A mathematical approach to beam matching
Abstract
Objective:
This report provides the mathematical commissioning instructions for the evaluation of beam matching between two different linear accelerators.
Methods:
Test packages were first obtained including an open beam profile, a wedge beam profile and a depth–dose curve, each from a 10×10 cm2 beam. From these plots, a spatial error (SE) and a percentage dose error were introduced to form new plots. These three test package curves and the associated error curves were then differentiated in space with respect to dose for a first and second derivative to determine the slope and curvature of each data set. The derivatives, also known as bandwidths, were analysed to determine the level of acceptability for the beam matching test described in this study.
Results:
The open and wedged beam profiles and depth–dose curve in the build-up region were determined to match within 1% dose error and 1-mm SE at 71.4% and 70.8% for of all points, respectively. For the depth–dose analysis specifically, beam matching was achieved for 96.8% of all points at 1%/1 mm beyond the depth of maximum dose.
Conclusion:
To quantify the beam matching procedure in any clinic, the user needs to merely generate test packages from their reference linear accelerator. It then follows that if the bandwidths are smooth and continuous across the profile and depth, there is greater likelihood of beam matching. Differentiated spatial and percentage variation analysis is appropriate, ideal and accurate for this commissioning process.
Advances in knowledge:
We report a mathematically rigorous formulation for the qualitative evaluation of beam matching between linear accelerators.
Radiotherapy centres with more than one linear accelerator of the same make, model and multileaf collimator (MLC) often desire “beam matching” for immediate interchange of patients between these accelerators, without having to replan, in the event that one machine is not functioning properly. A high level of similarity between beam profiles, depth–dose curves and other properties from two similar accelerators signifies beam matching. Beam matching can be defined as a methodology that needs to be performed between two or more linear accelerators to ensure the equivalence of dose delivery. A linear accelerator is characterised by numerous electrical, mechanical and dosimetric parameters. Proper evaluation of the dosimetric and the mechanical parameters is crucial for precision and accuracy in treatment delivery to patients. Some parameters relating to dose include the output factor, depth–dose, profile, energy and dose rate. Electromechanical parameters consist of the gantry, collimator and couch isocenters, MLC positional accuracy, MLC speed, jaw positional accuracy, gantry speed, etc. [1]. It is possible that many of these parameters will be close in value between those for an existing accelerator and a newly installed one to within some specified tolerance limit. However, the method for beam matching is not well described for commissioning purposes. The only study that describes the problem in a quantitative manner is from Hrbacek et al [2]. The vendor-defined criteria for beam matching are insufficient to ensure the interchangeability of the beams [2–4]. Other investigators described the problem in a more qualitative way [5–8]. There is no task group report that clearly indicates the acceptability criteria of two or more matched beams. Here, we describe the test package required for necessary and adequate testing as well as the process by which comparisons should be objectively made.
It is important to understand that instead of measuring the full set of dosimetric data for the test accelerator, comparison of only a limited set of baseline measurements between the reference linear accelerator and the test linear accelerator is required for dosimetric fidelity cross-check between them. If the reference beam and test beam are found to be in agreement within a specified tolerance limit, no further profile and depth–dose measurement for the new machine are required. Therefore, beam matching leads to a considerable reduction in the dosimetric workload [2]. Matched linear accelerators may then be represented by a common set of data used in the treatment planning system.
Accelerator manufacturers, such as Varian Medical Systems, Inc. (Palo Alto, CA) and Elekta (Stockholm, Sweden), offer beam matching for X-rays. Varian's specific criterion requires the depth of maximum dose (dmax) along the central axis as well as the beam energy specification to match within ±1.5 mm of the nominal specified value between machines. The relative dose at a depth of 10 cm should also match within ±1%. For profile matching, the dose at dmax and at a depth of 10 cm is required to be in agreement within ±1% and ±2%, respectively, in the flat region for field sizes larger than 10×10 cm2. Elekta's specified criterion is nearly the same. However, an assignment for dmax to specify the matching criteria for energy and beam flatness is disregarded. Only a 10-cm-depth specification is needed. Further, profile matching is only averaged over an arbitrary 1-cm region in the flat region of the profile [3,4]. Neither manufacturer has any specific criteria for output factor matching or profile penumbra matching. Match criteria is entirely based on a single field size and at only one or two depths of interest. Such limited criteria are insufficient to ensure the interchangeability of linear accelerators in clinical practice.
