Determination of the P rp and radial dose correction factor in reference dosimetry

Objectives. In an addendum to AAPM TG-51 protocol, McEwen et al, (DOI: 10.1118/1.4866223) introduced a new factor P rp to account for the radial dose distribution of the photon beam over the detector volume mainly in flattening filter free (FFF) beams. P rp and its extension to non-FFF beam reference dosimetry is investigated to see its impact in a clinical situation. Approches. The P rp was measured using simplified version of Sudhyadhom et al (DOI: 10.1118/1.4941691) for Elekta and Varian FFF beams with two commonly used calibration detectors; PTW-30013 and Exradin-A12 ion chambers after acquiring high resolution profiles in detectors cardinal coordinates. For radial dose correction factor, the ion chambers were placed in a small water phantom and the central axis position was set to center of the sensitive volume on the treatment table and was studied by rotating the table by 15-degree interval from −90 to +90 degrees with respect to the initial (zero) position. Main results. The magnitude of P rp varies very little with machine, detector and beam energies to a value of 1.003 ± 0.0005 and 1.005 ± 0.0005 for 6FFF and 10FFF, respectively. The radial anisotropy for the Elekta machine with Exradin-A12 and PTW-30013 detector the magnitudes are in the range of (0.9995±0.0011 to 1.0015±0.0010) and (0.9998±0.0007 to 1.0015±0.0010), respectively. Similarly, for the Varian machine with Exradin-A12 and PTW-30013 ion chambers, the magnitudes are in the range of (1.0004±0.0010 to 1.0018±0.0018) and (1.0006±0.0009 to 1.0027±0.0007), respectively. Significance. The P rp is ≤ 0.3% and 0.5% for 6FFF and 10FFF, respectively. The radial dose correction factor in regular beams also does not impact the dosimetry where the maximum magnitude is ±0.2% which is within experimental uncertainty.


Introduction
Clinical reference dosimetry is essential for establishing a metrological basis for delivering accurate prescribed radiation dose to patients for achieving desired radiotherapy outcomes (ICRU Report 50 1993).Both national and international standards have attempted to streamline this process through seminal guidelines such as AAPM TG-51 (Almond et al 1999) and IAEA TRS-398 (IAEA TRS 398 2000) for photon and electron beam dosimetry around the globe.The intent of such code of practice (CoP) is to have megavoltage machines calibrated to deliver ±2% dose or better as described in IAEA human health report (IAEA TCS No. 78 2023).This is required in the context that a good clinician can see the difference in clinical outcome when dose differences are −5% and +7% as stated in many references (Mijnheer et al 1987, Wittkämper et al 1987, Brahme et al 1988) and subsequently adopted by ICRU-50 (ICRU Report 50 1993).The IAEA report, Human Health Series No 31 has also provided a contemporary argument for tighter or better dosimetric criterion for clinical care (IAEA TCS No. 78 2023).
To improve dose calibration accuracy, AAPM upgraded TG-21 (Schultz et al 1983) to the water-based calibration protocol TG-51 (Almond et al 1999).This CoP is probably most reliable with minimum efforts with least uncertainty based on the scientific rigors and mostly Monte Carlo derived parameters.However, TG-51 was developed before the clinical adoption of advanced technologies such as small fields (Das, Ding x y z SSD P P P P P , , , 2

raw TP ion pol elec rp
Equation (1) is generic, indicating dose in water (D w ) for beam quality Q, M: the corrected detector reading, k Q : the chamber correction factor between Co-60 and the clinical beam quality Q and N DW : the calibration coefficient acquired from an accredited dosimetry calibration laboratory (ADCL) for the detector in Co-60.Most of these parameters are described in detail in TG-51 (Almond et al 1999).The conversion from M raw (raw reading) to M as in equation (2) depends on various correction factors (P) such as temperature and pressure (P tp ), ion recombination (P ion ), polarity (P pol ), electrometer factor (P elec ) and radial dose function (P rp ).The additional parameter, P rp is correction factor to account for any off-axis variation in dose profile of the radiation field over the finite size of the sensitive volume of the ion chamber usually in x and y ( ) Where integration is over projection of the detector area (A) in x, y orthogonal coordinates, OAR is off-axis ratio in x and y direction and w(x, y) is a weighting function representing detector's sensitive volume along the beam axis as a function of the beam lateral coordinates.However, detail is not available as how this integration should be carried out and how to get w (x, y).Sudhyadhom et al (Sudhyadhom et al 2016) have shown that for Farmer-type chambers, variation of w(x, y) and OAR(x, y) over the radius of the chamber is negligible compared to variation along the chamber axis and provided more information about P rp and simplified the integration process with an analytical form in one dimension.Since ion chamber detector is cylindrical with radius ∼0.3 cm, weighting function, w(y) can be approximately 1.0 since beams are normalized at central axis and there is very little variation in OAR over the radius of ion chamber.In this situation equation ( ) rp Where x dimension in centimeter along the axis of detector and can be equated to 0.29 L (equation ( 5)) after integration, if L is physical length of the Farmer type of ion chamber.In above discussion, it is assumed that in-plane and cross-plane profiles are identical.
The value of OAR needs to be averaged over ±x and also for cross-and in-plane profiles.However, it does not account for possible dosimetric variation in other radial directions.
In this investigation, we have provided an implication of P rp in reference dosimetry, and its magnitude in reference conditions with simplified approach (equation ( 6)) as well radial dose distribution (from standard and FFF beams) that has not been explored earlier.

