Monte Carlo calculation of quality correction factors based on air kerma and absorbed dose to water in medium energy x-ray beams

According to the formalism of the IAEA TRS-398 CoP (Andreo et al 2001), the reference dose D_w,Q in z₀ = 2 cm water depth for a radiation field of beam quality Q can be determined from the absorbed dose to water calibration coefficient $N_{D,w,Q_0}$ for a medium energy x-ray reference beam quality Q₀ using equation (1):

$$\begin{eqnarray} &&D_{w,Q} = M_{w,Q} \cdot N_{D,w,Q_0} \cdot k_{Q,Q_{0}} \end{eqnarray} \tag{ 1 }$$

where M_w,Q is the detector signal corrected for the influence quantities at the reference water depth z₀ in a x-ray beam of quality Q and $k_{Q,Q_{0}}$ is the detector specific beam quality correction factor, accounting for the different response of the ion chamber at the beam quality Q relative to the reference beam quality Q₀. By definition, $k_{Q,Q_{0}}$ corresponds to the ratio of the calibration coefficients N_D,w for both beam qualities Q and Q₀:

$$\begin{eqnarray} &&k_{Q,Q_0} \equiv \frac{N_{D,w,Q}}{{N_{D,w,Q_0}}} = \frac{\displaystyle ^{D_{w,Q}}/_{M_{w,Q}}} {\displaystyle ^{D_{w,Q_0}}/_{M_{w,Q_0}}} \approx \frac{\displaystyle ^{D_{w,Q}}/_{D_{det,w,Q}}} {\displaystyle ^{D_{w,Q_0}}/_{D_{{det},w,Q_0}}} \end{eqnarray} \tag{ 2 }$$

Equation (2) includes the assumption, that the detector signal M may be replaced by the detector dose D, i.e. $M_{w,Q}/M_{w,Q_0} \approx D_{w,Q}/D_{w,Q_0}$. This assumption is well justified if the mean energy W_air required to generate an ion pair in air is independent from beam quality Q. According to the ICRU Report 90 (Seltzer et al 2016), the relationship between the dose to the air cavity of an ion chamber D and the chamber reading M is given by

$$\begin{equation} D = \frac{M}{m_{\textrm{air}}} \left(\frac{W_{\textrm{air}}}{e}\right) k_w k_{ii} \end{equation} \tag{ 3 }$$

where the constant $W_{\textrm{air}} =$ 33.97 eV is the average energy to create an ion pair in dry air due to electrons and $m_{\textrm{air}}$ is the mass of the dry air in the cavity. The correction factor k_w accounts for deviations of $W_{\textrm{air}}$ from the value of 33.97 eV and the correction factor k_ii corrects for the inclusion of initially ionized electrons in the cavity. According to ICRU Report 90 both correction factors are unity for primary photon energies above 100 keV, but the product $k_w k_{ii}$ may have an impact for photon energies below 100 keV. Therefore, in preliminary investigations the impact of k_w on the beam quality correction factor $k_{Q,Q_0}$ was determined. Based on the k_w data for monoenergetic photons given in ICRU 90, spectrum averaged k_w,Q data were calculated for the photon fluence spectra applied in this study (see table 1). As to be expected, the correction factor k_w was largest for the CCRI-100 spectrum, but the deviation from unity was only 2 per mille, the effect on $k_{Q,Q_0}$, i.e. the ratio $k_{w,Q}/k_{w,Q_0}$ was even smaller (about 1 per mille). Moreover, according to equation (3) one has to consider the product of the two corrections $k_w \cdot k_{ii}$. As both corrections go in opposite directions ($k_w \gt 1, k_{ii} \lt1$) their product is closer to unity than their individual components, i.e. in any case the impact of the correction $k_w \cdot k_{ii}$ on the beam quality correction factor $k_{Q,Q_0}$ is well below one per mille and the assumption that the chamber reading M may be replaced by the dose value D is well justified even for the x-ray energies investigated here.

Table 1. Parameters of used x-ray spectra. The spectra were calculated with the software SpekCalc (Poludniowski 2007). In all cases the anode angle θ was 30° and a 50 cm air gap behind the anode was included.

