On the measurement of scaling factors in the RW3 plastic phantom during high energy electron beam dosimetry

Ionometric electron dosimetry inside water-equivalent plastic phantoms demands special considerations including determination of depth scaling and fluence scaling factors (cpl and hpl) to shift from in-phantom measurements to those relevant to water. This study evaluates these scaling factors for RW3 slab phantom and also introduces a new coefficient, k(RW3), for direct conversion from RW3 measurements to water without involving scaling factors. The RW3 solid phantom developed by the PTW Company was used and the corresponding scaling factors including cpl, hpl, and k(RW3) were measured for conventional electron energies of 4, 6, 9, 12, and 16 MeV. Separate measurements were performed in water and the RW3 slab phantom using the Advanced Markus chamber. The validity of the reported scaling factors was confirmed by comparing the direct and indirect percentage depth dose (PDD) measurements in water and in the RW3 phantom. The cpl values for the RW3 phantom were respectively equal to 0.915, 0.927, 0.934, 0.937, and 0.937 for 4, 6, 9, 12, and 16 MeV electron energies. The hpl and k(RW3) values were dependent on the depth of investigation and electron energy. Application of the cpl−hpl factors and k(RW3) coefficients to measured data inside the RW3 can reliably reproduce the measured PDD curves in water. The mean difference between the PDDs measured directly and indirectly in water and in the RW3 phantom was less than 1.2% in both approaches for PDD conversion (cpl−hpl coupling and the use of k(RW3)). The measured scaling factors and k(RW3) coefficients are sufficiently relevant to mimic water-based dosimetry results through indirect measurements inside the RW3 slab phantom. Nevertheless, employing k(RW3) is more straightforward than the cpl−hpl approach because it does not involve scaling and it is also less time-consuming.


Introduction
Cancer treatment using radiotherapy, often as an adjuvant modality after surgery, is now regarded as a standard approach for improved patient management [1,2]. Different types of ionizing radiations, including high energy photons (such as X-rays and gamma rays), high energy electrons, and hadron beams (e.g. proton or Carbon ion) can be employed for radiotherapy purposes [3][4][5]. The type of ionizing radiation selected for tumor irradiation depends on factors such as tumor location, size of the radiation field (region to be covered by the ionizing radiation), treatment phase (main treatment or boost), treatment sessions, etc.
High energy electron beams are widely used for radiotherapy using different techniques such as external irradiation of superficial tumors, intraoperative radiotherapy (IORT) of the tumor bed after surgery in the operating theater, and total skin irradiation [6][7][8][9][10]. Although the use of electron beam for IORT purposes has been promoted in recent years, the most common application of electron beams is external radiotherapy of superficially distributed tumors such as skin cancers [11,12]. In this regard, dedicated electron accelerators have been introduced that produce clinical electron beams in the energy range of 6-24 MeV [13].

3
Electron beams for clinical purposes require an appropriate ionometric calibration including dose rate determination (usually in terms of cGy per Monitor Unit (MU)) as well as the dosimetric characterization of the clinical beam such as depth dose distribution and transverse dose measurement using a proper ionization chamber dosimeter [14].
As recommended by dosimetry codes of practice [15,16], the reference dosimetry medium for absolute and relative dose measurements of the electron beam is water. On the other hand, employing water phantoms for dosimetry may encounter some practical issues which may limit their application in clinical electron dosimetry [17]. For example, when the positioning accuracy of the ionization chamber dosimeter during the movements inside water is the main concern (e.g. when the dosimetry is performed near the surface region), the results are always influenced by an inherent uncertainty regarding the surface tension effect as well as by the dosimeter size itself [18,19]. Besides, when the ionization chamber dosimeter is not water-proof, employing the water phantom for dosimetry would be impractical [19][20][21]. Finally, motorized water phantoms and their relevant accessories are generally expensive and some radiation oncology departments may not have access to such automatic dosimetry equipment because of economic restrictions. In such situations, alternative solid phantoms such as solid water, plastic water, virtual water, PMMA, polystyrene, etc. which are nominally equivalent to water may be employed [20,[22][23][24]. However, due to a slight difference in physical density and effective atomic number respect to water, the spatial, angular, and energy distribution of the electron beam inside the plastic phantom may be different from what would be observed in water at the same physical depth [16,19]. Indeed, these differences may affect the results of electron dosimetry which would be different when the electron dosimetry is performed in water compared to its equivalent solid phantom. To account for such discrepancies and obtain convergent results from different dosimetry media (water and solid phantom), two specific scaling factors namely the depth scaling factor (c pl ) and the fluence scaling factor (h pl ) have been introduced [16]. Accordingly, these two factors should be accurately determined for each solid phantom employed in electron dosimetry in order to obtain robust results which can be reliably attributed to the absorbed dose measurement in water. The c pl and h pl scaling factors have been quantitatively determined for different solid phantoms by adopting ionometric dosimetry and/or the Monte Carlo simulation approach [22,[25][26][27].
