Correction factors for ionization chamber measurements with the 'Valencia' and 'large field Valencia' brachytherapy applicators

The Monte Carlo (MC) system used been PENELOPE version 2014 (PEN14) (Sempau et al 1997, 2003, Salvat 2015), which simulates the transport of photons and electrons in any arbitrary material for energies in the range of interest for medical physics. This MC system, henceforth referred to as PEN14, has been extensively applied in the field of brachytherapy (Ballester et al 2015, Ma et al 2017). PEN14 photon cross-sections for Rayleigh scattering are extracted from the EPDL97 cross sections library (Perkins et al 1991, Cullen et al 1997). For incoherent scattering, PEN14 uses the Relativistic Impulse Approximation together with binding effects and Doppler broadening, i.e. for a given scattering angle, the cross section yields a photon energy distribution rather than the single photon energy resulting from the scattering of a photon with a free electron (Ribberfors 1975). Photoelectric cross-sections are from the database of Sabbatucci and Salvat (2016), which uses the same theory as in the re-normalized calculations by Scofield (1978), but implements more accurate numerical algorithms and an extended energy range. A comparison of mass energy-absorption coefficients for different materials using this photon dataset and values from other libraries has been described by Andreo et al (2012). Electron cross-sections are directly calculated by PEN14. In this work, all material compositions and mass densities were those recommended by ICRU Report 37 (ICRU 1984) with the exception of the updated mean excitation energies and mass density for water and carbon (Andreo et al 2013, ICRU 2016).

The mHDR-v2 source model was used in the study; its detailed geometry was adopted from Granero et al (2011). The VA and LFVA applicators were modelled according to the descriptions by Granero et al (2008) and Candela-Juan et al (2016), respectively, using the specifications provided by the manufacturer. A schematic drawing of both applicators is shown in figure 1.

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The ¹⁹²Ir photon spectrum was taken from the NuDat database (Brookhaven National Laboratory 2013). The electron spectrum includes β decay, internal conversion electrons (IC) and Auger electrons that were not considered in the simulations since their effect on the total dose is known to be less than 0.1% at distances greater than 0.15 cm from the source centre (Ballester et al 2009). On average, in each disintegration 2.299 22 photons Bq⁻¹ s⁻¹ are emitted.

Blueprints, composition by weight and densities of the chamber materials provided by the manufacturer allowed us to model accurately the geometry details of the PTW34013 chamber; the configuration used for the MC simulations is shown in figure 2. The effective point of measurement for this chamber, P_eff, is recommended to be at the centre of the inner side of the front wall (PTW 2017), in consistency with IAEA TRS-381 (Almond et al 1997).

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For the MC simulation, the applicators were located at the surface of a cubic $40\times40\times40$ cm³ water phantom surrounded by dry air. The depth z is defined with respect to the phantom surface, where the origin of coordinates is located.

It has been shown, for both the VA and LFVA applicators, that the depth dose along the central axis does not depend on the collimator used (Granero et al 2008, Candela-Juan et al 2016). Therefore, only the cases of 3 cm (VA) and 5 cm (LFVA) collimator diameters have been analysed in the present study.

The goal of the MC simulations is to compute quantities with 0.1% Type A uncertainty, and all the simulation parameters have been tailored to achieve such goal. Four different types of simulation have been done for each applicator to obtain the following quantities:

• (i)  
  Photon spectra arriving at the surface of the water phantom. This was scored at the phantom surface between 1 keV and 1200 keV with a $\Delta E = 1$ keV energy grid.
• (ii)  
  Photon energy-fluence distributions, differential in energy, as a function of depth. Cylindrical voxels of 0.2 cm radius and 0.034 cm height were used as scoring regions on the central axis to mimic the chamber active volume. The distributions were scored in water, between 1 keV and 1200 keV with a $\Delta E = 1$ keV energy grid.
• (iii)  
  Depth-dose distributions in the water phantom in the absence of the ionization chamber. The same cylindrical voxels as in (ii) were used. An energy transport cut-off of 1 keV was chosen for photons. Since, for the energy range involved, radiative losses in water are less than 0.07% (Salvat 2015), and charged-particle equilibrium exists in the phantom, an infinite energy transport cut-off was chosen for both electrons and positrons, i.e. charged particles were not tracked for the depth-dose calculations. The Type A uncertainties achieved were below 0.1% in all voxels (0.07% or lower for depths greater than 0.1 cm).
• (iv)  
  Absorbed dose in the active volume of the ionization chamber positioned at different depths. This absorbed dose was assigned to the P_eff recommended for the chamber (the centre of the inner side of the front wall of the active volume), so that the shift in P_eff (see section 2.2), is $\Delta P_{\rm eff} = 0$. The water surrounding the ionization chamber was divided into two regions:

  • (a)  
    An inner region, containing the chamber and a surrounding cylinder 0.05 cm larger than the chamber dimensions. The rationale for this distance is that it corresponds to the continuous slowing-down range (R_CSDA) in water of secondary electrons whose yield of higher-order photons is less than the uncertainty goal of 0.1%. Therefore, all chamber materials and water inside this region were characterized by a 1 keV energy transport cut-off for photons, electrons and positrons to ensure that all particles could arrive at the active volume. Additionally, an event-by-event simulation was chosen for the electrons and positrons in this region.
  • (b)  
    An outer region extending up to the phantom dimensions, where the electron and positron energy transport cut-off was increased to 200 keV, justified in terms of the radiation yield (0.1%) and $R_{\rm CSDA}=0.045$ cm in water at this energy. Hence, no electron with energy below this threshold can reach the inner region and its bremsstrahlung yield is within the uncertainty goal. Type A uncertainties were lower than 0.1% in all positions.

Various approaches can be found in the literature to the implementation of perturbation correction factors in the determination of absorbed dose to water using measurements performed with a plane-parallel ionization chamber.

The most straightforward of these, and probably less prone to error in implementation, is to assign the absorbed dose to the effective point of measurement recommended for this type of chamber (usually below the entrance foil) and to include all type of perturbation corrections in a global chamber correction factor p_ch (Andreo et al 2000). For a plane-parallel chamber, this global correction can be defined as the product of various perturbation factors, e.g. $p_{\rm ch} = p_{\rm dis}\, p_{\rm wall}\, p_{\rm fl}$, where p_dis accounts for the effect of replacing a volume of water with the chamber, $p_{\rm wall}$ accounts for the presence of non-water-equivalent materials in the chamber walls, and p_fl corrects for the difference in photon fluence between water and the ionization chamber air volume. In this case, the relation between the absorbed dose to water at a point in a given depth, $D_{\rm w}(z)$, and the mean absorbed dose scored in the active volume of the ionization chamber $\bar D_{\rm air} (z)$ can be written as

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{D_{\rm w}(z)}{\bar D_{\rm air} (z)} = \left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w,air}\,p_{\rm dis}\, p_{\rm wall}\, p_{\rm fl} = \left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w,air}\,p_{\rm ch} \label{eq:1} \nonumber \end{align} \tag{ 1 }$$

where $\left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w, air}$ is the ratio of mass energy-absorption coefficients of water and air, averaged over the photon energy-fluence at a given depth z. In what follows, this approach will be referred to as Method I.

An alternative approach is to replace the use of a p_dis perturbation factor, which assumes the chamber P_eff to be just below the entrance foil, by a P_eff shifted a distance $\Delta P_{\rm eff}$ together with a partial chamber correction factor given by $p'_{\rm ch} = p_{\rm wall}\, p_{\rm fl}$. In this case, one would have

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle \frac{D_{\rm w}(z+\Delta P_{\rm eff})}{\bar D_{\rm air} (z)} = \left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w,air}\, p_{\rm wall}\, p_{\rm fl} = \left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w,air}\,p'_{\rm ch}; \label{eq:2} \nonumber \end{align} \tag{ 2 }$$

this will be referred to as Method II.

