Realization of the absorbed dose to water for 70 kV electronic brachytherapy sources for surface applications

Objective. The purpose of this work is to establish a primary standard for the absorbed dose to water for surface applications with 70 kV electronic brachytherapy sources and thereby to investigate several reference conditions with respect to applicability in metrology institutes, calibration labs and clinics. Approach. A primary standard for the absorbed dose to water for LDR-Seeds (ipFAC) was utilized. For this, the method of evaluation was modified to account for the different geometries in interstitial and surface brachytherapy and for the different energy distributions of the radiation fields, respectively. The correction factors required to determine the absorbed dose to water were evaluated using Monte Carlo (MC) simulations. MC-calculations were also used to estimate the uncertainties of this method. Main results. It could be shown that determining the absorbed dose to water in 1 cm depth below the surface of a water phantom D w (1 cm) is feasible with the ipFAC. Thereby the method to determine D w (1 cm) when the source is placed at 50 cm distance and the beam is collimated turns out to be more robust to variations in the directional emittance of the source than placing the source applicator assembly directly on the phantom. Significance. The first method features significantly lower uncertainties, 1.4% compared to 3.7% (both for k = 2), for the second one. However, a calibration based on a transfer chamber calibrated with the second method is more practical and easier to implement in clinical routine.


Introduction
Incidences of skin cancer have been rising over the past decades.In Europe it has varied between (40-130)/ 100,000 persons-per-year for basal cell carcinoma (BCC) and (8-30)/100,000 persons-per-year for squamous cell carcinoma (SCC), respectively (Lomas et al 2012).
Various radiotherapy techniques have been developed to treat skin cancer: superficial and orthovoltage x-rays, electron and megavoltage photon treatment, and brachytherapy (BT) in all the modalities: low dose rate (LDR), high dose rate (HDR), pulsed dose rate (PDR), and electronic BT (eBT).Reviews of different modalities and details on the applicators from the physics point of view have been published by the american brachytherapy society (ABS) (Ouhib et al 2015) and the ESTRO Advisory Committee for Radiation Oncology Practice (ACROP) (Guinot et al 2018).
Current skin applications in brachytherapy are distinguished by superficial or contact brachytherapy and interstitial BT with the insertion of plastic tubes or rigid needles (Joslin andFlynn 2001, Skowronek 2015).In superficial BT the dose is usually prescribed at a depth of (3-5) mm below the skin surface (Guinot et al 2018).For interstitial BT the dose is prescribed according to ICRU 58 (ICRU 58 1997) where the reference depth for dosimetry is defined as 1 cm.
Currently, five electronic BT sources can be used for skin cancer in Europe.The Axxent TM (Xoft Inc., San Jose, USA) with 50 kV tube voltage.For skin treatments, the source is used in combination with special applicators with a flattening filter (Fulkerson et al 2014).The Intrabeam TM (Carl Zeiss, Oberkochen, Germany) is operated at 50 kV tube potential with an especially developed skin applicator (Schneider et al 2014, Goubert andParent 2015).The third system is Papillon+ TM (Ariane Medical Systems Ltd, Alfreton, UK) (Hensley 2017).System four is the Esteya TM (Elekta, Stockholm, Sweden) (Garcia-Martinez et al 2014, Pons et al 2014), dedicated specifically to skin BT.It is operated at 69.5 kV tube voltage.Surface applicators with a flattening filter are used to achieve a dose distribution similar to that of the Valencia 192-Ir HDR applicator from the same manufacturer.The fifth system is the 'iort-50' from Wolf-Medizintechnik (WOMED, part of BEBIG Medical GmbH, Berlin)1 .The ioRT 50 can be used for intercavitary and skin therapy.It features a hollow anode tube and is applied at a tube potential of 50 kV and a maximum tube current of 7 mA.The cylinder of the tube has an outer diameter of 22 mm.The entire tube body is water-cooled and the built-in radiation shielding is made of lead-free materials.The focal point is specified as having a diameter of 15 mm and the angular distribution of the emitted field of the bare tube as 180°× 360°.The dose reference point is located along the emitter axis of the tube.Currently, the 'iort-50' has approval up to 50 kV only, but technically can be operated at 70 kV.
Within the framework of the European Metrology Programme for Innovation and Research (EMPIR), six European National Metrology Institutes (NMIs) together with four additional partners from universities and clinics were collaborating in a Joint Research Project PRISM-eBT, Primary standards and traceable measurement methods for x-ray emitting electronic brachytherapy devices, to establish a harmonized, simplified, and traceable dosimetry for electronic brachytherapy (eBT) in terms of absorbed dose to water2 .
The aim of work package 2 of this project, 'Traceability for superficial (skin) external treatment', is to establish and validate dosimetric traceability for eBT devices used for superficial (skin) treatment in terms of absorbed dose to water.The developed methodologies should thereby be easy to apply.One route is based on a primary standard for air kerma (VSL/Ciemat) (de Prez et al 2023) and the other is based on a primary standard for the absorbed dose to water in 1 cm (PTB, this work).Doses at the depth of interest, such as on the surface of a phantom, in 70 μm depth or in several mm in a water phantom can be derived from both primary standards with the aid of MC-calculations.
In a future work, both realizations will be compared by means of transfer standards calibrated under different reference conditions.These comparisons will then provide information on the most suitable reference conditions in terms of operability, robustness and accuracy.Additionally, two systems will be compared, an Esteya TM system located at VSL and the ioRT-50 ® operated at 70 kV tube potential located at PTB.This will provide additional information if calibrations with a specific tube can be performed by another.