	$E_{\textrm{min}}$	$E_{\textrm{max}}$	Filtration in mm			HVL
Spectrum	in keV	in keV	Al	Cu	Sn	in mm Cu
CCRI-100	10.0	100	3.82	—	—	0.1461
CCRI-135	13.5	135	2.225	0.23	—	0.4708
CCRI-180	18.0	180	3.16	0.49	—	0.9863
CCRI-250	25.0	250	3.26	1.68	—	2.515
H-200	20.0	200	4.26	1.11	—	1.648
H-250	25.0	250	4.18	1.63	—	2.504
W-150	15.0	150	4.16	—	1.0	1.869
W-200	20.0	200	4.19	—	2.0	3.125
X-1	11.0	110	2.225	—	—	0.1062
X-2	11.0	110	2.225	0.0575	—	0.1809
X-3	11.0	110	2.225	0.1150	—	0.2485
X-4	11.0	110	2.225	0.1725	—	0.3075
X-5	11.0	110	2.225	0.23	—	0.3587
X-6	12.0	120	2.225	0.23	—	0.4030

According to the TRS-398 CoP (Andreo et al 2001) the connection between air-kerma based and dose to water based formalisms for medium-energy x-rays is given as

$$\begin{eqnarray} &&\frac{N_{D,w,Q}} {N_{K,a,Q}} = \left[\left(\frac{\bar{\mu}_{en}}{\rho} \right)_{w,a}\right]^{z = 2 \text{cm}}_{Q} ~ p_{Q} \end{eqnarray} \tag{ 4 }$$

where N_K,a,Q is the air kerma calibration coefficient free in air for the beam quality Q and $\left(\bar{\mu}_{en}/ \rho \right)_{w,a}$ is the ratio, water to air, of the mass-energy absorption coefficients, averaged for each medium over the photon energy fluence spectrum of radiation quality Q at the reference water depth z. The perturbation factor takes into account the effects on the chamber response of the difference in spectra at the chamber position between the air kerma and dose to water formalism. Furthermore, the perturbation factor accounts for the replacement of water by air and the impact of the ionization chamber material. In order to calculate the perturbation factor p_Q using Monte Carlo simulations, the ratio of the calibration coefficients N_D,w,Q and N_K,a,Q must be calculated according to the following equation:

$$\begin{eqnarray} &&\frac{N_{D,w,Q}} {N_{K,a,Q}} = \frac{ \displaystyle ^{D_{w,Q}}/_{M_{w,Q}} } {\displaystyle ^{K_{a,Q}}/_{M_{a,Q} }} = \frac{ \displaystyle ^{D_{w,Q}}/_{D_{det,w,Q}} } {\displaystyle ^{K_{a,Q}}/_{D_{det,a,Q} }} \end{eqnarray} \tag{ 5 }$$

where K_a,Q is the air kerma free in air, M_a,Q the signal of the detector positioned free in air for the beam quality Q and D_det,a,Q the absorbed dose in the sensitive volume of the detector free in air. Equation (5) is also based on the assumption, that the chamber readings M_w,Q and M_a,Q can be replaced by the dose values D_det,w,Q and D_det,a,Q . In a recent study Bancheri et al (2020) have shown, that the ratios of the correction factors $\left(k_w k_{ii}\right)_{w,Q}/\left(k_w k_{ii}\right)_{a,Q}$ are indeed unity for spectra in the same energy range as investigated here, i.e. the impact of the correction $k_w \cdot k_{ii}$ on the perturbation correction p_Q is also negligible.

In summary, using equations (4) and (5), the perturbation correction p_Q can be calculated by Monte Carlo simulations as follows:

$$\begin{eqnarray} &&p_Q = \left(\frac{D_{w,Q}}{K_{a,Q}}\right)\left(\frac{D_{det,a,Q}}{D_{det,w,Q}}\right)/\left[\left(\frac{\bar{\mu}_{en}}{\rho} \right)_{w,a}\right]^{z = 2 \text{cm}}_{Q}. \end{eqnarray} \tag{ 6 }$$

The radiation fields of the medium energy photon beams were generated by an isotropic x-ray source emitting a limited cone beam, resulting in a circular field size of 10.5 cm in diameter at a distance of 100 cm. The spectral fluence distributions Φ_E of the used beams were calculated with the software SpekCalc (Poludniowski 2007), the applied parameters are summarized in table 1. The CCRI spectra (CCEMRI 1972) were adapted by setting a suitable value for the fixed filtration thickness, so that the resulting HVL matches those of the radiation sources used by ENEA-INMRI for calibration purposes. The characteristics of the H- and W-qualities are defined according to ISO 4037-1 (1996). The spectra named X-1 to X-6 are no reference beam qualities as described by ISO or CCEMRI, they were included to fill the gap between the HVL of the CCRI-100 and CCRI-135 spectrum.