The quantitative evaluation of the dose distribution must be much less generous. This is one of the single most attributable causes of intensity-modulated radiotherapy (IMRT) delivery quality assurance failures for patients who are required to be switched to a different machine after completing treatment planning for a particular linear accelerator. To elaborate this point, we consider IMRT quality assurance. The integral dose given from a beam passing through segmented fields must be evaluated for accuracy in any dynamic plan [2]. As encountered by many investigators, a systematic error is observable in energy fluence comparisons of the computerised plan and the actual delivery when the MLC motion is not identical [2,5–8]. The same difficulty arises when the therapy is planned on one machine and delivered on another that is not acceptably matched.
The best methodology available at present for a quantitative evaluation of the dose distribution for beam matching is based on the gamma index (γ), distance to agreement or percentage dose difference criteria [2,9,10]. Each of these methods has disadvantages. They are sensitive to normalisation and do not accentuate the dose gradient [11]. However, all beam matching methodologies described are based on an one-dimensional gamma analysis and have the same characteristic demerit of not appreciating the dose gradient [2,9–11]. It is noteworthy that prior published literature examined only Varian accelerators. [2,5–8]. No prior study has yet been reported for Elekta linear accelerators.
To address the deficiencies of beam matching methods commonly used at present, a new mathematical technique for the evaluation of dosimetric parameters is proposed in this report. The intent of this study is to demonstrate how to evaluate quantitatively a beam from a new accelerator that was manufactured to match an existing linear accelerator beam having the same energy. We emphasise the importance of beam match accuracy by demonstrating how to apply this method uniformly for all beam energies, whether the emission is bremsstrahlung X-rays or electrons, and to any accelerator model.
MATERIALS AND METHODS
Accelerator and software
The machines evaluated are identical to Elekta Model Precise linear accelerators. Before proceeding for dosimetric beam matching, all electromechanical parameters of both the linear accelerators were verified using the TG-142 [1] protocol. The vendor-defined “fine beam matching” (also referred to as “factory matching”) was conducted at the Elekta factory. The reference accelerator was installed 5 years before the delivery of this new machine. The machine to be installed was purchased to include a beam matching for 6-MV X-rays. In carrying out beam matching quality assurance test as part of the customer acceptance test by the manufacturers, it was found that the dose difference between the two machines for the flat region of the open beam profile was slightly higher than 2% (2.04%) [3]. Although this result does not strictly conform to the beam matching criterion specified by the manufacturer, the machine was accepted as beam matched, considering the mismatch as statistically insignificant. Moreover, because we planned to develop our own methodology with necessary tweaking on profile and depth–dose curves, vendor-specified criteria were taken only as initial values. Our beam matching requirements must also be satisfied. In this need, it was necessary to calibrate the machine for absolute dose as stipulated in standard documents in medical physics; AAPM TG-51 or International Atomic Energy Agency Technical Report Series 398 (IAEA TRS-398) [12,13]. Immediately following this, profile and depth–dose measurements were completed as recommended by AAPM TG-106 [14]. All relative measurements were taken in 1-mm increments with a PTW (Freiburg, Germany) dosimetry system, then exported from the scanning software to a spreadsheet. All the depth–dose curves and profiles were compared in absolute dose values (cGy MU−1) to avoid any ambiguity in normalisation [5]. Acceptability of the new linear accelerator, with respect to beam matching with the reference linear accelerator, was evaluated in the Nucletron (an Elekta company) Model Plato Sunrise treatment planning system.
