On the Monte Carlo Calculation of the Air-Kerma Strength for Brachytherapy Sources

1. Introduction

TG-43 protocol for the dosimetric characterization of brachytherapy sources (Nath et al. 1995) and its update TG-43U1 (Rivard et al. 2004) define the air-kerma strength as:

$$S_K={\dot{K}}_{\delta }^{\mathrm{ref}}·d^2,$$

(1)

where ${\dot{K}}_{\delta }^{\mathrm{ref}}$ represents the air-kerma rate, in vacuum, due to photons with energy greater than $\delta$, at a reference point located in the plane perpendicular to the source axis passing through its geometric center, and at a distance d from this source center. Furthermore, it is recommended that d be $1$m, a distance sufficiently large compared to the dimensions of the source, which allows assuming that the source behaves as a point source. In vacuum here means that, after measuring the air-kerma rate, it must be corrected to account for the effects (photon attenuation and scattering) of the air, or any other medium, present between the source and the detector, as well as any objects (walls, floors, or ceilings) surrounding the source. Also, the energy limit $\delta$ permits ignoring contributions from low-energy photons originating from the source metal coating, which are not relevant to the dose at distances greater than a few millimeters from the source when it is immersed in water or tissue in clinical practice. A value of $\delta =5$keV is typically adopted for sources that emit low-energy photons. For mean and high energies, which are those of interest in the present work, the HEBD-WG report recommended, however, $\delta =10$keV (Pérez-Calatayud et al. 2012a, 2012b).

To avoid many of the difficulties that arise when trying to measure them, air-kerma rates can be calculated based on an appropriate configuration. Assuming that the z-axis coincides with the axis of symmetry of the source along its length, the air-kerma rate in vacuum at a point within air with polar coordinates $(d,\theta )$ is given by (Seltzer et al. 2011):

$${\dot{K}}_{\delta }(d,\theta )={\int }_{\delta }^{E_{\max }}\mathrm{d}E{\displaystyle \frac{{\mu }_{\mathrm{tr}}\left(E\right)}{\rho }}E{\dot{\Phi }}_E(d,\theta ).$$

(2)

Here E labels the photon energy, $E_{\max }$ refers to the maximum energy of the photons emitted, ${\mu }_{\mathrm{tr}}\left(E\right)/\rho$ is the mass energy transfer coefficient for air, and ${\dot{\Phi }}_E(d,\theta )$ is the fluence rate of photons with energy E, exiting the source. It is worth noting that this expression for the air-kerma rate as a function of the fluence rate permits to deal with the kerma rate for a given material at a point within a different material; in our case,

$${\dot{K}}_{\delta }^{\mathrm{ref}}={\int }_{\delta }^{E_{\max }}\mathrm{d}E{\displaystyle \frac{{\mu }_{\mathrm{tr}}\left(E\right)}{\rho }}E{\dot{\Phi }}_E^{\mathrm{ref}},$$

(3)

where ${\dot{\Phi }}_E^{\mathrm{ref}}\equiv {\dot{\Phi }}_E(d=1\mathrm{m},\theta ={90}^{\mathrm{o}})$ is the fluence rate in reference conditions.

An approach that has been extensively used in this context is the Monte Carlo estimation of ${\dot{K}}_{\delta }(d,\theta )$. The procedure assumes that the source is in vacuum, surrounded by a very thin spherical shell, of a few microns, filled with air and with an inner radius of $1$m. In the simulation, the spectrum of the photons arriving to the shell is scored to determine the corresponding photon fluence rate at the point of interest, ${\dot{\Phi }}_E(d,\theta )$. This is the approach used by Almansa et al. (2011, 2017); Borg and Rogers (1999); Guerrero et al. (2014); Taylor and Rogers (2008a, 2008b) and Safigholi et al. (2023). Another way to calculate air-kerma rate used the dose distribution due to the source when it is embedded in air, which is corrected for the air effects. This second approach was considered in the works by Ballester et al. (2009); Daskalov et al. (1998); Granero et al. (2011) and Vijande et al. (2012).

However, and independently of the approach considered, these Monte Carlo estimations included several approximations that were applied to simplify the simulations. The first of them relates to the calculation of the photon fluence rate: a common practice has been to perform the simulations neglecting the electrons emitted by the radioisotopes present in the source (usually $\beta$ emitters). In fact, the only mention to electrons in TG-43U1 refers to secondary electrons produced in the interactions of the photons emitted by the radionuclides and establishes that no electron transport is required.

In the case of medium- and high-energy HDR brachytherapy sources, in particular those containing ¹⁹²Ir and ⁶⁰Co radioisotopes, the electrons emitted in the decay process may have energies of up to $1.45$MeV and $2.82$MeV, respectively. It has been argued (Ballester et al. 2009; Baltas et al. 2001; Wang and Li 2002) that the electrons emitted in the beta decay of these sources, as well as those coming from internal conversion and Auger emissions, are absorbed practically within the source core and encapsulation, eventually contributing to the total dose in water below 1% at distances very close to the source. On the other hand, Safigholi et al. (2023) mention that the transport of the electrons in the sources can be neglected because air-kerma measurements are carried out with ionization chambers that respond only to photon fluences. However, TG-43U1 (Rivard et al. 2004) highlights the need to account for the characteristic x-rays that may be produced following photoelectric absorption in atoms with medium or high atomic number, particularly in sources containing Ag or Pd. Despite this, no quantitative evaluation of the overall electron effects on $S_K$ can be found in the literature.

