ACTAUNIVERSITATIS

UPSALIENSISUPPSALA

2016

Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Medicine 1188

Micro/nanometric Scale Study ofEnergy Deposition and its Impacton the Biological Response forIonizing Radiation

Brachytherapy radionuclides, proton and carbon ionbeams

FERNANDA VILLEGAS NAVARRO

ISSN 1651-6206ISBN 978-91-554-9495-7urn:nbn:se:uu:diva-279385

Dissertation presented at Uppsala University to be publicly examined in Skoogssalen,Akademiska Sjukhuset, Ing. 78-79, Uppsala, Friday, 22 April 2016 at 09:30 for the degree ofDoctor of Philosophy (Faculty of Medicine). The examination will be conducted in English.Faculty examiner: Dr. Carmen Villagrasa (Institut de Radioprotection et de Sûreté Nucléaire –IRNS).

AbstractVillegas Navarro, F. 2016. Micro/nanometric Scale Study of Energy Deposition and itsImpact on the Biological Response for Ionizing Radiation. Brachytherapy radionuclides,proton and carbon ion beams. Digital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Medicine 1188. 53 pp. Uppsala: Acta Universitatis Upsaliensis.ISBN 978-91-554-9495-7.

Research in radiotherapy for cancer treatment focuses on finding methods that can improve thecompromise between tumour cell inactivation versus damage to the surrounding healthy tissue.As new radiation modalities such as proton therapy become accessible for everyday clinicalpractice, a better understanding of the variation in biological response of the tumour and healthytissues would improve treatment planning to achieve optimal outcome. The development ofradiobiological models capable of accurate predictions of biological effectiveness is needed.

Existing radiation quality descriptors such as absorbed dose and LET are insufficient toexplain variation in biological effectiveness for different treatment modalities. The stochasticnature of ionizing radiation creates discrete patterns of energy deposition (ED) sites which cannow be analysed through sophisticated computer simulations (e.g. Monte Carlo track structurecodes). This opens the possibility to develop a nanometre characterization of radiation qualitybased on the spatial cluster patterns of ED.

The aim of this thesis is to investigate the track structure (ED spatial pattern) propertiesof several radiation qualities at a micro- and nanometric scale while exploring their influencein biological response through correlations with published experimental data. This work usestrack structure data simulated for a set of 15 different radiation qualities: 4 commonly usedbrachytherapy sources, 6 different proton energies, 4 different carbon ion energies, and 60Cophotons used as reference radiation for quantification of biological effectiveness.

At a micrometre level, the behaviour of the microdosimetric spread in energy deposition fortarget sizes of the order of cell nuclei was analysed. The degree of the influence it had in thebiological response was found to be negligible for photon sources but for protons and carbonions the impact increased with decreasing particle energy suggesting it may be a confoundingfactor in biological response.

Finally, this thesis outlines a framework for modelling the relative biological effectivenessbased on the frequency distribution of cluster order as a surrogate for the nanometreclassification for the physical properties of radiation quality. The results indicate that thisfrequency is a valuable descriptor of ionizing radiation. The positive correlation acrossthe different types of ionizing radiation encourages further development of the frameworkby incorporating the behavior of the microdosimetric spread and expanding tests to otherexperimental datasets.

Keywords: Ioninzing radiation, Monte Carlo track structure code, Microdosimetric spread,Energy deposition clustering, RBE

Fernanda Villegas Navarro, Department of Immunology, Genetics and Pathology, MedicalRadiation Science, Rudbecklaboratoriet, Uppsala University, SE-751 85 Uppsala, Sweden.

© Fernanda Villegas Navarro 2016

ISSN 1651-6206ISBN 978-91-554-9495-7urn:nbn:se:uu:diva-279385 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-279385)

A mis amados padres que con suesfuerzo y dedicación me han

abierto muchas puertas

List of Papers

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

I Villegas, F., Tilly, N., and Ahnesjö, A. (2013) Monte Carlo

calculated microdosimetric spread for cell nucleus-sized targets exposed to brachytherapy 125I and 192Ir sources and 60Co cell ir-radiation Phys. Med. Biol., 58:6149-63 i Villegas, F. and Ahnesjö, A. (2016) Reply to Comment on

‘Monte Carlo calculated microdosimetric spread for cell nu-cleus-sized targets exposed to brachytherapy 125I and 192Ir sources and 60Co cell irradiation’ Phys. Med. Biol. In print

II Villegas, F., Tilly, N., and Ahnesjö, A. (2015) Microdosimetric spread for cell-sized targets exposed to 60Co, 192Ir and 125I sources Radiat. Proct. Dos., 166:365-68

III Villegas, F., Bäckström, G., Tilly, N., and Ahnesjö, A (2014) Cluster pattern analysis of energy deposition sites for the brachytherapy sources 103Pd, 125I, 192Ir, 137Cs, and 60Co Phys. Med. Biol., 59:5531-43

i Villegas, F., Bäckström, G., Tilly, N., and Ahnesjö, A

(2016) Corrigendum to ‘Cluster pattern analysis of energy deposition sites for the brachytherapy sources 103Pd, 125I, 192Ir, 137Cs, and 60Co’ Phys. Med. Biol. Submitted

IV Villegas, F., Tilly, N., Bäckström, G., Shirin A. E. and Ah-

nesjö, A (2016) The impact on linear-quadratic parameteriza-tion of cell survival data from the microdosimetric spread of specific energy for different cell nuclei size distributions Manu-script

V Villegas, F., Bäckström, G., Shirin A. E., Tilly, N. and Ahnesjö A (2016) Energy deposition clustering as a functional radiation quality descriptor for modelling Relative Biological Effective-ness Med. Phys. Submitted

Reprints were made with permission from the respective publishers.

Contents

Introduction ........................................................................................... 9 1.1.1 Ionizing radiation in cancer treatment ............................................. 10 1.2 Ionizing radiation interactions with matter ..................................... 11 1.3 Radiobiological modelling .............................................................. 12 1.4 Aims of the thesis ............................................................................ 14

Monte Carlo (MC) track structure code ............................................... 16 2.2.1 LIonTrack code ............................................................................... 16 2.2 Simulation of the brachytherapy sources and a 60Co photon source ....................................................................................................... 18 2.3 Simulation of protons (H+) and carbon ions (C6+) ........................... 20

Microdosimetry ................................................................................... 22 3.3.1 Microdosimetric spread framework ................................................ 22 3.2 Other microdosimetric properties ................................................... 24 3.3 Paper I ............................................................................................. 26 3.4 Paper II ............................................................................................ 28 3.5 Influence of microdosimetric spread in biological outcome ........... 29 3.6 Paper IV .......................................................................................... 30

Nanometre characterization of radiation quality ................................. 32 4.4.1 Nearest Neighbour distributions ..................................................... 32 4.2 Frequency of cluster order .............................................................. 34 4.3 Paper III .......................................................................................... 36

RBE modelling .................................................................................... 38 5.5.1 Framework using cell survival data as endpoint ............................. 38 5.2 Experimental data ........................................................................... 40 5.3 Paper V ............................................................................................ 41 5.4 Future perspectives for the model ................................................... 43

Summary and Outlook ......................................................................... 45 6.

Acknowledgements.............................................................................. 47 7.

References ........................................................................................... 49 8.

Abbreviations

CO Cluster order CSDA Continuous slowing down approximation DNA Deoxyribonucleic acid EC Electron capture ED Energy deposition HDR High-dose rate IBT Ion beam therapy IC Internal conversion LDR Low-dose rate LET Linear energy transfer LQ Linear-quadratic model MC Monte Carlo NN Nearest neighbour RBE Relative biological effectiveness SF Surviving fraction

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Introduction 1.

Soon after the discovery of x-rays and radioactivity it became obvious that exposure to radiation affects human tissue, healthy as well as malign. Radio-biology, the study of the biological effects of ionizing radiation, has played a central role in radiotherapy as it aids in the transition from “trial and error” treatments to treatments with more controlled outcomes. As advancements in technology make possible the use of particles such as protons and carbon ions in radiotherapy treatments, radiobiological modelling can, if successful, contribute to the “translation” of gained empirical knowledge from e.g. pho-tons to particles.

Correct function of cellular processes such as DNA repair and the faithful transmission of the genetic information to the cell´s progeny are dependent on the integrity of the DNA molecule. Unrepaired damage or alterations of the DNA molecule at particular sensitive coding segments can eventually lead to cell death. The stochastic nature of the ionizing radiation interactions causes a discrete pattern of transfer points where energy is deposited locally in the irradiated medium. These transfer points referred to as energy deposi-tion (ED) sites in this thesis, form spatial patterns or clusters that may corre-late to their effectiveness in creating initial damage to DNA as evidence from molecular biology suggests (Nikjoo et al 1994, Hill 1999, Grosswendt 2006).

Consequentially, the description of ionizing radiation at a nanometre level as opposed to the commonly used macrodosimetric quantities of absorbed dose and linear energy transfer (LET), may prove more valuable for under-standing cell survival. Incorporation of such descriptors into radiobiological modelling would also improve the prediction of the observed variation in biological responses. However, the task of finding such descriptor is non-trivial. Nanodosimetric experiments still face enormous technical challenges for measuring cluster-size distributions although some attempts have been made (Pszona et al 2000, De Nardo et al 2002, Garty et al 2002, Schulte et al 2006). Computer Monte Carlo (MC) track structure simulations have thus become a complementary state of the art tool for the generation and analysis of ED patterns.

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1.1 Ionizing radiation in cancer treatment The almost simultaneous discovery of x-rays and radioactivity at the end of the 19th century quickly gave way to their use not only in diagnostic but also in treatment for cancer. Since then, several irradiation techniques have been developed by exploiting the physical properties of ionizing radiation to im-prove the probability of tumour eradication while minimizing as much as possible the damage to the surrounding healthy tissue.

