The fundamental dosimetric quantities were defined in detail in ICRU Report 60 [7] and were continually corrected in ICRU Report 85 [21] and Report 90 [8]. Concerning the dosimetric quantity absorbed dose, it is usually derived from the energy loss along a given particle track length and is directly related to the dosimetric quantity particle fluence Φ. The absorbed dose to a medium D _med due to charged particles can thus be calculated by integrating all differential fluences Φ _E,med in a medium med of a given particle type multiplied with the associated electronic collision stopping power L as defined in ICRU 85 [22]. D med = δ − C P E C med = ∫ Φ E , m e d L ρ med d E (1) If there is no delta-particle equilibrium (δ-CPE) the quantity determined is not necessary the absorbed dose but the medium cema C _med according to ICRU Report 60 [7, 17]. Because the cema is defined for all charged particles; such as electrons, positrons, protons, etc., the energy transfer per unit distance has to be specified for the existing particle type. In this study, the secondary electron and positron fluence were considered for the calculation of cema, but note that positrons and electrons are not always explicitly separated by word. In ICRU 90 [8] a modified concept; the restricted or reduced cema C _Δ,med with the correction of track-end terms is defined as: C Δ ,med = ∫ Δ Φ E ,med L Δ ρ med d E + Φ E ,med ( Δ ) L ( Δ ) ρ med Δ (2) Formally, L _Δ from Eq. 1 has to be replaced by the reduced collision stopping power Λ_Δ as described by Kellerer et al. [9]. Nevertheless, the restricted stopping power as an approach for the reduced collision stopping power offers a good approximation for the applied photon energies and materials used in this study [9, 10]. The restricted medium cema is especially interesting because the requirement that δ-CPE exists can be neglected (Eq. (1)). In regards to MC-applications, the quantity restricted medium cema is also particularly useful, because MC-calculated fluence distributions are derived on a discrete energy grid and also limited by the low-energy cut-off value Δ. Electrons with kinetic energies less than Δ were presented by the so-called track-end term T E  med = Φ E , m e d ( Δ ) [ L ( Δ ) / ρ ] med Δ  which accounts for local energy depositions.

According to Hartmann et. al. [10], the cema conversion factor f _cema is defined in analogy to the dose conversion factor f _dose as the ratio between the restricted water cema at the point of interest C _Δ,w and the volume-averaged mean restricted medium cema C ̄ Δ , m e d det in the cavity (-medium) of the detector: f cema = C Δ , w C ̄ Δ , m e d det (3) with C Δ , w = ∫ Δ E max Φ E , w L Δ ρ w d E + T E  w and C ̄ Δ , m e d det = ∫ Δ E max Φ ̄ E , m e d det L Δ ρ med d E + T E  med det where Φ ̄ E , m e d det denotes the spectral charged particle fluence averaged over the cavity volume filled with the medium “med” within the detector and Φ _E,w is the spectral charged particle fluence in water at the point of interest. Note that the first subscript after Δ from C ̄ Δ , m e d det denotes for the medium of the stopping power, i.e. the detector medium. The superscript refers to the surrounding medium. In case the surrounding medium consists of all detector components, a superscript “det” is utilized. If a bare volume is used instead, the superscript is denoted as “w”. If the surrounding medium is identical to the medium, in which the fluence is scored, the superscript can be omitted.

