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International Journal of Radiation Biology, December 2012; 88(12): 916–921© 2012 Informa UK, Ltd.ISSN 0955-3002 print / ISSN 1362-3095 onlineDOI: 10.3109/09553002.2012.666003

Calculations of absorbed fractions in small water spheres for low-energy monoenergetic electrons and the Auger-emitting radionuclides 123I and 125I

Christos Bousis1, Dimitris Emfietzoglou1 & Hooshang Nikjoo2

1Medical Physics Laboratory, University of Ioannina Medical School, Ioannina, Greece, and 2Radiation Biophysics Group, Department of Oncology-Pathology, Karolinska Institute, Stockholm, Sweden

Introduction

For the case of disseminated disease in targeted radionu-clide therapy short-range charged particles, such as Auger electrons, are known to offer some distinct advantages com-pared to the more used long-range b-electrons, such as, a reduced cross-fire irradiation of neighboring healthy cells

and a higher ionization density which is generally associated with high biological effectiveness (Kassis 2004, Buchegger et al. 2006, Nikjoo et al. 2008). Owing to their localized energy deposition pattern, it is recognized that the dosimetry of Auger electrons should be carried out at the single-cell level, with emphasis on sub-cellular structures – a domain typical to the fields of microdosimetry and track-structure theory (Humm et al. 1994, Bardiès and Pihet 2000). Most dosimet-ric studies for low-energy electrons at the cellular and sub- cellular level have been undertaken using Monte-Carlo (MC) track-structure codes in unit-density ( 1 g cm3) vapor water medium (Semenenko and Stewart 2006, Chen 2008, Emfietzoglou et al. 2008, Bousis et al. 2010, Liu et al. 2009, Uusijärvi et al. 2009). Given that water in cells is in liquid/solid phase, efforts have been made to perform more realis-tic calculations of electron energy deposition patterns using track-structure codes with interaction cross-section data specific for liquid water (Bernhardt et al. 2002, Emfietzoglou et al. 2003, Wilson et al. 2004, Tung et al. 2007, Dingfelder et al. 2008, Wiklund et al. 2011).

The aim of the present study was to quantify the energy deposition in terms of the absorbed fraction (AF) by low-energy electrons (0.1–10 keV) and the two Auger-emitting radionuclides 125I and 123I distributed uniformly in liquid water-medium suspended cellular volumes of radii from 10–1000 nm which corresponds to the spatial scale of criti-cal cellular targets. Calculations have been performed by our in-house liquid water MC code (Emfietzoglou et al. 2003, Bousis et al. 2008) which combines detailed- and condensed-history electron track simulation for low- and high-energy electrons, respectively. In order to investi-gate non-scalable density effects on the calculated AF, the results are compared with MC calculations for unit-density water vapor medium as well as with semi-analytic calcula-tions using the continuous-slowing-down-approximation (CSDA). The latter approach has been adopted by the Medi-cal Internal Radiation Dose (MIRD) Committee (Goddu et al. 1997) for dosimetry at the cellular level ( 1 mm).

Correspondence: Dr Hooshang Nikjoo, PhD, Radiation Biophysics Group, Department of Oncology-Pathology, Karolinska Institutet, Box 260, SE-171 76 Stockholm, Sweden. E-mail: Hooshang.nikjoo@ki.se

(Received 5 September 2011; revised 16 January 2012; accepted 7 February 2012)

AbstractPurpose: To calculate the absorbed fraction (AF) of low energy electrons in small tissue-equivalent spherical volumes by Monte Carlo (MC) track structure simulation and assess the influence of phase (liquid water versus density-scaled water vapor) and of the continuous-slowing-down approximation (CSDA) used in semi-analytic calculations. Methods: An event-by-event MC code simulating the transport of electrons in both the vapor and liquid phase of water using appropriate electron-water interaction cross sections was used to quantify the energy deposition of low-energy electrons in spherical volumes. Semi-analytic calculations within the CSDA using a convolution integral of the Howell range-energy expressions are also presented for comparison. Results: The AF for spherical volumes of radii from 10–1000 nm are presented for monoenergetic electrons over the energy range 100–10,000 eV and the two Auger-emitting radionuclides 125I and 123I. The MC calculated AF for the liquid phase are found to be smaller than those of the (density scaled) gas phase by up to 10–20% for the monoenergetic electrons and 10% for the two Auger-emitters. Differences between the liquid-phase MC results and the semi-analytic CSDA calculations are up to ∼ 55% for the monoenergetic electrons and up to ∼ 35% for the two Auger-emitters. Conclusions: Condensed-phase effects in the inelastic interaction of low-energy electrons with water have a noticeable but relatively small impact on the AF for the energy range and target sizes examined. Depending on the electron energies, the semi-analytic approach may lead to sizeable errors for target sizes with linear dimensions below 1 micron.

