deposition of a 234 7Li3+ ion beam in water. Heterogeneities Any change of material in the beam path will influence the physical proc-esses described in chapter 2.1 and in heterogeneous parts in the human body, dose perturbations will arise from the different multiple scatter and energy absorption properties of the irradiated tissues. A dose calculation algorithm has to be able to account for these perturbations. The GFE theory is limited in handling the effects of arbitrary finite lateral tissue heteroge-neities due to the semi-infinite slab approximation, but several approaches accounting for the influence of longitudinal heterogeneous media have been introduced and have shown to accurately describe the non-uniform dose distributions in heterogeneous media (e.g. Brahme and Lax 1983, Lax and Brahme 1985, Petti 1992, Hong et al 1996, Schaffner et al 1999, Rus-sell et al 2000, Szymanowski and Oelfke 2002, Ciangaru et al 2005). In Paper III, it has been shown that the presented analytical model can properly calculate the dose distributions, where heterogeneities, such as air and PMMA slabs in the beam line, were taken into account. The effect is estimated by analytical calculation of the mass scattering power of each material in the beam and by SHIELD-HIT07-calculated planar integral energy depositions with the heterogeneities introduced into the MC geome-try.

35

3. Modeling of radiobiological effects of light ions beams

Treatment planning for light ion therapy requires not only accurate knowl-edge of the physical processes of energy deposition but also of the likely response of tumor and normal cells. This necessitates development of ana-lytical or numerical radiobiological models that can properly evaluate the biological response of the tumor as well as of the normal tissues in order to describe the effects of light ion irradiation on cellular survival. The scope of the thesis, besides the calculations of physical absorbed dose, also includes studies of concepts for evaluation of biological effects in cells in vitro after exposure to light ions. Clonogenic cell survival One way of measuring the cellular response to radiation is to quantify the amount of surviving cells after irradiation. Puck and Marcus were the first to publish a clonogenic cell survival curve for mammalian cells in culture after irradiation and they defined survival as the ability to procreate colonies with more than 50 cells within a certain time (Puck and Marcus 1956). The sur-viving fraction is defined as the ratio of the plating efficiencies of treated and untreated cells, where the plating efficiency is the percentage of seeded cells that grow into colonies. The surviving fraction is a solid measure of the bio-logical effect of radiation, but since it is such a prolonged technique, it is difficult to use for individualized treatments. For this purpose, a predictive in vivo assay, i.e. a fast method for determination of an inherent individual radiosensitivity or radioresistivity of patients, is required (Brock 1988, Brahme 2003). Modeling of cell survival Due to the complexity of the cellular and subcellular mechanisms of dam-aged cells, a mathematical model that can properly describe and predict the relationship between physical dose and biological effects becomes very in-tricate. The existing relatively simple models provide approximate descrip-tions of these complex biological effects of ion beams. There are several existing models that can qualitatively describe the shape of a survival curve. These so-called phenomenological (parameter-driven) models are useful in

36

practical applications, but usually they do not aspire to properly explain the biophysical events that lead to biological damage (Hall and Giaccia 2006). This challenge is instead accepted by mechanistic models that are used in basic research to attempt to bring better understanding of the physical, chemical and biological processes that lead to biological damage and cell repair- and death mechanisms (Ottolenghi et al 2001). The aim of the collaborative study, presented in Paper IV, of some of the radiobiological models in use today, was to verify the ability of a few radio-biological models to adequately describe experimentally measured cell sur-vival data after light ion irradiation. The cell survival models used in this joint study are listed in table 1. These models are in essence both mechanis-tic and phenomenological in that they are parameter-driven, i.e. the model parameters are extracted from fitting to experimental cell survival data, but some biophysical significance can also be attributed to the derived parame-ters. The applied radiobiological models can be divided into predictive or descriptive models, where the predictive model is able to predict in vitro biological response and the purely descriptive models try to establish a cor-relation between the extracted model parameters and the biophysical events that take place after exposure to radiation. The parameters of the used models are extracted from fitting procedures to cell survival data of AA human melanoma cells, irradiated in vitro with 60Co γ-rays and with 10B5+ ions of three different LET values. All of the discussed models show fair representations of the experimental data. Deviations from the experimental data of the fitted survival curves show that, in the low dose region, i.e. at high survival levels, the different models used in the study show variable ability to properly describe the initial slope of the survival curve. This results in different RBE values at high survival levels, where small differences in the models can give large variations in RBE. For the high dose region (D > 4 Gy), the models show differences in the ability to describe the exponential shape of cell killing, i.e. where the cell survival curve is a straight line in a log-linear plot. An important aspect of the comparison of the performance and practicality is that the models are inherently different and are thus in essence difficult to compare. For instance, fitting procedures using descriptive models, such as the linear quadratic (LQ) model, the repairable-conditionally repairable (RCR) damage model and the probabilistic two-stage (P2S) model, generally obtain good fits to the experimental data. The predictive model used in the study, i.e. Katz’s track structure theory (TST) model, usually results in somewhat poorer representation of the experimental data, but could be of greater worth in light ion therapy research due to its predictive power, which enables prediction of the cellular response predicted after any high-LET and