Materials and methods
The P rp was studied on two commonly utilized machines, Elekta and Varian.Measurements were made in reference conditions for a field size of 10×10 cm 2 at a depth of 10 cm at 100 cm source to surface distance with two commonly used referenceclass ion chambers, PTW-30013 and Exradin-A12 with volume of 0.6 cm 3 and 0.64 cm 3 , respectively.The length and diameter of the detectors are 2.3 cm and 0.61 cm and 2.26 cm, 0.71 cm for PTW-30013 and Exradin-A12 chambers, respectively.
Beam qualities evaluated were 6 MV, 6FFF, 10 MV, and 10FFF at their respective dose rates: 600 MU/ min, 1200 MU/min, 600 MU/min, and 1600 MU/ min on the Elekta Versa HD and 600 MU/min, 1400 MU/min, 600 MU/min, and 2400 MU/min on the Varian TrueBeam.The beam profiles were collected in a large water tank with high resolution step of 0.1 mm in cross-and in-plane.To investigate the radial dose correction factor (anisotropy) in other planes, a small 1-D water tank was placed on the treatment table such that the beam isocenter was located at reference depth (10 cm) in water.Chambers were positioned such that the center of the sensitive volume coincides with the isocenter with gantry and collimator at 0 degree.The ion chamber was placed horizontally in the x-y plane and the position was fine-tuned to find the true position of central axis by finding the location of maximum signal, as suggested by Dieterich and Sherouse (Dieterich and Sherouse 2011).This was accomplished by small changes to the table's position in x and y directions.The off-axis variation is accommodated by finding the actual location of the maximum reading in cardinal angles due to variation in the intensity profile.Readings were normalized to zero position, where P rp is treated as 1.00.As shown in equation (4), P rp is only dependent on the OAR which was evaluated based on average x and y profiles for a length of 0.29 L of ion chamber for each energy and machine.
The radial dose distribution was determined by rotating the treatment table at 15-deg intervals from −90 to +90 degrees about central axis of the beam.The rotational anisotropy was computed at respective position to the zero position to see the radial dose distribution of beam over the sensitive volume of the detectors as shown in figure 1.A PTW Unidos Romeo electrometer (PTW, Freiburg, Germany) was used for the chamber readings.For each angular position, three sets of readings with 100 MU were taken from which an experimental mean and standard deviation (SD) were calculated for uncertainty analysis.In each chamber and machine combination the maximum SD and percent error (SD/average) were also calculated.

Results
The orthogonal profiles (cross-plane and in-plane) at 10 cm depth for both machines are shown in figure 2. For both machines profiles are nearly identical in both planes within (<0.1%) within central regions (rectangular box in figure 2).For 10FFF beams profiles are a bit steeper but magnitudes in both planes are relatively identical.Additionally, there is no difference in profiles between Elekta and Varian machines in the center.The P rp was calculated based on respective OAR values (averaged over in-plane and cross-plane profiles) with ± 0.29 L of the ion chambers and shown in table 1.It is imperative that for standard beams P rp is 1.0, e.g.beam is flat over the sensitive volume of the ion chamber.Our values are slightly lower to that of Sudhyadhom et al (Sudhyadhom et al 2016) who reported a value of 1.004 and 1.007 for 6FFF and 10FFF beams respectively for a TrueBeam STx machine.
The radial anisotropy is another factor that needs to be looked for FFF beams especially during machine commissioning.As discussed in TG-106 (Das, Cheng et al 2008) that some treatment planning requires radial dose profiles but did not specify the tolerance.Figure 3 shows data for Elekta and Varian machines for the detector orientation in x and y planes achieved by rotating the table hence the chamber, at finite angles mimicking motion of detector.The error (evaluated as standard deviation, SD) in measurements is usually very small but was relatively large for 10 FFF (figure 3 and table 2).The magnitude of maximum error (%SD/average) from −90 deg to +90 degree for Exradin-A12 ion chamber are 0.0010 (0.10%) and 0.0032 (0.32%) for Elekta and Varian machines, respectively.For the PTW-30013 ion chamber, the values are 0.0013 (0.13%) and 0.0023 (0.23%) for Elekta and Varian, respectively.It is noted (figure 3) that there are no patterns in radial dose for both machines and ion chambers in regular and FFF beams.Additionally, there are no correlations in beam   This study shows that the effect of P rp in reference dosimetry is relatively small as shown in table 1.The magnitude of radial dose correction factor that accounts for non-cardinal angles is also relatively small <0.2% for both machines for any beam energy and is well within the overall measurement uncertainty.This indicates that conical shape of the dose distribution is uniform in radial angles as shown in figure 3 in each panel.For clinical cases and references dosimetry, P rp seems to have a very limited impact on the calibration.This study is limited to the reference condition in a large field (10×10 cm 2 ), with reference chamber with relatively large volume.As P rp depends on the volume averaging, one could use small volume detector but such options are seldom used in reference dosimetry.There might be a larger impact in small fields, particularly with FFF beams that needs additional investigation (IAEA TRS 483 2017, Das et al 2021).
Additionally, this factor may be important in MR-Linacs which is compounded by radial dose as well as distortion due to magnetic field.

