Depth-dose measurement corrections for the surface electronic brachytherapy beams of an Esteya^® unit: a Monte Carlo study

The Esteya eBT unit (see figure 1) accelerates electrons at 69.5 keV impinging on a tungsten target. This system includes a set of conical applicators, with diameters between 10 mm and 30 mm, closed by a polyfenilsulfone plastic cap (applicator exit), which allows a minimum source-to-surface distance (SSD) of 60 mm (Garcia-Martinez et al 2014, Candela-Juan et al 2015b, Valdes-Cortez et al 2019). The applicators considered in this work were the smallest, 10 mm (APP$_{\textrm 10 {mm}}$), and the largest, 30 mm (APP$_{30{\mathrm{mm}}}$), so that the full range of field sizes could be covered. The kV photon beam spectrum has an average energy of 36.3 keV (Valdes-Cortez et al 2020).

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The parallel-plate ionization chamber type studied in this work is the 'soft x-ray' PTW T34013. The specifications and blueprints of the chamber have been provided by the manufacturer.

The chamber (see figure 1) has a nominal sensitive volume of 0.005 cm³ (1.45 mm radius and 0.9 mm height), surrounded by a guard ring at the same potential as the electrode. Reported manufacturing tolerances refer solely to the outer part of the chamber, i.e. distances between two external support structures and the external face of the entrance window. The chamber is not waterproof, being designed for use embedded in a solid phantom. The chamber is calibrated by PTW (traceable to PTB) in a TW30 beam in a 'plastic water' phantom (PW LR: CIRS, Norfolk, VA, USA). According to the manufacturer, the chamber EPoM is situated at the center of the inner surface of the entrance window (EPoM_man, top of the sensitive volume). The IAEA TRS-398 Code of Practice (Andreo et al 2000), however, specifies the reference point to be at the outside surface of the front window. It should be noted that there is a small air gap of about 0.15 mm between the entrance window of the chamber and the plastic cap of the applicator, being in contact with the top of the chamber body (see figures 1 and 2, the step from the top surface of the polycarbonate outer ring and the entrance window). This air gap has been incorporated into our MC study to fully reproduce the experimental conditions. When the EPoM_man is located at the phantom surface, a part of the chamber comes out of the phantom, preventing the contact between the applicator of the eBT with the phantom surface.

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We have used the PENELOPE-2018 MC system (henceforth denoted PEN18) (Salvat 2019) together with the penEasy v. 2019-09-21 code (Sempau et al 2011). PENELOPE simulates electron and photon transport from 50 eV to 1 GeV, used in the study of both electronic (Valdes-Cortez et al 2020, Valdes-Cortez et al 2019, Croce et al 2012) and radionuclide-based (Ballester et al 2015, Vijande et al 2013, Almansa et al 2017, Gimenez-Alventosa et al 2018) brachytherapy sources.

PEN18 obtains the photoelectric cross-sections from the PHOTACS database (Sabbatucci and Salvat 2016), by using the elementary theory of the atomic photoelectric effect (independent electron model) (Pratt et al 1973, Scofield 1973) to calculate tables of excitation and ionization cross-sections. The user can choose to include or exclude the so-called 'Pratt's renormalization screening' (PRS) (Seltzer et al 2014, Andreo et al 2012), which is enabled by default (Salvat 2019).

The Rayleigh scattering cross-sections are calculated using non-relativistic perturbation theory, obtaining the atomic form factors from EPDL97 (Cullen et al 1997). Compton interactions use the relativistic impulse approximation, which takes into account both binding effects and Doppler broadening (Ribberfors 1983). Furthermore, PEN18 simulates explicitly the emission of characteristic x-rays, Auger and Coster-Kronig electrons that result from vacancies produced in K, L, M and N shells, using transition probabilities extracted from the Evaluated Atomic Data Library (EADL) (Perkins et al 1991). The energy of the x-rays published in the EADL was updated, when available, with the K and L shell transitions from Deslattes et al (2003), and the M lines from Bearden (1967). Other transition energies are calculated from the energy eigenvalues of the Dirac–Hartree–Fock-Slater equations for neutral atoms (Perkins et al 1991).

