Floriane Poignant

Hela Charfi

Chen-Hui Chan

Elise Dumont

David Loffreda

Etienne Testa

Benoît Gervais

Michael Beuve
email: m.beuve@univ-lyon1.fr

Étienne Testa

Benoit Gervais

Michaël Beuve

Benoit Gervais

Michaël Beuve

Monte Carlo simulation of free radical production under keV photon irradiation of gold nanoparticle aqueous solution. Part I: Global primary chemical boost

Keywords: Validation, Investigation, Writing -Original -draft, Writing -Review and editing Hela Charfi, Software, Validation, Investigation, gold nanoparticles, photon irradiation, water radiolysis, radiation-chemical yields, Monte Carlo simulation

Introduction

The use of nanoparticles in cancer therapy has been of high interest since the pioneering work of Hainfeld et al. [START_REF] Hainfeld | The use of gold nanoparticles to enhance radiotherapy in mice[END_REF], demonstrating the efficacy of the combination of injected gold nanoparticles (GNPs) with X-ray irradiation for mice with EMT-6 carcinoma tumors. The mechanisms of nanoparticle enhanced radiotherapy may originate from complex physico-chemical and biological steps which depend on many parameters, such as the composition of the NP, the energy of the irradiation beam, and the biological system [START_REF] Her | Gold nanoparticles for applications in cancer radiotherapy: Mechanisms and recent advancements[END_REF]. The relative contribution of these different steps, and the connection between them, still needs to be clarified [START_REF] Brun | Actual questions raised by nanoparticle radiosensitization[END_REF]. As the time and spatial scales involved in the early mechanisms are very short, a clear experimental identification of the origin of the effect is not straightforward and remains under investigation. In this context, theoretical approaches, such as Monte Carlo (MC) simulations, may help to better understand how early physical and chemical stages could impact a biological system and lead to cell death enhancement.

Many MC studies have focused on the physical stage, to study the dose deposition distribution at tumor scale [START_REF] Cho | Estimation of tumour dose enhancement due to gold nanoparticles during typical radiation treatments: a preliminary monte carlo study[END_REF][START_REF] Cho | The dosimetric feasibility of gold nanoparticle-aided radiation therapy (gnrt) via brachytherapy using lowenergy gamma-/x-ray sources[END_REF][START_REF] Zhang | Quantifying tumor-selective radiation dose enhancements using gold nanoparticles: a monte carlo simulation study[END_REF][START_REF] Montenegro | Monte carlo simulations and atomic calculations for auger processes in biomedical nanotheranostics[END_REF][START_REF] Mesbahi | Effect of photon beam energy, gold nanoparticle size and concentration on the dose enhancement in radiation therapy[END_REF][START_REF] Amato | Gold nanoparticles as a sensitising agent in external beam radiotherapy and brachytherapy: a feasibility study through monte carlo simulation[END_REF][START_REF] Sinha | Brachytherapy application with in situ dose painting administered by gold nanoparticle eluters[END_REF][START_REF] Koger | A method for converting dose-to-medium to dose-totissue in monte carlo studies of gold nanoparticle-enhanced radiotherapy[END_REF][START_REF] Koger | Optimization of photon beam energies in gold nanoparticle enhanced arc radiation therapy using monte carlo methods[END_REF][START_REF] Mcmahon | Radiotherapy in the presence of contrast agents: a general figure of merit and its application to gold nanoparticles[END_REF], and at both cellular and sub-cellular scale [START_REF] Jones | Estimation of microscopic dose enhancement factor around gold nanoparticles by monte carlo calculations[END_REF][START_REF] Cai | Investigation of the effects of cell model and subcellular location of gold nanoparticles on nuclear dose enhancement factors using monte carlo simulation[END_REF][START_REF] Douglass | Monte carlo investigation of the increased radiation deposition due to gold nanoparticles using kilovoltage and megavoltage photons in a 3d randomized cell model[END_REF][START_REF] Mcnamara | Dose enhancement effects to the nucleus and mitochondria from gold nanoparticles in the cytosol[END_REF][START_REF] Kirkby | Targeting mitochondria in cancer cells using gold nanoparticle-enhanced radiotherapy: A monte carlo study[END_REF]. Both keV and MeV photon energies were investigated [START_REF] Mcmahon | Biological consequences of nanoscale energy deposition near irradiated heavy atom nanoparticles[END_REF][START_REF] Mcmahon | Nanodosimetric effects of gold nanoparticles in megavoltage radiation therapy[END_REF], and hadrontherapy combined with GNPs was also studied [START_REF] Lin | Gold nanoparticle induced vasculature damage in radiotherapy: comparing protons, megavoltage photons, and kilovoltage photons[END_REF]. Nanoparticles made of heavy atoms with high atomic number (Z) such as metals are more efficient than water at absorbing low energy (keV) X-rays. For energies up to ∼ 500 keV, the photo-electric effect dominates in metals, resulting in a particularly high probability of photon-metal inter-action compared to photon-water interaction, and consequently an increase in the macroscopic (tumor) dose deposition [START_REF] Cho | Estimation of tumour dose enhancement due to gold nanoparticles during typical radiation treatments: a preliminary monte carlo study[END_REF][START_REF] Cho | The dosimetric feasibility of gold nanoparticle-aided radiation therapy (gnrt) via brachytherapy using lowenergy gamma-/x-ray sources[END_REF][START_REF] Zhang | Quantifying tumor-selective radiation dose enhancements using gold nanoparticles: a monte carlo simulation study[END_REF][START_REF] Montenegro | Monte carlo simulations and atomic calculations for auger processes in biomedical nanotheranostics[END_REF][START_REF] Mesbahi | Effect of photon beam energy, gold nanoparticle size and concentration on the dose enhancement in radiation therapy[END_REF][START_REF] Amato | Gold nanoparticles as a sensitising agent in external beam radiotherapy and brachytherapy: a feasibility study through monte carlo simulation[END_REF][START_REF] Sinha | Brachytherapy application with in situ dose painting administered by gold nanoparticle eluters[END_REF][START_REF] Koger | A method for converting dose-to-medium to dose-totissue in monte carlo studies of gold nanoparticle-enhanced radiotherapy[END_REF][START_REF] Koger | Optimization of photon beam energies in gold nanoparticle enhanced arc radiation therapy using monte carlo methods[END_REF][START_REF] Mcmahon | Radiotherapy in the presence of contrast agents: a general figure of merit and its application to gold nanoparticles[END_REF]. In particular, studies showed that a significant dose enhancement may occur when the concentration of GNP inside the tumor is high enough. For example, 7 mg gold/g tumor could result in a dose enhancement as high as 110 % for 140 kVp and 70 % for I-125 gamma rays [START_REF] Schuemann | Roadmap to clinical use of gold nanoparticles for radiation sensitization[END_REF]. Strategies were discussed to use GNPs as contrast agents for human eye choroidal melanoma [START_REF] Asadi | Gold nanoparticle-based brachytherapy enhancement in choroidal melanoma using a full monte carlo model of the human eye[END_REF], prostate cancer brachytherapy [START_REF] Martinov | Heterogeneous multiscale monte carlo simulations for gold nanoparticle radiosensitization[END_REF], ARC therapy applied to brain tumors [START_REF] Koger | Optimization of photon beam energies in gold nanoparticle enhanced arc radiation therapy using monte carlo methods[END_REF] or as a tumor vascular disrupting agents (VDAs) [START_REF] Ngwa | Applying gold nanoparticles as tumor-vascular disrupting agents during brachytherapy: estimation of endothelial dose enhancement[END_REF][START_REF] Ngwa | Gold nanoparticle-aided brachytherapy with vascular dose painting: Estimation of dose enhancement to the tumor endothelial cell nucleus[END_REF][START_REF] Berbeco | Localized dose enhancement to tumor blood vessel endothelial cells via megavoltage x-rays and targeted gold nanoparticles: new potential for external beam radiotherapy[END_REF][START_REF] Amato | Monte carlo study of the dose enhancement effect of gold nanoparticles during x-ray therapies and evaluation of the anti-angiogenic effect on tumour capillary vessels[END_REF][START_REF] Lin | Gold nanoparticle induced vasculature damage in radiotherapy: comparing protons, megavoltage photons, and kilovoltage photons[END_REF]. Another study suggested the use of radioactive low-dose rate GNPs, to improve the current low-dose rate brachytherapy strategies [START_REF] Laprise-Pelletier | Intratumoral injection of low-energy photon-emitting gold nanoparticles: A microdosimetric monte carlo-based model[END_REF].