Brachytherapy is a form of radiation treatment that places radioactive sources as close as possible to the tumour. The most common source place-ment strategies are the surface-mould technique (on the body surface), inter-stitial (often in malignant tissue), and intracavitary (in a body cavity). The radioactive implants come in different forms such as pellets, seeds, wires, etc. The seeds are metal encapsulated radionuclides usually no larger than a few millimetres which are, in some cases, permanently implanted into the patient. The handling of seeds inevitably represents some degree of exposure to medial staff. To reduce this risk, an “afterloader” machine may be used to automatically load the radionuclides via wires into catheters or “applicators” previously inserted into the patient. Today, brachytherapy uses dose plan-ning systems to optimize the arrangement of the seeds or applicators that would yield the best dose coverage of the tumour volume. Brachytherapy, requiring much less capital investments than external beam accelerators, is a common choice of treatment for cervix, prostate, head and neck, breast, and skin cancers.

The invention of orthovoltage x-ray machines at end of the 1920´s and later on commercial linear electron accelerators (1950´s) together with the development of 60Co γ -ray therapy, opened the possibility to treat deep seated tumours without having invasive procedures. Nonetheless, treatment with x-rays and γ -rays often resulted in excessive exposure to healthy tissue including the patient´s skin. This was mediated by use of higher energy beams and later refined by the development of cross-firing techniques such as intensity-modulated radiation therapy (IMRT) (see review by Ahnesjö et al (2006)) and the volumetric arc therapy (VMAT). Today these techniques allow a high degree of conformity between the irradiated volume and the patient´s particular tumour shape, however they create a bath of low dose over large volumes of healthy tissue surrounding the tumour site.

The foundation for ion beam therapy (IBT) was first proposed by Robert Wilson in 1946 (ICRU 2007). He explained how the physical properties of proton interactions would result in a considerable reduction of exposure of healthy tissue. Upon entry, protons travel in almost straight lines across bio-logical tissue slowly increasing the ionization density until near the end of its range where the rest of the proton´s energy is delivered in a very narrow region. This characteristic is known as the Bragg peak in dose-depth curves (see figure 2.2). This implies that no dose is deposited beyond this peak spar-

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ing all tissue that lies beyond. Moreover, modulating the initial kinetic ener-gy of the proton would cause the Bragg peak to spread out making possible the coverage of the entire tumour volume in depth.

The first proton treatment on a benign target was performed in Berkley in 1954 after extensive pre-clinical work (Tobias et al 1956). The first treated tumour followed only 3 years later at the Gustaf Werner Institute in Uppsala, Sweden (Falkmer et al 1962). Investigation on the potential of therapy with other light ions (helium and carbon ions) soon followed. According to statis-tics gathered by the Particle Therapy Co-operation Group there are little more than 50 IBT operational facilities world-wide that have treated up until 2014 over 137 000 patients, out of which 15 000 were treated only in 2014 (www.ptcog.ch). The increasing interest of IBT motivates radiobiological research in order to fine-tune the procedures including radiobiological mod-elling to improve outcome and the patient´s quality of life.

1.2 Ionizing radiation interactions with matter Ionizing radiation can be characterized as direct or indirect (ICRU 1980). The first includes charged particles e.g. electrons, protons and heavier ions, which will mainly transfer their energy directly to the irradiated matter by Coulomb-force interactions with the atom’s electrons. The second involves x-rays,γ -rays, and neutrons whose interactions set secondary charged parti-cles in motion delivering energy to the media through a large number of interactions. The points in space where a discrete event of energy is trans-ferred to the medium will be referred to as energy deposition (ED) sites throughout the rest of this work.

For charged particles, Coulomb-force interactions can be characterized as soft or hard collisions depending on how close the charged particle passes from the nucleus of the atom. The result of a hard collision is the ejection of a δ electron with enough kinetic energy to undergo additional collisions far away from the atom. Radiative processes may also happen in which the en-ergy loss results in the emission of a bremsstrahlung photon. The stopping power S dT dx= is defined as the energy loss per unit of path length x of a charged particle traversing inside the media. The mass stopping power Sρ is the result of dividing S by the density of the media ρ , expressed usually in units of MeVcm2g-1. S can be divided into the collision (electronic) term

cS , and the radiative term rS . The mean free path of charged particles is defined as the mean distance (of the order of nanometres) between interac-tions and it its magnitude depends on particle energy as is inversely propor-tional to S .

For photons, the dominant processes within the energy ranges of interest for clinical use, are the photoelectric (~1 keV to 50 keV) and Compton effect (~50 keV to 20 MeV). The photoelectric effect consists in the absorption of

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the photon by transferring all its energy to an atom´s electron. In the Comp-ton effect the photon is scattered transferring part of its energy to an atom´s electron.

The absorbed dose D is the common quantification of the physical ef-fects as it is defined as the average of the stochastic quantity energy imparted ε per mass m at a point of interest (ICRU 1980),

dD

dm

ε= . (1)

The unit of D is Gray (Gy) where 1 Gy = 1 Joule kg-1. When using cS to calculate energy deposition in small volumes, the es-

cape of higher energy δ electrons will cause an overestimation of dose. To overcome this, the restricted stopping power known as the linear energy transfer (LET) is used. LET is defined as the fraction of cS that includes only those hard collisions resulting in δ electrons with energies less than a cut-off value Δ ,

( )dTL dxΔ Δ= . (2)

The unrestricted LET L∞ , is defined when Δ equals the maximum transfer energy possible in the collision hence cL S∞ ≡ .

LET classification of ionizing radiation has been widely spread in the are-as of medical physics and radiobiology. Despite the fact that LET is not de-fined for indirect ionizing radiation, it is generally agreed that photons are low-LET due to the sparse energy deposition pattern of the secondary elec-trons (set in motion by photons) when compared to the denser energy depo-sition pattern of the protons or heavier charged particles. The relationship between LET and kinetic energy of the particle is mainly inversely propor-tional.

However, LET cannot represent the stochastic discrete nature of energy transfers at the micro or nanometre scale. This becomes a problem particu-larly when trying to describe biological effects as the sizes of the target vol-umes of interest are of the order of cell nuclei or the DNA molecule. This has prompted the introduction of the microdosimetric formalism which uses concepts to describe the actions of ionizing radiation in more detail. In par-ticular, this thesis explores the classification of radiation quality based on the cluster formation of EDs at a nanometre scale.

1.3 Radiobiological modelling The early models in radiobiology focused on quantifying relationships be-tween the absorbed dose D and the biological effects at various endpoints. One of the most studied is the cell´s capacity of proliferation by counting the proportion of surviving colonies as a function of D delivered by a damaging

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agent (i.e. ionizing radiation). The survival fraction SF diminishes as D increases regardless of radiation quality or type of cell, but the rate of decay depends on many physical and biological variables which makes any me-chanical explanation a real challenge. The most common parameterization of SF curves is the linear-quadratic (LQ) model which is based on two parame-ters, one proportional to dose and the other proportional to the square of the dose,

( )2SF( ) expD D Dα β = − + (3)

Mathematically, the α parameter describes the initial slope of the curve at low doses whereas the β parameter describes the shoulder of the curve at higher doses. The ratio α β depends, among other properties, on the type of cell and is useful for an empirical quantification of tissue complications or radiosensitivity. Mechanistic biological explanations however, have been formulated and disregarded as more biological and physical knowledge is gathered (see review from Ballarini (2010)). The Dual Action Theory devel-oped by Rossi (1979) was the first to incorporate microdosimetry theory into radiobiological modelling and although now abandoned it has inspired other models such as the Microdosimetric-Kinetic Theory and others (Hawkins 1998). However, advancements in track structure computer simulations sug-gest that radiobiological modelling needs a detailed characterization of ioniz-ing radiation down to the nanometre scale. One of the most successful mod-els is the Local Effect Model developed at GSI in Germany (Scholz and Kraft 1996, Scholz et al 1997, Elsässer and Scholz 2007, Elsässer et al 2008, Elsässer et al 2010). In its latest version, the clustering of random strand breaks along the DNA molecule modelled from amorphous (but continuous) energy depositions is translated into an enhancement factor for determination of the cell inactivation at high local doses (D >10Gy) of x-ray exposure which is then used as input for the prediction of biological effectiveness.

In the context of cell survival (and hence also tumour control), the RBE is defined as the ratio of doses for the same survival level of a reference radia-tion RD and a test radiation QD ,

R

Q equal action

RBED

D= . (4)

The relationship between RBE and LET has been deduced from survival

curves measured from exposure to different particle energies from several types of radiation. As figure 1.1 shows, RBE increases with LET until a maximum is reached. These maxima vary with particle type (Sørensen et al 2011) being around 30 keV µm-1 for protons and around 100 keV µm-1 for other light ions. It is evident that this relationship is not univocal. Particular-ly, photon emitting brachytherapy sources are in some research areas still

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classified as low-LET radiation despite the increasing evidence that different radionuclides present very different RBE values for survival (Fritz et al 1996, Nath et al 2005), microdosimetric (Reniers et al 2004, Wuu et al 1996, Zellmer et al 1992) or yield of DSB (Hsiao and Stewart 2008, Reniers et al 2008) as endpoints. This supports the fact that LET alone fails to describe the radiation quality at this level and that track structure based classifications may be more appropriate (Hill 1999).

Figure 1.1 RBE at 10% survival as a function of LET. Published experimental Chi-nese hamster cell survival data for the brachytherapy sources (Nath et al 1995, Fritz et al 1996) , protons (Folkard et al 1996, Belli et al 1998)and carbon ions (Weyrather et al 1999) was used for calculation of RBE values with 60Co photons (Stenerlöw et al 1995) as reference radiation. For the photon sources, the LET pre-sented is calculated as the dose-weighted LET from figure 4 in Kellerer (2002).

One motive for using RBE is that it can be conveniently applied as a dose weighting factor in treatment planning systems (Wambersie et al 2002). In clinical proton therapy, an RBE equal to 1.1 (ICRU 2007) is currently rec-ommended regardless of the penetrating proton´s kinetic energy. There is now sufficient evidence that RBE values may be as high as 3 for in vitro studies (Sørensen et al 2011) at the low energies present at distal edge of the proton spread-out Bragg peak. The benefits of using a variable RBE correc-tion have been investigated in several works (Paganetti et al 1997, Tilly et al 2005, Wilkens and Oelfke 2004). An accurate method to predict RBE may thus help fine-tune clinical treatments improving the outcome.