37

mixed irradiation. Moreover, the fitting parameters of the descriptive models might contribute to a description and understanding of the cellular processes that are induced by radiation exposure and the fitted parameters may well assist in approximating tumor and normal tissue radiosensitivity. The results from the presented work clearly indicate that further model evaluation of the cell survival, with larger sets of cell survival data for other ion species, LET values and cell lines, is necessary. There are several other cell survival models not included in the joint study. A few examples are the Repair-Misrepair model (Tobias 1985), the lethal lesion/potentially lethal lesion (LPL) model (Curtis 1986), Saturable-repair models (Goodhead 1985) and the Inducible Repair (IR) model (Joiner and Johns 1988). A cell survival model used in treatment planning for 12C6+ ion beam radiotherapy at GSI, is the local effect model (LEM). This model combines principles of microdosimetry and track structure models and is based on the assumption that the biological effect is entirely determined by the spatial local dose distribution inside the cell nucleus, and that there is no principal difference between the action of local dose depositions of γ- or X-rays and that of charged particles (Scholz and Kraft 1996, Elsässer and Scholz 2007). A short presentation of the models included in the study (listed in table 1) will follow. Table 1. List of the radiobiological models applied in Paper IV

Model References

Linear quadratic (LQ) (Sinclair 1966)

Repairable-conditionally repairable (RCR) damage (Persson et al 2002, Lind et al 2003)

Probabilistic two-stage (P2S) (Kundrat et al 2005, Kundrat 2006, Kundrat 2007)

Katz track structure theory model (Katz’s TST model)

(Katz and Sharma 1973, Katz et al 1994)

The linear quadratic (LQ) model is the most common approach for modeling cell survival in vitro today, with parameters and α β being a measure of the amount of lethal and sub-lethal damages, respectively (Sinclair 1966). The model was at first an empirical model to describe the linear-quadratic shape of the survival curve, but a subsequent theories (Kellerer and Rossi 1972, Chadwick and Leenhouts 1973) proposed to explain the biophysical background of the model parameters, and α β , and the model assumes that there are two components of cell killing by radiation. The LQ model has

38

been in extensive use in radiobiology for the last decades and it agrees rea-sonably well with experimental data at clinically relevant doses of around 2 Gy. At dose levels above 8 Gy, the modeled cell survival curve shows a con-tinuous bending which would indicate a more effects per unit dose. This has, however, been shown not to be the case. Rather, the survival curve has an exponential shape at high dose levels. Another disadvantage of the LQ model is the inability to account for low-dose hypersensitivity, which is seen for some cell types at doses below approximately 0.5 Gy. The probabilistic two-stage (P2S) model (Kundrat et al 2005, Kundrat 2006) takes into account DNA damage with two levels of severity. Lethal lesions are assumed to be formed either through a single lesion or the combination of two less severe lesions. The outcome of cellular repair processes is repre-sented explicitly through the corresponding repair success probabilities, which are assumed to be independent of radiation quality. The model en-ables descriptions of a large variety of survival curves. These include e.g. linear and shouldered curves as well as curves showing low dose hypersensi-tivity. The model has shown to be a helpful tool for analysis of radiation response of different cell lines (Hromcikova et al 2006). Furthermore, the model works with single-particle effects, i.e. the model parameters are spe-cific for a given ion and cannot be used directly to predict a cell response to other radiation modalities. The repairable-conditionally reparable (RCR) damage model has been pro-posed to describe the cell survival after low, medium, and high doses (Persson et al 2002, Lind et al 2003). The model is based on the interaction between two Poisson processes with a separation of damages into two types of lesions: a potentially repairable but also potentially lethal lesion if non-repaired or misrepaired and a conditionally repairable lesion. Thus, there is a possibility that damage is repaired, but damage can also be lethal if not re-paired correctly. The induction of damages is expressed with a simple linear-exponential cell survival expression, where the first term describes the frac-tion of cells that have not been damaged and the second term is the fraction of cells that have been damaged and subsequently repaired In addition to the model comparison in Paper IV, more detailed studies of radiobiological effects using the Katz’s track structure theory (TST) model are presented in Papers V-VII. The Katz’s TST developed by Katz and co-workers (see Katz et al 1971) is a parametric phenomenological model able to quantitatively describe and predict the cell survival in vitro after ion irra-diation, using four model parameters that are extracted from fits to measured cell survival data (Katz et al 1994). The model is built on physical rather than biological assumptions and views the sensitive regions in the cell as “detectors” or specific targets for radiation damage so that the cell survival

39

after irradiation is related to the number of targets inactivated. It assumes that the response, per dose, after exposure to all types of ionizing radiation is the same and that the differences in response between sparsely and densely ionizing radiation lies in the radial distribution of the produced -rays. Av-erage values of the dose distributions over the whole cell nucleus is utilized and the response of irradiated cells in vitro is calculated using cellular pa-rameters specific for different types of cells and the biological endpoint ana-lyzed. To calculate the dose response of cells in vitro, the “detector” is de-scribed by four parameters , , and and the “radiation field” is described by the ion charge,

δ

m 0D 0σ κPZ , the ion fluence, F , the ion speed, β , and

the track segment LET. For a given cell type, cellular parameters are fitted simultaneously to the available data set of track segment in vitro survival curves together with a reference 60Co dose response curve. Once the four parameters are fitted, the response of that specific cell type can be predicted for any high-LET and mixed irradiation, in any track segment. The Katz’s TST model does not account for the energy loss straggling of the incident particles. In addition, the model neglects the loss of primary parti-cles with depth due to the production of secondaries. In further develop-ments of this approach, these processes will be evaluated. The results in Pa-pers V-VII show that, in spite of the mentioned simplifications, the model can accurately calculate the cell survival after ion exposure.