One of the introduced water-equivalent slab phantoms for electron beam dosimetry is the RW3 solid phantom. This phantom is structured in slab form with different thicknesses that mimic the water media for electron dosimetry purposes at different depths and lateral distances. This study aims to quantitatively evaluate the depth and fluence scaling factors for a water-equivalent slab phantom in different conventional electron energies through an ionometric dosimetry approach and following the recommendations of the IAEA TRS-398 reference dosimetry protocol. The measured data would be compared with the data reported previously for other waterequivalent solid phantoms.

Methodology
The RW3 slab phantom The RW3 phantom used in the current study was the type developed by the PTW Company. This solid state phantom has a slab structure and can be used for dosimetry in high energy photon and electron beams ranging from 60 Co to 25 MV and 4 to 25 MeV, respectively. The phantom is made of a water-equivalent martial known as Goettingen White Water. As declared by the manufacturer, this phantom is suitable for monitor calibration (dose-rate measurements in terms of cGy/MU) and quality assurance purposes such as depth dose measurements related to the clinical linacs [28].
This slab phantom is composed of Hydrogen (H), Carbon (C) Oxygen (O), and Titanium (Ti) with elemental mass percentages of 7.59%, 90.41%, 0.80%, and 1.2%, respectively [29]. Considering this elemental composition, the effective atomic number (Z eff ) for this solid state phantom would be equal to 7.13 according to the formula used in the Oshima et al. study [30].
The RW3 solid phantom includes one plate with a thickness of 1 mm, two plates that are 2 mm thick each, one plate that is 5 mm thick, and 29 plates having a thickness of 10 mm each. This slab thickness configuration allows for depth dose measurements down to a depth of 10 cm with 1 mm increments. Each slab is accurately machined to keep the thickness tolerance below 0.1 mm and is prepared in a square geometry with the dimensions of 30 × 30 cm 2 . Therefore, the whole size of the RW3 solid phantom would be equal to 30 × 30 × 30 cm 3 [28].
The phantom's physical density was determined according to the recommendation of the IAEA TRS-398 reference dosimetry protocol [16]. In this regard, all slabs were taken into account and their densities were measured by dividing the weight of each slab by its volume. A calibrated balance was used for weight measurements.

c pl and h pl measurement
The c pl is a factor that scales any depth inside the water to its equivalent depth in the solid phantom through the following equation [16]: where R 50,ion,w and R 50,ion,pl are the depths at which the percentage depth ionization (PDI) curve reaches its half maximum value in water and in the solid phantom (RW3 phantom), respectively. It is worth mentioning that R 50,ion,w and R 50,ion,pl should be characterized in terms of density thickness. The density thickness is obtained through multiplying the physical thickness by the material density (expressed as g.cm −2 ).
Using c pl , any depth inside the plastic phantom can be converted to its equivalent depth in water by Eq. (2) [16]: It should be noted that Z w and Z pl in Eq. 2 are considered in terms of g cm −2 .
Even though the depth scaling factor may result in equivalent depths in water and plastic phantom, the chamber readings at equivalent depths in water and in the solid phantom may not be the same. This issue is mainly linked to the differences between the electron fluence spectra at equivalent depths in water and water equivalent solid phantom. To take this effect into account, the fluence scaling factor (h pl ) was introduced. As explained by the IAEA TRS-398 dosimetry protocol [16], h pl can be described by the following equation: where M Q,w (Z w ) and M Q,w (Z pl ) are correspondingly the chamber reading (often in terms of nC) at the depth of Z w in water and the equivalent scaled depth of Z pl (as described in Eq. (2)) in the solid state phantom. By having the h pl at different depths inside the solid phantom, the user can covert the chamber response inside the solid phantom to the response measured at the equivalent depth in water by applying Eq. (3) and therefore, compensate for the effect of the electron fluence spectra variations in the final electron dosimetry results. Therefore, by having these two factors one can mimic the depth dose data obtained in water through the ionometric electron beam dosimetry inside the plastic phantom.