1.4 Aims of the thesis The general aim of this thesis is to investigate track structure properties of several radiation qualities at a micro- and nanometric scale, and to explore

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some of their correlations with biological response. To achieve this, the fol-lowing particular aims were set:

• To explore the behaviour of the microdosimetric spread for several types

of ionizing radiation for target volumes equivalent to mammalian cell nuclei (papers I and II).

• To investigate the influence of the microdosimetric spread in the biolog-ical response expressed by the linear-quadratic parameterization of cell survival. Particularly the microdosimetric spreads calculated for cell populations with varying sizes were analysed (paper IV).

• To analyse the distance distribution between EDs and the frequency of energy deposition cluster size for the most common brachytherapy sources to reveal structural differences between them (paper III).

• To set up a framework for prediction of RBE values based on the ED clustering for different radiation qualities (paper V).

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Monte Carlo (MC) track structure code 2.

The advancements in modern computational systems over the past 60 years have opened the possibility for scrutinizing ionizing radiation interactions at the nanometre scale. The Monte Carlo method, known since the 18th century (e.g. the Buffon´s needle problem), has been applied to a variety of mathe-matical and physical problems to obtain approximate solutions by perform-ing statistical sampling experiments. It wasn´t until 1963 when the physicist Martin Berger was the first to successfully implement a computer based MC method to simulate the transport of ionizing radiation by writing a complete coupled electron-photon transport code known as ETRAN (see review pa-pers Seltzer (1991) and Nikjoo et al (2006)). MC transport codes can be divided into three types: track structure MC where all interactions are explic-itly simulated (i.e. event-by-event); class I where all scattering interactions are merged into transport steps like ERTRAN (condensed history); and class II which combines simulation the energy loss of soft collisions (small an-gle/low energy transfer) and event-by-event of the hard collisions (large angle/high energy transfer). Class I and II codes are usually faster and are mainly used for macroscopic problems such as calculation of dose distribu-tions in radiotherapy planning. Several track structure MC codes have been developed (Nikjoo et al 1994, 1997, 2002, Uehara et al 2001, Champion et al 2005) in order to study the spatial pattern of energy deposition for ions, electrons and photons in a micro- or nanometric scale. In the following sec-tions the MC transport code used for this work is described as well as a brief description on how the simulation of the different types of ionizing radiation was done.

2.1 LIonTrack code In this work, computer simulations were performed with the MC transport code LIonTrack (Bäckström et al 2013). It is to be noted that the published papers I, II, and III contain results based on a previous version of LIonTrack that was found to have critical bugs. Corrigendum to the latter papers and results in papers IV and V are all based on data from the corrected version of LIonTrack.

LIonTrack consists of two packages: the first is dedicated to the transport of protons, and other light ions such as helium, lithium and carbon; and the

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second is a modified version of the MC transport code PENELOPE 2008 (Salvat et al 2008) that performs the transport of the secondary electron gen-erated by light ion and/or photon interactions. LIonTrack transports particles event-by-event in liquid water. The transport energy range for ions is 1 to 300 MeVu-1 and electrons can be followed down to 50 eV at which point their residual energy is handled as locally deposited.

For the transport of protons and light ions, LIonTrack uses cross sections based on the Continuum-Distorted-Wave formalism with the Eikonal-Initial-State approximation for the determination of the initial state for the elec-trons. Inelastic nuclear interactions and elastic scattering of the ions are not implemented resulting in treatment of ion tracks as having straight paths so only short segments can be viably used.

PENELOPE 2008 has previously been used for several microdosimetric studies (Stewart et al 2002, Mainardi et al 2004, Hsiao and Stewart 2008, Bernal and Liendo 2009) because it offers the possibility to be run in an event-by-event mode, nonetheless the appearance of some artefacts have also been reported (Bernal and Liendo 2009). A modified version of PE-NELOPE was developed by Fernández-Varea et al (2012) where the default electron inelastic cross sections were substituted by a table of cross sections computed according to the Dingfelder et al (1998) formalism which better describes the energy loss interactions of the electrons (Dingfelder et al 1998, Dingfelder et al 2008).

Figure 2.1 Comparison of published radial dose distributions for 1 MeVu-1 protons in liquid water with the corrected LIonTrack (purple line) utilized in this work. Orig-inal LIonTrack by Bäckström et al (2013) (blue line), MC by Emfietzoglou et al (2006) (red circles), GEANT4-DNA from Incerti et al (2014) (red line), and experi-mental data Wingate and Baum (1976) (black line).

The output of LIonTrack for proton and light ion transport consists of two sets of files, one set containing the energy loss information of the ion track

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and the other set contains the information on the starting position, angle and energy of the generated secondary electrons. The latter file set becomes the phase space input for the modified PENELOPE. For the case of photon and/or electron transport, only the modified PENELOPE is needed.

Figure 2.1 presents a brief benchmarking of the corrected LIonTrack code for radial dose distributions estimated for 1 MeVu-1 proton tracks. The cor-rected LIonTrack is in agreement with other MC codes such as GEANT4 DNA at radius larger than 10 nm. Overall it is clear from figure 1 that the original LIonTrack (Bäckström et al 2013) overestimated the length of the electron tracks. This was caused by erroneous sampling of the mean free path of the electrons and their initial angle after ion interaction. Both issues were addressed in the corrected LIonTrack used in this thesis, except for the original versions of papers I, II, and III which were based on the old version.

2.2 Simulation of the brachytherapy sources and a 60Co photon source

The radionuclides used in brachytherapy vary widely in physical properties such as specific activity, half-life, photon energy, and dose rate. Particularly, the latter is often used to classify the sources into a low dose rate (LDR) group, and a high dose rate (HDR) group; both of which can be applied in a continuous or pulsed modality. The choice of source, modality, and total dose used for a specific treatment mostly relies on the empirical knowledge accumulated through the years. Indeed, since the 80’s it has been shown that the dose rate affects cell survival for in-vitro cell cultures (Freeman et al 1982, Marchese et al 1990, Nath et al 2005).

Following the terminology used by Carlsson and Ahnesjö (2000), the primary photon spectrum for a brachytherapy source includes the isotope photon decay and the direct photon descendants of these that have interacted anywhere inside the source or its encapsulation. A brachytherapy source can then, for the investigations done in this thesis, be approximated as an iso-tropic point source generating photons with initial energy according to this spectrum. Consequently, the primary photon spectra were used as input to the modified PENELOPE to start the transport of the electrons in the centre of a 30x30x30 cm3 water phantom. The absorption energy cut-off was set to 50 eV, considering as locally deposited all electrons produced with less than this cut-off energy. Data for each track was stored in separate files to ease handling by post analysis codes. Details of how the primary photon spectra were obtained for the investigated nuclide sources are described below.

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103Pd Palladium seeds are often used for permanent implants for prostate cancer due to its very short half-life (16.991 days) and high specific activity. 103Pd decays via electron capture (EC) to an excited state of 103Rh which then de-cays to ground state mainly by internal conversion (IC) emitting characteris-tic x-rays. The mean photon energy is approximately 0.021 MeV.

The photon spectrum was obtained via a PENELOPE simulation in which the geometry of the seed was constructed according to the diagram presented by Camgoz and Kumru (2011) of a TheraSeed 200 (Bard Brachytherapy, Inc.). These seeds consist of two cylindrical graphite pellets coated with radioactive palladium and separated by a cylindrical lead marker. The encap-sulation is a thin titanium tube with an outer diameter of 0.826 mm. The total length of the seed is 4.5 mm. The emitted photons were scored in a cylinder with a radius of 1 mm with its axis parallel to the metal casing and centred along the long axis of the seed.

125I The 125I seeds are used both for temporal and permanent implants to treat mainly prostate cancer. It has a half-life of 59.49 days and the principal mode of decay is by EC to an excited state of 125Te and then by IC to the ground state. The mean photon energy is 0.028 MeV.

The photon spectrum for the Nucletron SelectSeed 130.002 was obtained from the Carleton Laboratory for Radiotherapy Physics (CLRP) database based on the work of Taylor and Rogers (2008). This seed consists of a cy-lindrical silver rod encapsulated in a titanium cylinder with an outer diameter of 0.85 mm and with a total length of 4.5 mm.

192Ir The 192Ir sources are used in HDR brachytherapy for which an afterloader machine is used. The treatment times may vary from a few minutes to hours depending on the dose per fraction and the dose rate of the sources. It has a half-life of 73.81 days which makes it eligible for temporary implants. The principal mode of decay is via β− decay to an excited state of 192Pt and then by gamma decay with emission of several γ -rays. The mean photon energy is 0.36 MeV.

The photon spectrum for a Nucletron microSelectron-v1 source was ob-tained from the CLRP database. The 3.5 mm long source is contained at the tip of an AlSl304 wire with a diameter of 1.1 mm. The wire was extended for another 4.75 mm in the geometry set up by Taylor and Rogers (2008).

137Cs The 137Cs sources were first introduced for treatment of gynaecological tu-mours and for interstitial treatments. The long half-life (30.07 years) permits clinical use over longer periods compared to other brachytherapy sources.

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The mode of decay is via β− decay to the second excited state of 137Ba and then to ground state primarily through gamma decay. The mean photon en-ergy is 0.62 MeV.

The photon spectrum for the 137Cs source (commercially built by Radia-tion Therapy Resources, Valencia, USA) was obtained after a PENELOPE simulation in which the geometry of the source follows the description given in Thomason et al (1991) which consists of an active core made out of gold surrounded by a Pt filter and encased in a stainless steel casing. The total length is of 5 mm with an outer diameter of 0.72 mm.