40

4. Conclusions and outlook

An analytical model based upon the generalized Fermi-Eyges (GFE) theory is introduced. This model, which exploits the accuracy of the Monte Carlo (MC) code SHIELD-HIT07 in combination with the efficiency of an analyti-cal model, permits a reliable and fast calculation of the spatial dose distribu-tions for light ion particles. The model evaluates the multiple scattering processes of light ion beams for both primary and secondary ions with a simultaneous analytical description of the energy loss straggling processes of the primaries. By means of three divergence parameters, the GFE theory provides beam transport equations for description of the distribution in angle and position of a charged particle beam traversing a medium. In the present work, accurate descriptions of the radial dose profiles of ions transported in tissue equiva-lent media are obtained through two model modifications. These provide estimation of the multiple scattering as well as of the three depth-independent divergence parameters of the GFE theory, explicitly for frag-ments. Analysis of the divergence parameter dependence on primary ion species and initial energy indicate a possible selection of universal diver-gence parameters for the fragments. Such a set of parameters would reduce the reliance upon time-consuming MC calculations which would increase the speed of the technique. At present, about 60% of the overall time can be saved by using the modified analytical model, instead of a full SHIELD-HIT07 run, for the calculation of e.g. a 468 MeV/u 16O8+ ion pencil beam dose distribution in a water phantom. The results of the analytical pencil beam model show good agreement with both experimental and SHIELD-HIT07-calculated radial dose distributions, also for heterogeneous media. Although the analytical model contains sev-eral simplifications regarding both the physics of ion interactions with matter and the parameterization procedure, it nevertheless provides a useful and accurate tool of clinical relevance for the calculation of light ion absorbed dose distributions. In addition to the physical beam characterization, a study of some current radiobiological models, used for evaluation of biological effects of cells in vitro after exposure to light ions, is performed. A comparison between the models show results of varying accuracy, depending on the levels of dose

41

and linear energy transfer (LET), when compared to experimental data. The Katz cellular track structure theory in conjunction with a simple track-segment representation of ion beam transport model is also used to predict the RBE-LET relationships and the depth distributions of cell survival for two different cell lines in vitro. This simple approach shows good agreement with experimental cell survival data after exposure to ions of low entrance fluences. In further analysis of the radiobiological models, using larger sets of cell survival data for various ion species, the potential of the models to assist biological optimization procedures in a TPS may be explored. Although a model description of the complex biological effects of ion beams is ap-proximate, a simple radiobiological model may still be useful when analyz-ing the biological response to radiation.

42

Summary of papers

The following is a short summary of the Papers included in the thesis. Paper I The aim of this study was to use an analytical method for fast calculation of the multiple scattering of light ions in water. For this purpose, the general-ized Fermi-Eyges (GFE) theory was used and described in some detail. The range straggling was also evaluated analytically, using straggling concepts from ICRU (1993). Experimental verification of the calculation of the mean square radial spread was made for primary protons and primary carbon ions. The paper intended to act as an introduction to a follow-up study where the multiple scattering and range straggling concepts were to be utilized for the calculation of the light ion pencil beam total dose distributions (Paper II). Paper II In Paper II, a modified analytical model was described. This model separates the primary and secondary ion dose contributions in a beam in order to de-rive a description of the scattering of the light and heavy secondaries in a beam. Analytically calculated light ion pencil beam total dose distributions in water were presented. The scattering properties of the beam-produced fragments were analyzed and a parameterization of the beam divergence parameters in the GFE theory was performed through a fitting procedure. The prospect of finding universal divergence parameters approximately de-scribing the radial profiles of beam-produced secondaries was discussed. A good agreement was found for the radial profiles of protons in water be-tween the analytical results and SHIELD-HIT07 results, as well as with ex-perimental data. Paper III In Paper III, further experimental verification of the modified analytical model introduced in Paper II, was presented. It was shown that the model can account for heterogeneities, such as air and PMMA, in the beam line. The non-uniform dose distributions were estimated by analytical calcula-tion of the mass scattering power of each material in the beam line and

43

accounting for the heterogeneities in the SHIELD-HIT07-calculated planar integral energy depositions. A good correlation was obtained between the analytically and SHIELD-HIT07-calculated radial profiles of 155 MeV and 180 MeV protons and the corresponding experimental data. Paper IV In Paper IV, a few of the currently used radiobiological models, applicable in ion therapy, were studied. The biological optimization procedure in a treatment planning system (TPS) requires accurate radiobiological input. For this purpose, a proper estimation of the tissue response of radiation exposure is required. There is a shortage of proper comparisons of existing radiobio-logical models and, thus, this paper aimed at finding possible correlations or disagreements of the applied radiobiological models. The hope was to facili-tate the development towards more robust and precise models by means of a working collaboration between different research groups. To accomplish this, large sets of cell survival data for other ion species and different LET values will be required. The models used to analyze the data in the study were Katz’s track structure theory (TST) model, the linear quadratic (LQ) model, the probabilistic two-stage (P2S) model and the repairable-conditionally repairable (RCR) damage model. The model parameters were estimated from the experimental data by fitting procedures. The cell line used as a benchmark in these studies was the human melanoma cell line AA (primary tumor, Radiumhemmet, Stockholm, Sweden), irradiated under aerobic conditions by 60Co and 10B5+ ions of dif-ferent values of LET (40, 80 and 160 keV μm ). Paper V A simple beam model based on the Katz’s TST model of Katz and co-workers was used to predict the RBE-LET relationship for two different in vitro cell lines (V79 cells and the human melanoma cell line AA) irradiated with 14N7+ and 10B5+ ions. The model results were suggested to offer some guidance in understanding the complicated relations between radiobiological and physical parameters. It was shown that the range straggling will domi-nate the biological effects at the distal end of the lightest ions (protons and He2+) but that the track-segment concept and lack of range straggling as-sumptions could perhaps be acceptable for heavier ions. It was further dis-cussed that the model can be improved by incorporating the effects of the range straggling and multiple scattering processes of the particles, facilitat-ing a possible combination with one of the existing analytical or Monte Carlo codes, that are used for the calculation of physical dose distributions, in order to provide radiobiological data for a TPS.

