Microdosimetry-based investigation of biological effectiveness of 252Cf brachytherapy source: TOPAS Monte Carlo study

Objective. To investigate biological effectiveness of 252Cf brachytherapy source using Monte Carlo-calculated microdosimetric distributions. Approach. 252Cf source capsule was placed at the center of the spherical water phantom and phase-space data were scored as a function of radial distance in water (R = 1–5 cm) using TOPAS Monte Carlo code. The phase-space data were used to calculate microdosimetric distributions at 1 μm site size. Using these distributions, Relative Biological Effectiveness (RBE), mean quality factor ( Q̅ ) and Oxygen Enhancement Ratio (OER) were calculated as a function of R. Main results. The overall shapes of the microdosimetric distributions are comparable at all the radial distances in water. However, slight variation in the bin-wise yield is observed with R. RBE, Q̅ and OER are insensitive to R over the range 1–5 cm. Microdosimetric kinetic model based RBE values are about 2.3 and 2.8 for HSG tumour cells and V79 cells, respectively, whereas biological weighting function-based RBE is about 2.8. ICRP 60 and ICRU 40 recommendation-based Q̅ values are about 14.5 and 16, respectively. Theory of dual radiation action based RBE is 11.4. The calculated value of OER is 1.6. Significance. This study demonstrates the relative insensitivity of RBE, Q̅ and OER radially away from the 252Cf source along the distances of 1–5 cm in water.


Introduction
Radiotherapy treatment using ionizing radiation having high or intermediate linear energy transfer (LET) demonstrates higher biological effectiveness as compared to conventional photon sources.The key advantages of using neutron sources for treatment are: (a) its high LET, and (b) unlike photons, the presence of oxygen in cells or tissues does not affect its biological effectiveness (Koh et al 1994, Srivastava et al 2014).Similar to photon-based brachytherapy sources such as 60 Co, 137 Cs, 192 Ir, 169 Yb, 125 I and 103 Pd etc, miniature 252 Cf source is a potential brachytherapy source for interstitial implantation, external surface applications and intra-cavitary applications (Abd El-Hafez et al 1997).In addition to the wide applicability of 252 Cf neutron sources in the field of research, medicine and industry (Maruyama andBeach 1986, Ghassoun et al 2010), it has also been used as a brachytherapy source since 1970 (Nagarajan et al 1971, Colvett et al 1972, Krishnaswamy 1972, Anderson 1973, 1974). 252Cf is a suitable neutron brachytherapy source because of its long half-life (2.6 years) and relatively high neutron yield (Wierzbicki et al 1997).The advantages and limitations of using 252 Cf source for various brachytherapy applications were reported elsewhere (Maruyama and Beach 1986, Maruyama et al 1991, Wierzbicki et al 1994, Abd El-Hafez et al 1997, Rivard 2000, Ghassoun et al 2010, Lei et al 2011, Ghassoun 2013, Yadollahpour et al 2015, Zhou et al 2016, Qian et al 2017, Xiong et al 2017, Rasouli et al 2020, Karimi-Shahri et al 2021).
Several published studies were centered on radiobiological experiment-based investigation of Relative Biologic Effectiveness (RBE) of 252 Cf brachytherapy sources (Hall et al 1974, Feola et al 1982, Maruyama et al 1983, Maruyama 1984, Maruyama et al 1991, Fairchald et al 1997, Wierzbicki 2012, Zhou et al 2016, Rasouli et al 2020, Karimi-Shahri et al 2021).Wang and Zhang (2004) developed a new formula based on the theory of compound-dual-radiation-action to predict the cell-survival fraction for a mixed-radiation field of 252 Cf.RBE of ionizing radiations can also be determined based on different models such as the Biophysical model, theory of dual radiation action (TDRA) and microdosimetric kinetic model (MKM) which use microdosimetric distributions as input parameters (Rossi 1979, Griffiths 1985, Zaider et al 1996, Wierzbicki 2012, Karimi-Shahri et al 2021).Microdosimetry is the evaluation of stochastic distribution of energy depositions at cellular and subcellular targets (Rossi 1979, Griffiths 1985, Zaider et al 1996).The lineal energy (y) is an important quantity for evaluation of radiation quality.As per ICRU Report 36 (Griffiths 1985), y is defined as / e = y l, where ε is the energy imparted to matter in the volume of interest by a single energy deposition event and l is the mean chord length of the target volume.is the dose probability density of y.The moments of y distribution ( ȳF and ȳD ) are indicative of RBE of ionizing radiation (Kellerer and Rossi 1974).
Although microdosmetric studies on 252 Cf neutron sources related to radiation protection are reported (Rollet et al 2004, Zhang et al 2014, Chattaraj et al 2019a), a similar study for a 252 Cf brachytherapy source has not been reported in the literature.The present study is aimed at calculating the biological effectiveness of a 252 Cf brachytherapy source as a function of radial distance (R) in water phantom using microdosimetric technique.For this purpose, microdosimetric distributions at 1 μm site size as a function of R in water are calculated using the Monte Carlo-based tool for particle simulation (TOPAS) and FLUKA codes.Biological effectiveness is quantified in terms of mean Quality Factor ( ̅ Q), RBE and Oxygen Enhancement Ratio (OER).In the present study, RBE is derived based on Biophysical model, MKM and TDRA.̅ Q is calculated based on ICRP 60 and ICRU 40 recommendations using the microdosimetric distributions.Additionally, ̅ Q is derived based on distributions of unrestricted LET ( ¥ L ) in water and compared with the microdosimetry-based ̅ Q values.The study also demonstrates an efficient method for calculating microdosimetric distributions.