The electron beam produced by a Varian Trilogy medical linac was employed for determining the depth and the fluence scaling factors. c pl and h pl were measured for nominal electron energies of 4, 6, 9, 12, and 16 MeV. All measurements were performed in the reference irradiation conditions of 100 cm SSD (source to surface distance), and a field size of 10 × 10 cm 2 (up to 12 MeV electron energy and 20 × 20 cm 2 for higher electron energies) at the phantom surface.
(1) c pl = R 50, ion,w R 50, ion,pl To measure the c pl , an Advanced Markus ionization chamber (TM34045, PTW, Germany) was used as a field detector. The ionization chamber was positioned inside an automatic MP3-M motorized water phantom in such a way that its reference point (which is on the inner surface of the chamber entrance window at the midpoint and 1.3 mm below the chamber protection cap) was accurately located at the phantom surface. Then, the ionization chamber was moved to further distances from the surface (different depths based on the employed electron energy during the irradiations) and PDI curves at different electron energies were measured through the upward movement of the chamber at 1 mm increments. The data acquired were then analyzed by Mephysto Navigator software (PTW, Germany) to obtain the R 50,w values at different electron energies of 4, 6, 9, 12, and 16 MeV. It should be noted that a Semiflex ionization chamber (TM31010, PTW, Germany) was also used as a reference detector (in the corner of the field) in order to compensate for the effects of accelerator output variations during the PDI measurements [31]. Both field and reference detector response readouts (in terms of collected electric charge) were measured by a dual-channel TANDEM electrometer. The operating voltage for both chambers was set to + 400 V. Advanced Markus chamber movements in the water phantom were manipulated by a TBA controller unit. PDI measurements were taken along the electron beam central axis according to the recommendations of the IAEA TRS-398 and AAPM TG-106 dosimetry protocols [16,31].
To measure the R 50,pl , the same procedure as that used in water was followed. The main difference with respect to the water measurements was that during the measurements inside the RW3 solid phantom the depths were changed manually. In this regard, the Advanced Markus ionization chamber was located inside its dedicated adoption plate and the PDI measurements at different electron energies were made from the depth of 5 mm, through the successive adding of 1 mm RW3 thicknesses above the ionization chamber dosimeter. It should be mentioned that a 10 cm thick slab phantom was also located beneath the ionization chamber in order to fully satisfy the electron backscattering conditions. In-plastic (RW3) measurements were performed by two separate digital electrometers. The first (Unidos Webline, PTW, Germany) was connected to the field detector (Advanced Markus) and the second (Unidos E, PTW, Germany) to the reference detector (Semiflex chamber). The data obtained at different electron energies were then analyzed with Mephysto Navigator software to find the corresponding R 50,pl (in terms of g.cm −2 ). Once the R 50,w and R 50,pl are known for various electron energies, the corresponding c pl values can finally be obtained according to Eq. (1).
To measure the h pl values at different depths, the relevant PDI curves, measured in water and in the RW3 solid phantoms, were employed. In this regard, the chamber readings at equivalent depths (by scaling the depths by the c pl factor) in water and in the RW3 phantoms were extracted and relevant h pl values were consequently determined according to Eq. (2).

Measurement of the k(RW3)
A novel scaling parameter, named k(RW3), was also introduced in this study to achieve the depth dose curves in water by measuring the PDI curves in the RW3 solid phantom. The k(RW3) was defined by the following equation: where M Q,w (Z) and M Q,w (Z) are the ionization chamber readouts in water and in the RW3 solid phantom respectively, at the same depth (Z). The key point in Eq. (4) is that the M Q values are the chamber readings at the same physical depth (in terms of cm) not the density thickness. With this simplification, there is no need to consider scaling factors for depth equivalence (no need for a depth scaling factor and density of the considered solid state phantom). In addition, there is also no need to consider the fluence scaling factor in this case because the deviations between measured M Q values at the same physical depths in water and in the RW3 slab phantom may directly reflect the differences between the corresponding electron fluence spectra. According to Eq. (4), if the measured ionization values at different depths inside the RW3 solid phantoms are multiplied by the relevant k(RW3) values, one can easily obtain the expected PDI curves in water.