60Co Simulations of 60Co photons consisted only of the emission of a monoener-getic 1.25 MeV photons which corresponds to the mean energy of the two γ -rays (1.17 and 1.332 MeV) emitted after β− decay to the ground state of 60Ni. The energy spectrum of the first generation electrons (emitted electrons after the first photon interaction) was scored from a PENELOPE simulation performed in a water phantom. One reason for using the Compton electron spectrum instead of the photon spectrum is that the probability of finding scattered photons within a volume of experimental relevance is low due to the long mean free path. 60Co sources are often used as reference radiation in applications under low scatter conditions. Our track structure simulations take this electron spectrum as a starter for electron tracks simulating a point source.

2.3 Simulation of protons (H+) and carbon ions (C6+) The energies of the proton (and lower energy carbon ions) analysed in this work are found at the distal edge of the Bragg peak of particles whose initial kinetic energy are typical in clinical use (see figure 2.2). The higher RBE values for these tracks (Sørensen et al 2011) were the main reason for their analysis. As this work focuses on radiation quality characterization at a mi-cro- and nanometre scale, it is of interest to score track segments that are of the order of typical mammalian cell nuclei sizes (e.g. spherical volumes with diameters between 1 to 12 µm), hence all ion tracks simulated here are at least 12 µm long unless otherwise stated.

All proton and carbon ion track segments were simulated in a semi-infinite water phantom following theδ electron tracks down to 50 eV. As many of our analyses consist of finding correlations with biological experi-mental data, we required that all particles have a nominal energy approxi-mately equal to the energy conditions at which experimental data was ac-quired (see table 2). The thickness of the segment of the simulated track varied according to the type of particle and its energy, so that at least a 12 µm thick layer (of the order of the diameter of cell nuclei) would provide

21

longitudinal particle equilibrium conditions for the δ electrons. Thus pad-ding layers were placed up and down-stream of this scoring layer that corre-spond to twice the continuous slowing down approximation (CSDA) range of the most energetic secondary electron emitted by the ion, accounting for both build-up and backscatter within the scoring layer (see figure 2.3).

Figure 2.2 Depth dose curve for a 200 MeVu-1 proton beam in water. The circles mark the mean energy in MeV of the particles as they traverse a slab of water.

Figure 2.3 Illustration of the padding and scoring layers for proton and carbon ion track simulations. Note the scale difference of the horizontal and vertical axes. The rectangle on the right side of the figure contains a lateral view of a 1 MeVu-1 proton track segment to scale.

22

Microdosimetry 3.

The stochastic behaviour of ionizing radiation naturally causes a spread of energy deposition in finite volumes. As the volume of interest decreases in size down to the micrometre scale relevant for living cells or cell-nuclei compartments, mean quantities such as dose or LET are no longer sufficient. Their analogous microdosimetric quantities, specific energy and lineal ener-gy, should be employed. In general, biological assays rely on the macroscop-ic dose as a reference to the amount of energy deposited and rarely consider the intrinsic microdosimetric spread which may be a confounding factor in biological response. A thorough study of this quantity is given by papers I, II and IV. The latter further concentrates on the impact of the microdosimetric spread on the biological outcome of cell survival assays.

3.1 Microdosimetric spread framework The specific energy z is defined as the direct quotient of ε divided by m. In the formalism by Kellerer and Chmelevsky (Kellerer and Chmelevsky 1975b), ε is the sum of the energy transferred to the medium via all the interactions with atoms of the medium of the primary particle and its sec-ondaries. In this work the collection of all EDs generated by interactions of one primary particle including all its secondary electron cascades is defined as a track.

The frequency distribution of the specific energy ( )f z describes the probability with which a certain amount of energy is imparted in a specified target within the irradiated medium (Kellerer and Chmelevsky 1975a). Therefore, the frequency distribution for exactly ν tracks ( )f zν can be ob-tained through a summation of the stochastic distributions by convolving the single track frequency distribution 1( )f z with itself ν times, i.e. for 1ν > we have

max

1 10

( ) ( ) ( )dz

f z f z z f z zν ν − ′ ′ ′= − , (5)

where maxz is the highest value in 1( )f z . The expectation value for 1ν = is( ) ( )1 1 1d dz zf z z f z z= and it represents the mean energy imparted per

track, which is characteristic of the radiation quality and the size of the vol-

23

ume of interest. Therefore, from the “dose receiving” point of view, the mean number of tracks n needed to yield a dose D is given by

1

Dn

z= . (6)

If a population of cells has been irradiated to a dose D , the z that a cell or cell nucleus-sized volume will receive varies with a distribution deter-mined by the fluctuations in both the number of tracks passing the target (which normally follows a Poisson distribution of mean value n ), and the energy deposited per track (e.g. ( )f zν ). This distribution is known as the dose dependent frequency distribution (Kellerer and Chmelevsky 1975a),

( )0

( , ) e!

n nf z D f z

ν

νν ν

∞−

== . (7)

By definition, the expectation value z of ( , )f z D equals D and its standard deviation zσ provides a measure of the microdosimetric spread for a group of cells or cell nuclei. For the cases where 1n ≤ , the Poisson proba-bility for 0ν = is the dominating factor in equation (7) and in order to main-tain normalization of ( , )f z D , 0 ( )f z should be set equal to a Dirac delta unity distribution positioned at 0z = .

As n increases with dose, the Poisson part of equation (7) resembles a normal distribution ( , )N n n with mean n and standard deviation n , which scaled to the energy variable becomes 1 1( , )N n z n z⋅ ⋅ . At this point, the convolutions of equation (5) also converge to a normal distribution of the form

11( , )zN zν ν σ⋅ ⋅ , and as only those frequencies in equation (7) with νclose to n will contribute, equation (7) may be replaced by

( )1

2 21( , ) ( , )zf z D N z n z σ≈ ⋅ + . (8)

In the following, distributions calculated according to equation (7) will be referred to as full convolution whereas distributions calculated with equation (8) will be referred to as the normal approximation.

The derivation of equation (7) indicates that the microdosimetric spread

zσ is dependent on both radiation quality and target volume. The square root of the relative variance is the relative microdosimetric spread rel

z z Dσ σ= , used in the experimental determination of other microdosimetric properties (e.g. variance and variance-covariance methods). Nonetheless, little has been published about the magnitude of rel

zσ and/or the trends of its behaviour. The relative variance defined in ICRU (1983) as rel 2( )z Dz Dσ = where the single–event dose mean specific energy Dz is defined as the second moment of 1( )f z divided by 1z , is consistent with the expression in the normal ap-proximation

1

2 21rel

1

1 zz

z

zD

σσ

+= ⋅ . (9)

24

In papers I and II we explore the behaviour of the ( , )f z D distributions for several photon sources commonly used in brachytherapy and provide data for rel

zσ across a wide range of doses and target volumes. Paper IV extends the study of the behaviour of the rel

zσ for several proton and carbon ion ener-gies.

3.2 Other microdosimetric properties The lineal energy is defined as y lε= where ε is the imparted energy by a single track on a convex target volume with mean chord length l . This quantity is of particular interest to radiobiologists as y represents the micro-dosimetry analogous of the linear energy transfer (LET) even though the relationship between them is far more complex (Lindborg and Nikjoo 2011). The frequency distribution of lineal energy ( )f y can be calculated from the

1( )f z distribution using the relationship ( )y z m l= . The shape of the 1( )f zdepends on several stochastic factors such as the track-length distribution, the LET distribution and the straggling of the ionizing particle (ICRU 1983). Under the conditions of constant stopping power and CSDA, the ( )f y distri-bution will deviate from a characteristic triangular pattern at the lower and higher ends of y values. The triangular shape comes as a result of the well-known chord-length distribution for spheres (see review on chord-length distributions by Bradley (2000)) whereas the deviations at the lower and higher y values are due to delta electrons hitting the target and straggling of the ion itself, respectively. In figure 3 of paper IV the triangular shape can be seen for the proton and carbon ions. For the photon sources this shape is not present as the torturous path of the electrons smoothen it out. By definition, the mean lineal energy F ( )dy yf y y= approaches the unrestricted LET L∞ , values as the target size increases.

Particularly, the dose-mean lineal energy Dy defined as

2

F

( )dD

y f y yy

y= , (10)

is commonly used for radiation quality characterization and thus has been the centre of attention for some radiobiological models (Lillhök et al 2007, Lindborg et al 2013, 2015). Figure 3.1 compares published Dy values either experimentally or MC derived for a range of sphere diameters irradiated with a 60Co photon source. Our MC derived Dy values are found within experi-mental uncertainty.

25

Figure 3.1 Dose mean lineal energy Dy values shown as a function of mean chord length l for a 60Co radiation experimentally obtained with a coin-shaped detector (black square) (error bars are equal to 15% of the Dy as estimated in Forsberg and Lindborg (1981)), a wall-less sphere detector (purple up triangle) (Lillhök et al 2007), PENELOPE 2011 calculation (blue circles) (Chiriotti et al 2014), and recal-culated with the corrected MC code (pink down triangles) for this work.

The distributions ( )yd y were also calculated (see figure 3.2) where

F( ) ( )d y yf y y= is the dose probability density of lineal energy. These dis-tributions provide a more detailed description of the radiation quality as op-posed to mean values such as Dy and are often used in radiation protection works for calculating quality factors Q as a function of y instead of L∞(ICRU 1983). For 60Co a double peak is seen in the ( )yd y distribution (ap-prox. at 0.2 and 1.2 keV µm-1) for a sphere diameter of 1 µm which converg-es to the expected y ≈ 0.3 keV µm-1, as the size of the target volume increas-es. This has been previously reported (Stewart et al 2002, Chiriotti et al 2014) in PENELOPE simulations.

Figure 3.2 Spectra ( )yd y for 1 µm (left panel) and 4 µm (right panel) diameter spheres for radiation from the brachytherapy radionuclides 125I (blue line), 192Ir (pink line), and a 60Co photon source (purple line).

26

3.3 Paper I The aim is to calculate the magnitude of rel

zσ for two of the most common radionuclides in brachytherapy 125I, 192Ir, and a 60Co photon source, usually the reference in biological effectiveness studies, for a range of volumes that are representative of mammalian cell nuclei.