Realization of D w (1 cm)
The quantity D w (1 cm) is realized using PTB's primary standard for the absorbed dose to water for low energy brachytherapy sources ipFAC (in phantom Free Air Chamber).
The entrance plates are made of a 12 mm thick polymethyl methacrylate (PMMA) disc and the back plates (a total thickness of 60 mm) are made of high-density polyethylene (PEHD).The cross section of the measuring volume is defined by means of an aperture with a radius of 10 mm placed at a distance of 158.5 mm from the reference plane of the measuring volume.Background and leakage measurements are performed with the source in place by inserting a 5 mm thick tungsten shutter.The entrance plate is graphitized on the inner side and biased at the potential U in order to act as the polarizing electrode and to serve as the reference plane for the measurement.In front of the back plate is a graphitized polyethylene foil, where a centre collecting electrode and an outer guard ring -both at ground potential -are located.The diameter of the collecting electrode amounts to 140 mm.The x-ray tube is positioned 500.8mm in front of the reference plane.

Method of evaluation
The principles of the method of evaluation were developed based on radiation transport theory and are described in detail in (Schneider andSelbach 2011, Schneider 2012).The method is based on a Monte Carlodetermined conversion factor C(x i , x i+1 ) to be applied to the difference of the ionization charge for the two plate separations x i and x i+1 .This factor is composed of quotients of kerma values calculated for different plate separations in the chamber.As discussed in Schneider (2012) the design of the experimental set-up allows the contribution due to the net energy fluence of the secondary electrons at the surface of the interaction volume to be neglected.
Additional terms must be considered to determine the absorbed dose to water at a point at 1 cm distance from the end of the applicator tube fitted to the x-ray unit within the water phantom.The equation of determination is: with the ionization constant (W/e), air r the density of air, r ref the reference distance from the focal spot to the reference point of measurement, r m distance of the focal spot to the reference point of the quantity D w (1 cm), k inv the correction for deviations from the inverse square law which is valid only for point sources.Furthermore, k div is the conversion from the cross-sectional area of the measuring area lateral to the beam axis to the point on this axis.Q(x i+1 ) and Q(x i ) are the ionization charges measured at the two plate separations x i+1 and x i , which must be greater than the range of the secondary electrons, i.e. at least 40 mm and 70 mm according to the maximum range of 50 keV and 70 keV electrons in air, respectively.Correction of the measured charges are subsumed under k i.Among these are the corrections for the saturation effect k sat , the corrections for electrons induced by a secondary interaction process k scat , in addition the temperature k T and pressure correction k p .
In equation (2), the conversion factor C(x i , x i+1 ) is given in more detail.The left form corresponds to its meaning whereas the right form shows the components which needs to be calculated: For the calculation the following components need to be determined: A = π r A 2 the area of the measuring volume lateral to the beam axis at zero plate separation, the ratio of the mean air Kerma in the phantom K V x a p i ( )at plate separation x i and at plate separation zero.By definition c(0 where the last term is component of c x .Figure 1 illustrates the main correction and conversion factors.The upper part shows a sketch of the measurement set-up and the chamber.The source with the applicator tube fitted is placed outside the chamber in front of a collimator that defines the cross-section