The high gradient penumbra region of the profile curve can be expressed in terms of a gauss error function for one side and a complementary error function on the contralateral side [9]. However, these two error functions cannot be feasibly combined to express the complete profile. A similar problem exists for the depth–dose curve. There is no simple fit for the increase in the depth of maximum dose that adequately follows the declination in dose beyond that point [15,16]. Owing to the inability in obtaining these data in an analytical form, numerical differentiation is required. Scanning data now present in spreadsheet software were operated on with simple calculus tools in OriginLab (Northampton, MA) Model Microcal Origin v. 6.0 mathematical software. It is this first derivative that defines the slope of the fitted profile or curve, whereas the second derivative represents the curvature. First and second derivatives together are ideally suited to discern any variance between the new beam and the existing beam.
Profile beam matching method
The profile is expressed in terms of the dose (D) per monitor unit (MU) with units cGy MU−1. The first derivative denoted D′=dD/dx is the rate of change of energy deposition in space or the slope of the curve. The second derivative termed D″=d2D/dx2 indicates the rate of change of slope or termed as curvature. For a dose profile curve, the first and the second derivatives are important to highlight the sharpness in the rise and fall of the penumbra region. There are two regions of the profile curve and its derivates that are of main interest; the derivative significant area (DSA) and the derivative insignificant area (DIA). Figure 1 illustrates half of the 10×10 cm2 open profile for a 6 MV beam, including the first derivative and second derivative in space. The nature of the other half of the profile is complementary. For critical visualisation, the profile curve was plotted on a log10 scale, whereas the derivatives are shown on a linear scale. Venselaar et al [17] indicated that dose differences in the penumbral region may be as high as 40% and predominates all other profile variations [9]. Therefore, it is only necessary to evaluate ΔD in the DIA region and to critically judge ΔD′ and ΔD″ in the DSA region. Although ΔD′ is insignificant in the DIA region, it can be compared to estimate the local fluctuation of the test profiles against reference profiles, as has been done here additionally for the reader.
Half of a standard profile and its first and second derivatives in space; the profile (cGy MU−1) plotted on a log10 scale; first and second derivatives are shown in a linear scale; the derivative significant area (DSA) and derivative insignificant area (DIA) are indicated (the contralateral side of the curve not shown exists complimentary to this side).
We considered the open wedged profile and percentage depth dose (PDD) for a 10×10 cm2 field from the reference machine as standard curve. A percentage dose error (%DE) and spatial error (SE) were then introduced to the standard curves. The errors chosen included a 1%, 2% and 3% dose error, a 1-, 2- and 3-mm SE as well as a mixed error (%DE and SE) of 1%/1 mm, 2%/2 mm and 3%/3 mm. The first and second derivatives of the standard curves and error-introduced curves were then computed. Finally, the dose differences (ΔD), first derivative differences (ΔD′) and second derivative differences (ΔD″) were calculated between each set of standard curve and error curve. These bandwidth values in the test package, expressed as ΔDBW, ΔDBW and ΔD″BW, were over both sides of the central axis.
While evaluating beam matching characteristics quantitatively, between the reference and test linear accelerators, a set of similar values for dose and derivative difference were defined for the reference accelerator. These values denoted as ΔDBM, ΔD′BM and ΔD″BM were compared against the ceiling values of the test package (ΔDBW, ΔD′BW and ΔD″BW). Tables 1 and 2 demonstrate the results for the open profile and wedged profile, respectively.