252 Cf brachytherapy source and phantom models
The present study simulated the applicator tube (AT) 252 Cf source capsule from Oak Ridge National Laboratory (ORNL).The source details were taken from Karimi-Shahri et al (2021) and schematic diagram of the source is shown in figure 1.The active core of the source capsule is a cylinder (1.5 cm length × 0.61 mm diameter) made of Cf 2 O 3 of density 12 g cm −3 .The cylindrical active core is surrounded by a primary capsule followed by a secondary capsule.Both the primary and secondary capsules are made of Pt/Ir-10% mass having density of 21.51 g cm −3 .The inner and outer diameters of the primary capsule are 1.35 and 1.75 mm, respectively and its inner and outer lengths are 15.50 and 17.78 mm, respectively.The secondary capsule has inner and outer diameters of 1.80 and 2.80 mm, respectively, and inner and outer lengths of 17.82 and 23.14 mm, respectively.The gap between the primary and secondary capsules is filled with air (0.0012 g cm −3 ).The rounded ends of capsules are welded.In addition, a bodkin evelet having diameter of 0.63 mm is embedded in the secondary capsule (Rivard et al 1999).The typical source strength of a remote after loading HDR capsule is 0.3-0.4mg 252 Cf and using advanced radiochemistry techniques, loading up to 1 mg of 252 Cf per mm 3 is feasible (Rivard et al 1999). 252Cf is produced in nuclear reactor.Cf source typically contains up to 85% of 252 Cf and the remaining 15% is 249 Cf, 250 Cf, and 251 Cf which have negligible dosimetric impact due to their long half-lives (Knauer and Martin 1997). 252Cf decays by spontaneous neutron fission (3.1%) and alpha emission (96.9%).The decay process of 252 Cf includes emission of beta, prompt gamma up to 6.5 MeV and photons from fission products up to 2 MeV (Wierzbicki et al 1997).The neutron and photon yield from one μg of 252 Cf are 2.31 × 10 6 neutrons s −1 (Permar 1976) and 1.332 × 10 13 photons s −1 (Wierzbicki et al 1997), respectively.For the encapsulated 252 Cf brachy source, alpha and beta will be absorbed by the source capsule materials and hence not considered in the present study.Neutrons and photons can penetrate the encapsulation material and contribute in the total dose deposited.Rivard et al (1999) and Wierzbicki (2012) showed that approximately, 33% of the total absorbed dose is due to photon emissions.However, in terms of RBE weighted dose (absorbed dose × RBE), the relative contribution from photons will be very less as compared to neutrons.Note that the RBE and ̅ Q are sensitive to neutron energy and greater than unity whereas for photons, value of these parameters is unity, independent of photon energy (Karimi-Shahri et al 2021).Rasouli et al (2020) demonstrated that the contribution of neutrons to the total dose is maximum and the equivalent gamma dose is negligible when compared to those of neutrons.The above findings showed that in terms of RBE-weighted dose or equivalent gamma dose, relative contribution from neutrons is maximum as compared to photons.Hence, the present study considers only the neutron components of 252 Cf.In the Monte Carlo calculations, the neutron energy spectrum of a 252 Cf source was modeled as a Watt fission spectrum (Brown et al 2002, Rasouli et al 2020), and it is given by the equation: , 1 2 where E is neutron energy in MeV.The fission neutron energy extends up to 20 MeV with most probable energy of 0.7 MeV and average energy of about 2.3 MeV.
The present study simulated a spherical phantom with a diameter of 30 cm and filled with liquid water of density 0.998 g cm −3 .The phantom dimensions and density are consistent with the published literature (Karimi-Shahri et al 2021).The AT 252 Cf brachytherapy source capsule with isotropic emission of neutrons was placed at the center of the water phantom.