In solution irradiation experiments with colloidal GNPs [START_REF] Sicard-Roselli | A new mechanism for hydroxyl radical production in irradiated nanoparticle solutions[END_REF][START_REF] Gilles | Gold nanoparticles functionalization notably decreases radiosensitization through hydroxyl radical production under ionizing radiation[END_REF][START_REF] Gilles | Quantification of hydroxyl radicals and solvated electrons produced by irradiated gold nanoparticles suggests a crucial role of interfacial water[END_REF][START_REF] Cheng | Chemical enhancement by nanomaterials under x-ray irradiation[END_REF][START_REF] Davidson | Average physical enhancement by nanomaterials under x-ray irradiation[END_REF][START_REF] Davidson | multiplication algorithm for combined physical and chemical enhancement of x-ray effect by nanomaterials[END_REF][START_REF] Chang | Electron paramagnetic resonance spectroscopy investigation of radical production by gold nanoparticles in aqueous solutions under x-ray irradiation[END_REF] or in presence of bio-molecules [START_REF] Foley | Enhanced relaxation of nanoparticle-bound supercoiled dna in x-ray radiation[END_REF][START_REF] Carter | Nanoscale energy deposition by x-ray absorbing nanostructures[END_REF][START_REF] Butterworth | Variation of strand break yield for plasmid dna irradiated with high-z metal nanoparticles[END_REF][START_REF] Brun | Parameters governing gold nanoparticle x-ray radiosensitization of dna in solution[END_REF][START_REF] Brun | Gold nanoparticles enhance the x-ray-induced degradation of human centrin 2 protein[END_REF][START_REF] Mcmahon | Energy dependence of gold nanoparticle radiosensitization in plasmid dna[END_REF] put in light the importance of the chemical stage. In particular, the use of plasmid DNA evidenced that the enhancement of DNA single strand breaks was dependent on the energy of the incident photon in the keV range [START_REF] Brun | Parameters governing gold nanoparticle x-ray radiosensitization of dna in solution[END_REF][START_REF] Mcmahon | Energy dependence of gold nanoparticle radiosensitization in plasmid dna[END_REF], the concentration of GNPs [START_REF] Brun | Parameters governing gold nanoparticle x-ray radiosensitization of dna in solution[END_REF], the chemical environment of the solution and in particular its scavenging ability [START_REF] Carter | Nanoscale energy deposition by x-ray absorbing nanostructures[END_REF][START_REF] Butterworth | Variation of strand break yield for plasmid dna irradiated with high-z metal nanoparticles[END_REF][START_REF] Mcmahon | Energy dependence of gold nanoparticle radiosensitization in plasmid dna[END_REF], and the proximity between the GNPs and the biomolecules [START_REF] Carter | Nanoscale energy deposition by x-ray absorbing nanostructures[END_REF][START_REF] Brun | Gold nanoparticles enhance the x-ray-induced degradation of human centrin 2 protein[END_REF]. Large increase of reactive oxygen species (ROS) production was also measured with in vitro probes for cells irradiated in presence of GNPs, compared to cells irradiated without GNPs [START_REF] Geng | Thio-glucose bound gold nanoparticles enhance radio-cytotoxic targeting of ovarian cancer[END_REF][START_REF] Jeynes | Investigation of gold nanoparticle radiosensitization mechanisms using a free radical scavenger and protons of different energies[END_REF], although it is unclear whether it is directly produced by the interaction of radiations with GNPs, or indirectly by the interaction of the GNPs with cellular components. Overproduction of free radicals during water radiolysis induce an oxidative stress which may result in cells failing to maintain normal physicological redox-regulated functions [START_REF] Fu | Mechanisms of nanotoxicity: generation of reactive oxygen species[END_REF].

Dramatic effects, such as oxidative modification of proteins, lipid peroxidation, DNA-strand break, etc. [START_REF] Fu | Mechanisms of nanotoxicity: generation of reactive oxygen species[END_REF], result in cell death.