In both methods, the purpose is to determine the chamber perturbation correction factor (p_ch or $p'_{\rm ch}$) and the relevant displacement of the effective point of measurement $\Delta P_{\rm eff}$ in a given photon field. The roadmap can be described as follows:

• (i)  
  A Type A uncertainty goal was specified. All simulations were required to conform to it.
• (ii)  
  The depth-dose distribution in water, $D_{\rm w}(z)$, was evaluated for the ¹⁹²Ir photon spectrum by means of a Monte Carlo simulation.
• (iii)  
  The mean absorbed dose in the active volume of the ionization chamber was scored positioning the chamber with its recommended P_eff (inner side of the chamber front wall) at various depths, $\bar D_{\rm air}(z_i)$.
• (iv)  
  The $\left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w, air}$-ratio was calculated for the photon energy-fluence at the depths where the ionization chamber was located.
• (v)  
  The global chamber perturbation correction factor was determined from equation (1), i.e.

  $$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle p_{\rm ch} = \frac{D_{\rm w}(z)}{\bar D_{\rm air}(z)\, \left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w,air}}. \label{eq:3} \nonumber \end{align} \tag{ 3 }$$
• (vi)  
  The shift $\Delta P_{\rm eff}$ was obtained by iteration, modifying its value until the ratio ${D_{\rm w}(z+\Delta P_{\rm eff})}/{\bar D_{\rm air} (z)}$ became as depth-independent as possible. This was fulfilled by minimizing the usual $\chi^2$ distribution, defined as $\chi^2 = \sum\limits_{i = 1}^N {\left(\,f_i - p \right){}^2}/{\Delta f_i^2}$, where N is the number of chamber positions,

  $$\begin{align*} \newcommand{\e}{{\rm e}} \displaystyle f_i =\frac{D_{\rm w}(z_i+\Delta P_{\rm eff})}{\bar D_{\rm air}(z_i)\, \left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w,air}},\nonumber \end{align*}$$

  and $\Delta f_i$ is the Type A uncertainty of the f_i ratios. This approach assumes that the partial chamber perturbation correction factor $p'_{\rm ch}$ is depth-independent. By minimizing $\chi^2$ with respect to p, one arrives at $p = \frac{s_1}{s_0}$ and $\chi^2= s_2 - \frac{s_1^2}{s_0}$, with $s_n = \sum\limits_{i = 1}^N {f_i^n}/{\Delta f_i^2}$⁶. The $\Delta P_{\rm eff}$ value can then be easily obtained from a $\chi^{2}$ versus $\Delta P_{\rm eff}$ plot. Further details of the minimizing procedure can be found in Kawrakow (2006).

The photon spectra arriving at the surface of the water phantom for the two applicators are shown in figure 3. Their average energies differ by approximately 10%, being 384 keV for the VA applicator and 343 keV for the LFVA. The expected hardening of the VA spectrum due to the flattening filter can be clearly observed both in the average energies and in the shape of the spectrum.

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Although both applicators are designed to yield flat absorbed dose distributions at the clinical prescription depth, the absorbed dose delivered by the two applicators at the prescription depth differs in value. The energy dependence and the different spatial distribution of the photons (those emitted from the VA are emitted from a single central position and then filtered, while those emitted by the LFVA are also emitted from lateral positions and unfiltered) produce different depth-dose distributions. This is shown in figure 4, where it can be observed that the VA depth-dose curve has a gradient of 10% mm⁻¹, while that of the LFVA is reduced to 5% mm⁻¹. In the following, it will be seen how these differences in shape and average energy impact the determination of the correction factors.

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Photon energy-fluence distributions were evaluated as described in section 2 for the depths indicated by symbols in figure 4. Their values were used to obtain the $\left(\overline{\mu_{\rm en}/\rho} \right)_{\rm w, air}$-ratios as a function of depth. It was found that, although the mass energy-absorption coefficients averaged over the local photon energy-fluence depend on the depth within the required uncertainty of 0.1%, their ratios are depth independent. The values obtained were $\left(\overline{\mu_{\rm en}/\rho} \right)_{\rm w, air}=1.109\pm 0.001$ for the VA applicator and $\left(\overline{\mu_{\rm en}/\rho} \right)_{\rm w, air}=1.104 \pm 0.001$ for the LFVA.

In 2015, the American Brachytherapy Society (ABS) published a report that included a literature summary and best practices used in non-melanoma skin cancer (NMSC) (Ouhib et al 2015). For superficial NMSC, the ABS Report indicated a typical prescription depth of 5 mm for moulds and flaps, and of 3 mm for the shielded applicators currently available—these values being based on reported clinical practice. The reason for this difference in prescription depth has been justified in an exploratory Monte Carlo study (Granero et al 2017). It shows that the percentage depth dose at 3 mm depth for shielded applicators is approximately the same as that at 5 mm depth for moulds and flaps, supporting the ABS prescription depth recommendations. These treatment modalities have various average energies and spectra, ranging from the hundreds of keV in the shielded applicators to the tens of keV in the electronic brachytherapy modalities. In spite of this, no difference in clinical outcome has been reported (Ouhib et al 2015).