of the measuring volume.In equation (2), with the help of c(x i ) Kerma-values at zero plate separation are obtained within the cross section of the measuring volume.k div converts this quantity to a point quantity and the mass energy transfer coefficients converts air to water Kerma.In the middle panel, k PhW converts the plastics into water with a 1 cm front layer.The absorbed dose to water obtained in this condition is also investigated in this work, it will be named in the following D 1cm, 50cm , w ( )i.e. the absorbed dose in 1 cm water depth at a distance of 50 cm from the source.This condition is far from treatment condition, which is usually in close contact to the skin, but establishing this condition as a reference condition for calibration has the positive effect that conventional x-ray units could be used for calibration purposes.Estimation of robustness and uncertainties are shown in sections 3.1 to 3.3.The corresponding quantity for this calibration results in: In the lower panel of figure 1, the source is virtually moved to the front plate by application of the inverse square law with (r ref /r m ) 2 and a MC determined correction factor k inv (section 2.3).The factor k inv accounts for the discrepancies from the ideal inverse square law which holds only for point sources, because of a specific focal spot size and scattering in the applicator tube.In this condition, the source and applicator tube are in direct contact with the phantom.This geometry corresponds to the conditions under which patients are treated.The absorbed dose to water obtained in this condition is termed D 1 cm, 1 cm w ( ) in the following.The ipFAC was originally designed for the radiation field of I-125 seeds (max.35.5 keV).For 70 keV photons, there is only a small region within the chamber where secondary equilibrium prevails in both sagittal and lateral direction-due to the rather large electron range in air of approximately 70 mm.Therefore, measurements were only performed for plate separations of 90 mm up to 125 mm instead of 50 mm up to 200 mm.

MC-tube models
While the energy distribution of the photons can directly be obtained by measurements, this is much more complex for the spatial and directional distribution.Therefore, in this work, the influence of the lack of knowledge of these distributions on the measurand absorbed dose to water was determined.Different models were investigated to evaluate the dependency of the dose quantity on the spatial and directional distribution of the photon field of the real source.The vendor's specification of the tube's focal spot is having a diameter of 15 mm and an angular distribution of the emitted field of 180°× 360°for the bare tube.The applicator tube has an inner diameter of 23 mm at the height of the tube focus.This diameter widens over a length of 30-30 mm.
The simplest model is based on a point source (point) on the axis of the tube.From this point source, photons are emitted in all directions and a huge portion of the photons hits the inner wall of the applicator tube in which they may be absorbed or scattered.
In the second model, an isotropic directional distribution is also assumed with the photons starting from a circular surface with 15 mm diameter (isotr).This model is closest to the technical specification of the vendor.In a third model, a pronounced forward direction is assumed: the photons start from the same circular surface, but the isotropic directional distribution is limited to 15°in the forward direction (isotr 15 ).In the case of isotr 15 , the photons hardly touch the applicator's tube inner wall, see figure 2.
For the determination of the uncertainty of k inv , additional tube models were investigated with focal spot sizes of 18 mm, 15 mm, 12 mm and 10 mm with an isotropical beam characteristic and a pronounced forward direction with an angle of 45°.