| Induced error | Rise/fall average BW: ΔD′BW | Induced error | Rise/fall average BW: ΔD″BW | Induced error | Plateau average BW: ΔDBW | Induced error | Plateau BW: ΔD′BW |
| 1% | 0.11 | 1% | 0.04 | 1% | 0.02 | 1% | 0.00 |
| 2% | 0.21 | 2% | 0.15 | 2% | 0.04 | 2% | 0.00 |
| 3% | 0.31 | 3% | 0.11 | 3% | 0.05 | 3% | 0.00 |
| 1%/1 mm | 3.83 | 1%/1 mm | 1.37 | 1 mm | 0.85 | 1 mm | 0.70 |
| 1 mm | 3.84 | 1 mm | 1.37 | 1%/1 mm | 0.87 | 1%/1 mm | 0.74 |
| 2 mm | 7.25 | 2%/2 mm | 2.55 | 2%/2 mm | 1.23 | 2 mm | 0.76 |
| 2%/2 mm | 7.28 | 2 mm | 2.60 | 2 mm | 1.46 | 2%/2 mm | 0.77 |
| 3 mm | 10.36 | 3%/3 mm | 3.58 | 3%/3 mm | 1.54 | 3%/3 mm | 0.78 |
| 3%/3 mm | 10.55 | 3 mm | 3.62 | 3 mm | 2.02 | 3 mm | 0.79 |
| Wedge thick end | Constant slope region | Wedge thin end | |||||||
| Induced error | Thick end BW: ΔD′BW | Thick end BW: ΔD″BW | Induced error | Constant slope BW: ΔDBW | Induced error | Constant slope BW: ΔD′BW | Induced error | Thin end BW: ΔD′BW | Thin end BW: ΔD″BW |
| 1% | 0.02 | 0.01 | 1% | 0.13 | 1% | 0.00 | 1% | 0.04 | 0.02 |
| 2% | 0.03 | 0.01 | 1 mm | 0.24 | 2% | 0.00 | 2% | 0.08 | 0.03 |
| 3% | 0.05 | 0.02 | 2% | 0.27 | 3% | 0.00 | 3% | 0.12 | 0.05 |
| 1 mm | 0.69 | 0.34 | 1%/1 mm | 0.28 | 1 mm | 0.14 | 1%/1 mm | 1.77 | 0.83 |
| 1%/1 mm | 0.69 | 0.34 | 2 mm | 0.39 | 3 mm | 0.14 | 1 mm | 1.78 | 0.83 |
| 2 mm | 1.32 | 0.62 | 3% | 0.40 | 1%/1 mm | 0.14 | 2 mm | 3.38 | 1.44 |
| 2%/2 mm | 1.32 | 0.63 | 2%/2 mm | 0.42 | 3%/3 mm | 0.14 | 2%/2 mm | 3.38 | 1.46 |
| 3 mm | 1.80 | 0.74 | 3%/3 mm | 0.42 | 2 mm | 0.19 | 3 mm | 4.62 | 1.88 |
| 3%/3 mm | 1.88 | 0.77 | 3 mm | 0.48 | 2%/2 mm | 0.19 | 3%/3 mm | 4.70 | 1.91 |
Depth–dose beam matching method
The depth–dose curve monotonically increases from the surface to the depth of maximum dose and then monotonically decreases as a function of depth at an exponential rate. The portion of the depth–dose curve beyond dmax has a simple and nearly constant slope. It then follows that the highest variations of the first and second derivatives is obtained in the build-up region since small spatial dose variation yields minimal derivatives beyond that point. Hence, for the depth–dose curve ΔD′ and ΔD″ were compared up to the depth dmax. The second derivative becomes insignificant beyond dmax. Therefore, only the ΔD was compared downstream from that point. An identical set of errors were introduced as described for the profile curves in Profile beam matching method section to obtain the bandwidth values ΔDBW, ΔD′BW and ΔD″BW for the test package. The bandwidths were then compared with the test package ceiling values. The test package table for depth–dose beam matching was not enumerated, but rather pictorially presented.