For the decomposition of the cema conversion factor f _cema, it is necessary to combine a stopping power medium with a different fluence medium, resulting in a hybrid version of restricted cema. Therefore a third subscript is used, which refers to the medium in which the fluence is calculated. If the fluence medium is identical to the stopping power medium, both subscripts are summarized into one subscript. The cema conversion factor f _cema can thus be decomposed into a product of sub-factors: f cema = C Δ , w C Δ , m e d , w ⋅ C Δ , m e d , w C ̄ Δ , m e d det = s w,med SA ⋅ C Δ , m e d , w C ̄ Δ , m e d det (4) where C _Δ,med,w is the restricted medium cema in water at the point of interest surrounded by water: C Δ , m e d , w = ∫ Δ E max Φ E , w L Δ ρ med d E + T E  med,w The first term of Eq. (4) is the Spencer-Attix restricted stopping power ratio water to the cavity medium s w,med SA . The second term can be denoted as the global fluence correction factor p _glf that includes all detector-specific changes of the spectral fluence in relation to the undisturbed spectral fluence in water: p glf = C Δ ,med,w C ̄ Δ ,med  det = ∫ Δ E max Φ E ,w d E + Φ E ,w ( Δ ) ∫ Δ E max Φ ̄ E ,med  det d E + Φ ̄ E ,med  det ( Δ ) (5) The global fluence correction factor can further be decomposed into p glf = C Δ , m e d , w C ̄ Δ , m e d w ⋅ C ̄ Δ , m e d w C ̄ Δ , m e d det (6) with the internal fluence correction factor p _int p int = C Δ , m e d , w C ̄ Δ , m e d w (7) which corrects for cavity-specific influences only and p _ext p ext = C ̄ Δ , m e d w C ̄ Δ , m e d det (8) for external fluence perturbations due to different detector components. The external fluence correction factor p _ext can also be decomposed into the correction factors from specific constructive detector components for example: p _cel for the perturbation caused by the central electrode in an ionization chamber, p _stem for the perturbation related to the detector stem and p _wall for the perturbation due to the wall (Figure 1). To investigate those specific external fluence correction factors, a decomposition technique introduced by Wulff et al. [18] and Zink and Wulff [22] can be applied. Note that this approach was originally developed for the calculation of dose conversion factors derived from dose ratios. In this study, the restricted cema ratios, respectively the fluence ratios were used. A schematic description of the decomposition technique is shown in Figure 1: The specific fluence correction factors are derived by calculating the cema in the sensitive volume of the ionization chamber while specific components are being removed. In the first step, the volume-averaged restricted medium cema in the detector cavity is calculated while all constructive components are being considered ( C ̄ Δ , m e d det ) . The second simulation is performed without the central electrode to determine the volume-averaged restricted medium cema in the sensitive detector volume surrounded by all detector components except the central electrode C ̄ Δ , m e d cel . The fluence correction factor p _cel can be calculated subsequently through the ratio of C ̄ Δ , m e d cel / C ̄ Δ , m e d med and accounts for the fluence perturbation caused by the central electrode. The chain of decomposition can be continued by stepwise removing other detector components and comparing the resulting cema values.

The internal fluence correction factor can be further decomposed into a product of fluence correction factors p _int = p _dis ⋅ p _vol ⋅ p _ρ , where p _dis corrects the effects due to the displacement of water, p _vol corrects the volume-averaging effect and p _ρ accounts for the perturbation due to the different physical density of medium “med” in comparison to water (Figure 2). If a measurement is performed with a shift of the effective point of measurement (EPOM), it can be assumed that p _dis is equal one [6]. Note that the fluence correction factor p _fl accounting for the influence of different atomic compositions on the charged particle fluence was not separately determined in this study [13]. To analyze the spectral behavior of the internal fluence correction factor p _int and its sub-factors p _vol and p _ρ under charged particle dis-equilibrium, a determination method by Wang and Rogers [19, 20] was used. Therefore, a simulation in a cylindrical volume with a length of 0.48 cm and different radii, placed with their shifted reference point at 0.5 cm and 1.5 cm depth in the water phantom, was performed. The first simulation contains a volume filled with “low-density water” (ldw), a virtual material with a physical density of air and a restricted stopping power of water. The second simulation was performed with a water-filled cavity. The spectral density correction factor p _ρ is thus given as the ratio between the calculated cema values from both simulations. A schematic description of the determination of internal fluence correction factors is shown in Figure 2. It should be pointed out that almost all simulations with a sensitive medium different than liquid water were performed with a shift of EPOM 0.5 ⋅ r _cav away from the focus, where r _cav is the internal radius of the chamber cavity, to compensate the influence of the displacement effect according to the DIN6800-2 [6]. In the case of the ldw-filled cylindrical volume as shown in Figure 2, the spectral fluence was once calculated with and once without the shift of EPOM to investigate its influence on the spectral fluence correction factors.