A full MC study of the Esteya unit has been performed elsewhere (Valdes-Cortez et al 2019). There, the photon energy spectrum produced by bremsstrahlung emission in the tungsten target and exiting the beryllium window of an Esteya unit was scored (see figure 1). To improve the calculation efficiency, in the present simulations this photon energy spectrum was used as a point-like source placed at the center of the lower surface of the tungsten target with a polar and azimuthal aperture of 18^° (enough to fully cover the primary collimator) and 360^°, respectively. Hence, the simulation ensemble includes the phantom, the chamber (when needed), and the full geometry of the Esteya eBT system.

Water has been recommended by the IAEA Code of Practice TRS-398 (Andreo et al 2000) and AAPM TG-253 (Fulkerson et al 2020) as the reference medium for the determination of the absorbed dose in kV photon beams. Therefore, in the present simulations the phantom considered was of liquid water with the composition recommended by ICRU Report 37 (Berger et al 1984) and the updated mean excitation energies and mass density given by ICRU Report 90 (Seltzer et al 2014).

The study of the chamber response with depth requires determining the perturbation correction factors at several depths along the beam axis in the water phantom. To accomplish that purpose, it is necessary to calculate the absorbed dose in water ($D_\mathrm{w}$), the absorbed dose in the sensitive volume of the chamber ($D_\mathrm{cav}$), and the ratios of the mean mass-energy absorption coefficients water to air ${(\overline{{\mu}_\mathrm{en}/\rho})}_\mathrm{w,air}$ at all the depths considered. Both $D_\mathrm{w}$ and $D_\mathrm{cav}$ have been scored directly by evaluating the energy imparted within the corresponding volume of interest, whereas the $\mu_\mathrm{en}$-ratios were determined from the MC-calculated photon spectra (see below).

A desirable dosimetry condition is to perform measurements in regions with charge particle equilibrium (CPE) (Ma et al 2001). That condition should imply depths larger than the expected range of the secondary electrons in the water phantom (the continuous slowing down approximation range in water for electrons of 70 keV is 0.08 mm). Standard treatment conditions for the Esteya eBT system require the use of a polyfenilsulfone plastic cap (see figure 2) with a typical thickness of about 0.5 mm, which is intended to be in contact with the treatment surface. Hence, complete CPE is achieved at all depths.

All the results were obtained through parallelized MC simulations. The processes were kept uncorrelated through proper management of the initial seeds (Badal and Sempau 2006). When electron transport was required (see below), it was simulated in detailed mode (PENELOPE transport parameters C1 = C2 = 0), i.e. without resorting to the mixed (Class II) algorithm option incorporated in PENELOPE. The variance reduction tools used were particle splitting and interaction forcing, with adequate use of the particle weight to maintain the simulation unbiased (Salvat 2019).

The selected energy cutoff for photons was 8 keV in all materials of the eBT device, and 1 keV in all structures embedded in the water phantom (i.e. ionization chamber and scoring volumes). The 8 keV threshold was selected because the amount of photons with energy lower than this value leaving the applicator (see figure 3) is negligible (Valdes-Cortez et al 2019). Due to the short range of secondary electrons in water, electron transport can be ignored, and its energy assumed to be deposited on the spot, i.e. an infinite energy cutoff for electrons was considered for $D_\mathrm{w}$. With respect to $D_\mathrm{cav}$, the energy cutoff of 1 keV for electrons was chosen in all chamber materials and in a water envelope region of 0.1 mm thick around the chamber. When the chamber is in contact with the plastic cap (first depth voxel), the same electron cutoff was considered both in the plastic cap and in the air gap.