To determine the beam quality correction factors $k_{Q,Q_0}$ and the perturbation factors p_Q , the following dosimetric quantities were calculated: (a) the absorbed dose to water D_w , (b) the absorbed dose to the detector's cavity in water D_det,w,Q and free in air D_det,a,Q , (c) the air kerma free in air K_a,Q , and (d) the ratio of the mean mass energy absorption coefficients for the media water and air $\left[\bar{\mu}_{en}/\rho\right]_{w,a}^{2~\mathrm{cm}}$ at the depth of measurement z₀ = 2 cm in water for quality Q.

The simulations for all ion chambers were performed under reference conditions according to the IAEA TRS-398 CoP, i.e. the source-detector distance was 100 cm. For the simulations in water, a water phantom sized 20 × 20 × 20 cm³ surrounded by air was modeled. As the x-ray spectra were calculated with a 50 cm air gap behind the anode (see table 1), the air volume in beam direction did start at 50 cm from the x-ray source (z = 0 cm), the surface of the water phantom was positioned at z  = 98 cm. For the simulations free in air, the water was simply replaced by air. In all cases the field size at the detector was 10.5 cm in diameter, as also used for the measurements at ENEA-INMRI. As reference beam quality Q₀ the spectrum CCRI-250 was chosen, because it is the beam quality at which primary standards of D_w realize their measurements with a relatively low uncertainty in comparison to other qualities. This holds for both water and graphite calorimeters.

Two different Monte Carlo code systems were used by three institutions independent from each other. The EGSnrc code system (Kawrakow et al 2019a) was used by ENEA-INMRI and THM. The PENELOPE code system (Salvat 2015) was used by IST. All presented dosimetric quantities were calculated by Monte Carlo simulations applying the renormalized photoelectric cross sections and recommendations for the materials water and graphite given in the ICRU Report 90 (Seltzer et al 2016).

For the PENELOPE calculations the main program penEasy v.2015-05-13 (compatible with the PENELOPE 2014 code package) (Salvat 2015, Sempau et al 2011) was applied. The tallyEnergyDeposition counter was used to calculate the energy deposited in the chamber cavity and in water. To improve the simulation efficiency simple particle splitting was implemented for all particles (photons and electrons) entering the water volume present up to 2 cm around the chamber with a factor of 20. The transport parameters used in the PENELOPE simulations are summarized in table 2.

For the EGSnrc simulations the code version EGSnrc 2018 was applied with the transport parameters summarized in table 3. The quantities D_w,Q , D_det,w,Q and D_det,a,Q were determined with the user code egs_chamber (Wulff et al 2008). The new EGSnrc application egs_kerma (Kawrakow et al 2019b) has been used to calculate kerma free in air at 100 cm source to detector distance in a cylindrical air voxel with 1 cm radius and 0.2 cm (THM) or 0.1 cm (ENEA-INMRI) height, placed symmetrically around the point of measurement. The variance reduction techniques implemented in the user code egs_chamber were used to improve the efficiency of the Monte Carlo simulations (Wulff et al 2008). The following variance reduction techniques have been used both by ENEA-INMRI and THM: intermediate phase space storage (IPSS); photon cross-section enhancement (XCSE) volume with a XCSE factor of 64 and the Russian Roulette range rejection technique with a survival probability of 1/64. The parameter ESAVE was set to 0.512 MeV.