44

Paper VI The Katz’s TST model introduced in Paper V was used in this work to pre-dict the depth distribution of cell survival of two different in vitro cell lines (V79 cells and the human melanoma cell line AA) irradiated with 12C6+ ions. The paper stated that a more correct method to convert physical dose to bio-logically effective dose, instead of using RBE as a weighting factor, would be to use the survival depth distributions. It was further argued that this could facilitate comparisons of the therapeutical advantages with light ion and conventional radiation therapy. The cell survival in the target volume, which is one of the main objects of interest in radiation therapy, would then be the common denominator between the different modalities. Paper VII Paper VII further develops the concepts introduced in Papers V and VI of Katz’s TST model for light ion radiation therapy. Model parameters repre-senting V79 and CHO cell cultures in vitro were extracted from the pub-lished data. By using a simple track-segment representation of an ion beam transported in water and applying the derived model parameters, the cell survival depth distributions or cell survival vs. dose, at different positions in a water phantom, could be evaluated and compared with published experi-mental data. Further evidence of the predictive power of the model was illus-trated by the good agreement between track-segment calculations and ex-periments obtained for a mixed-field irradiation of V79 cells. The results demonstrated possible application of the Katz’s TST model for evaluation of cell survival in the mixed-radiation field present in a spread out Bragg peak. It was discussed that the applied model approximation of disregarding pri-mary ion fluence loss has less influence on the cell survival calculations for beams of lower initial energies and lower entrance particle fluences. In fur-ther developments of this type of calculation, apart from generating the ap-propriate range straggling calculation to reproduce the correct Bragg peak, the losses of the primary particle fluences due to production of secondary particles will be evaluated. Calculations using the analytical TST model are fast, so the presented ap-proach could, after further improvement, have potential application in ion therapy planning. This enables evaluation and discussion of the depth distri-bution of cell survival in vitro, in a one-dimensional, single-particle track-segment approach. It is suggested that the predictive capability of the model may be useful in analyzing clinical results and in designing new strategies for ion beam radiation therapy.

45

Acknowledgements

Many are the people who have made my time as a PhD student an adven-ture. I want to give my sincere thanks, and to express my gratitude, to all friends and colleagues that have shown such compassion and consideration when needed the most. I’m indebted to everyone who made a contribution to the completion of this thesis – all those poor souls I forced to read pages and pages of manuscripts – and, naturally, to all my coauthors for your efforts and for interesting and rewarding collaboration. I wish to thank the Uppsala and Catana proton therapy groups for their work and interest in our joint projects.

Furthermore, I am especially grateful to the following people: Professor Anders Brahme, a visionary like no other, with an endless

stream of inspiring ideas. Thank you for giving me the opportunity to start and the trust to finish this project.

Docent Irena Gudowska, my supervisor, I want to give you a warm thank

you for all the time you didn’t have and you so generously gave me anyway, 24-seven. Your deep knowledge of physics, your attention to detail and your thoroughness have been a source of inspiration and education. Thanks for making me dare to go out into the deep waters!

Docent Bo Nilsson, thank you for inspiring teaching and for constantly

looking out for me. Your support kept me on track during all these years. Professor Dževad Belkić, my coauthor and involuntary mentor, thank you

for all the time you spent meticulously going through manuscripts and for making sure I knew what I was writing.

Professor Karen Belkić, my other involuntary mentor, thank you for your

sound advice and thoughts on life and health, work and science, family and friends. I’m very grateful for your help in linguistics.

46

Professor Mike Waligorski, coauthor, mentor and friend, and the enthusi-ast of the year! Thank you for your non-stop encouragement, pep talks and relentless effort to prepare our papers.

Dr Annelie Meijer, to have such a happy and sunny co-supervisor like

you should be every PhD student’s right! Thanks for always cheering me on and for being such an excellent roll model in motherhood, research and friendship.

Docent Margareta Edgren, always eager for discussions, thank you for in-

spiring me to look in other directions for answers, and for nice dinner com-pany at the department.

Docent Bengt Lind, one of my many co-supervisors, thank you for your

enthusiasm and encouragement and for being open to new ideas. Professor Nikolai Sobolevsky, thank you for your continuous efforts in

our fruitful collaboration and for your contribution to our joint publications. Lil Engström, Ann-Charlotte Ekelöf and Marianne Granström, always

ready to lend a hand, an ear or a shoulder to lean on. Thank you for that and for all your help with my thesis work.

All my colleagues at the department of Medical Radiation Physics, past

and present, who are, luckily for me, too many to mention here by name, thank you for being such wonderful friends! I am indebted to you for creat-ing such a warm, interesting and fun atmosphere. You’ve all made it a pleas-ure to go to work.

My friends outside work, thank you for your love, patience and under-

standing when I had to put life on hold for a while. I’m back among the liv-ing.

I want dedicate this thesis to Håkan and Freya, my parents and my entire

family - in Europe and overseas. I am so lucky to have such an amazing fam-ily, always supporting, always there, my soldiers in life. I love you!

Håkan and Freya, you make me the luckiest person alive! Thank you for your patience, encouragement and love and for making me feel ten feet tall.