RESULTS
Profile beam matching analysis—open fields
Comparison of the dose profiles of the reference and test machines showed that the plateau region of the two curves is matched for lower %DE/SE values than that for the penumbra region. This observation is in agreement with our hypothesis that beam matching for the entire dose profile curve cannot be ensured using a single number [2]. We deduced appropriate criteria by dividing the profile curve in these two regions separately. Open field results are shown in Table 3. Here, the “rise” and “fall” refer to the penumbra regions for x<0 and x>0, respectively. All results were within the specified limits of the introduced error, except for the ΔD″ values for 40×40 cm2 field size profile at dmax depth. The ΔD″ value obtained in this case was beyond any ceiling value, giving rise to the speculation that the beam scan should be repeated for validity. This was intentionally provided for the reader to observe where a failure might exist.
| Field size (cm2) | Direction and depth | Rise BW: ΔD′BM | Below ΔD′BWa | Rise BW: ΔD″BM | Below ΔD″BWa | Constant slope BW: ΔDBM | Below ΔDBWa | Constant slope BW: ΔD′BM | Below ΔDBWa | Fall BW: ΔD′BM | Below ΔDBWa | Fall BW: ΔD″BM | Below ΔD″BWa |
| 5×5 | x-dmax | 0.97 | 1%/1 mm | 0.35 | 1%/1 mm | 1.13 | 2%/2 mm | 0.23 | 1 mm | 0.46 | 1%/1 mm | 0.27 | 1%/1 mm |
| 10×10 | y-dmax | 2.18 | 1%/1 mm | 1.05 | 1%/1 mm | 0.05 | 3% | 0.56 | 1 mm | 1.28 | 1%/1 mm | 0.87 | 1%/1 mm |
| 15×15 | y-dmax | 1.54 | 1%/1 mm | 0.84 | 1%/1 mm | 1.71 | 3 mm | 0.44 | 1 mm | 0.98 | 1%/1 mm | 0.64 | 1%/1 mm |
| 20×20 | y-dmax | 2.29 | 1%/1 mm | 0.88 | 1%/1 mm | 1.17 | 2 mm | 0.31 | 1 mm | 2.46 | 1%/1 mm | 0.89 | 1%/1 mm |
| 25×25 | x-dmax | 3.86 | 1%/1 mm | 1.58 | 2%/2 mm | 1.95 | 3 mm | 0.51 | 1 mm | 2.57 | 1%/1 mm | 0.96 | 1%/1 mm |
| 30×30 | y-100 mm | 0.38 | 1%/1 mm | 0.30 | 1%/1 mm | 1.94 | 3% | 0.41 | 1 mm | 2.68 | 1%/1 mm | 1.09 | 1%/1 mm |
| 40×40 | y-dmax | 6.34 | 2 mm | 4.87 | * | 0.05 | 3% | 0.56 | 1 mm | 6.84 | 2 mm | 2.90 | 3%/3 mm |
Our overall assessment of the open profile analysis was that 71.4% of all the segments for ΔDBM, ΔD′BM and ΔD″BM (30 out of 42) matched within a bandwidth of 1%/1 mm. The dose difference (ΔDBM) in the constant slope region matched within a maximum value of 3%. However, the slope difference (ΔD′BM) for all the tested field sizes matched within 1 mm in that same region. With the point-to-point dose variation between the reference and test profiles being relatively high, the difference in the rate of spatial dose deposition (slope difference) was quite low. In the DSA region, both ΔD′BM and ΔD″BM matched within 1%/1 mm.
For a better understanding, Table 3 demonstrates the beam matching results (ΔDBM, ΔD′BM and ΔD″BM) for a 10×10 cm2 field size at a depth of dmax, and their respective bandwidths (ΔDBW, ΔD′BW and ΔD″BW) that matched. The first derivative difference (ΔD′BM) and dose difference (ΔDBM) in the plateau region matched within 3% for ΔDBW and within 1 mm for ΔD′BW. The first derivative difference matched within 1 mm in the plateau region and 1 mm in the fall/rise region. The second derivative difference (ΔD″BM) matched within 1%/1 mm for all data.