Another commonly used approximation is that, when the fraction of energy lost by charged particles through radiative processes is negligible, the mass energy transfer coefficient appearing in equation (2) can be approximated by the mass energy absorption coefficient, which is tabulated in various sources (see e. g., the NIST database (Hubbell and Seltzer 2004).

In the present work, the Monte Carlo estimation of $S_K$ is studied in detail, investigating the influence of including the beta emission spectra of the sources, as well as the electron transport. In particular the effects of the transport parameters used are analyzed. Calculations have been carried out using the Monte Carlo code penelope-2024 (Salvat 2025) for two HDR sources: the ¹⁹²Ir microSelectron mHDR-v2r and the ⁶⁰Co Flexisource Co-60 HDR sources. An aspect that has also been studied is the changes produced in $S_K$ when it is calculated, as indicated in equation (2), using the mass energy transfer coefficients, instead of the mass energy absorption coefficients usually considered. In connection with this point, the influence of the energy bin size used to score the photon fluence rates has been also analyzed. Finally, in case of the Flexisource Co-60 HDR source, and when the beta emission spectrum of ⁶⁰Co is not included, the impact of using a complete photon emission spectrum, as recommended in Pérez-Calatayud et al. (2012a, 2012b), instead of a minimum photon emission spectrum containing only the two most probable photon emissions, has been also studied. Also the influence of the air composition has been analyzed because HEBD-WG (Pérez-Calatayud et al. 2012b) recommended dry air whilst moist air had been recommended in TG-43 (Nath et al. 1995) and TG-43U1 (Rivard et al. 2004).

2. Materials and Methods

2.1. HDR Sources

In this work, the air-kerma strength, $S_K$ has been calculated for two HDR sources, whose schemes are drawn in Figure 1.

The first one is the Flexisource Co-60 HDR (Elekta, Stockholm). The radioisotope is ⁶⁰Co; it is distributed uniformly within a cylinder made of pure Co, with a diameter of $0.5$mm and a height of $3.5$mm. The simulation geometry shown in Figure 1 is the same used in previous works (Almansa et al. 2017; Gebremariam et al. 2023; Vijande et al. 2012). In the following, this source is referred to as “Flexisource”.

The second source is the microSelectron mHDR-v2r source (Elekta, Stockholm), which incorporates ¹⁹²Ir as radioisotope embedded uniformly in an Ir cylindric core with a diameter of $0.6$mm and a height of $3.5$mm. In Figure 1 the complete source geometry used in the simulations is sketched. This geometry is the same as that used by other authors (Almansa et al. 2011; Granero et al. 2011; Safigholi et al. 2023). Hereinafter, this source is denoted as “microSelectron”.

2.2. Calculation of $S_K$

The calculation of $S_K$ was carried out using equation (1) and to evaluate the air-kerma rate the integral in equation (2) was discretized as follows:

$${\dot{K}}_{\delta }(d,\theta )=b_E{\displaystyle \sum _{i=1}^N}{\displaystyle \frac{{\mu }_{\mathrm{tr}}\left(E_i\right)}{\rho }}{E}_i{\dot{\Phi }}_{E_i}(d,\theta )\Theta (E_i-\delta ).$$

(4)

Here $b_E=E_{\max }/N$ is the size of each of the N energy bins in which the photon energy interval has been divided, and $E_i$ is the central energy of the i-th bin. Furthermore, $\Theta \left(x\right)$ is the Heaviside function, which takes the value 0 or 1 when $x<0$ or $x\ge 0$, respectively.

In the simulations carried out to estimate the photon fluence rates ${\dot{\Phi }}_{E_i}(d,\theta )$, the source was in vacuum in the center of a spherical $10\mu$m thick shell, with an inner radius of $1$m, filled with air and acting as a photon detector spectrometer. The calculations have been done using the Monte Carlo code penelope-2024 (Salvat 2025). This code allows for the simulation of the interaction of electrons, positrons, and photons with matter for energies ranging from $50$eV to $1$GeV. Photons are simulated in detail: photon interactions are simulated one by one. Electrons and positrons are simulated within a mixed scheme in which collisions can be hard or soft, depending on whether the polar deviation angles and/or energy losses are above or below certain specific threshold values set by the user. While hard events are simulated in detail, all soft interactions occurring between any two consecutive hard events are simulated using a multiple scattering approximation, as a single virtual event called a “hinge”.