As pointed by Her et al. [START_REF] Her | Gold nanoparticles for applications in cancer radiotherapy: Mechanisms and recent advancements[END_REF] in a recent review, the chemical stage has yet not been fully investigated. While there have been many MC studies on the physical step, to our knowledge only a few MC studies have been conducted on the ROS enhancement for GNP-enhanced radiation therapy [START_REF] Xie | Simulation on the molecular radiosensitization effect of gold nanoparticles in cells irradiated by x-rays[END_REF][START_REF] Tran | Geant4 monte carlo simulation of absorbed dose and radiolysis yields enhancement from a gold nanoparticle under mev proton irradiation[END_REF][START_REF] Rudek | Radio-enhancement by gold nanoparticles and their impact on water radiolysis for x-ray, proton and carbon-ion beams[END_REF], although it was suggested to be a crucial step that may connect the physical effect to the biological consequences [START_REF] Brun | Actual questions raised by nanoparticle radiosensitization[END_REF][START_REF] Laprise-Pelletier | Intratumoral injection of low-energy photon-emitting gold nanoparticles: A microdosimetric monte carlo-based model[END_REF]. In particular, no systematic study of free radical production has been performed for GNP-enhanced radiotherapy in the range of keV photon energies.

The goal of this work is thus to investigate the impact of GNPs on the production of free radicals for keV photon irradiation in water by means of MC simulations. We shall compute the yield Y X of chemical species X, for a given concentration C NP of GNPs homogeneously distributed in a volume of water, irradiated by a beam of monoenergetic photons in the range 20-90 keV. As most of the probes of the aforementioned studies measured either hydroxyl radicals ( • OH) and hydrogen peroxide (H 2 O 2 ), this study focused on these two species.

In particular, we aim to estimate how the increase of dose deposition during the physical stage, including the Auger cascade, affects the chemical radiolytic yield, for an average photon impact. Our results provide values of the boost of chemical species delivered by primary photon impacts at short time following the impact, which we shall refer to as the primary boost. It may be used as a inputs for further macroscopic simulations, including cumulative track effects and potentially GNP chemical reactivity. This paper is organized as follows. In section 2, the MC tool and system considered are presented. The average dose deposition and the radical species production are studied in section 3 as a function of photon energy, size of the GNP and type of chemical species. Section 4 discusses and concludes on the results, with regard to theoretical and experimental results available in the literature.

Material and methods

General considerations

A. Simulation strategy. Our solution sample represents a volume of liquid water containing a homogeneous concentration C NP of GNPs. In such a sample, we consider a cubic sub-volume of interest V placed at a depth from the beam entrance so that the flux of charged particles (e.g. electrons) entering the volume is equal to the flux of charged particles leaving the volume. Such a condition is often referred to as the equilibrium of charged-particles. Under this constraint, the length of V is assumed sufficiently small with respect to the photon mean free path, and sufficiently close to the beam entrance to ensure that the photon beam remains monoenergetic. In such conditions, it is possible to neglect the photon beam transport and attenuation. Therefore, in our simulation, V is irradiated by a beam of monoenergetic photons, for which a prescribed reference dose D in water (i.e., without GNPs) can be defined. For such a volume, the mean number of interactions can directly be calculated both in water and in the GNPs, based on D. Photon impacts can be considered homogeneously distributed in each material, according to their total photon absorption cross section [START_REF] Cunha | Considerations on the miniaturization of detectors for in vivo dosimetry in radiotherapy: A monte carlo study[END_REF]. Such a volume is typically of what has been used for radiation chemistry [START_REF] Sicard-Roselli | A new mechanism for hydroxyl radical production in irradiated nanoparticle solutions[END_REF].

While desirable, a direct simulation of irradiation of a large volume containing a large number of GNPs is unfortunately not possible, because the computational cost of chemical simulations scales as the square of the number of chemical species.

We thus added the following approximations and considerations. First, we simulated the physical and chemical processes in V , applying periodic boundary conditions (see Fig. 1) to achieve charged-particle equilibrium, as defined above [START_REF] Cunha | Considerations on the miniaturization of detectors for in vivo dosimetry in radiotherapy: A monte carlo study[END_REF]. The dimensions of V were larger than a single-track extension, and we thus avoided the spurious interaction of a track with its periodic image.

Secondly, we calculated radical production track-by-track, ignoring the interaction between the radicals generated in two different tracks. This consideration is consistent with the results obtained with most standard experimental methods used to measure radiation-chemical yields. Such experiments consist in using scavengers to reduce inter-track effects, or in applying medium renewal at a rate that is fast enough compared to the dose rate. In such conditions, the concentration of radicals is proportional to the number of absorbed photons, and thus to the prescribed dose D. We thus defined the average yield of radicals as the yield obtained for 1 µs time t evolution in a single-photon track, and then averaged over a large number of tracks. We expect our approach to be valid under relatively low flux conditions. In this case the relative effect of GNP with respect to pure water is definitely well estimated. For more accuracy under high flux conditions, the track overlap should be taken into account.

Finally, we limited our study to a concentration C NP of GNPs sufficiently low to neglect the probability that a single track overlap simultaneously two GNPs.

For a GNP concentration of 1 mg • mL -1 , we typically find that the probability for a track created in water to hit a GNP is of a few percents. Another working hypothesis is that the GNP does not participate to the chemical stage. We assume that they are chemically inert. Moreover, up to 1 µs, the probability that chemical species react with a GNP is rather small, due to limited diffusion and to the fact that most of the particles are created in water far from any GNP for the considered concentration of 1 mg • mL -1 . It was therefore possible to consider only three kinds of single-track event following a photo-ionization: the case (W) where a photon interacted with water and did not lead to secondary particles interacting with the GNP, the case (W+NP) where a photon interacted with water and led to at least one secondary particle interaction with the GNP and the case (NP) where the photon directly interacted with the GNP. They are referred to as (1,2,3), corresponding respectively to (W,W+NP,NP), and illustrated in Fig. 1. As detailed in the Appendix A, the probability that a track hits two or more GNP is negligible. The final yield was obtained as a sum of these 3 contributions, weighted by their respective probability.