To accomplish this, the first step was to calculate the 1( )f z distributions for each radiation quality across a range of volumes. The code used for scor-ing energy deposition in Paper I applied cubical target volumes but after publication, this code was updated to score in spherical volumes as well. This was implemented in order to make fair comparisons to published mi-crodosimetric data from experiments commonly using spherical shaped de-tectors. We found that a change in target shape from cubic to spherical vol-ume causes a general decrease in microdosimetric values of about 15% (see discussion in the Reply to Comment on Paper I).

Briefly, the scoring of 1( )f z went as follows. Each electron track (primary and secondary interactions) is enclosed in a bounding box which is then sub-divided into a grid of cubic voxels of the desired cell nucleus-sized volume. By walking along the track we add the energy deposition of all EDs within each touched voxel. Normalization of the scored histogram yields the single-track frequency distribution of deposited energy 1( )f ε which then is trans-formed to 1( )f z by applying z mε= , where m is the mass of a cubic voxel. If the desired target volumes are spheres instead of cubes, then the grid of voxels is replaced by a (non-space filling) grid of spheres and only the EDs within each touched sphere will be scored. In order to avoid bias by placement of the electron´s first interaction, the initial position of the track was randomized which is essential for the lower photon energies as their electron tracks are of the order of the size of the target volume. The results in figures 3.3 and 3.4 were all calculated for spherical target volumes.

Subsequent corrections of the MC code led to an increase in 1z values (decreasing values of n to less than unity). Figure 3.3 demonstrates that the normal distributions become valid as either target size or dose increases.

The relzσ was calculated from the ( , )f z D distributions either by apply-

ing equation (7) for low doses or by the normal approximation (equation (8)) for higher doses. The results show that rel

zσ increases with decreasing pho-ton energy and decreasing volume, see figure 3.4.

A simple parameterization was derived for relzσ as a function of target

volume and dose

2

rel 1z k

k

V Dσ = , (11)

where 1k and 2k are parameters dependent on radiation quality (see table 1). This parameterization offers a fast estimation of rel

zσ values without the need for calculation or measurement of 1( )f z for a specific target volume.

27

Figure 3.3 Comparison of the ( , )f z D distributions at 0.05 and 1 Gy calculated with the full convolution formalism (purple dots) of equation (7) to the normal approxi-mation (pink line) of equation (8) for a 60Co irradiation of spherical target volumes of 2 and 4 µm diameter (top and bottom panels respectively).

Figure 3.4 Comparison of the relative microdosimetric spread rel

zσ as a function of dose D z= for the brachytherapy radionuclides 125I, 192Ir, and a 60Co photon source for a range of spherical volumes with diameters from 2 to 12 µm. The mean photon energy is given in the parenthesis.

The high values relzσ for the smaller sized target volumes suggests that sub-

groups of cells within an irradiated tissue may present very different biologi-cal responses and hence zσ is likely an important confounding factor in the estimation of biological outcome.

28

Table 1. Values of 1k and 2k for 60Co, 192Ir and 125I for volume V given in µm3 and dose D in Gy.

Parameter 60Co 192Ir 125I 3Gyµm1k

0.49 0.61 0.72

2k 0.40 0.39 0.38

3.4 Paper II This study is an extension of paper I as it presents further information about the 1( )f z distributions and of the range of validity for rel

zσ obtained with the normal approximation approach (equation (11)).

The results with the corrected MC show that the maximum z found in

1( )f z decreases with increasing volume regardless of radiation quality as expected (see figure 3.5). Resonance peaks are observed for 125I for a 12 µm diameter sphere as a result of a photoelectron being spent entirely within the target. This phenomenon was not seen with the faulty MC because it overes-timated mean free path.

Figure 3.5 Comparison of the single-track frequency distributions 1 ( )f z for the brachytherapy radionuclides 125I and 192Ir, and a 60Co photon source for spherical target volumes with diameters of 2 µm (dashed red line), 6 µm (blue line), and 12 µm (dotted purple line). The mean photon energy is given in parenthesis.

The Shapiro-Wilk test for normality applied to the full convolution ( , )f z D calculated for discrete doses within the range of 0.1 to 2 Gy showed that the dose threshold for normality increases with decreasing target volume, pre-senting higher values with increasing photon energy (see table 2 in the Reply to the comment). Nonetheless, for clinical dose ranges and for low dose ra-

diobiological experimentation, the relzσ parameterization is still valid as fur-

ther analysis revealed a 0.1% to 12% deviation from relzσ calculated directly

29

from the second moment of the full convolution. In low dose/low dose-rate experimental radiobiology, the skewness of ( , )f z D implicates a wide spread in specific energy which is likely to influence the biological response.

3.5 Influence of microdosimetric spread in biological outcome

An increment in relzσ is expected for the higher LET of protons and carbon

ions when compared to photon sources due to increased importance of the Poisson part of equation (7) for fewer tracks. Indeed this is illustrated in figure 5 of paper IV for the energy range of protons and carbon ions ana-lysed. The high rel

zσ values hint at the possibility that biological outcome from an in vitro biological experiment might be influenced by the uncertain-ty in z deposition from the cell´s point of view.

To find the degree of involvement of the microdosimetric spread in the LQ parameterization of biological response we hypothesize that the survival curve for a population of cell nuclei all receiving equal specific energy z can also be described by a LQ parameterization with the parameters a and b ,

( )2SF( ) expz az bz = − + . (12)

The macroscopic dose response SF( )D can be interpreted as the average of the individual SF( )z weighted by the ( , )f z D distribution for the cell popula-tion,

0

SF( ) ( , )SF( )dD f z D z z∞

= . (13)

In order to investigate the significance of ( , )f z D for an observed re-sponse given by published experimental α and β values, a and b can be determined by fitting SF( )z for a known ( , )f z D to the calculated SF( )D ,

{ }

( ) ( )fit fit

2

2 2

0

,

ArgMin ln ( , )exp d d

a b

D D f z D az bz z Dα β∞

=

− + − − +

(14)

It follows that the influence of zσ on biological effectiveness (e.g. RBE at 10% survival with 60Co as reference) can be extracted by factorizing the experimental RBE into a spread factor

zkσ and an RBEz under the theoreti-

cal assumption that each cell nucleus received equal specific energy,

RBE(10%)

RBEzz

kσ = . (15)

30

3.6 Paper IV The aim is to further analyse the impact on the biological response that zσmay have for cell populations of diverse sizes.

The scoring of 1( )f z for the proton and carbon ions has a slightly differ-ent approach to that of the photon sources (paper I and II). Briefly, a layer of spheres of volume V were placed on the plane perpendicular to the core´s axis in the middle of the scoring layer (see figure 2.3). By walking along the track (both core andδ electron branches), all EDs within any of the spheres were scored. In analogy with the 1( )f z scoring procedure for the photon sources, the track´s core was moved randomly across the projected area of the middle sphere. The 1( )f z distribution was estimated by repeating this procedure over 500 tracks per particle energy.

The results show that the value of z

kσ (equation (15)) is always less than unity suggesting that zσ has a dampening effect on RBE. However,

zkσ for

photon sources are close to unity due to the low values of relzσ (1 - 10%)

within the clinical dose range (1–10 Gy). The RBE dampening effect incre-ments with decreasing particle energy and decreasing target size, mostly following the trends of rel

zσ . When comparing RBEz results for single-sized to those for variable size targets (as for a cell nucleus distribution), it is clear that mixed target volumes will increase RBEz .

Figure 3.6 compares values of /α β ratios, for randomized mock exper-imental values of α and β , to /a b values from fitting according to equa-tion (14) assuming either a single-sized or variable sized cell nuclei popula-tion. The results further support that the degree of involvement of zσ in bio-logical outcome depends on size distribution of the target volumes and on radiation quality. Furthermore, the fact that /a b ratios are generally lower than their /α β counterparts suggests that even though a tissue that is classi-fied as an early responder (higher /α β values), may be composed of sub-populations of cell nuclei that are late responders (lower /a b values) pre-senting a wider range of radiosensitivities due only to the stochastic texture of energy deposition.

Interestingly, the results for 5.27 MeVu-1 carbon ions seem to challenge our hypothesis of the existence of a SF( )z when target volumes are small ( ≤7 µm) as the fitting procedure failed to find satisfying a and b values. In-deed, under these conditions zσ was found to be very high suggesting that most cells are not hit at all and those that are hit will have no chance of sur-vival as they would receive very high specific energy due to the passage of the track´s core. The fitting framework finds itself lacking in these “all or nothing” circumstances. As the size of the target increases, the probability of miss decreases and the model for a and b values enters the regime of valid-ity set by the experimental data.

31

Figure 3.6 Sample from figure 6 in paper IV that presents the relationship between [ ]randα β ratios chosen from random sets of { },α β and [ ]fit

a b ratios obtained with equation (14). Results for the single size distributions are shown on the left panels where as for the variable size distributions are on the right panels. The diagonal line shows a one-to-one relationship to guide the eye.

The results showed in this study should be taken as indicative of trends ra-ther than specific for the influence of the microdosimetric spread of real healthy or tumour tissues because of the lack of experimental data for cell and cell nucleus size distributions. However, the demonstrated target size dependence emphasizes the need for cell target size data for tumours and other tissues relevant for radiotherapy.

32

Nanometre characterization of radiation 4.quality

The ultimate target in cancer treatment with radiotherapy is the DNA mole-cule as it carries vital information on the functionality and reproduction of the cell and, if exposed to a damaging agent such as ionizing radiation, the normal functioning of the cell might become compromised. Radiotherapy modelling would thus benefit from finding an appropriate radiation quality descriptor that includes physical characteristics of ionizing radiation at a nanometre scale of DNA molecules. In the following sections we search for patterns in the spatial distribution of individual energy deposition transfer points (i.e. EDs) and sort them into clusters of different sizes (i.e. cluster order). For each radiation quality a frequency distribution of cluster order can be determined which can be thought of as the fingerprint of the interac-tion of that particle with matter (liquid water in this work). Paper III contains a broad analysis of the properties of the cluster patterns for the most com-monly used brachytherapy sources. Paper V extends this analysis to tracks from several energies that lie at the distal end of the Bragg peak of protons and carbon ions.