Interaction of neutrons
The interactions of neutrons with tissue or tissue equivalent medium are discussed in details elsewhere (Rivard et al 1999, Chattaraj et al 2019a).Neutrons are thermalized through elastic scattering with hydrogen nuclei of water.The thermal neutrons undergo capture reactions: (a) 1 H (n,γ = 2.23 MeV prompt γ) 2 H (0.33 b), and (b) 14 N(n,p = 580 keV) 14 C with nitrogen (1.83 b) and (c) 14 N(n,γ = 10.8MeV prompt γ) 15 N (0.08 b) (Burrows 2006).Fast neutrons interact with 12 C, 14 N and 16 O present in TEPC wall through: (a) elastic scattering producing recoil nuclei, (b) non-elastic process which releases α-particles, and (c) 12 C(n, nγ) 12 C and 16 O(n, nγ) 16 O inelastic scattering reactions.Note that 16 O is abundantly present in water and high density Cf 2 O 3 source core material.The neutrons also interact with the source capsule materials such as natural Platinum and Iridium.
TOPAS uses reference physics lists of the Geant4 toolkit (Perl et al 2012).In TOPAS, the hadronic interaction physics processes are modeled using 'g4h-phy_QGSP_BIC_HP' physics list (Agostinelli et al 2003, Allison et al 2006).Elastic scattering of hadrons is activated using 'g4h-elastic_HP' physics constructors.'g4ion-binary cascade' is used to model hadronic inelastic scattering for ions.The 'g4h-phy_QGSP_BIC_HP' uses the high precision neutron models and cross sections to describe elastic and inelastic scattering, capture and fission by neutrons below 20 MeV (Agostinelli et al 2003, Allison et al 2006).The G4NDL database is required for this physics list (Agostinelli et al 2003, Allison et al 2006).The productions and interactions of electrons and gamma are handled by 'g4em-livermore' physics module.
For neutrons below 20 MeV, FLUKA uses its own neutron cross section library (P5 Legendre angular expansion, 260 neutron energy groups) containing more than 250 different materials (Ferrari et al 2021).FLUKA includes Standard multi-group transport of neutrons with photon and fission neutron generation, detailed kinematics of elastic scattering of neutrons on hydrogen nuclei, transport of recoil protons and protons from 14 N(n,p) reaction.For neutron interactions with other than hydrogen nuclei, KERMA factors are used to calculate energy deposition (including from low-energy fission).FLUKA generates capture gamma according to the multigroup treatment, but these gammas are transported with the more accurate 'EMF' package which performs continuous transport in energy and allows for secondary electron generation (Ferrari et al 2021).

TOPAS Monte Carlo calculations
TOPAS (Perl et al 2012) is a Monte Carlo platform layered on top of Geant4 toolkit (Agostinelli et al 2003, Allison et al 2006).The present study used TOPAS 3.6.1 and its microdosimetric extension.The lineal energy scorer implemented in TOPAS for scoring microdosimetric lineal energy distributions in the spherical cavity of the TEPC is developed by Underwood et al (2017).The details of TOPAS microdosimetric extension is described elsewhere (Zhu et al 2019).Physics modules used in the present study for all the TOPAS-based simulations are 'g4em-livermore', 'g4h-phy_QGSP_BIC_HP', 'g4decay, g4ion-binary cascade', 'g4h-elastic_HP' and 'g4stopping'.

Calculation of neutron and secondary photon fluence spectra
The fluence spectra of neutrons and secondary photons (produced by the interaction of neutrons with water and source materials) in energy (E) were calculated at R = 1-5 cm in water phantom using TOPAS when the AT 252 Cf source was positioned at the centre of the phantom.The simulation was repeated by replacing the water with vacuum i.e. in the absence of water medium.Using the calculated fluence spectra, fluence-weighted average energy ( Ēfl ) for neutrons and secondary photons were calculated using the following formula: where E i and j i are the mid-point energy and fluence of neutrons or secondary photons in the i th bin, respectively.The fluence-weighted average energy of neutrons and secondary photons are denoted as Ēfl n and Ē , fl ph respectively.

Calculation of absorbed dose
Absorbed dose to water per neutron (D w ) were calculated as a function of R in water (up to 5 cm) along the transverse axis of the AT 252 Cf source using TOPAS (VOXEL size: 1 mm × 1 mm × 1 mm).The calculated values of D w were normalized with respect to the value at R = 1 cm.The TOPAS-calculated values of D w along the transverse axis of the source were converted to absorbed dose rate (  D w ) by multiplying the values by neutron yield of 2.31 × 10 6 n/(s • μg ).

Microdosimetry of 252 Cf source in vacuum
Microdosimetric distributions of a bare point 252 Cf source (hereafter referred as point 252 Cf source) and AT 252 Cf source capsule were calculated using TOPAS in the LET-1/2 TEPC (Far West Technology lnc.2022) filled with TE propane gas of density 7.874 × 10 −5 g cm −3 , which simulates 1 μm site size.The LET-1/2 TEPC consists of a spherical cavity of 1.27 cm diameter which is surrounded by a 1.27 mm thick wall made of A-150 plastic.In these simulations, the source and the TEPC were in vacuum and the distance between the source and the TEPC was 2 cm.In the case of point 252 Cf source, neutrons were modelled to emit in a cone (emission restricted to halfangle of 21°) covering the outer diameter of the TEPC.Whereas the AT 252 Cf source capsule emits neutrons isotopically.These calculations were carried out to investigate the effect of source encapsulation materials on the microdosimetric distributions.The schematic diagram of the simulation setup is shown in figure 2. Note that Chattaraj et al (2019b) adopted a similar approach while investigating the effect of encapsulation materials on the microdosimetric distributions of brachytherapy photon sources.

Phase-space scoring in water phantom
TOPAS-based microdosimetric calculation of a AT 252 Cf brachytherapy source as a function of R in water phantom is a two-step process: (a) calculation of phase-space in water, and (b) calculation of microdosimetric distributions in LET-1/2 TEPC using the phase-space as a source.In the simulations, spherical shells of inner and outer radii of 0.762 and 0.763 cm, respectively, made of water were placed at R = 1, 2, 3, 4 and 5 cm along the transverse axis of the water phantom as shown in figure 3. Note that inner radius of 0.762 cm for spherical phasespace corresponds to the outer radius of LET-1/2 TEPC.The phase-space was scored at the outer surface of the spherical shell.Particles such as neutron, alpha, deuteron, triton, proton, electron, and gamma were scored in the phase space.Note that these gamma photons are produced as a result of interaction of neutrons with water and source materials such as active core and encapsulations.Hereafter, these gamma photons are referred as secondary photons.The phase-space scored were saved in a binary format.Note that the co-ordinates of the phase-space are always with respect to the WORLD volume.The phase-space file stored the 'RunID', 'EventID', 'TrackID', 'ParentID' and 'TimeOfFlight' of all the particles.