The decomposition in external and internal effects as described above is very similar to Bouchard’s decomposition following a sequence of detector components. This formulation is consistent with the classical cavity theory-based factorization of the dose conversion f dose = D med det / D w for ionization chambers following [1]. 1 f dose = f cema = s w,med SA ⋅ ∏ i p i (9) A major advantage of the cema formulation is that all named correction factors can be expressed as a function of energy p _E,i . For example p E , g l f ≔ Φ E , w Φ ̄ E , m e d det (10) the energy-independent value of p _glf can thus be calculated through the fluence-weighted mean-value of p _E,glf, i.e., p glf = ∫ Δ E max p E , glf ⋅ Φ ̄ E , med det d E + { p E , glf ( Δ ) ⋅ Φ ̄ E , med det ( Δ ) } ∫ Δ E max Φ ̄ E , med det d E + Φ ̄ E , med det ( Δ ) (11)

Under non-reference conditions the determined dose has to be corrected by the conceptual foundation included in the current DIN 6800-2 dosimetry protocol [6]. The factor k _NR corrects any changes of the beam-quality correction factor k _Q due to different field size FS, off-axis distance R, depth z or source-to-surface distance SSD compared to the geometrical reference conditions with k NR = k Q z , F S , R , S S D k Q 10 c m , 10 c m × 10 c m , 0 c m , 100 c m (12) and can also be expressed by k NR = D w / D med NR D w / D med R . (13) Analogously to the dose conversion factor D _w/D _med, a cema conversion factor C _w/C _med, introduced by Hartmann et al. [10] can be used to calculate the non-reference correction factor k _NR: k NR = δ − C P E C w / C med NR C w / C med R (14) In this study, k _NR is calculated as a function of depth z.

The simulations were performed with a modified version of the EGSnrc user code egs_chamber [22]. The charged particle fluence scoring was adopted from the user code cavity [23] and extended using the track-end correction method III as described in [24]. A short overview of the fluence calculation method is given in [10, 24]. The choice of the Monte Carlo user code egs_chamber is motivated by high usability for dosimetric calculations of air-filled ionization chambers. In a previous study [25] and internal work reports the compatibility of all used variance-reduction techniques was confirmed. With the modified user code, the absorbed dose D, the spectral electron as well as the positron fluence Φ ̄ E were calculated for two thimble air-filled ionization chambers (PTW type 31021 Semiflex 3D and Sun Nuclear SNC125c™). Details on selected detector properties and cross sections of the used MC-models can be found on the manufacturer’s websites and in Figure 3. All quantities were also calculated for a small cubic water voxel of 1 mm side-length. The restricted spectral cema C _Δ,E was calculated using Eq. (2). Following the formalism introduced by Hartmann et al. [10], the global fluence correction factor p _E,glf (Eq. (5)), the cema conversion factor f _E,cema (Eq. 4) and its sub-factors p _i,E (as described in Figures 1, 2)- all differential in energy -were derived by the differential charged particle fluence Φ _E and the Spencer-Attix restricted stopping-power ratio s w,a SA at different depths in a cubic water phantom with 30 cm side length. Due to the fact that the positron fluence proportion is usually in the order of a few percent of the total charged particle fluence Φ, all shown energy-dependent p _E,i in Section 3 are calculated for electrons. The total fluence correction factors were calculated by taking into account the positron fluence according to Eq. (11). All detectors were oriented with their axis perpendicular to the beam axis and placed with their reference point at depth from 0.5 cm to 20 cm, considering the shift of the EPOM according to DIN6800-2 [6]. The phantom was irradiated with a 6 MV linear accelerator x-ray spectrum taken from Mohan [26] from a collimated point source with a field size of 10 cm × 10 cm and an SSD of 100 cm. The value of the common photon beam quality specifier TPR _20,10 at a 10 cm × 10 cm field size amounts to 0.671 using the approach Followill et al. [27]. The non-reference correction factors k _NR for different depths in water are calculated using Eq. (14).