Electron and positron simulation is driven by several user defined transport parameters. These parameters are the following: $C_1$ and $C_2$, which refer, respectively, to the average angular deflection and the maximum average fractional energy loss due to all soft interactions between two consecutive hard elastic events; $W_{\mathrm{CC}}$ and $W_{\mathrm{CR}}$, which fix, respectively, the cut-off energies for hard inelastic collisions and hard bremsstrahlung emission; the absorption energies, $E_{\mathrm{abs}}$, for photons, electrons and positrons, which establish the energy thresholds below which the simulation of the corresponding particle is stopped and its remaining energy is deposited locally, and finally, $s_{\max }$, which defines the maximum step between two successive interactions and is strongly relevant when the simulation geometry includes very thin elements (as it occurs in the case of the sources simulated in the present work). Users must choose the values of these parameters for each material present in the simulation geometry.

It is worth pointing out that if $C_1=C_2=W_{\mathrm{CC}}=0$ and $W_{\mathrm{CR}}<0$ are chosen, the simulation of electrons becomes fully detailed. The use of the transport parameters, as well as their optimal values, are described in the penelope user manual (Salvat 2025).

In this work, penmain was the main program employed to control the simulation process.

The air-kerma strength values obtained according to equation (1) with ${\dot{K}}_{\delta }$ given by equation (2) were labelled as $S_K^{\mathrm{tr}}$. If the mass energy absorption coefficient ${\mu }_{\mathrm{en}}/\rho$ is considered instead the mass energy transfer coefficient ${\mu }_{\mathrm{tr}}/\rho$, an approach assumed in many previous studies (Almansa et al. 2011, 2017; Borg and Rogers 1999; Guerrero et al. 2014; Taylor and Rogers 2008a, 2008b), the obtained air-kerma strengths were labelled as $S_K^{\mathrm{en}}$.

Two datasets were considered to keep the values of these mass coefficients. The first one was provided by the code MUTREN, included in the penelope distribution. The values obtained for both ${\mu }_{\mathrm{tr}}/\rho$ and ${\mu }_{\mathrm{en}}/\rho$ in this way are fully consistent with the calculations of the fluence rates. Different air compositions were considered to determine the mass coefficients. Table 1 shows these compositions. In the following sections, unless otherwise indicated, the calculations have been performed assuming TG-43U1 dry air.

A second set of data were taken from the NIST database (Hubbell and Seltzer 2004). In this case, a cubic spline interpolation was carried out in order to obtain the values required for the photon energies (according to the bin sizes of the fluence rates).

2.3. Simulations

Depending on the emission spectra considered, three types of simulations were carried out. In the first one, the complete disintegration schemes of ⁶⁰Co and ¹⁹²Ir, as given in the nucleide database (Bé et al. 2011), were used. These emission spectra include all electrons ($\beta$, internal conversion and Auger) and photons (gamma and characteristic x-rays) emitted by the isotopes. These simulations were labelled as full decay scheme.

In a second type of simulations, electrons were neglected. As recommended in the HEBD-WG report (Pérez-Calatayud et al. 2012b), in these simulations all photon emissions, with energies above $\delta$, were considered, and they were labelled as complete photon spectrum simulations. The energies of the photon emissions considered in these simulations are indicated in Table 2.

In case of Flexisource, also a spectrum including only the two main photon emissions (those with energies $1173.23$keV and $1332.49$keV), was considered. The corresponding simulations were labelled as minimum photon spectrum.

The effect of the detail with which the electron transport was performed has been also studied. In this case, those results that were obtained in a fully detailed simulation (see above), were compared to those found in mixed simulations in which $C_1=C_2=0.05$, $W_{\mathrm{CC}}=10$keV and $W_{\mathrm{CR}}=1$keV. These latter values are more restrictive than those recommended for “safe” simulations in the penelope manual (Salvat 2025). The corresponding results have been labelled as “detailed” and “mixed”, respectively.

In what refers to the absorption energies of electrons and positrons, $E_{\mathrm{abs}}\left(e^{\mp }\right)=100$eV was used for all materials in all simulations except in those where the electron transport was not simulated, in which case $E_{\mathrm{abs}}\left(e^{\mp }\right)=10$MeV. For photons, $E_{\mathrm{abs}}\left(\gamma \right)=1$keV, $5$keV and $10$keV were used.

The parameter $s_{\max }$ has been chosen as one tenth of the characteristic thickness of each component of the sources. This means $60\mu$m for the active core, $5\mu$m for the air separating this active core from the capsule, $10\mu$m for the capsule and $90\mu$m for the cable.

Finally, photon spectra at the detection sphere were scored for energies up to $2$MeV. No photons with higher energies have been detected. To analyze the effects linked to the size of the bins considered in equation (4) for the calculation of the integral defining the air-kerma rate, fluence spectra with bin sizes $b_E=10$keV (scored spectra with 200 bins) and $b_E=1$keV (scored spectra with 2000 bins) were calculated.

The number of histories followed was chosen according the bin size of the scored photon spectra. Specifically, ${10}^9$ histories were followed in all the simulations performed with a fluence bin size of $10$keV and ${10}^{10}$ for $1$keV fluence bins. Throughout the text, uncertainties of the results are shown with a coverage factor $k=3$.

The air-kerma strength unit employed in the present work was $1\mathrm{U}{\mathrm{GBq}}^{-1}$, where $1\mathrm{U}=1\mathrm{cGy}{\mathrm{cm}}^{-2}{\mathrm{h}}^{-1}$.