The measured PDI curves in water and in the RW3 solid phantom were employed to obtain the k(RW3) values at different depths. The responses of the Advanced Markus ionization chamber (M Q ) at the same depths in water and in the RW3 phantom were extracted from the measured PDI curves and the k(RW3) values at various depths were then determined following Eq. (4).

Validity of the measured scaling factors
In order to check the validity of the reported depth scaling factors, fluence scaling factors as well as the k(RW3) values, the directly measured percentage depth dose (PDD) in water were compared to those indirectly acquired through ionometric measurements inside the RW3 solid phantom by considering the c pl , hpl , and the k(RW3) factors.
To measure the PDD curves in water, at first the PDI curves were measured by the Advanced Markus ionization chamber for various electron energies. Then the acquired PDI curves were converted to PDD curves by accounting for the corresponding water-to-air stopping power ratios reported in the IAEA TRS-398 dosimetry protocol [16]. On the other hand, for the RW3 measurements, PDI curves at different nominal electron energies were obtained inside this slab phantom. Then, the acquired PDI curves inside the plastic phantom were shifted to water in two distinct ways: (1) by considering the c pl and h pl data simultaneously, and (2) through k(RW3), by going directly from measurements taken in the slab phantom at a given depth, to the measurement at the same physical depth in water. In all the measurements, a Semiflex ionization chamber (TM31010) was also employed as a reference detector, in the corner of the field. Finally, some PDD parameters including R 100 (depth of maximum dose), R 90 (depth at which the PDD curve reaches 90% of the maximum value), R 80 (depth at which the PDD curve reaches 80% of the maximum value), and R 50 (depth at which the PDD curve reaches half of the maximum value; this depth is also known as the electron beam quality index [16]) were extracted from the PDD curves measured directly and indirectly (those respectively measured in water and in the RW3 solid phantom) at different nominal electron energies and quantitatively compared against each other.

Results
Averaging the measured densities for all considered slabs demonstrated that the density of the employed RW3 phantom is equal to 1.045 g.cm −3 ± 0.015 (1.4%).
The acquired PDD data at different electron energies of 4, 6, 9, 12, and 16 MeV are shown in Fig. 1.
As illustrated in Fig. 1, the dose fall-off gradient at depth is sharper for low electron energies than it is for higher Fig. 1 Acquired PDD curves at different nominal electron energies of 4, 6, 9, 12, and 16 MeV through ionometric dosimetry inside the water energies. In this regard, almost no plateau region would be observed in the case of 4 and 6 MeV electron energy, while a wide plateau region is realized for 16 MeV electron energy.
The PDD parameters relevant to the measured PDD curves in water are reported in Table 1.
As demonstrated in Table 1, all PDD parameters increase as the electron energy increases.
The comparison between the measured PDI curves in water and in the RW3 solid phantom at different electron energies is illustrated in Fig. 2. The solid and dashed lines respectively indicate the obtained PDI curves in water and in the RW3 slab phantom. It should also be mentioned that the PDI measurements inside the RW3 solid phantom were initiated from a 5 mm depth due to positioning uncertainties at smaller depths. Furthermore, ionization measurements at lower depths were not considered significant for the purpose of this paper because (1) no clinical electron beam parameter along the central axis (i.e., the depth of maximum dose, R 90 , R 80 , and R 50 which have been reported in the current study) lays within this spatial range (0-5 mm depth), and (2) the fluence scaling factor (h pl ) values have been reported with 5 mm increments to be in accordance with the measured data by AAPM TG-25 protocol for other solid state materials. On the other hand, the h pl value is equal to unity at the surface of any solid phantom (this issue is also demonstrated by AAPM TG-25 protocol [19]) and therefore the measurements were started from 5 mm depth in RW3 solid phantom.
As indicated in Fig. 2, the accordance between the measured PDI curves in water and in the RW3 solid phantom is more evident at higher electron energies. Furthermore, the data reported in Fig. 2 show that the PDI curves measured in water and in the RW3 solid phantom are not matched exactly for all studied conventional electron energies. Instead, a slight difference is observed between the acquired PDI curves in the two studied dosimetry media. Despite the nominal water equivalence of the RW3 solid phantom, this finding explicitly demonstrates that the direct ionometric dosimetry measurements in the RW3 solid phantom cannot directly result in the values expected in a water phantom. Therefore, this proves the necessity of scaling factor measurements for this solid state phantom. The extracted R 50,ion values from the measured PDI curves in water and in the RW3 solid phantom at different electron energies are listed in Table 2.