For all calculations, a low-energy cut-off limit from 1 keV kinetic energy was used. The lower limit of the spectral fluence calculation was also set to 1 keV as well as the energy resolution of the spectral charged particle fluence but all calculated ratios in Figures 5–10 have been averaged over a logarithmic bin size from 1 keV to 100 keV in order to reduce statistical fluctuations. The Russian Roulette (RR) and cross-section enhancement (XCSE) were used to decrease the statistical uncertainties and gain efficiency due to reduced CPU runtimes. The RR-Parameter ESAVE was therefore set to the ECUT-value of 512 keV to produce an accurate spectral fluence distribution. For the RR—and XCSE-Region, the detector was covered with a cylindrical volume with a +1 cm shell thickness filled with water. For depths 0.5 cm and 1 cm the RR- and XCSE-region was extended as far as the surface of the water phantom was reached. The general simulation settings, according to the recommendations of AAPM TG-268 [29] are shown in Table 1.

In Figure 4A the undisturbed electron fluence spectrum Φ _E,w is shown at different depths in water evoked by the irradiation with a 6 MV linear accelerator x-ray spectrum taken from Mohan [26]. For clarity reasons, the spectral electron fluence in 2 cm, 5 cm and 20 cm water depth were omitted in Figure 4. Qualitatively, it can be seen that Φ _E,w is depth-dependent while the highest fluence occurs at a water depth of 1.5 cm. For energies above 4 MeV, the spectral electron fluence decreases by several orders of magnitude and converges to zero. The relative secondary electron fluence at depth z normalized to 1.5 cm water depth given in Figure 4B revealed that Φ _E,w changes differently in energy depending on depth in water: In the build-up region, the relative electron fluence spectrum systematically depends on energy; especially in the high-energy range whereas at 10 cm water depth only slight spectral fluctuations occur.

In Figure 5 the spectral internal fluence correction factor p _E,int for a “low-density water”-filled cavity of different radii and 0.48 cm length is shown as a function of charged particle energy in 0.5 cm water depth irradiated by a 6 MV linear accelerator x-ray spectrum [26]. Because the positron fluence proportion is > 2% of the total charged particle fluence Φ, all shown energy-dependent p _E,i in Section 3 are for electrons. Nevertheless, the positron fluence was included for the calculation of the energy-independent total fluence correction factors p _i as tabled in Table 2 and Table 3. It is important to note that a correction factor p _i smaller than unity indicates that the charged particle fluence in the bare volume Φ ̄ med w is greater compared to the undisturbed water fluence Φ_w. In Figure 5A the spectral internal fluence correction factor is shown for a cavity, whereby the reference point was shifted by 0.5 ⋅ r _cav away from the focus to compensate the influence due to the displacement of water. It can be seen that the internal fluence correction factor p _E,int for the energy spectrum from 0 to 1 MeV is smaller than unity for all cylinder radii (with only one exception for r = 0.4 cm, in the spectrum from 0 to 5 keV). For energy values above 1 MeV, the internal fluence correction factor is increasing. The largest deviation of p _E,int from unity can be observed for the cavity with the largest radius.

To verify the influence of the displacement effect in the build-up region, a simulation without the shift of the EPOM was performed. Comparing the spectral internal fluence correction factor p _E,int from Figure 5A, B, indicates that the displacement effect is nearly compensated by the positioning of the cavity 0.5 ⋅ r _cav away from the focus; in particular for energies below 1 MeV. The increase of p _E,int from Figure 5A for energies > 1 MeV hence can be seen inter alia as an effect of uncompensated displacement. Thus it appears that in the proper meaning the fluence correction factor for the density effect p _E,ρ given in Figure 6B and hence the internal fluence correction factor p _E,int shown in Figure 5A includes the correction of displacement too.