3. Results and Discussion

In this work, reference results were the $S_K^{\mathrm{tr}}$ values obtained in mixed simulations carried out by assuming the full decay scheme of the isotopes, the TG-43U1 dry air composition, an absorption energy for photons $E_{\mathrm{abs}}\left(\gamma \right)=1$keV and a bin size of the fluence rate spectrum $b_E=1$keV. The values obtained are those marked in bold in the various tables.

The comparison of the air-kerma strength value obtained in a given simulation, $S_K$, with the corresponding reference value, $S_K^{\mathrm{ref}}$, was carried out in terms of the absolute difference, $\Delta =|S_K^{\mathrm{ref}}-S_K|$, the relative difference, ${\Delta }_S=\Delta /\left(S_K^{\mathrm{ref}}\right)$, and the ratio ${\Delta }_{\sigma }=\Delta /{\sigma }_{\mathrm{ref}}$, which gives the number of times that $\Delta$ contains the uncertainty of the reference value, ${\sigma }_{\mathrm{ref}}$.

3.1. ${\mu }_{\mathrm{tr}}$ vs. ${\mu }_{\mathrm{en}}$, Role of $b_E$, and Detailed vs. Mixed Simulations

The first point to note concerns the effect of using ${\mu }_{\mathrm{en}}/\rho$ instead of ${\mu }_{\mathrm{tr}}/\rho$. In the case of Flexisource, this change results in a reduction in $S_K$ of $\Delta =1.2\mathrm{U}{\mathrm{GBq}}^{-1}$, that is, a relative reduction of ${\Delta }_S=0.39\%$, while for microSelectron is much smaller, amounting to $\Delta =0.08\mathrm{U}{\mathrm{GBq}}^{-1}$, which means a relative reduction of ${\Delta }_S=0.08\%$. These differences correspond to ${\Delta }_{\sigma }=7.9$, for Flexisource and ${\Delta }_{\sigma }=3.9$, for microSelectron.

The different behavior observed for the two sources can be explained by looking at Figure 2. There, the staggered lines show the accumulated air-kerma rate in reference conditions,

$$\begin{array}{ccc}\hfill {\dot{K}}_0^{\mathrm{ref}}\left(ϵ\right)& =& {\int }_0^{ϵ}\mathrm{d}E{\displaystyle \frac{{\mu }_{\mathrm{tr}}\left(E\right)}{\rho }}E{\dot{\Phi }}_E^{\mathrm{ref}},\hfill \end{array}$$

(5)

for the two sources, while the blue curve corresponds to the relative difference between both mass coefficients,

$${\displaystyle {\kappa }_{\mu }=\frac{{\mu }_{\mathrm{tr}}/\rho -{\mu }_{\mathrm{en}}/\rho }{{\mu }_{\mathrm{tr}}/\rho }.}$$

(6)

As can be seen, the difference between ${\mu }_{\mathrm{en}}/\rho$ and ${\mu }_{\mathrm{tr}}/\rho$ becomes greater as the energy of the photons increases, reaching about $1\%$ for $ϵ=2$MeV. On the other hand, the photon energies that contribute significantly to ${\dot{K}}_0^{\mathrm{ref}}$ in the case of the ⁶⁰Co Flexisource (see red dashed line) are much higher than those for the ¹⁹²Ir microSelectron (green solid line). Consequently, the ⁶⁰Co source is about five times more sensitive to the use of ${\mu }_{\mathrm{en}}/\rho$ instead of ${\mu }_{\mathrm{tr}}/\rho$ and the reduction in $S_K$ produced by this change is more evident in the case of the former source.

In any case, it is important to note that the difference observed when using ${\mu }_{\mathrm{en}}/\rho$ instead of ${\mu }_{\mathrm{tr}}/\rho$ results in a significant reduction of $S_K$ for both sources.

The second aspect to be analyzed is that linked to the fluence bin size, $b_E$. In Table 3 the results of the reference mixed calculation, which correspond to $b_E=1$keV, are compared to those found using $b_E=10$keV (see first two rows). When considering this later bin size, $S_K^{\mathrm{tr}}$ increases by ${\Delta }_S=0.22\%({\Delta }_{\sigma }=4.4)$ for the Flexisource. However, in the case of the microSelectron, it decreases by ${\Delta }_S=0.49\%({\Delta }_{\sigma }=23.6)$.

The variations are, once again, significant for both sources, particularly for the latter where the complex photon spectrum interferes with the $b_E$ value to produce the observed behavior. In fact, the spectrum of the photons reaching the detection sphere in the case of ⁶⁰Co is dominated by two lines at $1173.23$keV and $1332.49$keV, which are separated by more than $150$keV and situated in a region where the product $E{\mu }_{\mathrm{tr}}\left(E\right)/\rho$, entering in equation (2), which defines the air-kerma rate, varies slowly and smoothly. In this situation, substituting the energy of the emission line by the central energy of a bin $10$keV wide, instead of that of a bin $1$keV wide, only introduces a certain uncertainty in the calculation.