Keywords: Auger electron, Monte Carlo, liquid water, cellular dosimetry

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Electron energy deposition in small water spheres 917

Methods

Overview of the Monte Carlo codeAll MC calculations have been carried out using our in-house MC code (Emfietzoglou et al. 2000, 2003, Bousis et al. 2008) which can transport electrons in both the vapor and liquid phases of water by employing detailed-history simulation for electron energies below 10 keV and condensed-history simulation above 10 keV. For the two Auger-electron emitters examined the yield above 10 keV is less than 2% of the total yield. In the detailed-history scheme, the code simulates collision-by-collision all the elastic and inelastic interactions until all the electrons (primary and secondaries) slow down to energies below 1 Ry (13.6 eV) as described in Emfietzo-glou et al. (2000, 2003). In the condensed-history simula-tion, the code still uses detailed simulation for all the elastic collisions but a mixed-simulation scheme for the inelastic losses whereby d-ray production in ‘hard’ inelastic collisions as well as energy-loss straggling in ‘soft’ inelastic collisions are explicitly accounted for as described in detail by Bousis et al. (2008). A brief discussion of the liquid-water version of MC4 with emphasis on recent improvements on the interac-tion cross sections since the documentation of the original version follows.

Energy-loss differential cross sections and inelastic mean free paths are based on the dielectric response function of liquid water. For electron energies below 500 eV where the Born approximation becomes increasingly inaccurate exchange and second-order perturbation corrections are applied (Emfietzoglou et al. 2003). A major advance in the new version of our code is the implementation of the recent dielectric response function model for liquid water of Emfi-etzoglou et al. (2005). Compared to the previous model which was based on a Drude parameterization of the Oak Ridge optical data (Heller et al. 1974) and simple dispersion relations to arbitrary momentum transfer, the new model is based on more recent inelastic X-ray spectroscopy (IXS) data (Hayashi et al. 2000, Watanabe et al. 2000) which pro-vide a more complete information for the electronic excita-tion spectrum of liquid water over the energy-momentum plane. The IXS data are parameterized using a more correct extended-Drude scheme which accounts, for the first time, for many-body exchange-correlation effects at non-zero momentum transfer. Non-Born effects are considered in the same way with the original version by means of perturbation and exchange correction terms; however, these corrections are re-evaluated using the new dielectric response function. A more efficient implementation of these corrections has allowed the extension of the electron transport cutoff from ∼ 50 eV (used in the original version of our code) down to 1 Ry. As reported elsewhere (Emfietzoglou and Nikjoo 2007) the new dielectric response function results in substantially smaller inelastic and stopping cross sections (by ∼ 20–50%) in the sub-keV range while asymptotically converge at high energies to the stopping power values recommended by the International Commission of Radiation Units and Measure-ments (ICRU 1984).

For the vapor water version of MC4 we improve upon previous calculations (Emfietzoglou et al. 2000) by including

a model (Emfietzoglou et al. 2008) for the total and energy-differential ionization cross-section based on a Platzman plot analysis leading to a very good agreement ( 8%) with the recommended total ionization cross section values of Itikawa and Mason (2005). Finally, for both phases of water the following changes have been made with regard to angu-lar distributions:

(i) Following Wilson and Nikjoo (1999), the recipe of Grosswendt and Waibel (1978) for choosing the sec-ondary electron ejection angle has now been replaced by the angle-differential Bethe formula (Kim 1972) which makes use of the optical oscillator strength (for vapor) or the dielectric response function (for liquid) and, therefore, explicitly accounts for phase-effects. The agreement with the available experimental data on the ejection angle of low-energy secondaries is now improved.

(ii) The screening parameter in the screened Ruther-ford formula for the elastic cross section, originally obtained from the nitrogen data of Grosswendt and Waibel (1978), has now been replaced by the value of Uehara et al. (1992) which is based on experimental water data.

All the simulation results presented are average values over 1,000,000 primary electron tracks. The overall (statisti-cal) uncertainty is estimated to be less than 1%. For monoen-ergetic electrons, the primary electron energies cover the range from 100–1,000 eV with a step of 100 eV and from 1–10 keV with a step of 1 keV. The absorbed energy was scored in spherical shells 1 nm thick. The electron cut-off was set at 1 Ry ( 13.6 eV) due to limitations of our inelastic model below this value. For the two Auger radionuclides studied, the emission spectra (including Coster-Kronig and IC electrons) were taken from the American Association of Physicists in Medicine (AAPM) Nuclear Medicine Task Group Report (Howell 1992). Contrary to the organ level, the contribution of photons emitted by the two iodine isotopes (of about 3–4 and 27–30 keV) to cellular AF may, to a good approxima-tion, neglected. This is because photons with energies in the range of 3–30 keV have a mean free path of about 0.005–3 cm (Hubbel and Seltzer 2004), that is, orders of magnitude larger than the dimensions of the examined spherical targets (10–1000 nm). For modeling the uniform distribution of radio-activity the point of origin for each primary electron was dis-tributed randomly (proportional to the volume mass) inside the sphere. Note that the simulation follows electrons even if outside the target sphere since it is possible for the electrons to re-enter the scoring volume following large-angle scatter-ing events. The AF was deduced by calculating the ratio of the energy deposition in the target volume (which equals the sum of the energy deposition in the corresponding spherical shells) divided by the initial energy of the primary electron.