According to the procedure shown in Figure 2, the spectral internal fluence correction factor p _E,int from Figure 5A was factorized into the subfactors p _E,vol and p _E,ρ (Figure 6). The fluence correction factors for the volume-averaging effect p _E,vol shown in Figure 6A is approximately constant for all energies and varies with the particle’s energy likely by 1%–3%. Generally p _E,vol is greater than unity for all cylinder radii. As shown in Figure 6B the calculated density correction factor p _E,ρ is below unity for energies from 0 to 1 MeV and greater than unity for energies > 1 MeV for all investigated cylinder radii.

In Figure 7A the Monte Carlo calculated internal fluence correction factors p _int as well as the factorization into the subfactors p _ρ and p _vol are shown as a function of the cavity radius. For a water depth of 0.5 cm a systematic dependency on the cavity radius is observable: on the one hand p _vol is increasing with increasing cavity radii. Conversely p _ρ is decreasing almost by the same margin as p _vol is increasing. For a cavity radius of 0.5 cm both effects have a maximum deviation from unity of below 5‰. The volume-averaging correction factor p _vol prevails over the density effect correction factor p _ρ for all cavity radii. Consequently the internal fluence correction factor p _int is greater unity for all investigated cavity radii. The fluence correction factors for 1.5 cm water depth are shown in Figure 7B. All fluence correction factors differ from unity with a maximum deviation of 0.4%, while p _vol is greater; and p _ρ is below unity. The volume correction factor p _vol is only slightly increasing depending on the cavity radius whereby the internal fluence correction factor is increasing by a few tenths of a percentage too. The density effect correction factor p _ρ remains constant within the range of uncertainties.

In Figure 8 the global fluence correction factor p _E,glf and the cema conversion factor f _E,cema differential in energy at different depths in water are shown for cylindrical ionization chambers irradiated with a 6 MV linear accelerator x-ray spectrum taken from Mohan [26]. A value of p _E,glf below unity implements that the particles fluence in the ionization chamber Φ ̄ med det is greater compared to the undisturbed fluence in water Φ_w, which corresponds to an over-response of the detector. Concerning the global fluence correction factor p _E,glf as shown in Figure 8, a systematic dependency on secondary electron’s energy is observable: In the low-energy spectrum at reference conditions the global fluence correction factor p _E,glf is increasing until it reaches a maximum at 0.2 MeV. For the PTW type 31021 p _E,glf is decreasing for energies above 0.2 MeV, resulting in a relative minimum at approximately 1 MeV. Compared to the PTW 31021, the relative minimum is only slightly pronounced for the SNC125c™. For high energies, the spectral global fluence correction factor p _E,glf at reference conditions converges to unity. The detector response from ionization chambers is also depending on depth in water: For energies below 2 MeV, the over-response is increasing with decreasing water depth. At 0.5 cm water depth, p _E,glf for high energies is increasing resulting in an under-response of the ionization chamber.

As described in Section 2.2, the cema conversion factor f _E,cema is calculated by multiplying with the Spencer-Attix restricted stopping-power ratio s w,med SA differential in energy. For air-filled ionization chambers, f _E,cema is greater than unity for all energies, whereby it varies in a range of 0.2% in the low-energy spectrum.

In Figure 9A the specific fluence correction factors for constructive details from the PTW type 31021 as a function of energy are shown for a depth of 0.5 cm in water. The fluence correction factor for the perturbation caused by the central electrode p _E,cel is quite constant as a function of energy and differs from unity in a range of 2% for high electron energies. The wall perturbation correction factor p _E,wall changes marginally by ∼1% in the energy range from 0 to 1 MeV and decreases for energies below 1 MeV. On the contrary, the fluence correction factor for the influence of the stem p _E,stem is smaller than unity and shows a substantial decrease in the energy spectrum from 0.2 MeV to 1 MeV with a maximum deviation from unity from nearly 3%. As can be seen in Figure 9A, the internal fluence correction factor depends on energy and follows the course of p _E,int shown in Figure 6 with only one exception: An decrease occur in the low-energy spectrum.