As demonstrated in Table 2, the R 50, ion values in water and in the RW3 solid phantom are different at the same electron energy so that a maximum difference of about 9.3% was observed at 4 MeV electron energy.
The measured depth scaling factor (c pl ) values at the different electron energies of 4, 6, 9, 12, and 16 MeV are shown in Fig. 3.
The results presented in Fig. 3 illustrate that the c pl values for the RW3 solid phantom are always below unitary, due to the higher R 50,ion values with respect to the water at different electron energies (as reported in Table 2).
As shown in Fig. 3, the c pl value approaches the unity with an increase in electron energy, except when moving from 12 to 16 MeV electron energy where no remarkable change is observed in the c pl value. In this regard, the c pl value increases from 0.915 to 0.937 when shifting from 4 to 16 MeV electron energy.
The determined fluence scaling factor (h pl ) values for RW3 solid phantom at different combinations of the depth/ electron energy have been listed in Table 3. It should be mentioned that the selected depths for h pl measurement were the same as those considered by AAPM TG-25 clinical electron beam dosimetry protocol [19] at different energies.
The mean h pl value at different electron energies of 4, 6, 9, 12, and 16 MeV was obtained as 1.022, 0.995, 0.999, 0.998, and 1.009, respectively. A comparison between the measured mean h pl values for the RW3 solid phantom at 6, 12, and 16 MeV electron energy with those reported for other plastic phantoms in the literature [19] is presented in Table 4.
The measured k(RW3) factors at different depth/electron energy combinations are presented in Fig. 4.
The results presented in Fig. 4 indicate that the k(RW3) factors have greater values at higher electron energies.   Furthermore, the values of k(RW3) steadily decrease as the depth increases.
A comparison between the directly measured PDD curves in water at different electron energies and those indirectly obtained through simultaneous applying the c pl and h pl values to the RW3 solid phantom ionometric measurements is shown in Fig. 5.
As demonstrated in Fig. 5, there is a desirable agreement between the measured PDD curves in water and those acquired through indirect measurements inside the RW3 solid phantom. The dosimetric parameters of PDD curves at different electron energies have been quantitatively compared in Table 5.
As illustrated in Table 5 the maximum difference between the listed PDD parameters is equal to 2.46% which is observed for the R 100 parameter at 16 MeV electron energy.

Discussion
The results of Fig. 1 demonstrate that the plateau region of PDD curves gradually disappears with decreasing the electron energy. This issue is mainly due to the more pronounced multiple electron scattering as well as superior stopping power values at lower electron energies. Deflection angles from the original direction, due to electron scattering, are more evident in the case of lower electron energies. As a result, it can be expected that electron fluence along the beam central axis decrements more rapidly and consequently, a steeper dose gradient along the depth would be observed for low energy electron beams. Furthermore, the higher stopping power values of the electron beam at lower electron energies imply that a greater fraction of electron  energy would be lost and therefore, a higher dose fall-off gradient at depth is expected for low energy electron beams.
As shown in Fig. 1, the penetration depth increases at higher electron energies. The probability of inelastic interactions with atomic electrons (resulting in ionization and excitation processes) and atomic nuclei decreases (resulting in bremsstrahlung production) at higher electron energies which can finally lead to a decrease in the rate of electron energy loss [32]. Therefore, it can be expected that penetration depth and the parameters of the PDD curves will also increase.
The deviations between the PDI curves in water and in the RW3 solid phantom which are reported in Fig. 2 are mainly linked to the differences between the nominal physical density of water and RW3 solid phantom (respectively equal to 1 and 1.045 g.cm −3 ) as well as their effective atomic numbers (7.51 for water and 7.13 for RW3). These differences between the physical densities and effective atomic numbers may change the electron density values (number of electrons per gram). On the other hand, the electron density is the most effective parameter in electron interactions (elastic and inelastic collision and scattering) inside the medium [33]. Consequently, it can be expected that the energy and angular electron distribution in water differ from those in the RW3 phantom, a fact which can ultimately lead to different electron fluence spectra at the same depth in water and in the RW3 solid phantom.