Profile beam matching analysis—wedged fields
Results of beam matching analysis for two wedged fields at two different depths (total four) are shown in Table 4. Only the radial scan is demonstrated for these wedged data. For transverse scan across the wedge, the employed methodology is the same as demonstrated in case of open field. The terminology changes according to the wedge geometry; the wedge profile curves were subdivided according to the thick and thin end of the wedge, and the mid-portion was analysed as a constant slope region (not plateau). For these matching curves, 70.8% (17 out of 24) profile segments match well within the 1%/1 mm in the bandwidth.
| Field size (cm2) | Direction and depth | Thick end | Constant slope region | Thin end | |||||||||
| Thick end BW: ΔD′BM | Below ΔD′BWa | Thick end BW: ΔD″BM | Below ΔD″BWa | Constant slope region BW: ΔDBM | Below ΔDBWa | Constant slope region BW: ΔD′BM | Below ΔD′BWa | Thin end BW: ΔD′BM | Below ΔD′BWa | Thin end BW: ΔD″BM | Below ΔD″BWa | ||
| 10×10 | y-W-dmax | 0.15 | 3 mm | 0.130 | 1 mm | 0.42 | 2%/2 mm | 0.13 | 1 mm | 0.74 | 1%/1 mm | 0.43 | 1%/1 mm |
| 30×30 | y-W-dmax | 0.11 | 2 mm | 0.053 | 1 mm | 0.37 | 2 mm | 0.16 | 2 mm | 0.61 | 1%/1 mm | 0.33 | 1%/1 mm |
| 30×30 | y-W-10 cm | 0.07 | 1 mm | 0.043 | 1 mm | 0.33 | 2 mm | 0.09 | 1 mm | 0.54 | 1%/1 mm | 0.30 | 1%/1 mm |
| 10×10 | y-W-10 cm | 0.10 | 1 mm | 0.068 | 1 mm | 0.32 | 2 mm | 0.12 | 1 mm | 0.76 | 1%/1 mm | 0.44 | 1%/1 mm |
Depth–dose beam matching analysis
The results of depth–dose comparison are demonstrated in Figure 2a,b. Each of the first derivative bandwidths ΔD′BM and second derivative bandwidths ΔD″BM was compared from the surface up to dmax. For depths greater than dmax up to 300 mm, the dose difference ΔDBM was compared only. The slope of the depth–dose curve was constant beyond dmax. ΔD′BM and ΔD″BM were insignificant in this region and were within 2% of that for the reference curve. Up to the depth of dmax, the ΔD′BM and ΔD″BM curves matched within 1%/1 mm, while beyond dmax, the ΔDBM curve matched within 1.5%/1.5 mm. This is indicated in Figure 2b, where it is observed that 96.8% points (274 out of 283) fit within a window formed by an upper boundary of 0.5%/0.5 mm and a lower boundary of −1%/−1 mm. Only 3.2% of all points were found to fall outside the specified bandwidth. The output difference between the test and reference linear accelerators was found to match within a window of −0.2% to 0.7%.
(a) ΔD′BM (cGy MU−1 mm−1) and ΔD″BM (cGy MU−1 mm−2) values of the depth–dose curve in the build-up region, ΔD′BM and ΔD″BM for different square open field sizes from 5×5 cm2 to 40×40 cm2 and including a 10×10 cm2 wedged field matching within a bandwidth (ΔD′BW and ΔD″BW) of 1%/1 mm. (b) Difference (ΔDBM) in depth–dose between the reference and test linear accelerators beyond the depth of maximum dose: 96.8% points of ΔDBM for different square fields match within a bandwidth (ΔDBW) of 1.5%/1.5 mm.