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Table 1 summarizes the details concerning the MC simulations in this work following the recommendation of AAPM TG-268 (Sechopoulos et al 2018). Further information regarding the PENELOPE transport parameters and variance reduction tools can be found in its manual (Valdes-Cortez et al 2019, Salvat 2019, Sempau and Andreo 2006).

Table 1. Summary of the main characteristics of the Monte Carlo simulations used in this work.

Item	Description	References
Code	1) PENELOPE-2018	1) Salvat (2019)
	2) penEasy (v. 2019-09-21)	2) Sempau et al (2011)
	Intel® Fortran compiler 18.0.3
Validation	Previously validated	Ye et al (2004), Croce et al (2012),
		Valdes-Cortez et al (2019)
Timing	Average values for APP$_{10 {\mathrm{mm}}}$ (sum of
	parallel processes, Intel (R) Xeon(R)
	Gold 6154 CPU @3.00 GHz):
	- $D_\mathrm{cav}\$ obtained in 28 647 CPU hours
	and 7×1011 histories.	Parallel processes as of
	- $D_\mathrm{w}$ obtained in 629 CPUhours and	Badal and Sempau (2006)
	1010 histories.
	- ${(\overline{{\mu}_\mathrm{en}/\rho})}_\mathrm{w,air}$ obtained in 1330 CPU
	hours and 5×1010 histories.
Source description	Photon point collimated source. Spectrum
	from detailed modelling of the Esteya	Valdes-Cortez et al (2019)
	x-ray tube (GIS model, see reference)
Cross-sections	1) Photoelectric from PHOTACS;	1) Sabbatucci and Salvat (2016);
	2) Rayleigh scattering using	2) Sakurai (1967), Born (1969),
	non-relativistic perturbation theory;	Baym (1974), Cullen et al (1997);
	3) Compton from relativistic impulse	3) Ribberfors (1983);
	approximation; 4) Atomic relaxation with	4) Perkins et al (1991),
	EADL transition probabilities.	Deslattes et al (2003),
		Bearden (1967)
Transport parameters	Photon cut-off = 8 keV in all materials in
	eBT device, 1 keV in ionization chamber
	and water phantom; Electron cut-off =	Valdes-Cortez et al (2019),
	1 keV in all materials surrounding	Sempau and Andreo (2006),
	sensitive chamber volume and water	Salvat (2019).
	envelope (PENELOPE parameters C1 =
	C2 = 0); Electron transport disabled
	elsewhere.
Variance reduction tools	1) Interaction forcing: $D_\mathrm{w}$ scoring bins
	and sensitive chamber volume;	1) Salvat (2019);
	2) Splitting particles (penEasy rotational	2) Sempau et al (2011).
	option at the flattening filter).
Scored quantities	Absorbed dose in water and chamber
	cavity; photon fluence.
Statistical uncertainties	≤ 0.1% (k = 2)
Post-processing	None

2.4.1. Absorbed depth-dose and mass-energy absorption coefficient

The absorbed dose in water, $D_\mathrm{w}$, was scored between the water phantom surface (0 mm depth) and 20 mm depth, in cylinders of 1 mm radius and 0.1 mm height assuming CPE conditions. The absorbed dose in the sensitive volume of the chamber, $D_\mathrm{cav}$, was scored with the chamber EPoM$_\mathrm{man}$ positioned at the same depth as the upper boundary of the corresponding $D_{\textrm w}$ bin. The ratio of the mean mass-energy absorption coefficients water to air, ${(\overline{{\mu}_\mathrm{en}/\rho })}_\mathrm{w,air}$, averaged over the energy-fluence spectrum at each depth was calculated as (Ma et al 2001):

$$\begin{equation} {\left(\overline{\frac{{\mu}_\mathrm{en}}{\rho }}(z)\right)}_\mathrm{w,air} = \frac {\int^{E_\mathrm{max}}_0{E \,\Phi_{E,{\textrm w}}\left[{\left(\frac {{\mu}_{\mathrm{en}}(E)}{\rho }\right)}_\mathrm{w}\right]\ dE}} {\int^{E_\mathrm{max}}_0{E \,\Phi_{E,{\textrm w}}\left[{\left(\frac{{\mu}_\mathrm{en}(E)}{\rho }\right)}_{\mathrm{air}}\right]\ dE}}, \end{equation} \tag{ 1 }$$