Conclusions
The magnitude of P rp is very small 1.003 and 1.005 for 6 FFF and 10 FFF and independent on Farmer type of ion chambers and machine type, Elekta and Varian.Similarly, the radial dose correction for regular beams is also very small <0.2% due to relatively flat beam in the vicinity of central axis which is within the limit of experimental accuracy.These values should be evaluated during machine commissioning and could be incorporated in dose calibration.
et al 2008, IAEA TRS 483 2017, Das et al 2021) and flattening filter free (FFF) beams with very high dose rate (1400-2400 MU/min) (Georg et al 2011, Fogliata et al 2012).It was realized that some of the parameters in TG-51 may need revision in light of more scientific knowledge.An addendum to TG-51 was published in 2014 by McEwen et al (McEwen et al 2014) where a new factor P rp was introduced to account for radial nonuniformity of the photon beam dose over the detector volume that may be present in any beam but magnified mainly in FFF beams.This is due to the large volume typically 0.6 cm 3 of calibration detectors.According to the addendum, P rp appears in the formalism as indicated in the equation below.Pictorially, it is shown in figure 1 with conical dose distribution and finite size detector; coordinates.Even though, FFF beams have been used over 20 years (Titt et al 2006) and the P rp was introduced in 2014, (McEwen et al 2014) there is very little data on P rp on any detector and machines in the literature.Additional evaluation on TG-51 in the form of TG-374 (Muir et al 2022) provided additional addendum to and explored the P rp .It was indicated that P rp is equal to volume averaging (k vol ) as defined in IAEA TRS-483 (IAEA TRS 483 2017).

Figure 1 .
Figure 1.Schematic of the FFF conical dose.Imagine a finite size detector that is measuring dose in x and y planes with variable conical dose (if any) over sensitive volume of the detector.

Figure 2 .
Figure 2. Cross-plane and in-plane beam profiles for 6FFF and 10FFF beams from Elekta and Varian machines at 10 cm depth for a 10×10 cm 2 fields.The vertical solid lines indicate extent of ion chamber length needed for the averaging of the profiles for OAR computation for P rp .

Figure 3 .
Figure 3. Radial dose correction factor (P rp ) measured in various positions for two detectors and 2 machines.The maximum error bars are also shown in each panel.The zero value on the x axis is at zero degree collimator and zero deg table angle.

Table 1 .
Calculated values of P rp for two detectors and two machines.It is expected that the conical dose distributions found in FFF beams would yield values for P rp that may differ significantly from unity.The P rp that accounts for the radial dose distribution over the sensitive volume of the ion chamber is dependent on the OAR of the beam.This was introduced by McEwen et al(McEwen et al 2014)in the addendum of TG-51 but neither provided method as how to measure nor any estimate of the magnitude.It was further elaborated by Muir et al in TG-374 (Muir et al 2022) as equal to K vol .Unfortunately, these publications did not elaborate any methodology for determining this factor.Using analytical function and approximating the K vol , Sudhyadhom et al (Sudhyadhom et al 2016) provided method to compute P rp except it is not clear for a clinical physicist as how to measure the factor.We have provided a simplified approach to measure P rp .The key point is acquisition of high-resolution profiles in x and y directions.Based on equation (4) the data needs to be averaged over ±x and ±y to compute OAR.

Table 2 .
Average radial dose distribution data for Elekta and Varian machine with two reference detectors over range of measurements.