DISCUSSION
In this work, a methodology has been developed for quantitative evaluation of beam matching between two linear accelerators and demonstrated for Elekta machines. This method has two advantages over prior published works on beam matching using a gamma index. Firstly, Low and Dempsey [10] described that the method of gamma index (γ) does not appreciate the dose gradient [11]. It then follows that any unacceptable dose gradient may be overlooked by that methodology as demonstrated by Hrbacek et al [2] based on the γ index. The beam matching methodology presented in this work is free from this difficulty, because it considers dose profile matching in the high gradient region. Secondly, as indicated in Figure 2b, ΔDBM is negatively influenced about the y=0 boundary. This effect is accentuated when comparing the dose differences of depth–dose curves. For most of the field sizes considered in this work, the test depth–doses had higher values than the reference curves for the existing accelerator. This results in negative values for ΔDBM. To accommodate these findings, the upper boundary was taken as 0.5%/0.5 mm and the lower boundary taken as −1%/−1 mm. This is an interesting feature that cannot be identified or corrected for by the γ analysis method, since γ indices can only yield positive [2,9] results.
For an Elekta linear accelerator, the beam profile is steered by adjusting the transverse magnetic field induced by two pairs of inductor coils 1R–1T (at the origin of the waveguide) and 2R–2T (at the end of the waveguide where the electron beam is rotated by 270°). For a fixed beam geometry shaped by beam limiting devices like jaws and/or an MLC, the shape of the profile is adjusted by changing the 2R–2T coil current. The 1R–1T coil current changes in proportion with the 2R–2T coil current and cannot be adjusted independently. The change in the coil currents changes the transverse magnetic field and consequently the trajectory and spatial spread of the accelerated electron beam. Similarly for a depth–dose curve adjustment, the penetration characteristic corresponds to the beam energy, which is governed by the bending coil current. Adjustment of the 2R–2T coil current or the bending coil current may change the shape of dose profile curves or depth–dose curves for field sizes other than that used during tweaking. Thus, after the 2R–2T coil current is adjusted, we recommend that scanning of all profile curves and depth–dose curves be repeated before implementation of the final beam matching procedure in the commissioning process.
CONCLUSIONS
We have described a new method for quantitative evaluation of beam matching between two linear accelerators and demonstrated its application for two factory-matched Elekta linear accelerators. Our results show that there is good agreement of the dosimetric characteristics of two beams of these machines. More than 70% of the open and wedge profile points and 98% of the depth–dose curve points matched within a 1%/1-mm window. It can be shown that using a simple calculation for 6-MV beam energy, a 1% error in depth–dose leads to a mere 1.7% of treatment delivery error. This is well within the tolerance level recommended in the International Commission on Radiation Units and Measurements 24 (ICRU-24) report [18]. These findings prove that the present method is a more reliable one for beam matching analysis between two linear accelerators. The manufacturer defined criteria for beam matching is often too generous and should be evaluated as detailed here. It is possible that the interchange of one patient's plan between two machines will not result in an identical treatment if dose-gradient matching is not taken into account.
For open and wedge beam profile gradient matching, we recommend that the ΔD′ and ΔD″ be kept within 1%/1 mm. For plateau matching, we recommend a ceiling of 2%/2 mm. For depth–dose curve matching, ΔD′ and ΔD″ in the build-up region should lie within 1%/1 mm and with ΔD should be within 1.5%/1.5 mm for all depths at 95% agreement. The technique described in this work for the quantitative evaluation of beam matching using first-order and second-order derivatives is a generalised method. The method is applicable for all energies of both X-ray and electrons and can be used for other linear accelerators without restriction. We recommend that users generate their own test packages (for the bandwidth values) using the demonstrated methodology from their reference linear accelerator rather than from any manufacturer-provided golden beam data set.
All authors would like to sincerely thank Senior Medical Physicist Dr Sateesh Prasad of the National Centre for Cancer Care and Research, Hamad Medical Corporation, Doha, Qatar, for his suggestion and comments. We also sincerely thank Mr CSN Rao, Mr Dilip Kumar and Mr Nilendu Saha of Elekta Medical System (India) and Mr Gopinathan Chandrasekaran of Varian (India) for their technical inputs.
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