where $\Phi_{E,{\textrm w}}$ is the photon fluence differential in energy in water at a given depth z, and $\left[{\left({\mu}_\mathrm{en}\left(E\right)/\rho \right)}_\mathrm{w}\right]$ and $\left[{\left({\mu}_\mathrm{en}\left(E\right)/\rho \right)}_\mathrm{air}\right]$ are the mass energy-absorption coefficients for water and air, respectively. The photon fluence corresponds to spectra with a bin width of 0.1 keV scored in a cylinder of 1 mm height and 1 mm radius, with its upper surface positioned at the same depth in water as the EPoM$_{\textrm man}$. All the above-mentioned factors were consistently evaluated with PEN18.

2.4.2. Determination of the EPoM and correction factors

The absorbed dose to water at depth z, $D_\mathrm{w}(z)$, can be calculated from the mean absorbed dose in the sensitive volume assigned to a point at depth $z^{\prime}$, $D_\mathrm{cav}(z^{\prime})$, through the ratio

$$\begin{equation} \frac{D_\mathrm{w}(z)}{D_\mathrm{cav}(z^{\prime})} = \left[{\left(\overline{\frac{{\mu}_{\textrm en}}{\rho }}(z^{\prime})\right)}_\mathrm{w,air}\right]\ p, \end{equation} \tag{ 2 }$$

where ${(\overline{{\mu}_\mathrm{en}/\rho })}_\mathrm{w,air}$ is calculated in a volume centered at depth $z^{\prime}$, and p is an overall factor to correct for the perturbations created by the presence of the chamber in the water phantom.

To determine the dependence of the chamber response on depth and evaluate p, three scenarios have been analyzed following the methodology proposed by Gimenez-Alventosa et al (2018):

• (1)  
  A surface correction factor $(p = p_\mathrm{surf})$ obtained through Equation (2) at the water phantom surface using the EPoM$_\mathrm{man}$ provided by the manufacturer, i.e. the chamber is located with the EPoM$_\mathrm{man}$ positioned at the phantom surface:

  $$\begin{equation} p_\mathrm{surf} = \frac{D_\mathrm{w}(z = 0)}{D_{\mathrm{cav}}(z^{\prime} = 0)\left[{\left(\overline{\frac{{\mu}_\mathrm{en}}{\rho }}(z^{\prime} = 0)\right)}_\mathrm{w,air}\right]}. \end{equation} \tag{ 3 }$$

  This approach will indicate that a single measurement performed at the water phantom surface would suffice to correlate satisfactorily $D_\mathrm{cav}$ and $D_\mathrm{w}$ for all depths. Such procedure will resemble the methodology normally applied in kV the phantom surface:
• (2)  
  A global correction factor $(p = p_\mathrm{glob})$, calculated using the EPoM$_\mathrm{man}$. Such method will require measuring and/or simulating the chamber response at all depths. To do so one evaluates the ratio:

  $$\begin{equation} f_i = \frac{D_\mathrm{w}(z_i)}{D_{\mathrm{cav}}(z_i)\left[{\left(\overline{\frac{{\mu}_\mathrm{en}}{\rho }}(z_i)\right)}_{\mathrm{w,air}}\right]} \end{equation} \tag{ 4 }$$

  for all depths z_i . The correction factor $p_\mathrm{glob}$ is then obtained by minimizing the differences between $D_\mathrm{w}$ and $D_\mathrm{cav}$ using a 1D chi-square distribution

  $$\begin{equation} \chi^2\left(p_\mathrm{glob}\right) = \sum^N_{i = 1} \frac{{\left(f_i - p_{\mathrm{glob}}\right)}^2}{{\Delta f_i}^2}, \end{equation} \tag{ 5 }$$