Microdosimetric distributions in water
The following simulation setups were used for calculation of microdosimetric distributions as a function of R using the TOPAS code.For all the calculations described below, the electron range cut was set at 100 nm in the TEPC wall and cavity.
Case 1: Microdosimetric distributions at R = 1, 2, 3, 4 and 5 cm were calculated in LET-1/2 TEPC using the pre-calculated phase-space as source.The pre-calculated phase-space at a given R stored all the primary and scattered neutrons as well as secondaries produced from interactions of these neutrons with water and source materials (see section 2.3.4).Hence, TEPC was placed in free space surrounded by the phase-space source  (see figure 4(a)).In this simulation set up, the center of the LET-1/2 TEPC coincided with the co-ordinates of the center of spherical shell (with respect to the WORLD volume) as used for scoring the phase-space.
Case 2: to compare the microdosimetric distributions calculated using the phase-space, a full simulation consisting of AT 252 Cf source capsule positioned at the center of water phantom with LET-1/2 TEPC at R = 2 and 5 cm was carried out (see figure 4(b)).

FLUKA calculations
TOPAS is comparatively a new Monte Carlo code.Zhu et al (2019) compared the TOPAS microdosimetric extension for proton and carbon ion beams against FLUKA-and MCHIT-calculated as well as measured microdosimetric distributions.Chattaraj et al (2022) performed microdosimetric distributions of N-series photons and ISO beta sources using TOPAS.However, TOPAS-based microdosimetric distributions for neutron source are not available in the published literature.Hence, microdosimetric distributions of a 252 Cf brachytherapy source were calculated using FLUKA combined with Flair and compared with TOPAS-based distributions.Note that Flair is the Advanced Graphical User Interface for FLUKA code.FLUKA has been well benchmarked for calculation of the microdosimetric distributions of neutron sources (Fasso et al 2001, Rollet et al 2004, Zhang et al 2014, Chattaraj et al 2019a).The TOPAS-based microdosimetric calculations as described in Case 2 were repeated using FLUKA for R = 1-5 cm (see figure 4(b) for simulation setup).Additionally, absorbed dose to water and DoseQLET were calculated as a function of R in water along the transverse axis of the source using USRBIN scoring card (VOXEL size: All the simulations were performed in a 64 bit UBUNTU 20.04 LTS OS-based Work-stations which has 64 Intel(R) Xeon(R) Gold 6242 CPU@2.80GHzprocessors and 128 GB RAM.All the TOPAS simulations were performed in multithreaded mode.For TOPAS full simulation, each radial distance in water needed a separate simulation.In the case of phase-space approach, the phase-spaces at R = 1-5 cm were scored simultaneously in a  Results are reported without using any kind of filtrations Results are reported without using any kind of filtrations single run.To calculate microdosimetric distributions using this phase-space, separate simulation was required for each radial distance.In the FLUKA microdosimetric calculation at different R, transport cut-off for electron was set at 400 keV in the water phantom to speed up the simulation (see table 1).Note that the CSDA range of 400 keV electron is less than the wall thickness of the TEPC (Berger 1993) and hence electron below 400 keV will not be able to enter the sensitive volume of the TEPC (Chattaraj et al 2019b).

¯( )
¥ Q L at a given R is calculated by taking ratio of the FLUKA-calculated values of DoseQLET ( ¯( ) ) to absorbed dose to water.¯( ) where, ( ) d L is the dose probability density of ¥ L .̅ Q depends on LET and RBE.̅ Q converts absorbed dose to dose equivalent and thus takes care of the biological effectiveness of ionizing radiations.RBE depends on the observed biological effect, specific test organism and the experimental conditions in a complex way.Thus, judgement is required for using RBE value.Hence, ̅ Q is chosen as an alternative of the RBE (Kerr 1988).The suffix, 't' denotes test radiation which is neutron in the present study and the suffix 'x' is reference radiation which is 60 Co gamma rays.

Calculation of RBE
The saturation-corrected dose-mean lineal energy, y * , is defined by the equation: α 0 is the slope in the limit of LET = 0 and β is independent of the type of radiations i.e. b b = .
x t y 0 is the saturation parameter, which is used to correct the overkilling effect for high LET radiation.

MKM HSG
The TDRA-based RBE (RBE TDRA ) in the presence of saturation correction is described by Keller and Rossi (Kellerer and Rossi 1974) and is defined as where ȳD,ref is dose average lineal energy for reference radiation which is 200 kVp x-rays in the present study.