47

Bibliography

Ahnesjö A, Saxner M and Trepp A 1992 A pencil beam model for photon dose calculation Med Phys 19 263-73

Amaldi U and Kraft G 2007 European developments in radiotherapy with beams of large radiobiological effectiveness J Radiat Res (Tokyo) 48 Suppl A A27-41

Bajard M, Conto JMD and Remillieux J 2004 Status of the "ETOILE" project for a French hadrontherapy centre Radiother Oncol 73 Suppl 2 S211-5

Barashenkov VS 1993 Cross sections of interaction of particles and nuclei with nuclei, JINR, Dubna (in Russian)

Barashenkov VS and Polanski A 1994 Electronic guide for nuclear cross sections, JINR, E2-94-417, Dubna

Bethe HA, Rose M and Smith LP 1938 The multiple scattering of electrons Proceedings of the American Philosophical Society 78 573-85

Biaggi M, Ballarini F, Burkard W, Egger E, Ferrari A and Ottolenghi A 1999 Physical and biophysical characteristics of a fully modulated 72 MeV therapeutic proton beam: model predictions and experimental data Nucl Instrum Meth Phys B 159 89-100

Bichsel H 1968 Charged-particle interactions in Radiation Dosimetry, ed F H Attix and W C Roesch (New York: Academic Press) pp 157-228

Bortfeld T 1997 An analytical approximation of the Bragg curve for therapeutic proton beams Med Phys 24 2024-33

Botvina AS, Dementyev AV, Smirnova ON, Sobolevsky NM and Toneev VD 1997 MSDM – Multy-Stage Dynamical Model. International codes and model intercomparison for intermediate energy activation yields. NSC/DOC(97)-1, NEA/P&T No 14, OECD, ed R Michel and P Nagel (Paris) p 307

Boyle P and Ferlay J 2005 Cancer incidence and mortality in Europe, 2004 Ann Oncol 16 481-8

Bragg WH and Kleemann R 1905 On the α-particles of radium and their loss of range in passing through various atoms and molecules Phil. Mag. 10 318-40

Brahme A 1972 On the optimal choice of scattering foils for electron therapy, Internal Report, Royal Institute of Technology, TRITA-EPP-72-17

Brahme A 1975 Simple expressions for the penetration of high energy

electrons in matter, SSI: 1975-011, National Institute of Radiation Protection, Stockholm

48

Brahme A 2003 Biologically optimized 3-dimensional in vivo predictive assay-based radiation therapy using positron emission tomography-computerized tomography imaging Acta Oncol 42 123-36

Brahme A 2004 Recent advances in light ion radiation therapy Int J Radiat Oncol Biol Phys 58 603-16

Brahme A and Lax I 1983 Absorbed dose distribution of electron beams in uniform and inhomogeneous media Acta Radiol Suppl 364 61-72

Brahme A, Lax I and Andreo P 1981 Electron beam dose planning using discrete Gaussian beams. Mathematical background Acta Radiol Oncol 20 147-58

Brahme A, Lewensohn R, Ringborg U, Amaldi U, Gerardi F and Rossi S 2001 Design of a centre for biologically optimised light ion therapy in Stockholm Nuclear Instruments and Methods in Physics Research B 184 569-88

Brock WA 1988 Predictive assays in Innovations in Radiation Oncology, ed H R Withers and L J Peters (Berlin: Springer-Verlag)

Carlsson AK, Andreo P and Brahme A 1997 Monte Carlo and analytical calculation of proton pencil beams for computerized treatment plan optimization Phys Med Biol 42 1033-53

CERN Program Library: http://wwwasd.web.cern.ch/wwwasd/cernlib/download/2005_source/tar/ Chadwick KH and Leenhouts HP 1973 A molecular theory of cell survival

Physics in Medicine and Biology 18 78-87 Chu WT 1999 Hadron Therapy in Biomedical uses of Radiation, Part B

Therapeutic Applications ed W R Hendee (New York, Weinheim: Wiley VCH) pp 1055-131

Ciangaru G, Polf JC, Bues M and Smith AR 2005 Benchmarking analytical calculations of proton doses in heterogeneous matter Med Phys 32 3511-23

Curtis SB 1986 Lethal and potentially lethal lesions induced by radiation--a unified repair model Radiat Res 106 252-70

Dahlin H, Lamm IL, Landberg T, Levernes S and Ulso N 1983 User requirements on CT-based computed dose planning systems in radiation therapy Acta Radiol Oncol 22 397-415

Deasy JO 1998 A proton dose calculation algorithm for conformal therapy simulations based on Moliere's theory of lateral deflections Med Phys 25 476-83

Debus J, Haberer T, Schulz-Ertner D, Jäkel O, Wenz F, Enghardt W, Schlegel W, Kraft G and Wannenmacher M 2000 [Carbon ion irradiation of skull base tumors at GSI. First clinical results and future perspectives] Strahlenther Onkol 176 211-6

Dementyev AV and Sobolevsky NM 1997 SHIELD – Universal Monte Carlo hadron transport code: scope and applications. 3rd workshop on simulating accelerator radiation environments (SARE 3), (KEK, Tsukuba, Japan)

49

Dementyev AV and Sobolevsky NM 1999 SHIELD - Universal Monte Carlo hadron transport code: scope and applications Radiation Measurements 30 553-7

Elsässer T and Scholz M 2007 Cluster effects within the local effect model Radiat Res 167 319-29

Endo M, Koyama-Ito H, Minohara S, Miyahara N, Tomura H, Kanai T, Kawachi K, Tsujii H and Morita K 1996 HIPLAN—A heavy ion treatment planning system at HIMAC Jpn Soc Ther Radiol Oncol 8 231–8

Enghardt W, Crespo P, Fiedler F, Hinz R, Parodi K, Pawelke J and Pönisch F 2004 Charged hadron tumor therapy monitoring by means of PET Nucl. Instrum. Methods A 525 284-8

Enghardt W, Debus J, Haberer T, Hasch BG, Hinz R, Jäkel O, Krämer M, Lauckner K, Pawelke J and Pönisch F 1999 Positron emission tomography for quality assurance of cancer therapy with light ion beams Nucl. Phys. A 654 1047c-50c

Eyges L 1948 Multiple scattering with energy loss Phys Rev 74 1534 Fermi E 1940 The ionization loss of energy in gases and in condensed

materials Physics Review 57 485 Fippel M and Soukup M 2004 A Monte Carlo dose calculation algorithm for

proton therapy Med Phys 31 2263-73 Geithner O, Andreo P, Sobolevsky N, Hartmann G and Jäkel O 2006