4.1 Nearest Neighbour distributions The characterization of the tracks at a nanometre scale begins with scoring the distances between neighbouring EDs. This gives an overview of the spa-tial distribution of the location at which ionizations and excitations occurred which in turn expands the understanding of the organization of the tracks.

The algorithm for finding the first nearest neighbour distances ( NN1d ) is as follows: 1. Let m be the total number of EDs in a track. Calculate the distances

from EDi to EDk for all k between 1i + and m , starting with 1i = . 2. Find the minimum distance ( )NN1 i

d in this pool of distances. 3. Score ( )NN1 i

d in a histogram with the predefined distance bin-width dΔ .

4. Repeat from 1 where i increments by one in each iteration until 1i m= − .

33

Step 2 can be generalized to determine the second NN2d , third NN3d , etc. neighbour distances. Normalization of the resulting histograms by dΔ and by the total energy deposited ε yields the frequency distribution of the n-th nearest neighbour (NN) per deposited energy NN( ) /nf d ε . Hence, the proba-bility of finding the nearest ED per deposited energy within a distance d and d d+ Δ is given by NN( ) /nf d dε ⋅ Δ . These frequencies are independent of target volume geometry.

Figure 4.1 compares the NN1( ) /f d ε for 1 MeVu-1 protons to 98.75 MeVu-1 carbon ions for the 1st NN. Both of these radiation qualities have the same LET (25.7 keVµm-1). The NN1( ) /f d ε frequency for a 60Co photon source is also shown for comparison. It is clear that at a nanoscale, iso-LET radiations have different spatial patterns such that longer distances to the NN are more frequent for carbon ions. The EDs with inter-distances longer than 3 nm (same decay rate) belong mostly to the δ electron interactions for carbon ions. In contrast, the δ electrons from 1 MeVu-1 protons will have a shorter mean free path which means that the proton tracks are more densely ionizing than the carbon ions, despite having equal LET.

Figure 4.1 Comparison of the frequency of the first nearest neighbour per deposited energy for the iso-LET (25.7 keVµm-1) radiation qualities of 1 MeVu-1 protons and 98.75 MeVu-1 carbon ions. The frequency for 60Co photons is also shown for com-parison.

The NN distributions reveal interesting features of the patterns in energy deposition intrinsic to the radiation quality under study. Moreover, they can also be used to study inter or intra-track relations. For multiple tracks, the steps described above were applied to the union of all EDs within a target disregarding the track which it belongs to. Paper III compares NN( ) /nf d εdistributions calculated from single-track to multiple-track approaches for commonly used brachytherapy sources.

34

4.2 Frequency of cluster order A further step into the characterization of ionizing radiation at a nanometre scale is to group the EDs into spatial clusters. There is no unique way for doing this. Some published methods rely on determining the amount of EDs within a predetermined volume (Charlton et al 1985, Brenner and Ward 1992, Reniers et al 2004). For some radiation qualities where the EDs are densely packed across relatively larger volumes, such methods will bias the cluster count towards smaller sized clusters as the predetermined scoring volume may be too small in comparison. In contrast, the methodology used in this work depends on a single parameter; the cluster distance cd defined as the maximum allowed distance between two neighbouring EDs belonging to a cluster. A graphical representation of the cluster definition is given in fig-ure 4.2 and the algorithm for its determination from track data is as follows: 1. Start a new cluster by putting an unclassified ED into a clean cache. 2. Search for all unclassified neighbouring EDs at a distance cd d≤ , add

them to the cache. 3. Let i be the number of EDs cached in 2. If 1i ≥ repeat 2 for each of

them. 4. When no more neighbours at cd d≤ are found, close the cluster at cluster

order (CO) equal to its number of cached EDs. 5. Repeat from 1 until all EDs are classified. The histogram of the amount of clusters of certain CO normalized per im-parted energy yields the so called frequency of cluster order (CO) /

cdf ε . The choice of cd influences the shape of the (CO) /

cdf ε distribution as presented in figure 4.3. As expected, the frequencies will decrease with in-creasing CO at a slower rate with a larger cd . Even an increase in a few nm (from 2 to 5 nm) can increment the frequency of clusters of a given CO by one order of magnitude.

An important result from figure 4.3 is that the (CO) /cdf ε data differenti-

ates iso-LET radiations (1 MeVu-1 protons and 98.75 MeVu-1 carbon ions). This is in line with published suggestions that variation in biological effec-tiveness may be correlated to cluster frequencies (Kraft 2000).

35

Figure 4.2 Illustration of the cluster method used for calculation of the frequency of cluster order. This example uses c 2d = nm (length of arrows) to group a total of 13 EDs (represented by the stars) into two CO 1= clusters (black stars), one CO 3=cluster (purple stars), and one CO 8= cluster (blue stars). With this methodology the clusters are not restrained within a predefined scoring volume but rather intrinsic to the track´s texture.

Figure 4.3 Comparison of the frequency of cluster order calculated with the criterion

cd equal to 2 nm (left panel) and 5 nm (right panel) for the iso-LET (25.7 keVµm-1) radiation qualities of 1 MeVu-1 protons and 98.75 MeVu-1 carbon ions.

Part of the discussion in paper V focuses on comparing the size of some the DNA structures to a geometrical property of the clusters i.e. the mean cluster length COL . The clusters as such are just a collection of points in space and several definitions of its effective length are possible. In this work the cluster length is defined as the maximum distance between two EDs along the clus-ter´s main axis identified by means of the correlation matrix of the ED loca-tions. Translation and rotation of a sample of 100 clusters of the same CO

36

yielded a distribution of COL per radiation quality out of which COL could be estimated.

4.3 Paper III The aim is to study the differences in the spatial patterns of energy deposi-tion for the most commonly used brachytherapy radionuclides 103Pd, 125I, 192Ir, 137Cs, and a 60Co photon source. Analysis of the frequencies

NN( ) /nf d ε and (CO) /cdf ε under two conditions: multi-track which anal-

yses the accumulated EDs (regardless of the track they originated from) in a given volume irradiated with a dose D ; and single-track which analyses the EDs within a single track without the constriction of volume.

Publication of paper III was done before the error in the MC code was de-tected. Corrected results are given in the corrigendum and they did not change the major findings here outlined.

Figure 4.4 Comparison between the frequency distributions of the first nearest neighbour NN1( ) /f d ε normalized per deposited energy ε for 103Pd (left panel) and 60Co (right panel) photon sources. The frequencies have been calculated for 2 Gy (filled circles) and 5 Gy (filled triangles), and from single-track scoring (empty circles). For comparison the frequency for a uniform random distribution of points in space (empty squares) is also shown.

The NN( ) /nf d ε calculated with the single-track approach continually de-creases with distance as the mean free path increases when comparing the largely abundant track-enders to that for the first few electron interactions. As the dose increases (multi-track approach), the corresponding NN( ) /nf d εseparates from the single-track frequency as seen in the right panel of figure 4.4. This component resembles the shape of a uniform random point distri-bution. The location of the separation of the multi-track from the single-track

NN( ) /nf d ε frequencies happens at smaller distances as dose increases which can be interpreted as the distance at which inter-track interactions begin to occur. However, the random component is barely visible for the lowest energy photon source (see left panel in figure 4.4) which indicates

37

that higher doses (> 5 Gy) are needed to start observing inter-track interac-tions.

On the other hand, the results from the cluster analysis show that the(CO) /

cdf ε frequency was able to discern pattern differences between the studied radiation qualities. For instance, the frequencies for CO 3≥ for the lowest photon energy source (103Pd) are about 18% higher than the frequen-cies for 60Co for c 2d = nm and 40% for c 10d = nm, respectively. This dif-ference drops to about 3% and 6% when comparing the frequencies for 137Cs to those for 60Co. Particularly, the higher frequency of cluster order for

c 2d = nm for 125I and 103Pd agree with the observed higher biological effec-tiveness of these radionuclides, which suggests that (CO) /

cdf ε could be a good radiation quality descriptor for radiobiological purposes.

38

RBE modelling 5.

The growing use of protons and other light ions in cancer treatment radio-therapy motivate the development of more accurate predictors for the rela-tive biological effectiveness (RBE). RBE is a well-defined quantity obtaina-ble from radiobiological experiments. Once extracted, it can conveniently be applied as a weighting factor for dose in radiobiology-based treatment plan-ning systems. In the following section we describe the framework for an RBE prediction model based on the frequency of cluster order. Paper V suc-cessfully applies this framework to effectively correlate the predicted RBE with experimental RBE, proving the usefulness of the cluster order frequen-cy as a radiation quality descriptor.

5.1 Framework using cell survival data as endpoint One may assume that the mean number of lethal events (LE) to a cell due to the initial DNA damage is proportional to the absorbed dose. In the LQ model this proportionality is represented by the α parameter. It has also been proposed that the clustering of energy deposition sites characteristic of the stochastic nature of ionizing radiation is critical to the biological re-sponse (Uehara et al 2001, Champion 2003, Emfietzoglou et al 2006, Liamsuwan and Nikjoo 2013) e.g. the initial DNA damage. It follows that the probability for a linear dependence of LE versus dose could be reformu-lated to be proportional to the number of EDs per mass m , caused by the dose. According to our definition of the frequency of cluster order CO,Qf for radiation quality Q (note a slight change in notation from the previous chap-ter) the mean total number N of EDs for an imparted energy ε may be ex-pressed by

CO,Q

CO

CON fε= ⋅ ⋅ . (16)

The construction of a function for α employing equation (16) assumes the existence of a weighting function CO,w α independent of radiation quality that describes the cell’s sensitivity to the occurrence of a LE generated via a cluster of order CO. Under these circumstances the probability for causing an LE per cell mass m can be expressed as

39

CO, CO,Q

CO

COw f

Dm

αεα

⋅ ⋅ ⋅=

. (17)

By setting D mε= , one gets

( )Q CO, CO,QCO

COw fαα = ⋅ ⋅ . (18)

A direct analogy of equation (18) with an independent set of weights CO,w βwill be used for the quadratic term of LQ model

2

Q CO, CO,QCO

COw fββ

= ⋅ ⋅ . (19)

Both CO,w α and CO,w β have the dimension of mass and a simple interpretation is that they represent how much mass a cell (cell nucleus) uses to capture EDs from the dose per CO to cause an LE. If the weighting functions CO,w α and CO,w β exist, equations (18) and (19) constitute a simple way for predic-tion LQ parameters provided the knowledge of the “physics only” parameter i.e. CO,Qf for a given radiation quality.