Semi-analytic approachAt the cellular level the MIRD Committee has adopted a semi-analytic approach to calculate absorbed fraction and related quantities based on the continuous-slowing-down-approximation (CSDA) using analytic range-energy

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918 C. Bousis et al.

expressions (Goddu et al. 1997). With some restrictions, the MIRD scheme is applicable to any spatial scale of inter-est, i.e., from the organ down to the cell level (Kassis 1992, Howell 1994). In this scheme, the absorbed fraction fi k hr r←( ) , i.e., the fraction of energy emitted from the source region rh that is absorbed in the target region rk for the ith radiation component (i.e., type and energy), is obtained by the following convolution integral:

φi k h rk rhX Ei x

r r xE

dE

dXx( ) ( )

( )

← = ∫ ←−

Y1

d , (1)

where Yrk rhx← ( ) is the geometric reduction factor represent-

ing the mean probability that a randomly directed vector of length x starts from a random point within the source region rh and ends within the target region rk,

and dE

dX X Ei x( )−

is the stopping power evaluated at X E xi( ) − which is the residual range of a particle with initial energy Ei after pass-

ing a distance x through the medium. The MIRD commit-tee evaluates dE/dX from differentiation of the Howell et al. (1989) range-energy expressions, X(Ei), that represent a low-energy improvement over Cole’s original formula (Cole 1969).

Results

In Figure 1 we present our MC calculated AF for monoen-ergetic electrons uniformly distributed in various submicron spherical volumes of water along with the results of the CSDA convolution integral of Equation (1) (here rk rh, i 1) using the Howell range-energy formula. The MC results are denoted either as vapor water or as liquid water depending on whether interaction cross sections for the vapor or liquid phase have been used; in both cases a unit-density medium is assumed. The AF differences (in %) using the CSDA- Howell values as reference are displayed in the insets. In Table I we present AF for a uniform distribution of the two Auger-emitters, 125I and 123I, in spherical volumes of water obtained by the CSDA convolution integral (denoted as CSDA-Howell)

Figure 1. The AF for a uniform distribution of monoenergetic electrons as a function of their initial energy for spherical volumes of various sizes calculated by the CSDA convolution integral with the Howell range-energy formula as well as by MC simulation using inelastic scattering cross sections for either liquid water or (density scaled) water vapor. The inset displays the AF differences (in %) using the CSDA-Howell values as reference.

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Electron energy deposition in small water spheres 919

in a smaller fraction of the primary electron energy being deposited inside the sphere (also discussed in Emfietzoglou et al. 2008). As expected, the differences decrease and shift to higher energies with increasing sphere size. The deviation between the CSDA and the liquid-phase MC results is up to ∼ 55%. Interestingly, at sufficiently high energies the above trend is reversed with the CSDA calculated AF being larger. This is because, at sufficiently high primary electron ener-gies, a significant fraction of the more energetic secondary electrons (d-rays) becomes capable of escaping the target volume, thus, depositing more and more energy outside of it. As a result, with increasing electron energy the neglect of the finite d-ray range in the CSDA method leads, eventu-ally, to an overestimation of the AF (up to ∼ 10–40%). This ‘reverse’ effect is more evident for smaller target sizes where d-rays can more efficiently escape the target volume. It is important to notice that this d-ray effect is somewhat more pronounced in the liquid than in the gas phase due to the greater ionization yield and larger inelastic mean free paths (Emfietzoglou and Nikjoo 2007). For the target sizes exam-ined the AF difference between the liquid and gas phase does not exceed the 10–20% level with the values for the gas phase being systematically higher due to the absence of long-range screening resulting in a stronger electron-water interaction (i.e., smaller inelastic mean free paths) and, thus, to smaller electron penetration.

The results of Table I show that differences between MC and CSDA calculations of the AF for the two Auger emitters (123I and 125I) range from ∼ 7% for the largest sphere exam-ined to 36% (liquid phase) and 47% (gas phase) for the small-est sphere. On the other hand, the phase-effect is only 1–3% except for the smallest target size where it reaches 10%.