Calculation of OER
OER is defined as the ratio of doses required to achieve the same biological effect under anoxic and oxic conditions.In the absence of oxygen, the production of sub-lesions is reduced by a factor, ρ which may have a constant value of 0.62 or may be function of y which approaches 1 at higher value of y (see figure 5) (Kellerer and Rossi 1974).The detailed formalism of OER is described by Kellerer and Rossi (1974).As per TDRA, the OER in the presence of saturation correction and in the range of low dose is given by: presents TOPAS-calculated fluence spectra of neutrons and secondary photons at R = 1, 2 and 5 cm in water when the 252 Cf capsule is placed at the center of the water phantom.The figure also includes fluence spectra of neutrons and secondary photons when the 252 Cf capsule is placed in vacuum i.e. in the absence of water medium.Yield of low energy neutrons (1 eV -20 keV) increases with the increase of R in water.The overall shape of secondary photon fluence spectra at R = 1, 2 and 5 cm in water is comparable and the peak is at about 2.2 MeV which is due to 1 H (n,γ = 2.23 MeV) 2 H reaction.Note that although the peak position is at 2.2 MeV but the peak height is higher at larger R.For example, the peak heights are 0.31, 0.49 and 0.62 at R = 1, 2 and 5 cm, respectively.No photon peak is observed at 2.2 MeV in the absence of water medium.A small photon peak is observed at about 6.1 MeV in all cases which may be due to the 16  Cf source capsule is in vacuum.

Absorbed dose in water
Figure 7 presents TOPAS-calculated normalized absorbed dose to water in water phantom along the transverse axis of the 252 Cf source capsule. D w falls-off rapidly with R in water.For example, the values of  D w at R = 2 and 5 cm are, respectively, about 28% and 4% of the value at R

Microdosimetric distributions
The microdosimetric distribution is the plot of yd(y) on a linear scale versus y on a log-scale.The distribution is normalized such that ( ) ò = d y dy 1.In this type of plot, equal areas under the curve represent equal doses delivered at the intervals of y-values.The present study used 25 lineal-energy bins per decade for plotting the distributions.

Microdosimetric distributions in vacuum
Figure 8 presents TOPAS-calculated microdosimetric distributions at 1 μm site size for a point 252 Cf and AT 252 Cf source when the LET-1/2 TEPC is in vacuum to show the effect of source materials on the microdosimetric distributions.The overall shape of the distributions is comparable for both the cases.However, in the case of a AT 252 Cf source, bin-wise yield is slightly higher in the region y = 0.1-5 keV/μm when compared to the point source (see the inset of figure 8).This is due to the production of secondary photons in the source materials.Marginal difference in the bin-wise yield is observed between the two distributions in the region y > 100 keV/μm.This difference may be attributed to the fact that higher statistical uncertainties are associated with the bin-wise yields in the region y > 100 keV/μm.For the distributions in the region y > 100 keV/μm, main contributions are from heavy recoil nuclei which have lesser yield as compared to the secondary protons which are responsible for the distributions in the region y < 100 keV/μm and hence higher uncertainties in the region y > 100 keV/μm.Note that the encapsulation materials slightly modify the neutron spectrum ( Ēfl n = 2.3 and 2.12 MeV, respectively for bare and AT 252 Cf capsule).However, this may not affect the microdosimetric distributions in the region y > 100 keV/μm.The calculated ȳF and ȳD values are: (a) 32.411 ± 0.614 and 81.891 ± 4.993 keV/μm, respectively, for the point 252 Cf source, and (b) 19.159 ± 0.706 and 78.979 ± 8.511 keV/μm, respectively, for AT 252 Cf source.The presence of source materials alters the ȳF values.However, ȳD values remain unchanged (considering the statistical  uncertainty).Higher bin-wise yield in y = 0.1-5 keV/μm region in the case of aAT 252 Cf source results in smaller ȳF value as compared to that of point 252 Cf source (see the inset of figure 8).
The TOPAS-calculated ȳF and ȳD values for the point 252 Cf source are in good agreement with the corresponding Geant4-calculated values by Chattaraj and Selvam (2021).Note that values of ȳF and ȳD as reported by Chattaraj and Selvam (2021) are 32.774±0.025and 80.513±0.352keV/μm , respectively.The overall shape of the TOPAS-calculated microdosimetric distribution including peak-height and peak position for the point 252 Cf source at 1 μm site size shows reasonably good agreement with that calculated using Geant4 toolkit by Chattaraj and Selvam (2021) for a bare 252 Cf source.