B. Investigated parameters. Four GNP radii (5 nm, 12.5 nm, 25 nm and 50 nm) and energies from 20 to 90 keV were chosen to study the impact of the GNP size and photon energy on chemical species yields. The half length of V was therefore comprised between 15 µm at 20 keV to 160 µm at 90 keV. This ensures that the dimensions of V are lower than the photon mean free path, which is typically of the order of 1 cm at 20 keV. The different calculations were all performed for a GNP concentration of 1 mg • mL -1 . This concentration is in the range of reported in solution [START_REF] Sicard-Roselli | A new mechanism for hydroxyl radical production in irradiated nanoparticle solutions[END_REF][START_REF] Gilles | Gold nanoparticles functionalization notably decreases radiosensitization through hydroxyl radical production under ionizing radiation[END_REF][START_REF] Gilles | Quantification of hydroxyl radicals and solvated electrons produced by irradiated gold nanoparticles suggests a crucial role of interfacial water[END_REF][START_REF] Cheng | Chemical enhancement by nanomaterials under x-ray irradiation[END_REF][START_REF] Davidson | Average physical enhancement by nanomaterials under x-ray irradiation[END_REF] and in vitro [START_REF] Lechtman | A monte carlo-based model of gold nanoparticle radiosensitization accounting for increased radiobiological effectiveness[END_REF][START_REF] Lechtman | Interplay between the gold nanoparticle subcellular localization, size, and the photon energy for radiosensitization[END_REF][START_REF] Retif | Monte carlo simulations guided by imaging to predict the in vitro ranking of radiosensitizing nanoparticles[END_REF] experimental concentrations. For comparison with the results from the literature, we reported different quantities in Tab. 1. We define n(E) (respectively f (E)) the number 145 (respectively proportion) of ionized GNPs in a cubic volume of length equal to 10 µm (of the order of the size of a cell), for photon energy E, 1 Gy irradiation and GNP concentration of 1 mg • mL -1 . We found that n(E) was equal to 6 and 45 for E equal to 80 and 20 keV respectively. f (E) is extremely small for 5 nm GNPs regardless of the photon energy, while it is close to 50 % for 50 nm Table 1: Concentration of GNP for various metrics and various GNP sizes. The average distance between two GNPs was calculated assuming that GNPs are homogeneously distributed in a cubic network.

Monte Carlo simulation for dosimetry and water radiolysis

The irradiation of the volume V was modeled by 3 distinctive stages. During the physical stage, particles traveling through the media deposit energy, which provided us dosimetric quantities. This physical stage is followed by a physico-155 chemical and a chemical stage leading to the creation of many highly reactive chemical species. Up to the beginning of the chemical stage, the track structure is highly heterogeneous. Hence the chemical reactions cannot be described by simple first or second order kinetic equations [START_REF] Frongillo | Monte carlo simulation of fast electron and proton tracks in liquid water-ii. nonhomogeneous chemistry[END_REF]. Thus, at short time, Monte Carlo (MC) techniques are required to calculate radiation-chemical yields as a 160 function of time. We used the MC code MDM [START_REF] Gervais | Numerical simulation of multiple ionization and high LET effects in liquid water radiolysis[END_REF][START_REF] Poignant | Theoretical derivation and benchmarking of cross sections for low-energy electron transport in gold[END_REF][START_REF] Tessaro | Theoretical study of w-values for particle impact on water[END_REF][START_REF] Li | Intercomparison of dose enhancement ratio and secondary electron spectra for gold nanoparticles irradiated by x-rays calculated using multiple monte carlo simulation codes[END_REF], which performs an event-by-event tracking of electrons and ions. Formerly known as the LQD code [START_REF] Gervais | Numerical simulation of multiple ionization and high LET effects in liquid water radiolysis[END_REF], it was originally developed to track ions and low energy electrons and products of chemical reactions down to very low energies (thermalization) in pure liquid water. It was later modified to consider various heterogeneous domains such as metallic media, which has led to emergence of the new branch MDM, i.e. for MeDiuM [START_REF] Tessaro | Theoretical study of w-values for particle impact on water[END_REF][START_REF] Poignant | Theoretical derivation and benchmarking of cross sections for low-energy electron transport in gold[END_REF]. A full description of the MC models and constant of chemical reactions used in the present work are available elsewhere [START_REF] Gervais | Numerical simulation of multiple ionization and high LET effects in liquid water radiolysis[END_REF][START_REF] Poignant | Theoretical derivation and benchmarking of cross sections for low-energy electron transport in gold[END_REF][START_REF] Colliaux | Simulation of ion-induced water radiolysis in different conditions of oxygenation[END_REF], and are briefly presented in this section.

Physical stage

The physical stage occurs in the first few fs after each photon absorption. At the end of this stage, the water molecules may be either excited (H 2 O * ) or ionized

(H 2 O + , H 2 O 2+ and H 2 O -)
, and the medium contains thermalized electrons (e th ). Our model [START_REF] Gervais | Numerical simulation of multiple ionization and high LET effects in liquid water radiolysis[END_REF] consisted in an event-by-event tracking of particles in the different media (water or gold). The energy deposited inside GNPs was considered as lost energy, and vacancies or electrons that were contained within GNPs were no longer tracked at the end of this stage. The next paragraphs give a brief overview of the interactions considered for photons and electrons with the media of interest (water and gold).

A. Photons. Only the photoelectric effect and the Compton effect were taken into account as they are predominant at these energies. The photoelectric cross sections were taken from the XCOM database [START_REF] Berger | XCOM: Photon cross section database (version 1.5[END_REF]. In our simulation, the angular distribution of the ejected photoelectron was assumed to be isotropic.

For the Compton effect, the cross section was calculated using the differential cross section based on the Klein-Nishina formula [START_REF] Leo | Techniques for nuclear and particle physics experiments: a how-to approach[END_REF], given by,

dσ dE kin = πr 2 e m e c 2 γ 2 2 + s 2 γ 2 (1 -s) 2 + s 1 -s s - 2 γ , s = E kin hν (1)
with E kin the energy of the recoil electron, γ = hν mec 2 , r e = 2.818 × 10 -15 m is the classical radius of the electron, hν is the energy of the incoming photon and m e c 2 = 511 keV is the electron mass energy. The total cross section was then obtained by integrating Eq. 1 from 0 to the maximum energy transfer E kin,max , given by

E kin,max = hν 2γ 1 + 2γ . (2)

B. Electrons. A full description of the model for gold-electron interactions is

given in the study by Poignant et al. [START_REF] Poignant | Theoretical derivation and benchmarking of cross sections for low-energy electron transport in gold[END_REF], and for water-electron interactions in the study by Gervais et al. [START_REF] Gervais | Numerical simulation of multiple ionization and high LET effects in liquid water radiolysis[END_REF]. Briefly, we modeled the full electronic cascade following the primary photon interaction down to thermalization, accounting for inelastic (ionizations, electron and vibrational excitations, electronic attachment) and elastic interactions in both gold and water. Regarding ionization, a solid state formalism was applied for outer-shell electrons of gold, while an atomic model was applied for both water and inner-shell electrons of gold. We emphasize that in this study, the cross sections do not depend on the size and shape of the nanoparticles. In particular, plasmons resonances correspond to the ones of bulk gold. The possible effects of surface plasmon resonance modification due to the GNP shape and size or due to GNP coupling with the surrounding medium are not taken into account. In particular, plasmons resonances correspond to the ones of bulk gold. For an accurate evaluation of radiationchemical yields, it is mandatory to account for de-excitation and recombination processes. We considered Auger de-excitation both in water and in gold. For gold [START_REF] Poignant | Theoretical derivation and benchmarking of cross sections for low-energy electron transport in gold[END_REF], data were taken from the EADL library. Emitted fluorescence photons