The spectrum of ¹⁹²Ir, on the other hand, consists of several dozens of lines spread from $10$keV to about $1.4$MeV, with the bulk of the emission concentrated between $295$keV and $613$keV and an additional, intense, group of X-rays characteristic of Pt and Os, between $61$keV and $78$keV. Below about $100$keV, the product $E{\mu }_{\mathrm{tr}}\left(E\right)/\rho$ for air changes rapidly with energy, so that grouping lines whose ${\mu }_{\mathrm{tr}}/\rho$ values differ appreciably into a single $10$keV interval, and evaluating the coefficient at the center of the interval, results in a much larger uncertainty. As a result of this, the air-kerma strength for Flexisource increases when $b_E=10$keV is used instead of $b_E=1$keV, with ${\Delta }_S=0.22\%$, while for microSelectron a reduction is observed with ${\Delta }_S=0.49\%$. The sign of the effect is opposite for the two sources simply due to the local curvature of $E{\mu }_{\mathrm{tr}}\left(E\right)/\rho$ in the regions where the respective spectra are concentrated.

It is worth noting again that the effect is significant for both sources, though it is much larger in case of microSelectron than for Flexisource. Therefore, it is recommended to use fluence bins with size $b_E=1$ keV, mainly for ¹⁹²Ir sources. Besides, it is mandatory to indicate the bin size employed in the simulations.

The variations in $S_K$ resulting from running detailed simulations (see second data group in Table 3) instead of the mixed reference simulations were also analyzed. The detailed simulations slightly increase $S_K$. The largest variation occurs in case of the Flexisource with $b_E=10$keV, where the $S_K$ value obtained in the detailed simulation is larger than that found in the mixed simulation with ${\Delta }_S=0.10\%$, although the two results agree within $1.7\sigma$. These results indicate that mixed simulations permit a correct calculation of the air-kerma strength.

It is worth pointing out that the same effects observed for the reference results due to the use of ${\mu }_{\mathrm{en}}/\rho$ instead of ${\mu }_{\mathrm{tr}}/\rho$ are seen in the other cases analyzed: mixed simulations with $b_E=10$keV and detailed simulations with the two $b_E$ values considered. Besides, the same effect observed for the mixed simulations when using $b_E=10$keV instead of $b_E=1$keV is also seen for the detailed simulations.

3.2. Electron Effects

The first contribution due to electrons is that linked to the $\beta$ emissions produced in the decay of the ⁶⁰Co and ¹⁹²Ir isotopes. To estimate these contributions the reference results have been compared to those obtained in simulations where the complete photon spectra of both ¹⁹²Ir and ⁶⁰Co isotopes, and also the minimum photon spectrum in case of the latter, are considered. The results are quoted in Table 4.

The consideration of the complete photon spectra of the isotopes, as recommended in the HEBD-WG report (Pérez-Calatayud et al. 2012b), instead of the full decay schemes produces a slight increase of the air-kerma strength in the case of the Flexisource (${\Delta }_S=0.07\%;{\Delta }_{\sigma }=1.3$), while $S_K^{\mathrm{tr}}$ reduces by ${\Delta }_S=0.19\%$ (${\Delta }_{\sigma }=9.3$) for the microSelectron. Again a different behavior is observed between the two sources and the results point out that being the variations produced in the two sources almost the same in absolute value, $\Delta =0.20\mathrm{U}{\mathrm{GBq}}^{-1}$ and $\Delta =0.19\mathrm{U}{\mathrm{GBq}}^{-1}$, the fact that the uncertainties are much smaller for the microSelectron than for the Flexisource makes the difference being significant for the former and not for the latter for which the electron contributions associated to $\beta$ decay are not at all relevant.

In order to rule out any electronic contribution, simulations were carried out in which only the photon spectra were taken into account and, in addition, very high absorption energies $E_{\mathrm{abs}}\left(e^{\mp }\right)=10$MeV were chosen for electrons and positrons. This ensured that any electron (and, where applicable, positron) that might be generated as a secondary particle would be stopped immediately afterwards, ignoring its transport. In case of the complete photon spectra, $S_K^{\mathrm{tr}}$ increases by ${\Delta }_S=0.10\%$ (${\Delta }_{\sigma }=2.0$) for the Flexisource, while reduces by ${\Delta }_S=0.29\%$ (${\Delta }_{\sigma }=14.1$) for the microSelectron. Again, electron effects are more relevant in the case of the microSelectron source, showing up as almost negligible for the Flexisource.

In what refers to simulations carried out with the minimum photon spectrum of ⁶⁰Co for the Flexisource (last data block in Table 4), it can be seen that they produce the same $S_K^{\mathrm{tr}}$ value as the reference simulation when $E_{\mathrm{abs}}\left(e^{\mp }\right)=100$eV is considered. But if electrons are completely stopped by using $E_{\mathrm{abs}}\left(e^{\mp }\right)=10$MeV, ${\Delta }_S=0.03\%$ (${\Delta }_{\sigma }=0.7$), also agreeing with the reference value within the uncertainty. Finally, it is worth noting that the consideration of the minimum photon spectrum instead of the complete one does not produce significant effects, with ${\Delta }_S=0.07\%$ (${\Delta }_{\sigma }=1.4$).