The depth dose at the effective point of measurement of the ionization chamber located at different depths z differ from the corresponding depth doses in water at the same depth, as shown in figure 4. These differences are up to  −8% for the VA applicator and  −4% for the LFVA. Thus, the need to incorporate a correction factor and/or a displacement of the effective point of measurement becomes evident.

The formalism described in section 2 was applied to the VA and LFVA applicators, obtaining the $\chi^{2}$ per degree of freedom distributions shown in figure 5. Note that for the case of the LFVA applicator (blue curve) such distribution is practically flat in the vicinity of the entrance window, i.e. $\Delta P_{\rm eff} = 0$, starting to increase for values of $\Delta P_{\rm eff} \approx -0.02$ cm.

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The values obtained for the ionization chamber perturbation correction factors and the shift of the effective point of measurement obtained with each method are given in table 1 for the two applicators, along with the respective values of the average ratios of mass energy-absorption coefficients. Since $\Delta P_{\rm eff} = 0$ corresponds to the position of P_eff recommended by both TRS-381 (Almond et al 1997) and the manufacturer, Methods I and II coincide for the LFVA applicator. This is not the case for the VA, where Method II provides an effective point of measurement located close to the bottom of the active volume.

Table 1. Perturbation correction factors (p_ch and $p'_{\rm ch}$) and shift of the effective point of measurement ($\Delta P_{\rm eff}$), relative to the centre of the inner side of the front wall, for the PTW34013 plane-parallel ionization chamber using the VA and LFVA applicators with an ¹⁹²Ir mHDR-v2 source. Values of the $\left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w, air}$ for the corresponding spectra are also given.

	Applicator	VA	LFVA
Method I	Global correction factor, pch	0.976	0.943
	Shift of the effective point of measurement, $\Delta P_{\rm eff}$ (cm)	—	—
Method Ib	Global correction factor, pch	$p_{\rm ch}(z)$a	0.943
	Shift of the effective point of measurement, $\Delta P_{\rm eff}$ (cm)	—	—
Method II	Partial correction factor, $p'_{\rm ch}$	0.922	0.943
	Shift of the effective point of measurement, $\Delta P_{\rm eff}$ (cm)	0.057	0.0
	$\left(\overline{\mu _{\rm en}/\rho} \right)_{\rm w, air}$	1.104	1.109

^a$p_{\rm ch}(z) = 0.986{\rm{-}}0.023\, z~({\rm cm})$.

To determine the adequacy of the calculated values, a residual analysis was performed. Ratios between the corrected ionization chamber dose and depth-dose values in water are shown in figure 6 for the LFVA (a) and VA (b)–(d) applicators.

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As discussed above, for the LFVA applicator both methods agree and the corresponding residual analysis shows data consistent with the uncertainty goal of the present work. For the VA applicator, although the residuals for Method II are well within the present uncertainty goal at all depths, for Method I discrepancies are in the range [−0.8%, +1.1%]. These differences represent an improvement over the  −8% observed between depth dose in water and uncorrected ionization chamber results, but they are large for clinical use.

Considering that the relevant depths for the clinical applications of the VA applicator are less than 1 cm, one possible modification while maintaining the effective point of measurement recommended by the manufacturer is to allow for a depth-dependent correction factor $p_{\rm ch}(z)$. We term this alternative Method Ib. The simplest choice is a linear depth-dependence, the best parametrization being $p_{\rm ch}(z) = 0.986{\rm{-}}0.023 z$ (with z in cm). As can be observed in figure 6(d), this modification improves the differences observed in Method I.

To conclude, figure 7 shows the ionization chamber absorbed doses compared to the absorbed dose in the water phantom at various depths for each applicator and method. These results illustrate the energy dependence of the shift of the effective point of measurement and clearly emphasize the need for detailed state-of-the-art MC simulations before using this ionization chamber with different shielded applicators.

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