were not tracked and therefore did not lead to additional energy deposition in the volume. Although these photons might carry a large amount of the initial photon energy after a GNP ionization, most of them had an energy larger than 10 keV. Hence, their mean free path in water was larger than the dimensions of the volume of interest, meaning their contribution to the radiation-chemical yield was negligible. The small probability of thermalized electron recombination with a water vacancy was taken into account, resulting in an excited water molecule [START_REF] Ouerdane | Radiolysis of water confined in porous silica: a simulation study of the physicochemical yields[END_REF]. To ensure a consistent tracking of electron kinetic energy when changing medium, both materials were described by a mesoscopic potential of values equal to -1.30 eV for water and -10.04 eV for gold. The kinetic energy of the electron was modified when crossing the water-GNP interface, on the basis of the mesoscopic potential difference between the two media.

Physico-chemical stage in water

This stage models the relaxation of water molecules leading to the production of primary chemical species, occurring from 10 -15 to 10 -12 s after the interaction of a photon with the media. The molecules may dissociate, in which case the two products were generated at a given distance from one to another to account for their kinetic energy. The different branching ratios of each molecular rearrangement are provided in another work [START_REF] Gervais | Numerical simulation of multiple ionization and high LET effects in liquid water radiolysis[END_REF]. They were not influenced by the presence of the GNP. At the end of this stage, several chemical species were produced, with a majority of e - aq , H 3 O + and • OH. Note that chemical species did not diffuse during this stage.

Chemical stage in water

At 10 -12 s, the primary chemical species start to diffuse and may react with each other. Our simulation tracked all chemical species up to 10 -6 s. At the end of the chemical stage, the so-called steady state is reached and the radiolytic yields reach asymptotic values [START_REF] Gervais | Numerical simulation of multiple ionization and high LET effects in liquid water radiolysis[END_REF][START_REF] Frongillo | Monte carlo simulation of fast electron and proton tracks in liquid water-ii. nonhomogeneous chemistry[END_REF]. Our method used to calculate the yields is described elsewhere [START_REF] Gervais | Numerical simulation of multiple ionization and high LET effects in liquid water radiolysis[END_REF][START_REF] Colliaux | O2 and glutathione effects on water radiolysis: a simulation study[END_REF][START_REF] Colliaux | Simulation of ion-induced water radiolysis in different conditions of oxygenation[END_REF]. The simulation parameters were taken from the work by Frongillo et al. [START_REF] Frongillo | Monte carlo simulation of fast electron and proton tracks in liquid water-ii. nonhomogeneous chemistry[END_REF]. More than fifty reactions were possible, and are listed in the work by Colliaux et al. [START_REF] Colliaux | Simulation of ion-induced water radiolysis in different conditions of oxygenation[END_REF]. During this stage, interaction of chemical species with the GNP were not considered.

Calculated quantities 2.3.1. Radiation-chemical yield in water

Defining R X (D) the mean number of species X produced in a volume V for a dose of irradiation D, we decomposed R X (D) as the sum of the 3 contributions as illustrated in Fig. 1,

R X (D) = R X,1 (D) + R X,2 (D) + R X,3 (D) (3)
Given our hypotheses introduced in section 2.1, each contribution R X (D) is the dose weighted average over a large number of single-tracks, associated to the

selected event i, R X,i (D) = n i (D)R 1 X,i , with i ∈ [1, 2, 3]. (4)
R 1 X,i is the average number of chemical species X obtained for a single-track event i. n i (D) is the mean number of absorbed photon leading to event i for the given dose D. Note that n 1 (D) = n 2 (D), as both represent the mean number of photon interactions in water. They are expressed as follows,

     n 1 (D) = n 2 (D) = F (D)σ W ρ W V (1 -C NP V NP ) n 3 (D) = F (D)σ NP ρ NP V C NP V NP , (5)
with F (D) the fluence of the beam which is linear with the dose D, V the total volume, V NP the volume of one nanoparticle, C NP the GNP concentration, σ x the total cross section for photon-interaction in the medium x, ρ x the density of the medium x.

Each number of chemical species R 1 X,i does not depend on the dose but depends on the chosen event i, and as such depends on the GNP concentration, as detailed in Appendix A. It is obtained by averaging the number of chemical species obtained per single-track event i over a large number of single-track events. The most straightforward strategy to obtain such events is to generate a photon impact in a sample consisting of a random array of GNPs distributed in water.

The yield Y X in nmol • J -1 is deduced from R X (D) as follows,

Y X = 10 12 DV × ρ W × N a R X (D) (6)
with Y X expressed in nmol • J -1 , D the dose (J • kg -1 ), V the total volume (cm 3 ), ρ W the water density (g•cm -3 ), N a the Avogadro number. Note that, as the chemical species number was proportional to D, the resulting yields are independent of the dose D. Moreover, since the concentration of GNPs is sufficiently low, the contributions R X,2 (D) and R X,3 (D) are proportional to C NP , as shown in Appendix A. This would be different if the concentration was larger. Alternatively, as events of type 2 (W+NP) are rare events, it is possible to fasten the simulation by using a biasing method. Our strategy consisted in first generating a track in pure water. We then calculated a position for which the track interacted with the GNP, along with the probability for this track to interact with the GNP. The track was re-generated, using the same seed, but translating it to force the interaction of secondary particles to occur.

The probability for the track to interact with the GNP was then used as a weight coefficient, as detailed in Appendix A. Note that replacing the GNPs by nanoparticles made of water led to the same yield as for pure water.

Energy deposition in water

After irradiating a volume of water containing C NP GNPs, dosimetric outputs consisted in the spatial distribution of all low-energy electrons and ionized or excited water molecules. The track-by-track method, previously introduced for the calculation of the average yield of chemical species, was also applied for the calculation of the dose deposition. For a concentration C NP of GNPs and a prescribed dose D, we find a dose deposition D ,

D = i n i (D)D 1 i with i ∈ [1, 2, 3] (7)
with D 1 i the average dose for the track type i. As for the calculation of chemicalradiation yields, the volume of interest V used to obtain the average dose varied from 30 3 to 320 3 µm 3 , according to the photon energy.