Concerning the spectral specific fluence correction factors p _E,i at 10 cm water depth as shown in Figure 9B, the wall and central electrode correction factors stay nearly constant as a function of energy compared to 0.5 cm water depth. The energy-independent total correction factors are differing from their values at 0.5 cm water depth by only 0.8%. The deviation from unity from the stem correction factor is decreasing, resulting in a fluence-weighted mean-value of p _stem with an amount of 0.998 while the decrease in the energy spectrum from 0.2 MeV to 1 MeV is less noticeable. The spectral internal fluence correction factor p _E,int for a water depth of 10 cm converges to unity with increasing energy, leading to a total p _int of ∼1.

The spectral specific fluence correction factors for the perturbation caused by constructive details from the SNC125c™  are shown in Figure 10. Especially in the case of the fluence disturbance due to the detector stem, spectral differences between the Semiflex 3D and the SNC125c™  are observable: While the stem perturbation correction factor p _E,stem for the PTW 31021 depends on the energy, it is approximately constant as a function of energy for the SNC125c™. On the contrary, the spectral fluence perturbation caused by the wall and the central electrode looks quite similar. All resulting specific fluence correction factors p _i are shown in Table 2. Note that the internal correction factor p _int based on the dose conversion factors includes the Spencer-Attix restricted stopping power ratio water to air s w,air SA whereas it is excluded in the fluence-based approach on the right-hand side.

All calculated values for cylindrical air-filled ionization chambers are summarized in Table 3. Note that p _glf is numerically equal to p _Q. The k _Q value for the PTW 31021 was calculated using the cema approximation k _Q = f _cema(TPR = 0.671)/f _cema(⁶⁰Co) in 10 cm depth with f _cema(⁶⁰Co) = 1.127. The result amounts to 0.986 and agrees within the uncertainty of 0.3% given for the data of the consensus k _Q values for MV photon beams updated in the IAEA TRS-398 [32].

In Table 3 the calculated p _glf and k _NR (see Eq. (14)) for air-filled cylindrical ionization chambers are shown as a function of depth. The global fluence correction factor p _glf for the PTW 31021 varies within a range of 3%, the non-reference correction factor increases depending on the chamber type roughly by the same percentage. In comparison, the SNC125c™  have a slighter influence on the undisturbed charged particle fluence Φ_w in water at 0.5 cm depth, whereas it changes only by an amount of < 2%. The resulting non-reference correction factor k _NR is depth-dependent and changes in the order of ∼ 3 %.

In the following, it can be summarized that considerable changes in the spectral electron fluence in water occur in the low-energy spectrum due to the presence of the investigated detectors, especially in the build-up region. According to Eq. (2), the restricted medium cema C _Δ,med is calculated by integrating the spectral electron and positron fluence in the medium Φ_med weighted by the associated restricted stopping powers L _Δ,med. Hence disturbances in the spectral electron fluence particularly in the lower energy range are more meaningful with respect to the determined cema.

Our results have shown that even small changes in depth can have major effects on the differential charged particle fluence in the ionization chamber in the build-up region, mainly due to internal effects. The internal fluence correction factor can be further reduced to the product of the density effect p _E,ρ and the volume-averaging effect p _E,vol. It could also be shown that the displacement effect of water is nearly compensated by the positioning of the chamber 0.5 ⋅ r _cav away from the focus.