Calculation of stopping power ratios for carbon ion dosimetry Phys Med Biol 51 2279-92

Goitein M 1992 3D treatment planning for heavy charged particles Radiat Environ Biophys 31 241-5

Goitein M and Miller T 1983 Planning proton therapy of the eye Med Phys 10 275-83

Goodhead DT 1985 Saturable repair models of radiation action in mammalian cells Radiat Res Suppl 8 S58-67

Goodhead DT 1994 Initial events in the cellular effects of ionizing radiations: clustered damage in DNA Int J Radiat Biol 65 7-17

Gudowska I, Sobolevsky N, Andreo P, Belkić Dž and Brahme A 2004 Ion beam transport in tissue-like media using the Monte Carlo code SHIELD-HIT Phys Med Biol 49 1933-58

Gudowska I and Sobolevsky NM 2006 Calculations of particle and heavy ion interactions with space shielding materials using the SHIELD-HIT transport code Radiation Measurements 41 1091-6

Gunzert-Marx K, Schardt D and Simon RS 2004 Fast neutrons produced by nuclear fragmentation in treatment irradiations with 12C beam Radiat Prot Dosimetry 110 595-600

Haberer T, Debus J, Eickhoff H, Jäkel O, Schulz-Ertner D and Weber U 2004 The Heidelberg Ion Therapy Center Radiother Oncol 73 Suppl 2 S186-90

Hall EJ and Giaccia AJ 2006 Radiobiology for the radiologist (Philadelphia, Pa: Lippincott Williams & Wilkins)

50

Heilbronn L, Iwata Y, Iwase H, Murakami T, Sato H, Nakamura T, Ronningen RM, Ieki K, Gudowska I and Sobolevsky NM 2006 Secondary neutron-production cross sections from heavy-ion interactions in composite targets Phys Rev C74

Herault J, Iborra N, Serrano B and Chauvel P 2007 Spread-out Bragg peak and monitor units calculation with the Monte Carlo code MCNPX Med Phys 34 680-8

Hill MA 2004 The variation in biological effectiveness of X-rays and gamma rays with energy Radiat Prot Dosimetry 112 471-81

Hogstrom KR, Mills MD and Almond PR 1981 Electron beam dose calculations Phys Med Biol 26 445-59

Hong L, Goitein M, Bucciolini M, Comiskey R, Gottschalk B, Rosenthal S, Serago C and Urie M 1996 A pencil beam algorithm for proton dose calculations Phys Med Biol 41 1305-30

Hromcikova H, Kundrat P and Lokajicek M 2006 Detailed analysis of the response of different cell lines to carbon irradiation Radiat Prot Dosimetry 122 121-3

IAEA 2008 Relative biological effectiveness in ion beam therapy, Technical report series no 461, Vienna

ICRU 1984 Radiation Dosimetry: Electron Beams with energies between 1 and 50 MeV, Int. Commission on Radiation Units and Measurements. Report 35, MD, ICRU, Bethesda

ICRU 1993 Stopping powers and ranges for protons and alpha particles, Int. Commission on Radiation Units and Measurements. Report 49, MD, ICRU, Bethesda

ICRU 2005 Stopping of Ions Heavier Than Helium, Int. Commission on Radiation Units and Measurements. Report 73, MD, ICRU, Bethesda

Joiner MC and Johns H 1988 Renal damage in the mouse: the response to very small doses per fraction Radiat Res 114 385-98

Jung B, Montelius A, Dahlin H, Ekström P, Ahnesjö A, Hogström B and Glimelius B 1997 The conceptual design of a radiation oncology planning system Comput Methods Programs Biomed 52 79-92

Jäkel O, Krämer M, Karger CP and Debus J 2001 Treatment planning for heavy ion radiotherapy: clinical implementation and application Phys Med Biol 46 1101-16

Kanematsu N, Akagi T, Takatani Y, Yonai S, Sakamoto H and Yamashita H 2006 Extended collimator model for pencil-beam dose calculation in proton radiotherapy Phys Med Biol 51 4807-17

Kanematsu N, Yonai S and Ishizaki A 2008 The grid-dose-spreading algorithm for dose distribution calculation in heavy charged particle radiotherapy Med Phys 35 602-7

Kase Y, Kanematsu N, Kanai T and Matsufuji N 2006 Biological dose calculation with Monte Carlo physics simulation for heavy-ion radiotherapy Phys Med Biol 51 N467-75

Katz R, Ackerson B, Homayoonfar M and Sharma SC 1971 Inactivation of cells by heavy ion bombardment Radiat Res 47 402-25

51

Katz R and Sharma SC 1973 Response of cells to fast neutrons, stopped pions and heavy ion beams Nuclear Instruments and Methods 111 93-116

Katz R, Zachariah R, Cucinotta FA and Zhang C 1994 Survey of cellular radiosensitivity parameters Radiat Res 140 356-65

Kellerer AM and Rossi HH 1972 The theory of dual radiation action Current Topics in Radiation Research Quarterly 8 85-158

Kempe J and Brahme A 2008 Energy-range relation and mean energy variation in therapeutic particle beams Med Phys 35 159-70

Kempe J, Gudowska I and Brahme A 2007 Depth absorbed dose and LET distributions of therapeutic 1H, 4He, 7Li, and 12C beams Med Phys 34 183-92

Kimstrand P, Traneus E, Ahnesjö A, Grusell E, Glimelius B and Tilly N 2007 A beam source model for scanned proton beams Phys Med Biol 52 3151-68

Kraft G 2000 Tumor therapy with heavy charged particles Progress in Particle and Nuclear Physics 45 S473-S544