  where N is the number of the chamber positions, and Δf_i is the Type A uncertainty of the ratio f_i given in equation (4).
• (3)  
  A shift correction factor $(p = p_\mathrm{shift})$ including a modification on the EPoM. We have used the approach proposed by Kawrakow (2006) to find the EPoM that minimizes the effect of depth in the correction factors. This consists on a refinement of case (ii) considering the EPoM as a free parameter to be specified within the ionization chamber sensitive volume.Starting from a depth z = EPoM$_\mathrm{man}$, we found the Δz value for which the ratio

  $$\begin{equation} f_i = \frac{D_\mathrm{w}(z_i+\Delta z)}{D_{\mathrm{cav}}(z_i)\left[{\left(\overline{\frac{{\mu}_\mathrm{en}}{\rho }}(z_i)\right)}_{\mathrm{w,air}}\right]}, \end{equation} \tag{ 6 }$$

  is as independent of depth as possible. Hence, Δz will provide an improved EPoM value. To accomplish that condition, it is necessary to minimize a 2D chi-square distribution

  $$\begin{equation} \chi^2\left(p_\mathrm{shift},\Delta z\right) = \sum^N_{i = 1} \frac{{\left(f_i-p_{\mathrm{shift}}\right)}^2}{{\Delta f_i}^2}, \end{equation} \tag{ 7 }$$

  where N is the number of chamber positions, and Δf_i is the Type A uncertainty of the ratio f_i given in equation (6). Further details of the minimization process can be found in e.g. Kawrakow (2006) and Gimenez-Alventosa et al (2018).

As mentioned earlier, the vendor does not report manufacturing tolerances in the chamber sensitive volume. The possible role, played by such tolerances on the absorbed-dose determination, has been evaluated by performing a set of depth-dose measurements using three different PTW T34013 chambers (SN 000 810, 000 311, and 000 146). These measurements were performed by a single person, using a single eBT Esteya unit and a Plastic Water Low Range (PW LR: CIRS, Norfolk, VA, USA) phantom. For the Esteya unit, the water equivalence of plastic phantoms was studied by (Garcia-Martinez et al 2014, Candela-Juan et al 2015b, Valdes-Cortez et al 2019), reporting differences between the absorbed dose to a water voxel located at the surface of a water-equivalent plastic phantom and the absorbed dose to a water voxel located at the surface of a water phantom to be less than 0.2%.

Plastic Water LR phantom consists out of slabs with different thickness from 1 mm up to 20 mm. One 20 mm slab has a groove for inserting the T34013 chamber and its cable (see figure 2, left), so that the top surfaces of the chamber and the slab are aligned. Five 20 mm slabs were placed under the chamber to provide full backscatter conditions. Two kinds of measurements were carried out: reference absorbed dose ($D_\mathrm{ref}$) at 3 mm depth (typical reference depth used in clinical practice) to establish the output and relative depth-dose curves (PDD), normalized at 3 mm depth (Ouhib et al 2015, Guinot et al 2018). $D_\mathrm{ref}$ was calculated according to the TRS-398 Code of Practice, using the method described in (Candela-Juan et al 2015b) and EPoM$_\mathrm{man}$. All measurements were done for the two applicator diameters, APP$_{10 {\mathrm{mm}}}$ and APP$_{30 {\mathrm{mm}}}$, used throughout this work. The differences between the three chambers were evaluated through $\{\left[{\textrm PDD}(z)_{a} \times D_{{\textrm ref},a}\right]/\left[{\textrm PDD}(z)_{b} \times D_{{\textrm ref},b}\right]-1\}\times 100$, where the subscripts a and b denote different chambers.