In the build-up region, the volume-averaging effect p _E,vol is leading to a constant lower electron fluence Φ ̄ E , w in the cavity compared to the fluence Φ _E,w in water for all energies especially with increasing cavity radii. On the contrary, the lower density of air compared to water is leading to an over-response in the fluence spectrum below 1 MeV. This over-response might be explained by the lateral scattering of secondary particles in the cavity and the significantly greater range of charged particles in air compared to water. For high electron energies, an under-response occurs, which might be explained by (1) the depth-depending variations in the electron spectrum in water (Figure 4) in combination with (2) the partially uncompensated displacement effect.

1. As shown in Figure 3, distinctive fewer electrons in water exist in the energy range above 0.5 MeV in the build-up region compared to the spectrum at 1.5 cm water depth. The interpretation of the resulting leak of high-energy electrons is not so intuitive but might be explained in a qualitative manner by the energy-dependent existence of charged particle equilibrium (CPE): Per definition, CPE exists for a volume if each charged particle of a given type and energy leaving the cavity is replaced by an identical particle with the same physical properties [2]. Thus CPE can only be achieved at a depth z equal or greater compared to the linear continuous slowing-down range R _CSDA of the electron with energy E. The existence of CPE hence in fact depends on the kinetic energy E of secondary charged particles and the given medium. Considering a water-filled cavity the fluence distribution consists of electrons that are getting created inside the cavity (“starters”) and those entering the cavity from outside (“crossers”) [12]. If the bare volume is filled with air, the charged particle fluence is merely due to particles entering the cavity from outside and crossing it completely (Bragg-Gray condition). The increased CSDA-range R _CSDA due to the finite size of the air-filled cavity and the assumption that no electrons are created inside might lead to a lower fluence in the high-energy spectrum in the air-filled cavity compared to the water-filled one.

Next to this effect, the depth-dependent secondary charged particle fluence spectrum is leading to variations in the restricted stopping power ratios s w,air SA from ∼ 0.3 % like already demonstrated by [6, 18].

The internal fluence correction factor as discussed in Section 4.1 was calculated for a volume with approximately similar dimensions compared to the size of the sensitive volume from the PTW type 31021 Semiflex 3D. It is therefore possible to compare the course of both internal fluence correction factors as a function of energy at 0.5 cm water depth (compare p _E,int for r _cav = 0.25 cm shown in Figure 5A with p _E,int from Figure 9A). If the volume is filled with air (Figure 9A), p _E,int increases in the energy range up to 100 keV. As can be seen in Figure 5A, where the volume was filled with low-density water the increase of the internal fluence correction factor is neglected and p _E,int is approximately constant. The only difference between both simulations is that the stopping-powers (on the one hand L _Δ,w and on the other hand L _Δ,air) differ from each other due to diverse electron densities n (∝ Z/A) and mean excitation potentials I (∝ Z) of the used media. Concerning the physical density and the size of the volume, they were quasi identical. Bouchard et al. [13] pointed out that various atomic compositions of the detector medium will cause the energy absorption to differ due to differences in the interaction cross-sections. This effect primarily influences the fluence of low-energy charged particles and was denoted by Bouchard as p _fl. The influence of different stopping-powers and atomic compositions (i.e., p _fl) on the fluence calculation might be a possible explanation for the increase of p _E,int from Figure 9A. However, the conclusions about concerning p _E,int from Section 4.1 can also be applied to air-filled ionization chambers.