One may expect that CO,w α and CO,w β depend on cell type since they can be thought of as representative of the cell’s radiosensitivity. In a clinical context, this can be overcome by assuming that the sensitivity for various cell lines scales equally across the CO,Qf frequencies and that the experi-mental LQ values for the reference radiation (i.e. x-rays or 60Co) are known for the cell line in question. Eventually equation (18) could then be used to estimate the α value for any radiation quality from

( )( )exp

CO, CO,QCO

Q RCO, CO,R

CO

CO

CO

w f

w f

α

αα α

⋅ ⋅=

⋅ ⋅

, (20)

where the sole dependency on cell line is through the experimentalexpRα

value. However, such development of the framework is beyond the current scope and for now this thesis concentrates on the determination of the CO,w αand CO,w β functions by applying a fitting procedure for a specific set of pub-lished LQ experimental data.

From the LQ equation the expression for RBE as a function of survival fraction SF follows as

40

( )

( )QQ

2R R

2R RR

SF2

Q Q2

Q

4Log SF

RBE4

Log SF

α αβ ββ

α αβ ββ

− − = − −

. (21)

By substituting the proposed surrogate functions for α and β parameters

(equations (18) and (19)) into equation (21) we obtain a straightforward function for relating RBE to CO,w α and CO,w β values. This approach will be used as the core of an empirical RBE model based on track structure proper-ties of the ionizing radiation.

5.2 Experimental data Clonogenic cell survival assays provide a simple method for measuring a cell´s capacity of proliferation as a function of dose after exposure to ioniz-ing radiation. Survival curves can be influenced not only by the radiation quality but also by many other variables such as dose rate, cell cycle phase or temperature at time of exposure. The comparison between published data becomes a problem as the biological assay protocol is often modified to adapt specific experimental conditions. This makes the study of biological effectiveness challenging.

Nonetheless, for our analysis purposes we have selected a group of 15 in vitro experimental survival data for the cell line V79 (Chinese hamster lung cells) that were judged to have used similar enough protocols. Table 2 con-tains experimental LQ parametersα and β for the brachytherapy radionu-clides 103Pd, 125I, 192Ir, and 137Cs; 6 proton energies in the range of 0.91 to 5 MeVu-1 and 4 carbon ion energies in the range of 4 to 77 MeVu-1, and a 60Co photon source for reference purposes.

However, for 103Pd, and 125I, data was obtained from CCL16 (also known as DON) hamster cells which are equivalent to V79 except that CCL16 is susceptible to the vesicular stomatitis virus as reported by the General Cell Collection division from the Public Health England. This should not yield any impediment for our analysis. It is also worth to point out that Nath et al (2005) studied the variability of α values as a function of dose rate. We chose to use the results for the lowest dose rate 6.9 cGy h-1 studied by them. Since the COf is representative of single tracks, the damage inflicted by the low dose rate sources seem more appropriate.

41

Table 2. Published LQ and parameters from experimentally obtained survival data for V79 cell line. The calculated RBE for the commonly used 2 Gy fraction dose with 60Co as reference is also given. The colour code will be used for some plots.

Radiation Quality (MeV)

Colour Code

α (Gy-1) β (Gy-1) RBE2Gy Refernces

103Pd (0.021) ● 0.300+0.050 0.000+ns 1.22+0.21 Nath et al (2005)* 125I (0.028) ● 0.220+0.040 0.000+ns 0.89+0.17 Nath et al (2005)* 192Ir (0.360) ● 0.330+0.013 0.0093+ns 1.39+0.05 Fritz et al (1996) 137Cs (0.615) ● 0.301+0.011 0.000+ns 1.22+0.04 Fritz et al (1996) 60Co (1.250) ● 0.188+0.019 0.029+0.002 1.00+0.06 Stenerlöw et al (1995)

(MeVu-1)

H+ (0.91) ▲ 0.740+0.025 0.011+0.004 3.03+0.10 Folkard et al (1996) H+ (1.40) ▲ 0.469+0.029 0.043+0.009 2.07+0.10 Belli et al (1998) H+ (1.72) ▲ 0.450+0.035 0.028+0.006 1.94+0.13 Folkard et al (1996) H+ (3.18) ▲ 0.372+0.032 0.036+0.009 1.68+0.13 Belli et al (1998) H+ (3.59) ▲ 0.320+0.058 0.039+0.011 1.51+0.22 Folkard et al (1996) H+ (4.97) ▲ 0.289+0.023 0.024+0.006 1.32+0.09 Belli et al (1998) C6+ (4.06) ■ 0.639+0.140 0.039+0.035 2.71+0.48 Weyrather et al (1999) C6+ (5.27) ■ 0.990+0.081 -0.022+0.02† 4.02+0.36 Weyrather et al (1999) C6+ (10.95) ■ 0.910+0.096 0.044+0.033 3.79+0.41 Weyrather et al (1999) C6+ (76.90) ■ 0.337+0.073 0.025+0.011 1.50+0.26 Weyrather et al (1999)

* CCL16 hamster cell line was used in these experiments † In this thesis work the β value is taken equal to zero ns = not stated in reference

RBE values can be calculated from LQ data for any survival fraction as de-scribed by equation (21). Nonetheless, to allow a straightforward inter-comparison between the biological outcome of the reference (generally 60Co) and test radiations (i.e. protons or carbon ions) it has been suggested that RBE values be calculated for a dose of 2 Gy. 2GyRBE thus becomes repre-sentative of the late tolerance of normal tissues (Wambersie 1999, Wambersie et al 2002, Dale et al 2009).

5.3 Paper V The aim is to explore the use of the frequency of CO to construct a function for the calculation of RBE with cell survival as an endpoint. To predict α and β values with our model framework, the data in table 2 will be used as a training set for determination of the weighting function COw . An objective function simultaneously incorporates values for α , β and 2GyRBE over all radiation qualities Q is

42

{ }( )

( )( )( )( )

CO, CO,

2exp2Gy,Q

1,2Gy,Q CO, CO,

CO, CO, CO,,

2exp2, QQ

2eQ

Cxp

3, Q ,Q O

,

, argmi

BE

R

n

R

BEc

c

c

d

d

wd

w

w w

w w w

wα β

α β

α β α

β

α α

β β

− Ω

=

+Ω −

+ −

Ω

, (22)

where iΩ are (somewhat arbitrarily chosen) importance weights for each term type. The modα and modβ are given by equations (18) and (19), re-spectively. The mod,2GyRBE is given by equation (21) where Rα , Qα , Rβ , and Qβ are also expressed in terms of equations (18) and (19).

The results indeed demonstrate that the weighting functions COw for both theα and β are independent of radiation quality. As explained in section 4.2, progressively decreasing the value of cd causes the regrouping of EDs into clusters of smaller orders, e.g. “breaking” clusters of higher order into small-er cluster elements. This suggests that an optimum cd (or range of cd ) may exist such that the grouping of EDs have the highest chance for causing damage. Indeed, by performing the above fitting with COf frequencies calcu-lated for a range of cd ( 0.5 3.0cd≤ ≤ nm) that covers the vicinity of DNA´s double strand we were able to find that COL for clusters with cd = 2.5 nm, “resonate” with some of the DNA structures and hence are prone to inflict lethal damage (see figure 5.1). Moreover, the relative residuals for the corre-lation of α and RBE values for the interval 2.0 3.0cd≤ ≤ were the lowest compared to the rest of the cd studied range.

Figure 5.1 Distribution of the modelledα components as a function of CO for the highest and lowest particle energy analysed in this work. For comparison also the estimated mean cluster length COL is shown.

43

Figure 5.1 shows that all radiation qualities peak at CO = 3 ( CO 2L ≈ nm) making them candidates for DNA double strand breakage and their abun-dance makes them all the more relevant. A second peak is visible at about CO = 25 ( CO 11L ≈ nm about the size of the nucleosome) being particularly prominent for the lowest proton energy (0.91 MeVu-1). Clusters of higherCO ( CO > 20 ), although far less abundant, still play a role in DNA damage suggesting that limiting nanodosimetry to volumes of the size of the double helix of the DNA will not be enough to accurately estimate biological re-sponse.

The satisfactory correlation between the predicted α and 2GyRBE to their experimental counterparts across several types of radiation (see figure 5.2) further supports our hypothesis that a predictive RBE model based on the clustering properties of the discrete energy deposition sites is possible.

Figure 5.2 Correlation of the modelled Qα and QRBE calculated with equations (18) and (21) to their experimental counterparts given in table 2. The colour and symbol code is also given in table 2.

5.4 Future perspectives for the model The positive results of paper V encourage further development of the pro-posed framework through various tests described below.

An extended test for its applicability and robustness would be to use it for a wider proton energy range covering the entire Bragg peak and the entrance channel. Protons with energies above 100 MeVu-1 would have RBE ratios very close to unity, forming an important test point.

This work has used experimental data for V79 cells as a training database for estimating the COw function. It would be of interest to test this framework with sets of experimental data form other cell lines. This would enable more accurate determination and mapping of the applicability of the COw function. Indeed, the assumption that the radiosensitivity of the cell scales across fre-

44

quency of clusters (equation (20)) would be tested. Unfortunately mapping of high quality survival data across radiation qualities represents a real chal-lenge because of numeral experimental difficulties. The lack of reliable sets of data does represent a hinder for the development of this framework thus dedicated radiobiological experiments is strongly encouraged.