Statistical uncertainty

For the contribution i and the chemical species X, associated with N i the number of tracks of type i simulated, the standard deviation σ R X (D) of the number

of chemical species R X (D) is, σ R X,i (D) = 1 N i Ni t=1 n i (D)R t X,i -R X,i (D) 2 (8)
with R X,i t , the number of chemical species X obtained for the track t. The final statistical uncertainty, for the number of chemical species, reads,

σ R X (D) = σ R X,1 √ N 1 2 + σ R X,2 √ N 2 2 + σ R X,3 √ N 3 2 (9)
Similar uncertainties were obtained for the energy deposition, the yields and radial quantities. Each results were displayed with error bars that were set to ± σ R X . statistical effect. The increase of the dose due to GNPs is comprised between ∼ 6 % and ∼ 16 % depending on the photon primary energy. For energy larger than 20 keV, there is an increase of the dose deposition with increasing photon energy, as the photo-electron energy is higher and lose therefore less energy within the GNP. Then, the relative increase goes down to 6 % at 80 keV as the ratio between gold and water photon absorption cross section decreases. The maximum is reached around 40-50 keV. At 90 keV, there is small increase of the dose due to the increase of the photo-electric cross section of gold by K-shell ionization. It can be noted that the increase is relatively independent of the GNP radius, except for low energy photons, in which case the energy deposited in the GNP is significantly more important for large GNPs. At 20 keV, GNP with a 50 nm (respectively 5 nm) radius absorbs ∼ 25 % (respectively 5 %) of the total photon energy following a photo-ionization. It is equal to less than 10 % for all radii beyond 40 keV. Moreover, the events W+NP do not induce much energy loss by energy absorption in the GNPs, because the track initiated in water leads mostly to dose deposition in water even though a GNP has been hit. Thus, on average, at these photon energies, the contribution coming from photon interaction with water (t W+NP + t W ) can be replaced by a contribution of pure water. Finally, we noticed that Auger electrons emitted from a GNP contribute to less than 4 % of the total dose deposition at 1 mg • mL -1 . As detailed in Appendix A, the relative increase of the dose may be linearly extrapolated at larger GNP concentrations, up to few tens of mg • mL -1 . For electron irradiation, there is first a sharp increase of H 2 O 2 production at very low primary electron energies (< 300 eV). Then, H 2 O 2 yield decreases for primary electron energies increasing up to ∼ 20 keV, due a lower recombination probability with increasing energy. Finally, it reaches a plateau, whose value at 90 keV is ∼ 70 % of the maximum value. On the contrary, • OH yield first strongly decreases down to 50 % of its maximal value, before increasing again and reaching a plateau that is nearly equal to the maximal value at 90 keV.

Results

Energy deposition

Radiation-chemical yields in water

This variation for electron impact reflects the change of the track structure as the projectile energy changes.

For photon irradiation without GNP, there is a small fluctuation of the radiationchemical yields depending on the photon energy. From 20 to 35 keV, • OH yield increases from 203.0 ± 0.4 nmol • J -1 up to 215.2 ± 0.8 nmol • J -1 . At these energies, the photo-electric effect remains non negligible. Consequently, the primary electron generated following a photon interaction in water has an energy that is nearly equal to the photon energy. We therefore obtain yields that are very close to those obtained for electron irradiation. As the photon energy increases, the Compton effect starts dominating [START_REF] Berger | XCOM: Photon cross section database (version 1.5[END_REF]. Following a Compton event in water at such energies, the primary electron energy distribution is rather flat up to a maximum value significantly smaller than the incoming photon energy (typically ∼ 20 keV for a 90 keV photon). Consequently, many primary electrons are emitted in the 0-20 keV range, for which • OH production is reduced.

We therefore observe a decrease of • OH yield down to 199.5 ± 0.9 nmol • J -1

and an increase of H 2 O 2 production. Then, for a photon energy of 90 keV, the primary electron energy tends to be higher on average, and the proportion of primary electrons emitted at very low energies is reduced in proportion, thus favoring back • OH production. This result for pure water is consistent with a previous study reported in the literature [START_REF] Yamaguchi | A spur diffusion model applied to estimate yields of species in water irradiated by monoenergetic photons of 50 ev-2 mev[END_REF] Fig It may also be noted that, just as for the dose enhancement, the radius of the GNP does not impact much the radiolytic enhancement. Although, at 20 keV, the enhancement significantly varies from ∼ 9 to ∼ 12 % with radius from 50 to 5 nm, this difference becomes negligible at higher energies. This is simply due to energy absorption within the GNP, which decreases with increasing photon energy.

Discussion and conclusion

Over the past two decades, experimental and theoretical studies have shown the ability of gold nanoparticles (GNPs) to increase the efficiency of radiation therapy. For low-energy (keV) photons, this enhancement partially results from an increase of the dose deposition, due to a higher cross section for gold-photon interaction compared to water-photon interaction. While Monte Carlo (MC)

simulations have widely been used to quantify this dose boost [START_REF] Cho | Estimation of tumour dose enhancement due to gold nanoparticles during typical radiation treatments: a preliminary monte carlo study[END_REF][START_REF] Cho | The dosimetric feasibility of gold nanoparticle-aided radiation therapy (gnrt) via brachytherapy using lowenergy gamma-/x-ray sources[END_REF][START_REF] Zhang | Quantifying tumor-selective radiation dose enhancements using gold nanoparticles: a monte carlo simulation study[END_REF][START_REF] Montenegro | Monte carlo simulations and atomic calculations for auger processes in biomedical nanotheranostics[END_REF][START_REF] Mesbahi | Effect of photon beam energy, gold nanoparticle size and concentration on the dose enhancement in radiation therapy[END_REF][START_REF] Amato | Gold nanoparticles as a sensitising agent in external beam radiotherapy and brachytherapy: a feasibility study through monte carlo simulation[END_REF][START_REF] Sinha | Brachytherapy application with in situ dose painting administered by gold nanoparticle eluters[END_REF][START_REF] Koger | A method for converting dose-to-medium to dose-totissue in monte carlo studies of gold nanoparticle-enhanced radiotherapy[END_REF][START_REF] Koger | Optimization of photon beam energies in gold nanoparticle enhanced arc radiation therapy using monte carlo methods[END_REF][START_REF] Mcmahon | Radiotherapy in the presence of contrast agents: a general figure of merit and its application to gold nanoparticles[END_REF][START_REF] Jones | Estimation of microscopic dose enhancement factor around gold nanoparticles by monte carlo calculations[END_REF][START_REF] Cai | Investigation of the effects of cell model and subcellular location of gold nanoparticles on nuclear dose enhancement factors using monte carlo simulation[END_REF][START_REF] Douglass | Monte carlo investigation of the increased radiation deposition due to gold nanoparticles using kilovoltage and megavoltage photons in a 3d randomized cell model[END_REF][START_REF] Mcnamara | Dose enhancement effects to the nucleus and mitochondria from gold nanoparticles in the cytosol[END_REF][START_REF] Kirkby | Targeting mitochondria in cancer cells using gold nanoparticle-enhanced radiotherapy: A monte carlo study[END_REF][START_REF] Mcmahon | Biological consequences of nanoscale energy deposition near irradiated heavy atom nanoparticles[END_REF][START_REF] Mcmahon | Nanodosimetric effects of gold nanoparticles in megavoltage radiation therapy[END_REF][START_REF] Lin | Gold nanoparticle induced vasculature damage in radiotherapy: comparing protons, megavoltage photons, and kilovoltage photons[END_REF], the impact of GNPs on radical species production has yet hardly been investigated. In the present study, we investigated the average primary boost of chemical species yields (H 2 O 2 and