The uncertainties of MC-derived quantities were evaluated according to the GUM recommendations (BIPM 2010), assuming normal distributions. Uncertainties are expressed with a coverage factor k = 2 as recommended by the AAPM TG-138 (DeWerd et al 2011). Type A uncertainties were smaller than 0.1% for $D_\mathrm{w}$, $D_\mathrm{cav}$, ${\left({\mu}_\mathrm{en}/\rho \right)}_\mathrm{w}$, and ${\left({\mu}_\mathrm{en}/\rho \right)}_\mathrm{air}$. Type B and combined uncertainties were estimated as follows:

• (1)  
  For ${(\overline{{\mu}_\mathrm{en}/\rho })}_\mathrm{w,air}$, Andreo et al (2012) reported a combined uncertainty of 0.2% (group II for an x-ray beam with an average energy of 34.1 keV).
• (2)  
  Ratio $D_\mathrm{w}/D_\mathrm{cav}$: Photoelectric cross-section uncertainties (predominant interaction for photons of 30 keV, considering all materials along their path) have been estimated between 2% and 3% for photons below 100 keV, mostly due to the implementation of the Pratt's renormalization screening (PRS) in the different MC codes (Seltzer et al 2014). $D_\mathrm{w}/D_{\mathrm{cav}}$ was evaluated at selected depths (0 mm, 3 mm, and 10 mm) with and without PRS to estimate the Type B uncertainty associated, obtaining a value of about 0.2%.
• (3)  
  The impact on the uncertainty estimation arising from the simplified source model describing the Esteya unit (see section 2.4) was estimated by performing additional MC simulations at selected depths using the complete description given in Valdes-Cortez et al (2019). Differences between the two source types were less than 0.02% for ${(\overline{{\mu}_{\textrm en}/\rho })}_\mathrm{w,air}$ and less than 0.2% for $D_\mathrm{w}/D_\mathrm{cav}$ at all depths.
• (4)  
  $p_\mathrm{surf}$, $p_\mathrm{glob}$ and $p_\mathrm{shift}$: Their combined Type B uncertainties were obtained by adding in quadrature the uncertainty values estimated in (i), (ii), and (iii).

Figure 3 shows the fluence spectra of the APP$_{10 {\mathrm{mm}}}$ and APP$_{30 {\mathrm{mm}}}$ applicators at different depths. The maximum variation of ${(\overline{{\mu}_\mathrm{en}/\rho })}_\mathrm{w,air}$ from 0 to 20 mm depth was 0.06%. In addition, the largest difference between the APP$_{10 {\mathrm{mm}}}$ and APP$_{30 {\mathrm{mm}}}$ cases, for the entire range of depths, was 0.04%. Hence, ${(\overline{{\mu}_{\textrm en}/\rho})}_\mathrm{w,air}$ was considered in the following a depth- and applicator-independent quantity, with a value of 1.018 and an uncertainty of 0.2% (see table 2). These values are in good agreement with the data and uncertainty estimates published by (Ma et al 2001, Andreo 2019, Valdes-Cortez et al 2020).

Table 2. Estimated relative uncertainties (k = 2) for the correction factors and the quantities used in their calculation.

		Uncertainty (%)
	Component	Type A	Type B
${(\overline{{\mu}_\mathrm{en}/\rho})}_\mathrm{w,air}$	Grouping II (50 kV) in Andreo et al (2012) a		0.2
	Effect of point vs full x-ray source		0.02
	MC statistics	0.1
	$u_\mathrm{c}$	0.2
$D_\mathrm{w}/D_\mathrm{cav}$	Effect of photoelectric cross-section		0.2
	Effect of point vs full x-ray source		0.2
	MC statistics	0.1
	$u_\mathrm{c}$	0.3
Correction factors	$u_\mathrm{c}$	0.4

^a Effect of the photoelectric cross-section uncertainties is also taken into account.

Figure 4 shows the ratio $D_\mathrm{w}/D_\mathrm{cav}$ for both applicators, APP$_{10 {\mathrm{mm}}}$ and APP$_{30 {\mathrm{mm}}}$. These ratios were obtained with a Type A uncertainty of 0.1%. Type B uncertainties were evaluated incorporating the sources of uncertainty listed in section 2.6 (ii) and (iii), leading to a combined uncertainty of about 0.3% (see table 2).