An issue that remains to be solved is the poor correlation between pre-dicted and experimental β values (see figures 5 and 6 in paper V). A possi-ble cause is lack of variability in CO,w β (see figure 3 in paper V). Thus func-tion for the prediction of β (equation (19)) needs some fine-tuning for which the general approach of setting equal values for all radiation qualities may be a way to overcome this shorcomming.

Finally, the results in paper IV point to a potential benefit of including the microdosimetric spread into radiobiological modelling. The framework pro-posed here may account for variation in specific energy of the cell popula-tion by incorporation of a weighting factor for zσ into the predictive func-tion for α (equation (18)). Perhaps the use of a and b values (see equation (12)) instead of α and β for estimation of COw for a specific energy z might help shed some light to this issue.

Overall, this framework offers the possibility for fast RBE calculations by simple interpolation of COf from pre-calculated look-up tables provided the knowledge of the energy fluence at the target site. This is of particular inter-est in the context of mixed radiation beams where prediction of the variation in RBE may be most useful for improving the treatment outcome.

45

Summary and Outlook 6.

Ionizing radiation is commonly characterized by the linear energy transfer (LET) defined as the quotient of the mean energy spent per travelled dis-tance of the primary particle in a medium. LET has been found to correlate with biological response from in vitro experiments, although with limited accuracy (see e.g. reviews by Sørensen et al (2011) and Paganetti (2014)). There is increasing evidence that the micro- and/or nanometric distributions of energy depositions can yield better correlation to biological effectiveness (Nikjoo et al 2006, Kraft 2000). Integration of an accurate model for relative biological effectiveness (RBE) undeniably would improve clinical treatment planning. However, there is still a need for more reliable learning data for such tools that would aid in the understanding of the relationship between appropriate quality descriptors and the biological response for different radi-ation qualities.

The aim of this thesis is to investigate the track structure properties of several radiation qualities at a micro- and nanometric scale and explore their influence with respect to biological response. Particularly, the use of the frequency of energy deposition cluster size as basis of the framework for constructing a semi-mechanistic RBE predictive model has been investigat-ed. This thesis was based on Monte Carlo (MC) track structure simulations of a set of 15 different radiation qualities including commonly used brachy-therapy sources, protons and carbon ions. Particularly, the MC code LIonTrack (Bäckström et al 2013) corrected for two critical errors found within the code, was used to simulate the ionizing radiation tracks.

The microdosimetric properties presented in papers I, II, and IV required the development of custom-made analysis codes based on the formalism of Kellerer and Chmelevsky (Kellerer and Chmelevsky 1975b, 1975a). Particu-larly, an extensive analysis of the quantification of the stochastic fluctuation of energy deposition expressed by the specific energy was carried out. Pa-pers I and II demonstrate that distribution of specific energy for a given dose D, whose standard deviation is equal to the microdosimetric spread, could be calculated with a less cumbersome approach as it approximates a normal distribution after a certain dose threshold. This led to a convenient parame-terization of the microdosimetric spread as a function of dose and target vol-ume size, consistent with the expression for relative variance of specific energy given by ICRU (1983). The results in paper I show that the magni-tude of the spread increases with decreasing dose and target volume regard-

46

less of radiation quality. Moreover, the spread increased with decreasing particle energy. Although paper I and II analyse only 125I, 192Ir and 60Co pho-tons, equal trends were also observed for proton and carbon ion beams (pa-per IV).

The influence of the microdosimeteric spread on the linear-quadratic pa-rameterization of biological response was studied in paper IV. Since there is a strong dependence on size of the target, this paper focused on the impact of the added target volume variability within a tissue which is usually neglected in radiobiological and microdosimetric analysis. The results suggest that the microdosimetric spread dampens the differences in biological effectiveness of the ionizing radiation. The degree of the dampening effect follows the relationship between the microdosimetric spread and particle type and ener-gy; with carbon ions being the most influential. A conclusion of paper IV was that the influence of the configuration of cell sizes, expressed through the resulting microdosimetric spread, could be a confounding factor for the development of radiobiological modelling.

Papers III and V are based on a nanometric scale characterization of the spatial cluster patterns of EDs. A single parameter clustering method was used to calculate the frequency of cluster order for a given radiation quality. Paper III concentrates on an analysis of the cluster patterns of several brachytherapy sources (103Pd, 125I, 192Ir, and 137Cs) and 60Co photons. The cluster characterization was able to detect a difference up to 20 % between the cluster frequencies of the lower energy sources and that of the 60Co pho-tons.

A framework for the prediction of RBE based entirely on the nanometric cluster characterization has been proposed in paper V. The framework ex-plores the possibility of assuming the existence of a radiation quality inde-pendent weighting function COw that quantifies a cluster´s capacity to pro-duce a lethal event in the cell. A fitting procedure was developed using a set of 15 Chinese hamster cell survival data from different radiation qualities. Overall, the modelled RBE values correlated to their experimental counter-parts. The findings of paper V prove that the cluster frequency is a quality descriptor that can be used for radiobiological modelling.

The simplicity of the proposed framework for RBE prediction makes it appealing for radiobiological modelling, as it relies on the knowledge of only two functions, one describing the physics of the impinging radiation and another describing, to a certain extent, the sensitivity of the cell´s viabil-ity towards ED clusters of different sizes. Furthermore, the framework´s positive results obtained from the collective modelling RBE for photons, protons and carbon ions indicate its uniqueness among other radiobiological models that concentrate usually on one type of particle.

Future work will concentrate on the derivation and analysis of the COwfunctions for different human tissues as well as investigating the incorpora-tion of other microdosimetric properties into the predictive model.

47

Acknowledgements 7.

There is no greatest gift than that of knowledge for any young researcher and I am forever grateful to have received such a gift from my two supervisors Anders Ahnesjö and Nina Tilly. Anders, thank you for having your door always open for answering my numerous questions and for the all the mind-boggling discussions. I admire your way of looking at any scientific problem from a 1000 angles; surely one is bound to solve the problem! Nina, your strength and your hunger for life are and will be an eternal source of inspira-tion not only for me but for all the lives you have touched. You have shown me that anything is possible if you set your mind to it and for that I am very grateful.

To José my unofficial supervisor, thank you for encouraging me to con-tinue my scientific journey up north, surely your love for Sweden rubbed off on me. Gracias por las gratas tertulias en Barcelona, Lund y Uppsala.

To Shirin and Piotr, thank you for all the scientific discussions which surely improved my thesis work.

To Dr. María-Ester Brandan and Dr. Patricia Ostrosky my former super-visors and mentors, thank you for sharing your passion of physics and biolo-gy and encouraging me at every step of my academic trajectory.

Jag skulle vilja tacka till alla mina gamla och nuvarande kollegor på Sjukhusfysik Avdelningen. Tack för era talamod när jag försökte prata svenska med er och för ett chans att lära mig språket och kultur under fika. Sverige har blivit mitt ny hem.

To my fellow PhD students Calle, Kenneth, David T., and Gloria (now graduated) I am grateful for your kindness and for receiving me with open arms when I first arrived and for the many scientific and not so scientific discussions always with a touch of laughter and good beer at hand. To the newest PhD accomplice Eric and fellow researcher David B. thanks for lis-tening to my continuous ranting about my project and for your words of encouragement.

My dearest PhD ladies Maryam, Vicky, Tonge, Ruta, and Samir truly your presence in my life has made my journey through this PhD lighter and full of joy. Thank you for your unconditional friendship.

I have no words to express my deepest gratitude to my Mexican/Swedish family Brunåker. Alicia y Svante les agradezco que me hayan abierto no solo su casa sino su corazón. Ivonne, tu amistad la llevaré por siempre conmigo y

48

recordaré con mucho cariño las ricas cenas que siempre me hicieron sentir-me como en casa.

To all my international friends in Uppsala I thank you for letting me get to know all your cultures, truly you have made my time in Sweden unforget-table.

A mis queridas Valeria, Damaris, Rosalía, Jacaranda y Camila gracias por compartir estos años conmigo. Su amistad ha sido esencial en mi vida lejos de casa ya que con ustedes siempre pude compartir tanto mis sueños como mis miedos recibiendo siempre consejos. Las quiero enormemente chicas.

Adiv, Marcelino, Xoch y Hermes, gracias por la amistad que me han brindado desde hace ya más de … bueno muchos años. Fue increíble com-partir con ustedes la experiencia de hacer un doctorado en Europa.

A mi amada gente en México, Carina, Gina, Eduardo, Cyn y Fer que me han demostrado que no importa la cantidad de kilómetros que nos separan, el cariño y apoyo siempre han estado ahí.

To my beloved Miloš, I am so unbelievably happy that our lives have crossed, your love has given me strength to continue pushing me to be a better person. I cannot thank you enough for your patience because truly, the completion of this thesis would have been impossible without you.

To my lovely sister, thank you for being there and I hope to share your many achievements as you also start to fly.

Papá y mamá, how in the world will I ever thank you for your love and support in any endeavour I take. All that I am is because of you. Papá gracias por ser esa voz de aliento y gracias mamá por ser mi gran fuente de inspira-ción.

49

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Acta Universitatis UpsaliensisDigital Comprehensive Summaries of Uppsala Dissertationsfrom the Faculty of Medicine 1188

Editor: The Dean of the Faculty of Medicine

A doctoral dissertation from the Faculty of Medicine, UppsalaUniversity, is usually a summary of a number of papers. A fewcopies of the complete dissertation are kept at major Swedishresearch libraries, while the summary alone is distributedinternationally through the series Digital ComprehensiveSummaries of Uppsala Dissertations from the Faculty ofMedicine. (Prior to January, 2005, the series was publishedunder the title “Comprehensive Summaries of UppsalaDissertations from the Faculty of Medicine”.)

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