The more desirable accordance between the measured PDI curves at higher electron energies, as depicted in Fig. 2, may be attributed to the decreased probability of elastic and inelastic electron collision and scattering interactions at higher electron energies, a fact which may entail greater convergence of the electron fluence spectra at the same physical depth in water and in the RW3 solid phantom. Therefore, it can be deduced that there would be a better accordance between the measured PDI curves in water and in the RW3 solid phantom at higher nominal electron energies.
The deviations between the obtained R 50,ion inside the water and RW3 solid phantom (reported in Table 2) are because of the slight differences in electron interactions probability within the studied dosimetry media as well as differences in physical density.
The less significant increment of c pl value with increasing the electron energy from 12 to 16 MeV in comparison with those observed at lower energy range (as illustrated in Fig. 3) may be justified by the reduced differences between R 50,ion (water) and R 50,ion (RW3) at higher electron energies (e.g., the relative difference would be equal to 9.3% at 4 MeV electron energy, while a relative difference of 6.7% is observed at 16 MeV electron energy).
The mean c pl value of 0.930 for the RW3 solid phantom was calculated by averaging over the measured depth scaling factor values at different electron energies. On the other hand, this value for solid water (WT1), plastic water, virtual water, PMMA, clear polystyrene, and white polystyrene has been reported to be 0.949, 0.982, 0.946, 0.941, 0.922, and 0.922, respectively [16]. Although the mean c pl value for the RW3 solid phantom lies within the reported c pl range for other plastic phantoms, the deviations in physical density and effective atomic number can lead to different c pl values for various water-equivalent solid phantoms which are commonly employed for radiation dosimetry purposes. In addition, the obtained c pl value for RW3 is closer to the c pl value of polystyrene among the above-mentioned solid phantoms.
As reported in Table 3, the h pl values will change with the variations of depth for all considered nominal electron energies. The main contributing parameter to the h pl value is the angular distribution of the electron beam inside the depth. Changing the depth can vary the electron scattering power (resulting from the elastic and inelastic scattering from atomic electrons and atomic nuclei) and consequently affect the h pl value.
The results reported in Table 4 show that the mean h pl values of the RW3 solid phantom at 6 MeV and 12 MeV electron energies are more similar to PMMA, so that the corresponding differences were obtained as 2.1% and 0.6%. On the other hand, the best similarity between the mean h pl values at 16 MeV electron energy was found for white polystyrene, such that its difference with RW3 solid phantom was about 0.1%. In addition, the results of mean h pl values for listed plastic phantoms in Table 4 reveal that the accordance of h pl values increments with increasing the electron energy. In this regard, the mean differences among the h pl values of list solid phantoms at 6, 12, and 16 MeV electron energies were obtained as 2.7%, 1.2%, and 0.4%, respectively. The decrement of k(RW3) factors with increasing the depth of measurement is related to the chamber response reduction at higher depths and would be more evident at lower electron energies (especially 4 and 6 MeV energies). These properties are similar to the trend of tissue-air ratio (TAR) variations with changing the depth of measurement and beam energy. Finding such similarity between the k(RW3) and TAR concept is expected because TAR indicates the ratio of the absorbed dose by the tissue (medium) to that of a small mass of water in air at the same point and identical irradiation conditions [32] and k(RW3) is the ratio of ionization chamber reading inside the water to that of RW3 plastic phantom at the same depth and irradiation conditions. Nevertheless, k(RW3) has some advantages over the TAR concept including 1) much more simplicity in the practical measurement of k(RW3) respect to the TAR concept and 2) measuring the TAR concept at high beam energies is almost impossible because the required size of ionization chamber buildup cap for providing the electronic equilibrium conditions is so large. On the other hand, due to the fully in-phantom measurements in the case of k(RW3), the electronic equilibrium conditions would be easily fulfilled by the small sizes of buildup cap of employed ionization chambers.
The listed data in Table 5 illustrate that the mean difference between the PDD parameters at electron energies of 4, 6, 9, 12, and 16 MeV is equal to 1.2%, 1.1%, 1.0%, 0.4%, and 0.8%, respectively. Accordingly, the mean difference between the measured PDD parameters, accounting for all studied energies, would be equal to 0.9%. Therefore, it can be deduced that the acquired scaling factor values (c pl and h pl ) for the RW3 solid phantom at different electron energies can be accurately employed to mimic the measured depth dose distribution in water through indirect ionometric dosimetry within this water-equivalent slab phantom.