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For the applicator APP$_{10 {\mathrm{mm}}}$, a variation in $D_\mathrm{w}/D_\mathrm{cav}$ of about 1% can be observed between the water phantom surface and at a depth of 10 mm, from 1.376 to 1.390. The corrections factors $p_\mathrm{surf}$ and $p_{\mathrm{glob}}$ obtained were 1.351 and 1.359, respectively.

With respect to the third proposed method, $p_\mathrm{shift}$(EPoM), equation 7 reaches a minimum at Δz = 0.4 mm (see figure 5), with a value of $p_\mathrm{shift}$ = 1.324. Figure 6 shows the differences between $D_\mathrm{w}$ and the absorbed dose to water calculated using the three proposed methods, $p_\mathrm{shift}$(EPoM$_\mathrm{man}$), $p_\mathrm{glob}$(EPoM$_\mathrm{man}$), and $p_\mathrm{shift}$(EPoM). It can be seen that the $p_\mathrm{glob}$(EPoM$_\mathrm{man}$) method reduces the differences to within [-0.4%, +0.6%], while $p_\mathrm{shift}$(EPoM) (with an EPoM at Δz = 0.4 mm) reduces the differences to within ± 0.4% from 0 mm to 15 mm depth.

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In the case of the APP$_{30 {\mathrm{mm}}}$ applicator, the maximum variation of $D_\mathrm{w}/D_\mathrm{cav}$ was 0.6% at 6 mm depth, from 1.348 to 1.356. The values of $p_\mathrm{surf}$ and $p_\mathrm{glob}$ were 1.324 and 1.330, respectively.

With respect to $p_\mathrm{shift}$(EPoM), the optimum EPoM value was also found at Δz = 0.4 mm below EPoM$_\mathrm{man}$ (see figure 5 and Table 3), with $p_\mathrm{shift}$ equal to 1.300. The use of $p_{\mathrm{glob}}$ (EPoM$_\mathrm{man}$) reduces the differences between $D_\mathrm{cav}$ and $D_\mathrm{w}$ to within 0.5%, while $p_\mathrm{shift}$ (EPoM) (EPoM at Δz = 0.4 mm) reduces the differences to within 0.3%.

Table 3. Summary of the values for the three correction factors proposed along with their combined estimated uncertainties (k = 2). The fifth column shows the differences [minimum, maximum] between the absorbed dose obtained from equation 2, using $D_\mathrm{cav}$ and the perturbation correction factors of the third column, and the simulated $D_\mathrm{w}$. The Uncorrected row corresponds to the differences between $D_{\mathrm{cav}}$ and $D_\mathrm{w}$ before applying any of the proposed methods.

		Perturbation correction
Applicator	Method	factors (p)	Δz (mm)	Differences (%)
APP$_{10{\mathrm{mm}}}$	Uncorrected	1	–	[−28.0, −27.3]
	$p_\mathrm{surf}$	1.351 ± 0.005	–	[−1.0, 0.0]
	$p_\mathrm{glob}$	1.360 ± 0.005	–	[−0.4, +0.6]
	$p_\mathrm{shift}$	1.324 ± 0.005	0.4	[−0.5, +0.4]
APP$_{30{\mathrm{mm}}}$	Uncorrected	1	–	[−26.3, −25.8]
	$p_\mathrm{surf}$	1.324 ± 0.005	–	[−0.6, 0.0]
	$p_\mathrm{glob}$	1.330 ± 0.005	–	[−0.1, +0.5]
	$p_\mathrm{shift}$	1.300 ± 0.005	0.4	[−0.2, +0.3]

For both applicators, $p_\mathrm{surf}$ (EPoM$_\mathrm{man}$), $p_\mathrm{glob}$ (EPoM$_\mathrm{man}$), and $p_\mathrm{shift}$(EPoM) were obtained with a combined uncertainty smaller than 0.4%. Table 2 shows the uncertainties estimated in the calculation of the various correction factors. Differences of about 2% were observed between the corresponding correction factors depending on the applicator considered, 10 mm or 30 mm in diameter. Hence, a conservative approach leads us to assume the same value for other applicators with diameters in-between 10 and 30 mm.