Microdosimetric distributions in water
Figure 9 presents the microdosimetric distributions of a AT 252 Cf source in water at 1 μm site size.The TOPAScalculated microdosimetric distributions include: (a) distributions as a function of R in water using the precalculated phase-spaces (see figures 9(a)), and (b) distributions at R = 2 cm and contributions from the secondary particles using full simulation (see figure 9(b)).Figure 9(c) shows the FLUKA-calculated microdosimtric distributions as a function R in water.The overall shape of the microdosimetric distributions at R = 1-5 cm is almost comparable (see figures 9(a) and (c)).However, slight variations in peak-height and peak positions are observed with change in R which is explained below.As R increases: (a) peak-height decreases and  peak positions shift slightly towards higher y values, and (b) yield in the region y = 0.1-5 keV/μm increases which is due to increase in the relative contributions from secondary photons at larger R. Note that at R = 1 cm, the fluences of neutrons and secondary photons are 95% and 5% of the total fluence, respectively, whereas these are 75% and 25% at R = 5 cm.A comparison of TOPAS and FLUKA-based distributions (see figures 9(a) and (c)) show that at a given R: (a) both distributions are comparable in the region y = 5-100 keV/μm (except slight difference in the peak-height), (b) bin-wise yields are different in the regions y = 0.1-5 keV/μm and y = 100-1000 keV/μm.These differences may be attributed due to differences in the physics models used in TOPAS and FLUKA.In FLUKA, event-by-event transportation is not possible for secondary α particles and recoil heavy nuclei produced from neutrons and these charged secondaries deposit their energy at the point of production as per the KERMA factor (Ferrari et al 2021), whereas TOPAS transports each particle on event-byevent basis.This may be the possible reason for differences in the region y = 100-1000 keV/μm.TOPAS and FLUKA used different neutron cross-section libraries due to which there may be difference in (n, γ) productions and hence, difference is observed in the region y = 0.1-5 keV/μm.Table 4 compares TOPAS-and FLUKA-calculated ȳ , F ȳD and y * values for a AT 252 Cf source at 1 μm site size as a function of R. TOPAS-based values are calculated using the phase-space.The ȳF values decrease with the increase in R. Considering the statistical uncertainty, the ȳD values calculated in TOPAS or FLUKA are almost insensitive to the radial distance in water.The variation of y * with R also follows a similar trend as that of ȳ .D A comparison of ȳF values of TOPAS and FLUKA shows a difference of about 9% at R = 1-2 cm and good agreement at R = 3-5 cm.TOPAS-calculated ȳD value at a given R is higher than that calculated using FLUKA (about 10%-20% depending upon R).The difference between TOPAS and FLUKA-calculated ȳF or ȳD values is due to the difference in the microdosimetric distributions (see figures 9(a) and (c)).The ȳD values are sensitive to the yield at the tail part of the distributions (100-1000 keV/μm) whereas, ȳF values are sensitive to the yield in the region 0.1-5 keV/μm of the distributions.As y * values are calculated considering y 0 = 150 keV/μm, the binwise yield with y > 150 keV/μm has less influence on the y * values.Considering the associated statistical uncertainties, TOPAS-and FLUKA-calculated y * values are comparable.Note that MKM-based RBE depends on the y * value (see section 2.7).TOPAS-calculated full simulation-based ȳF and ȳD values at R = 2 cm are in good agreement with the corresponding TOPAS-calculated values using phase space (see table 4).

RBE, Q and OER
¥ Q L and RBE TDRA are insensitive to R. At a given R, QICRU40 is about 9% higher than QICRP60 and ¯( ) This may be attributed to the differences in the algorithms.Note that both ¯( ) ¥ Q L and QICRP60 are calculated based on ICRP 60 recommendation.However, calculation of QICRP60 assumes » ¥ y L whereas ¯( ) Values of RBE TDRA are  The calculated value of OER is about 1.6 when ρ is a function of y and OER is about 2.17 when ρ has a constant value of 0.62.Assuming ρ has a constant value, the calculated value of OER = 2.17 is markedly higher than the observed OER values of 1.5-1.7 for inactivation of mammalian cells (Kellerer and Rossi 1974).

Computational efficiency
At smaller R in water (up to R = 2 cm), satisfactory statistical uncertainty on ȳF and ȳD can be achieved at the expense of large CPU time (about 135 h at R = 2 cm).As the abundance of neutron fluence decreases with R, it is difficult to achieve satisfactory statistical uncertainty on the calculated microdosimetric distribution at larger R.For example, at R = 5 cm, even after involving 180 h CPU time for tracking 4 × 10 9 primary particles, the statistical uncertainty on ȳD is about 40%.However, using the pre-calculated phase-space results in improving the efficiency of the calculations (smaller CPU time) with satisfactory statistical uncertainty (see table 1).