For both Monte Carlo codes the absorbed dose to water D_w in 2 cm water depth was calculated in a small cylindrical voxel with 1 cm radius and 0.025 cm height, placed symmetrically around the point of measurement. To calculate the detector dose D_det , detailed models of the ionization chambers were created based on blueprints made available by the manufacturers. Monte Carlo based models of the Farmer-type chamber NE 2571 were developed independently for both Monte Carlo codes. As the chamber is not waterproof, a sleeve made of 1 mm thick PMMA was included in the simulations. The other chambers were only build for the code EGSnrc. The Farmer-type chamber PTW 30 013 was modeled independently by two partners (THM, ENEA-INMRI). Regarding the chamber FC65-G from vendor IBA, two different models were build: the type FC65-G, which is currently offered by IBA and the type FC65-GX, which is a further development of the FC65-G chamber type. Both models differ in the diameter of the central electrode, in case of the FC65-GX the diameter is about 10% larger. Moreover, the ion chambers PTW 31 010 and PTW 31 013 were included in this study. The models developed by the different partners showed some minor geometrical differences in small details like the guard ring and the stem. Exemplary cross sections of the chambers are given in figure 1.

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The mean water-to-air mass energy absorption coefficient ratios $\left({\bar{\mu}_{en}}/{\rho} \right)_{w,a}$ for all investigated beam qualities were calculated with EGSnrc, using two different approaches. One approach is to calculate the spectral photon fluence Φ_E at 2 cm water depth using the user code FLURZnrc and the monoenergetic mass energy absorption coefficients µ_en (E)/ρ for water and air with the user code g. The mean water-to-air mass energy absorption coefficient ratio $\left({\bar{\mu}_{en}}/{\rho} \right)_{w,a}$ can then be calculated as

$$\begin{eqnarray} &&\left(\frac{\bar{\mu}_{en}}{\rho} \right)_{w,a} = \frac{\displaystyle \sum^{N}_{i = 1} E_i \Phi_{E_i} \left( \frac{{{\mu}_{en}}(E_i)}{{\rho} } \right)_{w} \Delta E}{\displaystyle \sum^{N}_{i = 1} E_i \Phi_{E_i} \left( \frac{{{\mu}_{en}}(E_i)}{{\rho} } \right)_{a} \Delta E} \end{eqnarray} \tag{ 7 }$$

where E_i and ΔE are the photon energy and the energy bin width of the calculated spectrum, respectively.

On the other hand, the user code egs_kerma can be applied to calculate $\left({\bar{\mu}_{en}}/{\rho} \right)_{w,a}$ values in 2 cm water depth. Initially the user code was designed to calculate the medium kerma K_m according to the following equation:

$$\begin{eqnarray} &&K_m = \displaystyle \sum \limits_{i = 1}^{n} \frac{l_i}{V} E_i \left( \frac{{{\mu}_{en}}(E_i)}{{\rho} } \right)_{m} \end{eqnarray} \tag{ 8 }$$

where l_i denotes the particle track length through the volume of interest V, E_i the energy of the ith photon and n the number of photons. The user must provide data containing $E_i \left({{\mu}_{en}}(E_i)/{\rho} \right)_{w}$ values for the given medium m on a logarithmic energy scale (Kawrakow et al 2019b). The individual contributions l_i of photons crossing the volume of interest V are calculated by Monte Carlo simulations of the particle transport in the simulation geometry.

However, in this study egs_kerma was used to calculate the mean water-to-air mass energy absorption coefficient ratios $\left({\bar{\mu}_{en}}/{\rho} \right)_{w,a}$ in the volume of interest V. For that reason, kerma was calculated twice in the volume of interest: First kerma in water was calculated according to equation (8) by providing $E_i \left({{\mu}_{en}}(E_i)/{\rho} \right)_{w}$ values for water, subsequently the calculations were repeated, whereby $E_i \left({{\mu}_{en}}(E_i)/{\rho} \right)_{a}$ values for air were provided to the simulation. The ratio of both calculated kerma values results in the mean water-to-air mass energy absorption coefficient ratio $\left({\bar{\mu}_{en}}/{\rho} \right)_{w,a}$ according to the following equation:

$$\begin{eqnarray} &&\left( \frac{\bar{\mu}_{en}} {\rho} \right)_{w,a} = \frac{\displaystyle \sum \limits_{i = 1}^{n} \frac{l_i}{V} E_i \left( \frac{{{\mu}_{en}}(E_i)}{{\rho} } \right)_{w}}{\displaystyle \sum \limits_{i = 1}^{n} \frac{l_i}{V} E_i \left( \frac{{{\mu}_{en}}(E_i)}{{\rho} } \right)_{a}}. \end{eqnarray} \tag{ 9 }$$

As in the first approach, the monoenergetic mass energy absorption coefficients for water and air were calculated with the user code g.