Krämer M, Jäkel O, Haberer T, Kraft G, Schardt D and Weber U 2000 Treatment planning for heavy-ion radiotherapy: physical beam model and dose optimization Phys Med Biol 45 3299-317

Krämer M and Scholz M 2000 Treatment planning for heavy-ion radiotherapy: calculation and optimization of biologically effective dose Phys Med Biol 45 3319-30

Kundrat P 2006 Detailed analysis of the cell-inactivation mechanism by accelerated protons and light ions Phys Med Biol 51 1185-99

Kundrat P 2007 A semi-analytical radiobiological model may assist treatment planning in light ion radiotherapy Phys Med Biol 52 6813-30

Kundrat P, Lokajicek M and Hromcikova H 2005 Probabilistic two-stage model of cell inactivation by ionizing particles Phys Med Biol 50 1433-47

Landau L 1944 On the energy loss of fast particles by ionization Journal of Physics USSR 8 201-5

Larsson B 1967 Radiological properties of beams of high-energy protons Radiat Res Suppl 7 304-11

Lawrence JH, Tobias CA, Born JL, McCombs R, Roberts JE, Anger HO, Low-Beer BV and Huggins CB 1958 Pituitary irradiation with high-energy proton beams: a preliminary report Cancer Res 18 121-34

Lax I and Brahme A 1985 Electron beam dose planning using Gaussian beams. Energy and spatial scaling with inhomogeneities Acta Radiol Oncol 24 75-85

Lax I, Brahme A and Andreo P 1983 Electron beam dose planning using Gaussian beams. Improved radial dose profiles Acta Radiol Suppl 364 49-59

Lee M, Nahum AE and Webb S 1993 An empirical method to build up a model of proton dose distribution for a radiotherapy treatment-planning package Phys Med Biol 38 989-98

52

Lind BK, Persson LM, Edgren MR, Hedlöf I and Brahme A 2003 Repairable-conditionally repairable damage model based on dual Poisson processes Radiat Res 160 366-75

Maccabee HD, Raju MR and Tobias CA 1968 Fluctuations of energy loss by heavy charged particles in thin absorbers Phys. Rev. 165 469-74

Matsufuji N, Fukumura A, Komori M, Kanai T and Kohno T 2003 Influence of fragment reaction of relativistic heavy charged particles on heavy-ion radiotherapy Phys Med Biol 48 1605-23

Matsufuji N, Komori M, Sasaki H, Akiu K, Ogawa M, Fukumura A, Urakabe E, Inaniwa T, Nishio T, Kohno T and Kanai T 2005 Spatial fragment distribution from a therapeutic pencil-like carbon beam in water Phys Med Biol 50 3393-403

Maughan RL, Yudelev M, Kota C, Burmeister J, Porter AT, Forman JD, Blosser H, Blosser E and Blosser G 1999 Neutron radiation therapy: application of advanced technology to the treatment of cancer Nuclear Physics A654 1068c-71c

Mejaddem Y, Belkić Dž, Hyödynmaa S and Brahme A 2001 Calculations of electron energy loss straggling Nuclear Instruments and Methods in Physics Research B 173 397-410

Newhauser W, Fontenot J, Zheng Y, Polf J, Titt U, Koch N, Zhang X and Mohan R 2007 Monte Carlo simulations for configuring and testing an analytical proton dose-calculation algorithm Phys Med Biol 52 4569-84

Newhauser W, Koch N, Hummel S, Ziegler M and Titt U 2005 Monte Carlo simulations of a nozzle for the treatment of ocular tumours with high-energy proton beams Phys Med Biol 50 5229-49

Niita K, Sato T, Iwase H, Nose H, Nakashima H and Sihver L 2006 PHITS-a particle and heavy ion transport code system Radiation Measurements 41 1080-90

Oelfke U and Bortfeld T 2001 Inverse planning for photon and proton beams Med Dosim 26 113-24

Ottolenghi A, Ballarini F and Biaggi M 2001 Mechanistic and phenomenological models for the estimate of radiation-induced biological damage Physica Medica 17

Paganetti H 2004 Four-dimensional Monte Carlo simulation of time-dependent geometries Phys Med Biol 49 N75-81

Paganetti H, Jiang H and Trofimov A 2005 4D Monte Carlo simulation of proton beam scanning: modelling of variations in time and space to study the interplay between scanning pattern and time-dependent patient geometry Phys Med Biol 50 983-90

Parodi K, Ferrari A, Sommerer F and Paganetti H 2007 Clinical CT-based calculations of dose and positron emitter distributions in proton therapy using the FLUKA Monte Carlo code Phys Med Biol 52 3369-87

Pedroni E, Scheib S, Bohringer T, Coray A, Grossmann M, Lin S and Lomax A 2005 Experimental characterization and physical

53

modelling of the dose distribution of scanned proton pencil beams Phys Med Biol 50 541-61

Persson LM, Edgren MR, Stenerlöw B, Lind BK, Hedlöf I, Jernberg AR, Meijer AE and Brahme A 2002 Relative biological effectiveness of boron ions on human melanoma cells Int J Radiat Biol 78 743-8

Petti PL 1992 Differential-pencil-beam dose calculations for charged particles Med Phys 19 137-49

Pfeiffer K and Bendl R 2001 Real-time dose calculation and visualization for the proton therapy of ocular tumours Phys Med Biol 46 671-86

Press W, Flannery B, Teukolsky S and Vetterling W 1986 Numerical Recipes (New York: Cambridge University Press)

Pshenichnov I, Mishustin I and Greiner W 2005 Neutrons from fragmentation of light nuclei in tissue-like media: a study with the GEANT4 toolkit Phys Med Biol 50 5493-507