According to these results, it can be stated that electrons may modify $S_K$ through the bremsstrahlung photons and the characteristic X-rays that they can emit while traveling inside the source active core, in the capsule, or in the cable. What has been observed in our calculations is that the consideration of electrons reduces $S_K$ for Flexisource, and increases it, by a bigger amount, in case of microSelectron.

This may seem surprising, since, the magnitude of this indirect contribution must increase with the energy of the electrons and, according (Bé et al. 2011), the value of $Q_{\beta }$ for ⁶⁰Co ($2823.07$ keV) is approximately twice that of ¹⁹²Ir ($1459.7$ keV). However, the excited state of ⁶⁰Ni that is most likely populated following the decay of ⁶⁰Co has a high excitation energy ($2505.75$ keV); consequently, the electrons emitted by ⁶⁰Co in that transition have a maximum energy of $317.32$ keV, with an average energy of $95.6$ keV only. On the other hand, the $\beta$ decay of ¹⁹²Ir primarily populates two excited states of ¹⁹²Pt, with excitation energies of $784.58$keV and $920.92$keV, and the electrons emitted in these transitions have maximum energies of $675.1$keV and $538.8$keV, and average energies of $209.9$keV and $162.1$keV, respectively. Since the energies of the electrons emitted in the $\beta$ decay of ¹⁹²Ir are, on average, higher than those of the electrons emitted from the $\beta$ decay of ⁶⁰Co, the former produce more radiation photons that, therefore, result in a relatively more significant effect.

Neglecting the ${\beta }^{-}$ emission spectrum, along with the transport of photons that may produce, results in a significant variation in the calculation of $S_K$, and this variation is relatively greater in the case of ¹⁹²Ir sources.

3.3. Other Calculations

A common method for determining the values of $S_K$ is to calculate the fluence rates by Monte Carlo and use the ${\mu }_{\mathrm{en}}/\rho$ coefficients provided by NIST, adequately interpolated at the required energies to perform the numerical calculation in equation (4) (Almansa et al. 2011; Borg and Rogers 1999; Taylor and Rogers 2008a, 2008b).

Table 5 compares the reference $S_K^{\mathrm{tr}}$ results (in boldface) with the $S_K^{\mathrm{en}}$ obtained by using the ${\mu }_{\mathrm{en}}/\rho$ coefficients provided by NIST (Hubbell and Seltzer 2004) and assuming either the full decay scheme or the complete photon spectrum for the two sources analyzed. Apart from the values obtained with $b_E=1$keV, also those found with $b_E=10$keV are shown to check whether the interpolation induces some additional effect.

In the case of Flexisource, the use of the NIST mass energy absorption coefficients produces a significant reduction of $S_K$, with ${\Delta }_S=0.18\%$ (${\Delta }_{\sigma }=3.5$). However, the reduction for the microSelectron is almost negligible (${\Delta }_S=0.01\%$; ${\Delta }_{\sigma }=0.6$). It is worth noting that the differences observed when the penelope${\mu }_{\mathrm{en}}/\rho$ coefficients were used (see above) were larger, being significant even in the case of microSelectron. On the other hand, if instead of the full decay scheme, the complete photon spectrum is assumed in the sources, these differences reduce for Flexisource (${\Delta }_S=0.11\%$; ${\Delta }_{\sigma }=2.1$), but they increase significantly for microSelectron, where ${\Delta }_S=0.21\%$ (${\Delta }_{\sigma }=10.3$). Finally, utilizing $b_E=10$keV instead of $b_E=1$keV introduces larger differences between the corresponding $S_K$ values, except in this latter case, in which ${\Delta }_S=0.24\%$ and ${\Delta }_{\sigma }=8.5$ are obtained.

Air composition may also induce some effects on the calculated air-kerma strength. In this respect, it is worth pointing out that in TG-43 (Nath et al. 1995) it was recommended to use 40% moist air, whose composition, together with that of the dry air included in the penelope database, is shown in Table 1. The reference $S_K^{\mathrm{tr}}$ values have been calculated again by using the ${\mu }_{\mathrm{tr}}/\rho$ coefficients determined with the penelope code MUTREN for these two new air compositions. The results obtained are shown in Table 6.

Moist air increases $S_K^{\mathrm{tr}}$ by ${\Delta }_S=0.16\%$ (${\Delta }_{\sigma }=3.2$), in the case of Flexisource, and by ${\Delta }_S=0.09\%$ (${\Delta }_{\sigma }=4.3$), in the case of microSelectron, and the corresponding results do not agree within $3\sigma$ uncertainty. If the penelope dry air is considered, the $S_K^{\mathrm{tr}}$ obtained agree with the reference value, the relative differences being ${\Delta }_S=0.13\%$ (${\Delta }_{\sigma }=2.5$) and ${\Delta }_S=0.05\%$ (${\Delta }_{\sigma }=2.2$) for Flexisource and microSelectron, respectively.