MC determination of c x i ( )
In figure 3, the function c x i ( )is presented for the three source models (point, isotr and isotr 15 ) to estimate the uncertainty of the c x i ( )function due to incomplete knowledge of the emitting characteristic of the source.K V x a p i ( )was calculated for the following plate separations (all in mm): 1, 5, 10, 20, 30, 40, 50, 60, 80, 100, 120,  140, 160, 180 and 200.Specific tracking volumes need to be defined since fluence or kerma calculations in the EGSnrc code are based on track length estimations.A model function composed of two exponential terms has been fitted to these values and K 0 a p ( ) is obtained from the value of the model function at zero plate separation.In the upper panel of figure 3, throughout the range of plate separations, only little discrepancies between the functions c x i ( )for the individual source models can be observed.Statistical uncertainties of the values are below 0.1% (k = 1).From the lower panel, in which the ratio of c x i ( )of one of the isotropic sources to that of the point source is presented, it can be seen that the discrepancies are below 0.5% in the range from 90 to 125 mm plate separation in which the measurements were performed.
The model function to fit c x i ( )for a point source was determined to y = y 0 +A 1 with y 0 = 0.516654, A 1 = 0.13014, t 1 = 14.64757,A 2 = 0.353206 and t 2 = 80.03475.

MC determination of k scat
A part of the secondary electrons induced by the radiation field does not result from the primary interaction process but from a secondary interaction process.Electrons resulting from these interactions are not included in the Kerma definition and therefore the measured charges must be corrected by this amount.The correction factor was determined with the EGSnrc user code CAVRZnrc (Rogers et al 2010).The values presented in figure 4 vary between 0.9995 and 0.9965.For the plate separations at which the measurements were performed, i.e. at plate separations from 90 mm up to 125 mm, k scat is between 0.998 and 0.997.The model function to fit k scat was determined to y = y 0 +A 1 * exp(x i /t 1 ) + A 2 * exp(x i /t 2 ) with y 0 = 0.99639, A 1 = 3.72585 * 10 −4 , t 1 = 30.03777,A 2 = 2.96 * 10 −3 and t 2 = 664.07759.The overall uncertainty of k scat is in the order of 0.1%.

Determination of the remaining correction factors
The divergence correction factor was obtained analytically by k div = (R/d) 2 /ln[1+(R/d) 2 ] = 1.0004 according to Seltzer et al (2003), with the beam diameter R = 1.453 cm and a distance from the radiation source of The mean value amounts to 3.343 10 −4 Gy s −1 with a standard deviation of 3 10 −7 Gy s −1 .From this value,  D 1 cm, 50 cm w ( ) = 3.344 10 −4 Gy s −1 is obtained by application of k .div 3.5.MC determination of k inv Firstly, k inv was calculated for the three source models.The values obtained were 1.056 for the isotropical source, 1.114 for the point sour and 0.926 for the isotr 15 source.Based on measurements with GAFchromic films provided by the manufacturer 4 , other, more realistic tube models were selected.k inv was therefore calculated for 8 models, varying the beam diameter from 18 to 10 mm and simulating for each of these diameters a source radiating completely isotropically and one radiating isotropically only at 45°around the axis (table 1).
In table 1 the values for k inv are given.In the left columns the focus diameters are stated for which k inv has been calculated.In the middle column values of k inv for isotropic beam characteristic are given and in the right column those for an isotropic source limited to 45°around the axis.k inv is at maximum for the model based on the specification: focus 15 mm, isotropic beam shape.Other values are lower up to 3%.Based on this study the uncertainty distribution is selected as rectangular with a half with of 3%.The value for k inv is that of the model according to the specification and amounts to 1.056.3.6.Determination of D 1 cm, 1 cm w ˙( )and uncertainty budgets of D 1 cm, 1 cm w ˙( )and D 1 cm, 50 cm w ˙( ) In this section, the uncertainty budgets of  D 1 cm, 1 cm w ( )and  D 1 cm, 50 cm w ( ) are determined.Both quantities are determined experimentally as the average over corrected charge differentials at different plate separations (see equation ( 4)).The last fraction in this equation represents the average of the experimental data and will be denoted Exp in the following: The uncertainty of Exp correspond to the statistical uncertainty determined in section 3.4.With application of Exp,  D 1 cm, 1 cm w ( )becomes )is determined to 45.5 mGy s −1 with an uncertainty of 3.7% (k = 2).It can clearly be seen that k inv has the largest uncertainty contribution.
Using Exp  D 1 cm, 50 cm w ( ) becomes: It was determined to  D 1 cm, 50 cm w ( ) = 3.34 10 −4 Gy s −1 with an uncertainty of 1.4% (k = 2).Its uncertainty budget is presented in table 3. Due to the absence of k , inv the uncertainty of  D 1 cm, 50 cm w ( ) is much smaller than the one of  D 1 cm, 1 cm , w ( )having its largest contribution in the uncertainty of the beam diameter at the measuring volume, r A , due to the uncertainty of the beam aperture.