Puck TT and Marcus PI 1956 Action of x-rays on mammalian cells J Exp Med 103 653-66

Rossi B 1952 High Energy Particles. (London: Constable and Company Ltd)

Rossi B and Greisen K 1941 Cosmic-ray theory Rev Mod Phys 13 265-68 Russell KR, Grusell E and Montelius A 1995 Dose calculations in proton

beams: range straggling corrections and energy scaling Phys Med Biol 40 1031-43

Russell KR, Isacsson U, Saxner M, Ahnesjö A, Montelius A, Grusell E, Dahlgren CV, Lorin S and Glimelius B 2000 Implementation of pencil kernel and depth penetration algorithms for treatment planning of proton beams Phys Med Biol 45 9-27

Sandison GA and Chvetsov AV 2000 Proton loss model for therapeutic beam dose calculations Med Phys 27 2133-45

Sandison GA, Lee CC, Lu X and Papiez LS 1997 Extension of a numerical algorithm to proton dose calculations. I. Comparisons with Monte Carlo simulations Med Phys 24 841-9

Schaffner B, Pedroni E and Lomax A 1999 Dose calculation models for proton treatment planning using a dynamic beam delivery system: an attempt to include density heterogeneity effects in the analytical dose calculation Phys Med Biol 44 27-41

Schardt D, Schall I, Geissel H, Irnich H, Kraft G, Magel A, Mohar MF, Munzenberg G, Nickel F, Scheidenberger C, Schwab W and Sihver L 1996 Nuclear fragmentation of high-energy heavy-ion beams in water Adv Space Res 17 87-94

Scholz M and Kraft G 1996 Track structure and the calculation of biological effects of heavy charged particles Adv Space Res 18 5-14

Schulz-Ertner D, Jäkel O and Schlegel W 2006 Radiation therapy with charged particles Semin Radiat Oncol 16 249-59

Scott WT 1963 The theory of small-angle multiple scattering of fast charged particles Reviews of modern physics 35 231-313

54

Sinclair WK 1966 The shape of radiation survival curves of mammalian cells cultured in vitro, Biophysical aspects of radiation quality, Technical Reports series No. 58, IAEA Lyon

Sobolevsky NM 2001 The SHIELD Transport Code: a Tool for Computer Study of Interaction of Particles and Nuclei with Complex Media. Proc. of the 3rd Yugoslav Nuclear Society Int. Conf. YUNSC 2000. The VINCA Institute, (Belgrade) pp 539-64

Sommerer F, Parodi K, Ferrari A, Poljanc K, Enghardt W and Aiginger H 2006 Investigating the accuracy of the FLUKA code for transport of therapeutic ion beams in matter Phys Med Biol 51 4385-98

Soukup M, Fippel M and Alber M 2005 A pencil beam algorithm for intensity modulated proton therapy derived from Monte Carlo simulations Phys Med Biol 50 5089-104

Stenerlöw B, Blomquist E, Grusell E, Hartman T and Carlsson J 1996 Rejoining of DNA double-strand breaks induced by accelerated nitrogen ions Int J Radiat Biol 70 413-20

Striganov SI 2005 On the theory and simulation of multiple Coulomb scattering of heavy-charged particles Radiat Prot Dosimetry 116 293-6

Svensson H, Ringborg U, Näslund I and Brahme A 2004 Development of light ion therapy at the Karolinska Hospital and Institute Radiother Oncol 73 Suppl 2 S206-10

Szymanowski H, Mazal A, Nauraye C, Biensan S, Ferrand R, Murillo MC, Caneva S, Gaboriaud G and Rosenwald JC 2001 Experimental determination and verification of the parameters used in a proton pencil beam algorithm Med Phys 28 975-87

Szymanowski H and Oelfke U 2002 Two-dimensional pencil beam scaling: an improved proton dose algorithm for heterogeneous media Phys Med Biol 47 3313-30

Tobias CA 1985 The repair-misrepair model in radiobiology: comparison to other models Radiat Res Suppl 8 S77-95

Tobias CA, Roberts JE, Lawrence JH, Low-Beer BV, Anger HO, Born JL, McCombs R and Huggins C 1955 Irradiation hypophysectomy and related studies using 340 MeV protons and 190 MeV deuterons. International Conference on the Peaceful Uses of Atomic Energy, (Geneva) pp 95-106

Tourovsky A, Lomax AJ, Schneider U and Pedroni E 2005 Monte Carlo dose calculations for spot scanned proton therapy Phys Med Biol 50 971-81

Tsujii H, Mizo J, Kamada T, Baba M, Tsuji H, Kato H, Kato S, Yamada S, Yasuda S, Ohno T, Yanagi T, Imai R, Kagei K, Kato H, Hara R, Hasegawa A, Nakajima M, Sugane N, Tamaki N, Takagi R, Kandatsu S, Yoshikawa K, Kishimoto R and Miyamoto T 2007 Clinical Results of Carbon Ion Radiotherapy at NIRS J Radiat Res (Tokyo) 48 Suppl A A1-A13

Wambersie A 1992 Neutron therapy: from radiobiological expectation to clinical reality Radiation Protection Dosimetry 44 379-95

55

56

Ward JF 1988 DNA damage produced by ionizing radiation in mammalian cells: identities, mechanisms of formation, and reparability Prog Nucleic Acid Res Mol Biol 35 95-125

Varian 2005 Planning reference guide for beam configuration—Eclipse, Varian Report PN B401642R01G, (Palo Alto, CA)

Vavilov PV 1957 Ionization losses of high-energy heavy particles Soviet Physics JETP 5 749-51

Wilson RR 1946 Radiobiological use of fast protons Radiology 47 487-91 Yang CN 1951 Actual path length of electrons in foils Physics Review 84

599-600