The effects due to the consideration of the larger fluence bin size in the case of these two new air compositions are similar to those observed for the reference TG-43U1 dry air composition, though it is again worth noting the significant reduction in the air-kerma strength that this bin size produces in the case of the microSelectron source, with ${\Delta }_{\sigma }>20.0$.

The role of the air composition considered deserves also a comment. The original TG-43 protocol (Nath et al. 1995) recommended to consider moist air with a 40% degree of humidity. This recommendation was initially maintained in the protocol revision TG-43U1 (Rivard et al. 2004), but it was changed for dry air in the report HEBD-WG (Pérez-Calatayud et al. 2012a, 2012b). The difference in the value of $S_K$ between both air compositions is significant, with ${\Delta }_{\sigma }>3$ in both sources.

Another parameter playing a role in this type of calculations is $\delta$, the threshold used as a lower limit in the calculation of the air-kerma rate defined in equation (2). Table 7 shows the comparison between the air-kerma strengths calculated with $\delta =10$keV (reference values), and those found for $\delta =5$keV and $\delta =0$. In what refers to this parameter, Flexisource and microSelectron show, again, a very different behavior. In the former source, the three calculations agree within $1\sigma$, with differences ${\Delta }_S\le 0.03\%$.

In case of microSelectron, going from $\delta =10$keV to $\delta =5$keV increases $S_K$ by $0.10\%$, while reducing even more the threshold down to $\delta =0$ adds $0.01\%$ only, and the $S_K^{\mathrm{tr}}$ obtained with these latter two values agree within $1\sigma$. In other words, almost the entire effect comes from the $5-10$keV range, which contains precisely the characteristic X-rays of the K-shell of the elements that constitute the AISI encapsulation of the sources ($5.4$keV for Cr, $6.4$keV for Fe and $7.5$keV for Ni). These X-rays are produced in much greater abundance by the ¹⁹²Ir source whose spectrum induces a photoelectric component larger than that of ⁶⁰Co source, in which Compton scattering dominates.

To finish, Figure 3 shows the dependence of the air-kerma rate with the angle $\theta$. The results of the Flexisource (red solid circles) and the microSelectron (green open squares) are shown. To permit the comparison between both sources, the values have been normalized as follows:

$${\displaystyle R\left(\theta \right)=\frac{{\dot{K}}_{\delta }(d=1\mathrm{m},\theta )}{{\dot{K}}_{\delta }(d=1\mathrm{m},\theta ={90}^{\mathrm{o}})}.}$$

(7)

It is evident that microSelectron shows a much larger angular dependence than the Flexisource. This behavior had been already pointed out in previous works for the ⁶⁰Co BEBIG Co0.A86 source (Guerrero et al. 2014), for the Flexisource itself (Almansa et al. 2017; Vijande et al. 2012), and also in the comparison between HDR sources of ⁶⁰Co and ¹⁹²Ir carried out by Gebremariam et al. (2023).

3.4. Comparison with Previous Calculations

To finish it is worth comparing the air-kerma strength obtained in this work with those from previous studies on the same sources analyzed here and on other sources similar to them.

For Flexisource, Almansa et al. (2017) obtained $S_K=305.1\left(3\right)$ U GBq⁻¹ in a calculation done with penelope-2014, using the complete photon spectrum of ⁶⁰Co, without electron transport, with fluence bins of $1$keV, an absorption energy for photons of $10$keV and the ${\mu }_{\mathrm{tr}}/\rho$ coefficients obtained with MUTREN. The value obtained in the present work in the same conditions produced a value of $305.2\left(4\right)$U GBq⁻¹ (see Table 4, second data block, $E_{\mathrm{abs}}\left(e^{\mp }\right)=10$MeV). Both values agree within the uncertainties, differing by $0.03$% only.

Furthermore, by using ${\mu }_{\mathrm{en}}/\rho$ instead of ${\mu }_{\mathrm{tr}}/\rho$, Almansa et al. (2017) observed a $0.39$% reduction in air-kerma strength, the same decrease that we have obtained in our calculations (see Table 3), even though both studies used two versions of the code published more than a decade apart.

Another ⁶⁰Co source different to Flexisource is the BEBIG Co0.A86 source (Eckert & Ziegler BEBIG GmbH, Germany). For this source, Guerrero et al. (2014), in a simulation similar to that described just above, quoted $S_K=304.6\left(7\right)$ U GBq⁻¹, which is again compatible with the values obtained for Flexisource. The main reason for this is that $S_K$ is essentially determined by the isotope emission spectrum, while it is slightly sensitive to the encapsulation details that, in any case, are rather similar in both sources.

It is possible to find in the literature several studies on the microSelectron. For a model with a geometry slightly different from that studied in this work, Borg and Rogers (1999) reported a value of $S_K=97.3\left(1\right)$ U GBq⁻¹ using EGS4, and Berenguer Serrano et al. (2006) obtained a value of $S_K=98.2\left(3\right)$ U GBq⁻¹ using Geant4. More recently, Almansa et al. (2011) obtained $S_K=98.6\left(2\right)$ U GBq⁻¹. In the latter case, the simulation was performed using penelope-2008 and considering the same geometry shown in Figure 1. In our case, $S_K=98.95\left(6\right)$ U GBq⁻¹, which represents a difference of $0.35\%$ compared to the result by Almansa et al., although both agree within a margin of $1.7\sigma$. Tsuji et al. (2024) obtained $S_K=98.29\left(20\right)$ U GBq⁻¹ in simulations performed with the EGS5 Monte Carlo code, a value from which our result differs significantly.