Discussions
Even in teletherapy greater uncertainties than in many other fields of radiotherapy are generally accepted for skin irradiations.In this respect, the uncertainties in the realization of D 1 cm, 50 cm w ( ) are very small with 1.4% (k = 2).This quantity turns out to be very robust to variations in the directional emittance of the source (3.1).This robustness indicates that this quantity can be transferred by using different kind of sources, probably even conventional x-ray tubes.
However, the drawback of this approach is that to meet the reference condition of the calibration a tube, a collimator, and an ionization chamber with a phantom centered and perpendicular to a symmetry axis needs to be set-up.This is a laborious work since suitable equipment is hardly available in clinics.
During the PRISM-eBT project it became apparent that clinical users would like to see a calibration setup in which the transfer chamber is fixed to the tube using special mounts, so that calibration is practical and safe.The reference condition of D 1 cm, 1 cm w ( )is more suitable for this approach.With an uncertainty of 3.7% (k = 2) this method is sufficient even when little knowledge about the tube is available.Smaller uncertainty can be achieved by a detailed characterization of the source and the creation of a precise MC-model.
It is not clear today if the application of the method for D 1 cm, 1 cm w ( )requires the specific source to be present at the calibration lab.Or if other sources can be used, probably even conventional tubes.If the specific source needs to be present in the calibration lab, this will restrict the dissemination of this quantity.This task will be investigated in future work, especially the comparison with VSL and Ciemat that has been mentioned.

Conclusion
In this work the quantity D 1 cm, 50 cm w ( ) for a 70 kV eBT x-ray source was realized with an uncertainty below 1.4% (k = 2).This quantity is robust against variations and thus imperfect knowledge of the emission characteristic of the source.Additionally, the quantity D 1 cm, 1 cm w ( )was realized.This quantity is more practical for clinical use but more knowledge of the emission characteristic of the source is needed.This quantity was realized in this work with an uncertainty of 3.7% (k = 2).

i()
Since secondary electron equilibrium prevails at plate separation zero, the air kerma, K a , is converted to water kerma, K w , by application of the corresponding mass energy transfer coefficients, photon spectrum at the reference point.k PhW achieves the conversion from the plastic phantom to a water phantom with a 'front plate' of 1 cm thickness.
2.3.Monte Carlo calculations 2.3.1.Software The Monte-Carlo Calculations to determine the correction factors and function were performed with the EGSnrc toolkit (Kawrakow et al 2000).The EGSnrc application 'egs_kerma' was used to determine most of the values (Kawrakow et al 2021) 3 besides k scat which was determined with the User Code CAVRZnrc (Rogers et al 2010).ICRU90 cross-sections implemented in the toolkit were used.The x-ray fluence spectrum of the tube used for the calculations was obtained spectrometrically.

Figure 1 .
Figure 1.Visualization of the applied correction factors: upper panel: measurement set-up; middle panel: the plastic phantom at plate separation zero is replaced by a water phantom; lower panel: the source with applicator tube fitted directly in contact with the water phantom.

Figure 2 .
Figure 2. Sketches of three source models with applicator tube fitted, generated by EGSnrc: left-point source, middle-isotropic source over the area of the focal spot with 15 mm diameter right-iso-tropical source from the same area with a pronounced forward direction limited to 15°from the axis.

Figure 3 .
Figure 3. Function c x i( ) for the three source models: upper panel: point, isotr and isotr 15 ; lower panel: ratio of c x i ( ) relative to the point source.The dashed line represents a 0.5% interval.

Figure 4 .
Figure 4. Calculated values of k scat with their statistical uncertainties (k = 1) from the simulations and model function fitted. i

Figure 5 .
Figure 5. Measured water kerma rate in the phantom of the ipFAC at plate separation zero for seven different combinations of plate separations.

Table 1 .
Calculation of k inv for 4 focus diameters and two beam shapes.