Due to the similar results, the graphical comparison between the directly measured PDD curves in water and those indirectly acquired by applying the k(RW3) factors to measured ionometric data within the RW3 solid phantom is not presented here. It is worth noting that a good agreement was also found between the obtained in-water PDD data and measured ones employing k(RW3) factors, so that the mean difference between the measured PDD parameters was within 1.2% for electron energies under investigation. Therefore, reliable results can be obtained through the conversion using the k(RW3) approach, although the accuracy of simultaneously employing the c pl and h pl for this purpose may give more reliable results due to the lower relative difference between the measured PDD parameters by these scaling factors and the values obtained by measuring the PDD curves in water.
Although less formal, the direct application of the k(RW3) factors seems to be more practical compared to the coupling between the c pl and h pl scaling factors. Indeed, there is no need to scale the measurement depths inside the solid state phantom, where confusion may arise during the movement from plastic to water through considering the depth scaling factor and physical density of the solid phantom employed for radiation dosimetry. In addition, coupling the c pl and h pl for the PDD conversion is a more sophisticated process than employing the k(RW3) factors, because it requires the determination of three distinct parameters including c pl , h pl, and the physical density of the batch of the solid phantom. On the other hand, only the k(RW3) coefficients are essential in shifting from in-phantom measurements to those relevant to water in the case of the second approach.
As discussed previously, the mean difference between the measured PDD parameters inside the water and those acquired in RW3 solid phantom by applying the k(RW3) factors was equal to 1.2%. This finding explicitly demonstrates that the introduced k(RW3) factors for dosimetric characterization of electron beam in clinical practice can introduce a deviation of about 1.2% in determining the clinical parameters such as therapeutic range (R 90 ) which is considered as the interested clinical depth for covering the distal end of the tumor volume during patient irradiation with the electron beam.
Due to the employment of plastic phantoms for dosimetry in the reference conditions [16], it is recommended that the introduced k(RW3) factors should be used for electron dosimetry at the reference conditions where the radiation field size has a predefined value (10 × 10 cm 2 for electron energies up to 12 MeV and 20 × 20 cm 2 for higher ones). Nevertheless, due to the similarity between the TAR concept and k(RW3) factor, it is expected that the k(RW3) factor increments with increasing the radiation field size since a direct relationship is found between the TAR value and radiation field size [32]. Therefore, separate measurements are required to determine the k(RW3) factors for the field sizes other than the reference radiation field.

Conclusions
The depth and fluence scaling factors (c pl and h pl ) for a water equivalent solid phantom, the RW3 slab phantom, were measured at different conventional electron energies of 4, 6, 9, 12, and 16 MeV through an ionometric radiometry approach. Besides, a new factor, named k(RW3), was also introduced in this study for direct conversion from measured data inside the RW3 phantom to the data expected in water, without any need for further scaling factors.
Our results demonstrated that the measured c pl and h pl scaling factors for the RW3 solid phantom can be reliably used to mimic the expected depth dose distribution data in water at different electron energies through indirect ionometric dosimetry within this dedicated plastic phantom. Both scaling factors were dependent on the electron energy.
The dosimetric relevance of measured k(RW3) factors at different depth/electron energy combinations was also confirmed, such that they can be operationally employed for depth dose measurements inside the RW3 solid phantom with deviations of about 1.2% with respect to the data directly measured in water at different conventional electron energies.
Finally, our findings demonstrate that the k(RW3) coefficients are more straightforward and less time-consuming than the coupling of the c pl and h pl scaling factors for electron dosimetry inside the RW3 plastic phantom and relevant conversion, while the associated uncertainty with PDD measurements was almost the same for both considered approaches. Funding The authors declare that no funds, grants, or other support were received during the preparation of this manuscript.

Author contributions
Data availability All data generated or analysed during this study are included in this published article.

Declarations
Conflict of interest Hamid Reza Baghani, Stefano Andreoli, and Mostafa Robatjazi have no conflict of interest to declare.
Ethical approval This article does not contain any studies with human participants or animals performed by any of the authors.
Informed consent None declared.