A summary of the correction factors and their estimated uncertainties is given in table 3.

Watson et al (2017) explored the response of the T34013 chamber with an unfiltered beam of 50 kV at the same depth intervals as in this work. They reported a modification of about 2.5% in the value of ${(\overline{{\mu}_{\textrm en}/\rho })}_\mathrm{w,air}$. When using EPoM$_\mathrm{man}$ and the nominal geometry provided by the chamber manufacturer, the ratio $D_\mathrm{w}/D_\mathrm{cav}$ was estimated to be 1.33 (approximated value) at 3 mm depth (first point of the published series), decreasing nearly by 5% at 15 mm depth. On the other hand, when the authors consider possible manufacturing tolerances in the height of the sensitive volume (using EPoM$_\mathrm{man}$), they found differences in the ratio $D_\mathrm{w}/D_\mathrm{cav}$ with depth of about 12%. The authors conclude that changing the EPoM from the manufacturer's recommendation to the midpoint of the sensitive volume may reduce these variations in the chamber response. However, that change would increase the depth dependence of the correction factors of the chamber, from 5% to 15% at 15 mm depth.

Gimenez-Alventosa et al (2018) simulated the response of the T34013 chamber for the case of the Valencia and Large Field Valencia applicators. Both surface applicators use a brachytherapy ¹⁹²Ir source. The authors found spectral variations due to differences in the design of the applicators (i.e. presence of a flattening filter in one of the applicators, different diameters, etc). In that work, each applicator showed a constant ${(\overline{{\mu}_{\textrm {en}}/\rho })}_\mathrm{w,air}$ value (within the statistical uncertainties) for all depths investigated, with a variation of 0.4% in the value calculated between applicators, and 3% in $p_\mathrm{glob}$ (method I in that work). Furthermore, they found different Δz values for each model of applicator. The filtered beam produces a value Δz = 0.57 mm, while for the unfiltered beam the optimum Δz was found at EPoM$_\mathrm{man}$ (i.e. Δz = 0).

The average absorbed dose rates at the clinical reference depth (3 mm) were 2.31 Gy min⁻¹ and 2.64 Gy min⁻¹ for the APP$_{10 {\mathrm{mm}}}$ and APP$_{30 {\mathrm{mm}}}$, respectively. The absorbed dose rates differences observed between the three chambers were within 1.5% for both applicators. Furthermore, for depth-dose measurements the differences in the relative measurements (up to 20 mm depth) were smaller than 2%, with an average difference within 1% for both applicators. Therefore, depth-dose measurements uncertainties associated with chamber-to-chamber differences can be estimated to be at most of 2%, a value that also includes the uncertainties related with the experimental setup (e.g. alignment and positioning, among others).

Watson et al (2017) evaluated differences due to manufacturing tolerances for an unfiltered 50 kV eBT by performing different MC simulations. They reported that the effect of manufacturing tolerances on the T34013 chamber may generate differences in the absorbed dose of about 5% at 20 mm depth. It can be concluded that the effect of the manufacturer tolerances on the chamber response for an Esteya unit seems to be lower than estimations made in the literature for eBT 50 kV beams.

This study relies on state-of-the-art MC simulations for a particular eBT system and an ionization chamber. Therefore, the obtained results depend strongly on the precise description of both the Esteya unit and the PTW34013 ionization chamber provided by the manufacturers. Any subsequent major structural modification implemented by any of the vendors will require repeating this study to rule out any unforeseen change in the correction factors and the effective point of measurement. Hence, the results reported here cannot be directly extrapolated to any other eBT system and/or ionization chamber, for which separate machine- and chamber-specific studies, including a faithful description of the systems involved, will be required.