• OH) under keV photon irradiation of water containing a concentration C NP of GNPs by mean of MC simulation. The variation of radiation-chemical yields was studied with regard to photon energy (20 to 90 keV) and GNP radius (5 to 50 nm), at 1 µs following the primary photon interaction.

For 1 mg • mL -1 of gold, we obtained a relative dose increase which varied from ∼ 6 to ∼ 16 %, depending mostly on the photon energy but also, to a lesser extent, on the GNP radius for the lowest photon energies. Most of the energy was deposited by the photo-electron, while the contribution of Auger electrons overall remained below 4 %. For sufficiently low GNP concentration, this increase was linear with the GNP concentration up to a few tens of mg • mL -1 . Values obtained for 1 mg•mL -1 may therefore be linearly extrapolated at other GNP concentrations. The obtained dosimetric results are consistent with those reported in the literature. For instance, Lechtman et al. [START_REF] Lechtman | Implications on clinical scenario of gold nanoparticle radiosensitization in regards to photon energy, nanoparticle size, concentration and location[END_REF] reported that, for 5-100 nm diameter GNPs and a monoenergetic source of Pd-103 (20.48 keV), ∼ 6-8 mg • mL -1 of GNPs would be required to double the dose. In comparison, at 20 keV and assuming a linear extrapolation of our results with regard to GNP concentration, we find that 8.8-11 mg • mL -1 would be necessary to double the dose deposition for 10-100 nm diameter GNPs. In another work [START_REF] Cho | Estimation of tumour dose enhancement due to gold nanoparticles during typical radiation treatments: a preliminary monte carlo study[END_REF], a dose enhancement factor of 3.8 to 5.6 was reported for an atomic mixture of goldwater, and gold concentrations varying from 7 to 30 mg • mL -1 , at 140 kVp (mean energy 57.9 keV). For our smallest GNP and an energy of 60 keV, we find enhancement ratios from 1.8 to 4.6 for 7-30 mg • mL -1 , which is slightly lower. Besides the difference in physical models, this difference is partly due to the energy absorption within the GNP, which cannot be reproduced with an atomic mixture. Recently, Martinov et al [START_REF] Martinov | Heterogeneous multiscale monte carlo simulations for gold nanoparticle radiosensitization[END_REF] found, for a uniform mixture of 20-100 nm diameter GNPs and water, an enhancement ratio of 3.1 and 2.7 at 20 mg • mL -1 , 20 keV. In comparison, for a 100 nm GNP, we reached a dose increase of 2.8.

We found that the increase of primary radiolytic yield of H 2 O 2 and • OH is of the same order of magnitude as the dose increase, comprised between 6 to 14 % for 1 mg • mL -1 of gold, and proportional to the gold concentration. It varied mostly with the photon energy mainly because of the evolution of the photoelectric and Compton cross sections with energy, and was maximum at 40-50 keV. Variations of the yields of a few percents were obtained with regard to the other parameters (GNP size and chemical species type). As expected, the highest yields were obtained for the smallest GNPs, as the energy lost in the GNP was minimized. At this scale, we did not observed a particularly high H 2 O 2 radiolytic yield, despite the dense clusters of ionization following an Auger cascade. It must be emphasized however that the Auger contribution to the total dose is rather small. Such behavior was expected as the chemical species that are produced directly derive from the dose deposition in water.

The small enhancement of global radical production predicted at short time by our simulations suggests that the apparent overproduction obtained experimentally using coumarin scavengers [START_REF] Sicard-Roselli | A new mechanism for hydroxyl radical production in irradiated nanoparticle solutions[END_REF][START_REF] Cheng | Chemical enhancement by nanomaterials under x-ray irradiation[END_REF][START_REF] Davidson | Average physical enhancement by nanomaterials under x-ray irradiation[END_REF][START_REF] Davidson | multiplication algorithm for combined physical and chemical enhancement of x-ray effect by nanomaterials[END_REF][START_REF] Gilles | Quantification of hydroxyl radicals and solvated electrons produced by irradiated gold nanoparticles suggests a crucial role of interfacial water[END_REF] arise from more complex mechanisms operating over longer time. Further studies, involving in particular chemical reactions with GNPs, need to be undertaken to better understand the experimental observations. We are currently investigating such questions using MC and analytical tools, and quantum mechanical calculation [START_REF] Chan | A water solvation shell can transform gold metastable nanoparticles in the fluxional regime[END_REF] to better understand the kinetic of chemical reactions that may occur at the surface of GNPs, and the interaction of coumarin with GNP surface. Another perspective concerns the spatial distribution of the chemical species. While our simulations showed a minor contribution of Auger cascades to the global production of radicals, one may question whether a boost of radicals may appear close to the GNPs. This will be the presented in the second part of this work. generator. p I0,s,1 and R 1 X,I0,s,1 (respectively R 1 X,I0,s,2 ) are determined for each couple I 0 , s, by generating randomly numerous GNP positions in the volume V . On average and for a GNP concentration of 1 mg • mL -1 , we found P I0,s,2 to be of the order of a few percents. Following the same approach, the probability that a generated track hit two GNPs is of the order of P 2 I0,s,2 /2. For concentrations of the order of 10 mg • mL -1 , we therefore find a probability to hit one GNP of the order of 21 %, and a probability to hit two GNPs of the order of 3 %. Besides, as shown in the results section, when a track hit a GNP, the amount of energy lost in the GNP is small, and most of the energy remains deposited in water. Small probabilities combined with a small energy loss allows us to neglect the probability to hit two or more GNPs for one track and to extrapolate our results linearly with the GNP concentration up to a few tens of mg • mL -1 .