Discussions
RBE of neutrons depends on the neutron energy spectrum whereas RBE is unity for photons, independent of its energy (Karimi-Shahri et al 2021).However, the present study involving a AT 252 Cf source model shows that although the neutron fluence spectrum varies with R in water (see figure 6), the calculated biological effectiveness in terms of ̅ Q or RBE is independent of R (see tables 5 and 6).This implies that in the case of brachytherapy treatment using AT 252 Cf source, a constant RBE value can be applied up to the treatment distance of 5 cm from the source.
The existing dosimetry protocol that uses the concept of RBE for the 252 Cf brachytherapy sources contains large uncertainties (Maruyama et al 1991).Experimentally evaluated RBE depends on a number of factors such as: (a) type of tissue irradiated, (b) distance from the source (the neutron spectrum changes with the distance), (c) absorbed dose and dose rate, (d) tissue hypoxia and (e) reference radiation etc (Wierzbicki 2012).RBE MKM HSG and RBE r show reasonably good agreement with most of the published RBE values (within range of 2.1-3) at high dose.In Medical Radiological Research Center, Obninsk, the RBE value calculated using empirical formula for fractionated exposure is 3 for a mixed neutron and γ field (Wierzbicki 2012) which is also comparable to RBE MKM V79 and RBE r calculated in the present study.Maruyama and Feola (1983) reported RBE of a 252 Cf for normal tissues, tumours and cells and showed that RBE at anoxic cells is higher than that at oxic cells.The calculation algorithms used for RBE MKM and RBE r do not contain dose or dose-rate dependency term and also do not consider tissue hypoxia whereas experimentally obtained RBE includes all these factors.The calculated RBE values in the present study may be a good indicator to show that RBE does not change substantially over the range R = 1-5 cm but should not be used to derive absolute RBE values as they are very specific to the biological end point (Coutrakon et al 1997).Thus, RBE MKM and RBE r values are not expected to give clinicians firm RBE values for treatment planning (Coutrakon et al 1997).However, when microdosimetric distribution is known, these models (see section 2.7) can be used in conjunction with other cell, organ, and tissue response to radiation to arrive at a judiciously chosen RBE value for treatment planning (Coutrakon et al 1997).The probable reasons for exhibiting higher RBE value in the case of fast neutrons are: (a) neutron irradiation produces higher proportion of irreversible or unrepairable changes as compared to photons (Wierzbicki 2012) (b) dependence of the sensitivity of cells on the phase of their division cycle is less pronounced for neutron irradiation (Rodney Withers et al 1974, Wierzbicki 2012), and (c) OER value for neutron is less than that of gamma (Wierzbicki 2012).Drew et al (1972) and Djordijevic et al (1973) showed that for neutron component of a 252 Cf, OER = 1.5 to 1.7 and for γ radiation, OER = 2.5-3.5.Hall et al 1974 showed that OER = 1.42 for 252 Cf.The calculated OER value of 1.6 for neutron component of a 252 Cf source (assuming ρ is function of y) in the present study agrees well with the values reported by Drew et al (1972), Djordijevic et al (1973) and Hall et al (1974).The damage of cancerous tissue by photons is mostly due to production of free radicals and hence the hypoxic cells present in malignant tumours become radioresistant under photon irradiation (Karimi-Shahri et al 2021).However, the biological effectiveness of neutrons is less sensitive to the presence of oxygen (lesser OER than for γ radiation) because neutron interacts with tissue through nuclear interaction (Karimi-Shahri et al 2021).

Conclusions
In the present study, biological effectiveness of a AT 252 Cf brachytherapy source as a function of radial distance in water is calculated using TOPAS-based microdosimetric distributions at 1 μm site size.The study shows that using the pre-calculated phase-space as source for calculating microdosimetric distributions in water is efficient in terms of CPU time as compared to full simulation.TOPAS-calculated microdosimetric parameters such as ȳ , F ȳD and y * agree reasonably well to the corresponding parameters which are based on FLUKA.The study shows that microdosimtric parameter, ȳF is sensitive to distance from the source in water whereas ȳD and y * are insensitive.Although neutron fluence spectrum varies over the distance of 1-5 cm in water, the values of ̅ Q, RBE r, RBE TDRA and RBE MKM are insensitive to distance.The calculated values of RBE r, RBE MKM V79 and RBE MKM HSG are about 2.8, 2.8 and 2.3, respectively.Depending on the calculation models, ̅ Q is in the range of 11.4-16.2.RBE TDRA compares reasonably well with the ̅ Q values.The calculated value of OER is about 1.6 which is less than that of γ radiation due to which the treatment outcome of a 252 Cf-based brachytherapy has less dependency on tissue hypoxia.
mean lineal energy.( ) f y represents the number of events having event size between y and y + dy and ( )

Figure 1 .
Figure 1.Schematic diagram of the AT 252 Cf source capsule.Red color: active core; green color: primary and secondary capsules; yellow color: air; blue color: bodkin evelet.

Figure 2 .
Figure 2. Schematic diagram of the simulation setup for calculation of microdosimetric distribution in vacuum: (a) point 252 Cf source; (b) AT 252 Cf source capsule.White sphere surrounded by green sphere represents TEPC and red cylinder represents a AT 252 Cf source capsule emitting neutrons isotropically.

Figure 3 .
Figure 3. Simulation geometry of spherical phase-space in TOPAS.The cylindrical part at the centre represents AT 252 Cf source capsule.The large aqua coloured sphere represents 'spherical water phantom'.The green-coloured spheres represent various positions of 10 μm thick spherical shells filled with water for scoring phase-space.Spheres at the right side of the source represents R = 1, 3 and 5 cm and at the left side represents R = 2 and 4 cm.
w 2.5.Monte Carlo parameters The summary of the Monte Carlo parameters used in the calculations as per the recommendations of AAPM TG-286 (Sechopoulos et al 2018) is shown in table 1.