Although the maximum deviation between MC and CSDA results for the monoenergetic electrons is rather similar for the three spheres, it shifts at higher energies with increasing sphere size (i.e., at ∼ 0.5 keV for the 10 nm sphere, at 2–3 keV for the 100 nm sphere and at 8–9 keV for the 1000 nm sphere). On the other hand, the deviations in the case of the Auger-emitters generally increase with decreasing sphere size being maximum for the 10 nm sphere. This is because of the high yield of sub-keV electrons (∼ 90%) which, as noted above, tend to increase the discrepancies for the smallest sphere (10 nm).

Conclusion

Condensed-phase effects in electron-water inelastic scat-tering models used in MC simulations have a noticeable but

as well as by our MC code using the two different sets of interaction cross sections that correspond to the vapor or liquid phase. Values in the parentheses are the differences (in %) of the MC calculations from CSDA predictions.

Discussion

The purpose of the present study is the comparison of AF for low-energy electrons distributed in subcellular-size spheri-cal volumes using Monte Carlo calculations with interaction cross sections corresponding to either the vapor or the liquid phase of water. In addition, the MC results are compared against the semi-analytic CSDA calculations that underline the MIRD cellular dosimetry model. By this comparison we examine both the limits of applicability of the CSDA semi-analytic method as well as the sensitivity of the results to the so-called phase effect, that is, to the influence of non-scalable density effects in the electron–water interaction models used as input in MC simulation.

The semi-analytic CSDA calculations have proven very practical and sufficiently accurate under many circum-stances. However, their extension from the cellular (several microns) to the subcellular (nm-mm) level may be questioned on the grounds that at the submicron scale the ‘discrete’ nature of interactions that is responsible for such effects as energy-loss straggling, angular deflections and secondary electron production cannot be neglected. On the other hand, MC track-structure codes simulating collision-by-collision the slowing down process of all generations of particles can account in a self-consistent manner for the ‘discrete’ nature of interactions, thus, overcoming the deficiencies of the CSDA approach. However, the majority of MC track-structure calculations for low-energy electrons have been carried out in a unit-density water vapor medium. Thus, the accuracy of such calculations depends upon the degree of which the so-called gas-phase approximation for water is valid in this case. This problem has been studied in various contexts in track structure theory over the past 30 years; see for example, Turner et al. (1982), LaVerne and Mozumber (1986), Nikjoo et al. (1994), Bigildeev and Michalik (1996), Emfietzoglou and Nikjoo (2005). In general, given that cells are in a liquid/solid-like state, one expects that the use of material-specific interac-tion cross-sections (pertinent to liquid/solid water) will be a step towards more accurate electron transport calculations.

It may be seen from Figure 1 that the MC calculated AF are generally higher than the CSDA results. This is because the Howell formula predicts a higher electron penetration in water than the present MC simulations, thus, resulting

Table I. AF for a uniform distribution of the two Auger-emitters, 125I and 123I, in spherical volumes of various sizes in both phases of water medium calculated by the CSDA convolution integral using the Howell range-energy formula as well as by MC simulation using interaction cross sections for either (density scaled) vapor water or liquid water. Values in the parentheses are the differences (in %) of the MC calculations from CSDA-Howell predictions.

Radius (nm)

Absorbed fraction (AF)125I 123I

MC-vapor MC-liquid CSDA-Howell MC-vapor MC-liquid CSDA-Howell

10 0.0729 (44.6%) 0.0677 (34.3%) 0.0504 0.0305 (46.6%) 0.0283 (36.1%) 0.020850 0.131 (14.9%) 0.128 (12.3%) 0.114 0.0523 (15.2%) 0.0511 (12.6%) 0.0454

100 0.178 (21.9%) 0.172 (17.8%) 0.146 0.0671 (19.8%) 0.0658 (17.5%) 0.0560500 0.420 (15.1%) 0.415 (13.7%) 0.365 0.134 (12.6%) 0.133 (11.8%) 0.119

1000 0.485 (7.9%) 0.481 (7.0%) 0.450 0.151 (7.1%) 0.150 (6.4%) 0.141

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relatively small effect (less than 10–20%) in the AF for the sphere sizes (10–1000 nm radius) and electron energies and Auger radionuclides examined here. The AF for the liquid phase are found to be always smaller than those of the vapor phase due to a larger ionization yield and a weaker electron-water inelastic interaction in the condensed-phase. The differences between the semi-analytic CSDA calculations using the Howell range-energy expression and the liquid-phase MC simulations can be sizeable (up to ∼ 35 for the Auger emitters and ∼ 55% for the monoenergetic electrons) and may limit the applicability of the semi-analytic calcu-lations for target sizes with linear dimensions smaller than about 1 micron.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

CB and DE acknowledge financial support by the Euro-pean Union FP7 ANTICARB (HEALTH-F2 - 2008-201587) research program.

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