With increasing complexity of the simulation geometry, it becomes difficult to derive a specific effect due to the superimposition of several influencing factors on the spectral charged particle fluence. As described in Figure 1 the decomposition into specific external effects can help to analyze the perturbation caused by different detector components on the spectral fluence in the detector. As shown in Figure 8 the spectral global fluence correction factor p _E,glf is smaller than unity in the energy range from 0 to 100 keV leading to an over-response in the low-energy fluence spectrum. Figure 9, 10 showed that the presence of the central electrode is resulting in a constant higher secondary charged particle fluence in the detector cavity Φ ̄ E , a i r det in comparison to the undisturbed water fluence Φ _E,w. The over-response in the low-energy fluence spectrum is also enhanced by the production of secondary charged particles in the wall and especially in the stem. In the medium energy range from 100 keV to 2 MeV a decrease of the global fluence correction factor p _E,glf occurs, which can be assigned to the production of medium energy electrons due to the stem and the scattering of those into the sensitive volume. Also the fluence correction factors p _E,cel and p _E,wall are decreasing in the build-up region, whereby they remained constant within the range of uncertainties at 10 cm water depth. Interestingly, the fluence perturbation caused by the stem seems to be smaller for the SNC125c™  compared to the PTW type 31021 Semiflex 3D. The smaller length of the sensitive volume from the Semiflex 3D in comparison to the SNC125c™  might be the reason for the higher influence of the stem on the charged particle fluence distribution in the PTW type 31021 Semiflex 3D. In the high-energy spectrum with energies greater than 2 MeV, the global fluence correction factor in the build-up region is mainly affected by the described internal effects mentioned in Section 4.1. At 10 cm depth, p _E,int for electron energies greater than 0.6 MeV is nearly constant for both investigated ionization chambers and differs from unity by only 0.2%.

Concerning the resulting total stem correction factor p _stem for the PTW type 31021 calculated in this study, it is 0.998 at 10 cm which is in good agreement with findings from Wulff et al. [15, 18] for the PTW type 31010 Semiflex. Furthermore, the relatively high proportion of aluminum in the PTW 31021 stem model in comparison to the idealized stem used by Wulff et al. [15, 18] seems to have no influence on the total stem correction factor at all. At 0.5 cm depth, the stem correction factor p _stem showed an amount of 0.987 and is next to the internal fluence correction factor one of the main reasons for the significantly pronounced over-response in the build-up region. The total fluence correction factor for the central electrode at 0.5 cm depth is in good agreement with simulated values from Wulff et al. [15]. In addition, the total fluence correction factor for the influence of the chamber wall on the charged particle fluence was found to be 0.2% larger.

Multiple studies have been investigating the influence of non-reference conditions especially in the case of small fields [33–35]. A changed detector response for the determination of depth-dose curves or relative distributions for photon beams with ionization chambers is usually not taken into account and the correction factor k _NR is assumed to be unity. There are also several studies [36, 37] that reported differences in the detector response in the build-up region depending on the choice of the EPOM. The study of Pena et al. [36] assumed that if a measurement of a percentage depth-dose curve is performed without a shift of EPOM, further depth-dependent conversion factors should be used accounting for the variations in the depth-dependent detector response. Wegener et al. [37] could reduce the influence of different detector responses on relative measurements by using detector-individually derived effective points of measurement.

This study showed that the non-reference correction factor k _NR is depth-depending and varies likely by 3% for the Semiflex 3D and 2% for the SNC125c™  due to different values of the specific fluence correction factors p _i from constructive details and due to the depth-depending variations in the Spencer-Attix restricted stopping power ratios in an order of approximately 0.3% (Table 3). Mainly, the spectral differences in the build-up region are due to the energy-dependent dis-equilibrium and the involved steep gradients which are responsible for the variations in the detector response under non-equilibrium conditions. Under equilibrium conditions in depths behind the dose maximum, k _NR for air-filled ionization chambers varies likely by 0.2% with increasing water depth. Concerning the non-reference correction factors k _NR for air-filled ionization chambers found in this study displayed good agreement with findings from Wulff et al. [15]. In further studies, the existence of δ-CPE in the build-up region and thus the validity of Eq. (14) should be investigated.

The results of this study have shown that in high-energy photon beams, the shift of EPOM is always a compromise solution and does not offer an ideal correction for all depths at the build-up region. Due to the non-linear depth-dependency of p _glf or k _NR shown in Table 3, it could make sense to determine the absorbed dose to water at the build-up region without the shift of EPOM and use instead detector-individually derived depth-dependent correction factors as described by Pena et al. [36]. Those depth-dependent correction factors can be obtained for individual depths and beam qualities via Monte Carlo simulations.