4. Conclusions

In this work, a systematic analysis of the various quantities contributing to the air-kerma strength defined in the TG-43 protocol has been carried out by using the Monte Carlo code penelope. Simulations have been performed for two high-energy HDR brachytherapy sources: Flexisource and microSelectron. The results have permitted to analyze the effects of using ${\mu }_{\mathrm{en}}/\rho$ instead of ${\mu }_{\mathrm{tr}}/\rho$, the modifications introduced by the bin size of the energy fluences used to calculate the air-kerma rate, the changes produced by carrying out detailed simulations instead of the mixed ones (that have been considered as a reference in the present study), the effects linked to the electrons emitted in the ${\beta }^{-}$ decays of the isotopes of the sources and their transport, the changes due to air composition and the use of either dry or moist air, the role played by the energy threshold used to calculate the air-kerma rate, and the angular dependence of this latter magnitude.

To the best of our knowledge, this is the most exhaustive analysis concerning the calculation of the TG-43 air-kerma strength that has been carried out to date. And perhaps the most striking feature of the results obtained is that the two sources respond very differently to the various calculation scenarios analyzed.

The overall contribution of electrons to $S_K$ is small, but not negligible. Completely eliminating electron transport changes $S_K$ by $0.10\%$ (${\Delta }_{\sigma }=2.0$) for Flexisource, and by $0.29\%$ (${\Delta }_{\sigma }=14.1$) for microSelectron. The effect is greater for the ¹⁹²Ir source, even though $Q_{\beta }$ is approximately twice as large for ⁶⁰Co. The reason is that the dominant ${\beta }^{-}$ branch of ⁶⁰Co populates a high-energy excited state of ⁶⁰Ni and delivers electrons with an average energy of only $95.6$keV, whereas the electrons emitted in the $\beta$ decay of ¹⁹²Ir have average energies of $162.1$keV and $209.9$keV.

Mixed and fully detailed electron transport models provide kerma strengths in air that agree within $1.7\sigma$. The mixed simulations, which are much less computationally expensive, are therefore suitable for this type of calculation.

Replacing the mass energy transfer coefficient with the mass energy absorption coefficient when calculating the kerma rate in air reduces $S_K$ by $0.39\%$ for Flexisource, and by $0.08\%$ for microSelectron, the modifications being significant in both cases. This approximation is therefore not recommended and ${\mu }_{\mathrm{tr}}/\rho$ should be used in these calculations.

The size of the energy bins used to score photon fluences is the most influential factor among those analyzed in the case of microSelectron, in which changing $b_E=1$keV to $b_E=10$keV causes $S_K$ to decrease by $0.49\%$. For Flexisource, this change leads to a $0.22\%$ increase in $S_K$. Intervals of $1$keV or less are recommended, especially for sources with complex photon spectra.

Using the 40% moist air recommended in the original TG-43 or the dry air recommended in the HEBD-WG report results in differences in the value of $S_K$ of $0.16\%$ and $0.09\%$ for Flexisource and microSelectron, respectively, both disagreements being larger than $3\sigma$.

The photon energy cutoff threshold $\delta$ is irrelevant for the ⁶⁰Co source, but it modifies $S_K$ by $0.10\%$ for the ¹⁹²Ir source.

None of the contributions analyzed, when considered individually, alters the reference $S_K^{\mathrm{tr}}$ by more than $0.5\%$, but they all are systematic rather than random, do not have the same sign for the two sources, and their combined effect could be on the order of $1\%$. Since $S_K$ normalizes the entire TG-43U1 dataset, a systematic bias in $S_K$ is directly propagated to the dose rate constant, $\Lambda =\dot{D}(r_0,{\theta }_0)/S_K$, and, therefore, to the absolute dose administered to the patient.

This makes it advisable that any Monte Carlo estimation of the air-kerma strength be accompanied by a careful choice of the value of $\delta$, the fluence bin size, $b_E$, the air composition, a consistent set of mass coefficients, and whether ${\mu }_{\mathrm{tr}}/\rho$ or ${\mu }_{\mathrm{en}}/\rho$ is used, without neglecting to include the emitted electrons and, of course, their transport. In addition, it is recommended that these values be explicitly stated along with any Monte Carlo estimates of the air-kerma strength, so that meaningful comparisons can be made between the various calculations.

Finally, it should be noted that, although only two high-energy HDR sources have been studied in the present work, in the case of low-energy sources, such as those based on ¹²⁵I or ¹⁰³Pd, for which $\delta =5$ keV is typically chosen and whose fluence is concentrated in the energy region where ${\mu }_{\mathrm{tr}}/\rho$ varies most rapidly, the sensitivity with respect to $b_E$, $\delta$, and the chosen set of mass coefficients is expected to be as large as that observed here or even higher.