Acknowledgment

For the track type NP (i.e. 3), the determination of its contribution was more straightforward. The calculation of R 1 X,3 consisted of generating a sufficient number of primary photons interaction within the GNP to reach a statistically accurate value of radiation-chemical yields.

Figure 1 :

 1 Figure 1: Schematic view of the cubic volume of interest V considered for radical species simulation. The number of photon impacts are computed in each medium (water and GNPs) and randomly distributed in each corresponding volume. Sub-contributions are represented in blue (W), orange (W+NP) and green (NP).

  150

- 3 9. 9 ×∼ 5 ×

 395 10 13 6.3 × 10 12 7.9 × 10 11 9.9 × 10 10 volume : cube of 10 × 10 × 10 µm3 10 -5

Fig. 2

 2 Fig. 2 represents dosimetric quantities for 1 mg • mL -1 of GNPs and for a prescribed dose D = 1 Gy. The variation of the dose for pure water is a purely

Figure 2 :

 2 Figure 2: Dose D as a function of the photon energy E photon , for C NP = 1 mg •mL -1 , various GNP radii, a prescribed dose to water D =1 Gy.

Figure 3 :

 3 Figure 3: Radiation-chemical yields of H 2 O 2 and • OH as a function of the primary electron (solid line) or photon (squaredots) energy in pure water, at 1 µs, normalized to the respective maximal values.

Fig. 3

 3 Fig. 3 represents the normalized radiation-chemical yields of H 2 O 2 and • OH obtained for electron and photon irradiation without GNPs, as a function of the incident particle energy at 1 µs following the primary interaction. Both yields are correlated as the production of H 2 O 2 is mainly due to the recombination of two • OH. The yields are normalized to the maximal values: 109 nmol • J -1 for H 2 O 2 and 246 nmol • J -1 for • OH. The electron and photon results would be nearly identical if only the photoelectric effect took place.

  Fig. 4 displays the absolute ((a) and (b)) and relative to pure water ((c) and (d)) yields of H 2 O 2 and • OH at 10 -6 s, as a function of the primary photon energy (in keV), for a GNP concentration of 1 mg • mL -1 , for different R NP .Regarding the yields in the presence of GNPs, both H 2 O 2 and • OH yields follow similar trends with regard to the photon primary energy variation and behave alike the dose. There is first an increase of the yield enhancement from ∼ 11 to ∼ 14 % for H 2 O 2 and ∼ 8 to ∼ 14 % for • OH, when the photon energy increases

  This work was supported by the LABEX PRIMES (ANR-11-LABX-0063) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). This work has been supported by the Fondation ARC pour la recherche sur le cancer.

Figure 4 :

 4 Figure 4: H 2 O 2 and • OH yield dependence upon primary photon energy for different R NP and pure water at 10 -6 s. The GNP concentration C GNP was set at 1 mg • mL -1 . On top, yields are in nmol • J -1 while on the bottom, relative enhancements with regard to the yield in the absence of GNPs are in %.

Appendices

Appendix A Track by track methodology While the expression of the coefficients n i (D) is straightforward (see Eq. 5), the calculation of R 1 X,i requires MC simulations. Each track t may be described by (I 0 , → r 0 , s), where I 0 is the original event, defined by the photo-electron velocity and the subsequent Auger electrons velocities, → r 0 is the position of the primary photon interaction, and s is a series of random number determining totally the Markov chain. As mentioned in section 2.3.1, we decomposed the production of chemical species X, for a given irradiation dose D, as a linear combination of the chemical species produced by isolated tracks of types i, R 1 X,i , where i refers either to a track type W, W+NP and NP (i.e. 1, 2 and 3). We expose thereafter the method we used to obtain R 1 X,i , first for a photon interacting in water and then interacting in a GNP.

We define V W = V -X NP V NP , the volume occupied by water for a total volume V containing X NP (number of GNP per cm 3 ) of volume V NP . Then, the average number of chemical species per photon interaction in water is given by,

σ I0 is the cross section for a photon interaction that leads to an initial event I 0 , σ W is the total cross section of photon interaction in water, and, by definition,

is the probability distribution that the initial event I 0 occurs at the position → r 0 . We assume the distribution uniform, thus ρ(

r0,s is the number of chemical species X induced by the track t defined by (I 0 , → r 0 , s).

To differentiate the contributions W and W+NP, we introduce two functions,

As a photon interaction in water leads either to the track type 1 or 2 (i.e. the track either interacts with a GNP or not), we find,

We introduce,

For a track defined by an origin I 0 and the series of numbers s defining all the 475 secondary particles, P I0,s,1 (respectively P I0,s,2 ) represents the probability that any track characterized by I 0 ,s has no secondary electron hitting a GNP (respectively has at least one electron hitting a GNP). Note that P I0,s,1 + P I0,s,2 = 1.

We further introduce,

Then Eq. A3 becomes,

with

The estimation of the probability that a secondary particle hits a GNP is not straightforward when having a given number of GNPs, C NP (number of particle per cm 3 ). We introduced the number of nanoparticles X NP in the volume V (X NP = C NP V , and the probability that a track hits a GNP in a large dilution limit, p I0,s,2 . Given a low concentration of GNP, and assuming that they are homogeneously distributed, the probability of not hitting a GNP, in the presence of X NP GNPs, follows a binomial distribution,

Considering a small dose (low track density), and a low radical diffusion (t ≤ 10 -6 s), the probability that a track interacts with the GNP is low, (p I0,s,2 1).

Thus, one has,

Therefore, for X NP GNPs in the volume V , P I0,s,1 and P I0,s,2 may be estimated with the probability that a track I 0 , s hits one GNP as follows,

For each track (I 0 , s) generated, we estimated p I0,s,2 , from which we derived P I0,s,1 and P I0,s,2 . Concretely, to estimate R 1 X,1 (respectively R 1 X,2 ), we generated many primary events I 0 . s is described by the seed of the pseudo-random

• We calculate radiolitic yields of free radicals for low-energy (20-90 keV) photon irradiation of gold nanoparticle water solution. • 1mg/mL of GNPs increases radical production from 6 to 14 %.

• Radical production enhancement is similar to the dose enhancement.

Declaration of interests

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: