FEM design and simulation of a short, 10 MV, S-band Linac with Monte Carlo dose simulations Devin Baillie, J. St. Aubin, B. G. Fallone, and S. Steciw Citation: Medical Physics 42, 2044 (2015); doi: 10.1118/1.4915953 View online: http://dx.doi.org/10.1118/1.4915953 View Table of Contents: http://scitation.aip.org/content/aapm/journal/medphys/42/4?ver=pdfcov Published by the American Association of Physicists in Medicine Articles you may be interested in Skin dose in longitudinal and transverse linac-MRIs using Monte Carlo and realistic 3D MRI field models Med. Phys. 39, 6509 (2012); 10.1118/1.4754657 Effect of longitudinal magnetic fields on a simulated in-line 6 MV linac Med. Phys. 37, 4916 (2010); 10.1118/1.3481513 An integrated 6 MV linear accelerator model from electron gun to dose in a water tank Med. Phys. 37, 2279 (2010); 10.1118/1.3397455 The design of a simulated in-line side-coupled 6 MV linear accelerator waveguide Med. Phys. 37, 466 (2010); 10.1118/1.3276778 High resolution entry and exit Monte Carlo dose calculations from a linear accelerator 6 MV beam under the influence of transverse magnetic fields Med. Phys. 36, 3549 (2009); 10.1118/1.3157203
FEM design and simulation of a short, 10 MV, S-band Linac with Monte Carlo dose simulations Devin Baillie Department of Oncology, Medical Physics Division, University of Alberta, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada
J. St. Aubin Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada and Department of Oncology, Medical Physics Division, University of Alberta, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada
B. G. Fallone Department of Physics, University of Alberta, 11322 – 89 Avenue, Edmonton, Alberta T6G 2G7, Canada; Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada; and Department of Oncology, Medical Physics Division, University of Alberta, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada
S. Steciwa) Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada and Department of Oncology, Medical Physics Division, University of Alberta, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada
(Received 24 October 2014; revised 7 March 2015; accepted for publication 10 March 2015; published 30 March 2015) Purpose: Current commercial 10 MV Linac waveguides are 1.5 m. The authors’ current 6 MV linear accelerator–magnetic resonance imager (Linac–MR) system fits in typical radiotherapy vaults. To allow 10 MV treatments with the Linac–MR and still fit within typical vaults, the authors design a 10 MV Linac with an accelerator waveguide of the same length (27.5 cm) as current 6 MV Linacs. Methods: The first design stage is to design a cavity such that a specific experimental measurement for breakdown is applicable to the cavity. This is accomplished through the use of finite element method (FEM) simulations to match published shunt impedance, Q factor, and ratio of peak to mean-axial electric field strength from an electric breakdown study. A full waveguide is then designed and tuned in FEM simulations based on this cavity design. Electron trajectories are computed through the resulting radio frequency fields, and the waveguide geometry is modified by shifting the first coupling cavity in order to optimize the electron beam properties until the energy spread and mean energy closely match values published for an emulated 10 MV Linac. Finally, Monte Carlo dose simulations are used to compare the resulting photon beam depth dose profile and penumbra with that produced by the emulated 10 MV Linac. Results: The shunt impedance, Q factor, and ratio of peak to mean-axial electric field strength are all matched to within 0.1%. A first coupling cavity shift of 1.45 mm produces an energy spectrum width of 0.347 MeV, very close to the published value for the emulated 10 MV of 0.315 MeV, and a mean energy of 10.53 MeV, nearly identical to the published 10.5 MeV for the emulated 10 MV Linac. The depth dose profile produced by their new Linac is within 1% of that produced by the emulated 10 MV spectrum for all depths greater than 1.5 cm. The penumbra produced is 11% narrower, as measured from 80% to 20% of the central axis dose. Conclusions: The authors have successfully designed and simulated an S-band waveguide of length of 27.5 cm capable of producing a 10 MV photon beam. This waveguide operates well within the breakdown threshold determined for the cavity geometry used. The designed Linac produces depth dose profiles similar to those of the emulated 10 MV Linac (waveguide-length of 1.5 m) but yields a narrower penumbra. C 2015 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4915953] Key words: linac-MR, S-band, linear accelerator, x-ray production, radiation therapy 1. INTRODUCTION A large group of researchers at the Cross Cancer Institute (Edmonton, AB, Canada) is currently developing a hybrid Linac–MR system, where a medical linear accelerator (Linac) and a magnetic resonance (MR) imager are coupled together 2044
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into one device for radiation therapy.1 This Linac–MR system offers the unique ability to track tumors using MR in real time2 while sculpting the radiation field to the tumor as it moves. Less normal tissue is therefore irradiated, allowing for escalated tumor doses, which is expected to increase tumor control and survival rates in cancer patients. The current Linac–MR
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design at the Cross Cancer Institute uses a short 27.5 cm, single-energy 6 MV x-ray waveguide positioned parallel to the treatment direction.1 In this orientation, the electron beam from the Linac is directly incident on the target and bend magnets are not required to redirect the electron beam into the treatment direction. The lack of bend magnets simplifies the integration with the MR imager. Though 6 MV is suitable for many treatments, more optimal dose coverage can be achieved through the use of higher energies for larger patients with deep seated tumors,3–5 up to 10 MV for modulated therapies. Beyond 10 MV, for modulated therapies, there is increased whole body neutron dose.6 To achieve higher energies, a high-energy waveguide is needed, however these waveguides are approximately 1.5 m long, much longer than the 27.5 cm 6 MV waveguides used in current Linac–MR systems. These high-energy waveguides are designed to produce energies up to 18 MV or 22 MV, and no midenergy (10 MV) clinical Linacs are currently available. Conventionally, these long high-energy waveguides are positioned perpendicular to the treatment direction, and the electron beam is redirected onto the x-ray target using bend magnets. The addition of bend magnets in close proximity to the MR imaging magnet would affect the field homogeneity. As the entire unit rotates around the patient, these longer high energy waveguides would require a huge (3 m + 2 × source-axis distance high) treatment vault if positioned parallel to the treatment direction, making the vault renovation costs for the system prohibitive. Current high-energy waveguides also use steering and focusing coils, further complicating its integration with an imaging magnet as they would need to be modified to account for deflections due to the fringe magnetic fields. A compact midenergy waveguide (without bend magnets, buncher cavities, or focusing coils) would allow simpler designs for 10 MV Linac–MR systems and conventional Linacs that still fit in current radiotherapy vaults. Our goal is to design a waveguide that is the same length, 27.5 cm, as the currently used 6 MV waveguide. It would then be a direct replacement within our current Linac–MR system. In order to produce a 10 MV photon beam from a 27.5 cm waveguide, we must increase the energy imparted to the electron beam, which involves increasing the radio frequency (RF) field strength within the waveguide by increasing the input power. The major concern when increasing the waveguide’s input power is electric breakdown within the waveguide. This is caused when microscopic imperfections (such as a small protrusion of material) result in a local enhancement of the electric field by factors of 100 or more.7 This microscopic field enhancement results in a huge current density from the region, up to 1012 A/m2, which results in extreme resistive heating of the metal protrusion and the formation of a plasma. The plasma then arcs across the waveguide, absorbing most of the power from the standing RF wave. Small amounts of liquid metal are formed at the source of the plasma and create additional imperfections which become the source of future arcs. Over time, this damage accumulates and deforms the cavity sufficiently to detune the waveguide until it no longer resonates at the correct frequency and the waveguide will not function.7 In a linear accelerator cavity, the largest electric Medical Physics, Vol. 42, No. 4, April 2015
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fields are at the nose cones (the structure in the red box in Fig. 1), therefore any arcing is between the nose cones in the cavity. Since electric breakdown is caused by imperfections in the surface of the accelerating waveguide, which are random in nature, it is difficult to predict in advance whether a given cavity will break down or not. Threshold electric field strengths are highly dependent on the surface finishing and RF processing techniques, where the RF field strength is gradually increased in order to smooth any microimperfections without damaging the waveguide, used in the construction of the waveguide.7 Fewer and smaller imperfections result in less enhancement of the RF field, which in turn means higher power can be used before breakdown occur. The breakdown threshold is also geometry dependent, as the specific dimensions of the cavity affect the ability for the plasma to arc across the cavity. The RF also affects the threshold field strength, as this affects how long the field emitted current is “on,” which determines the amount of heating as well as the time for the plasma to arc across the cavity;7 higher frequencies therefore have much higher thresholds. The process is not completely understood, which makes it impossible to derive a specific breakdown threshold for a specific cavity geometry, instead we must depend on experimental measurements or phenomenological models of electric breakdown. Several models have been proposed to determine thresholds for RF breakdown within a standing wave cavity, each with its own benefits and drawbacks. Most of these models are phenomenological in nature, fitting various parameters to a function in an attempt to interpolate or extrapolate from the experimentally determined thresholds for specific known cavities. The first attempt to quantify a threshold was done by Kilpatrick in 1957. Kilpatrick extrapolated from low frequency and DC results to produce an expression for a threshold electric field, below which breakdown does not occur, f = 1.64 · E p(th)2 exp(−8.5/E p(th)),
(1)
where f is the frequency in MHz and E p(th) is the peak electric field within the cavity in MV/m. At our frequency, 2997 MHz, this results in a maximum electric field of 46.8 MV/m before breakdown occurs. In additional to the use of only low frequency results, the poor quality vacuum used (10−1–10−5 Pa, compared to 10−8 Pa inside modern accelerators) and the lack of surface finishing or RF processing of the electrodes used allow this threshold to be exceeded by a large margin.8 Wang and Leow published a study which addresses many of the issues with Kilpatrick’s work.9 They used more modern vacuum systems and finishing techniques, and produced a new expression by fitting multiple experiments, some of which were in the frequency range of interest to us, E p(th) = 195 · ( f /1000)1/2,
(2)
where f and E p(th) are as defined above, which gives a maximum electric field of 338 MV/m. This threshold is much more relevant for the current work and was used in our previous feasibility study;10 however, because it was a fit to multiple cavities over an extremely wide frequency range (from 2.8 to 11.4 GHz), it neglects the geometric dependence of breakdown
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F. 1. An axisymmetric cross-section of a single accelerator cavity. The dimensions varied to match the cavity parameters for cavity 1 in Table I are the nose cone length, width, and outer radius of curvature; L NC, WNC, and R NC, respectively. The cavity radius and width, and the inner and outer radii of curvature; R Cav, WCav, R In, and R Out, respectively. Because the electric fields are highest at the nose cones, this is where breakdown can occur, and the dimensions inside the red box are expected to determine breakdown thresholds. The beam hole radius was kept at 2.5 mm for consistency with the Linac compared against.
thresholds that is present even for cavities operating at the same frequency,11 let alone different frequencies. We therefore require either a threshold that is geometry independent or that is specific to the particular cavity geometry we are using. More recently, it was proposed by Dolgashev et al. that surface magnetic field strength could be a geometry independent threshold for electric breakdown.12 Unfortunately for our application, these experiments were all done at much higher frequencies in the x-band (11.424 GHz), so the applicability to our work is questionable. Due to the uncertainties involved in applying a specific breakdown threshold to a different cavity geometry, the breakdown threshold must be determined individually for each cavity geometry. Tanabe, in 1983, published an experimental measurement of breakdown for three cavities,11 along with the shunt impedance, Q factor, and ratio of surface to axial electric field strength (Table I). This experiment is extremely relevant to the current work as the three cavities operate at the same frequency as our intended design. Improved vacuum and modern surface finishing suggest that breakdown thresholds for modern cavities matching this experiment would be higher than those published by Tanabe. Tanabe used a vacuum pressure of 10−5 Pa,11 compared to 10−8 Pa inside modern accelerators. Tanabe also used only 1 h RF processing time with low peak-power11 compared to the 3–14 h of gradually increasing Medical Physics, Vol. 42, No. 4, April 2015
RF power required before maximum field strength achievable was no longer improved in the more recent experiments by Wang and Leow,13 suggesting that the RF processing could be improved. We do not depend on these improvements for the results in this paper, though they strongly suggest that the threshold is quite conservative for modern Linacs. Our group previously published a feasibility study10 using earlier simulations and comparing against the threshold predicted by Eq. (2). This study showed that there was a margin T I. Cavity properties and breakdown thresholds for one of the cavities in the experiment published by Tanabe (Ref. 11): quality factor, Q, effective shunt impedance per unit length, ZT 2 (where Z is the shunt impedance per unit length and T is the transit-time factor), ratio of peak-surface to axial electric field strength, E p /E 0 [axial field E 0, as defined in Karzmark (Ref. 14) or Wangler (Ref. 15)], maximum surface electric field at breakdown, E p(th), and axial field strength at breakdown, E 0(th). Cavity Q ZT 2(MΩ/m) E p /E 0 E p(th)(MV/m) E 0(th)(MV/m)
1
2
3
18 520 104 3.61 239.4 66.3
18 411 117.1 6.04 263.1 43.6
16 835 130.2 8.08 246.4 30.5
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of more than 30% between the fields required to produce a 10 MV photon beam and the breakdown threshold. As discussed above, however, Eq. (2) does not account for the geometric dependence of the breakdown threshold. Our study also used field scaling within the waveguide to approximate optimizations, rather than actually designing an optimized waveguide geometry. The goal of the present research is therefore to design a new accelerator waveguide, based on the experimental numbers published by Tanabe.11 The new waveguide is to be the same length as the Varian 600C waveguide currently used in the Linac–MR project at the Cross Cancer Institute but capable of producing energies up to 10 MV without exceeding the breakdown threshold determined by Tanabe.
2. METHODS 2.A. Waveguide design and RF solution
The first stage of the waveguide design was to model an accelerating cavity based on the experiment published by Tanabe.11 The axial field at breakdown, E0(th), for the three cavities investigated by Tanabe is 66.3 MV/m, 43.6 MV/m, and 30.5 MV/m, respectively. In order to produce a shorter, higherenergy accelerator, we require the highest possible axial field at breakdown, E0(th), as this is the field that accelerates electrons. The high accelerating field in the first cavity (Table I) required for breakdown makes it the most promising candidate for use as a higher energy accelerator. As Tanabe did not publish the cavity geometry, we simulated a single cavity within COMSOL Multiphysics using an axisymmetric geometry and the eigenfrequency solver to calculate the shuntimpedance, Q factor, and field ratio in order to match the published parameters. A stochastic optimization was then run by varying the dimensions from Fig. 1. The dimensions from
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the existing Linac model16 were used as initial conditions, and new dimensions were chosen randomly within a window about the current “best solution” while the window width was reduced by 5% every 10 iterations. The “best solution” was the minimum of the summation of the absolute values of the percent errors in each of the three cavity metrics (Q, ZT 2, and E p /E0). The termination condition was chosen to be 0.1% sum of the absolute values of the errors. This cavity was then used as the basis for our multicavity accelerating waveguide to ensure the applicability of the 239 MV/m breakdown threshold to our accelerating waveguide. A full waveguide was then designed by coupling six cavities together through side coupling cavities, where the first cavity was a half cavity for electron capture14 (AC1 in Fig. 2). The length of each cavity must be one half the RF wavelength so that the electrons which are traveling at ultrarelativistic speeds (v ≈ c) traverse the cavity in one half RF period, arriving in the next cavity as the electric field becomes negative, allowing them to be accelerated through each of the cavities. The half cavity allows the electrons which are not yet traveling at ultrarelativistic speeds to traverse the cavity within one half RF period.14 The single cavity was designed to resonate at 2.997 GHz; however, the irises between the side coupling cavities and the accelerating cavities decreased the cavity inductance and increased the resonant frequency.17 The port to couple power from the RF source was placed in the first full accelerating cavity (AC2 in Fig. 2) instead of the third as in our previous work.16 This was done to allow the ability to vary the beam energy in the future. The coupling port in the first full cavity will allow the fields in the first half-cavity and the fields in the remaining cavities (AC2–AC6 in Fig. 2) to be independently varied using energy switches. The port dimensions were unchanged from the previous work, 17.1 × 23.5 mm with 3.4 mm radius of curvature on the corner.10
F. 2. A full accelerator constructed by coupling multiple accelerating and side-coupling cavities together. The electrons enter from the electron gun into AC1 and are accelerated through AC1–AC6. Power enters the waveguide through the port in AC2. The cavities are tuned by adjusting the diameter in the accelerating cavities (ACx) and the postlength in the coupling cavities (CCx). The first coupling cavity (CC1) is offset by varying amounts to control RF power flow into AC1. The length of the posts (LP) in each coupling cavity is adjusted to tune the resonant frequency. Medical Physics, Vol. 42, No. 4, April 2015
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The input port again decreased the cavity inductance, increasing the resonant frequency. The resonant frequency of a particular cavity was determined by adding copper spheres to detune the other cavities. This process is analogous to the tuning of physical waveguides by adding metal spheres to detune certain cavities. For our simulations, spheres with a diameter of 0.9 times the gap size (distance between nose cones for the accelerating cavities, distance between the posts in the coupling cavities) with perfectly conducting boundary conditions were used. These are much larger than would be physically possible in an actual waveguide where the sphere size is limited by the size of the openings. The use of larger spheres saved significant computational expense because the interior of the spheres was not meshed. COMSOL was then used to calculate the eigenfrequency of the sphere-filled waveguide. Because of the lack of symmetry due to the side coupling cavities and the input port, this required a full 3D model with 90 000–120 000 3rd order isoparametric elements with cubic interpolation functions, a minimum element quality of approximately 0.09 and an element volume ratio of approximately 2.8×10−6, and took approximately 1 h on a computer with four 12 core 2.2 GHz AMD Opteron 6174 CPUs and 224 GB of RAM. In order to restore the resonant frequency of the accelerating cavities after the addition of the coupling cavities and the input port, each cavity had to be individually tuned. In our previous work16,18 to emulate the Varian 600C, we had accomplished this by adjusting the nose cone length (L NC in Fig. 1) to increase or decrease the cavity capacitance in order to adjust the frequency. Because the nose cone is the source of electric breakdown, we wished to leave the dimensions of the nose cone unchanged from those resulting from matching the cavity parameters in the Tanabe breakdown study. Instead, we adjusted the cavity radius (RCav in Fig. 1) to control the inductance, which allowed us to control the frequency. The coupling cavities were tuned by adjusting the length of the posts (L P in Fig. 2) until each coupling cavity resonated at the same frequency as the accelerating cavities. The resonant frequency of a single cavity was calculated, the diameter adjusted, and the resonant frequency recalculated until the cavity resonated at 2997 ± 0.1 MHz. This process was repeated for each of the accelerating cavities. After fully tuning each cavity of the waveguide, we needed to calculate the RF fields within the waveguide. We used the COMSOL RF Module PARDISO solver to find the steady state (harmonic) solution when power is applied at the port. In our previous work,16,18 2.3 MW excitation at the port accurately matched the output from the 2.5 MW magnetron. Tripling this gave 6.9 MW excitation at the port to simulate a 7.5 MW klystron power source. This model had approximately 470 000 3rd order isoparametric elements, solving with cubic interpolation functions, a minimum element quality of approximately 0.25, an element volume ratio of approximately 1.1 × 10−4, and took approximately 2 h on the same computer as the tuning operation. The peak surface fields calculated within the waveguide were then compared with Eth, from Table I, to determine whether electric breakdown would become an issue. Our previous work has shown that the field strength in the first half accelerating cavity (AC1 in Fig. 2) has a large Medical Physics, Vol. 42, No. 4, April 2015
2048 T II. Cavity dimensions to match the parameters published by Tanabe. All dimensions are in mm. L NC WNC R NC WCav R Cav R Out R In
8.79 mm 3.20 mm 1.89 mm 43.63 mm 39.39 mm 16.28 mm 4.35 mm
effect on the resulting electron spectrum18 and therefore need to be precisely optimized to improve the electron (and photon) beam properties. These fields were controlled by offsetting the first coupling cavity (CC1 in Fig. 2) toward the electron gun end of the waveguide, which shrinks the coupling iris and reduces the power coupled into the half cavity. Because the size of the coupling irises changed with this offset, the resonant frequencies of the first two accelerating cavities and the first coupling cavity also changed. These cavities therefore had to be retuned using the same method as the initial tuning for each coupling cavity shift. Tuning the second accelerating cavity (AC2 in Fig. 2) affected the resonant frequency of the second coupling cavity (CC2 in Fig. 2), so this cavity also needed to be retuned. After retuning, the RF solution was recalculated. Reduced power in the first half cavity resulted in higher field strengths in the remaining cavities, so the peak fields within the waveguide were compared to the threshold for each shift investigated. The electron trajectories within the waveguide were then recalculated for each coupling cavity shift, producing one electron phase space incident on the x-ray target for each cavity shift. 2.B. Electron dynamics
PARMELA was then used to calculate the electron trajectories as they are accelerated by the RF fields calculated in the waveguide. As in previous work, the harmonic RF solution from COMSOL was interpolated onto a grid for use with PARMELA.19 The electron phase space injected into the waveguide was produced by the same Opera 3D/SCALA simulated electron gun as our group’s previous work.20 The PARMELA reference particle was started at a phase of −45◦, to ensure it remained in the center of the traveling bunch, and RF phase was
T III. Accelerating cavity radii (R Cav) and coupling cavity postlengths (L P ) after tuning the unshifted and the 1.45 mm shifted (denoted by *) accelerator models. All units are mm. Cavity number
R Cav
LP
R Cav*
L P*
1 2 3 4 5 6
39.338 39.106 39.338 39.338 39.338 39.363
9.626 9.628 9.623 9.624 9.622 N/A
39.315 39.115 39.338 39.338 39.338 39.363
9.624 9.629 9.623 9.624 9.622 N/A
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F. 3. Electron energy spectrum produced by an emulated Varian 10 MV Linac (gray) compared to the spectrum produced by the new waveguide design (black) for the unshifted model (a) and the 1.45 mm shift (b). The dashed line is the 7 MeV cutoff, below which electrons were not included in the beam energy or beam current calculations.
increment by 3.6◦ (3.3 ps) for each step. Three-dimensional space charge effects were calculated on a 32 × 32 × 256 grid. The injected electrons were uniformly spread over two RF periods, which produced leading and trailing half-bunches, which were discarded during postprocessing. Spreading the input electrons over two full periods instead of one accounted for effects of leading and trailing bunches on the central bunch, removing simulation end effects. For each waveguide model, PARMELA was run until 45×106 were kept (i.e., not counting those discarded during processing). In order to evaluate the electron energy spectrum produced by our Linac model, we compared the energy and energy spectrum width of the resulting electron beam with those of an electron energy spectrum emulating a Varian 10 MV Linac, published by Sheik-Bagheri and Rogers.21 The full width at half maximum (FWHM) of the electron energy spectrum was compared with the value published by Sheik-Bagheri and Rogers to emulate a Varian 10 MV Linac. Quantifying the
F. 4. Electron beam FWHM as the coupling cavity is shifted (black points) compared with that of the spectrum of an emulated Varian 10 MV Linac (dashed gray line). Medical Physics, Vol. 42, No. 4, April 2015
energy of the beam was not entirely straightforward due to the differently shaped spectra. Spectrum produced by actual linear accelerators or full simulations (especially in the absence of bend magnets) tend to be binodal (with 2 peak energies) with a low energy tail which does not contribute significantly to the eventual photon beam. Sheik-Bagheri and Rogers produced electron spectrums by using a Gaussian electron energy distribution to match photon output; because the low energy tail does not contribute to the photon beam, it is not accounted for in the emulated Varian 10 MV electron. Instead of taking a mean of all energies, we took the mean of the energies above 7 MeV, which was expected to be more comparable to the Gaussian spectrum used in Sheik-Bagheri and Rogers. A cutoff of 7 MeV was chosen to remove the tail without removing any portion of the peak (relative intensity >5%) for any of the spectra produced. The choice of cutoff energy can slightly change the specific mean energies computed, but values above
F. 5. Electron beam energy as the coupling cavity is shifted (black points) compared with that of an emulated Varian 10 MV Linac (dashed gray line). Electron beam energy is defined here as the mean energy of the electrons with energies greater than 7 MeV.
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F. 6. Waveguide target current as the coupling cavity is shifted (black points) compared against the target current measured on a Varian 10 MV Linac (dashed gray line). The target current only includes those electrons with energies greater than 7 MeV.
6 MeV produce very similar results. However, a cutoff above 7.8 MeV will remove a portion of the peak region, more significantly affecting the results for some of the spectra. While the FWHM and the mean energy do not fully characterize the binodal distributions, as the FWHM approaches 0.5 MeV, the two peaks overlap and the FWHM and mean are expected to provide a reasonable comparison to the Gaussian distribution in this region. 2.C. Monte Carlo dose calculations
EGSnrc was then used to calculate dose distributions produced by the new waveguide models in a water phantom for comparison with an existing 10 MV Linac. A 10 MV Linac head (target, primary collimator, flattening filter, monitor chamber, mirror, and jaws) was simulated in BEAMnrc following Varian specifications,22 and the electron phase spaces produced by PARMELA were converted to an input phase space and used as input.10 For each coupling cavity shift, three field sizes were simulated: 4 × 4, 10 × 10, and 20 × 20 cm,
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producing three phase spaces for input into DOSXYZnrc. DOSXYZnrc was then used to calculate depth dose curves and beam penumbras for each shift and field size. The depth dose curves were calculated with voxel sizes of 5 × 5 × 1 mm for the first 3.5 cm depth and 5 × 5 × 5 mm until 30 cm depth, with the phantom modeled to 40 cm depth to ensure accurate backscatter. As in our previous work,10 we used 100 × 106 histories for the BEAMnrc simulations and 15 × 109 for the DOSXYZnrc simulations. For comparison purposes, the BEAMnrc and DOSXYZnrc simulations were repeated using the Varian 10 MV electron beam properties from SheikBagheri and Rogers.21 The depth of maximum dose, d max, and the ratio of doses at 10 and 20 cm, D10/20, were calculated for each field size, plotted against the cavity offset, and compared to the results for the Varian 10 MV spectrum. These results were used to further adjust the coupling cavity offset for a better agreement.
3. RESULTS AND DISCUSSIONS The stochastic optimization to match the cavity parameters from Tanabe resulted in the dimensions in Table II. Running the stochastic optimization multiple times with different initial conditions consistently produced less than 1% variation in the nose cone dimensions, while variations up to approximately 5% were produced in the remaining dimensions. The diameter of the accelerating cavities after retuning, as well as the postlength in the coupling cavities, are shown with no coupling cavity shift and with a coupling cavity shift of 1.45 mm in Table III. The maximum electric field, EP , within the simulated waveguide for each coupling cavity shift is between 12% and 15% below the breakdown threshold of 239 MV/m. Because the breakdown threshold of 239 MV/m is already conservative due to advances in RF processing and vacuum systems since the experiment was done, electric breakdown is not expected to be a problem for any of the simulated waveguide models. The electron energy spectrum is compared against the Varian 10 MV spectrum from Sheik-Bagheri and Rogers for the simulated waveguide with no coupling cavity shift [Fig. 3(a)] and with a shift of 1.45 mm [Fig. 3(b)]. The FWHMs of the
F. 7. Transverse cross sections of the electron phase space incident on the x-ray target. The divergences (x ′ = p x /p and y ′ = p y /p) are plotted along the ordinate axis while the displacement from the position of maximum intensity is on the abscissa axis. Medical Physics, Vol. 42, No. 4, April 2015
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F. 8. Electron focal spot distribution on the x-ray target, normalized to and centered on the maximum intensity. The darkest region shown corresponds to 50% of the maximum intensity and has a diameter of approximately 0.07 mm. The smaller focal spot with 10% intensity at about y = −0.15 cm corresponds to the similarly positioned asymmetric feature in the phase space plot Fig. 7(b).
unshifted model, the 1.45 mm shift model and the Varian spectrum are 3.12, 0.347, and 0.315 MeV, respectively. The mean energies of the peak (defined as the mean energy of all particles with energies above 7 MeV) are 9.67, 10.53, and 10.5 MeV for the unshifted model, the 1.45 mm shift model, and the Varian spectrum, respectively. The fraction of the electron beam with energy below 7 MeV is 0.12 for the unshifted model and 0.13 for the 1.45 mm shifted model. Figure 4 shows how the FWHM varies as the coupling cavity is shifted and compares against the Varian spectrum, and Fig. 5 shows the mean energy of the peak region, again compared to the Varian spectrum. Taken together, we see that with a shift of 1.45 mm, we have an electron spectrum that is very similar to the Varian spectrum. Figure 6 shows the electron beam current incident on the target, which is more than double that measured on a Varian 10 MV Linac (40 mA) for shifts less than 0.25 mm, and more than triple for shifts greater than 0.25 mm. This was not a specific design goal, but a result of the electron gun used. One
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concern with an increased beam current is heating in the target, which will have an impact on target longevity. A detailed investigation into target design and methods to mitigate target heating issues is required, which is beyond the scope of this work. The uncertainty in Figs. 4–6 is not statistical uncertainty in the particle simulations (which is negligible) but rather a result of the imperfection of the tuning procedure. The x and y transverse cross sections of the electron phase space are plotted in Fig. 7 for the waveguide with the 1.45 mm coupling cavity shift, and the electron focal spot on the target is shown in Fig. 8. The FWHM of the electron focal spot (Fig. 8) is 0.07 mm. The small focal spot will result in high power per unit area in the target which could further contribute to target heating issues caused by the increased beam current. Investigations into the effect of the higher power per unit area in the target, potential cooling techniques, and other mitigation strategies will be investigated in the future but are beyond the scope of this investigation. The relatively large asymmetry in the y direction [Figs. 7(b) and 8] is due to the input coupling port causing asymmetries in the RF fields18 in the second accelerating cavity (AC2 in Fig. 2). Due to this asymmetry, the y direction RMS emittance, ε y,rms, is much larger than the x direction, ε x,rms. In the y direction, the RMS emittance is 0.397 π mm mrad, while in the x direction it is 0.122 π mm mrad. The depth dose profile is compared against one from an identical simulation with a Varian 10 MV spectrum, again for the waveguide model without any coupling cavity shift [Fig. 9(a)] and with a shift of 1.45 mm [Fig. 9(b)]. The higher and narrower energy spectrum of the 1.45 mm shifted waveguide produces a depth dose profile that is within 1% of the Varian 10 MV profile for all points deeper than 1.5 cm, showing that we have achieved a beam energy equivalent to existing Varian 10 MV. Figure 10 shows the same comparisons for the beam penumbra, which are narrower than that of the Varian 10 MV. The width of the penumbra [Fig. 10(b)] for the new Linac design is smaller than that of the Varian 10 MV. If we define the penumbra width as the distance over which the dose drops from 80% to 20% of the central axis dose, the new Linac without coupling cavity shift [Fig. 10(a)] produces a penumbra
F. 9. Depth dose profile from an emulated Varian 10 MV Linac (dashed gray) compared to the profile produced by the new waveguide design (solid black) for the unshifted model (a) and the 1.45 mm shift (b). Each field size has been normalized at 10 cm depth and then independently scaled for clarity (scaling factors of 0.9, 1.0, and 1.1 for field sizes of 4 × 4, 10 × 10, and 20 × 20 cm2, respectively). Medical Physics, Vol. 42, No. 4, April 2015
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F. 10. Beam penumbra from an emulated Varian 10 MV Linac (dashed gray) compared to the penumbra produced by the new waveguide design (solid black) for the unshifted model (a) and the 1.45 mm shift (b). Each field is normalized to the central axis dose. Due to the y direction asymmetry of the beam shown in Figs. 7 and 8, these penumbra are taken in the y direction to demonstrate the worst case scenario.
width of 6.4 mm, which is 14% larger than the Varian penumbra width of 5.6 mm. However, with a coupling cavity shift of 1.45 mm, the penumbra width is reduced by 22% to 5.0 mm, which is 11% smaller than the Varian’s. The FWHM of the focal spot with the 1.45 mm shifted model is 0.07 mm, while that published for the Varian 10 MV is 1.5 mm. While this FWHM is about 20 times smaller than the published value, it does not fully describe the size and shape of the focal spot (Fig. 8). If we artificially reduce the FWHM of the Varian 10 MV focal spot to 0.07 mm and repeat the simulation, the penumbra width is reduced to 5.3 mm, indicating that most of the difference between the 1.45 mm shifted model and the Varian 10 MV Linac is due to the focal spot size. The remaining difference may be due to the shape of the electron distribution on the target or the differences between the energy spectra. The depth of maximum dose, d max, for a 10 × 10 cm field varies with the coupling cavity shift between 22.3 and 23.4 cm. The ratio of dose at 10 and 20 cm depth, D10/20, for a 10×10 cm field varies between 1.59 and 1.61. At a shift of 1.45 mm, there is a d max of 23.40 mm, within 0.5 mm of the 123.73 mm d max for the Varian 10 MV and a D10/20 of 1.595, within 0.5% of the Varian 10 MV, at 1.589.
than 3 times greater than that measured on a Varian 10 MV. The depth of maximum dose is within 0.5 mm of that produced by the Varian 10 MV Linac, and the ratio of dose at 10 cm depth to the dose at 20 cm depth is within 0.5%. For all depths greater than 1.5 cm, the depth dose profiles produced by our new Linac agree to within 1% to that of the Varian 10 MV Linac. The penumbra of the beam produced by our new Linac is 11% smaller than that produced by the Varian 10 MV Linac, which is expected to result in improved dose distributions. All the results obtained here are substantially similar to those obtained in our previous feasibility study, with the addition of dimensions for specific waveguide geometry. This suggests that the field scaling approach used in our feasibility study was an accurate approximation to the coupling cavity offset used in this paper.
4. CONCLUSIONS
a)Author
We have simulated the design of a linear accelerator waveguide capable of producing a 10 MV photon beam, with the waveguide length of 27.5 cm (same as a 600C waveguide). We computed the maximum electric field strengths inside the waveguide to be more than 12% below the breakdown threshold specific to our cavity design. It should be noted that the threshold is already conservative due to advances in vacuum and surface finishing since the publication of the breakdown threshold.7 With a first-coupling-cavity shift of 1.45 mm, the electron beam energy and spectrum FWHM are nearly identical to that of an emulated Varian 10 MV Linac, while the electron beam current incident on the target is more Medical Physics, Vol. 42, No. 4, April 2015
ACKNOWLEDGMENTS Funding for this work was provided by the Alberta Cancer Foundation, Western Economic Diversification Canada, and Alberta Innovates—Health Solutions.
to whom correspondence should be addressed. Electronic mail: [email protected] 1B. G. Fallone et al., “First MR images obtained during megavoltage photon irradiation from a prototype integrated linac-MR system,” Med. Phys. 36(6), 2084–2088 (2009). 2J. Yun, K. Wachowicz, M. Mackenzie, S. Rathee, D. Robinson, and B. G. Fallone, “First demonstration of intrafractional tumor-tracked irradiation using 2D phantom MR images on a prototype linac-MR,” Med. Phys. 40(5), 051718 (12pp.) (2013). 3A. Pirzkall, M. P. Carol, B. Pickett, P. Xia, M. Roach III, and L. J. Verhey, “The effect of beam energy and number of fields on photon-based IMRT for deep-seated targets,” Int. J. Radiat. Oncol., Biol., Phys. 53(2), 434–442 (2002). 4M. Pasler, D. Georg, H. Wirtz, and J. Lutterbach, “Effect of photon-beam energy on VMAT and IMRT treatment plan quality and dosimetric accuracy for advanced prostate Cancer,” Strahlenther. Onkol. 187(12), 792–798 (2011).
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5M.
Hussein, S. Aldridge, T. Guerrero Urbano, and A. Nisbet, “The effect of 6 and 15 MV on intensity-modulated radiation therapy prostate cancer treatment: Plan evaluation, tumour control probability and normal tissue complication probability analysis, and the theoretical risk of secondary induced malignancies,” Br. J. Radiol. 85, 423–432 (2012). 6D. S. Followill, F. Nüsslin, and C. G. Orton, “IMRT should not be administered at photon energies greater than 10 MV,” Med. Phys. 34(6), 1877–1879 (2007). 7J. Wang and G. Loew, Field emission and rf breakdown in high-gradient room-temperature linac structures, SLAC-PUB-7684, 1997. 8W. D. Kilpatrick, “Criterion for vacuum sparking designed to include both rf and dc,” Rev. Sci. Instrum. 28(10), 824–826 (1957). 9J. W. Wang, “RF properties of periodic accelerating structures for linear colliders,” SLAC-Report-339, 1989. 10D. Baillie, J. S. Aubin, B. G. Fallone, and S. Steciw, “Feasibility of producing a short, high energy s-band linear accelerator using a klystron power source,” Med. Phys. 40(4), 041713 (5pp.) (2013). 11E. Tanabe, “Voltage breakdown in s-band linear accelerator cavities,” IEEE Trans. Nucl. Sci. 30(4), 3551–3553 (1983). 12V. Dolgashev, S. Tantawi, Y. Higashi, and B. Spataro, “Geometric dependence of radio-frequency breakdown in normal conducting accelerating structures,” Appl. Phys. Lett. 97, 171501 (2010).
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A. Loew and J. W. Wang, “RF breakdown studies in room temperature electron linac structures,” SLAC-PUB-4647, 1988. 14C. J. Karzmark, Medical Electron Accelerators (McGraw-Hill, Inc., Health Professions Division, New York, 1993). 15T. P. Wangler, Rf Linear Accelerators (Wiley-VCH, Weinheim, 2008). 16J. St. Aubin, S. Steciw, C. Kirkby, and B. Fallone, “An integrated 6 MV linear accelerator model from electron gun to dose in a water tank,” Med. Phys. 37, 2279–2288 (2010). 17D. E. Nagle, E. A. Knapp, and B. C. Knapp, “Coupled resonator model for standing wave accelerator tanks,” Rev. Sci. Instrum. 38(11), 1583–1587 (2004). 18J. St Aubin, S. Steciw, and B. Fallone, “The design of a simulated in-line side-coupled 6 MV linear accelerator waveguide,” Med. Phys. 37, 466–476 (2010). 19B. J. H. and L. M. Young, Parmela, LA-UR-96–1835, Rev, 2005. 20J. St Aubin, S. Steciw, and B. Fallone, “Effect of transverse magnetic fields on a simulated in-line 6 MV linac,” Phys. Med. Biol. 55, 4861–4869 (2010). 21D. Sheikh-Bagheri and D. W. O. Rogers, “Sensitivity of megavoltage photon beam Monte Carlo simulations to electron beam and other parameters,” Med. Phys. 29, 379–390 (2002). 22“Varian medical systems Monte Carlo data package,” available under nondisclosure-agreement from Varian medical systems.
FEM design and simulation of a short, 10 MV, S-band Linac with Monte Carlo dose simulations Devin Baillie Department of Oncology, Medical Physics Division, University of Alberta, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada
J. St. Aubin Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada and Department of Oncology, Medical Physics Division, University of Alberta, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada
B. G. Fallone Department of Physics, University of Alberta, 11322 – 89 Avenue, Edmonton, Alberta T6G 2G7, Canada; Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada; and Department of Oncology, Medical Physics Division, University of Alberta, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada
S. Steciwa) Department of Medical Physics, Cross Cancer Institute, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada and Department of Oncology, Medical Physics Division, University of Alberta, 11560 University Avenue, Edmonton, Alberta T6G 1Z2, Canada
(Received 24 October 2014; revised 7 March 2015; accepted for publication 10 March 2015; published 30 March 2015) Purpose: Current commercial 10 MV Linac waveguides are 1.5 m. The authors’ current 6 MV linear accelerator–magnetic resonance imager (Linac–MR) system fits in typical radiotherapy vaults. To allow 10 MV treatments with the Linac–MR and still fit within typical vaults, the authors design a 10 MV Linac with an accelerator waveguide of the same length (27.5 cm) as current 6 MV Linacs. Methods: The first design stage is to design a cavity such that a specific experimental measurement for breakdown is applicable to the cavity. This is accomplished through the use of finite element method (FEM) simulations to match published shunt impedance, Q factor, and ratio of peak to mean-axial electric field strength from an electric breakdown study. A full waveguide is then designed and tuned in FEM simulations based on this cavity design. Electron trajectories are computed through the resulting radio frequency fields, and the waveguide geometry is modified by shifting the first coupling cavity in order to optimize the electron beam properties until the energy spread and mean energy closely match values published for an emulated 10 MV Linac. Finally, Monte Carlo dose simulations are used to compare the resulting photon beam depth dose profile and penumbra with that produced by the emulated 10 MV Linac. Results: The shunt impedance, Q factor, and ratio of peak to mean-axial electric field strength are all matched to within 0.1%. A first coupling cavity shift of 1.45 mm produces an energy spectrum width of 0.347 MeV, very close to the published value for the emulated 10 MV of 0.315 MeV, and a mean energy of 10.53 MeV, nearly identical to the published 10.5 MeV for the emulated 10 MV Linac. The depth dose profile produced by their new Linac is within 1% of that produced by the emulated 10 MV spectrum for all depths greater than 1.5 cm. The penumbra produced is 11% narrower, as measured from 80% to 20% of the central axis dose. Conclusions: The authors have successfully designed and simulated an S-band waveguide of length of 27.5 cm capable of producing a 10 MV photon beam. This waveguide operates well within the breakdown threshold determined for the cavity geometry used. The designed Linac produces depth dose profiles similar to those of the emulated 10 MV Linac (waveguide-length of 1.5 m) but yields a narrower penumbra. C 2015 American Association of Physicists in Medicine. [http://dx.doi.org/10.1118/1.4915953] Key words: linac-MR, S-band, linear accelerator, x-ray production, radiation therapy 1. INTRODUCTION A large group of researchers at the Cross Cancer Institute (Edmonton, AB, Canada) is currently developing a hybrid Linac–MR system, where a medical linear accelerator (Linac) and a magnetic resonance (MR) imager are coupled together 2044
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into one device for radiation therapy.1 This Linac–MR system offers the unique ability to track tumors using MR in real time2 while sculpting the radiation field to the tumor as it moves. Less normal tissue is therefore irradiated, allowing for escalated tumor doses, which is expected to increase tumor control and survival rates in cancer patients. The current Linac–MR
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© 2015 Am. Assoc. Phys. Med.
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design at the Cross Cancer Institute uses a short 27.5 cm, single-energy 6 MV x-ray waveguide positioned parallel to the treatment direction.1 In this orientation, the electron beam from the Linac is directly incident on the target and bend magnets are not required to redirect the electron beam into the treatment direction. The lack of bend magnets simplifies the integration with the MR imager. Though 6 MV is suitable for many treatments, more optimal dose coverage can be achieved through the use of higher energies for larger patients with deep seated tumors,3–5 up to 10 MV for modulated therapies. Beyond 10 MV, for modulated therapies, there is increased whole body neutron dose.6 To achieve higher energies, a high-energy waveguide is needed, however these waveguides are approximately 1.5 m long, much longer than the 27.5 cm 6 MV waveguides used in current Linac–MR systems. These high-energy waveguides are designed to produce energies up to 18 MV or 22 MV, and no midenergy (10 MV) clinical Linacs are currently available. Conventionally, these long high-energy waveguides are positioned perpendicular to the treatment direction, and the electron beam is redirected onto the x-ray target using bend magnets. The addition of bend magnets in close proximity to the MR imaging magnet would affect the field homogeneity. As the entire unit rotates around the patient, these longer high energy waveguides would require a huge (3 m + 2 × source-axis distance high) treatment vault if positioned parallel to the treatment direction, making the vault renovation costs for the system prohibitive. Current high-energy waveguides also use steering and focusing coils, further complicating its integration with an imaging magnet as they would need to be modified to account for deflections due to the fringe magnetic fields. A compact midenergy waveguide (without bend magnets, buncher cavities, or focusing coils) would allow simpler designs for 10 MV Linac–MR systems and conventional Linacs that still fit in current radiotherapy vaults. Our goal is to design a waveguide that is the same length, 27.5 cm, as the currently used 6 MV waveguide. It would then be a direct replacement within our current Linac–MR system. In order to produce a 10 MV photon beam from a 27.5 cm waveguide, we must increase the energy imparted to the electron beam, which involves increasing the radio frequency (RF) field strength within the waveguide by increasing the input power. The major concern when increasing the waveguide’s input power is electric breakdown within the waveguide. This is caused when microscopic imperfections (such as a small protrusion of material) result in a local enhancement of the electric field by factors of 100 or more.7 This microscopic field enhancement results in a huge current density from the region, up to 1012 A/m2, which results in extreme resistive heating of the metal protrusion and the formation of a plasma. The plasma then arcs across the waveguide, absorbing most of the power from the standing RF wave. Small amounts of liquid metal are formed at the source of the plasma and create additional imperfections which become the source of future arcs. Over time, this damage accumulates and deforms the cavity sufficiently to detune the waveguide until it no longer resonates at the correct frequency and the waveguide will not function.7 In a linear accelerator cavity, the largest electric Medical Physics, Vol. 42, No. 4, April 2015
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fields are at the nose cones (the structure in the red box in Fig. 1), therefore any arcing is between the nose cones in the cavity. Since electric breakdown is caused by imperfections in the surface of the accelerating waveguide, which are random in nature, it is difficult to predict in advance whether a given cavity will break down or not. Threshold electric field strengths are highly dependent on the surface finishing and RF processing techniques, where the RF field strength is gradually increased in order to smooth any microimperfections without damaging the waveguide, used in the construction of the waveguide.7 Fewer and smaller imperfections result in less enhancement of the RF field, which in turn means higher power can be used before breakdown occur. The breakdown threshold is also geometry dependent, as the specific dimensions of the cavity affect the ability for the plasma to arc across the cavity. The RF also affects the threshold field strength, as this affects how long the field emitted current is “on,” which determines the amount of heating as well as the time for the plasma to arc across the cavity;7 higher frequencies therefore have much higher thresholds. The process is not completely understood, which makes it impossible to derive a specific breakdown threshold for a specific cavity geometry, instead we must depend on experimental measurements or phenomenological models of electric breakdown. Several models have been proposed to determine thresholds for RF breakdown within a standing wave cavity, each with its own benefits and drawbacks. Most of these models are phenomenological in nature, fitting various parameters to a function in an attempt to interpolate or extrapolate from the experimentally determined thresholds for specific known cavities. The first attempt to quantify a threshold was done by Kilpatrick in 1957. Kilpatrick extrapolated from low frequency and DC results to produce an expression for a threshold electric field, below which breakdown does not occur, f = 1.64 · E p(th)2 exp(−8.5/E p(th)),
(1)
where f is the frequency in MHz and E p(th) is the peak electric field within the cavity in MV/m. At our frequency, 2997 MHz, this results in a maximum electric field of 46.8 MV/m before breakdown occurs. In additional to the use of only low frequency results, the poor quality vacuum used (10−1–10−5 Pa, compared to 10−8 Pa inside modern accelerators) and the lack of surface finishing or RF processing of the electrodes used allow this threshold to be exceeded by a large margin.8 Wang and Leow published a study which addresses many of the issues with Kilpatrick’s work.9 They used more modern vacuum systems and finishing techniques, and produced a new expression by fitting multiple experiments, some of which were in the frequency range of interest to us, E p(th) = 195 · ( f /1000)1/2,
(2)
where f and E p(th) are as defined above, which gives a maximum electric field of 338 MV/m. This threshold is much more relevant for the current work and was used in our previous feasibility study;10 however, because it was a fit to multiple cavities over an extremely wide frequency range (from 2.8 to 11.4 GHz), it neglects the geometric dependence of breakdown
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F. 1. An axisymmetric cross-section of a single accelerator cavity. The dimensions varied to match the cavity parameters for cavity 1 in Table I are the nose cone length, width, and outer radius of curvature; L NC, WNC, and R NC, respectively. The cavity radius and width, and the inner and outer radii of curvature; R Cav, WCav, R In, and R Out, respectively. Because the electric fields are highest at the nose cones, this is where breakdown can occur, and the dimensions inside the red box are expected to determine breakdown thresholds. The beam hole radius was kept at 2.5 mm for consistency with the Linac compared against.
thresholds that is present even for cavities operating at the same frequency,11 let alone different frequencies. We therefore require either a threshold that is geometry independent or that is specific to the particular cavity geometry we are using. More recently, it was proposed by Dolgashev et al. that surface magnetic field strength could be a geometry independent threshold for electric breakdown.12 Unfortunately for our application, these experiments were all done at much higher frequencies in the x-band (11.424 GHz), so the applicability to our work is questionable. Due to the uncertainties involved in applying a specific breakdown threshold to a different cavity geometry, the breakdown threshold must be determined individually for each cavity geometry. Tanabe, in 1983, published an experimental measurement of breakdown for three cavities,11 along with the shunt impedance, Q factor, and ratio of surface to axial electric field strength (Table I). This experiment is extremely relevant to the current work as the three cavities operate at the same frequency as our intended design. Improved vacuum and modern surface finishing suggest that breakdown thresholds for modern cavities matching this experiment would be higher than those published by Tanabe. Tanabe used a vacuum pressure of 10−5 Pa,11 compared to 10−8 Pa inside modern accelerators. Tanabe also used only 1 h RF processing time with low peak-power11 compared to the 3–14 h of gradually increasing Medical Physics, Vol. 42, No. 4, April 2015
RF power required before maximum field strength achievable was no longer improved in the more recent experiments by Wang and Leow,13 suggesting that the RF processing could be improved. We do not depend on these improvements for the results in this paper, though they strongly suggest that the threshold is quite conservative for modern Linacs. Our group previously published a feasibility study10 using earlier simulations and comparing against the threshold predicted by Eq. (2). This study showed that there was a margin T I. Cavity properties and breakdown thresholds for one of the cavities in the experiment published by Tanabe (Ref. 11): quality factor, Q, effective shunt impedance per unit length, ZT 2 (where Z is the shunt impedance per unit length and T is the transit-time factor), ratio of peak-surface to axial electric field strength, E p /E 0 [axial field E 0, as defined in Karzmark (Ref. 14) or Wangler (Ref. 15)], maximum surface electric field at breakdown, E p(th), and axial field strength at breakdown, E 0(th). Cavity Q ZT 2(MΩ/m) E p /E 0 E p(th)(MV/m) E 0(th)(MV/m)
1
2
3
18 520 104 3.61 239.4 66.3
18 411 117.1 6.04 263.1 43.6
16 835 130.2 8.08 246.4 30.5
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of more than 30% between the fields required to produce a 10 MV photon beam and the breakdown threshold. As discussed above, however, Eq. (2) does not account for the geometric dependence of the breakdown threshold. Our study also used field scaling within the waveguide to approximate optimizations, rather than actually designing an optimized waveguide geometry. The goal of the present research is therefore to design a new accelerator waveguide, based on the experimental numbers published by Tanabe.11 The new waveguide is to be the same length as the Varian 600C waveguide currently used in the Linac–MR project at the Cross Cancer Institute but capable of producing energies up to 10 MV without exceeding the breakdown threshold determined by Tanabe.
2. METHODS 2.A. Waveguide design and RF solution
The first stage of the waveguide design was to model an accelerating cavity based on the experiment published by Tanabe.11 The axial field at breakdown, E0(th), for the three cavities investigated by Tanabe is 66.3 MV/m, 43.6 MV/m, and 30.5 MV/m, respectively. In order to produce a shorter, higherenergy accelerator, we require the highest possible axial field at breakdown, E0(th), as this is the field that accelerates electrons. The high accelerating field in the first cavity (Table I) required for breakdown makes it the most promising candidate for use as a higher energy accelerator. As Tanabe did not publish the cavity geometry, we simulated a single cavity within COMSOL Multiphysics using an axisymmetric geometry and the eigenfrequency solver to calculate the shuntimpedance, Q factor, and field ratio in order to match the published parameters. A stochastic optimization was then run by varying the dimensions from Fig. 1. The dimensions from
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the existing Linac model16 were used as initial conditions, and new dimensions were chosen randomly within a window about the current “best solution” while the window width was reduced by 5% every 10 iterations. The “best solution” was the minimum of the summation of the absolute values of the percent errors in each of the three cavity metrics (Q, ZT 2, and E p /E0). The termination condition was chosen to be 0.1% sum of the absolute values of the errors. This cavity was then used as the basis for our multicavity accelerating waveguide to ensure the applicability of the 239 MV/m breakdown threshold to our accelerating waveguide. A full waveguide was then designed by coupling six cavities together through side coupling cavities, where the first cavity was a half cavity for electron capture14 (AC1 in Fig. 2). The length of each cavity must be one half the RF wavelength so that the electrons which are traveling at ultrarelativistic speeds (v ≈ c) traverse the cavity in one half RF period, arriving in the next cavity as the electric field becomes negative, allowing them to be accelerated through each of the cavities. The half cavity allows the electrons which are not yet traveling at ultrarelativistic speeds to traverse the cavity within one half RF period.14 The single cavity was designed to resonate at 2.997 GHz; however, the irises between the side coupling cavities and the accelerating cavities decreased the cavity inductance and increased the resonant frequency.17 The port to couple power from the RF source was placed in the first full accelerating cavity (AC2 in Fig. 2) instead of the third as in our previous work.16 This was done to allow the ability to vary the beam energy in the future. The coupling port in the first full cavity will allow the fields in the first half-cavity and the fields in the remaining cavities (AC2–AC6 in Fig. 2) to be independently varied using energy switches. The port dimensions were unchanged from the previous work, 17.1 × 23.5 mm with 3.4 mm radius of curvature on the corner.10
F. 2. A full accelerator constructed by coupling multiple accelerating and side-coupling cavities together. The electrons enter from the electron gun into AC1 and are accelerated through AC1–AC6. Power enters the waveguide through the port in AC2. The cavities are tuned by adjusting the diameter in the accelerating cavities (ACx) and the postlength in the coupling cavities (CCx). The first coupling cavity (CC1) is offset by varying amounts to control RF power flow into AC1. The length of the posts (LP) in each coupling cavity is adjusted to tune the resonant frequency. Medical Physics, Vol. 42, No. 4, April 2015
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The input port again decreased the cavity inductance, increasing the resonant frequency. The resonant frequency of a particular cavity was determined by adding copper spheres to detune the other cavities. This process is analogous to the tuning of physical waveguides by adding metal spheres to detune certain cavities. For our simulations, spheres with a diameter of 0.9 times the gap size (distance between nose cones for the accelerating cavities, distance between the posts in the coupling cavities) with perfectly conducting boundary conditions were used. These are much larger than would be physically possible in an actual waveguide where the sphere size is limited by the size of the openings. The use of larger spheres saved significant computational expense because the interior of the spheres was not meshed. COMSOL was then used to calculate the eigenfrequency of the sphere-filled waveguide. Because of the lack of symmetry due to the side coupling cavities and the input port, this required a full 3D model with 90 000–120 000 3rd order isoparametric elements with cubic interpolation functions, a minimum element quality of approximately 0.09 and an element volume ratio of approximately 2.8×10−6, and took approximately 1 h on a computer with four 12 core 2.2 GHz AMD Opteron 6174 CPUs and 224 GB of RAM. In order to restore the resonant frequency of the accelerating cavities after the addition of the coupling cavities and the input port, each cavity had to be individually tuned. In our previous work16,18 to emulate the Varian 600C, we had accomplished this by adjusting the nose cone length (L NC in Fig. 1) to increase or decrease the cavity capacitance in order to adjust the frequency. Because the nose cone is the source of electric breakdown, we wished to leave the dimensions of the nose cone unchanged from those resulting from matching the cavity parameters in the Tanabe breakdown study. Instead, we adjusted the cavity radius (RCav in Fig. 1) to control the inductance, which allowed us to control the frequency. The coupling cavities were tuned by adjusting the length of the posts (L P in Fig. 2) until each coupling cavity resonated at the same frequency as the accelerating cavities. The resonant frequency of a single cavity was calculated, the diameter adjusted, and the resonant frequency recalculated until the cavity resonated at 2997 ± 0.1 MHz. This process was repeated for each of the accelerating cavities. After fully tuning each cavity of the waveguide, we needed to calculate the RF fields within the waveguide. We used the COMSOL RF Module PARDISO solver to find the steady state (harmonic) solution when power is applied at the port. In our previous work,16,18 2.3 MW excitation at the port accurately matched the output from the 2.5 MW magnetron. Tripling this gave 6.9 MW excitation at the port to simulate a 7.5 MW klystron power source. This model had approximately 470 000 3rd order isoparametric elements, solving with cubic interpolation functions, a minimum element quality of approximately 0.25, an element volume ratio of approximately 1.1 × 10−4, and took approximately 2 h on the same computer as the tuning operation. The peak surface fields calculated within the waveguide were then compared with Eth, from Table I, to determine whether electric breakdown would become an issue. Our previous work has shown that the field strength in the first half accelerating cavity (AC1 in Fig. 2) has a large Medical Physics, Vol. 42, No. 4, April 2015
2048 T II. Cavity dimensions to match the parameters published by Tanabe. All dimensions are in mm. L NC WNC R NC WCav R Cav R Out R In
8.79 mm 3.20 mm 1.89 mm 43.63 mm 39.39 mm 16.28 mm 4.35 mm
effect on the resulting electron spectrum18 and therefore need to be precisely optimized to improve the electron (and photon) beam properties. These fields were controlled by offsetting the first coupling cavity (CC1 in Fig. 2) toward the electron gun end of the waveguide, which shrinks the coupling iris and reduces the power coupled into the half cavity. Because the size of the coupling irises changed with this offset, the resonant frequencies of the first two accelerating cavities and the first coupling cavity also changed. These cavities therefore had to be retuned using the same method as the initial tuning for each coupling cavity shift. Tuning the second accelerating cavity (AC2 in Fig. 2) affected the resonant frequency of the second coupling cavity (CC2 in Fig. 2), so this cavity also needed to be retuned. After retuning, the RF solution was recalculated. Reduced power in the first half cavity resulted in higher field strengths in the remaining cavities, so the peak fields within the waveguide were compared to the threshold for each shift investigated. The electron trajectories within the waveguide were then recalculated for each coupling cavity shift, producing one electron phase space incident on the x-ray target for each cavity shift. 2.B. Electron dynamics
PARMELA was then used to calculate the electron trajectories as they are accelerated by the RF fields calculated in the waveguide. As in previous work, the harmonic RF solution from COMSOL was interpolated onto a grid for use with PARMELA.19 The electron phase space injected into the waveguide was produced by the same Opera 3D/SCALA simulated electron gun as our group’s previous work.20 The PARMELA reference particle was started at a phase of −45◦, to ensure it remained in the center of the traveling bunch, and RF phase was
T III. Accelerating cavity radii (R Cav) and coupling cavity postlengths (L P ) after tuning the unshifted and the 1.45 mm shifted (denoted by *) accelerator models. All units are mm. Cavity number
R Cav
LP
R Cav*
L P*
1 2 3 4 5 6
39.338 39.106 39.338 39.338 39.338 39.363
9.626 9.628 9.623 9.624 9.622 N/A
39.315 39.115 39.338 39.338 39.338 39.363
9.624 9.629 9.623 9.624 9.622 N/A
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F. 3. Electron energy spectrum produced by an emulated Varian 10 MV Linac (gray) compared to the spectrum produced by the new waveguide design (black) for the unshifted model (a) and the 1.45 mm shift (b). The dashed line is the 7 MeV cutoff, below which electrons were not included in the beam energy or beam current calculations.
increment by 3.6◦ (3.3 ps) for each step. Three-dimensional space charge effects were calculated on a 32 × 32 × 256 grid. The injected electrons were uniformly spread over two RF periods, which produced leading and trailing half-bunches, which were discarded during postprocessing. Spreading the input electrons over two full periods instead of one accounted for effects of leading and trailing bunches on the central bunch, removing simulation end effects. For each waveguide model, PARMELA was run until 45×106 were kept (i.e., not counting those discarded during processing). In order to evaluate the electron energy spectrum produced by our Linac model, we compared the energy and energy spectrum width of the resulting electron beam with those of an electron energy spectrum emulating a Varian 10 MV Linac, published by Sheik-Bagheri and Rogers.21 The full width at half maximum (FWHM) of the electron energy spectrum was compared with the value published by Sheik-Bagheri and Rogers to emulate a Varian 10 MV Linac. Quantifying the
F. 4. Electron beam FWHM as the coupling cavity is shifted (black points) compared with that of the spectrum of an emulated Varian 10 MV Linac (dashed gray line). Medical Physics, Vol. 42, No. 4, April 2015
energy of the beam was not entirely straightforward due to the differently shaped spectra. Spectrum produced by actual linear accelerators or full simulations (especially in the absence of bend magnets) tend to be binodal (with 2 peak energies) with a low energy tail which does not contribute significantly to the eventual photon beam. Sheik-Bagheri and Rogers produced electron spectrums by using a Gaussian electron energy distribution to match photon output; because the low energy tail does not contribute to the photon beam, it is not accounted for in the emulated Varian 10 MV electron. Instead of taking a mean of all energies, we took the mean of the energies above 7 MeV, which was expected to be more comparable to the Gaussian spectrum used in Sheik-Bagheri and Rogers. A cutoff of 7 MeV was chosen to remove the tail without removing any portion of the peak (relative intensity >5%) for any of the spectra produced. The choice of cutoff energy can slightly change the specific mean energies computed, but values above
F. 5. Electron beam energy as the coupling cavity is shifted (black points) compared with that of an emulated Varian 10 MV Linac (dashed gray line). Electron beam energy is defined here as the mean energy of the electrons with energies greater than 7 MeV.
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F. 6. Waveguide target current as the coupling cavity is shifted (black points) compared against the target current measured on a Varian 10 MV Linac (dashed gray line). The target current only includes those electrons with energies greater than 7 MeV.
6 MeV produce very similar results. However, a cutoff above 7.8 MeV will remove a portion of the peak region, more significantly affecting the results for some of the spectra. While the FWHM and the mean energy do not fully characterize the binodal distributions, as the FWHM approaches 0.5 MeV, the two peaks overlap and the FWHM and mean are expected to provide a reasonable comparison to the Gaussian distribution in this region. 2.C. Monte Carlo dose calculations
EGSnrc was then used to calculate dose distributions produced by the new waveguide models in a water phantom for comparison with an existing 10 MV Linac. A 10 MV Linac head (target, primary collimator, flattening filter, monitor chamber, mirror, and jaws) was simulated in BEAMnrc following Varian specifications,22 and the electron phase spaces produced by PARMELA were converted to an input phase space and used as input.10 For each coupling cavity shift, three field sizes were simulated: 4 × 4, 10 × 10, and 20 × 20 cm,
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producing three phase spaces for input into DOSXYZnrc. DOSXYZnrc was then used to calculate depth dose curves and beam penumbras for each shift and field size. The depth dose curves were calculated with voxel sizes of 5 × 5 × 1 mm for the first 3.5 cm depth and 5 × 5 × 5 mm until 30 cm depth, with the phantom modeled to 40 cm depth to ensure accurate backscatter. As in our previous work,10 we used 100 × 106 histories for the BEAMnrc simulations and 15 × 109 for the DOSXYZnrc simulations. For comparison purposes, the BEAMnrc and DOSXYZnrc simulations were repeated using the Varian 10 MV electron beam properties from SheikBagheri and Rogers.21 The depth of maximum dose, d max, and the ratio of doses at 10 and 20 cm, D10/20, were calculated for each field size, plotted against the cavity offset, and compared to the results for the Varian 10 MV spectrum. These results were used to further adjust the coupling cavity offset for a better agreement.
3. RESULTS AND DISCUSSIONS The stochastic optimization to match the cavity parameters from Tanabe resulted in the dimensions in Table II. Running the stochastic optimization multiple times with different initial conditions consistently produced less than 1% variation in the nose cone dimensions, while variations up to approximately 5% were produced in the remaining dimensions. The diameter of the accelerating cavities after retuning, as well as the postlength in the coupling cavities, are shown with no coupling cavity shift and with a coupling cavity shift of 1.45 mm in Table III. The maximum electric field, EP , within the simulated waveguide for each coupling cavity shift is between 12% and 15% below the breakdown threshold of 239 MV/m. Because the breakdown threshold of 239 MV/m is already conservative due to advances in RF processing and vacuum systems since the experiment was done, electric breakdown is not expected to be a problem for any of the simulated waveguide models. The electron energy spectrum is compared against the Varian 10 MV spectrum from Sheik-Bagheri and Rogers for the simulated waveguide with no coupling cavity shift [Fig. 3(a)] and with a shift of 1.45 mm [Fig. 3(b)]. The FWHMs of the
F. 7. Transverse cross sections of the electron phase space incident on the x-ray target. The divergences (x ′ = p x /p and y ′ = p y /p) are plotted along the ordinate axis while the displacement from the position of maximum intensity is on the abscissa axis. Medical Physics, Vol. 42, No. 4, April 2015
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F. 8. Electron focal spot distribution on the x-ray target, normalized to and centered on the maximum intensity. The darkest region shown corresponds to 50% of the maximum intensity and has a diameter of approximately 0.07 mm. The smaller focal spot with 10% intensity at about y = −0.15 cm corresponds to the similarly positioned asymmetric feature in the phase space plot Fig. 7(b).
unshifted model, the 1.45 mm shift model and the Varian spectrum are 3.12, 0.347, and 0.315 MeV, respectively. The mean energies of the peak (defined as the mean energy of all particles with energies above 7 MeV) are 9.67, 10.53, and 10.5 MeV for the unshifted model, the 1.45 mm shift model, and the Varian spectrum, respectively. The fraction of the electron beam with energy below 7 MeV is 0.12 for the unshifted model and 0.13 for the 1.45 mm shifted model. Figure 4 shows how the FWHM varies as the coupling cavity is shifted and compares against the Varian spectrum, and Fig. 5 shows the mean energy of the peak region, again compared to the Varian spectrum. Taken together, we see that with a shift of 1.45 mm, we have an electron spectrum that is very similar to the Varian spectrum. Figure 6 shows the electron beam current incident on the target, which is more than double that measured on a Varian 10 MV Linac (40 mA) for shifts less than 0.25 mm, and more than triple for shifts greater than 0.25 mm. This was not a specific design goal, but a result of the electron gun used. One
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concern with an increased beam current is heating in the target, which will have an impact on target longevity. A detailed investigation into target design and methods to mitigate target heating issues is required, which is beyond the scope of this work. The uncertainty in Figs. 4–6 is not statistical uncertainty in the particle simulations (which is negligible) but rather a result of the imperfection of the tuning procedure. The x and y transverse cross sections of the electron phase space are plotted in Fig. 7 for the waveguide with the 1.45 mm coupling cavity shift, and the electron focal spot on the target is shown in Fig. 8. The FWHM of the electron focal spot (Fig. 8) is 0.07 mm. The small focal spot will result in high power per unit area in the target which could further contribute to target heating issues caused by the increased beam current. Investigations into the effect of the higher power per unit area in the target, potential cooling techniques, and other mitigation strategies will be investigated in the future but are beyond the scope of this investigation. The relatively large asymmetry in the y direction [Figs. 7(b) and 8] is due to the input coupling port causing asymmetries in the RF fields18 in the second accelerating cavity (AC2 in Fig. 2). Due to this asymmetry, the y direction RMS emittance, ε y,rms, is much larger than the x direction, ε x,rms. In the y direction, the RMS emittance is 0.397 π mm mrad, while in the x direction it is 0.122 π mm mrad. The depth dose profile is compared against one from an identical simulation with a Varian 10 MV spectrum, again for the waveguide model without any coupling cavity shift [Fig. 9(a)] and with a shift of 1.45 mm [Fig. 9(b)]. The higher and narrower energy spectrum of the 1.45 mm shifted waveguide produces a depth dose profile that is within 1% of the Varian 10 MV profile for all points deeper than 1.5 cm, showing that we have achieved a beam energy equivalent to existing Varian 10 MV. Figure 10 shows the same comparisons for the beam penumbra, which are narrower than that of the Varian 10 MV. The width of the penumbra [Fig. 10(b)] for the new Linac design is smaller than that of the Varian 10 MV. If we define the penumbra width as the distance over which the dose drops from 80% to 20% of the central axis dose, the new Linac without coupling cavity shift [Fig. 10(a)] produces a penumbra
F. 9. Depth dose profile from an emulated Varian 10 MV Linac (dashed gray) compared to the profile produced by the new waveguide design (solid black) for the unshifted model (a) and the 1.45 mm shift (b). Each field size has been normalized at 10 cm depth and then independently scaled for clarity (scaling factors of 0.9, 1.0, and 1.1 for field sizes of 4 × 4, 10 × 10, and 20 × 20 cm2, respectively). Medical Physics, Vol. 42, No. 4, April 2015
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F. 10. Beam penumbra from an emulated Varian 10 MV Linac (dashed gray) compared to the penumbra produced by the new waveguide design (solid black) for the unshifted model (a) and the 1.45 mm shift (b). Each field is normalized to the central axis dose. Due to the y direction asymmetry of the beam shown in Figs. 7 and 8, these penumbra are taken in the y direction to demonstrate the worst case scenario.
width of 6.4 mm, which is 14% larger than the Varian penumbra width of 5.6 mm. However, with a coupling cavity shift of 1.45 mm, the penumbra width is reduced by 22% to 5.0 mm, which is 11% smaller than the Varian’s. The FWHM of the focal spot with the 1.45 mm shifted model is 0.07 mm, while that published for the Varian 10 MV is 1.5 mm. While this FWHM is about 20 times smaller than the published value, it does not fully describe the size and shape of the focal spot (Fig. 8). If we artificially reduce the FWHM of the Varian 10 MV focal spot to 0.07 mm and repeat the simulation, the penumbra width is reduced to 5.3 mm, indicating that most of the difference between the 1.45 mm shifted model and the Varian 10 MV Linac is due to the focal spot size. The remaining difference may be due to the shape of the electron distribution on the target or the differences between the energy spectra. The depth of maximum dose, d max, for a 10 × 10 cm field varies with the coupling cavity shift between 22.3 and 23.4 cm. The ratio of dose at 10 and 20 cm depth, D10/20, for a 10×10 cm field varies between 1.59 and 1.61. At a shift of 1.45 mm, there is a d max of 23.40 mm, within 0.5 mm of the 123.73 mm d max for the Varian 10 MV and a D10/20 of 1.595, within 0.5% of the Varian 10 MV, at 1.589.
than 3 times greater than that measured on a Varian 10 MV. The depth of maximum dose is within 0.5 mm of that produced by the Varian 10 MV Linac, and the ratio of dose at 10 cm depth to the dose at 20 cm depth is within 0.5%. For all depths greater than 1.5 cm, the depth dose profiles produced by our new Linac agree to within 1% to that of the Varian 10 MV Linac. The penumbra of the beam produced by our new Linac is 11% smaller than that produced by the Varian 10 MV Linac, which is expected to result in improved dose distributions. All the results obtained here are substantially similar to those obtained in our previous feasibility study, with the addition of dimensions for specific waveguide geometry. This suggests that the field scaling approach used in our feasibility study was an accurate approximation to the coupling cavity offset used in this paper.
4. CONCLUSIONS
a)Author
We have simulated the design of a linear accelerator waveguide capable of producing a 10 MV photon beam, with the waveguide length of 27.5 cm (same as a 600C waveguide). We computed the maximum electric field strengths inside the waveguide to be more than 12% below the breakdown threshold specific to our cavity design. It should be noted that the threshold is already conservative due to advances in vacuum and surface finishing since the publication of the breakdown threshold.7 With a first-coupling-cavity shift of 1.45 mm, the electron beam energy and spectrum FWHM are nearly identical to that of an emulated Varian 10 MV Linac, while the electron beam current incident on the target is more Medical Physics, Vol. 42, No. 4, April 2015
ACKNOWLEDGMENTS Funding for this work was provided by the Alberta Cancer Foundation, Western Economic Diversification Canada, and Alberta Innovates—Health Solutions.
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