Figure 4 .
Figure 4. Schematic diagram of the simulation geometries.(a) Case 1 and (b) Case 2. White sphere surrounded by green sphere represents TEPC; red cylinder represents AT 252 Cf source capsule; red-colored spherical shell surrounding the TEPC represents phase-space source; blue color sphere represents water phantom RBE can be calculated by combining microdosimetric distributions with biological weighting function, r(y)(Pihet et al 1990, Coutrakon et al 1997, Paganetti et al 1997) or using MKM(Coutrakon et al 1997, Hawkins 2003, Kase et al 2006) or TDRA(Kellerer and Rossi 1974).The r (y)-based approach described byPihet et al (1990) is a biophysical model which was used to predict the RBE(Coutrakon et al 1997, Paganetti et al 1997).The r(y)-based RBE can be estimated according to the relation: was derived experimentally using neutron therapy beams with different radiation qualities by the iterative procedure considering acute effects in the intestinal crypt cells of mice as the biological end point(Pihet et al 1990, Coutrakon et al 1997).The y versus r(y) curve has similar shape to the LET versus RBE curve for the in vitro human kidney cell and V79 hamster cell irradiation(Coutrakon et al 1997).Coutrakon et al (1997) demonstrated that r(y)-based RBE shows reasonably a good agreement with the experimentally obtained RBE at the 10% survival of V79 hamster cells in the absorbed dose range of 2-12 Gy of protons and 60 Co (reference radiation).The MKM is a phenomenological model which was first introduced byHawkins (2003) and then modified byKase et al (2006).The MKM-based RBE at 10% survival level can be calculated using the following equation (Newpower et al 2019):
02cGy g w which shows reasonably good agreement with the value of 1.873 cGy/ h ⋅ μg calculated by Rivard et al (1999) using MCNP code.At R = 2 cm, Burmeister et al (2005) reported measured value of  D w = 4.23±0.53mGy/h ⋅ μg which

Figure 6 .w
Figure 6.TOPAS-calculated fluence spectra of (a) neutrons and (b) secondary photons at R = 1, 2 and 5 cm in water phantom when the 252 Cf source is placed at the center of water phantom.Also shown are fluence spectra of neutrons and secondary photons when the 252 Cf source capsule is in vacuum.

Figure 7 .
Figure 7. TOPAS-calculated normalized absorbed dose to water in the water phantom along the transverse axis of the AT 252 Cf source.The dose is normalized at R = 1 cm along the transverse axis of the source.

Figure 8 .
Figure 8.Comparison of TOPAS-calculated microdosimetric distributions of a point 252 Cf source and AT 252 Cf source capsule at 1 μm site size.In this simulations LET-1/2 TEPC was placed in vacuum.

Figure 9 .
Figure 9. Calculated microdosimetric distributions of a AT 252 Cf source at 1 μm site size as a function of R in water: (a) TOPAScalculated using phase-space, (b) TOPAS-based full simulations at R = 2 cm in water and secondary particle contributions and (c) FLUKA calculated.

Figure 9
(b)  shows that the contributions are from: (a) electrons in y = 0.1-20 keV/μm (negligible yield), (b) protons in y = 5-100 keV/μm, and (c) recoil carbon and oxygen nuclei in y = 100-1000 keV/μm.The combined microdosimetric distribution from all the particles (see legend: 'total' in figure 9(b)) is in good agreement with the corresponding distribution in figure 9(a).This comparison justifies the use of phase-spacebased approach.

Table 1 .
Summary of parameters used in the Monte Carlo calculations as per the recommendations of AAPM TG-286 (Sechopoulos et al 2018).
(Burrows 2006)action with the source active core material (Cf 2 O 3 ) as well as in water medium.Note that 16 O(n, nγ)16O reaction with fast neutrons yield 6.13 MeV characteristic γ rays(Burrows 2006, Elsheikh 2018).19192Ir and 195 Pt(n,γ)196Pt yield spectrum of gamma photons(Burrows 2006).191 has significant cross section for (n, γ) reactions for neutrons below 100 keV(Burrows 2006).Note that the multiple scattering of neutrons from the water phantom produce low energy neutrons (<100 keV) near the source capsule which can interact with 191 Ir and 195 Pt.

Table 3 .
TOPAS-calculated values of Ēfl n and Ēfl ph as a function of R in water.R(cm) Ēfl n (MeV)

Table 5
where only single-track action is considered.At low dose, RBE has the maximum value and is comparable to ̅ Q (Kyriakou et al 2021, Chattaraj et al 2022).The TOPASbased microdosimetric distributions calculated using phase-space as source are used to calculate Q , w ) ICRP60 QICRU40 and RBE TDRA whereas ¯( )

Table 4 .
TOPAS-and FLUKA-calculated ȳ , F ȳD and y * values (in keV/μm) presented for a AT 252 Cf brachytherapy source at 1 μm site size as a function of R in water.at all R. Considering the statistical uncertainties, RBE TDRA shows reasonably good agreement with QICRP60 and Q .
¥ Q L ICRU40 Table 6 presents TOPAS-based values of RBE r , RBE MKM V79 and RBE MKM HSG as a function of R in water.These RBE values are insensitive to R.Although the calculation formalisms of RBE r and RBE MKM V79 are different, the values of RBE r compare well with that of RBE .point are same.Values of RBE r and RBE MKM V79 are about 22% higher than RBE MKM HSG which is due to the fact that the biological end-points are different.

Table 5 .
Wierzbicki 2012the published values of RBE of a 252 Cf source including the biological end-points and reference radiations considered to achieve the RBE value.Depending on the experimental conditions, wide variation is observed in the reported RBE values of a 252 Cf neutron source (see table7).At dose rates less than 0.1 Gy h −1 , RBE is 7-8 (Abd El-Hafez et al 1997,Wierzbicki 2012) which shows reasonably good agreement with the RBE TDRA calculated in the present study, considering the statistical uncertainties in the range of 16%-18%.Note that RBE TDRA is defined at low doses.In the present study, the calculated values of RBE MKM V79 and RBE MKM HSG are 2.8 and 2.3, respectively whereas the RBE r Values of Q and RBE TDRA as a function of R in water phantom.
¥Q L